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1,108,101,564,515 | arxiv | \section{INTRODUCTION}
\label{sec:INTRO}
Quantum and general relativistic field theories have been studied
for many years. The metaphysical differences between these
have not yet been resolved and a combined
theory is not available. A number of enigmas in present day
thought suggest that something is wrong. Among these is
the persistence of wave-particle duality, the unending discussions of
measurement theory, the problem of the quantization
of gravity, and the general difficulties relating to
a principle of quantization.
An analysis by geometrical methods suggests that the basic difficulties
are caused by an incorporated internal inconsistency.
According to the rules of logic, the presence
of an internal contradiction produces arbitrary conclusions.
Even if a concealed inconsistency is not explicitly identified, it may
occur because experimental results do not agree with accepted beliefs.
To indicate the possible consequences for other theories, a short discussion
of the implications is given here. A more complete account of the geometrical
theories will be presented later.
Apparent inconsistencies often appear in
physical theories. These occur not for failure of logic but
because of the way knowledge is obtained. Much of physics
knowledge comes from experiment. Unfortunately, such experiments
do not directly supply a consistent set of premises.
Notwithstanding
errors in the experimental method, which are not the
subject of this article, logical consistency is often lost
because
of oversimplification, chance concurrence, or
mistaken perception. In practice, the study of an
inconsistency or paradox is often enlightning and may reveal the
interrelationship of difficult ideas. A more serious concern is
the existence of a real fundamental inconsistency, undiscovered,
in accepted beliefs.
These may not produce any immediate sign until analysis
or new measurement uncovers trouble in a distant or isolated
setting. Any denial of a real inconsistency is not
productive and leads to interminable difficulties.
\footnote{A concealed inconsistency allows the proliferation of
sophistical arguments.
If the premises contain a contradiction,
a politically desirable result can always be demonstrated and the
criticism of opponents can always be disproved. This leads to a
situation in which truth is determined by politics.}
The central argument of this paper concerns the construction of
differential equations. To be able to calculate with
confidence, a consistent conceptual origin of the derivative is necessary.
The fundamental derivative is
defined by the process of taking limits.
\begin{equation}
{{df \over dx}=\lim_{\epsilon \to 0}{f(x+\epsilon)-f(x) \over \epsilon}}
\end{equation}
This definition implicitly describes a field in a
continuous space with a coordinate system. Any derivative that is used for
calculation should be related to this definition.
\section{DERIVATIVES IN QUANTUM MECHANICS}
\label{sec:DQM}
In quantum mechanics, derivatives are usually
introduced by more subtle methods.
The accepted construction contains essential algebraic steps.
It is useful to follow the historical context.
Starting with Newton, motion is described by the derivative of
spatial position with respect to time.
\begin{equation}
{ F=m{d^{2}x \over dt^{2}}}
\end{equation}
This temporal equation has evolved into the more sophisticated
mathematical form that uses lagrangians and hamiltonians.
These facilitate the solution of problems containing constraints.
\footnote{Such constraints and contact forces
are now recognized as the result of
quantum phenomenology and should properly be explained by the quantum
theory of materials.} One derives
\begin{equation}
{ {\partial \over \partial t}
\left( {\partial L \over \partial \dot x} \right) -
{\partial L \over \partial x} =0 }
\end{equation}
and
\begin{equation}
{H(p,x)=p \dot x - L \quad p={\partial L \over \partial \dot x} }
\end{equation}
where differentiations are made with respect to
to space, momentum, and other physical quantities.
The derivation of quantum equations proceeds by the process of first
quantization. Many subtle variations and refinements exist
but the elementary scheme that follows is sufficient for the argument.
Specific suitable hamiltonians ${H=H(p,x)}$ are chosen.
These are rewritten, ${H=H(p,x)}$,
having identical appearance, but with $x$ and $p$ taken as
algebraic objects rather than as fields.
Into this expression, which must be carefully arranged,
derivative operators are inserted in place of the momentum and energy
symbols. These derivatives
${p \to { \hbar \over i}{\partial \over \partial x }}$ and
${E \to i \hbar {\partial \over \partial t} }$
are introduced algebraically
without consideration for the elementary definition~\cite{DD}.
They must not appear in difficult places and they must be normal ordered
by a tenuous set of rules. This
assembled operator can then be applied to an hypothesized wave function.
\begin{equation}
{i \hbar {\partial \over \partial t}\psi
=H\left( {\hbar \over i}{\partial \over \partial x},x\right) \psi}
\end{equation}
The resulting equations agree well with experiment, and in an historical
context,demonstrate unequivocal success.
\section{DERIVATIVES IN GENERAL RELATIVITY}
\label{sec:DGR}
In contrast, derivatives in general relativity are developed from
a sense of displacement that is equivalent to the elementary
definition~\cite{ANDERSON}.
To obtain mathematical consistency in
curvilinear coordinate systems, a coefficient of connection
${\Gamma^{\nu}_{\beta \mu}}$ must be defined and used.
\begin{equation}
V^{\nu}( x^{\mu}+\Delta x^{\mu}) =
V^{\nu}( x^{\mu})+{\partial V^{\nu} \over \partial x^{\mu}}\Delta x^{\mu}+
\Gamma^{\nu}_{\beta \mu} V^{\beta}\Delta x^{\mu}.
\end{equation}
The connection ${\Gamma^{\nu}_{\beta \mu}}$ is set equal to the
christoffel symbol in general relativity
and in other Riemannian geometries.
\begin{equation}
\Gamma^{\nu}_{\beta\mu}={1 \over 2}g^{\nu \lambda}
\left( {\partial g_{\mu \lambda} \over \partial x^{\beta}}+
{\partial g_{\lambda \beta} \over \partial x^{\mu}}-
{\partial g_{\beta \mu} \over \partial x^{\lambda}}\right)
\end{equation}
This construction
is essential because of the use of geodesics as a fundamental
description of gravitational motion.
To define a coefficient of connection, common derivatives
of a metric or other tensor
are essential and unavoidable.
For non-Riemannian geometries, an additional tensor is added to the
christoffel symbol in forming the connection.
\section{COMBINATIONS OF QUANTUM AND RELATIVISTIC DERIVATIVES}
\label{sec:QAR}
The quantum and relativistic methods are antagonistic.
A classical theory of mechanics, suitable
for the application of first quantization, must have a momentum vector
${p^{\mu}}$ for even one
simple particle. This classical momentum, a first order relativistic
vector, must transform correctly under arbitrary coordinate transformations.
Consequently, it has a displacement which must be defined with the aid of a
coefficient of connection.
Upon application of the first quantization procedure, a physical
momentum is replaced by a momentum
operator. That operator has already been used in the definition of the
classical covariance of the quantity it repaces. Moreover,
the derivatives associated with the connection have undefined
properties during the process of constructing an algebraic hamiltonian.
Those contained within the definition of the coefficient
of connection cannot be treated as algebraic
quantities. Following this, the selection of a completed operator
for the wave function seems problematical.
The introduction of derivatives in different ways, one
from the quantum operator, one from the coefficient of connection,
does not establish their equivalence.
Nor is it reasonable to suppose that the result
can be made covariant by any proper mathematical manipulation. The
absence of a unified concept of physical derivative remains an impasse
to progress. Many attempts to combine quantum mechanics and
general relativity seem to fail because of this inconsistency in
their respective mathematical foundations.
\section{IS A UNIFIED THEORY POSSIBLE}
\label{sec:UFT?}
The coexistence of separate incompatible concepts of differentiation
has been accepted since the 1920's. The mathematical problem has
not been resolved nor fully discussed.
In principle it may be possible to avoid these problems by supposing
an exclusively algebraic structure without
any initial geometrical presumptions.
Geodesics, tensors, curvatures and
all but the simplest sorts of coordinates would be avoided.
Then, after algebraic
first quantization, the phenomenology of general relativity would have to be
derived in some approximation by appropriate manipulations.
Such a method may be possible, but is not yet a convincing reality.
The alternative approach suggested here is to construct quantum
theories without using first quantization.
There is no presupposed classical theory.
One studies suitable
geometrical invariants to find such expressions as might
describe quantum mechanics with or without
other fields and interactions. In principle,
such a theory may be derivable from a lagrangian; but, without the
availability of a classical momentum, the standard rules for defining
such a lagrangian are inadequate. The usual methods for defining models
fail and an entirely different approach is required.
\section{A GEOMETRICAL THEORY OF QUANTUM MECHANICS}
\label{sec:GTQM}
As an example of such a direct quantum construction,
a theory that is consistent with
these conditions can be constructed from general relativity.
Using the minimal substitution as a transformation of derivatives,
${{\partial \over \partial x^{\mu}} \to
{\partial \over \partial x^{\mu} }-
{ie \over \hbar c}A_{\mu}}$
and applying it naively to the usual Riemannian connections
of general relativity, there results
\begin{equation}
\Gamma^{\prime\nu}_{\mu \beta} =
{\nu \brace \mu\beta}-
{ie \over 2 \hbar c}\left(
\delta^{\nu}_{\mu}A_{\beta}+
\delta^{\nu}_{\beta}A_{\nu}-
g_{\mu \beta}g^{\nu \lambda}A_{\lambda}\right) .
\end{equation}
There is no apriori reason to believe that this will work an generate
as useful theory. However,
if the change is applied uniformly, then at least
the continued consistency of the infinitesimal displacements is assured.
The result is non-Riemannian but precisely of the form supposed by
Weyl~\cite{WEYL} in his early unified field theory.
The numerical factor is suitable for quantum
scale events. The known properties of Weyl's theory guarantee
covariance. A more complete investigation
shows that a fairly complete quantum structure is contained within
the geometry. Quantum phenomena are predicted without quantization.
\section{BETTER UNIFIED FIELD THEORIES}
\label{sec:BUFT}
More complicated but also more complete field theories are possible.
One can at least hope to
include field and motion equations for quantum mechanics,
general relativity and electrodynamics. Such a system should
have geodesic trajectories and consistent displacement operators.
The known fundamental geometrical theories incorporate constructions
that cause special problems. The derivatives in the connections are
equivalent to momentum operators.
A further direct application of quantization to these theories
is likely to lead to over quantization. The degree of
the differential equations will be too high leading at least to
ghost solutions and extraneous interdependency of physical
effects~\cite{OQFT}.
A consistent interpretation then becomes difficult.
A terrifying set of conclusions is suggested. All classical relativistic
theories that incorporate curvilinear coordinates must be incorrect because
they contain suppressed quantum fields. None of them can be properly
quantized by any conventional procedure. Quantum general relativistic
theories cannot be constructed from classical physics but require a separate
self-consistent set of premises. IT is suggested that the
hamiltonian approach proposed by Dirac~\cite{DIRAC} and pursued by
others~\cite{HQ} will continue to incur difficulties.
The internal construction of general relativity suggests that any other
general covariant theory of interaction will likely be of
comparable complexity. These theories must contain connections
and derivatives, implicit or explicit. Such a relativistic quantum
theory must imply connections and consequently
cannot have a classical precursor. The general accepted methods to search
for covariant quantum field theories fail systematically. A particular
construction my
succeed fortuitously, but a better method is to use consistent geometrical
derivations from the beginning.
\section{INAPPLICABILITY OF VON NEUMANN'S THEOREM}
\label{sec:IVNT}
It is important to note that
Von Neumann's theorem concerning hidden variables does not apply to
fully geometrical theories of this type~\cite{SUSQM}. Because
the momentum operator is not a physical quantity but
a plain mathematical symbol,
the epistemological association with classical physics
is dropped and observations are based on geodesics or other geometrical
quantities. There is no conventional
measurement theory. Such a
theory, if it were to have hidden variables, could only contain such
quantities as transform properly. Bare
differential operators, ${p_{(op)}}$, do not qualify.
More generally, any theory that
does not associate physical quantities with a momentum operator
or that does not attach hidden variables to differential quantum
observables, abrogates the conditions of
Von Neumann's theorem.
The physical identification of curvilinear coordinates with particle
motion avoids measurement theory. The requirement of mathematical
consistency for the derivatives bypasses conventional quantization.
\section{SUMMARY AND CONCLUSION}
\label{sec:CONC}
The historical failure to combine quantum mechanics
with relativistic theories is argued to be due to fundamental
failure in the meta-mathematical structure of derivatives.
While the synthesis of gravitation, quantum mechanics,
and electrodynamics may be possible,
the equations should be written in terms of geometrical invariants.
They must use displacement operators self-consistently.
Measurement theory is avoided and Von Neumann's theorem does not
apply. Such geometrical theories
cannot be constructed quantum free nor can they be derived
by first quantization.
It may also be possible to have a geometry of interaction.
Noting the work of Kaluza~\cite{KALUZA}, five dimensional theories may
have the potential to incorporate geometrically based inhomogeneous field
equations. Various authors have attributed either the field or
motion equations to five dimensional effects~\cite{FDT}.
Although these are not widely
accepted, some of the difficulties may be because of the irreconcilable
differences between latent quantum effects and classical interpretations.
The interaction constants may be contained in the geometry. Those
describing the gravitational and electromagnetic field should have
geometrical interpretations. Unfortunately, all workable theories
of this type challenge the formalism and interpretation of conventional
quantum mechanics. The investigation of these remarkable
mathematical structures may lead to predictions that otherwise cannot
be formulated.
\acknowledgements
I would like to thank Dr. E. von Meerwall for translating an occasional German
article and B. Galehouse for helping with the typesetting.
|
1,108,101,564,516 | arxiv | \section{Supplementary Information}
\subsection{Fisher derivation for the uniform distribution $p(\mu)$}
Let us assume the standard Zeno approximations $q(\mu)\approx 1-\Delta^2H_{\Pi} \mu^2$, $\mathcal{P}^{\star}=1$, $(1-\mathcal{P}^{\star})=m\Delta^2H_{\Pi}\chi_2$, and consider the uniform probability distribution $p(\mu)= (\Theta(\mu-\mu_1)-\Theta(\mu-\mu_2))/(\mu_2-\mu_1)$, where $\Theta$ is the Heaviside function. The second moment of this distribution is given by
\begin{equation*}
\chi_2=\int_{\mu_1}^{\mu_2}p(\mu)\mu^2 d\mu=\frac{\mu_2^3-\mu_1^3}{3(\mu_2-\mu_1)}=\frac{\mu_1^2+\mu_1\mu_2+\mu_2^2}{3}.
\end{equation*}
Accordingly, the $k$-th moment is given by
\begin{equation*}
\chi_k=\frac{\mu_2^{k+1}-\mu_1^{k+1}}{(k+1)(\mu_2-\mu_1)}.
\end{equation*}
If we change $\mu_2$, then the partial derivatives of the statistical moments are
\begin{equation*}
\frac{\partial \chi_k}{\partial \mu_2}=\frac{(k+1)\mu_2^{k}(\mu_2-\mu_1)-(\mu_2^{k+1}-\mu_1^{k+1})}{(k+1)(\mu_2-\mu_1)^2} \; .
\end{equation*}
In particular, for the second moment, we have
\begin{equation*}
\frac{\partial \chi_2}{\partial \mu_2}=\frac{3\mu_2^{2}(\mu_2-\mu_1)-(\mu_2^3-\mu_1^3)}{3(\mu_2-\mu_1)^2}
=\frac{-3\mu_2^{2}\mu_1+ 2\mu_2^3+\mu_1^3}{3(\mu_2-\mu_1)^2}.
\end{equation*}
The probability density function, instead, changes by $\delta p^{(f)}(\mu)=\delta c\, f(\mu)$ with $f(\mu)=\frac{\delta_{Dirac}(\mu-\mu_2)-p(\mu)}{\mu_2-\mu_1}$, leading to the Fisher information
\begin{equation*}
\begin{split}
&F_{f}=\frac{\mathcal{P}^{\star}}{1-\mathcal{P}^{\star}}\left(m\int_{\mu}\frac{\delta_{Dirac}(\mu-\mu_2)-p(\mu)}{\mu_2-\mu_1} \ln q(\mu)d\mu\right)^2 \\
&\approx \frac{\mathcal{P}^{\star}}{1-\mathcal{P}^{\star}}\left(m\Delta^2 H_{\Pi}\int_{\mu}\frac{\delta_{Dirac}(\mu-\mu_2)-p(\mu)}{\mu_2-\mu_1} \mu^{2}d\mu\right)^2 \\
&\approx m\Delta^2H_{\Pi}\frac{(\mu_2^2-\chi_2)^2}{\chi_2(\mu_2-\mu_1)^2}.
\end{split}
\end{equation*}
Conversely, if we treat $\chi_2$ as the estimation parameter $\theta$, the Fisher information reads
\begin{equation*}
\begin{split}
&F(\chi_2)=\frac{\mathcal{P}^{\star}}{1-\mathcal{P}^{\star}}\left(m\frac{\partial}{\partial\theta}\int_{\mu}p(\mu)\ln q(\mu)d\mu\right)^{2} \\
&\approx\frac{\mathcal{P}^{\star}}{1-\mathcal{P}^{\star}}\left(m\frac{\partial}{\partial\chi_2}(\Delta^2 H_{\Pi}\chi_2)\right)^2 \\ &=\frac{\mathcal{P}^{\star}}{1-\mathcal{P}^{\star}}\left( m\Delta^2 H_{\Pi}\right)^2 \approx m\Delta^2 H_{\Pi}/\chi_2.
\end{split}
\end{equation*}
This is also the respective matrix element of the Fisher information matrix. The two results differ because we put a contraint on the shape of the probability distribution. In fact, at the second order (since we neglect higher moments) they differ by the square of derivative of the second moment as the Fisher information follows a chain rules for the derivative:
\begin{equation*}
\begin{split}
&\frac{F(f(\mu))}{F(\chi_2) (\frac{\partial}{\partial \mu_2}\chi_2)^2}=
\frac{(\mu_2^2-\chi_2)^2}{(\mu_2-\mu_1)^2}\left(\frac{3(\mu_2-\mu_1)^2}{-3\mu_2^{2}\mu_1+ 2\mu_2^3+\mu_1^3}\right)^2 \\
&=\frac{(3(\mu_2-\mu_1)\mu_2^2-(\mu_2^3-\mu_1^3))^2}{9(\mu_2-\mu_1)^4}\left(\frac{3(\mu_2-\mu_1)^2}{-2\mu_2^{2}\mu_1+ \mu_2^3+\mu_1^3}\right)^2 \\
&=\left(\frac{ 3(\mu_2-\mu_1)\mu_2^2-(\mu_2^3-\mu_1^3) }{-3\mu_2^{2}\mu_1+ 2\mu_2^3+\mu_1^3}\right)^2=1 \; .
\end{split}
\end{equation*}
\end{document}
|
1,108,101,564,517 | arxiv | \section{Introduction}
Underwater object detection is a branch of object detection research. Underwater is rich in mineral and biological resources, waiting for human to explore at the same time, there are many unknown dangers. In order to explore more safely and comprehensively, the research of underwater object detection and underwater robot has been put on the agenda. The underwater robot is designed to replace human beings to go deep into the seabed and complete the underwater scientific and military mission investigation, marine resource assessment, seabed geological testing and other dangerous tasks. Underwater object detection is the key basic technology to help the underwater robot to complete these underwater tasks better.
However, the underwater environment is very different from the land environment, which makes the object recognition more difficult. Due to the absorption and scattering characteristics of light in the water and other reasons, there will be some problems affecting the image quality, such as color distortion, blurring, contrast distortion and so on, when there is not enough light source in the water, resulting in the low accuracy of underwater image object detection. In the reality that the recognition accuracy of underwater object detection algorithm is still relatively low, the research of underwater object detection algorithm is very necessary.
\section{Related Work}
In this part we introduce our baseline model with standard Cascade-RCNN and its deformable convolution operation,as well as the Feature Pyramid Network[7] neck.
\subsection{Cascade RCNN}
While the ideas proposed in this work can be applied to
various detector architectures, we focus on the popular two-stage architecture of the Faster R-CNN, shown in Fig. 1
(a). The first stage is a proposal sub-network, in which the
entire image is processed by a backbone network, e.g. ResNet
[27], and a proposal head (“H0”) is applied to produce
preliminary detection hypotheses, known as object proposals. In the second stage, these hypotheses are processed by
a region-of-interest detection sub-network (“H1”), denoted
as a detection head. A final classification score (“C”) and a
bounding box (“B”) are assigned per hypothesis. The entire
detector is learned end-to-end, using a multi-task loss with
bounding box regression and classification components.
\\
\quad In order to generate high quality detection, we use Cascade-RCNN. Following the original implementation, we set the IoU thresholds to 0.5, 0.6 and 0.7 for each RCNN stage respectively. We also try different IoU thresholds, and find that the default setting yields best
performance.
\begin{center}
\begin{figure}[H]
\centering
\includegraphics[width=0.5\textwidth]{fig/Cascade.jpg}
\caption{Cascade-RCNN} \label{img}
\end{figure}
\end{center}
\subsubsection{bounding box regression}
\quad\\
A bounding box $\mathbf{b}=\left(b_{x}, b_{y}, b_{w}, b_{h}\right)$ contains the four coordinates of an image patch $\mathrm{x}$. Bounding box regression aims to regress a candidate bounding box $\mathbf{b}$ into a target bounding box $\mathbf{g}$, using a regressor $f(\mathbf{x}, \mathbf{b})$. This is learned from a training set $\left(\mathbf{g}_{i}, \mathbf{b}_{i}\right)$, by minimizing the risk
$$
\mathcal{R}_{\text{loc }}[f]=\sum_{i} L_{\text{loc }}\left(f\left(\mathbf{x}_{i}, \mathbf{b}_{i}\right), \mathbf{g}_{i}\right)
$$
As in Fast R-CNN [21],
$$
L_{l o c}(\mathbf{a}, \mathbf{b})=\sum_{i \in\{x, y, w, h\}} \text{ smooth }_{L_{1}}\left(a_{i}-b_{i}\right)
$$
where
$$
\text{ smooth }_{L_{1}}(x)=\left\{\begin{array}{cl}
0.5 x^{2}, & |x|<1 \\
|x|-0.5, & \text { otherwise }
\end{array}\right.
$$
is the smooth $L_{1}$ loss function. To encourage invariance to scale and location, smooth $_{L_{1}}$ operates on the distance vector $\Delta=\left(\delta_{x}, \delta_{y}, \delta_{w}, \delta_{h}\right)$ defined by
$$
\begin{array}{c}
\delta_{x}=\left(g_{x}-b_{x}\right) / b_{w}, \quad \delta_{y}=\left(g_{y}-b_{y}\right) / b_{h} \\
\delta_{w}=\log \left(g_{w} / b_{w}\right), \quad \delta_{h}=\log \left(g_{h} / b_{h}\right)
\end{array}
$$
\subsection{Deformable Convolution Network}
The DCN consists of two parts of operation.The first is deformable convolution. It adds 2D offsets to the regular grid sampling locations in the standard convolution. It enables free form deformation of the
sampling grid. It is illustrated in Figure 2. The offsets
are learned from the preceding feature maps, via additional
convolutional layers. Thus, the deformation is conditioned
on the input features in a local, dense, and adaptive manner.
The second is deformable RoI pooling. It adds an offset
to each bin position in the regular bin partition of the previous RoI pooling. Similarly, the offsets are learned from the preceding feature maps and the RoIs, enabling
adaptive part localization for objects with different shapes.
Both modules are light weight. They add small amount of parameters and computation for the offset learning. They can readily replace their plain counterparts in deep CNNs and can be easily trained end-to-end with standard backpropagation. The resulting CNNs are called deformable
convolutional networks, or deformable ConvNets.
\begin{center}
\begin{figure}[H]
\centering
\includegraphics[width=0.5\textwidth]{fig/DCN.jpg}
\caption{Deformable Convolution } %
\label{img}
\end{figure}
\end{center}
\begin{center}
\begin{figure}[H]
\centering
\includegraphics[width=0.5\textwidth]{fig/DCN2.jpg}
\caption{Deformable RoIPooling Network}
\label{img}
\end{figure}
\end{center}
\subsection{Feature Pyramid Network}
Our method takes a single-scale image of an arbitrary
size as input, and outputs proportionally sized feature maps
at multiple levels, in a fully convolutional fashion. This process is independent of the backbone convolutional architectures, and in this paper we present results using ResNext. The construction of our pyramid involves a bottom-up pathway, a top-down pathway, and lateral connections, as introduced in the following. Bottom-up pathway. The bottom-up pathway is the feedforward computation of the backbone ConvNet, which computes a feature hierarchy consisting of feature maps at several scales with a scaling step of 2. There are often many layers producing output maps of the same size and we say these layers are in the same network stage. For our feature pyramid, we define one pyramid level for each stage. We choose the output of the last layer of each stage as our reference set of feature maps, which we will enrich to create our pyramid. This choice is natural since the deepest layer of each stage should have the strongest features.Specifically, for ResNext we use the feature activations output by each stage’s last residual block. We denote the output of these last residual blocks as {C2, C3, C4, C5} for conv2, conv3, conv4, and conv5 outputs, and note that they have strides of {4, 8, 16, 32} pixels with respect to the input image. We do not include conv1 into the pyramid due to its large memory footprint.The structure can be displayed as Fig.4
\begin{center}
\begin{figure}[H]
\centering
\includegraphics[width=0.5\textwidth]{fig/FPN.jpg}
\caption{Feature Pyramid Network}
\label{img}
\end{figure}
\end{center}
\section{Approach}
\subsection{Network Structure}
The whole network can be divided into two parts as the traditional two-stage detection algorithm.The first part is the feature extraction backbone and the second part is feature fusion neck and detection predition head.
\subsubsection{Backbone}\quad \\
In this case we apply ResNext101 as our powerful backbone,because ResNext101 combines the advantages of both ResNet and InceptionNet,which maintains residual connection between convolution blocks and also have a wide forward block like InceptionNet.At the same time we pretrained this part of the network on coco dataset and freeze the parameters of the first convolution block and trained the other parts with our underwater dataset end-to-end.
ResNeXt101 Backbone model could be seen as Fig.5.\\
\begin{center}
\begin{figure}[H]
\centering
\includegraphics[width=0.5\textwidth]{fig/X101.jpg}
\caption{. Equivalent building blocks of ResNeXt. (a): Aggregated residual transformations right. (b): A block equivalent to (a), implemented as early concatenation. (c): A block equivalent to (a,b), implemented as grouped convolutions. Notations in bold
text highlight the reformulation changes. A layer is denoted as input channels, filter size, output channels}
\label{img}
\end{figure}
\end{center}
Except from the basic feature extraction backbone model,we also applied some plugins to this part.First we use non-local structure to replace the traditional average pooling and added global context information to our extracted feature map.We called this part the \textit{gcb plugins}.\\
After the basic backbone and the gcb plugins,the feature map is positional-encoded and fed into another structure called attention block.This kind of structure is totally the same as the structure being mentioned in the paper attention is all you need.We cannot use the traditional qkv+postional encoding because of the limitness of GPU resource,so we just use single-head attention with positional encoding instead of the multi-head attention.In this way we renamed this structure called attention plugins.
\subsubsection{Neck}\quad \\
Traditional FPN does not have a feature fusion operation after the backbone layers.We rethink the FPN structure according to the paper YOLOF you only look on one-level feature.In this paper the author mentioned that the effectness of FPN does not come from the multi-level faeture structure but the encoder=decoder structure ,for implementation details of Feature Pyramid Neural Network please infer to the first section of this paper.\\
Instead of the traditional FPN we apply BFP struture as the Libra RCNN and also we extract the advantage of NASFPN[8],this structure is shown in Fig.6.For the encoder- decoder structure is really an important idea in the Feature Pyramid Network,so we just simply do a fusion operation at the end of the Multi-level feature and encoded these feature maps into an single-level feature map.The we did a sequence of up-sampling operation and reconstruct the structure as U-Net,then we predict the anchor on each level of the feature maps as Cascade RCNN.
\begin{center}
\begin{figure}[H]
\centering
\includegraphics[width=0.5\textwidth]{fig/NAS-FPN.jpg}
\caption{NAS-FPN}
\label{img}
\end{figure}
\end{center}
\subsubsection{Detection Head}\quad \\
The Structure here is basically the same as original Cascade-RCNN,we also tried some other types of loss function,such as GIoU Loss,CIoU Loss and DIoU Loss.But none of them perform as well as the simple SmoothL1 Loss function.Also,in terms of the type of the sampler,we tried InstanceBalance Sampler which was introduced in Libra-RCNN and solved the problem of disparity between different classes of objects.Our dataset consists of pictures of four different types of underwater creatures,holothurian,echinus,scallop and starfish.In our case the amount of scallop is quite low and we did a simple count that we fould the number of the starfish is ten times of that of the starfish.So the InstanceBalance Sampler should have solved this kind of problem,but to my surprise it have not.And GIoU loss have the same problem.In the orginal paper GIoU performed quite well on COCO Dataset,but in our case GIoU and its improved versions CIoU and DIoU all lost their effect.So we just used the original Cascade detection head and applied soft-nms at the end of the algorithm instead of the simple nms.The detection head Structure is shown as following section.
\\
The initial hypotheses distribution produced by the RPN is heavily tilted towards low quality. For example, only 2.9\% of examples are positive for an IoU threshold u = 0.7. This makes it difficult to
train a high quality detector. The Cascade R-CNN addresses
the problem by using cascade regression as a resampling
mechanism. This is inspired by Faster-RCNN, where nearly
all curves are above the diagonal gray line, showing that
a bounding box regressor trained for a certain u tends
to produce bounding boxes of higher IoU. Hence, starting
from examples, cascade regression successively resamples an example distribution of higher IoU.
This enables the sets of positive examples of the successive
stages to keep a roughly constant size, even when the detector quality u is increased. Figure 4 illustrates this property,
showing how the example distribution tilts more heavily
towards high quality examples after each resampling step.\\
At each stage t, the R-CNN head includes a classifier
$ h_{t} $ and a regressor $ f_{t} $ optimized for the corresponding IoU threshold $ u_{t}$, where $ u_{t} > u_{t−1} $ . These are learned with loss:\\
\quad \\
\begin{center}
$ L\left(\mathbf{x}^{t}, g\right)=L_{c l s}\left(h_{t}\left(\mathbf{x}^{t}\right), y^{t}\right)+\lambda\left[y^{t} \geq 1\right] L_{l o c}\left(f_{t}\left(\mathbf{x}^{t}, \mathbf{b}^{t}\right), \mathbf{g}\right) $\\
\end{center}
\quad \\
where $ b
t = f_{t−1} (x_{
t−1}, b_{t−1}) $, g is the ground truth object for
$ x_{t} $
, $\lambda$ = 1 the trade-off coefficient, $ y_{t} $ is the label of $ x_{t} $ under the $u_{t}$ criterion, [·] is the indicator function.
Note that the use of [·] implies that the IoU threshold u
of bounding box regression is identical to that used for
classification. This cascade learning has three important
consequences for detector training. First, the potential for
overfitting at large IoU thresholds u is reduced, since positive examples become plentiful at all stages.Second, detectors of deeper stages are optimal for higher IoU thresholds. Third, because some outliers are removed as the IoU threshold increases,the learning
effectiveness of bounding box regression increases in the
later stages. This simultaneous improvement of hypotheses
and detector quality enables the Cascade R-CNN to beat
the paradox of high quality detection. At inference, the
same cascade is applied. The quality of the hypotheses
is improved sequentially, and higher quality detectors are
only required to operate on higher quality hypotheses, for
which they are optimal.
\section{Training Policy}
\subsection{Hyperparameter}
The entire model is trained with 4 Nvidia RTX 3090 GPU end2end with memory 64GB.We used SGD as optimizer and set the learning rate to 0.005(0.00125*number of gpus).The learning rate would drop on epoch 8 and 11 to 0.0005 and 0.0001 and have a warmup process at the beginning of epoch 1 with 500 iterations,this training policy has been proved to be effective with most computer vision tasks.We set the soft-nms threshold to 0.7 with rpn and 0.5 with rcnn so that the anchors would be filtered with suitable constraints.\\
We also changed the default confidence threshold from 0.3 to 0.0001,in this way the integration value of recall and accuracy will not lose too much points.
\subsection{Network Architecture}
We tried some ways to improve our ResNext backbone,as introduced in the previous sections.The context block gcb and attention block perform quite well with a huge improvement in map value.DCN is the part of the baseline model,and we must admit that this part is the key of many effective tricks.We also tried BFP[9] in the Feature Pyramid Network,but it seems not so useful on our dataset as the orginal coco dataset experiment.And GIoU[10] loss,CIoU[11] loss,DIoU[11] loss all did not work at all which is an astonishing phenomenon.The experiment result could be seen in the next section.
\\
\subsection{Data Augumentation Tricks}
We also implemented several training tricks to augument our pictures.Including RandomRotate 90 degrees,Random Flipping,Vertical Flipping,Cutout,Mixup and Multi-scale training and testing.At the final stage of B list.The only trick that we kept is RandomRotate,while the others will lead to worse robustness of the model.
\section{Result on Underwater Dataset}
In this section the testing result on the A list will be presented in
the form and we will see the final result on the B list.Note only the essential part of the evaluation is given.Some of the results has been removed because the adjustment of simple hyperparameters is not so intereseting.
\begin{table}[H]
\resizebox{260pt}{100pt}{
\begin{tabular}{|l|l|}
\hline
model & map@50:95 \\ \hline
baseline:Cascade+FPN+DCN+X101+Random90+Multi-scale & 0.523 \\ \hline
baseline without dcn & 0.527 \\ \hline
baseline+dcn pretrained on coco & 0.549 \\ \hline
baseline+dcn pretrained+gcb & 0.561 \\ \hline
baseline+dcn pretrained+gcb+data cleansing & 0.563512 \\ \hline
baseline+dcn pretrained+gcb+data cleansing+GIoU & 0.553 \\ \hline
baseline+dcn pretrained+gcb+data cleansing+CIoU & 0.554 \\ \hline
baseline+dcn pretrained+gcb+data cleansing+Cutout & 0.556 \\ \hline
baseline+dcn pretrained+gcb+data cleansing+MixUp & 0.557 \\ \hline
baseline+dcn pretrained+gcb+data cleansing+InstanceBalanceSampler & 0.5604 \\ \hline
baseline+dcn pretrained+gcb+data cleansing+BFP & 0.5604 \\ \hline
baseline+dcn pretrained+gcb+data cleansing+attention & 0.563542 \\ \hline
baseline+dcn pretrained+gcb+data cleansing+BBoxJitter & 0.563662 \\\hline
baseline+dcn pretrained+gcb+data cleansing+attention+BBoxJitter & 0.567 \\ \hline
\end{tabular}
}
\caption{Testing Result}
\end{table}
It is noteworthy that we add a new type of data augumentation called Bounding Box Jitter-BBoxJitter.This trick aims to adjust the bounding box of ground truth labels.Because the given training dataset contains Labeling noise which means that some of the locations of ground truth boxes are given by mistake on purpose.This trick only performs well in this typical circumstances.The offcial datasets would not have such labeling noises so do NOT try this tricks on them!!!
\\
\quad \quad \\Our test result on B list will come soon,please refer to the ofiicial heywhale website.
\section{Conclusion}
In this paper we present a new model called CDNet in the field of underwater detection challenge.This model is not perfect but effective in this kind of environment.We achieved acceptable result though there's still a long way to go until the state-of-art algorithm.According to my conprehension of this task,tiny object and overlapping object detection is the most vital direction of future works with wide range of improvement.At last thanks to supervisor Prof.Dr. Dong Wang and Prof.Dr. Yifan Wang.They kindly gave me efficient support with GPU resource and I'm really grateful for other kinds of contribution.
|
1,108,101,564,518 | arxiv | \section{INTRODUCTION}
\subsection{Motivation and literature overview}
Noncooperative generalized games over networks is currently a very active research field, due to the spreading of multi-agent network systems in modern society. Such type of games emerge in several application domains, such as smart grids \cite{dorfer:simpson-porco:bullo:16,parise:colombino:grammatico:lygeros:14}, social networks \cite{grammatico:18tcns} and robotics \cite{martinez:bullo:cortes:frazzoli:07}.
In a game setup the players, or agents, have a private and local objective function that depends on the decisions of some other players, which shall be minimized while satisfying both local and global, coupling, constraints. Typically each agent defines its decision, or strategy, based on some local information exchanged with a subset of other agents, called neighbors. One popular notion of solution for these games is a collective equilibrium where no player benefits from changing its strategy, e.g. a \textit{generalized Nash equilibrium} (GNE). Various authors proposed solutions to this problem \cite{pavel2017:distributed_primal-dual_alg,grammatico:18tcns,facchinei:fischer:piccialli:07}. These works propose only synchronous solutions for solving noncooperative games.
So, all the agents shall wait until the slowest one in the network completes its update, before starting a new operation. This can slow down the convergence drammatically, especially in large scale and heterogeneous systems. On the other hand, adopting an asynchronous update reduces the idle times, increasing efficiency. In addition, it can also speed up the convergence, facilitate the insertion of new agents in the network and even increse robusteness w.r.t. communication faults \cite{BERTSEKAS:1991:Survey_Asynch}.
The pioneering work of Bertsekas and Tsitsiklis \cite{bertsekas:1989:parallel_optimization} can be considered the starting point of the literature on parallel asynchronous optimization.
During the past years, several asynchronous algorithms for distributed convex optimization were proposed \cite{recht:2011:hogwild,Combettes:2015:stoch_quadi_fejer,liu:2015:asynchronous_parallel_stoch_coord_desc,nedic:2011:asynchronous_broadcast-based_convex_opt}, converging under different assumptions. The novel work in \cite{peng2016arock}, provides a simple framework (ARock) to develop a wide range of iterative fixed point algorithms based on nonexpansive operators and it is already adopted in \cite{Pavel:Yi:2018:Asynch} to seek variational GNE seeking under equality constraints and using edge variables.
In this paper, we propose an extension of the work in \cite{Pavel:Yi:2018:Asynch}. Specifically, we consider inequality coupling constraints and use a restricted set of auxiliary variables, namely, associated with the nodes rather than with the edges. Especially this latter upgrade is non-trivial and presents technical challenges in the asynchronous implementation of the algorithm, which we overcome by analyzing the influence of the delayed information on the update of the auxiliary variables. The use of node variables only, rather than edge variables, preserves the scalability of the algorithm, with respect to the number of nodes.
\subsection{Structure of the paper}
The paper is organized as follows: Section~\ref{sec:problem_formulation} formalizes the problem setup and introduces the concept of \textit{variational} v-GNE. In Section~\ref{sec:synch_case} the iterative algorithm for v-GNE seeking is derived for the synchronous case. The asynchronous counterpart of the algorithm is presented in Section~\ref{sec:asynch_case}. Section~\ref{sec:simulations} is dedicated to the simulation results for the problem of Cournot competition in networked markets. Section~\ref{sec:conclusion} ends the paper presenting the conclusions and the outlooks of this work.
\section{NOTATION}
\label{sec:notations}
\subsection{Basic notation}
The set of real, positive, and non-negative numbers are denoted by $\mathbb{R}$, $\mathbb{R}_{>0}$, $\mathbb{R}_{\geq 0}$, respectively; $\overline{\mathbb{R}}:=\mathbb{R}\cup \{\infty\}$. The set of natural numbers is $\mathbb{N}$. For a square matrix $A \in \bR^{n\times n}$, its transpose is $A^\top$, $[A]_{i}$ is the $i$-th row of the matrix and $[A]_{ij}$ represents the elements in the row $i$ and column $j$. $A\succ 0 $ ($A\succeq 0 $) stands for positive definite (semidefinite) matrix, instead $>$ ($\geq$) describes element wise inequality. $A\otimes B$ is the Kronecker product of the matrices $A$ and $B$. The identity matrix is denoted by~$I_n\in\bR^{n\times n}$. $\bld 0$ ($\bld 1$) is the vector/matrix with only $0$ ($1$) elements. For $x_1,\dots,x_N\in\mathbb{R}^n$, the collective vector is denoted as $\boldsymbol{x}:=\mathrm{col}(x_1,\dots,x_N)=[x_1^\top,\dots ,x_N^\top ]^\top$.
$\diag(A_1,\dots,A_N)$ describes a block-diagonal matrix with the matrices $A_1,\dots,A_N$ on the main diagonal. The null space of a matrix $A$ is $\mathrm{ker}(A)$.
The Cartesian product of the sets $\Omega_i$, $i=1,\dots ,N$ is $\prod^N_{i=1} \Omega_i$.
\subsection{Operator-theoretic notation}
The identity operator is by~$ \textrm{Id}(\cdot)$. The indicator function $\iota_\mathcal{C}:\bR^n\rightarrow[0,+\infty]$ of $\mathcal{C}\subseteq \bR^n$ is defined as $\iota_\mathcal{C}(x)=0$ if $x\in\mathcal{C}$; $+\infty$ otherwise.
The set valued mapping $N_{\ca C}:\bR^n\rightrightarrows \bR^n$ denotes the normal cone to the set $\mathcal{C}\subseteq \bR^n$, that is $N_{\ca C}(x)= \{ u\in\bR^n \,|\, \mathrm{sup}\langle \ca C-x,u \rangle\leq 0\}$ if $x \in \ca C$ and $\varnothing$ otherwise. The graph of a set valued mapping $\ca A:\ca X\rightrightarrows \ca Y$ is $\mathrm{gra}(\ca A):= \{ (x,u)\in \ca X\times \ca Y\, |\, u\in\ca A (x) \}$. For a function $\phi:\bR^n\rightarrow\overline{\mathbb{R}}$, define $\mathrm{dom}(\phi):=\{x\in\bR^n|f(x)<+\infty\}$ and its subdifferential set-valued mapping, $\partial \phi:\mathrm{dom}(\phi)\rightrightarrows\bR^n$, $\partial \phi(x):=\{ u\in \bR^n | \: \langle y-x|u\rangle+\phi(x)\leq \phi(y)\, , \: \forall y\in\mathrm{dom}(\phi)\}$. The projection operator over a closed set $S\subseteq \bR^n$ is $\textrm{proj}_S(x):\bR^n\rightarrow S$ and it is defined as $\textrm{proj}_S(x):=\mathrm{argmin}_{y\in S}\lVert y - x \rVert^2$. A set valued mapping $\ca F:\bR^n\rightrightarrows \bR^n$ is $\ell$-Lipschitz continuous with $\ell>0$, if $\lVert u-v \rVert \leq \ell \lVert x-y \rVert$ for all $(x,u)\, ,\,(y,v)\in\mathrm{gra}(\ca F)$; $\ca F$ is (strictly) monotone if $\forall (x,u),(y,v)\in\mathrm{gra}(\ca F)$ $\langle u-v,x-y\rangle \geq (>)0$ holds true, and maximally monotone if it does not exist a monotone operator with a graph that strictly contains $\mathrm{gra}(\ca F)$. Moreover, it is $\alpha$-strongly monotone if $\forall (x,u),(y,v)\in\mathrm{gra}(\ca F)$ it holds $\langle x-y, u-v\rangle \geq \alpha \lVert x-y \rVert^2$. The operator $\ca F$ is $\eta$-averaged ($\eta$-AVG) with $\eta\in(0,1)$ if $\lVert \ca F(x)-\ca F(y) \rVert^2 \leq \lVert x-y\rVert^2-\frac{1-\eta}{\eta}\lVert ( \textrm{Id}-\ca F)(x)-( \textrm{Id}-\ca F)(y) \rVert^2$ for all $x,y\in\bR^n$; $\ca F$ is $\beta$-cocoercive if $\beta\ca F$ is $\frac{1}{2}$-averaged, i.e. firmly nonexpansive (FNE).
The resolvent of an operator $\ca A:\bR^n\rightrightarrows \bR^n$ is $\mathrm{J}_{\ca A} :=( \textrm{Id}+\ca A)^{-1}$.
\section{Problem Formulation}
\label{sec:problem_formulation}
\subsection{Mathematical formulation}
\label{subsec:math_formulation}
We consider a set of $N$ agents (players), involved in a noncooperative game subject to coupling constraints. Each player $i\in\ca N:=\{1,\dots,N\}$ has a local decision variable (strategy) $x_i$ that belongs to its private decision set $\Omega_i\subseteq \bR^{n_i}$, the vector of all the strategies played is $\bld x:= \mathrm{col}(x_1,\dots,x_{N})\in\bR^{n}$ where $n=\sum_{i\in\ca N} n_i$, and $\bld x_{-i}=\mathrm{col}(x_1,\dots,x_{i-1},x_{i+1},\dots,x_{N})$ are the decision variables of all the players other than $i$. The aim of each agent $i$ is to minimize its local cost function $f_i(x_i,\bld x_{-i}):\Omega_i\times \Omega_{-i}\rightarrow\overline \bR$, where $\bld \Omega=\prod_{i\in\ca N} \Omega_i \subseteq \bR^{n}$, that leads to a coupling between players, due to the dependency on both $x_i$ and the strategy of the other agents in the game. In this work we assume the presence of affine constraints between the agent strategies. These shape the collective feasible decision set
\begin{equation}
\label{eq:collective_feasible_dec_set}
\ca{\bld X} := \bld \Omega \cap \left\{\bld x\in\bR^{n} \,|\, A\bld x\leq b \right\}\, ,
\end{equation}
where $A\in\bR^{m\times n}$ and $b\in\bR^m$. Then, the feasible set of each agent $i\in\ca N$ reads as
\begin{equation*}
\label{eq:feasible_dec_set}
\ca{X}_i(\bld{x}_{-i}) := \left\{ y\in\Omega_i \,|\, A_i y -b_i \leq \textstyle{\sum_{j\in \ca N\setminus\{ i\} }} b_j -A_jx_j \right\}\, ,
\end{equation*}
where $A = [A_1,\dots,A_N]$, $A_i\in\bR^{m\times n_i}$ and $\sum_{j=1}^N b_j =b$.
We note that both the local decision set $\Omega_i$ and how the player $i$ is involved in the coupling constraints, i.e. $A_i$ and $b_i$, are private information, hence will not be accessible to other agents.
Assuming affine constraints is common in the literature on noncooperative games \cite{Paccagnan_Gentile2016:Distributed_computation_GNE,pavel2017:distributed_primal-dual_alg}.
In the following, we introduce some other common assumptions over the aforementioned sets and cost function.\smallskip
\begin{stassumption}[Convex constraint sets]
\label{ass:convex_constr_set}
For each player $i\in\ca N$, the set $\Omega_i$ is convex, nonempty and compact. The feasible local set $\ca X_i(\bld x_{-i})$ satisfies Slater's constraint qualification.
\hfill \QEDopen
\end{stassumption}
\smallskip
\begin{stassumption}[Convex and diff. cost functions]
\label{ass:convex_diff_function}
For all $i\in\ca N$, the cost function $f_i$ is continuous, $\beta$-Lipschitz continuous, continuously differentiable and convex in its first argument.
\hfill \QEDopen
\end{stassumption}
\smallskip
In compact form, the game between players reads as
\begin{equation}\label{eq:game_formulation}
x_i\in \argmin_{y\in\bR^n} f_i(y,\bld x_{-i})\quad \textup{s.t.} \quad y\in \ca{X}_i(\bld{x}_{-i})\:.
\end{equation}
In this paper, we are interested in the \textit{generalized Nash equilibia} (GNE) of the game in \eqref{eq:game_formulation}.
\smallskip
\begin{definition}[Generalized Nash equilibrium]
\label{def:GNE}
A collective strategy $\bld x^*$ is a GNE if, for each player $i$, it holds
\begin{equation}
\label{eq:GNE_best_resp}
x_i^*\in \argmin_{y\in\bR^n} f_i(y,\bld x_{-i}^*)\quad \textup{s.t.} \quad y\in \ca{X}_i(\bld{x}_{-i}^*)\:.
\end{equation} \hfill\QEDopen
\end{definition}
\subsection{Variational GNE}
\label{sec:v_GNE}
Let us introduce an interesting subset of GNE, the set of so called \textit{variational GNE} (v-GNE), or \textit{normalized equilibrium point}, of the game in \eqref{eq:game_formulation} referring to the fact that all players share a common penalty in order to meet the constraints. This is a refinement of the concept of GNE that has attracted a growing interest in recent years - see \cite{kulkarni:shanbhag:12} and references therein.
This set can be rephrased as solutions of a variational inequality (VI), as in \cite{facchinei:fischer:piccialli:07}.
First, we define the \textit{pseudo-gradient} mapping of the game \eqref{eq:game_formulation} as
\begin{equation}
\label{eq:pseudo_grad}
F(\bld x)=\col\left( \{\nabla_{x_i}f_i(x_i,\bld x_{-i})\}_{i\in\ca N}\right)\,,
\end{equation}
that gathers all the subdifferentials of the local cost functions of the agents. The following are some standard technical assumptions on $F$, see \cite{Facchinei:2011:KKT_and_GNE,BELGIOIOSO:2017:convexity_and_monotonicity_aggr_games}.
\smallskip
\begin{stassumption}
\label{ass:subgrad_lipschitz_strong_mon}
The pseudo-gradient $F$ in \eqref{eq:pseudo_grad} is $\ell$-Lipschitz continuous and $\alpha$-strongly monotone, for some $\ell,\alpha>0$. \hfill\QEDopen
\end{stassumption}
\smallskip
Standing Assumption~\ref{ass:convex_diff_function} implies that $F$ is a single valued mapping, hence one can define VI($F,\bld X$) as the problem:
\begin{equation}
\label{eq:VI_GNE}
\textup{find } \bld x^*\in\bld X,\:\textup{s.t.}\: \langle F(\bld x^*),\bld x-\bld x^*\rangle \geq 0\,, \quad \forall \bld x \in \bld X \:.
\end{equation}
Next, let us define the KKT conditions associated to the game in \eqref{eq:game_formulation}. Due to the convexity assumption, if $\bld x^*$ is a solution of \eqref{eq:game_formulation}, then there exist $N$ dual variables $\lambda^*_i\in\bR^m_{\geq 0}$, $\forall i\in \ca N$, such that the following inclusion is satisfied:
\begin{equation} \label{eq:KKT_game}
\begin{split}
\bld 0 &\in \nabla_{x_i}f_i(x_i)+A_i^\top\lambda_i^*+N_{\Omega_i}(x_i^*)\, , \; \forall i\in\ca N \\
\bld 0 &\in b-A\bld x^*+ N_{\bR^m_{\geq 0}}( \lambda^*_i)\:.
\end{split}
\end{equation}
While in general the dual variables $\{\lambda_i\}_{i\in\ca N}$ can be different, here we focus on the subclass of equilibria sharing a common dual variable, i.e., $\lambda^*=\lambda_1^* = \dots =\lambda_N^*$.
In this case, the KKT conditions for the VI($F,\bld X$) in \eqref{eq:VI_GNE} (see \cite{facchinei:fischer:piccialli:07,facchinei:pang}) read as
\begin{equation} \label{eq:KKT_VI}
\begin{split}
\bld 0 &\in \nabla_{x_i}f_i(x_i)+A_i^\top\lambda^*+N_{\Omega_i}(x_i^*)\, , \; \forall i\in\ca N \\
\bld 0 &\in b-A\bld x^* + N_{\bR^m_{\geq 0}}( \lambda^*) \:.
\end{split} \:.
\end{equation}
By \eqref{eq:KKT_game} and \eqref{eq:KKT_VI}, we deduce that every solution $\bld x^*$ of VI($F,\bld X$) is also a GNE of the game in \eqref{eq:game_formulation}, \cite[Th.~3.1(i)]{facchinei:fischer:piccialli:07}. In addition, if the pair $(\bld x^*,\lambda^*)$ satisfies the KKT conditions in \eqref{eq:KKT_VI}, then $\bld x^*$ and the vectors $\lambda_1^*=\dots=\lambda_N^*=\lambda^*$ satisfy the KKT conditions for the GNE, i.e. \eqref{eq:KKT_game} \cite[Th.~3.1(ii)]{facchinei:fischer:piccialli:07}.
Note that under Standing Assumptions~\ref{ass:convex_constr_set}--\ref{ass:subgrad_lipschitz_strong_mon} the set of v-GNE is guaranteed to be a singleton \cite[Cor.~2.2.5;~Th.~2.3.3]{facchinei:pang}.
\section{Synchronous Distributed GNE Seeking}
\label{sec:synch_case}
In this section, we describe the \textit{\underline{S}ynchronous \underline{D}istributed \underline{G}N\underline{E} Seeking Algorithm with \underline{No}de variables} (SD-GENO). First, we outline the communication graph supporting the communication between agents, then we derive the algorithm via an operator splitting methodology.
\subsection{Communication network}
\label{sec:com_network}
The communication between agents is described by an \textit{undirected and connected} graph $\cal G =(\ca N,\ca E)$ where $\ca N $ is the set of players and $\ca E \subseteq \ca N \times\ca N$ is the set of edges. We define $|\ca E |= M$, and $|\ca N |= N$. If an agent $i$ shares information with $j$, then $(i,j)\in\ca E$, then we say that $j$ belongs to the neighbours of $i$, i.e., $j\in\ca N_i$ where $\ca N_i$ is the neighbourhood of $i$.
Let us label the edges $e_l$, for $l\in\{1,\dots,M\}$. We denote by $E\in\bR^{M\times N}$ the \textit{incidence matrix}, where $[E]_{li}$ is equal to $1$ (respectively $-1$) if $e_l=(i,\cdot)$ ($e_l=(\cdot,i)$) and $0$ otherwise. By construction, $E\bld 1_N = \bld 0_N$. Then, we define $\ca E_i^{\mathrm{out}}$ (respectively $\ca E_i^{\mathrm{in}}$) as the set of all the indexes $l$ of the edges $e_l$ that start from (end in) node $i$, moreover $\ca E_i=\ca E_i^{\mathrm{out}} \cup \ca E_i^{\mathrm{in}}$ .
The \textit{node Laplacian }$L\in\bR^{N\times N}$ of an undirected graph is a symmetric matrix and can be expressed as $L=E^\top E$, \cite[Lem.~8.3.2]{Godsil:algebraic_graph_theory}. In the remainder of the paper, we exploit the fact that the Laplacian matrix is such that $ L\bld 1_N = \bld 0_N $ and $\bld 1^\top_N L = \bld 0^\top_N $.
\subsection{Algorithm design}
\label{sec:alg_develop_synch}
Now, we present a distributed algorithm with convergence guarantees to the unique v-GNE of the game in \eqref{eq:game_formulation}. The KKT system in \eqref{eq:KKT_game}, can be cast in compact form as
\begin{equation} \label{eq:compact_KKT_game}
\begin{split}
\bld 0 &\in F(\bld x)+\Lambda^\top \bld \lambda+N_{\bld \Omega}(\bld x) \\
\bld 0 &\in \bar b-\Lambda\bld x + N_{\bR^{mN}_{\geq 0}}(\bld \lambda)
\end{split} \:,
\end{equation}
where $\bld \lambda = \col(\lambda_1,\dots,\lambda_N)\in\bR^{mN}$, $\Lambda= \diag(A_1,\dots,A_N)\in\bR^{mN\times n}$ and $\bar b = \col (b_1,\dots,b_N)\in\bR^{mN}$.
As highlighted before, for an agent $i$, a solution of its local optimality conditions is given by the strategy $x_i$ and the dual variable $\lambda_i$. To enforce consensus among the dual variables, hence obtain a v-GNE, we introduce the auxiliary variables $\sigma_l,\, l\in\{1,\dots,M\}$, one for every edge of the graph. Defining $\bld \sigma=\col (\sigma_1,\dots,\sigma_M)\in\bR^{mM}$ and using $\bld E = E\otimes I_m \in \bR^{mM\times mN}$, we augment the inclusion in \eqref{eq:compact_KKT_game} as
\begin{equation} \label{eq:mod_compact_KKT_game}
\begin{split}
\bld 0 &\in F(\bld x)+\Lambda^\top \bld \lambda+N_{\bld \Omega}(\bld x) \\
\bld 0 &\in \bar b-\Lambda\bld x + N_{\bR^{mN}_{\geq 0}}(\bld \lambda) +\bld E^\top \bld \sigma\\
\bld 0 &\in -\bld E \bld \lambda \:.
\end{split}
\end{equation}
The variables $\{\sigma_l\}_{l\in \{1\dots M\}}$ are used to simplify the analysis, but we will show how we decrease their number to one for each node, increasing the scalability of the algorithm, especially for dense networks.
From an operator theoretic perspective, a solution $\varpi^* = \col (\bld x^*,\bld \sigma^*,\bld \lambda^*)$ to \eqref{eq:mod_compact_KKT_game} can be interpreted as a zero of the sum of two operators, $\ca A$ and $\ca B$, defined as
\begin{equation}
\label{eq:operators_def}
\begin{split}
\ca A : \varpi \mapsto &\begin{bmatrix}
0 & 0 & \Lambda^\top \\
0 & 0 & -\bld E \\
-\Lambda & \bld E^\top & 0
\end{bmatrix}+\begin{bmatrix}
N_{\bld \Omega} (\bld x)\\
0\\
N_{\bR^{mN}_{\geq 0}} (\bld \lambda)
\end{bmatrix}\\
\ca B : \varpi \mapsto &\begin{bmatrix}
F(\bld x)\\
0\\
\bar b
\end{bmatrix}\,.
\end{split}
\end{equation}
In fact, $\varpi^*\in\mathrm{zer}(\ca A+\ca B)$ if and only if $\varpi^*$ satisfies \eqref{eq:mod_compact_KKT_game}.
Next, we show that the zeros of $\ca A+\ca B$ are actually the v-GNE of the initial game.
\smallskip
\begin{proposition}\label{prop:zer_AB_are_vGNE}
Let $\ca A$ and $\ca B$ be as in \eqref{eq:operators_def}. Then the following hold:
\begin{enumerate}[(i)]
\item $\mathrm{zer}(\ca A +\ca B)\not = \varnothing$ ,
\item if $ \col (\bld x^*,\bld \sigma^*,\bld \lambda^*)\in\mathrm{zer}(\ca A +\ca B)$ then $(\bld x^*,\lambda^*)$ satisfies the KKT conditions in \eqref{eq:KKT_VI}, hence $\bld x^*$ is the v-GNE for the game in \eqref{eq:game_formulation}.
\hfill\QEDopen
\end{enumerate}
\end{proposition}
\smallskip
The proof exploits the property of the incidence matrix $E$ of having the same null space of $L$, i.e. $\mathrm{ker}(E)=\mathrm{ker}(L)$, and the assumption that the graph is connected. It can be obtained via an argument analogue to the one used in \cite[Th.~4.5]{pavel2017:distributed_primal-dual_alg}, hence we omit it.
The problem of finding the zeros of the sum of two monotone operators is widely studied in literature and a plethora of different splitting method can be used to iteratively solve the problem \cite{eckstein1989:splitting}, \cite[Ch.~26]{bauschke2011convex}.
A necessary first step is to prove the monotonicity of the defined operators.
\smallskip
\begin{lemma}\label{lem:max_mon_of_operators}
The mappings $\ca A$ and $\ca B$ in \eqref{eq:operators_def} are maximally monotone. Moreover, $\ca B $ is $\frac{\alpha}{\ell^2}$-cocoercive. \hfill\QEDopen
\end{lemma}
\smallskip
The splitting method chosen here to find $\mathrm{zer}(\ca A+\ca B)$ is the \textit{preconditioned forward-backward} splitting (PFB), which can be applied thanks to the properties stated in Lemma~\ref{lem:max_mon_of_operators}.
\smallskip
\begin{remark}
The choice is driven by two main features simplicity and implementability. In fact, the PFB requires only one round of communication between agents at each iteration, minimizing in this way the most demanding operation in multi-agent algorithms, i.e., information sharing.
\end{remark}
\smallskip
The iteration of the algorithm takes the form of the so called Krasnosel'ski\u i iteration, namely
\begin{equation}
\label{eq:Krasno_iter_synch}
\begin{split}
\tilde\varpi^k &= T \varpi^k \\
\varpi^{k+1}&=\varpi^k+\eta (\tilde\varpi^k-\varpi^k)
\end{split}
\end{equation}
where $\varpi^k = \col(\bld x^k,\bld\sigma^k,\bld\lambda^k)$, $\eta>0$ and $T$ is the PFB splitting operator
\begin{equation}
\label{eq:T_PFB_operator}
T = \mathrm{J}_{\gamma\Phi^{-1}\ca A}\circ( \textrm{Id}-\gamma\Phi^{-1}\ca B)\, ,
\end{equation}
where $\gamma>0$ is a step size. The so-called preconditioning matrix $\Phi$ is defined as
\begin{equation}
\label{eq:preconditioning_matrix}
\Phi:=\begin{bmatrix}
\bld \tau^{-1} & 0 & -\Lambda^\top\\
0 & \delta^{-1}I_{mM} & \bld E \\
-\Lambda & \bld E^\top & \bld \varepsilon^{-1}
\end{bmatrix}
\end{equation}
where $\delta\in\bR_{>0}$, $\bld \varepsilon = \diag(\varepsilon_1,\dots,\varepsilon_N)\otimes I_m $ with $\varepsilon_i>0,\:\forall i\in \ca N$ and $\bld \tau$ is defined in a similar way.
From \eqref{eq:T_PFB_operator}, we note that $\fix(T)=\mathrm{zer}(\ca A + \ca B)$, indeed $\varpi \in\fix(T) \Leftrightarrow \varpi \in T \varpi \Leftrightarrow 0 \in \Phi^{-1}(\ca A+\ca B)\varpi \Leftrightarrow \varpi\in\zer(\ca A +\ca B)$, \cite[Th.~26.14]{bauschke2011convex}. Thus, the zero-finding problem is translated into the fixed point problem for the mapping $T$ in \eqref{eq:T_PFB_operator}.
At this point, we calculate from \eqref{eq:Krasno_iter_synch} the explicit update rules of the variables. We first focus on the first part of the update, i.e., $\tilde \varpi^k=T\varpi^k$. It can be rewritten as $\tilde \varpi^k \in \mathrm{J}_{\gamma\Phi^{-1}\ca A}\circ( \textrm{Id}-\gamma\Phi^{-1}\ca B)\varpi^k \Leftrightarrow \Phi(\varpi^k -\tilde\varpi^k) \in \ca A \tilde \varpi^k +\ca B \varpi^k$ and finally
\begin{equation}\label{eq:inclusion_synch}
\bld 0\in \ca A \tilde \varpi^k +\ca B \varpi^k + \Phi(\tilde\varpi^k -\varpi^k)\:,
\end{equation}
here $\tilde \varpi^k :=\col (\tilde {\bld x}^k,\tilde {\bld\sigma}^k, \tilde {\bld\lambda}^k)$. For ease of notation, we drop the time superscript $k$. By solving the first row block of \eqref{eq:inclusion_synch}, i.e.
$\bld 0 \in F(\bld x) +N_{\bld \Omega}(\tilde{\bld x}) + \bld \tau^{-1}(\tilde{\bld x} -\bld x) + \Lambda^\top\bld \lambda$, we obtain
\begin{equation}\label{eq:row3_sync}
\tilde{\bld x} = \mathrm{J}_{N_{\bld \Omega}} \circ\big(\bld x-\bld \tau(F(\bld x)+\Lambda^\top \bld \lambda ) \big)\,.
\end{equation}
The third row block of \eqref{eq:inclusion_synch} instead reads as $\bld 0 \in \bar b +N_{\bR^{mN}_{\geq 0}}(\tilde{\bld \lambda}) +\Lambda(2\tilde{\bld x}-\bld x) + \bld E^\top(2\tilde{\bld \sigma}-\bld \sigma)+ \bld \varepsilon^{-1}(\tilde{\bld\lambda}-\bld\lambda)$ that leads to
\begin{equation}\label{eq:row2_sync}
\tilde{\bld \lambda} = \mathrm{J}_{N_{\bR^{mN}_{\geq 0}}}\circ \big(\bld \lambda-\bld \varepsilon( \Lambda (2\tilde{\bld x}-\bld x) -\bar b - \bld E^\top(2\tilde{\bld \sigma}-\bld\sigma )) \big)\,.
\end{equation}
The second row block of \eqref{eq:inclusion_synch} defines the simple update $\tilde{\bld \sigma} = \bld \sigma + \delta \bld E\bld \lambda $. We note that in the update \eqref{eq:row2_sync} of $\tilde{\bld \lambda}$, only $\bld E^\top\bld \sigma $ is used, hence an agent $i$ needs only an aggregated information over the edge variables $\{\sigma_l\}_{l\in\ca E_i}$, to update its state and the dual variables. We exploit this property by replacing the edge variables with $\bld z = \bld E^\top \bld \sigma\in\bR^{Nm}$. In this way, the auxiliary variables are one for each agent, instead of being one for each edge. Using the property $\bld E^\top\bld E =L\otimes I_m=\bld L$, we cast the update rule of these new auxiliary variables as
\begin{equation}\label{eq:row1_mod_sync}
\begin{split}
&\tilde{\bld z}^k = \bld z^k + \delta \bld L \bld \lambda^k \\
&\bld z^{k+1} = \bld z^k +\eta (\tilde{\bld z}^k -\bld z^k) \, .
\end{split}
\end{equation}
By introducing $\bld z$ in \eqref{eq:row2_sync}, we then have
\begin{equation}\label{eq:row2_sync_mod}
\tilde{\bld \lambda} = \mathrm{J}_{N_{\bR^{mN}_{\geq 0}}}\circ \big(\bld \lambda+\bld \varepsilon( \Lambda ( 2\tilde{\bld x}-\bld x) -\bar b -2\tilde{\bld z} +\bld z ) \big)\,.
\end{equation}
The next theorem shows that an equilibrium of the new mapping is a v-GNE.
\smallskip
\begin{theorem}\label{th:eq_are_vGNE_mod_map}
If $\col(\bld x^*, \bld z^*, \bld \lambda^* )$ is a solution to the equations \eqref{eq:row3_sync}, \eqref{eq:row1_mod_sync} and \eqref{eq:row2_sync_mod}, with $\bld 1^\top\bld z^*=0$, then $\bld x^*$ is a v-GNE.
\hfill\QEDopen
\end{theorem}
\smallskip
\begin{remark}
The change of auxiliary variables, from $\bld \sigma$ to $\bld z$, is particularly useful in large non-so-sparse networks and it is in general convenient when the number of edges higher than the number of nodes. In fact, for dense networks, we have one auxiliary variable for each player, hence the scalability of the algorithm is preserved.
\end{remark}
\subsection{Synchronous, distributed algorithm with node variables (SD-GENO)}
\begin{algorithm}[t]
\DontPrintSemicolon
\textbf{Input:} $k=0$, $\bld x^0 \in \bR^{n} $, $\bld \lambda^0 \in\bR^{mN}$, $\bld z^0=\bld 0_{mN}$, and chose $\eta,\, \delta,\, \bld \varepsilon,\, \bld \tau$ as in Theorem~\ref{th:convergence_sync}. \;
\For{$i\in\ca N$}{
$\tilde{ x}_{i}^k=\mathrm{proj}_{\Omega_i} \big( x_{i}^k-\tau_i(\nabla_i f_i(x_{i}^k,\bld x_{-i}^k)+ A_i^\top \lambda_{i}^k ) \big) $ \;
$\tilde{z}_{i}^k = z_{i}^k +\delta \sum_{j\in\ca N_i } (\lambda_{i}^k - \lambda_{j}^k) $\;
$\tilde{\lambda}_{i}^k = \mathrm{proj}_{\bR^{m}_{\geq 0}} \big( \lambda_{i}^k+\varepsilon_i\left( A_i(2\tilde{x}_{i}^k - x_{i}^k) \right. $\;
$\hspace{4cm }\left. - b_i+z_{i}^k -2\tilde{z}_{i}^k \right) \big)$\;
$x_{i}^{k+1} = x_{i}^k +\eta(\tilde{ x}_{i}^k - x_{i}^k)$\;
$z_{i}^{k+1} = z_{i}^k +\eta(\tilde{ z}_{i}^k -z_{i}^k)$ \;
$\lambda_{i}^{k+1} = \lambda_{i}^k +\eta(\tilde{ \lambda}_{i}^k - \lambda_{i}^k)$\;
$k\leftarrow k+1$\;
}
\caption{SD-GENO}
\label{alg:synch_alg}
\end{algorithm}
We are now ready to state the update rules defining the synchronous version of the proposed algorithm.
The update rule is obtained by gathering \eqref{eq:row3_sync}, \eqref{eq:row1_mod_sync}, \eqref{eq:row2_sync_mod} and by modifying the second part of \eqref{eq:Krasno_iter_synch} via the auxiliary variables $\bld z$:
\begin{equation}\label{eq:update_compact_complete_alg}
\begin{split}
&\tilde{\bld x}^{k} = \mathrm{proj}_{\bld \Omega} \big(\bld x^k-\bld \tau(F(\bld x^k)+\Lambda^\top \bld \lambda^k ) \big) \\
&\tilde{\bld z}^{k} = \bld z^k + \delta \bld L \bld \lambda^k\\
&\tilde{\bld \lambda}^{k} = \mathrm{proj}_{\bR^{mN}_{\geq 0}} \big(\bld \lambda^k+\bld \varepsilon( \Lambda (2\tilde{\bld x}^k-\bld x^k) -\bar b -2\tilde{\bld z}^k +\bld z^k ) \big)\\
&\bld x^{k+1} = \bld x^k +\eta(\tilde{\bld x}^k -\bld x^k)\\
&\bld z^{k+1} = \bld z^k +\eta(\tilde{\bld z}^k -\bld z^k)\\
&\bld \lambda^{k+1} = \bld \lambda^k +\eta(\tilde{\bld \lambda}^k -\bld \lambda^k) \:,\\
\end{split}
\end{equation}
See also Algorithm~\ref{alg:synch_alg}, for the local updates.
The convergence of Algorithm~\ref{alg:synch_alg} to the v-GNE of the game in \eqref{eq:game_formulation} is guaranteed by the following theorem.
\smallskip
\begin{theorem}\label{th:convergence_sync}
Let $\vartheta>\frac{\ell^2}{2\alpha}$, $\bld \varepsilon$, $ \delta$, $\bld \tau>0$ such that $\Phi-\vartheta I\succeq 0$ and $\eta\in(0,\frac{4\alpha\vartheta-\ell^2}{2\alpha\vartheta})$. Then, Algorithm~\ref{alg:synch_alg} converges to the v-GNE of the game in \eqref{eq:game_formulation}.
\hfill\QEDopen
\end{theorem}
\smallskip
\section{Asynchronous Distributed Algorithm }
\label{sec:asynch_case}
In this section, we present the main contribution of the paper, the \textit{\underline{A}synchronous \underline{D}istributed \underline{G}N\underline{E} Seeking Algorithm with \underline{No}de variables} (AD-GENO), namely, the asynchronous counterpart of Algorithm~\ref{alg:synch_alg}. As in the previous section, we first define a preliminary version of the algorithm using the edge auxiliary variables $\bld \sigma$, and then we derive the final formulation via the variable $\bld z$. To achieve an asynchronous update of the agent variables, we adopt the ``ARock" framework \cite{peng2016arock}
\smallskip
\subsection{Algorithm design}
We modify the update rule in \eqref{eq:Krasno_iter_synch} to describe the asynchronism, in the local update of the agent $i$, as follows
\begin{equation}\label{eq:Krasno_asynch}
\varpi^{k+1}=\varpi^k+\eta \Upsilon_i(T\varpi^k-\varpi^k)\,,
\end{equation}
where $\Upsilon_i$ is a real diagonal matrix of dimension $n+(N+M)m$, where the element $[\Upsilon_{i}]_{jj}$ is $1$ if the $j$-th element of $\col( \bld x,\: \bld\sigma,\:\bld \lambda)$ is an element of $\col(x_i,\:\{\sigma_l\}_{l\in\ca E ^{\mathrm{out}}_i},\: \lambda_i)$ and $0$ otherwise
We assume that the choice of which agent performs the update at iteration $k\in\bN_{\geq 0}$ is ruled by an i.i.d. random variable $\zeta^k$, that takes values in $\bld \Upsilon:=\{ \Upsilon_i \}_{i\in\ca N}$. Given a discrete probability distribution $(p_1,\dots,p_N)$, let $\mathbb{P}[\zeta^k = \Upsilon_i] = p_i$, $\forall i\in \ca N$. Therefore, the formulation in \eqref{eq:Krasno_asynch} becomes
\begin{equation}\label{eq:Krasno_asynch_zeta}
\varpi^{k+1}=\varpi^k+\eta \zeta^k(T\varpi^k-\varpi^k)\,.
\end{equation}
We also consider the possibility of delayed information, namely the update \eqref{eq:Krasno_asynch_zeta} can be performed with outdated values of $\varpi^k$. We refer to \cite[Sec.~1]{peng2016arock} for a more complete overview on the topic. Due to the structure of the $\Upsilon_i$, the update of $x_i$, $\lambda_i$ and $\{\sigma_l\}_{l\in\ca E ^{\mathrm{out}}_i} $ are performed at the same moment, hence they share the same delay $\varphi_i^k$ at $k$.
We denote the vector of possibly delayed information at time $k$ as $\hat\varpi^k$, hence the reformulation of \eqref{eq:Krasno_asynch_zeta} reads as
\begin{equation}\label{eq:Krasno_asynch_final}
\varpi^{k+1}=\varpi^k+\eta \zeta^k(T- \textrm{Id})\hat\varpi^k\,.
\end{equation}
Now, we impose that the maximum delay is uniformly bounded.
\smallskip
\begin{stassumption}[Bounded maximum delay]\label{ass:bounded_delay}
The delays are uniformly upper bounded, i.e. there exists $\bar\varphi>0$ such that $\sup_{k\in\bN_{\geq 0}}\max_{i\in\ca N}\{ \varphi_i^k \}\leq \bar\varphi<+\infty$. \hfill\QEDopen
\end{stassumption}
\smallskip
From the computational perspective, we assume that each player $i$ has a public and a private memory. The first stores the information obtained by the neighbours $\ca N_i$. The private is instead used during the update of $i$ at time $k$ and it is an unchangeable copy of the public memory at iteration $k$. The local update rules in Algorithm~\ref{alg:E-ADAGNES} are obtained similarly to Sec.~\ref{sec:alg_develop_synch} for SD-GENO, hence by using the definition of $T$. The obtained algorithm resembles ADAGNES in \cite[Alg.~1]{Pavel:Yi:2018:Asynch}, therefore we name it E-ADAGNES.
\begin{algorithm}[h!]
\DontPrintSemicolon
\textbf{Input:} $k=0$, $\bld x^0 \in \bR^{n} $, $\bld \lambda^0 \in\bR^{mN}$, $\bld \sigma^0=\bld 0_{mM}$, chose $\eta,\, \delta,\, \bld \varepsilon,\, \bld \tau$ as in Theorem~\ref{th:convergence_sync}. \;
\hrule
\smallskip
\textbf{Iteration $k$:} Select the agent $i^k$ with probability $\mathbb{P}(\zeta^k=\Upsilon_{i^k})=p_{i^k}$\;
\textbf{Reading:} Agent $i^k$ copies in its private memory the current values of the public memory, i.e. $x_{j}^{k-\varphi^k_j}$, $\lambda_{j}^{k-\varphi^k_j}$ for $j\in\ca N_{i^k}$ and $\sigma_{l}^{k-\psi^k_l}, \: \forall l\in\ca E_{i^k}$ \;
\textbf{Update:}\;
$\tilde{ x}_{i^k}^k = \mathrm{proj}_{\Omega_{i^k}} \big( x_{i^k}^{k-\varphi^k_{i^k}}-\tau_{i^k}(\nabla_{i^k} f_{i^k}(x_{i^k}^{k-\varphi^k_{i^k}},\hat{\bld x}_{-i^k}^k) + A_{i^k}^\top \lambda_{i^k}^{k-\varphi^k_{i^k}} ) \big) $ \;\bigskip
$\tilde{\sigma}_{l}^k = \sigma_{l}^{k-\varphi^k_{i_k}} + \delta ([E]_{l}\otimes I_m )\hat{\bld\lambda}^{k} \:,\quad \forall l\in \ca E_{i^k}^{\mathrm{out}}$\;\bigskip
$\tilde{\lambda}_{i^k}^k = \mathrm{proj}_{\bR^{m}_{\geq 0}} \big( \lambda_{i^k}^{k-\varphi^k_{i^k}}+\varepsilon_{i^k}( A_{i^k}(2\tilde{x}_{i^k}^k - x_{i^k}^{k-\varphi^k_{i^k}}) - b_{i^k} - ([E^\top]_{i^k}\otimes I_m)(2 \tilde{\bld \sigma}^{k} - \hat{\bld \sigma}^{k} )) \big)$\;\bigskip
$x_{i}^{k+1} = x_{i}^{k-\varphi^k_{i^k}} +\eta(\tilde{ x}_{i}^k - x_{i}^k-\varphi^k_{i^k})$\;
$\sigma_{l}^{k+1} = \sigma_{l}^{k-\varphi^k_{i_k}} +\eta(\tilde{ \sigma}_{l}^k -\sigma_{l}^{k-\psi^k_l})\:,\quad \forall l\in \ca E_{i^k} $ \;
$\lambda_{i}^{k+1} = \lambda_{i}^{k-\varphi^k_{i^k}} +\eta(\tilde{ \lambda}_{i}^k - \lambda_{i}^{k-\varphi^k_{i^k}})$\;
\textbf{Writing:} Agent $i^k$ writes in the public memories of $j\in\ca N_{i^k}$ the new values of $x_{i^k}^{k+1}$, $\lambda_{i^k}^{k+1}$ and $\{\sigma_{l}^{k+1}\}_{l\in\ca E_{i^k}^{\mathrm{out}}}$\; \smallskip
$k\leftarrow k+1$\;
\caption{E-ADAGNES}
\label{alg:E-ADAGNES}
\end{algorithm}
The convergence of the update \eqref{eq:Krasno_asynch_final} is proven relying on the theoretical results provided in \cite{peng2016arock} for the Krasnosel'ski\u i asynchronous iteration.
\begin{theorem}\label{th:convergence_asynch_sigma}
Let $\eta\in(0,\frac{4\alpha\vartheta-\ell^2}{\alpha\vartheta}\frac{cNp_{\min}}{4\bar\varphi\sqrt{p_{\min}}+1}]$, where $p_{\min}:=\min \{p_i\}_{i\in\ca N}$ and $c\in(0,1)$. Then, the sequence $\{\bld x^k\}_{k\in\bN_{\geq 0}}$ defined by Algorithm~\ref{alg:E-ADAGNES} converges to the v-GNE of the game in \eqref{eq:game_formulation} almost surely. \hfill\QEDopen
\end{theorem}
\subsection{Asynchronous, distributed algorithm with node variables (AD-GENO)}
We complete the technical part of the paper by performing the change from auxiliary variables over the edges to variables over the nodes, attaining in this way the final formulation of our proposed algorithm. With Algorithm~\ref{alg:E-ADAGNES} as starting point, we show that this change does not affect the dynamics of the pair $(\bld x,\bld \lambda)$, thus preserving the convergence.
However, in this case, we need to introduce an extra variable for each node $i$, i.e., $\mu_i\in\bR^{m}$. This is an aggregate information that groups all the changes of the neighbours dual variables from the previous update of $i$ to the present iteration. We highlight that these variables are updated during the writing phase of the neighbours, therefore they do not require extra communications between the agents.
\smallskip
\begin{remark}
The need for $\mu_i\in\bR^{m}$ arises from the different update frequency between $\{\sigma_l\}_{l\in\ca E_i}$ and $ z_i$.
Therefore, we cannot characterize the dynamics of $\bld \sigma$, if we define $\bld z=\bld E^\top\bld \sigma$ only.
\end{remark}
\smallskip
Algorithm~\ref{alg:AD-GENO} presents AD-GENO, where $\mu_i$ are rigorously defined.
\smallskip
\begin{algorithm}[h!]
\DontPrintSemicolon
\textbf{Input:} $k=0$, $\bld x^0 \in \bR^{n} $, $\bld \lambda^0 \in\bR^{mN}$, $\bld z^0=\bld 0_{mN}$, chose $\eta,\, \delta,\, \bld \varepsilon,\, \bld \tau$ as in Theorem~\ref{th:convergence_sync}. For all $i\in\ca N$ and $\mu_i=\bld 0_m$. \;
\hrule
\smallskip
\textbf{Iteration $k$:} Select the agent $i^k$ with probability $\mathbb{P}[\zeta^k=\Upsilon_{i^k}]=p_{i^k}$\;
\textbf{Reading:} Agent $i^k$ copies in its private memory the actual values of the public memory, i.e. $x_{j}^{k-\varphi^k_j}$, $\lambda_{j}^{k-\varphi^k_j}$, $z_{j}^{k-\varphi^k_j}$ for $j\in\ca N_{i^k}$ and $\mu_{i}$. Reset the public values of $\mu_{i}$ to $\bld 0_m$.\;
\textbf{Update:}\;
$\tilde{ x}_{i^k}^k = \mathrm{proj}_{ \Omega_{i^k}} \big( x_{i^k}^{k-\varphi^k_{i^k}}-\tau_{i^k}(\nabla_{i^k} f_{i^k}(x_{i^k}^{k-\varphi^k_{i^k}},\hat{\bld x}_{-i^k}^{k})+ A_{i^k}^\top \lambda_{i^k}^{k-\varphi^k_{i^k}} ) \big) $ \;\smallskip
$\tilde{z}_{i^k}^k = z_{i^k}^{k-\varphi^k_i} +\delta\eta\mu_{i^k} $\;\smallskip
$\tilde{\lambda}_{i^k}^k = \mathrm{proj}_{\bR^{m}_{\geq 0}} \left( \lambda_{i^k}^{k-\varphi^k_{i^k}}+\varepsilon_{i^k}( A_{i^k}(2\tilde{x}_{i^k}^k - x_{i^k}^{k-\varphi^k_{i^k}}) \right. - b_{i^k}-\tilde z_{i^k}^{k-\varphi^k_i}\left. -2\delta\sum_{j\in\ca N_i\setminus \{i\}} (\lambda_{i^k}^{k-\varphi^k_{i^k}}-\lambda_{j}^{k-\varphi^k_{j}}) \right)$\;\bigskip
$x_{i}^{k+1} = x_{i^k}^{k-\varphi^k_{i^k}} +\eta(\tilde{ x}_{i^k}^k - x_{i^k}^{k-\varphi^k_{i^k}})$\;
$z_{i^k}^{k+1} = \tilde z_{i^k}^{k} + \eta\delta\sum_{l\in\ca E_{i_k}^{\mathrm{out}}} ([E]_l\otimes I_m )\hat{\bld \lambda}^k $ \;
$\lambda_{i^k}^{k+1} = \lambda_{i^k}^{k-\varphi^k_{i^k}} +\eta(\tilde{ \lambda}_{i^k}^k - \lambda_{i^k}^{k-\varphi^k_{i^k}})$\;
\textbf{Writing:} Agent $i^k$ writes in the public memories of $j\in\ca N_{i^k}$ the new values of $x_{i^k}^{k+1}$ and $\lambda_{i^k}^{k+1}$, for $j\in\ca N_{i^k}\setminus\{i^k\}$ the player $i^k$ also overwrites $\mu_{j}$ as\;
$\mu_{j} \leftarrow \mu_{j} + \lambda_{j}^{k-\varphi^k_{j}} - \lambda_{i}^{k-\varphi^k_{i^k}}$\;\smallskip
$k\leftarrow k+1$\;
\caption{AD-GENO}
\label{alg:AD-GENO}
\end{algorithm}
\smallskip
The convergence of AD-GENO is proven by the following theorem. Essentially, we show that introducing $\bld z$ does not change the dynamics of $(\bld x,\bld \lambda)$.
\smallskip
\begin{theorem}\label{th:convergence_AD-GENO}
Let $\eta\in(0,\frac{4\alpha\vartheta-\ell^2}{\alpha\vartheta}\frac{cNp_{\min}}{4\bar\varphi\sqrt{p_{\min}}+1}]$ with $p_{\min}:=\min \{p_i\}_{i\in\ca N}$ and $c\in(0,1)$. Then, the sequence $\{\bld x^k\}_{k\in\bN_{\geq 0}}$ defined by Algorithm~\ref{alg:AD-GENO} converges to the v-GNE of the game in \eqref{eq:game_formulation} almost surely. \hfill\QEDopen
\end{theorem}
\section{Simulation}
\label{sec:simulations}
This section presents the implementation of AD-GENO to solve a network Cournot game, that models the interaction of $N$ companies competing over $m$ markets. The problem is widely studied and we adopt a set-up similar to the one in \cite{yu:2017distributed,pavel2017:distributed_primal-dual_alg}. We chose $N=8$ companies, each operating $4$ strategies, i.e., $x_i\in\bR^4$, $\forall i\in\ca N$. It ranges in $0\leq x_i \leq \Omega_i$, where $\Omega_i\in\bR^{4}$ ans its elements are randomly drawn from $[10,45]$. The markets are $ m = 4 $, named $A$, $B$, $C$ and $D$. In Figure~\ref{fig:markets_and_network}, an edge between a company and a market is drawn, if at least one of that player's strategies is applied to that market. Two companies are neighbors if they share a market.
\begin{figure}
\centering
\subfloat[][]
{\includegraphics[scale = 0.15]{MarketsInteractiona.pdf}\label{fig:Markets}} \quad
\subfloat[][]
{\includegraphics[scale = 0.15]{AgentNetwork.pdf}\label{fig:Network}} \\
\caption{(a) Interactions of the players $\{1,\dots,8\}$ with the markets $A$, $B$, $C$, $D$, (b) Communication network between players arising from the competition.}
\label{fig:markets_and_network}
\end{figure}
The constraint matrix is $\bld A=[A_1,\dots,A_N]\in \bR^{4\times 32}$ and the columns $k$ of $A_i$ have a nonzero element in position $j$ if the $k$-th strategy of player $i$ is applied to market $j$. The nonzero values are randomly chosen from $[0.6,1]$. The elements of $b\in\bR^{4}$ are the markets' maximal capacities and are randomly chosen from $[20,100]$. The arising inequality coupling constraint is $\bld{Ax}\leq b$. The local cost function is $f_i(x_i,\bld x_{-i})=c_i(\bld x) -P(\bld x)^\top A_ix_i$, where $c_i(\bld x)$ is the cost of playing a certain strategy and $P(\bld x)$ the price obtained by the market. We define the markets price as a linear function $P(\bld x)=\bar P -DA\bld x$, where $\bar P\in\bR^4 $ and $D\in\bR^{4\times 4}$ is a diagonal matrix, the values of their elements are randomly chosen respectively from $[250,500]$ and $[1,5]$. The cost function is quadratic $c_i(\bld x) = x_i^\top Q_i x_i + q_i^\top x_i$, where the elements of the diagonal matrix $Q_i\in\bR^{4\times 4}$ and the vector $q_i\in\bR^4$ are randomly drawn respectively from $[1,8]$ and $[1,4]$.
\begin{figure}
\centering
\subfloat[][]
{\includegraphics[width=.35\textwidth]{NormXVal_hold.pdf}\label{fig:NormXVal}} \\
\subfloat[][]
{\includegraphics[width=.35\textwidth]{LambdaError_hold.pdf}\label{fig:LambdaErr}} \\
\subfloat[][]
{\includegraphics[width=.35\textwidth]{ConstrFeas_hold.pdf}\label{fig:Constr}}\\
\caption{Communication in alphabetic order (blue) versus random communication (red): (a) Normalized distance from equilibrium,(b) Norm of the disagreement vector, (c) Averaged constraints violation (the negative values are omitted).}
\label{fig:simulations}
\end{figure}
We propose two setups, the case of communication over a ring graph with alphabetic order and the case of random communication (in Figure~\ref{fig:simulations}, respectively the blue and red trajectories). In the latter, we only ensure that every $20N$ iterations all the agents performed a similar number of updates. The edges of the graph are arbitrarily oriented. We assume that the agents update with uniform probability, i.e., $P[\zeta^k=\Upsilon_i]=\frac{1}{N}$. The step sizes $\delta,\:\varepsilon,\:\tau$ in AD-GENO are randomly chosen, the first from $[0.5,0.2]$ and the others from $[0.5,0.03]$, in order to ensure $\Phi\succ 0$ and $\eta = 0.35$. The maximum delay is assumed $\bar \varphi = 4$, therefore $\hat\varpi^k$ in \eqref{eq:Krasno_asynch_final} is $\hat\varpi^k = \col(\hat \varpi_1^k,\dots,\hat \varpi_N^k)$ where each $\hat \varpi_i^k$ is randomly chosen from $\{\hat \varpi_i^{k-\bar \varphi},\dots,\hat \varpi_i^k\}$.
The results of the simulations are shown in Figure~\ref{fig:simulations}. In particular, Figure~\ref{fig:NormXVal} presents the convergence of the collective strategy $\bld x^k$ to the v-GNE $\bld x^*$. Furthermore, Figure~\ref{fig:LambdaErr} highlights the convergence of the Lagrangian multipliers to consensus. We noticed that a simple update sequence, as the alphabetically ordered one, leads to a faster convergence than a random one. In general, the more the agents' updates are well mixed the faster the algorithm converge.
\section{Conclusion}
\label{sec:conclusion}
This work propose a variant of the forward-backward splitting algorithm to solve generalized Nash equilibrium problems via asynchronous and distributed information exchange, that is robust to communication delays.
A change of variables based on the node Laplacian matrix of the information-exchange graph allows one to preserve the scalability of the solution algorithm in the number of nodes (as opposed to the number of edges).
Full theoretical and numerical comparison between the proposed solution algorithm and that in \cite{Pavel:Yi:2018:Asynch} is left as future work.
Another interesting topic is the adaptation of the algorithm to the case of changing graph topology, in fact the independence from the edge variables makes this approach more suitable to address this problem.
|
1,108,101,564,519 | arxiv | \section{Introduction}
MixUp is a data augmentation technique originated from computer vision, where convex interpolations of input images and their corresponding labels are used as additional training source. It has empirically been shown to improve the performance of image classifiers, and leads to decision boundaries that transition linearly between classes, providing a smoother estimate of uncertainty ~\citep{Zhang:18}.
Adapting MixUp to language is non-trivial and not well explored. Unlike with images, one can not interpolate texts in the input space directly. Several studies get around this issue by interpolating sentence or word embeddings, and show that MixUp can improve model performance for sentence classification \citep {Guo:19, Sun:20}.
We investigate whether MixUp can be extended to a wider range of NLU tasks including sentiment classification, multiclass classification, sentence acceptability, natural language inference (NLI), and question answering (QA). In addition, we propose a more diverse set of MixUp methods at the 1) Input, 2) Manifold, and 3) sentence-embedding levels for BERT \citep{Devlin:19}, and compare their performance for different task types at different resource settings. At last, we study the effect of MixUp on BERT's calibration, which tells us about the quality of a model's predictive uncertainty \citep{Guo:17}.
We find that 1) MixUp improves model performance over baselines for IMDb and AGNews, particularly at lower resource settings. 2) MixUp can significantly reduce the test loss and calibration error of BERT by up to $50\%$ without sacrificing performance. 3) While MixUp of sentence embeddings can be universally applied to a diverse set of NLU tasks, MixUp methods that interpolate individual tokens, such as Input and Manifold MixUp, are more appropriate for simpler tasks that do not necessarily require syntactic knowledge (eg. sentiment analysis).
\section{Background}
\subsection{MixUp}
In MixUp, convex interpolations of training input and their labels are used instead of the original data for training, with the effect of improved generalization and robustness against adversarial attacks. The following describes the MixUp process:
\begin{equation}\label{eq:1}
\begin{gathered}
\widetilde{x} = \lambda x_i + (1-\lambda)x_j \\
\widetilde{y} = \lambda y_i + (1-\lambda)y_j
\end{gathered}
\end{equation}
where $x_i, x_j$ are the input vectors, $y_i, y_j$ their one-hot coded labels, and $\lambda \in [0,1]$ is sampled from the Beta distribution: $\lambda \sim \beta(\alpha, \alpha)$ during every mini-batch, where $\alpha$ is a hyperparameter. Lower $\alpha$ leads to more even proportions between $\lambda$ and $1-\lambda$.
The application of MixUp can also be extended to the hidden representations. This is referred to as Manifold MixUp \citep{Verma:19}.
\subsection{MixUp for Language}
\citet{Guo:19} investigate two strategies to apply MixUp to language using an RNN architecture: 1) sentence-level MixUp interpolates the sentence embeddings of two sentences, while 2) word-level MixUp interpolate individual word embeddings at each position between two sentences. Both strategies have been shown to improve accuracy and regularize the training for simple sentence classification tasks such as sentiment analysis.
In a similar work done concurrently at the time of our experiments, \citet{Sun:20} applied sentence embedding MixUp to finetune transformers on GLUE tasks \citep{Glue}, and reported improved predictive performance for most tasks.
While \citep{Sun:20} focused on sentence embedding MixUp as the augmentation strategy and predictive performance as the evaluation metric, we evaluated a more diverse set of MixUp methods, including those operating on the input and manifold levels, on additional metrics including negative log-likelihood and model calibration. Unlike the previous work, we did not find significant predictive improvement for the shared GLUE tasks. On the other hand, we showed that MixUp can significantly improve model calibration and reduce overfitting for transformers, and that Input and Manifold MixUp methods work better than sentence embedding MixUp for content-based tasks.
\subsection{Model Calibration}
As neural networks become more widely used in high risk fields such as medical diagnosis and autonomous vehicles, the quality of their predictive uncertainty becomes an important feature. Models should not only be accurate, but also know when they are likely to be wrong.
To quantitatively evaluate a model's predictive uncertainty, \citet{Guo:17} defines calibration as the degree to which a model's predictive scores are indicative of the actual likelihood of correctness. \citet{Thulasidasan:19} show that MixUp can improve calibration for CNN based models. \citet{Desai:20} evaluate the calibration of pretrained transformers and finds it to substantially deteriorate under data shift.
\section{Methods}
\subsection{Architecture}
We used the \texttt{bert-base-uncased} pretrained model based on Hugging Face's implementation \citep{Wolf2019HuggingFacesTS}.
\subsection{Training}
Baseline models are trained using Empirical Risk Minimization (no MixUp) with cross entropy loss. Hyperparameters are given in Appendix \ref{sec:hyper}.
MixUp models are trained using the MixUp methods discussed below, with otherwise identical architecture, hyperparameters, and training dataset (prior to augmentation).
Results are reported for the epoch with the best performance metric (accuracy or MCC).
\subsection{Datasets}
We choose to test on a diverse set of tasks in terms of both difficulties and task types. In particular:
\begin{itemize}
\item \textbf{IMDb}: Sentiment classification for positive and negative movie reviews ~\citep{IMDb}
\item \textbf{AGNews}: Multi-class classification for 4 topics of News articles. ~\citep{AGNews}
\item \textbf{CoLA}: Classification for grammatically correct/incorrect sentences ~\citep{Glue}
\item \textbf{RTE}: Natural language inference task, with entailment and no-entailment classes. ~\citep{Glue}
\item \textbf{BoolQ}: Question-answering task that matches a short passage with a yes / no question about the passage. \citep{SuperGLUE}
\end{itemize}
\begin{table}[h!]
\small
\centering
\begin{tabular}{llrrr}
\toprule \textbf{Task} & \textbf{Metric} & \textbf{Train} & \textbf{Dev} & \textbf{Test} \\
\midrule
IMDb & Accuracy & 25000 & 3200 & 21800 \\
AGNews & Accuracy & 120000 & 2400 & 5200 \\
CoLA & Matt Corr & 8551 & 1043 & 1064 \\
RTE & Accuracy & 2491 & 278 & 3000 \\
BoolQ & Accuracy & 9427 & 3270 & 3245 \\
\bottomrule
\end{tabular}
\caption{\label{font-table} Datasets sizes (unit: sentence)}
\label{table:datasets}
\end{table}
We simulated low resource settings for IMDb and AGNews by training only using a random sample of 32 training examples.
\subsection{Metrics}
We evaluate our models based on 1) Accuracy, 2) Cross Entropy Loss, and 3) Expected Calibration Error (ECE).
Classifiers capable of reliably forecasting their accuracy are considered to be well-calibrated. For instance, a calibrated classifier should be correct $80\%$ of the time on examples to which it assigns $80\%$ confidence (the winning probability). Let the classifier's confidence for its prediction $\hat{Y}$ be $C$. Then, the Expected Calibration Error is the expectation of the absolute difference between the accuracy at a given confidence level and the actual confidence level:
\begin{equation}\label{eq:2}
\begin{gathered}
\mathrm{E_C}[|P(Y=\hat{Y}|C=c) - c|]
\end{gathered}
\end{equation}
Details on how to empirically estimate this quantity can be found in Appendix \ref{sec:calib}.
\subsection{MixUp for Transformers}
We investigate three variants of MixUp that can be applied to the transformer architecture, in particular BERT: 1) CLS MixUp, 2) input-token MixUp, and 3) Manifold MixUp. As we shall see, the last is a generalization of the former two.
\subsubsection{CLS MixUp}
Unlike in computer vision, where inputs are preprocessed to the same dimensions, it is not straightforward to interpolate texts in the input space directly. Instead, we can perform MixUp on the sentence embeddings, which in the specific case of BERT are represented by the pooled CLS tokens from the final layer. Formally, given a pair of training input sentences $x_1, x_2$, their one hot label vectors $y_1, y_2$, and the sentence embeddings $f(x_1), f(x_2)$ generated using the BERT encoder, we interpolate their embeddings and the labels:
\begin{equation}\label{eq:3}
\begin{gathered}
\widetilde{x} = \lambda\: f(x_1) + (1-\lambda)\:f(x_2) \\
\widetilde{y} = \lambda y_1 + (1-\lambda)y_2 \\
\widetilde{y_p} = \mathrm{Classifier}(\widetilde{x})
\end{gathered}
\end{equation}
MixUp samples, in this case embeddings and labels, are generated by interpolating each sample in the batch with another randomly selected sample. We feed the interpolated embeddings into the classifier (while discarding the unmixed embeddings, thus maintaining the same batch size) and minimize the cross entropy loss between the predictions $\tilde{y_p}$ and the interpolated labels $\tilde{y}$ in a given batch by summing their losses and backpropagating it through the entire computational graph.
We repeat this process stochastically for every batch, where a new $\lambda$ is sampled every time (Eq \ref{eq:1}).
\subsubsection{Input MixUp}
Though less intuitive, we can also interpolate individual input token embeddings at each index between two sentences.
Given sentences $x_1=\left[x_1^1, x_1^2, \cdots, x_1^m \right]$, $x_2 = \left[x_2^1, x_2^2, \cdots, x_2^n\right]$ and their one hot label vectors $y_1$ and $y_2$, where $m \geq n$ and $x^i$ represents the $i^{\mathrm{th}}$ input token embedding, we pad $x_1$ and $x_2$ to the same length using the \texttt{SEP} token from BERT vocabulary (see Appendix \ref{sec:padding} for an evaluation of different padding options). We can then generate a new sentence $\widetilde{x}$ by taking a MixUp of individual input token embeddings from each position:
\begin{equation}\label{eq:4}
\begin{gathered}
\widetilde{x}^i = \lambda x_1^i + (1 - \lambda) x_2^i \ ; \ i \in (1,...,m) \\
\widetilde{x} = [\widetilde{x}^1, \widetilde{x}^2, ..., \widetilde{x}^m] \\
\widetilde{y} = \lambda y_1 + (1-\lambda)y_2 \\
\widetilde{y_p} = \mathrm{Classifier}(f(\widetilde{x}))
\end{gathered}
\end{equation}
Where $f(x)$ is the sentence embedding returned by the BERT encoder given input tokens $x$.
\subsubsection{Manifold MixUp}
\citet{Verma:19} demonstrated that applying MixUp to the hidden represenations, in addition to the inputs, yields additional benefit in reducing test errors in computer vision. Given that BERT consists of multiple layers, we also investigate the effect of this technique on its training.
We proceed as follows: for each batch, select a random layer $k$ from a set of eligible layers $S$ in the BERT with $n$ encoder layers. This set may include the input layer $0$ as well as the pooled sentence embedding layer $n+1$. We apply MixUp on the hidden representation embeddings, before feeding the interpolated embeddings to the next layer to continue with training. The loss is computed using the logits and the interpolated labels (Figure \ref{fig:manifold}).
When $S$ only contains $0$, Manifold MixUp reduces to Input MixUp. When $S$ only contains $n+1$, it reduces to CLS MixUp.
\begin{figure}[t]
\centering
\includegraphics[width=\columnwidth]{manifold_higher_res.png}
\caption{Schematics of Manifold MixUp. We mix the layer $k$ embeddings and the one-hot labels of two training samples using the same proportions $\lambda$ and $(1-\lambda)$ (See eq 1). }
\label{fig:manifold}
\end{figure}
\section{Results}
Our IMDB (Table \ref{table:1}) and AGNews (Table \ref{table:2}) results indicate that Manifold, Input, and CLS MixUp methods improve model robustness and calibration across a range of data regimes; the improvements are more significant when the amount of training data is scarce, which matches our expectation since MixUp can be seen as a form of data augmentation.
Manifold MixUp offers the best improvement for model calibration; in fact, the Manifold MixUp model trained on merely 32 samples is equally or more calibrated in comparison to the baseline model trained using the full training set.
Both sentiment analysis on IMDB and news category classification on AGNews are relatively simple tasks that do not necessarily require the model to have syntactic knowledge; instead, they are likely to be solved by only inspecting the sentiment or classification of the individual words. For example, a movie review with more negative words than positive ones is likely to be negative overall. Therefore, we expect Input and Manifold MixUp, which interpolate embeddings at the token level, to work well here; indeed, they outperform baseline and CLS MixUp in these simpler scenarios.
\begin{table}[t]
\small
\centering
\begin{tabular}{llrcr}
\toprule \textbf{Size} & \textbf{Model} & \textbf{Acc} & \textbf{Loss} & \textbf{ECE} \\
\midrule
Full & Base & $91.8 \pm 0.1$ & $61 \pm 8$ & $7.7 \pm 0.1$ \\
& CLS & $92.0 \pm 0.1$ & $29 \pm 1$ & $5.8 \pm 0.2$ \\
& Input & $92.1 \pm 0.1$ & $37 \pm 2$ & $6.8 \pm 0.2$ \\
& Mani & $91.8 \pm 0.1$ & $27 \pm 1$ & $4.4 \pm 0.6$ \\
\midrule
$32$ & Base & $68.2 \pm 1.6$ & $180 \pm 19$ & $29.0 \pm 1.5$ \\
& CLS & $69.6 \pm 1.7$ & $86 \pm 4$ & $21.0 \pm 2.8$ \\
& Input & $70.8 \pm 1.2$ & $61 \pm 7$ & $8.9 \pm 2.3$ \\
& Mani & $72.3 \pm 1.0$ & $55 \pm 2$ & $3.4 \pm 0.9$ \\
\bottomrule
\end{tabular}
\caption{\label{font-table} MixUp experiments for IMDB using various models (baseline, cls, input, manifold), train sizes. Standard errors over five runs. }
\label{table:1}
\end{table}
\begin{table}[t]
\small
\centering
\begin{tabular}{llrcr}
\toprule \textbf{Size} & \textbf{Model} & \textbf{Acc} & \textbf{Loss} & \textbf{ECE} \\
\midrule
Full & Base & $93.7 \pm 0.1$ & $36 \pm 7$ & $5.0 \pm 0.6$ \\
& CLS & $93.7 \pm 0.1$ & $28 \pm 1$ & $4.3 \pm 0.3$ \\
& Input & $93.8 \pm 0.1$ & $25 \pm 1$ & $4.1 \pm 0.3$ \\
& Man & $93.7 \pm 0.1$ & $23 \pm 5$ & $3.0 \pm 0.2$ \\
\midrule
$32$ & Base & $81.9 \pm 0.6$ & $72 \pm 9$ & $9.9 \pm 1.9$ \\
& CLS & $83.4 \pm 0.6$ & $55 \pm 2$ & $7.2 \pm 0.7$ \\
& Input & $82.9 \pm 0.3$ & $53 \pm 2$ & $6.2 \pm 2.4$ \\
& Mani & $82.3 \pm 0.4$ & $52 \pm 1$ & $4.8 \pm 1.8$ \\
\bottomrule
\end{tabular}
\caption{\label{font-table} MixUp experiments for AGNews using various train sizes. Standard errors over five runs. }
\label{table:2}
\end{table}
Unlike IMDB and AGNews, the remaining tasks CoLA (acceptability), RTE (NLI), and BoolQ (QA) rely on syntactic knowledge. Therefore, we would expect Input and Manifold MixUp to work relatively poorly for these tasks, since they can disrupt sentence syntax (eg. by interpolating nouns with verbs). This is empirically confirmed when we applied the three MixUp methods to training on the CoLA dataset (Table \ref{table:3}), where the Input and Manifold MixUp models under-perform baseline.
\begin{table}[t]
\small
\centering
\begin{tabular}{llrcr}
\toprule \textbf{Task} & \textbf{Model} & \textbf{MCC/Acc} & \textbf{Loss} & \textbf{ECE} \\
\midrule
CoLA & Base & $52.5 \pm 0.6$ & $56 \pm 4$ & $11.5 \pm 0.3$ \\
& CLS & $52.8 \pm 0.6$ & $48 \pm 1$ & $9.7 \pm 0.1$ \\
& Input & $50.8 \pm 0.8$ & $48 \pm 8$ & $10.1 \pm 1.4$ \\
& Mani & $51.0 \pm 1.0$ & $45 \pm 2$ & $9.0 \pm 0.9$ \\
\midrule
RTE & Base & $63.8 \pm 1.4$ & $125 \pm 20$ & $27.6 \pm 1.9$ \\
& CLS & $63.8 \pm 1.0$ & $80 \pm 4$ & $20.8 \pm 1.3$ \\
\midrule
BoolQ & Base & $73.4 \pm 0.3$ & $170 \pm 7$ & $24.8 \pm 0.3$ \\
& CLS & $73.1 \pm 0.3$ & $89 \pm 1$ & $21.8 \pm 0.3$ \\
\hline
\end{tabular}
\caption{\label{font-table} MixUp experiments for CoLA, RTE, and BoolQ. CoLA is evaluated using Test Matthew Correlation Coefficient (MCC), others using Test Accuracy. Standard errors over five runs.}
\label{table:3}
\end{table}
For the three more advanced tasks, while CLS MixUp exerts negligible effect on the model's predictive performance, it still significantly reduces test loss and improves calibration without sacrificing performance. The evolution of test losses across training steps for MixUp and baseline models are plotted in Appendix \ref{sec:loss}, demonstrating the ability of MixUp to reduce model overfitting. We note that in contrast to our experiments, \citet{Sun:20} were able to obtain improvements using CLS MixUp for CoLA and RTE using a different training regimen, where MixUp is only applied during the last half of training epochs.
\section{Conclusion}
In this work, we propose CLS, Input, and Manifold MixUp methods for the transformer architecture and apply them to a range of NLU tasks including sentiment analysis, multi-class classification, acceptability, NLI, and QA. We find that MixUp can improve the predictive performance of simpler tasks. More generally, MixUp can significantly reduce model overfitting and improve model calibration.
|
1,108,101,564,520 | arxiv | \section*{References}}
\newtheorem{theorem}{Theorem}[section]
\theoremstyle{definition}
\newtheorem{problem}[theorem]{Problem}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{algorithm}[theorem]{Algorithm}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{rrule}[theorem]{Reduction Rule}
\newtheorem{transformation}[theorem]{Transformation}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{example}[theorem]{Example}
\newcommand{\mathds{N}}{\mathds{N}}
\newcommand{\mathds{Z}}{\mathds{Z}}
\newcommand{\text{col}}{\text{col}}
\DeclareRobustCommand{\NoKernelAssume}{\ensuremath{\text{NP}\subseteq \text{{coNP/poly}}}}
\numberwithin{equation}{section}
\numberwithin{figure}{section}
\numberwithin{table}{section}
\newcommand{\decprob}[3]{
\pagebreak[3]
\begin{problem}\textsc{#1}
\setlength{\topsep}{0pt}
\begin{compactdesc}
\item[\it Input:] #2
\item[\it Question:] #3
\end{compactdesc}
\end{problem}
}
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\pagebreak[3]
\begin{problem}\textsc{#1}
\setlength{\topsep}{0pt}
\begin{compactdesc}
\item[\it Input:] #2
\item[\it Goal:] #3
\end{compactdesc}
\end{problem}
}
\title{Parameterized algorithms for power-efficient connected symmetric wireless sensor
networks}
\author[1]{Matthias Bentert}
\author[2,3]{René van Bevern}
\author[1]{André Nichterlein}
\author[1]{Rolf~Niedermeier}
\affil[1]{Institut f\"ur Softwaretechnik und Theoretische Informatik, TU~Berlin, Germany, \texttt{\{matthias.bentert,andre.nichterlein,rolf.niedermeier\}@tu-berlin.de}}
\affil[2]{Novosibirsk State University, Novosibirsk, Russian Federation, \texttt{[email protected]}}
\affil[3]{Sobolev Institute of Mathematics of the Siberian Branch of the Russian Academy of Sciences, Novosibirsk, Russian Federation}
\date{}
\newcommand{\textsc{MinPSC}}{\textsc{MinPSC}}
\newcommand{\textsc{MinPCCS}}{\textsc{MinPCCS}}
\newcommand{\textsc{MinPSC-AL}}{\textsc{MinPSC-AL}}
\newcommand{\(d\)-\textsc{PSC-AL}}{\(d\)-\textsc{PSC-AL}}
\newcommand{$k$-\textsc{PSC}}{$k$-\textsc{PSC}}
\newcommand{\textsc{\miPoSyCo}}{\textsc{\textsc{MinPSC}}}
\newcommand{\textsc{Annotated \miPoSyCo{}}}{\textsc{Annotated \textsc{MinPSC}{}}}
\newcommand{\textsc{Set Cover}}{\textsc{Set Cover}}
\newcommand{\textsc{Minimum Set Cover}}{\textsc{Minimum Set Cover}}
\DeclareMathOperator{\opt}{Opt}
\newcommand{margin}{margin}
\pagestyle{plain}
\begin{document}
\maketitle
\begin{abstract}
\noindent
We study an NP-hard problem motivated by
energy-efficiently maintaining the connectivity of
a symmetric wireless sensor communication network.
Given an edge-weighted \(n\)-vertex graph,
find a connected spanning subgraph of minimum cost,
where the cost is determined by letting each vertex
pay the most expensive edge incident to it in the subgraph.
We provide an algorithm
that works in polynomial time
if one can find a set of obligatory edges that
yield a spanning subgraph
with \(O(\log n)\)~connected components.
We also provide a linear-time algorithm
that reduces any input graph that consists of a tree
together with \(g\)~additional edges
to an equivalent graph with \(O(g)\)~vertices.
Based on this, we obtain a polynomial-time algorithm for~\(g\in O(\log n)\).
On the negative side, we show that
\(o(\log n)\)-approximating the difference~\(d\) between the optimal solution cost
and a natural lower bound is NP-hard
and that there are presumably no exact algorithms
running in \(2^{o(n)}\)~time
or in \(f(d)\cdot n^{O(1)}\)~time for any computable function~\(f\).
\end{abstract}
\noindent
\textbf{Keywords:}
monitoring areas and %
backbones,
parameterized complexity, %
%
color coding,
%
data reduction,
%
parameterization above lower bounds,
approximation hardness,
spanning trees
\section{Introduction}
We consider a well-studied graph problem arising in the context of saving
power in maintaining the connectivity of
symmetric wireless sensor communication networks.
Our problem,
which falls into the category of survivable network design~\cite{Pan11},
is formally defined as follows
(see \cref{fig:exampleMPSC} for an example).
\optprob{\textsc{Min-Power Symmetric Connectivity} (\textsc{MinPSC}{})%
}%
{\label{prob:miposyco}A connected undirected graph~$G = (V,E)$ with \(n\)~vertices, \(m\)~edges,
and edge weights (costs)~$w\colon E \to \mathds{N}$.}%
{Find a connected spanning subgraph~$T=(V,F)$ of~$G$ that minimizes
\begin{align*}
\sum_{v \in V} \max_{\{u,v\}\in F}w(\{u,v\}).\label{gf}
\end{align*}
}
\noindent
We denote the minimum cost of a solution to an \textsc{MinPSC}{} instance~\(I=(G,w)\) by~\(\opt(I)\).
Throughout this work,
\emph{weights} always refer to edges
and
\emph{cost} refers to vertices or subgraphs.
For showing hardness results,
we will also consider the \emph{decision version}
of \textsc{MinPSC}{}, which we call $k$-\textsc{PSC}{}.
Herein the problem is to decide
whether an \textsc{MinPSC}{} instance~\(I=(G,w)\)
satisfies \(\opt(I)\leq k\).
\begin{figure}[t]
\centering
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\node[circle,draw, label=$3$] (v6) [below left= of v3,yshift=0.5cm] {$v_6$};
\path[draw, ultra thick] (v1)-- node [above]{$5$}(v2);
\path[draw, ultra thick] (v2)-- node [above]{$6$}(v3);
\path[draw, ultra thick] (v3)-- node [above]{$5$}(v4);
\path[draw] (v2)-- node [below,xshift=-0.1cm]{$4$}(v5);
\path[draw, ultra thick] (v3)-- node [below,xshift=0.1cm]{$3$}(v6);
\path[draw, ultra thick] (v5)-- node [below]{$1$}(v6);
\end{tikzpicture}
\caption{A graph with positive edge weights and an optimal solution (bold edges).
Each vertex pays the most expensive edge incident to it in the solution (the numbers next to the vertices).
The cost of the solution is
the sum of the costs paid by each vertex.
Note that the optimal
solution has cost~$26$ while a minimum spanning tree
(using edge~\(\{v_2,v_5\}\) instead of~$\{v_2,v_3\}$)
has cost~$27$ (as a \textsc{MinPSC}{} solution).}
\label{fig:exampleMPSC}
\end{figure}
\cref{fig:exampleMPSC} reveals that computing
a minimum-cost spanning tree typically does not yield an optimal
solution for \textsc{MinPSC}{} (also see \citet{EPS13}
for a further
discussion concerning the relationship to minimum-cost spanning trees).
In this work, we provide a refined %
computational complexity analysis %
by initiating parameterized complexity studies
of \textsc{MinPSC}{} (and its decision version). In this way, we complement previous findings mostly concerning %
polynomial-time approximability~\cite{ACM+06,CPS04,EPS13}, heuristics and
integer linear programming~\cite{ACM+06,EMP17,MG05}, and computational
complexity analysis for special cases~\cite{CK07,CPS04,EPS13,HW16}.
\paragraph{Our contributions.}
Our work is driven by asking when small
input-specific parameter values allow for fast (exact) solutions
in practically relevant special cases.
Our two fundamental ``use case scenarios'' herein are
monitoring areas and infrastructure backbones.
Performing a parameterized complexity analysis,
we obtain new
encouraging exact algorithms together with new hardness results,
all summarized in \cref{tab:results}:
\begin{table}[t]
\caption{Overview on our results, using the following terminology:
\(n\)---number of vertices,
\(m\)---number of edges,
\(g\)---size of a minimum feedback edge set,
\(d\)---difference between optimal solution cost and a lower bound (see \cref{prob:mpscal}),
\(c\)---connected components of subgraph consisting of obligatory edges (see \cref{def:oblig}).
\textsc{MinPSC-AL}{} is the problem of computing the minimum value of~\(d\) (\cref{prob:mpscal}),
\(d\)-\textsc{PSC-AL}{} is the corresponding decision problem.}
\label{tab:results}
\setlength{\tabcolsep}{2.8pt}
\renewcommand{\arraystretch}{1.25}
\begin{tabularx}{\textwidth}{crXl}
\toprule
& problem & result & reference\\
\midrule
\multirow{4}{*}{\rotatebox[origin=c]{90}{Sec.~\ref{sec:ccs}}}
& \textsc{\miPoSyCo}
& solvable in~$O(\ln(1/\varepsilon) \cdot (36e^2/\sqrt{2\pi})^c \cdot n^4/\sqrt{c})$ time with error probability at most~$\varepsilon$
& \cref{thm:ccs}\\
& \textsc{\miPoSyCo} & solvable in~$c^{O(c\log c)} \cdot n^{O(1)}$ time & \cref{thm:ccs}\\
& \textsc{\miPoSyCo} & solvable in~$O(3^n \cdot (n+m))$ time & \cref{prop:exalg}\\
\midrule
\multirow{2}{*}{\rotatebox[origin=b]{90}{\!\!Sec.~\ref{sec:fes}}}
& \textsc{\miPoSyCo} &
linear-time data reduction algorithm that guarantees at most
\(40g-26\) vertices and \(41g-27\) edges
& \cref{thm:feskernel} \\
\midrule
\multirow{3}{*}{\rotatebox[origin=c]{90}{Sec.~\ref{sec:hardness}}}
& \textsc{MinPSC-AL} & NP-hard to approximate within a factor of~$o(\log n)$ &
\cref{thm:hard}\eqref{part:inapprox}\\
& \(d\)-\textsc{PSC-AL} & $W[2]$-hard when parameterized by~$d$ & \cref{thm:hard}\eqref{part:w[2]-compl}\\
& $k$-\textsc{PSC} & not solvable in~$2^{o(n)}$~time unless ETH fails& \cref{thm:hard}\eqref{part:eth-hard}\\
\bottomrule
\end{tabularx}
\end{table}
\looseness=-1
In \cref{sec:ccs},
we provide an (exact) algorithm for \textsc{MinPSC}{}
that works in polynomial time
if one can find a set of \emph{obligatory} edges that can be added
to any optimal solution and
yield a spanning subgraph with \(O(\log n)\) connected components.
In particular, this means that we show fixed-parameter tractability
for \textsc{MinPSC}{} with respect to the parameter ``number~\(c\) of connected
components in the spanning subgraph consisting of obligatory edges''.
Cases with small~\(c\) occur, for example,
in grid-like sensor arrangements,
which arise when monitoring
areas~\cite{ZH05,ZEAC09}.
In \cref{sec:fes},
we provide a linear-time algorithm
that reduces any input graph consisting of a tree
with \(g\)~additional edges
to an equivalent graph with \(O(g)\)~vertices and edges
(a partial kernel in terms of
parameterized complexity,
since the edge weights remain unbounded).
Combined with the previous result,
this yields fixed-parameter tractability with respect to the
parameter~\(g\) (also known as the feedback edge number of a graph),
and, in particular,
a polynomial-time algorithm for \(g\in O(\log n)\).
Such tree-like graphs occur when monitoring
backbone infrastructure
or pollution levels along waterways.
We provide some negative (that is, intractability) results in
\cref{sec:hardness}:
We show that
\(o(\log n)\)-approxima\-ting the difference~\(d\) between
the minimum solution cost
and a natural lower bound is NP-hard.
Moreover, we prove W[2]-hardness with respect to the parameter~$d$, that is,
there is presumably no algorithm
running in time \(f(d)\cdot n^{O(1)}\) for any computable function~\(f\).
Finally, assuming the Exponential Time Hypothesis (ETH), we show that
there is no \(2^{o(n)}\)-time algorithm for \textsc{MinPSC}{}.
\section{Parameterizing by the number of connected components
induced by obligatory edges}
\label{sec:ccs}
This section presents an algorithm that solves \textsc{MinPSC}{}
efficiently
if we can find \emph{obligatory edges}
that can be added to any optimal solution
and yield
a spanning subgraph with few connected components.
This is the case, for example,
when sensors are arranged in a grid-like manner,
which saves energy when monitoring areas \citep{ZH05,ZEAC09}.
To find obligatory edges,
we use a lower bound~\(\ell(v)\)
on the cost paid by each vertex~\(v\)
in the goal function of \textsc{MinPSC}{} (\cref{prob:miposyco}).
\begin{definition}[vertex lower bound{}s] \label{def:plb}
\emph{Vertex lower bound{}s} are a function~\(\ell\colon V\to\mathds{N}\)
such that,
for any solution~\(T=(V,F)\) of \textsc{MinPSC}{} and any vertex~\(v\in V\), it
holds that
\[
\max_{\{u,v\}\in F} w(\{u,v\})\geq \ell(v).
\]
\end{definition}
\begin{example}\label{ex:nn}
\looseness=-1
A trivial vertex lower bound{}~\(\ell(v)\) is given by the weight
of the lightest edge incident to~\(v\) because \(v\)~has to be
connected to some vertex in any solution.
Moreover,
since any edge~\(\{u,v\}\)
incident to a degree-one vertex~$u$ will be part of any solution,
one can choose~\(\ell\)
so that \(\ell(u) = w(\{u,v\})\) and \(\ell(v) \geq w(\{u,v\}\).
\end{example}
\noindent
Clearly, coming up with good vertex lower bound{}s
is a challenge on its own.
Once we have vertex lower bound{}s,
we can compute an \emph{obligatory subgraph},
whose edges
we can add to any solution
without increasing its cost:
\begin{definition}[obligatory subgraph]\label{def:oblig}
The \emph{obligatory subgraph~\(G_\ell\)} induced by
vertex lower bound{}s~\(\ell\colon V\to\mathds{N}\) for a graph~\(G=(V,E)\) consists
of all vertices of~\(G\) and all \emph{obligatory edges} \(\{u,v\}\) such that
\(\min\{\ell(u),\ell(v)\}\geq w(\{u,v\})\).
\end{definition}
\noindent
The better the vertex lower bound{}s~\(\ell\),
the more obligatory edges they potentially induce,
thus reducing the number~\(c\) of connected components
of~\(G_\ell\).
Yet already the simple vertex lower bound{}s in \cref{ex:nn}
may yield obligatory subgraphs
with only a few connected components in some applications:
\begin{example}
Consider the vertex lower bounds~\(\ell\) from \cref{ex:nn}.
If we arrange sensors in a grid,
which is the most energy-efficient arrangement of sensors
for monitoring areas \citep{ZH05,ZEAC09},
then \(G_\ell\) has only one connected component.
The number of connected components
may increase due to sensor defects
that disconnect the grid
or
due to varying sensor distances within the grid.
The worst case
is if the sensors have pairwise distinct
distances. Then, \(G_\ell\) has only one edge and
\(n-1\)~connected components.
\end{example}
\noindent
\looseness=-1
The number~\(c\) of connected components in~\(G_\ell\)
can easily be exploited in an exact \(O(n^{2c})\)-time
algorithm for \textsc{MinPSC}{},\footnote{To connect the \(c\)~components of~\(G_\ell\),
one has to add \(c-1\)~edges.
These have at most \(2c-2\)~end points.
One can try all \(n^{2c-2}\)~possibilities
for choosing these end points and
check each resulting graph for connectivity in \(O(n+m)\subseteq O(n^2)\)~time.}
which runs in polynomial time for constant~\(c\),
yet is inefficient already for small values of~\(c\).
We will show, among other things, a randomized algorithm
that runs in polynomial time for \(c\in O(\log n)\):
\begin{theorem}\label{thm:ccs}
\textsc{MinPSC}{} with vertex lower bound{}s~\(\ell\) is solvable
\begin{enumerate}[(i)]
\item\label{thm:ccs1} in \(O(\ln1/\varepsilon \cdot (36e^2/\sqrt{2\pi})^c\cdot 1/\sqrt{c}\cdot n^4)\)~time by a randomized algorithm with error probability at most~\(\varepsilon\)
for any given \(\varepsilon\in(0,1)\), and
\item\label{thm:ccs2} in \(c^{O(c\log c)}\cdot n^{O(1)}\)~time by a deterministic algorithm,
\end{enumerate}
where \(c\)~is the number of connected components of the obligatory subgraph~\(G_\ell\).
\end{theorem}
\begin{remark}
The deterministic algorithm
in \cref{thm:ccs}\eqref{thm:ccs2}
is primarily of theoretical interest,
because it classifies \textsc{MinPSC}{}
as \emph{fixed-parameter tractable} parameterized by~\(c\).
Practically,
the randomized algorithm
in \cref{thm:ccs}\eqref{thm:ccs1}
seems more promising.
The number of connected components
of obligatory subgraphs
has recently also been exploited
in fixed-parameter algorithms for problems
of servicing links in transportation networks
\citep{GWY17,SBNW11,SBNW12,BKS17},
which led to practical results.
\end{remark}
\noindent
The rest of this section
proves
\cref{thm:ccs}.
The proof also yields the following
deterministic algorithm for \textsc{MinPSC}{},
which will be
interesting in combination with
the data reduction algorithm in \cref{sec:fes}.
It is much faster than
the trivial algorithm
enumerating all of the possibly \(n^{n-2}\)~spanning trees:
\begin{proposition}\label{prop:exalg}
\textsc{MinPSC}{} can be solved in \(O(3^{n}\cdot(m+n))\)~time.
\end{proposition}
\noindent
Like some known approximation algorithms
for \textsc{\miPoSyCo}{} \citep{HW16,ACM+06},
our algorithms in \cref{thm:ccs}
work by adding edges to~\(G_\ell\)
in order to connect its \(c\)~connected components.
In contrast to these approximation algorithms,
our algorithms will find an \emph{optimal} set of edges to add.
To this end,
they work on a \emph{padded} version~\(G_{\ell}^{\bullet}{}\) of the input graph~\(G\),
in which each connected component of~\(G_\ell\) is turned into a clique.
Then,
it is sufficient to search for connected subgraphs of~\(G_{\ell}^{\bullet}{}\)
that contain at least one vertex of each connected component of~\(G_\ell\):
We can always add the edges in~\(G_\ell\) to such subgraphs
in order to obtain a connected spanning subgraph of~\(G\).
\begin{definition}[padded graph, components]\label{def:padded}
Let \(\ell\colon V\to\mathds{N}\) be vertex lower bound{}s for a graph~\(G=(V,E)\).
We denote the \(c\)~connected components
of the obligatory subgraph~\(G_\ell\) by~\(G_\ell^1,G_\ell^2,\dots,G_\ell^c\).
The \emph{padded graph}~\(G_{\ell}^{\bullet}{}=(V,E_{\ell}^{\bullet}{})\) with edge weights~\(w_{\ell}^{\bullet}{}\colon E_{\ell}^{\bullet}{}\to\mathds{N}\)
is obtained from~\(G\) with edge weights~\(w\colon E\to\mathds{N}\)
by adding zero-weight edges
between each pair of
non-adjacent vertices in~\(G_\ell^i\)
for each \(i\in\{1,\dots,c\}\).
\end{definition}
\noindent
To solve a \textsc{\miPoSyCo}{} instance~\((G,w)\)
with vertex lower bound{}s~\(\ell\colon V\to\mathds{N}\),
we have to add \(c-1\)~edges to~\(G_\ell\)
in order to connect its \(c\)~connected components.
These edges have at most \(2c-2\)~endpoints.
Thus,
we need to find a minimum-cost
connected subgraph in~\(G_{\ell}^{\bullet}{}\) that
\begin{itemize}
\item contains at most \(2c-2\)~vertices,
\item contains at least one vertex of each connected component of~\(G_\ell\),
\item such that each of its vertices~\(v\)
pays at least the cost~\(\ell(v)\)
that it would pay in any optimal solution to the \textsc{\miPoSyCo}{} instance~\((G,w)\).
\end{itemize}
We will do this using the color coding technique introduced by \citet{AYZ95}:
randomly color the vertices of~\(G_{\ell}^{\bullet}{}\)
using at most \(2c-2\)~colors
and then search for connected subgraphs of~\(G_{\ell}^{\bullet}{}\)
that contain exactly one vertex of each color.
Formally,
we will solve the following auxiliary problem on~\(G_{\ell}^{\bullet}{}\).
\optprob{Min-Power Colorful Connected Subgraph (\textsc{MinPCCS}{})}%
{\label[problem]{prob:mcpu}A connected undirected graph~$G = (V,E)$, edge weights~$w\colon E \to \mathds{N}$, vertex colors~\(\text{col}\colon V\to\mathds{N}\), a function~\(\ell\colon V\to\mathds{N}\) and a color subset~\(C\subseteq\mathds{N}\).}%
{Compute a connected subgraph~\(T=(W,F)\) of~\(G\) such that
\(\text{col}\) is a bijection between \(W\) and~\(C\) and such that $T$~minimizes
\[
\sum_{v \in W} \max \Bigl\{ \ell(v), \max_{\{u,v\}\in F}w(\{u,v\}) \Bigr\}.
\]
}
\noindent
Note that,
in the definition of \textsc{MinPCCS}{},
the function \(\ell\colon V\to \mathds{N}\)
does not necessarily give vertex lower bound{}s,
but makes sure
that each vertex~\(v\in V\)
pays at least~\(\ell(v)\) in any feasible solution
to \textsc{MinPCCS}{}.
In contrast to the usual way of applying color coding,
we cannot simply color
the vertices of our input graph~\(G\)
\emph{completely} randomly
and then apply an algorithm for \textsc{MinPCCS}{}:
One component of~\(G_\ell\) could
contain all colors and, thus,
a connected subgraph containing all colors
does not necessarily connect the components of~\(G_\ell\).
Instead,
we employ a trick that was previously applied
mainly heuristically in algorithm engineering in order to
increase the success probability of color coding algorithms \citep{BBF+11,BHK+10,DSG+08}:
Since we know that our sought subgraph
contains at least one vertex of each connected
component of~\(G_\ell\),
we color the connected components of~\(G_\ell\)
using
pairwise disjoint color sets.
Herein,
we first ``guess'' how many vertices~\(c_i\)
of each connected component~\(G_\ell^i\) of~\(G_\ell\)
the sought subgraph will contain
and use \(c_i\)~colors to color each component~\(G_\ell^i\).
We thus arrive at the following algorithm for \textsc{\miPoSyCo}{}:
\pagebreak[3]
\begin{algorithm}[for \textsc{\miPoSyCo}{}]%
\leavevmode
\small
\label[algorithm]{alg:ccs}
\begin{compactdesc}
\item[\it Input:] A \textsc{\miPoSyCo}{} instance~\(I=(G,w)\),
vertex lower bound{}s~\(\ell\colon V\to\mathds{N}\) for~\(G=(V,E)\), an upper bound~\(\varepsilon\in(0,1)\) on the error probability.
\item[\it Output:] A solution for~\(I\) that is optimal with probability at least~\(1-\varepsilon\).
\end{compactdesc}
\begin{compactenum}
\item \(c\gets{}\)number of connected components of the obligatory subgraph~\(G_\ell\).
\item\label{lin:comploop} \textbf{for each} \(c_1,c_2,\dots,c_c\in\mathds{N}^+\) such that \(\sum_{i=1}^cc_i\leq 2c-2\) \textbf{do}
\item\label{lin:colchoice} \quad choose pairwise disjoint~\(C_i\subseteq\{1,\dots,2c-2\}\)
with \(|C_i|=c_i\) for \(i\in\{1,\dots,c\}\).
\item\label{lin:repeat} \quad\textbf{repeat} \(t:=\ln\varepsilon/\ln(1-\prod_{i=1}^cc_i!/c_i^{c_i})\) \textbf{times}
\item\label{lin:randcol} \qquad \textbf{for} \(i\in\{1,\dots,c\}\), randomly color the vertices of component \(G_\ell^i\) of~\(G_\ell\)
\qquad\quad using colors from~\(C_i\), let the resulting coloring be \(\text{col}\colon V\to\mathds{N}\).
\item\label{lin:onemcpusol} \qquad Solve \textsc{MinPCCS}{} instance \((G_{\ell}^{\bullet},w_{\ell}^{\bullet},\text{col},\ell,C)\) using dynamic programming.
\item\label{lin:mcpusol} let \(T=(W,F)\) be the best \textsc{MinPCCS}{} solution found in any of the repetitions.
\item\label{lin:consF}\label{lin:retgraph} \textbf{return} \(T'=(V,(F\cap E)\cup E_\ell)\).
\end{compactenum}
\end{algorithm}
|
1,108,101,564,521 | arxiv | \section{Introduction}
\label{intro}
Many scientific domains use statistical models with constraints on parameters which often results in difficult estimation and inference problems. For example, in genetics, simplex constraints are used to account for compositional data \citep{wang2019bulk,lu2019generalized}. In risk and survival analysis, a monotonic baseline function is used account for cumulating risk over time \citep{cox1972regression}. We focus on the survival context, which is further compounded by the fact that the monotonic function is unknown. To address nonparametric regression, we propose a slice sampling Gibbs algorithm that is applicable to a broad class of generalized linear mixed models (GLMMs) and generalized additive models (GAMs) with linear inequality and shape constraints \citep{hastie2017generalized}. Our algorithm is flexible and customizable to many different settings, as well as being computationally and algebraically tractable.
In event time modeling, two popular classes of models: Cox proportional hazards (PH) and semiparametric proportional odds (PO) models, are often used to study censored outcomes \citep{rossini1996semiparametric,shen1998propotional,murphy1997maximum}. These two methods share a common structure of a monotonic increasing nonparametric baseline function of time in their regression equation. In the case of the Cox PH, the partial likelihood can be used to obtain consistent estimates of regression parameters without directly modeling the baseline hazard \citep{cox1975partial}. However, when either the baseline odds or baseline hazards is of interest, rigorous estimation and inference frameworks are necessary to study these two nonparametric functions \citep{zeng2007maximum}.
In semiparametric PO regression, we observe current status (whether a failure has occurred) at a monitoring time and a set of covariates related to failure status. From this information, we know whether a failure occurred before a monitoring time (failure time is before the monitoring time) or is censored (failure time is after the monitoring time). \cite{rossini1996semiparametric,huang1995maximum} showed that the likelihood can be simplified with an independence assumption, failure time and covariates are independent of monitoring time, leading to an ancillary statistic being removed from the likelihood during estimation. The resulting likelihood is equivalent to a logistic regression likelihood with an unknown monotonic baseline function of monitoring time. The baseline function can be dealt with using monotonic regression and be replaced by a linear combination basis functions, resulting a potentially high-dimension logistic regression \citep{hothorn2018most,shen1998propotional,hanson2007bayesian,rossini1996semiparametric}. As noted in \cite{lin2010semiparametric,wang2011semiparametric}, shrinkage priors can be used to address non-descriptive basis functions. \cite{ramsay1988monotone} outlined the use of the I-spline system in conjunction with constrained optimization for monotonic regression. Similar constraints can be used to enforce convexity and other shapes into nonparametric effects \citep{meyer2015bayesian,ghosal2023shape}. Incorporating basis functions into the data matrix, our Gibbs sampler obtains Bayesian inference of regression coefficients and the monotonic baseline function, while being able to incorporate information through a multivariate Gaussian prior and random effects \citep{wand2008semiparametric,vallejos2017bayesian,polson2013bayesian}.
Under a Markov chain Monte Carlo (MCMC) framework, uniform ergodicity is one of the prerequisite condition for central limit theorem (CLT) inference for MCMC estimators. In addition to uniform ergodicity, finite second moments of the posterior distribution are needed to ensure CLT properties of posterior MCMC samples \citep{roberts2004general,jones2004markov}. We prove uniform ergodicity of posterior MCMC samples and the existence of a moment generating function (MGF) for our posterior distributions to obtain Markov chain CLT results. We expand on our CLT results and propose joint bands, multiplicity adjusted inference for nonparametric effects and monotonic baseline functions \citep{ruppert2003semiparametric,lee2018bayesian,meyer2015bayesian,morris2015functional}.
Constrained model estimation and inference is a difficult problem in statistical modeling. Traditional convex optimization based algorithms such as \cite{lu2019generalized}, uses a complicated descent algorithm paired with a de-biased covariance for estimation and large sample theory inference when dealing with constraints. Furthermore, use of large sample theory may not be valid in modest sample size settings such as causal inference and clinical trials. Our Bayesian MCMC approach addresses a shortcoming in the literature by simultaneously obtaining estimation and inference for finite sample settings. Our slice sampler is well suited for estimation and inference, due to its CLT properties for the posterior samples while being able to ensure linear inequality and shape constraints. Furthermore, our proof approach and Gibbs algorithm can be modified and recycled to handle similarly parameterized problems such as Bayesian variable selection and hierarchical models. Given the theoretical properties and computationally succinct formulation of our Gibbs algorithm, we believe that our slice sampler is an attractive method for numerous statistical modeling problems
\section{Methods}
\label{methods}
\subsection{Proportional odds regression with current status data and nonparametric effects}
Using the derivation of the proportional odds model from \cite{shen1998propotional,rossini1996semiparametric,huang1995maximum}, we collect censoring time $T_i$ for each subject $i\in\{1,\dots,N\}$. We observe data: $W_i=(T_i, Y_i, \mathbf{x}_i) \in \mathbb{R}^{+} \times\{0,1\} \times \mathbb{R}^{p}$ where $Y_i=\mathbb{I}\left(T^*_i \leq T_i\right)$, indicating whether event, denoted by event time $T^*_i$, has occurred or not, i.e. the current status. We do not observed the true event time $T^*_i$, but know the current status at time $T_i$. Here $p$ covariates are given as $\mathbf{x}_i$. The semiparametric proportional odds model is defined as is defined as
$$
\mathbb{E}(y_i \mid t_i, \mathbf{x}_i)= \mathrm{Pr}(y_i \mid t_i, \mathbf{x}_i) =\frac{\exp \left(\alpha(t_i) + \mathbf{x}_i^\top \boldsymbol{\beta} \right)}{1+\exp \left(\alpha(t_i) + \mathbf{x}_i^\top \boldsymbol{\beta} \right)} .
$$
Using Bayes' rule, we have
$$
\mathrm{Pr}(W_i=w_i | \alpha, \boldsymbol{\beta}) = \mathrm{Pr}(y_i \mid t_i, \mathbf{x}_i, \alpha, \boldsymbol{\beta})\mathrm{Pr}(t_i, \mathbf{x}_i| \alpha, \boldsymbol{\beta}) = \frac{\exp \left(y_i \left(\alpha(t_i)+\mathbf{x}_i^\top \boldsymbol{\beta} \right)\right)}{1+\exp \left(\alpha(t_i)+\mathbf{x}_i^\top \boldsymbol{\beta} \right)} h(t_i, \mathbf{x}_i)
$$
where $\mathbf{x}_i, Y_i$ is assumed independent of $T_i$ and $h(t_i, \mathbf{x}_i)$ is the joint density of $(Y_i, \mathbf{x}_i)$ which does not depend on $(\alpha(t), \boldsymbol{\beta})$. Therefore $h(t_i, \mathbf{x}_i)$ is an ancillary statistic and can be omitted from the estimation. As a result, we obtain $\operatorname{logit} ( F(t_i \mid \mathbf{x}_i ) )=\alpha(t_i)+\mathbf{x}_i^{\top} \boldsymbol{\beta}$
and
$$
\frac{F(t_i \mid \mathbf{x}_i)}{1-F(t_i \mid \mathbf{x}_i)}=\frac{F_{0}(t_i)}{1-F_{0}(t_i)} \exp \left(\mathbf{x}_i^{\top} \boldsymbol{\beta} \right), \quad \alpha(t_i) = \log \left( \frac{F_{0}(t_i)}{1-F_{0}(t_i)} \right) .
$$
The logit function is monotonic on $(0,1)$; CDF $F_{0}(t_i)$ is monotonic and unknown; and $\alpha(t)$ is monotonic. Thus we have the following likelihood contribution for subject $i$,
\begin{equation} \label{eq0}
\mathrm{Pr} (W_i=w_i | \alpha, \boldsymbol{\beta}) \propto \left[ \frac{\exp \left( \alpha(t_i)+\mathbf{x}_i^{\top} \boldsymbol{\beta} \right)}{1+\exp \left( \alpha(t_i)+\mathbf{x}_i^{\top} \boldsymbol{\beta} \right)} \right]^{y_i} \left[ \frac{1}{1+\exp \left( \alpha(t_i)+\mathbf{x}_i^{\top} \boldsymbol{\beta} \right)} \right]^{1-y_i}.
\end{equation}
We can also write a general regression model with basis functions as $ \mathbf{m}_i^\top \boldsymbol{\eta} = \mathbf{x}_i^{\top} \boldsymbol{\beta} + \alpha(t_i) + \mathcal{S}(\mathcal{X}_i)$, where $\mathcal{S}(\mathcal{X}_i)$ is a nonparametric covariate effect.
\subsubsection{Monotonic regression with constrained coefficients}
\label{monotone}
Using I-splines ${I}_{m}(t)$, to construct a monotonic semiparametric regression using $M-2$ knots, we obtain the following regression
$$
\alpha(t) = \sum_{m=1}^{M+2} u_{\alpha,m} {I}_{m}(t)
$$
with $\mathbf{u}_\alpha$ as basis coefficients and can be expressed as $\mathbf{Z}_{\alpha} \mathbf{u}_{\alpha}$ in matrix form and the intercept $\beta_0$ being estimated without constraints \citep{ramsay1988monotone,meyer2008inference}. Note that constraints $\mathbf{u}_\alpha \geq \mathbf{0}$, guarantee a monotonic $\alpha(t)$ which can be achieved in a Gibbs sampler by sampling from a half-normal distribution. In line with Bayesian methodology for PO models, we impose shrinkage prior $\mathbf{u}_{\alpha} \sim \mathrm{N}(\textbf{0}, \tau^{-1}_{\alpha} \mathbf{I}_{M+2})$, on I-spline coefficients \citep{wang2011semiparametric}.
\subsubsection{Semiparametric regression with O'Sullivan penalized B-splines}
\label{gam}
We may represent nonparametric effects $S (\mathcal{X})$, using cubic B-splines with $M$ number of knots
\begin{equation} \label{eq1}
S (\mathcal{X}) = \sum_{m=1}^{M+4} b_{m} \mathcal{B}_{m}(\mathcal{X}) + e_i
\end{equation}
with $b_{m}$ as basis coefficients and $e_{i} \sim \mathrm{N}\left(0, \sigma^2_{\mathcal{B}} \right)$. For each basis, $b_{m}$ are B-spline coefficients and $\mathcal{B}_{m}(\mathcal{X}), m=1, \ldots, M+4$ are basis functions defined by the knots $\psi_{1}, \ldots, \psi_{M+8}$ where,
$$
\begin{aligned} L &=\psi_{1}=\psi_{2}=\psi_{3}=\psi_{4}<\psi_{5}<\cdots<\psi_{M+4}=\psi_{M+5} \\ &=\psi_{M+6}=\psi_{M+7}=\psi_{M+8}= U \end{aligned}
$$
and $L$ and $U$ are boundary knots \citep{hastie2009elements}.
Writing \eqref{eq1} in matrix form, we get $\boldsymbol{S} = \boldsymbol{\mathcal{B}} \boldsymbol{b}+\boldsymbol{e}
$ where $\boldsymbol{S}=\left[ S_1, \ldots, S_N \right]^{\top}$, $\boldsymbol{\mathcal{B}}$ is the $N \times(M+4)$ B-spline matrix, $\boldsymbol{b} =\left[ b_1, \ldots, b_{M+4} \right]^{\top}$ and $\boldsymbol{e}=\left[ e_{1}, \ldots, e_{N} \right]^{\top} \sim$
$\mathrm{N}\left(\mathbf{0}, \sigma^2_{\mathcal{B}} \mathbf{I}_{N}\right)$. The O'Sullivan penalize B-splines (O-splines), defines a second order penalty term, curvature smoothness penalty, $\lambda_{\mathcal{B}} \int \left\{S^{{\prime \prime}}(\mathcal{X})\right\}^{2} d \mathcal{X}$, which is reasonable and often desirable property in semiparametric regression \citep{o1986statistical}. As noted in \cite{wand2008semiparametric}, this penalty is equivalent to assuming a prior distribution on the coefficients to be $\boldsymbol{b} \sim \mathrm{N}\left(\mathbf{0}, {q}^{-1}_{\mathcal{B}} \boldsymbol{\Lambda} \right)$, $\left[ \boldsymbol{\Lambda} \right]_{mm^{\prime}} = \int {\boldsymbol{\mathcal{B}}}_m^{\prime \prime}(\mathcal{X}) {\boldsymbol{\mathcal{B}}}_{m^\prime}^{\prime \prime}(\mathcal{X}) d \mathcal{X}$. Coefficient estimation is analogous with a ridge regression, resulting the estimates $\hat{\boldsymbol{b}}=\left({ \boldsymbol{\mathcal{B}} }^{\top} { \boldsymbol{\mathcal{B}} }+\lambda_{\mathcal{B}} \boldsymbol{\Lambda} \right)^{-1} { \boldsymbol{\mathcal{B}} }^{\top} \boldsymbol{S}$ with $\lambda_{\mathcal{B}} = \sigma^2_{\mathcal{B}} q_{\mathcal{B}}$ and $\widehat{\boldsymbol{S}} = \boldsymbol{\mathcal { B }} \widehat{\boldsymbol{b}}$. Spectral analysis of the penalty reveals that $\text{rank}(\boldsymbol{\Lambda})=M+2$, meaning that $M+2$ covariates from $\boldsymbol{\mathcal{B}}$ are penalized, resulting in two fixed effects and $M+2$ random effects. The spectral decomposition yields $\boldsymbol{\Lambda}= \mathbf{P D P}^{\top}$, where $\mathbf{D}=\operatorname{diag}\left( 0,0, d_{1}, \ldots, d_{M+2} \right)$, $\mathbf{P}^{\top} \mathbf{P} =\mathbf{I}_{M+4}$ and $\mathbf{P}=\left( \mathbf{X}_{\Lambda}, \mathbf{Z}_{\Lambda} \right)$. We can write the penalized projection matrix of $\widehat{\boldsymbol{\alpha}}$ as
\begin{equation} \label{eq2}
\begin{array}{rl}
{ \boldsymbol{\mathcal{B}} } \left({ \boldsymbol{\mathcal{B}} }^{\top} { \boldsymbol{\mathcal{B}} }+\lambda_{\mathcal{B}} \boldsymbol{\Lambda} \right)^{-1} { \boldsymbol{\mathcal{B}} }^{\top}
&=
{ \boldsymbol{\mathcal{B}} } \mathbf{P D}^{-1}_* \mathbf{D}_* \mathbf{P}^{\top} \left( { \boldsymbol{\mathcal{B}} }^{\top} { \boldsymbol{\mathcal{B}} } + \lambda_{\mathcal{B}} \boldsymbol{\Lambda} \right)^{-1} \mathbf{P D}_* \mathbf{D}^{-1}_* \mathbf{P}^{\top} { \boldsymbol{\mathcal{B}} }^{\top} \\
&=
{ \boldsymbol{\mathcal{B}} } \mathbf{P D}^{-1}_* \left( \mathbf{D}^{-1}_* \mathbf{P}^{\top} \left( { \boldsymbol{\mathcal{B}} }^{\top} { \boldsymbol{\mathcal{B}} } + \lambda_{\mathcal{B}} \boldsymbol{\Lambda} \right) \mathbf{P D}^{-1}_* \right)^{-1} \mathbf{D}^{-1}_* \mathbf{P}^{\top} { \boldsymbol{\mathcal{B}} }^{\top} \\
&=
\mathbf{C} \left( \mathbf{C}^\top \mathbf{C} + \lambda_{\mathcal{B}} \operatorname{diag}\left(0,0, \mathbf{I}_{M+2} \right) \right)^{-1} \mathbf{C}^\top
\end{array}
\end{equation}
where $\mathbf{D}_*=\operatorname{diag}\left(1,1, \sqrt{d_{1}}, \ldots, \sqrt{d_{M+2}}\right)$, $\mathbf{C} = { \boldsymbol{\mathcal{B}} } \mathbf{P D}^{-1}_* = (\mathbf{X}_{\mathcal{B}}, \mathbf{Z}_{\mathcal{B}})$, $\mathbf{X}_{\mathcal{B}}={ \boldsymbol{\mathcal{B}} } \mathbf{X}_{\Lambda}$, and $\mathbf{Z}_{\mathcal{B}}= { \boldsymbol{\mathcal{B}} } \mathbf{Z}_{\Lambda} \operatorname{diag}\left(d_{1}^{-1 / 2}, \ldots, d_{M+2}^{-1 / 2}\right)$.
Equation \eqref{eq2}, follows the BLUP form of a mixed effect model with random effects on covariates $\mathbf{Z}_{\mathcal{B}}$ \citep{robinson1991blup,speed1991blup}. In addition, $\mathbf{X}_{\mathcal{B}} \in \operatorname{Span}\left( \left[ \mathbf{1}, \boldsymbol{\mathcal{X}} \right] \right)$, allowing us to substitute $\mathbf{X}_{\mathcal{B}}$ with the original design matrix: $\left[ \mathbf{1}, \boldsymbol{\mathcal{X}} \right]_i = (1,\mathcal{X}_i)$ of an intercept and continuous predictor $\mathcal{X}$. Here, $\mathbf{Z}_{\mathcal{B}}$ is a Demmler-Reinsch (DR) matrix corresponding to the random effects \citep{demmler1975oscillation}. Alternatively, we can replace $\mathbf{D}_*$ with $\mathbf{D}_{\dagger}=\mathbf{D}^{-1}_{\mathcal{X}} \oplus \operatorname{diag}\left(\sqrt{d_{1}}, \ldots, \sqrt{d_{M+2}}\right)$, where $\mathbf{D}_{\mathcal{X}}$ is a $2\times2$ matrix with column vectors as regression coefficients that map from $\mathbf{X}_{\mathcal{B}}$ to $\left[ \mathbf{1}, \boldsymbol{\mathcal{X}} \right]$, $\mathbf{D}_{\mathcal{X}} = \left( \mathbf{X}^\top_{\mathcal{B}} \mathbf{X}_{\mathcal{B}}\right)^{-1} \mathbf{X}^{\top}_{\mathcal{B}} \left[ \mathbf{1}, \boldsymbol{\mathcal{X}} \right] $. Here, $\oplus$ is a direct sum which concatenates matrices into a block diagonal matrix. Equation \eqref{eq1} can be represented as linear mixed effect models, which can be expressed as $\widehat{\boldsymbol{S}} = \boldsymbol{\mathcal{X}} \widehat{ {\beta} }_{\mathcal{X}} + \mathbf{Z}_{\mathcal{B}} \widehat{ \mathbf{u} }_{\mathcal{B}}$ with $\mathbf{u} _{\mathcal{B}} \sim \mathrm{N}(\textbf{0}, \tau^{-1}_{\mathcal{B}} \mathbf{I}_{M+2})$ and the intercept being assigned to $\alpha(t)$.
\subsection{Connection with Bayesian GLMMs}
\label{GLMM}
Use our previous derviation, we can write $ \mathbf{m}_i^\top \boldsymbol{\eta} = \mathbf{x}_i^{\top} \boldsymbol{\beta} + \alpha(t_i) + \mathcal{S}(\mathcal{X}_i) = \mathbf{x}_i^\top \boldsymbol{\beta} + \mathbf{z}_{\alpha,i}^\top \mathbf{u}_{\alpha} + \mathbf{z}_{\mathcal{B},i}^\top \mathbf{u}_{\mathcal{B}} $ in regression matrix form. We can also write the analogous penalized negative log likelihood for GLMMs as
\begin{equation} \label{pllglm}
- \log L(\boldsymbol{\eta} \mid \mathbf{y}, \mathbf{M} ) + \tau_{\alpha} \| \mathbf{u}_{\alpha} \|^2_2 + \tau_{\mathcal{B}} \| \mathbf{u}_{\mathcal{B}} \|^2_2
\end{equation}
such that $\mathbf{u}_\alpha \geq \mathbf{0}$. Uniformly ergodic Gibbs samplers have been proposed for Bayesian mixed logistic regression \citep{polson2013bayesian,choi2013polya,wang2018analysis,rao2021block}. We derive a slice sampler that ensures monotonicity of $\alpha(t)$ and can be applied to the general class of Bayesian GLMMs and GAMs. We prove CLT properties for MCMC estimators which allows us to simultaneously ensure monotonicity when estimating $\alpha(t)$ and construct joint bands on functions $\alpha(t)$ and $\mathcal{S}(\mathcal{X})$.
\subsubsection{Truncated gamma and truncated normal distributions}
A Bayesian analog of the mixed effect model are priors $\mathbf{u} \sim \mathrm{N}(\textbf{0}, \tau^{-1} \mathbf{I}_{M+2})$, and $\tau \sim \mathrm{T G} \left(a_0, b_0, \tau_{0}\right)$ where $\tau$ follows a truncated gamma distribution, $\pi \left(\tau \mid a_{0}, b_{0}, \tau_{0}\right)=c_1 \left(\tau_{0}, a_{0}, b_{0}\right)^{-1} \tau^{a_{0}-1} \exp \left(-b_{0} \tau \right) \mathbb{I}\left(\tau \geq \tau_{0}\right)$ where $c_1 \left(\tau_{0}, a_{0}, b_{0}\right)=\int_{\tau_{0}}^{\infty} \tau^{a_{0}-1} \exp \left(-b_{0} \tau\right) d \tau$. In practice $\mathrm{Pr}(\tau_j \leq \tau_0)$ is negligibly small and we set $\tau_0=1000^{-1}$ for our analysis. Bayesian analysis uses the data through the likelihood, to update the prior information; we modify the prior to reflect the monotonicity constraint by means of a truncated normal distribution \citep{li2015efficient}: $\mathbf{u} \sim \mathrm{TN}(\boldsymbol{\mu}, \boldsymbol{\Sigma}, \mathbf{R}, \mathbf{c}, \mathbf{d})$,
$$
\pi (\mathbf{u} \mid \boldsymbol{\mu}, \boldsymbol{\Sigma}, \mathbf{R}, \mathbf{c}, \mathbf{d} )
=
\frac{\exp \left\{-\frac{1}{2}(\mathbf{u}-\boldsymbol{\mu})^{\top} \boldsymbol{\Sigma}^{-1}(\mathbf{u}-\boldsymbol{\mu})\right\}}
{c_2 (\boldsymbol{\mu}, \boldsymbol{\Sigma}, \mathbf{R}, \mathbf{c}, \mathbf{d})}
\mathbb{I}(\mathbf{c} \leq \mathbf{R} \mathbf{u} \leq \mathbf{d})
$$
where $c_2 (\boldsymbol{\mu}, \boldsymbol{\Sigma}, \mathbf{R}, \mathbf{c}, \mathbf{d}) = \oint_{\mathbf{c} \leq \mathbf{R} \mathbf{u} \leq \mathbf{d}} \exp \left\{-\frac{1}{2}(\mathbf{u}-\boldsymbol{\mu})^{\top} \boldsymbol{\Sigma}^{-1}(\mathbf{u}-\boldsymbol{\mu})\right\} d \mathbf{u}$ and $\mathbf{R}$ is a rotation matrix. Note that half-normal distribution prior of $\mathbf{u}_{\alpha} \sim \mathrm{TN}(\mathbf{0}, \tau^{-1}_\alpha \mathbf{I}_{M+2}, \mathbf{I}_{M+2}, \mathbf{0}, \boldsymbol{\infty} )$ preserves the conjugacy of $\tau_\alpha$
$$
\pi (\mathbf{u} \mid \mathbf{0}, \tau^{-1}_\alpha \mathbf{I}_{M+2}, \mathbf{I}_{M+2}, \mathbf{0}, \boldsymbol{\infty} )
=
\left( {2 \tau_\alpha}/{\pi} \right)^{(M+2)/2}
{\exp \left( -\frac{ \tau_\alpha }{2} \mathbf{u}^{\top} \mathbf{u}\right) }
\mathbb{I}(\mathbf{0} \leq \mathbf{u} ) .
$$
Because the distribution is zero centered and the covariance matrix is isotropic, each marginal half-normal kernel has half the volume of the normal kernel. A normalization factor of 2 is multiplied to each marginal normal PDF in order to obtain the PDF of the half-normal distribution.
\subsubsection{Block slice sampler}
\label{slice}
The likelihood of exponential family GLMs is given by
$$
L(\boldsymbol{\eta} \mid \mathbf{y}, \mathbf{M} ) = \exp \left( \mathbf{y}^{\top} \mathbf{M} \boldsymbol{\eta} \right) \exp \left( -\sum_{i=1}^{N} \xi \left(\mathbf{m}_{i}^{\top} \boldsymbol{\eta}\right) \right)
$$
where $\xi (v)=e^{v}$ for Poisson regression, $\xi (v)=\log \left(1+e^{v}\right)$ for logistic regression, $\xi (v)=v^{2} / 2$ for linear model, etc. \citep{ghosal2022bayesian}. For the exponential proportion hazards model we have $\xi (v,t)=\exp(\log(t)+v)$, $\exp \left( \mathbf{y}^{\top} \mathbf{X} \boldsymbol{\beta} \right) \exp \left( -\sum_{i=1}^{N} \exp \left( \log(t_i) + \mathbf{x}_{i}^{\top} \boldsymbol{\beta}\right) \right)$. Note that, $\xi (v) \geq 0$ and $\xi (v)$ is convex.
The posterior kernel with normal priors for $\boldsymbol{\eta} \sim \mathrm{TN}( \mathbf{b}, \mathbf{A}(\boldsymbol{\tau})^{-1}, \mathbf{R}_\alpha, \mathbf{c}_\alpha, \boldsymbol{\infty} )$ is
$$
\pi(\boldsymbol{\eta} \mid \mathbf{y}) \propto L(\boldsymbol{\eta} \mid \mathbf{y}, \mathbf{M} ) \pi (\boldsymbol{\eta} \mid \mathbf{b}, \mathbf{A}(\boldsymbol{\tau})^{-1}, \mathbf{R}_\alpha, \mathbf{c}_\alpha, \boldsymbol{\infty} )
$$
where $\mathbf{A}(\boldsymbol{\tau})^{-1}$ is the covariance matrix and $\boldsymbol{\tau}=\{ \tau_\alpha, \tau_\mathcal{B} \}$. If we order $\mathbf{M}=[\mathbf{X}, \mathbf{Z}_\alpha, \mathbf{Z}_\mathcal{B} ]$, $\boldsymbol{\eta}=[\boldsymbol{\beta}^{\top}, \mathbf{u}^{\top}_\alpha, \mathbf{u}^{\top}_\mathcal{B}]^{\top}$, then $\mathbf{A}(\boldsymbol{\tau}) = \boldsymbol{\Sigma}^{-1} \oplus \tau_\alpha \mathbf{I}_{M+2} \oplus \tau_{\mathcal{B}}\mathbf{I}_{M+2}$ is the prior precision. The prior mean is $\mathbf{b} = [\boldsymbol{\mu}^\top, \mathbf{0}_{M+2}^\top, \mathbf{0}_{M+2}^\top ]^{\top}$. We denote the rotation $\mathbf{R}_\alpha = \mathbf{I}_{p} \oplus \mathbf{I}_{M+2} \oplus \mathbf{I}_{M+2}$ and lower bound as $\mathbf{c}_\alpha = \left[ -\boldsymbol{\infty}^\top_{p}, \mathbf{0}^\top_{M+2}, -\boldsymbol{\infty}^\top_{M+2} \right]^\top$ inorder to ensure $\mathbf{u}_\alpha \geq \mathbf{0}$. Any other linear inequality constraints on the fixed effect coefficients $\boldsymbol{\beta}$ can be concatenated into $\{ \mathbf{R}_\alpha, \mathbf{c}_\alpha \}$, e.g. simplex constraint $\mathbf{1}^\top \boldsymbol{\beta} \geq 1$, $-\mathbf{1}^\top \boldsymbol{\beta} \geq -1$, $\boldsymbol{\beta} \geq \mathbf{0}$.
We introduce uniformly distributed latent auxiliary variables $\omega_i \sim \mathrm{U}(0,1)$ and inequality constraints on $ \mathbf{u}_\alpha $ to obtain joint posterior
\begin{equation} \label{SS_joint}
\begin{array}{rl}
\pi(\boldsymbol{\eta}, \boldsymbol{\omega} \mid \mathbf{y}) \propto&
L(\boldsymbol{\eta} \mid \mathbf{y}, \mathbf{M} ) \pi (\boldsymbol{\eta} \mid \mathbf{b}, \mathbf{A}(\boldsymbol{\tau})^{-1}, \mathbf{R}_\alpha, \mathbf{c}_\alpha, \boldsymbol{\infty} )
\pi(\boldsymbol{\omega} \mid \boldsymbol{\eta}) \\
\propto &
\exp \left\{\mathbf{y}^{\top} \mathbf{M} \boldsymbol{\eta}-
\frac{1}{2}\left(\boldsymbol{\eta}- \mathbf{b} \right)^{\top}
\mathbf{A}(\boldsymbol{\tau})
\left(\boldsymbol{\eta}- \mathbf{b} \right)\right\} \\
&\times
\mathbb{I} ( \mathbf{c}_\alpha \leq \mathbf{R}_\alpha \boldsymbol{\eta} )
\prod_{i=1}^{N} \mathbb{I} \left( \omega_i \leq \exp \left(- \xi \left( \mathbf{m}^\top_i \boldsymbol{\eta} \right) \right) \right) .
\end{array}
\end{equation}
When the joint distribution \eqref{SS_joint} is integrated with respect to $\omega_i$, we obtain the marginal distribution $\pi(\boldsymbol{\eta} \mid \mathbf{y})$ and see that we have $\omega_i|\boldsymbol{\eta} \sim \mathrm{U} \left( 0, \exp \left(- \xi \left( \mathbf{m}^\top_i \boldsymbol{\eta} \right) \right) \right)$ where truncated uniform sampling gave rise to the name slice sampler \citep{mira2002efficiency,damlen1999gibbs,neal2003slice}. In our two variable example for $\{ \boldsymbol{\eta}, \boldsymbol{\omega} \}$, we can show that the mean and conditional distribution of $\boldsymbol{\eta}$ is another truncated normal with the kernel of $\boldsymbol{\eta} | \boldsymbol{\omega}, \boldsymbol{\tau} \sim \mathrm{N} \left( \mathbf{b} + \mathbf{A}(\boldsymbol{\tau})^{-1} \mathbf{M}^{\top} \mathbf{y}, \mathbf{A}(\boldsymbol{\tau})^{-1} \right)$ such that $\prod_{i=1}^{N} \mathbb{I} \left( \omega_i \leq \exp \left(- \xi \left( \mathbf{m}^\top_i \boldsymbol{\eta} \right) \right) \right)$ and $\mathbb{I} ( \mathbf{c}_\alpha \leq \mathbf{R}_\alpha \boldsymbol{\eta} )$ are satisfied.
Incorporating our random effect structure using a gamma distribution truncated below at $\tau_0$, $\tau_j \sim \mathrm{T G} \left(a_0, b_0, \tau_{0} \right)$, our joint distribution becomes
$$
\begin{array}{rl}
\pi(\boldsymbol{\eta}, \boldsymbol{\omega}, \boldsymbol{\tau} \mid \mathbf{y}) \propto&
L(\boldsymbol{\eta} \mid \mathbf{y}, \mathbf{M} ) \pi (\boldsymbol{\eta} \mid \mathbf{b}, \mathbf{A}(\boldsymbol{\tau})^{-1}, \mathbf{R}_\alpha, \mathbf{c}_\alpha, \boldsymbol{\infty} )
\pi(\boldsymbol{\omega} \mid \boldsymbol{\eta})
\pi \left( \boldsymbol{\tau} \mid a_{0}, b_{0}, \tau_{0}\right) \\
\propto &
\exp \left\{\mathbf{y}^{\top} \mathbf{M} \boldsymbol{\eta}-
\frac{1}{2}\left(\boldsymbol{\eta}- \mathbf{b} \right)^{\top}
\mathbf{A}(\boldsymbol{\tau})
\left(\boldsymbol{\eta}- \mathbf{b} \right)\right\} \\
&\times \prod_{j \in \{\alpha, \mathcal{B}\}} \tau_j^{a_{0}+M/2} e^{-b_{0} \tau_j} \mathbb{I}\left(\tau_j \geq \tau_{0}\right) \\
&\times
\mathbb{I} ( \mathbf{c}_\alpha \leq \mathbf{R}_\alpha \boldsymbol{\eta} )
\prod_{i=1}^{N} \mathbb{I} \left( \omega_i \leq \exp \left(- \xi \left( \mathbf{m}^\top_i \boldsymbol{\eta} \right) \right) \right) .
\end{array}
$$
The conditional distributions $\tau_j \mid \boldsymbol{\eta} \sim \mathrm{T G} \left(a_{0}+({M+2})/{2}, b_{0}+{ \mathbf{u}_j^{\top} \mathbf{u}_j}/{2}, \tau_{0}\right)$ can be derived from the joint distribution. Prior for $\tau_\alpha$ induces shrinkage on $\alpha(t)$ and prior $\tau_{\mathcal{B}}$ induces smoothness on $\mathcal{S}(\mathcal{X})$ based on the second derivative. Note the joint distribution of $\{ \boldsymbol{\eta}, \boldsymbol{\tau} \}$ is given by
$$
\pi(\boldsymbol{\eta}, \boldsymbol{\tau} \mid \mathbf{y}) \propto
L(\boldsymbol{\eta} \mid \mathbf{y}, \mathbf{M} ) \pi (\boldsymbol{\eta} \mid \mathbf{b}, \mathbf{A}(\boldsymbol{\tau})^{-1}, \mathbf{R}_\alpha, \mathbf{c}_\alpha, \boldsymbol{\infty} ) \pi \left( \boldsymbol{\tau} \mid a_{0}, b_{0}, \tau_{0}\right) .
$$
Here $\xi(v)$ is convex and lower bounded at 0 which allows us to obtain linear inequality constraints for $\boldsymbol{\eta}$ given $\omega_i \in (0,1)$
\begin{align*}
\omega_i &\leq \exp( - \xi \left( \mathbf{m}^\top_i \boldsymbol{\eta} \right) ) = \exp( - \log (1 + \exp(\mathbf{m}^\top_i \boldsymbol{\eta}) ) ) < 1 \\
\omega^{-1}_i &\geq 1 + \exp \left( \mathbf{m}^\top_i \boldsymbol{\eta} \right) \\
\log(\omega_i^{-1} - 1) &\geq \mathbf{m}^\top_i \boldsymbol{\eta} .
\end{align*}
We can concatenate $\mathbf{u}_\alpha \geq \mathbf{0}$ and $-\mathbf{m}^\top_i \boldsymbol{\eta} \geq -\log(\omega_i^{-1} - 1)$ for all $i$ as a rotation matrix inequality, $\mathbf{R}_\omega \boldsymbol{\eta} \geq \mathbf{c}_\omega$. Here we stack $-\mathbf{M}$ on top of $\mathbf{R}_\alpha$ to get $\mathbf{R}_\omega$. We define vector ${[\mathbf{c}_*]}_i = -\log(\omega_i^{-1} - 1)$ and stack it on top of $\mathbf{c}_\alpha$ to get $\mathbf{c}_\omega$. Note that $\mathbf{c}_\omega$ is a function of $\boldsymbol{\omega}$. Our block Gibbs sampler is given as
\begin{equation} \label{S_sampler}
\begin{array}{c}
\boldsymbol{\eta} \mid \boldsymbol{\omega}, \boldsymbol{\tau} \sim \mathrm{TN} \left( \mathbf{A}(\boldsymbol{\tau})^{-1}\boldsymbol{\mu}_*, \mathbf{A}(\boldsymbol{\tau})^{-1}, \mathbf{R}_\omega, \mathbf{c}_\omega, \boldsymbol{\infty} \right) \\
\omega_i \mid \boldsymbol{\eta} \sim \mathrm{U} \left( 0, \exp( - \xi (\mathbf{m}^\top_i \boldsymbol{\eta}) ) \right) \\
\tau_j \mid \boldsymbol{\eta} \sim \mathrm{T G} \left(a_{0}+\frac{M+2}{2}, b_{0}+\frac{ \mathbf{u}^{\top}_j \mathbf{u}_j}{2}, \tau_{0}\right)
\end{array}
\end{equation}
where $\xi(v) = \log(1+e^v)$ and $\boldsymbol{\mu}_* = \mathbf{M}^{\top} \mathbf{y} + \mathbf{A}(\boldsymbol{\tau}) \mathbf{b} = \mathbf{M}^{\top} \mathbf{y} + \left[ \left[ \boldsymbol{\Sigma}^{-1} \boldsymbol{\mu} \right]^{\top}, \mathbf{0}^\top_{2M+4} \right]^{\top}$. Note that \eqref{S_sampler} is a general Gibbs sampler for linear inequality constrained Bayesian GLMMs and GAMs which can be customized based on $\xi(v)$ and shape determining basis system \citep{meyer2008inference,ghosal2023shape}.
\section{Uniform ergodicity and Markov chain central limit theorem}
We establish uniform ergodicity and that posterior samples are square integrable i.e., the second moment exist which guarantees central limit theorem (CLT) results for posterior averages and consistent estimators of the associated asymptotic variance \citep{jones2004markov}. This Markov chain CLT result is a special case of martingale CLT \citep{kurtz1981central,meyn2012markov}. Our posterior samples for our functions $\alpha(t)$ and $\mathcal{S}(\mathcal{X})$ are matrix multiplication of the posterior coefficient samples, $\mathcal{Z}_\alpha \mathbf{u}_\alpha$ and $\mathcal{Z}_\mathcal{B} \mathbf{u}_\mathcal{B}$ and have the same CLT properties. Here $\mathcal{Z}_\alpha$ and $\mathcal{Z}_\mathcal{B}$ are matrix representation of the continuous I-spline and Demmler-Reinsch bases.
First, we show uniform ergodicity of $\boldsymbol{\eta}$ from Gibbs sampler \eqref{S_sampler}, taking advantage of the truncated gamma \citep{wang2018analysis}. A key feature of this strategy is that by truncating the gamma distribution at a small $\tau_0 = \epsilon$ results in useful inequalities related to ergodicity and in practice $\mathrm{Pr}(\tau_j \leq \epsilon)$ is negligibly small. We denote the $\boldsymbol{\eta}$-marginal Markov chain as $\Psi \equiv \{ \boldsymbol{\eta}(n) \}^\infty_{n=0}$ and Markov transition density (Mtd) of $\Psi$ as
$$
k\left( \boldsymbol{\eta} \mid \boldsymbol{\eta}^{\prime}\right)
=
\int_{\mathbb{R}_{+}} \int_{\mathbb{H}^{N}} \pi( \boldsymbol{\eta} \mid \boldsymbol{\omega}, \boldsymbol{\tau}, \mathbf{y}) \pi\left(\boldsymbol{\omega}, \boldsymbol{\tau} \mid \boldsymbol{\eta}^{\prime}, \mathbf{y}\right) d \boldsymbol{\omega} d {\tau}
$$
where $\boldsymbol{\eta}^{\prime}$ is the current state and $\boldsymbol{\eta}$ is the next state, with $\mathbb{H}^{N}=[0,1]^N$ is a hypercube. We show the Mtd of satisfies the following minorization condition: $k\left( \boldsymbol{\eta} \mid \boldsymbol{\eta}^{\prime}\right) \geq \delta h(\boldsymbol{\eta})$, where there exist a $\delta > 0$ and density function $h$, to prove uniform ergodicity \citep{roberts2004general}. Uniform ergodicity is defined as bounded and geometrically decreasing bounds for total variation distance to the stationary distribution in number of Markov transitions $n$, $\left\|K^{n}( \boldsymbol{\eta}, \cdot)-\Pi(\cdot \mid \mathbf{y})\right\|:=\sup _{A \in \mathscr{B}}\left|K^{n}( \boldsymbol{\eta}, A)-\Pi(A \mid \mathbf{y})\right| \leq V r^{n}$.
Here $\mathscr{B}$ denotes the Borel $\sigma$-algebra of $\mathbb{R}^{p+2M+4}$, $K(\cdot, \cdot)$ be the Markov transition function for the Mtd $k(\cdot, \cdot)$
$$
K\left( \boldsymbol{\eta}^{\prime}, A\right)=\operatorname{Pr}\left( \boldsymbol{\eta}^{(j+1)} \in A \mid \boldsymbol{\eta}^{(j)}=\boldsymbol{\eta}^{\prime}\right)=\int_{A} k\left( \boldsymbol{\eta} \mid \boldsymbol{\eta}^{\prime}\right) d \boldsymbol{\eta},
$$
and $K^{n}\left( \boldsymbol{\eta}^{\prime}, A\right)=\operatorname{Pr}\left( \boldsymbol{\eta}^{(n+j)} \in A \mid \boldsymbol{\eta}^{(j)}=\boldsymbol{\eta}^{\prime}\right)$
We denote $\Pi(\cdot \mid \mathbf{y})$ as the probability measure with density $\pi(\boldsymbol{\eta} \mid \mathbf{y})$, $V$ is bounded above and $r\in(0,1)$.
\begin{theorem} \label{thm1}
Assume that $a_0 > 0$, $b_0 > 0$, $\xi ( \mathbf{m}_i^{\top} \boldsymbol{\eta} ) \geq 0$ and $\xi(v)$ is convex, then the Markov chain $\Psi$ of \eqref{S_sampler} for constrained Bayesian GLMMs and GAMs is uniformly ergodic.
\end{theorem}
\begin{theorem} \label{thm2}
For any fixed $\mathbf{t} \in \mathbb{R}^{p+2M+4}$, $\int_{\mathbb{R}^{p+2M+4}} e^{\boldsymbol{\eta}^{\top} \mathbf{t}} \pi(\boldsymbol{\eta} \mid \mathbf{y}) d \boldsymbol{\eta}<\infty$. Hence, the moment generating function of the posterior distribution exists.
\end{theorem}
The prior precision is lower bounded, making the variances of mixed effect components upper bounded and contained in a hypercube. By transiting our parameters of interest through a flexible auxiliary variable space, bounded in a finite volume hypercube, we obtain desirable integral properties for our Markov chain. We leave the proofs to the Proofs section. Note that the proof requires $a_0 + (M+2)/2 \geq 1$ which is implied by the knot selection and construction of the spline bases. This condition is necessary when adapting our slice sampler to a general mixed model setting.
\subsection{Posterior inference: Joint bands, SimBaS and GBPV}
Our CLT properties extend to continuous functional predictors, allowing us to construct joint bands while accounting for a Bayesian false discovery rate. Using $\alpha(t)$ as an example, the Simultaneous Band Scores (SimBaS) and joint bands found in \cite{meyer2015bayesian} and \cite{ruppert2003semiparametric} are a direct corollary of our CLT results. Suppose $\alpha^{(r)}(t) = \beta^{(r)}_{0} + \mathcal{Z}_\alpha \mathbf{u}_\alpha^{(r)}$ for $r\in \{1,\dots,R\}$ is a sample from the posterior, then intervals for $\alpha(t)$ are given as
$$
I_{u}(t)=\hat{\alpha}(t) \pm q_{(1-u)}[\widehat{\operatorname{St.Dev}}\{\hat{\alpha}(t)\}]
$$
where the variable $q_{(1-u)}$ is the $(1-u)$ quantile taken over $R$ of the quantity
$$
Z^{(r)}= \max_{ t \in \mathcal{T} } \left|\frac{\alpha^{(r)}(t)-\hat{\alpha}(t)}{\widehat{\operatorname{St.Dev}}\{\hat{\alpha}(t)\}}\right|
$$
and multiplicity adjusted probability score at values of $t$ are given as
$$
P_{\operatorname{SimBaS}}(t)=\frac{1}{R} \sum_{r=1}^{R} \mathbb{I} \left\{\left|\frac{\hat{\alpha}(t)}{\widehat{\operatorname{St} \cdot \operatorname{Dev}}\{\hat{\alpha}(t)\}}\right| \leq Z^{(r)}\right\} .
$$
For each $t$, $\mathrm{P}_{\operatorname{SimBaS}}(t)$ can be used as local probability scores that have multiple testing adjusted global properties. For example, we can flag domain $\left\{t: \mathrm{P}_{\operatorname{SimBaS}}(t)<u\right\}$ as significant. From these we can compute $P_{\text{Bayes}}=\min_t\left\{P_{\operatorname{SimBaS}}(t)\right\}$, which denote global Bayesian p-values (GBPV) such that we reject the global hypothesis that $\alpha(t) \equiv 0$ whenever $P_{\text{Bayes}}<u$.
\section{Illustrative examples}
We use the \texttt{R} package \texttt{cascsim} \cite{cascsim} to sample from truncated gamma and \texttt{tmvtnsim} \cite{tmvtnsim} efficiently sample from truncated normal. The package \texttt{tmvtnsim} is a C++ implementation of algorithms found in \cite{li2015efficient}. In addition, we follow the construction of I-splines found in \cite{meyer2008inference} and are normalized such that $\mathbf{Z}_{\alpha}^\top \mathbf{y} = \mathbf{0}$ and are plotted in Figure \ref{fig:1}.
\begin{figure}
\centering
\includegraphics[width=6in]{bases}
\caption{Examples of I-Spline and DR-Spline bases for $\alpha(t)$ and $\mathcal{S}(\mathcal{X})$. }
\label{fig:1}
\end{figure}
We simulate and fit two Poisson models for our illustrative examples: a) monotonic regression with log mean, $\log(\mu(t)) = \alpha(t) = \log( 0.005(t-10)^3+10 )$; b) nonparametric regression with log mean, $\log(\mu(t)) = \mathcal{S}(t) = \sin(t)$. For model a), we maximize the unpenalized constrained likelihood of \eqref{pllglm} using \texttt{R} package \texttt{CVXR} \citep{fu2020cvxr} and penalized version of the likelihood using the slice sampler and observe that the slice sampler shrinks the monotonic covariate effect towards the intercept. Our hierarchical modeling of I-spline coefficients adaptively shrinks the monotonic nonparametric effect. For our second illustrative example, model b), we compare the slice sampler to the widely used GAMs implemented in \texttt{mgcv} \citep{wood2022mgcv}. Both \texttt{mgcv} and the slice sampler uses a second derivative smoothness penalty and we observed similar model fits for both methods. We plot our results in Figure \ref{fig:2} and used vague priors for our analyses.
\begin{figure}
\centering
\includegraphics[width=6in]{examples}
\caption{Poisson models fitted with I-splines and DR-bases: a) unpenalized vs penalized I-splines using the convex optimization and slice sampler; b) second derivative penalty with GAMs and slice sampler. Slice sampler shrinks the monotonic effect towards the intercept. Smoothed GAM and slice sampler have similar fits. Posterior samples are plotted with gray lines.}
\label{fig:2}
\end{figure}
\section{Real data analysis}
We fit a current status proportional odds model for concussion recovery data from a study conducted at the Children's Hospital of Pennsylvania. A sample of $N=74$ participants from a large prospective observational cohort study assessing diagnostic measures of concussion were used in our analysis. In this example, $Y$ is the whether a subject recovered by the last clinical visit and $T$ is the time from concussion to last clinic visit. Recovery, $Y$ is defined as a SCAT5 score less than 5 at the time of last clinic visit \citep{echemendia2017sport}. We include linear effects for King-Devick (KD) test completion time \citep{galetta2015adding} and pupil eye pain binary variable, both recorded at the last clinic visit. In addition, we include age as a nonparametric effect. Age and KD completion time were mean centered and we used knots based on quantiles of the raw data distribution to generate $M+2=7$ splines bases for mixed effect components. We fit $\operatorname{logit} (F(t_i \mid \mathbf{x}_i ) )=\alpha(t_i) + \mathcal{S}(\mathcal{X}_{\text{age}_i}) + {x}_{\text{pain}} \beta_{\text{pain}} + {x}_{\text{KD}} \beta_{\text{KD}}$ using our slice sampler with results for nonparametric effects plotted in Figure \ref{fig:3}. We expect a positive intercept and $\alpha(t)>0$ due to 73\% of our cohort being recovered cases. However, we observed steep increases in the probability of recovery as a function of time, before 28 days. In addition, we observed that recovery odds, $\mathcal{S}(\mathcal{X}_{\text{age}_i})$ remains relatively constant until age 16 then decreases with age. These results are aligned with known factors associated with concussion recovery with nonparametric regression elucidating the shape of these trends \citep{desai2019factors}. In addition, the posterior mean and 0.95 credible interval for pupil eye pain effect is $-3.267(-5.632, -1.214)$ and $-0.043(-0.111, 0.019)$ for KD completion time effect. Our model suggest that pupil eye pain is highly correlated with having a concussion and pupil eye pain is already widely used as a concussion indicator. Longer time spent completing the KD test is associated with a higher probability of concussion; however, this relationship remains marginal in our analysis.
\begin{figure}
\centering
\includegraphics[width=6in]{concussion}
\caption{Global Bayesian p-values (GBPVs), posterior mean, 0.95 joint bands for monotonic time effect and nonparametric age effect and GLM estimates. Logistic regression GLM was fitted without nonparametric effects for comparison. Steep increases in the probability of recovery as a function of time occurs before 28 days. Relatively constant odds of recovery are observed for age 12-16 with odds of recovery decreasing after age 16.}
\label{fig:3}
\end{figure}
\section{Discussion}
Our Markov chain CLT results establishes rigorous inference with finite sample data and settings with shape and linear inequality constraints. Our MCMC procedure allows for simultaneous estimation and inference unlike traditional constrained convex optimization methods. In addition, we derived second derivative smoothness penalties through random effect parameterization which allows GAMs to be fitted with our slice sampler. The hierarchical modeling of random effects also allow us to adaptively induce shrinkage on nonparametric monotone effects. Our mixed effect slice sampler can be applied to canonical exponential family models which includes GLMM, GAM and Exponential PH models.
One limitation of our approach is sampling from high dimensional truncated normal distribution can be slow. Possible resolutions for high dimensional settings are parallel MCMC methods \citep{xing2014asymp}. In addition, we hope that increase use of slice samplers can lead to new research in truncated normal sampling. Our Gibbs sampler and proof techniques can be adapted to related settings such as Bayesian variable selection and is a promising direction for future work.
\section*{Proofs}
\begin{proof}Proof of Theorem \ref{thm1}.
Here we rely on the properties of slice samplers and useful inequalities due to the truncated gamma prior. Note that $\mathbf{A}\left(\tau_{0}\right) = \boldsymbol{\Sigma}^{-1} \oplus \tau_0 \mathbf{I}_{M+2} \oplus \tau_0\mathbf{I}_{M+2}$, $\mathbf{A}\left(\boldsymbol{\tau}\right) \succeq \mathbf{A}\left(\tau_{0}\right)$, $\exp \left( -\xi \left( \mathbf{m}^\top_i \boldsymbol{\eta} \right) \right) \leq 1$, $g(\boldsymbol{\eta} ) \mathbb{I}( \mathbf{c}_\alpha \leq \mathbf{R}_\alpha \boldsymbol{\eta} ) \leq g(\boldsymbol{\eta} )$ and $\oint_{ \mathbf{R}_\alpha \boldsymbol{\eta} \geq \mathbf{c}_\alpha } g( \boldsymbol{\eta} ) d \boldsymbol{\eta} = \int_{\mathbb{R}^{p+2M+4}} g( \boldsymbol{\eta} ) \mathbb{I}( \mathbf{c}_\alpha \leq \mathbf{R}_\alpha \boldsymbol{\eta} ) d \boldsymbol{\eta} \leq \int_{\mathbb{R}^{p+2M+4}} g(\boldsymbol{\eta} ) d \boldsymbol{\eta}$.
We have
$$
\begin{array}{rl}
\pi( \boldsymbol{\eta} \mid \boldsymbol{\omega}, \boldsymbol{\tau}, \mathbf{y}) \pi\left(\boldsymbol{\omega} \mid \boldsymbol{\eta}^{\prime}, \mathbf{y}\right)
= &
\exp \left\{-\frac{1}{2}( \boldsymbol{\eta}- \mathbf{A}(\boldsymbol{\tau})^{-1}\boldsymbol{\mu}_* )^{\top} \mathbf{A}(\boldsymbol{\tau})( \boldsymbol{\eta}- \mathbf{A}(\boldsymbol{\tau})^{-1}\boldsymbol{\mu}_* )\right\} \\
& \times
\frac{\mathbb{I}(\mathbf{c}_\omega \leq \mathbf{R}_\omega \boldsymbol{\eta} )}
{c_2 ( \mathbf{A}(\boldsymbol{\tau})^{-1}\boldsymbol{\mu}_*, \mathbf{A}(\boldsymbol{\tau})^{-1}, \mathbf{R}_\omega, \mathbf{c}_\omega, \infty)} \\
& \times
\prod_{i=1}^{N} \mathbb{I} \left( \omega_i \leq \exp \left(- \xi \left( \mathbf{m}^\top_i \boldsymbol{\eta}^\prime \right) \right) \right) \exp \left( \xi \left( \mathbf{m}^\top_i \boldsymbol{\eta}^\prime \right) \right) \\
= &
\exp \left[-\frac{1}{2} \left( \boldsymbol{\eta}^{\top} \mathbf{A}(\boldsymbol{\tau}) \boldsymbol{\eta} - 2 \boldsymbol{\eta}^{\top} \boldsymbol{\mu}_* \right) - \frac{1}{2} \boldsymbol{\mu}^{\top}_* \mathbf{A}(\boldsymbol{\tau})^{-1} \boldsymbol{\mu}_* \right] \\
& \times
\frac{ \mathbb{I}(\mathbf{c}_\omega \leq \mathbf{R}_\omega \boldsymbol{\eta} ) }
{c_2 ( \mathbf{A}(\boldsymbol{\tau})^{-1}\boldsymbol{\mu}_*, \mathbf{A}(\boldsymbol{\tau})^{-1}, \mathbf{R}_\omega, \mathbf{c}_\omega, \infty)} \\
& \times
\prod_{i=1}^{N} \mathbb{I} \left( \omega_i \leq \exp \left(- \xi \left( \mathbf{m}^\top_i \boldsymbol{\eta}^\prime \right) \right) \right) \exp \left( \xi \left( \mathbf{m}^\top_i \boldsymbol{\eta}^\prime \right) \right) \\
= &
\exp \Bigg[ -\frac{1}{2} \left( \boldsymbol{\beta}^{\top} \boldsymbol{\Sigma}^{-1} \boldsymbol{\beta} + \sum_{j \in \{ \alpha, \mathcal{B} \}} \tau_j \mathbf{u}_j^{\top} \mathbf{u}_j - 2 \boldsymbol{\eta}^{\top} \boldsymbol{\mu}_* \right) \\
& - \frac{1}{2} \boldsymbol{\mu}^{\top}_* \mathbf{A}(\boldsymbol{\tau})^{-1} \boldsymbol{\mu}_* \Bigg] \\
& \times
\frac{ \mathbb{I}(\mathbf{c}_\omega \leq \mathbf{R}_\omega \boldsymbol{\eta} ) }
{c_2 ( \mathbf{A}(\boldsymbol{\tau})^{-1}\boldsymbol{\mu}_*, \mathbf{A}(\boldsymbol{\tau})^{-1}, \mathbf{R}_\omega, \mathbf{c}_\omega, \infty)} \\
& \times
\prod_{i=1}^{N} \mathbb{I} \left( \omega_i \leq \exp \left(- \xi \left( \mathbf{m}^\top_i \boldsymbol{\eta}^\prime \right) \right) \right) \exp \left( \xi \left( \mathbf{m}^\top_i \boldsymbol{\eta}^\prime \right) \right) \\
\end{array}
$$
where
$$
\begin{array}{rl}
& c_2 \left(\mathbf{A}(\boldsymbol{\tau})^{-1} \boldsymbol{\mu}_*, \mathbf{A}(\boldsymbol{\tau})^{-1}, \mathbf{R}_\omega, \mathbf{c}_\omega, \infty \right) \\
&= \oint_{ \mathbf{R}_\omega \boldsymbol{\eta} \geq \mathbf{c}_\omega } \exp
\left[
-\frac{1}{2}( \boldsymbol{\eta} - \mathbf{A}(\boldsymbol{\tau})^{-1} \boldsymbol{\mu}_* )^{\top}
\mathbf{A}(\boldsymbol{\tau})
( \boldsymbol{\eta} - \mathbf{A}(\boldsymbol{\tau})^{-1} \boldsymbol{\mu}_*)
\right]
d \boldsymbol{\eta} \\
&\leq \oint_{ \mathbf{R}_\omega \boldsymbol{\eta} \geq \mathbf{c}_\omega } \exp
\left[
-\frac{1}{2}( \boldsymbol{\eta} - \mathbf{A}(\boldsymbol{\tau})^{-1} \boldsymbol{\mu}_* )^{\top}
\mathbf{A}(\tau_0)
( \boldsymbol{\eta} - \mathbf{A}(\boldsymbol{\tau})^{-1} \boldsymbol{\mu}_\omega)
\right]
d \boldsymbol{\eta} \\
&< \int_{\mathbb{R}^{p+2M+4}} \exp
\left[
-\frac{1}{2}( \boldsymbol{\eta} - \mathbf{A}(\boldsymbol{\tau})^{-1} \boldsymbol{\mu}_\omega )^{\top}
\mathbf{A}(\tau_0)
( \boldsymbol{\eta} - \mathbf{A}(\boldsymbol{\tau})^{-1} \boldsymbol{\mu}_\omega)
\right]
d \boldsymbol{\eta} \\
&= (2 \pi)^{(p+2M+4)/2} \left| \mathbf{A}(\tau_0) \right|^{-1/2} .
\end{array}
$$
Note that $\mathbb{I}(\mathbf{c}_\omega \leq \mathbf{R}_\omega \boldsymbol{\eta} ) = \mathbb{I}(\mathbf{c}_\alpha \leq \mathbf{R}_\alpha \boldsymbol{\eta} ) \prod_{i=1}^{N} \mathbb{I} \left( \omega_i \leq \exp \left(- \xi \left( \mathbf{m}^\top_i \boldsymbol{\eta} \right) \right) \right)$.
From \cite{mira2002efficiency} (Theorem 7), we have
$$
\begin{array}{rl}
&\displaystyle\int_{\mathbb{H}^{N}} \pi( \boldsymbol{\eta} \mid \boldsymbol{\omega}, \boldsymbol{\tau}, \mathbf{y}) \pi\left(\boldsymbol{\omega} \mid \boldsymbol{\eta}^{\prime}, \mathbf{y}\right)
d \boldsymbol{\omega} \\
& \geq
\exp \left[-\frac{1}{2} \left( \boldsymbol{\eta}^{\top} \mathbf{A}(\boldsymbol{\tau}) \boldsymbol{\eta} - 2 \boldsymbol{\eta}^{\top} \boldsymbol{\mu}_* \right) - \frac{1}{2} \boldsymbol{\mu}^{\top}_* \mathbf{A}(\boldsymbol{\tau})^{-1} \boldsymbol{\mu}_* \right] \\
& \times
\frac{ \mathbb{I}( \mathbf{c}_\alpha \leq \mathbf{R}_\alpha \boldsymbol{\eta} ) \prod_{i=1}^{N} \exp \left( -\xi \left( \mathbf{m}^\top_i \boldsymbol{\eta} \right) \right) }{ \prod_{i=1}^{N} \sup_{\boldsymbol{\eta}} \exp \left( -\xi \left( \mathbf{m}^\top_i \boldsymbol{\eta} \right) \right) } (2 \pi)^{-(p+2M+4)/2} \left| \mathbf{A}(\tau_0) \right|^{1/2} \\
& \geq
\exp \left[-\frac{1}{2} \left( \boldsymbol{\eta}^{\top} \mathbf{A}(\boldsymbol{\tau}) \boldsymbol{\eta} - 2 \boldsymbol{\eta}^{\top} \boldsymbol{\mu}_* \right) - \frac{1}{2} \boldsymbol{\mu}^{\top}_* \mathbf{A}(\boldsymbol{\tau})^{-1} \boldsymbol{\mu}_* \right] \\
& \times
{ \mathbb{I}( \mathbf{c}_\alpha \leq \mathbf{R}_\alpha \boldsymbol{\eta} ) \prod_{i=1}^{N} \exp \left( -\xi \left( \mathbf{m}^\top_i \boldsymbol{\eta} \right) \right) } (2 \pi)^{-(p+2M+4)/2} \left| \mathbf{A}(\tau_0) \right|^{1/2}
\end{array}
$$
where $\mathbb{H}^{N}=[0,1]^N$ is a hypercube.
Note that $
\exp \left[ - \frac{1}{2} \boldsymbol{\mu}^{\top}_* \mathbf{A}(\boldsymbol{\tau})^{-1} \boldsymbol{\mu}_* \right]
\geq
\exp \left[ - \frac{1}{2} \boldsymbol{\mu}^{\top}_* \mathbf{A}({\tau_0})^{-1} \boldsymbol{\mu}_* \right]
$. We have
$$
\begin{array}{rl}
& \displaystyle\int_{\mathbb{R}_{+}^2} \displaystyle\int_{\mathbb{H}^{N}}
\pi( \boldsymbol{\eta} \mid \boldsymbol{\omega}, \boldsymbol{\tau}, \mathbf{y}) \pi\left(\boldsymbol{\omega} \mid \boldsymbol{\eta}^{\prime}, \mathbf{y}\right)
d \boldsymbol{\omega}
\pi\left(\boldsymbol{\tau} \mid \boldsymbol{\eta}^{\prime}, \mathbf{y}\right)
d \boldsymbol{\tau} \\
& \geq
\displaystyle\int_{\mathbb{R}_{+}^2} \Bigg\{
\exp \left[-\frac{1}{2} \boldsymbol{\beta}^{\top} \boldsymbol{\Sigma}^{-1} \boldsymbol{\beta} - \sum_{j \in \{ \alpha, \mathcal{B} \}} \frac{\tau_j}{2} \mathbf{u}_j^{\top} \mathbf{u}_j +\boldsymbol{\eta}^{\top} \boldsymbol{\mu}_* -\frac{1}{2} \boldsymbol{\mu}_*^{\top} \mathbf{A}\left(\tau_{0}\right)^{-1} \boldsymbol{\mu}_* \right] \\
& \times
{ \mathbb{I}( \mathbf{c}_\alpha \leq \mathbf{R}_\alpha \boldsymbol{\eta} ) \prod_{i=1}^{N} \exp \left( -\xi \left( \mathbf{m}^\top_i \boldsymbol{\eta} \right) \right) } (2 \pi)^{-(p+2M+4)/2} \left| \mathbf{A}(\tau_0) \right|^{1/2} \\
& \times
\prod_{j \in \{ \alpha, \mathcal{B} \}}
\tau_j^{a_{0}+ M/2} \exp \left[-\left(b_{0}+ {\mathbf{u}_j^{\prime}}^{\top} \mathbf{u}_j^{\prime} / 2\right) \tau_j \right]
\frac{ \mathbb{I} \left(\tau_j \geq \tau_{0}\right) }{c_1\left(\tau_{0}, a_{0}+(M+2) / 2, b_{0}+{\mathbf{u}_j^{\prime}}^{\top} \mathbf{u}_j^{\prime} / 2\right)}
\Bigg\} d \boldsymbol{\tau} \\
& \geq
\exp \left[-\frac{1}{2} \boldsymbol{\beta}^{\top} \boldsymbol{\Sigma}^{-1} \boldsymbol{\beta} +\boldsymbol{\eta}^{\top} \boldsymbol{\mu}_* -\frac{1}{2} \boldsymbol{\mu}_*^{\top} \mathbf{A}\left(\tau_{0}\right)^{-1} \boldsymbol{\mu}_* \right] \\
& \times
{ \mathbb{I}( \mathbf{c}_\alpha \leq \mathbf{R}_\alpha \boldsymbol{\eta} ) \prod_{i=1}^{N} \exp \left( -\xi \left( \mathbf{m}^\top_i \boldsymbol{\eta} \right) \right) } (2 \pi)^{-(p+2M+4)/2} \left| \mathbf{A}(\tau_0) \right|^{1/2} \\
& \times
\displaystyle\int_{\mathbb{R}_{+}^2} \Bigg\{
\prod_{j \in \{ \alpha, \mathcal{B} \}}
\tau_j^{a_{0}+ M/2} \exp \left[-\left(b_{0} + {\mathbf{u}_j^{\prime}}^{\top} \mathbf{u}_j^{\prime} / 2 +{\mathbf{u}_j}^{\top} \mathbf{u}_j / 2 \right) \tau_j \right] \\
& \times
\frac{ \mathbb{I} \left(\tau_j \geq \tau_{0}\right) }{c_1\left(\tau_{0}, a_{0}+(M+2) / 2, b_{0} +{\mathbf{u}_j^{\prime}}^{\top} \mathbf{u}_j^{\prime} / 2\right)}
\Bigg\} d \boldsymbol{\tau} .
\end{array}
$$
In addition to what has been noted in \cite{wang2018analysis}, we show with u-substitution, second fundamental theorem of calculus and chain rule or Leibniz integral rule,
\begin{equation} \label{TG_inequality}
\begin{array}{rl}
&\frac{1}{c_1\left(\tau_{0}, a_{0}+ (M+2) / 2, b_{0}+ {\mathbf{u}_j^{\prime}}^{\top} \mathbf{u}_j^{\prime} / 2\right)}
\displaystyle\int_{\tau_{0}}^{\infty} \tau_j^{a_{0}+ M / 2} \exp \left[-\left(b_{0}+ {\mathbf{u}_j^{\prime}}^{\top} \mathbf{u}_j^{\prime} / 2+\mathbf{u}_j^{\top} \mathbf{u}_j / 2\right) \tau_j \right] d \tau_j \\
&=
\frac{
\left(b_{0}+ {\mathbf{u}_j^{\prime}}^{\top} \mathbf{u}_j^{\prime} / 2\right)^{a_{0}+ (M+2) / 2}
}{
\left(b_{0}+ {\mathbf{u}_j^{\prime}}^{\top} \mathbf{u}_j^{\prime} / 2+\mathbf{u}_j^{\top} \mathbf{u}_j / 2\right)^{a_{0}+ (M+2) / 2}
}
\frac{
\displaystyle\int_{\left(b_{0}+ {\mathbf{u}_j^{\prime}}^{\top} \mathbf{u}_j^{\prime} / 2+\mathbf{u}_j^{\top} \mathbf{u}_j / 2\right) \tau_{0}}^\infty x^{a_{0}+ M/ 2} \exp (-x) d x
}{
\displaystyle\int_{\left(b_{0}+ {\mathbf{u}_j^{\prime}}^{\top} \mathbf{u}_j^{\prime} / 2\right) \tau_{0}}^{\infty} x^{a_{0}+ M / 2} \exp (-x) d x
} \\
& \geq
\left(\frac{b_{0}}{b_{0}+\mathbf{u}_j^{\top} \mathbf{u}_j / 2}\right)^{a_{0}+ (M+2)/ 2}
\frac{
\displaystyle\int_{\left(b_{0}+ {\mathbf{u}_j^{\prime}}^{\top} \mathbf{u}_j^{\prime} / 2+\mathbf{u}_j^{\top} \mathbf{u}_j / 2\right) \tau_{0}}^\infty x^{a_{0}+ M/ 2} \exp (-x) d x
}{
\displaystyle\int_{\left(b_{0}+ {\mathbf{u}_j^{\prime}}^{\top} \mathbf{u}_j^{\prime} / 2\right) \tau_{0}}^{\infty} x^{a_{0}+ M / 2} \exp (-x) d x
} \\
& =
\left(\frac{b_{0}}{b_{0}+\mathbf{u}_j^{\top} \mathbf{u}_j / 2}\right)^{a_{0}+ (M+2)/ 2}
\frac{
g_1({\mathbf{u}_j^{\prime}}^{\top} \mathbf{u}_j^{\prime} / 2)
}{
g_2({\mathbf{u}_j^{\prime}}^{\top} \mathbf{u}_j^{\prime} / 2)
} \\
& \geq
\left(\frac{b_{0}}{b_{0}+\mathbf{u}_j^{\top} \mathbf{u}_j / 2}\right)^{a_{0}+ (M+2)/ 2} \exp \left(-\tau_{0} \mathbf{u}_j^{\top} \mathbf{u}_j / 2\right).
\end{array}
\end{equation}
Here $g_3(v) = g_1(v) - \exp \left(-\tau_{0} \mathbf{u}_j^{\top} \mathbf{u}_j / 2\right) g_2(v)$, where $v = {\mathbf{u}_j^{\prime}}^{\top} \mathbf{u}_j^{\prime} / 2$ and
$$
\begin{array}{rl}
\frac{d}{dv} g_3(v) =&
- \left[ \left( \left( b_{0}+ {\mathbf{u}_j^{\prime}}^{\top} \mathbf{u}_j^{\prime} / 2+\mathbf{u}_j^{\top} \mathbf{u}_j / 2\right) \tau_{0} \right)^{a_{0}+ M/ 2} \exp \left( - \left(b_{0}+ {\mathbf{u}_j^{\prime}}^{\top} \mathbf{u}_j^{\prime} / 2+\mathbf{u}_j^{\top} \mathbf{u}_j / 2\right) \tau_{0} \right) \right] \tau_0 \\
& + \exp \left(-\tau_{0} \mathbf{u}_j^{\top} \mathbf{u}_j / 2\right) \left[ \left( \left(b_{0}+ {\mathbf{u}_j^{\prime}}^{\top} \mathbf{u}_j^{\prime} / 2\right) \tau_{0} \right)^{a_{0}+ M/ 2} \exp \left( -\left(b_{0}+ {\mathbf{u}_j^{\prime}}^{\top} \mathbf{u}_j^{\prime} / 2\right) \tau_{0} \right) \right] \tau_0 \\
=&
\left[
\left( \left(b_{0}+ {\mathbf{u}_j^{\prime}}^{\top} \mathbf{u}_j^{\prime} / 2\right) \tau_{0} \right)^{a_{0}+ M/ 2}
- \left( \left(b_{0}+ {\mathbf{u}_j^{\prime}}^{\top} \mathbf{u}_j^{\prime} / 2+\mathbf{u}_j^{\top} \mathbf{u}_j / 2\right) \tau_{0} \right)^{a_{0}+ M/ 2}
\right] \\
& \times
\exp \left( - \left(b_{0}+ {\mathbf{u}_j^{\prime}}^{\top} \mathbf{u}_j^{\prime} / 2+\mathbf{u}_j^{\top} \mathbf{u}_j / 2\right) \tau_{0} \right) \tau_0 \\
<& 0
\end{array}
$$
because of $\left[
\left( \left(b_{0}+ {\mathbf{u}_j^{\prime}}^{\top} \mathbf{u}_j^{\prime} / 2\right) \tau_{0} \right)^{a_{0}+ M/ 2}
- \left( \left(b_{0}+ {\mathbf{u}_j^{\prime}}^{\top} \mathbf{u}_j^{\prime} / 2+\mathbf{u}_j^{\top} \mathbf{u}_j / 2\right) \tau_{0} \right)^{a_{0}+ M/ 2}
\right] < 0$. We showed that $g_3(v)$ is a decreasing function. Together with the limit $\lim_{v \rightarrow \infty} g_3(v) = 0$, we see that $g_3(v) \geq 0$ and
$$
\begin{array}{rl}
g_3(v) &\geq 0 \\
g_1(v) &\geq \exp \left(-\tau_{0} \mathbf{u}_j^{\top} \mathbf{u}_j / 2\right) g_2(v) \\
g_1(v)/g_2(v) &\geq \exp \left(-\tau_{0} \mathbf{u}_j^{\top} \mathbf{u}_j / 2\right) \\
\frac{
\displaystyle\int_{\left(b_{0}+ {\mathbf{u}_j^{\prime}}^{\top} \mathbf{u}_j^{\prime} / 2+\mathbf{u}_j^{\top} \mathbf{u}_j / 2\right) \tau_{0}}^\infty x^{a_{0}+ M/ 2} \exp (-x) d x
}{
\displaystyle\int_{\left(b_{0}+ {\mathbf{u}_j^{\prime}}^{\top} \mathbf{u}_j^{\prime} / 2\right) \tau_{0}}^{\infty} x^{a_{0}+ M / 2} \exp (-x) d x
}
&\geq \exp \left(-\tau_{0} \mathbf{u}_j^{\top} \mathbf{u}_j / 2\right) .
\end{array}
$$
Thus, from \eqref{TG_inequality} we have
$$
\begin{array}{rl}
k\left( \boldsymbol{\eta} \mid \boldsymbol{\eta}^{\prime}\right)
&\geq
\exp \left[-\frac{1}{2} \boldsymbol{\beta}^{\top} \boldsymbol{\Sigma}^{-1} \boldsymbol{\beta} +\boldsymbol{\eta}^{\top} \boldsymbol{\mu}_* -\frac{1}{2} \boldsymbol{\mu}_*^{\top} \mathbf{A}\left(\tau_{0}\right)^{-1} \boldsymbol{\mu}_* \right] \\
& \times
{ \mathbb{I}( \mathbf{c}_\alpha \leq \mathbf{R}_\alpha \boldsymbol{\eta} ) \prod_{i=1}^{N} \exp \left( -\xi \left( \mathbf{m}^\top_i \boldsymbol{\eta} \right) \right) }
(2 \pi)^{-(p+2M+4)/2} \left| \mathbf{A}(\tau_0) \right|^{1/2} \\
& \times
\prod_{j \in \{ \alpha, \mathcal{B} \}}
\left({b_{0}}/{(b_{0}+\mathbf{u}_j^{\top} \mathbf{u}_j / 2)}\right)^{a_{0}+ (M+2)/ 2} \exp \left(-\tau_{0} \mathbf{u}_j^{\top} \mathbf{u}_j / 2\right) \\
& \geq \delta h(\boldsymbol{\eta})
\end{array}
$$
where
$$
\begin{array}{rl}
h( \boldsymbol{\eta} ) =& \exp \left[-\frac{1}{2} \boldsymbol{\beta}^{\top} \boldsymbol{\Sigma}^{-1} \boldsymbol{\beta} +\boldsymbol{\eta}^{\top} \boldsymbol{\mu}_* -\frac{1}{2} \boldsymbol{\mu}_*^{\top} \mathbf{A}\left(\tau_{0}\right)^{-1} \boldsymbol{\mu}_* \right] \\
& \times
\frac{ \mathbb{I}( \mathbf{c}_\alpha \leq \mathbf{R}_\alpha \boldsymbol{\eta} ) }{c_{4}( \mathbf{M}, \mathbf{y})}
{ \prod_{i=1}^{N} \exp \left( -\xi \left( \mathbf{m}^\top_i \boldsymbol{\eta} \right) \right) } \\
& \times
\prod_{j \in \{ \alpha, \mathcal{B} \}}
\left({b_{0}}/{(b_{0}+\mathbf{u}_j^{\top} \mathbf{u}_j / 2)}\right)^{a_{0}+ (M+2)/ 2} \exp \left(-\tau_{0} \mathbf{u}_j^{\top} \mathbf{u}_j / 2\right) \\
\delta = & (2 \pi)^{-(p+2M+4)/2} \left| \mathbf{A}(\tau_0) \right|^{1/2} { c_{4}( \mathbf{M}, \mathbf{y}) } < 1
\end{array}
$$
and
$$
\begin{array}{rl}
c_{4}( \mathbf{M}, \mathbf{y}) =& \exp \left[-\frac{1}{2} \boldsymbol{\mu}_*^{\top} \mathbf{A}\left(\tau_{0}\right)^{-1} \boldsymbol{\mu}_* \right]
\oint_{ \mathbf{u}_\alpha \geq \mathbf{0} }
\exp \left[-\frac{1}{2} \boldsymbol{\beta}^{\top} \boldsymbol{\Sigma}^{-1} \boldsymbol{\beta} + \boldsymbol{\eta}^{\top} \boldsymbol{\mu}_* \right] \\
& \times
{ \prod_{i=1}^{N} \exp \left( -\xi \left( \mathbf{m}^\top_i \boldsymbol{\eta} \right) \right) }\\
& \times
\prod_{j \in \{ \alpha, \mathcal{B} \} }
\left(\frac{b_{0}}{b_{0}+\mathbf{u}_j^{\top} \mathbf{u}_j / 2}\right)^{a_{0}+ (M+2) / 2} \exp \left(-\tau_{0} \mathbf{u}_j^{\top} \mathbf{u}_j / 2\right) d \boldsymbol{\eta} \\
\leq &
\exp \left[-\frac{1}{2} \boldsymbol{\mu}_*^{\top} \mathbf{A}\left(\tau_{0}\right)^{-1} \boldsymbol{\mu}_* \right]
\oint_{ \mathbf{u}_\alpha \geq \mathbf{0} }
\exp \left[-\frac{1}{2} \boldsymbol{\beta}^{\top} \boldsymbol{\Sigma}^{-1} \boldsymbol{\beta} + \boldsymbol{\eta}^{\top} \boldsymbol{\mu}_* \right] \\
& \times
\prod_{j \in \{\alpha, \mathcal{B} \} }
\exp \left(-\tau_{0} \mathbf{u}_j^{\top} \mathbf{u}_j / 2\right) d \boldsymbol{\eta} \\
=&
\oint_{ \mathbf{c}_\alpha \leq \mathbf{R}_\alpha \boldsymbol{\eta} }
\exp \left[-\frac{1}{2} \left( \boldsymbol{\eta} - \mathbf{A}(\tau_0)^{-1} \boldsymbol{\mu}_* \right)^{\top} \mathbf{A}(\tau_0) \left( \boldsymbol{\eta} - \mathbf{A}(\tau_0)^{-1} \boldsymbol{\mu}_* \right) \right] d \boldsymbol{\eta} \\
<&
\int_{\mathbb{R}^{p+2M+4}}
\exp \left[-\frac{1}{2} \left( \boldsymbol{\eta} - \mathbf{A}(\tau_0)^{-1} \boldsymbol{\mu}_* \right)^{\top} \mathbf{A}(\tau_0) \left( \boldsymbol{\eta} - \mathbf{A}(\tau_0)^{-1} \boldsymbol{\mu}_* \right) \right] d \boldsymbol{\eta} \\
=&
(2 \pi)^{(p+2M+4)/2}\left| \mathbf{A}\left(\tau_{0}\right)\right|^{-1 / 2} <\infty .
\end{array}
$$
This concludes the proof for uniform ergodicity.
\end{proof}
\begin{proof}Proof of Theorem \ref{thm2}.
Recall that
$$
\int_{\mathbb{R}^{2}_+} \int_{\mathbb{H}^{N}} \pi(\boldsymbol{\eta}, \boldsymbol{\omega}, \boldsymbol{\tau} \mid \mathbf{y}) d \boldsymbol{\omega} d \boldsymbol{\tau} =\pi(\boldsymbol{\eta} \mid \mathbf{y} )
$$
and normalizing constant $c( \mathbf{y} )$,
$$
\begin{array}{c}
\pi(\boldsymbol{\eta}, \boldsymbol{\tau} \mid \mathbf{y}) = c( \mathbf{y} )^{-1} L(\boldsymbol{\eta} \mid \mathbf{y}, \mathbf{M} ) \pi (\boldsymbol{\eta} \mid \mathbf{b}, \mathbf{A}(\boldsymbol{\tau})^{-1}, \mathbf{R}_\alpha, \mathbf{c}_\alpha, \boldsymbol{\infty} )
\pi \left( \boldsymbol{\tau} \mid a_{0}, b_{0}, \tau_{0}\right)\\
c( \mathbf{y} ) =
\displaystyle\int_{\mathbb{R}^{2}_+} \displaystyle\int_{\mathbb{R}^{p+2M+4}}
{
L(\boldsymbol{\eta} \mid \mathbf{y}, \mathbf{M} )
\pi (\boldsymbol{\eta} \mid \mathbf{b}, \mathbf{A}(\boldsymbol{\tau})^{-1}, \mathbf{R}_\alpha, \mathbf{c}_\alpha, \boldsymbol{\infty} )
\pi \left( \boldsymbol{\tau} \mid a_{0}, b_{0}, \tau_{0}\right) } d \boldsymbol{\eta} d \boldsymbol{\tau} .
\end{array}
$$
Note that
$$
\begin{array}{rl}
\pi(\boldsymbol{\eta}, \boldsymbol{\omega}, \boldsymbol{\tau} \mid \mathbf{y} ) &=
\pi(\boldsymbol{\eta}, \boldsymbol{\tau} \mid \mathbf{y}) \pi \left( \boldsymbol{\omega} \mid \boldsymbol{\eta} \right) \\
&=
c( \mathbf{y} )^{-1} L(\boldsymbol{\eta} \mid \mathbf{y}, \mathbf{M} )
\pi (\boldsymbol{\eta} \mid \mathbf{b}, \mathbf{A}(\boldsymbol{\tau})^{-1}, \mathbf{R}_\alpha, \mathbf{c}_\alpha, \boldsymbol{\infty} )
\pi \left( \boldsymbol{\tau} \mid a_{0}, b_{0}, \tau_{0}\right) \pi \left( \boldsymbol{\omega} \mid \boldsymbol{\eta} \right).
\end{array}
$$
Thus,
$$
\begin{array}{rl}
\pi(\boldsymbol{\eta}, \boldsymbol{\omega}, \boldsymbol{\tau} \mid \mathbf{y})
= &
c( \mathbf{y} )^{-1} c_1 \left(\tau_{0}, a_{0}, b_{0}\right)^{-2} |\mathbf{A}(\boldsymbol{\tau})|^{1/2} 2^{M+2} (2\pi)^{-(p+2M+4)/2} \mathbb{I} ( \mathbf{c}_\alpha \leq \mathbf{R}_\alpha \boldsymbol{\eta} ) \\
&\times
\exp \left\{\mathbf{y}^{\top} \mathbf{M} \boldsymbol{\eta}-
\frac{1}{2}\left(\boldsymbol{\eta}- \mathbf{b} \right)^{\top}
\mathbf{A}(\boldsymbol{\tau})
\left(\boldsymbol{\eta}- \mathbf{b} \right) \right\} \\
&\times \prod_{j \in \{\alpha, \mathcal{B}\}} \tau_j^{a_{0}-1} e^{-b_{0} \tau_j} { \mathbb{I}\left(\tau_j \geq \tau_{0}\right) } \\
&\times
\prod_{i=1}^{N} \left[ \exp \left(- \xi \left( \mathbf{m}^\top_i \boldsymbol{\eta} \right) \right) \pi \left( {\omega_i} \mid \boldsymbol{\eta} \right) \right] \\
=&
c( \mathbf{y} )^{-1} c_1 \left(\tau_{0}, a_{0}, b_{0}\right)^{-2} |\mathbf{A}(\boldsymbol{\tau})|^{1/2} 2^{M+2} (2\pi)^{-(p+2M+4)/2} \mathbb{I} ( \mathbf{c}_\alpha \leq \mathbf{R}_\alpha \boldsymbol{\eta} ) \\
&\times
\exp \left( \mathbf{y}^{\top} \mathbf{M} \boldsymbol{\eta} \right) \exp \left\{
- \frac{1}{2}\left(\boldsymbol{\eta}- \mathbf{b} \right)^{\top}
\mathbf{A}(\boldsymbol{\tau})
\left(\boldsymbol{\eta}- \mathbf{b} \right) \right\} \\
&\times \prod_{j \in \{\alpha, \mathcal{B}\}} \tau_j^{a_{0}-1} e^{-b_{0} \tau_j} { \mathbb{I}\left(\tau_j \geq \tau_{0}\right) } \\
&\times
\prod_{i=1}^{N} \left[ \exp \left(- \xi \left( \mathbf{m}^\top_i \boldsymbol{\eta} \right) \right) \pi \left( {\omega_i} \mid \boldsymbol{\eta} \right) \right] \\
=&
c( \mathbf{y} )^{-1} c_1 \left(\tau_{0}, a_{0}, b_{0}\right)^{-2} |\mathbf{A}(\boldsymbol{\tau})|^{1/2} 2^{M+2} (2\pi)^{-(p+2M+4)/2} \mathbb{I} ( \mathbf{c}_\alpha \leq \mathbf{R}_\alpha \boldsymbol{\eta} ) \\
&\times
\exp \left( \mathbf{y}^{\top} \mathbf{M} \boldsymbol{\eta} \right) \exp \left\{
- \frac{1}{2}\left(\boldsymbol{\eta}- \mathbf{b} \right)^{\top}
\mathbf{A}(\boldsymbol{\tau})
\left(\boldsymbol{\eta}- \mathbf{b} \right) \right\} \\
&\times \prod_{j \in \{\alpha, \mathcal{B}\}} \tau_j^{a_{0}-1} e^{-b_{0} \tau_j} { \mathbb{I}\left(\tau_j \geq \tau_{0}\right) } \prod_{i=1}^{N} \pi\left( \omega_j \right) \mathbb{I} \left( \omega_j \leq \exp \left( -\xi \left(\mathbf{m}_{i}^{\top} \boldsymbol{\eta} \right) \right) \right)\\
\leq&
c( \mathbf{y} )^{-1} c_1 \left(\tau_{0}, a_{0}, b_{0}\right)^{-2} |\mathbf{A}(\boldsymbol{\tau})|^{1/2} 2^{M+2} (2\pi)^{-(p+2M+4)/2} \\
&\times
\exp \left( \mathbf{y}^{\top} \mathbf{M} \boldsymbol{\eta} \right) \exp \left\{
- \frac{1}{2}\left(\boldsymbol{\eta}- \mathbf{b} \right)^{\top}
\mathbf{A}(\boldsymbol{\tau})
\left(\boldsymbol{\eta}- \mathbf{b} \right) \right\} \\
&\times \prod_{j \in \{\alpha, \mathcal{B}\}} \tau_j^{a_{0}-1} e^{-b_{0} \tau_j} { \mathbb{I}\left(\tau_j \geq \tau_{0}\right) } \pi (\boldsymbol{\omega}) \prod_{i=1}^{N} \mathbb{I} \left( \omega_j \leq \exp \left( -\xi \left(\mathbf{m}_{i}^{\top} \boldsymbol{\eta} \right) \right) \right) \\
=&
c( \mathbf{y} )^{-1} c_1 \left(\tau_{0}, a_{0}, b_{0}\right)^{-2} 2^{M+2} \exp \left( \mathbf{y}^{\top} \mathbf{M} \boldsymbol{\eta} \right) \pi(\boldsymbol{\eta} | \mathbf{b}, \mathbf{A}(\boldsymbol{\tau})^{-1} ) \\
&\times \prod_{j \in \{\alpha, \mathcal{B}\}} \tau_j^{a_{0}-1} e^{-b_{0} \tau_j} { \mathbb{I}\left(\tau_j \geq \tau_{0}\right) } \pi (\boldsymbol{\omega}) \prod_{i=1}^{N} \mathbb{I} \left( \omega_j \leq \exp \left( -\xi \left(\mathbf{m}_{i}^{\top} \boldsymbol{\eta} \right) \right) \right)
\end{array}
$$
where $\pi(\boldsymbol{\eta} | \mathbf{b}, \mathbf{A}(\boldsymbol{\tau})^{-1} )$ is the PDF of $\boldsymbol{\eta} \sim \mathrm{N} (\mathbf{b}, \mathbf{A}(\boldsymbol{\tau})^{-1})$. We can integrate out $\boldsymbol{\omega}$ and upper bound using $\exp \left( -\xi \left(\mathbf{m}_{i}^{\top} \boldsymbol{\eta} \right) \right) \leq 1$,
$$
\begin{array}{rl}
\int_{\mathbb{H}^{N}} \pi(\boldsymbol{\eta}, \boldsymbol{\omega}, \boldsymbol{\tau} \mid \mathbf{y}) d \boldsymbol{\omega}
=&
\pi(\boldsymbol{\eta}, \boldsymbol{\tau} \mid \mathbf{y}) \\
\leq&
c( \mathbf{y} )^{-1} c_1 \left(\tau_{0}, a_{0}, b_{0}\right)^{-2} 2^{M+2} \exp \left( \mathbf{y}^{\top} \mathbf{M} \boldsymbol{\eta} \right) \pi(\boldsymbol{\eta} | \mathbf{b}, \mathbf{A}(\boldsymbol{\tau})^{-1} ) \\
&\times \prod_{j \in \{\alpha, \mathcal{B}\}} \tau_j^{a_{0}-1} e^{-b_{0} \tau_j} { \mathbb{I}\left(\tau_j \geq \tau_{0}\right) } \\
&\times \prod_{i=1}^{N} \exp \left( -\xi \left(\mathbf{m}_{i}^{\top} \boldsymbol{\eta} \right) \right) \\
\leq&
c( \mathbf{y} )^{-1} c_1 \left(\tau_{0}, a_{0}, b_{0}\right)^{-2} 2^{M+2} \exp \left( \mathbf{y}^{\top} \mathbf{M} \boldsymbol{\eta} \right) \pi(\boldsymbol{\eta} | \mathbf{b}, \mathbf{A}(\boldsymbol{\tau})^{-1} ) \\
&\times \prod_{j \in \{\alpha, \mathcal{B}\}} \tau_j^{a_{0}-1} e^{-b_{0} \tau_j} { \mathbb{I}\left(\tau_j \geq \tau_{0}\right) } .
\end{array}
$$
Here we set $\mathbf{z} = \mathbf{t} + \mathbf{M}^\top \mathbf{y}$ in order to facilitate the proof, where $\int \exp( \boldsymbol{\eta}^{\top} \mathbf{z} ) \pi(\boldsymbol{\eta} | \mathbf{b}, \mathbf{A}(\boldsymbol{\tau})^{-1} ) d\boldsymbol{\eta} $ is of the form of a normal MGF.
We have
$$
\begin{array}{rl}
\int_{\mathbb{R}^{2}_+}
\int_{\mathbb{R}^{p+2M+4}} e^{\boldsymbol{\eta}^{\top} \mathbf{t} }
\pi(\boldsymbol{\eta}, \boldsymbol{\tau} \mid \mathbf{y})
d \boldsymbol{\eta} d \boldsymbol{\tau}
\leq &
\displaystyle\int_{\mathbb{R}^{2}_+}
\displaystyle\int_{\mathbb{R}^{p+2M+4}} \Bigg\{
c( \mathbf{y} )^{-1} c_1 \left(\tau_{0}, a_{0}, b_{0}\right)^{-2} 2^{M+2} \\
&\times \exp( \boldsymbol{\eta}^{\top} \mathbf{t} ) \exp \left( \mathbf{y}^{\top} \mathbf{M} \boldsymbol{\eta} \right) \\
&\times \pi(\boldsymbol{\eta} | \mathbf{b}, \mathbf{A}(\boldsymbol{\tau})^{-1} ) \prod_{j \in \{\alpha, \mathcal{B}\}} \tau_j^{a_{0}-1} e^{-b_{0} \tau_j} \mathbb{I}\left(\tau_j \geq \tau_{0}\right)
\Bigg\} d \boldsymbol{\eta} d \boldsymbol{\tau} \\
= &
\displaystyle\int_{\mathbb{R}^{2}_+}
\displaystyle\int_{\mathbb{R}^{p+2M+4}} \Bigg\{
c( \mathbf{y} )^{-1} c_1 \left(\tau_{0}, a_{0}, b_{0}\right)^{-2} 2^{M+2} \exp( \boldsymbol{\eta}^{\top} \mathbf{z} ) \\
&\times \pi(\boldsymbol{\eta} | \mathbf{b}, \mathbf{A}(\boldsymbol{\tau})^{-1} ) \prod_{j \in \{\alpha, \mathcal{B}\}} \tau_j^{a_{0}-1} e^{-b_{0} \tau_j} { \mathbb{I}\left(\tau_j \geq \tau_{0}\right) }
\Bigg\} d \boldsymbol{\eta} d \boldsymbol{\tau} \\
= &
c( \mathbf{y} )^{-1} c_1 \left(\tau_{0}, a_{0}, b_{0}\right)^{-2} 2^{M+2} \\
&\times \displaystyle\int_{\mathbb{R}^{2}_+} \Bigg\{
\exp \left( \mathbf{b}^{\top} \mathbf{z} + \frac{1}{2} \mathbf{z}^{\top} \mathbf{A}(\boldsymbol{\tau})^{-1} \mathbf{z} \right) \\
&\times
\prod_{j \in \{\alpha, \mathcal{B}\}} \tau_j^{a_{0}-1} e^{-b_{0} \tau_j} { \mathbb{I}\left(\tau_j \geq \tau_{0}\right) }
\Bigg\} d \boldsymbol{\tau} \\
\leq &
c( \mathbf{y} )^{-1} c_1 \left(\tau_{0}, a_{0}, b_{0}\right)^{-2} 2^{M+2}
\exp \left( \mathbf{b}^{\top} \mathbf{z} + \frac{1}{2} \mathbf{z}^{\top} \mathbf{A}({\tau_0})^{-1} \mathbf{z} \right) \\
&\times
\displaystyle\int_{\mathbb{R}^{2}_+} \Bigg\{
\prod_{j \in \{\alpha, \mathcal{B}\}} \tau_j^{a_{0}-1} e^{-b_{0} \tau_j} { \mathbb{I}\left(\tau_j \geq \tau_{0}\right) }
\Bigg\} d \boldsymbol{\tau}\\
< & \infty
\end{array}
$$
and the moment generating function exist. This concludes the proof.
\end{proof}
\backmatter
\bibliographystyle{biom}
|
1,108,101,564,522 | arxiv | \section{}\label{s1}
\section{Introduction}
\label{sec:introduction}
\subsection{Background}
At its core, Neuro-Symbolic AI (NeSy) is ``the combination of deep learning and symbolic reasoning" \cite{Garcez_Lamb_2020}. The goal of NeSy is to address the weaknesses of each of symbolic and sub-symbolic approaches while preserving their strengths (see figure \ref{fig:capabilities}). Thus NeSy promises to deliver a best-of-both-worlds approach which embodies the ``two most fundamental aspects of intelligent cognitive behavior: the ability to learn from experience, and the ability to reason from what has been learned"\cite{Garcez_Lamb_2020, Valiant_2003}.
Remarkable progress has been made on the learning side, especially in the area of Natural Language Processing (NLP) and in particular with deep learning architectures such as the transformer \cite{vaswani2017attention}. However, these systems display certain intrinsic weaknesses which some researchers argue cannot be addressed by deep learning alone; and that in order to do even the most basic reasoning, we need rich representations which enable precise, human interpretable inference via mathematical logic.
\begin{figure}
\includegraphics[scale=0.3]{figures/eps/nesy-table-diagram.eps}
\caption{Symbolic vs Sub-Symbolic strengths and weaknesses. Based on the work of \cite{Garcez_Gori_Lamb_Serafini_Spranger_Tran_2019} }
\label{fig:capabilities}
\end{figure}
Historically, rivalry between symbolic and connectionist, or sub-symbolic, AI research has stymied collaboration across these fields. Although it should be acknowledged that not everyone thought the two were incompatible. As early as 1991, Marvin Minsky presciently asked ``Why is there so much excitement about Neural Networks today, and how is this related to research on Artificial Intelligence? Much has been said, in the popular press, as though these were conflicting activities. This seems exceedingly strange to me, because both are parts of the very same enterprise” \cite{minsky1991logical}.
More recently, a discussion between Gary Marcus and Yoshua Bengio at the 2019 Montreal AI Debate prompted some passionate exchanges in AI circles, with Marcus arguing that ``expecting a monolithic architecture to handle abstraction and reasoning is unrealistic", while Bengio defended the stance that ``sequential reasoning can be performed while staying in a deep learning framework"\cite{AIDebate2019}.
Spurred by this discussion, and almost ironically, by the success of deep learning (and ergo, the clarity into its limitations), research into hybrid solutions has seen a dramatic increase (see figure \ref{fig:allNeSy}). At the same time, discussion in the AI community has culminated in ``violent agreement" \cite{Kautz} that the next phase of AI research will be about ``combining neural and symbolic approaches in the sense of NeSy AI [which] is at least a path forward to much stronger AI systems" \cite{Sarker_Zhou_Eberhart_Hitzler_2021}. Much of this discussion centers around the ability (or inability) of deep learning to reason, and in particular, to reason outside of the training distribution. Indeed, at IJCAI 2021, Yoshua Bengio affirms that ``we need a new learning theory to deal with Out-of-Distribution generalization" \cite{IJCAI2021}. Bengio's talk is titled ``System 2 Deep Learning: Higher-Level Cognition, Agency, Out-of-Distribution Generalization and Causality". Here, System 2 refers to the System 1/System 2 (S1/S2) dual process theory of human reasoning developed by psychologist and Nobel laureate Daniel Kahneman in his 2011 book ``Thinking, Fast and Slow" \cite{kahneman2011thinking}. AI researchers have drawn many parallels between the characteristics of sub-symbolic and symbolic AI systems and human reasoning with S1/S2. Broadly speaking, sub-symbolic (neural, deep-learning) architectures are said to be akin to the fast, intuitive, often biased and/or logically flawed S1. And the more deliberative, slow, sequential S2 can be thought of as symbolic or logical. But this is not the only theory of human reasoning as we'll discuss later in this paper.
\begin{figure}[htbp]
\centering
\includegraphics[scale=0.55]{figures/eps/allNesyByYear.eps}
\caption{Number of Neuro Symbolic articles published since 2010, normalized by the total number of all Computer Science articles published each year. The figure represents the unfiltered results from Scopus given the search keywords described in section \ref{search_process}.}
\label{fig:allNeSy}
\end{figure}
\subsection{Reasoning \& Language}
``Language understanding in the broadest sense of the term, including question answering that requires commonsense reasoning, offers probably the most complete application area of neurosymbolic AI"\cite{Garcez_Lamb_2020}. This makes a lot intuitive sense from a linguistic perspective. If we accept that language is compositional, with rules and structure, then it should be possible to obtain its meaning via logical processing. Compositionality in language was formalized by Richard Montague in the 1970s, in what is now referred to as \textit{Montague grammar}. ``The key idea is that compositionality requires the existence of a homomorphism between the expressions of a language and the meanings of those expressions."\footnote{\url{https://plato.stanford.edu/entries/compositionality/}} in other words, there is a direct relationship between syntax and semantics. This is in line with Noam Chomsky's \textit{Universal grammar} which states that there is a structure to natural language which is innate and universal to all humans regardless of cultural differences. The challenge lies in representing this structure in a way that both captures the semantics and is computationally efficient.
On the one hand, distributed representations are desirable because they can be efficiently processed by gradient descent (the backbone of deep learning). On the other, the meaning embedded in a distributed representation is difficult if not impossible to decompose. So while a large language model (LLM) may be very good at making certain types of predictions, it is not able to provide an explanation of how it got there. We've also seen that the larger the model - more parameters as well as more training data - the better the predictions. But even as these models get infeasibly large, they still fail on tasks requiring basic commonsense. The example in Figure \ref{fig:gpt3fumble}, given by Marcus and Davis in \cite{Marcus_Davis} is a case in point.
\begin{figure}[htbp]
\footnotesize
\begin{zitat}{}
You are having a small dinner party. You want to serve dinner in the living room. The dining room table is wider than the doorway, so to get it into the living room, you will have to \textbf{remove the door. You have a table saw, so you cut the door in half and remove the top half.}
\end{zitat}
\caption{GPT3 text completion example. The prompt is rendered in regular font, while the GPT3 response is shown in bold \cite{Marcus_Davis}.}
\label{fig:gpt3fumble}
\end{figure}
On the other hand, traditional symbolic approaches have also failed to capture the essence of human reasoning. We don't need a scholar to confirm that everyday commonsense reasoning is nothing like the rigorous mathematical logic whose goal is validity. But even when the objective isn't commonsense, but rather tasks which require precise, deterministic answers such as expert reasoning or planning, traditional symbolic reasoners are slow, cumbersome, and computationally intractable at scale. Description Logics (DLs) such as OWL, for example, are used to reason over ontologies and knowledge graphs (KGs) on the Web. However, one must accept a harsh trade-off between expressivity and complexity when choosing a DL flavor. Improving the performance of reasoning over ontologies and knowledge graphs that power search and information retrieval across the Web is particularly relevant to the Semantic Web community. Hitzler et al. report on the current research in this area \cite{Hitzler_Bianchi_Ebrahimi_Sarker_2020}.
\subsection{Contributions}\label{sec:contributions}
Several surveys have already been conducted which cover the overall NeSy landscape going as far back as 2005, and as recently as 2021, so we will not attempt to replicate that here \cite{Garcez_Lamb_2020,Garcez_Gori_Lamb_Serafini_Spranger_Tran_2019, Sarker_Zhou_Eberhart_Hitzler_2021, Bader_Hitzler_2005, Hammer_Hitzler_2007, Garcez_Lamb_Gabbay_2009, Besold_Kuhnberger_2015, Gabrilovich_Guha_McCallum_Murphy_2015, garcez2015neural, besold2017neural, Belle_2020, Lamb_Garcez_Gori_Prates_Avelar_Vardi_2020, von_Rueden_Mayer_Beckh_Georgiev_Giesselbach_Heese_Kirsch_Walczak_Pfrommer_Pick_etal_2021, Yu_Yang_Liu_Wang_2021, Zhang_Chen_Zhang_Ke_Ding_2021, Tsamoura_Hospedales_Michael_2021}. In fact, our understanding of the field is guided by the works of these scholars. For a succinct overview we refer the reader to \cite{Sarker_Zhou_Eberhart_Hitzler_2021}. And for a more in-depth analysis we recommend \cite{Garcez_Gori_Lamb_Serafini_Spranger_Tran_2019}. Our aim is to synthesize recent work implementing NeSy in the language domain, and to verify if the goals of NeSy are being realized, what the challenges are, and future directions. To our knowledge this is the first attempt at this specific task. In the following sub sections, we briefly describe each of the following goals. These are similar to the benefits described in \cite{Sarker_Zhou_Eberhart_Hitzler_2021}:
\begin{enumerate}
\item Out-of-distribution Generalization
\item Interpretability
\item Reduced size of training data
\item Transferability
\item Reasoning
\end{enumerate}
\subsubsection{Out-of-distribution (OOD) Generalization}
OOD generalization refers to the ability of a model to extrapolate to phenomena not previously seen in the training data. The lack of OOD generalization in LLMs is often demonstrated by their inability perform commonsense reasoning, as in the example in figure \ref{fig:gpt3fumble}.
\subsubsection{Interpretability}
\label{IandE}
As Machine Learning (ML) and AI become increasingly embedded in daily life, the need to hold ML/AI accountable is also growing. This is particularly true in sensitive domains such as healthcare, legal, and some business applications such as lending, where bias mitigation and fairness are critical. These concerns, among others, are the province of Explainable AI (XAI). XAI can be broken down into three main categories:
\begin{enumerate}
\item \textit{Explainability} - the facility for an expert human to see how a model arrived at a prediction or inference - often represented as a set of rules derived from the model. It seeks to answer the question: \textit{How does the model work?}
\item \textit{Interpretability} - the facility for a non-expert human to see how the data used by the model led to a prediction or inference, usually in the form of a cause and effect articulation. It seeks to answer the question: \textit{Why did the model come to this conclusion?}
\item \textit{Interactivity} - the facility for a human to interrogate the model about counterfactuals. It seeks to answer the question: \textit{What would happen if the data was different?}
\end{enumerate}
In the literature, sometimes the term interpretability is used in place of explainability and vice versa, however, for our purposes, all three categories are subsumed under the general notion of interpretability. XAI is made possible by invoking an explicit reasoning module post hoc, or building interpretability into the system to begin with.
\subsubsection{Reduced size of training data}
SOTA language models utilize massive amounts of data for training. This can cost in the thousands or even millions of dollars, take a very long time, and is neither environmentally friendly nor accessible to most researchers or businesses. The ability to learn from less data brings obvious benefits. But apart from the practical implications, there is something innately disappointing in LLMs' `bigger hammer' approach. Science rewards parsimony and elegance, and NeSy promises to deliver results without the need for such massive scale.
\subsubsection{Transferability}
Transferability is the ability of a model which was trained on one domain, to perform similarly well in a different domain. This can be particularly valuable, when the new domain has very few examples available for training. In such cases we might rely on knowledge transfer similar to the way a human might rely on abstract reasoning when faced with an unfamiliar situation.
\subsubsection{Reasoning}\label{sec:intro_reasoning}
According to Encyclopedia Britannica, ``To reason is to draw inferences appropriate to the situation" \cite{Britanica}. Reasoning is not only a goal in its own right, but also the means by which the other above mentioned goals can be achieved. Not only is it one of the most difficult problems in AI, it is one of the most contested. In section \ref{discussion:reasoning} we examine the uses of the term reasoning in more depth.
The remainder of this manuscript is structured as follows. Section \ref{methods} describes the research methods employed for searching and analysing relevant studies. In Section \ref{results} we analyze the results of the data extraction, how the studies reviewed fit into Henry Kautz's NeSy taxonomy, and propose a simplified nomenclature for describing Kautz's NeSy categories. Section \ref{discussion} discusses various existing implementation challenges. Section \ref{future} presents limitations of the work and future directions for NeSy in NLP, followed by the conclusion in Section \ref{conclusion}.
\section{Methods}\label{methods}
Our review methodology is guided by the principles described in \cite{Kitchenham07guidelinesfor, Pare_Trudel_Jaana_Kitsiou_2015, Page_McKenzie_Bossuyt_Boutron_Hoffmann_Mulrow_Shamseer_Tetzlaff_Akl_Brennan_et_al._2021}.
\subsection{Research Questions}
\begin{enumerate} [RQ1:]
\item What are the existing studies on neurosymbolic AI (NeSy) in natural language processing (NLP)?
\item What are the current applications of NeSy in NLP?
\item How are symbolic and sub-symbolic techniques integrated and what are the advantages/disadvantages?
\item What are the challenges for NeSy and how might they be addressed (including existing proposals for future work)?
\item What areas of NLP might be likely to benefit from the NeSy approach in the future?
\end{enumerate}
\subsection{Search Process}\label{search_process}
We chose Scopus to perform our initial search, as Scopus indexes all the top journals and conferences we were interested in. This obviously precludes some niche publications and it is possible we missed some relevant studies. As our aim is to shed light on the field generally, our assumption is that the top journals are a good representation of the research area as a whole. Since we were looking for studies which combine neural and symbolic approaches, our query consists of combinations of neural and symbolic terms as well as variations of neuro-symbolic terms, listed in table \ref{t1}:
\begin{table*}[htbp]
\caption{Search Keywords} \label{t1}
\centering
\begin{tabular}{@{}lll}
\toprule
Neural Terms & Symbolic Terms & Neuro-Symbolic Terms\\
\midrule
sub-symbolic & symbolic & neuro-symbolic\\
machine learning & reasoning & neural-symbolic \\
deep learning & logic & neuro symbolic \\
& & neural symbolic \\
& & neurosymbolic \\
\bottomrule
\end{tabular}
\end{table*}
The initial query was restricted to peer-reviewed English language journal articles from the last 10 years and conference papers from the last 3 years, which produced 2,412 results. The query and additional details can be found in our github repository \footnote{\url{https://github.com/kyleiwaniec/neuro-symbolic-ai-systematic-review}}
\subsection{Study selection process}\label{s3}
We further limit the journal articles to those published by the top 20 publishers as ranked by Scopus's CiteScore, which is based on number of citations normalized by the document count over a 4 year window\footnote{\url{https://service.elsevier.com/app/answers/detail/a_id/14880/kw/citescore/supporthub/scopus/}}, and SJR (SCImago Journal Rank), a measure of prestige inspired by the PageRank algorithm over the citation network\footnote{\url{https://service.elsevier.com/app/answers/detail/a_id/14883/supporthub/scopus/related/1/}}, the union of which resulted in 29 publishers, and eliminated 669 articles, for a total of 1,510 journal articles and 232 conference papers for screening. Two researchers independently screened each of the 1,742 studies (articles and conference papers), based on the inclusion/exclusion criteria in Table \ref{tbl:inclusion}. An overview of the selection process can be seen in Figure \ref{fig:figure1}.
\begin{table*}[htbp]
\caption{Inclusion/Exclusion Criteria} \label{tbl:inclusion}
\renewcommand{\arraystretch}{1.5}
\begin{tabular}{@{}>{\raggedright\arraybackslash}p{6.5cm}>{\raggedright\arraybackslash}p{6.5cm}}
\toprule
Inclusion & Exclusion \\
\midrule
Input format: unstructured or semi structured text
& Input format: not text data, namely: images, speech, tabular data, categorical data, etc. \\
Output format: Any
& Application: Theoretical Papers, Position Papers, Surveys (in other words, not implementations) \\
Application: Implementation
& The search keywords match, but the actual content does not \\
Language: English
& Full text not available (Authors were contacted in these cases) \\
\bottomrule
\end{tabular}
\end{table*}
\begin{figure}[htbp]
\centering
\includegraphics[width=1\textwidth]{figures/eps/screening-diagram.eps}
\caption{Selection Process Diagram}
\label{fig:figure1}
\end{figure}
The first round of inclusion/exclusion was performed on the titles and abstracts of the 1,742 studies (1,510 articles and 232 papers) from the above identification step. The inclusion criteria at this stage was intentionally broad, as the process itself was meant to be exploratory, and to inform the researchers of relevant topics within NeSy.
This unsurprisingly led to some significant researcher disagreement on inclusion, especially since studies need not have been explicitly labeled as neuro-symbolic to be classified as such. Agreement between researchers can be measured using the Cohen Kappa statistic, with values ranging from [-1,1], where 0 represents the expected kappa score had the labels been assigned randomly, -1 indicates complete disagreement, and 1 indicates perfect agreement. Our score at this stage came to a rather low 0.33. Since this measure is not particularly intuitive, we include a Venn diagram of the number of studies included by each researcher - see Figure \ref{fig: agreement}.
\begin{figure}[htbp]
\centering
\includegraphics[scale=0.55]{figures/eps/researcher_agreement.eps}
\caption{Researcher Agreement Overlap.}
\label{fig: agreement}
\end{figure}
To better understand the disagreement and researcher biases, we do an analysis on the term frequency - inverse document frequency (TF-IDF) of one-, bi-, and tri-grams on each of the three areas of the venn diagram: studies included by researcher 1 only, studies included by researcher 2 only, and studies included by both researchers. We calculated the TF-IDF from the abstracts of each of the three groupings, and generated word clouds. At a glance, it appears that researcher 1 chose to include documents where terms related to the symbolic dimension such as ``symbolic", ``rules", and general terms such as ``framework", and ``data" appeared more frequently, whereas researcher 2, leaned towards terms along the neural dimension such as ``deep learning", ``neural", and ``networks". Given the academic background of each researcher, we reasoned that the discrepancy was due to bias towards each individual's area of research. In the third grouping, where both researchers agreed on inclusion, a more balanced distribution can be seen with the terms ``symbolic", ``artificial", and ``neural" carrying similar weight. (See Figure \ref{fig: word-cloud}.)
\begin{figure}[htbp]%
\centering
\subfloat[\centering Researcher 1]{{\includegraphics[scale=0.35]{figures/eps/R1_cloud.eps} }}%
\qquad
\subfloat[\centering Intersection]{{\includegraphics[scale=0.35]{figures/eps/R12_cloud.eps} }}%
\qquad
\subfloat[\centering Researcher 2]{{\includegraphics[scale=0.35]{figures/eps/R2_cloud.eps} }}%
\caption{TF-IDF Word Clouds}%
\label{fig: word-cloud}%
\end{figure}
We observed that it was not always clear from the abstract alone whether the sub-symbolic and symbolic methods were integrated in a way that meets the inclusion criteria, which may also have led to disagreement.
To facilitate the next round of review, we kept a shared glossary of symbolic and sub-symbolic concepts as they presented themselves in the literature. We each reviewed all 337 studies again, this time skimming the studies themselves. Any disagreement at this stage was discussed in person with respect to the shared glossary. This process led to the elimination of many studies for a final count of 75 studies marked for the next round of review.
\subsection{Quality Assessment}\label{quality}
The quality of each study was determined through the use of a nine-item questionnaire. Each question was answered with a binary value, and the study's quality was determined by calculating the ratio of positive answers. Studies with a quality score below 50\% were excluded. \\
\textbf{Quality Questions:}
\begin{enumerate} [Q1.]
\item Is there a clear and measurable research question?
\item Is the study put into context of other studies and research, and design decisions justified accordingly? (Number of references in the literature review/ introduction)
\item Is it clearly stated in the study which other algorithms the study’s algorithm(s) have been compared with?
\item Are the performance metrics used in the study explained and justified?
\item Is the analysis of the results relevant to the research question?
\item Does the test evidence support the findings presented?
\item Is the study algorithm sufficiently documented to be reproducible? (independent researchers arriving at the same results using their own data and methods)
\item Is code provided?
\item Are performance metrics provided? (hardware, training time, inference time)
\end{enumerate}
\begin{figure}[htbp]
\centering
\includegraphics[scale=0.45]{figures/eps/study-quality.eps}
\caption{Study quality}
\label{fig:studyQuality}
\end{figure}
More than 88\% of the studies satisfy the requirements listed from Q1 to Q6. However, nearly 82\% of the studies fail to provide source code or details related to the computing environment which makes the system difficult to reproduce. This leads to an overall reduction of the average quality score to 76.3\% - see Figure \ref{fig:studyQuality}.
Finally, each of the 34 studies selected for inclusion was evaluated, classified, and data extraction was performed for each of the features outlined in Table \ref{tab:features}. For acceptable values of individual features see \ref{sec:allowedvalues:appendix}. The lists of neural and symbolic terms referenced in the table constitute the glossary items learned from conducting the selection process.
Figure \ref{fig:publication}(a) shows the breakdown of conference papers vs journal articles, and Figure \ref{fig:publication}(b) shows the number of studies published each year. As evidenced by the graph, interest in NeSy has increased significantly since 2019 for NLP even more dramatically than the much more steady incline of interest in NeSy overall.
\begin{figure}[htbp]%
\centering
\subfloat[\centering Publication type]{{\includegraphics[scale=0.65]{figures/eps/num_pub.eps} }}%
\qquad
\subfloat[\centering Published year]{{\includegraphics[scale=0.55]{figures/eps/pub_year.eps} }}%
\caption{Publications selected for inclusion}%
\label{fig:publication}%
\end{figure}
\begin{table}[htbp]
\footnotesize
\centering
\caption{Data extraction features} \label{tab:features}
\renewcommand{\arraystretch}{1.5}
\begin{tabular}{@{}>{\raggedright\arraybackslash}p{4.5cm}>{\raggedright\arraybackslash}p{7.5cm}}
\toprule
\textbf{Feature} & \textbf{Description} \\ \midrule
Business application & Real world NLP task of the proposed study \\
Technical application & Type of model output - illustrated in Figure \ref{fig:problem_type} \\
Type of learning & Indicates learning method (supervised, unsupervised, etc.) \\
Knowledge representation & One of four categories described in Section \ref{sec:glossary} \\
Type of reasoning & Indicates whether knowledge is represented implicitly (embedded) or explicitly (symbolic) \\
Language structure & Indicates whether linguistic structure is leveraged to facilitate reasoning \\
Relational structure & Indicates whether relational structure is leveraged to facilitate reasoning \\%\midrule
Symbolic terms & List of symbolic techniques used by the models\\
Neural terms & List of neural architectures used by the models\\
Datasets & List of all datasets considered \\
Model description & Describes model architecture schematically \\
Evaluation Metrics & Evaluation metrics reported by the authors \\
Reported score & Model performance reported by the authors \\
Contribution & Novel contribution reported by the authors\\
Key-intake & Short description of the study \\
isNeSy & Indicates whether the authors label their study as Neuro-Symbolic \\
NeSy goals & For each of the goals listed in Section \ref{sec:introduction}, indicates whether the goal is met as reported by the authors \\
Kautz category & List of categories from Kautz's taxonomy \\
NeSy category & List of categories from the proposed nomenclature \\
Study quality & Percentage of positive answers in the quality assessment questionnaire \\
\bottomrule
\end{tabular}
\end{table}
\section{Results, Data analysis, Taxonomies}\label{results}
In this section, we perform quantitative data analysis based on the extracted features in Table \ref{tab:features}. Each study was labeled with terms from the aforementioned glossary, and each term in the glossary was classified as either symbolic, or neural. A bi-product of this process are two taxonomies built bottom-up of concepts relevant to the set of studies under review. The two taxonomies are a reflection of the definition of NeSy provided earlier: ``the combination of deep learning and symbolic reasoning". Thus on the learning side, we have neural architectures (described in Section \ref{sec:nn_taxonomy}), and on the symbolic reasoning side we have knowledge representation (described in Section \ref{sec:kr_taxonomy}). These results are rendered in Table \ref{table:nexytaxonomy}, with the addition of color representing a simple metric, or \textit{promise score}, for each study. The promise score is simply the number of goals reported to have been satisfied by the model(s) in the study.
\begin{table*}[htbp]
\caption{
\centering
\text{Neural \& Symbolic Combinations}
\hspace{\textwidth}
\colorbox{purple_1}{\color{black} 1} \colorbox{purple_2}{\color{white} 2} \colorbox{purple_3}{\color{white} 3} \colorbox{purple_4}{\color{white} 4}
\text{Number of NeSy goals satisfied out of the 5 described in Section \ref{sec:contributions}.}
\hspace{0.5\textwidth}
\textit{Note: some studies use multiple techniques.}
}
\label{table:nexytaxonomy}
\begin{tabular}{@{}llllll}
\toprule
\multicolumn{2}{@{}l}{} & \multicolumn{4}{c}{{Knowledge Representation}} \\
\cmidrule{3-6}
\multicolumn{2}{@{}l}{\multirow{2}{*}{{}}} &
{Frames} &
{Logic} &
{Rules} &
{\begin{tabular}[c]{@{}l@{}}Semantic \\ network\end{tabular}} \\
\cmidrule[.3pt]{1-6}
Linear Models &
{SVM} &
\colorbox{purple_2}{\color{white}{\cite{Diligenti2017143}}} &
&
\begin{tabular}[c]{@{}l@{}}\cite{Silva2011137}\colorbox{purple_2}{\color{white}{\cite{Belkebir201643}}} \\ \colorbox{purple_1}{\cite{DSouza201990}}\end{tabular}
&
\colorbox{purple_1}{\cite{Hussain20181662}}\\
\cmidrule[.3pt]{1-6}
{\multirow{2}{*}{{\begin{tabular}[c]{@{}l@{}}Early \\ Generations\end{tabular}}}} &
{MLP} &
\cite{Cui2021419} &
\colorbox{purple_1}{\cite{Bounabi2021229, Es-Sabery202117943}} &
\begin{tabular}[c]{@{}l@{}} \cite{Mehler2012159} \colorbox{purple_3}{\color{white}\cite{Yao201842}} \\ \colorbox{purple_1}{\cite{Chen2021328}}\end{tabular}
&
\\
\cmidrule[.05pt]{2-6}
& {CNN} &
\cite{Tato2019623} &
\cite{Chaturvedi2019264}\colorbox{purple_1}{\cite{Es-Sabery202117943}} &
\colorbox{purple_1}{\cite{Ayyanar2019}} &
\cite{Ezzat2020111, Gong202030885} \\
\cmidrule[.3pt]{1-6}
\multirow{2}{*}{{\begin{tabular}[m]{@{}l@{}}Graphical\\Models\end{tabular}}} &
{DBN} &
&
\cite{Chaturvedi2019264} &
&
\\
\cmidrule[.05pt]{2-6}
&
{GNN} &
\colorbox{purple_1}{\cite{Lemos2020647}} &
&
\colorbox{purple_3}{\color{white}{\cite{Huo2019159}}} &
\colorbox{purple_1}{\cite{Zhou20212015}}\\
\cmidrule[.3pt]{1-6}
\multirow{2}{*}{{\begin{tabular}[m]{@{}l@{}}Sequence-\\to-Sequence\end{tabular}}} & {RNN} & \begin{tabular}[c]{@{}l@{}}\cite{Brasoveanu2019656, Tato2019623} \\
\colorbox{purple_3}{\color{white}{\cite{Chen20201544}}}
\end{tabular} &
\begin{tabular}[c]{@{}l@{}}\cite{Graziani2019185}\colorbox{purple_2}{\color{white}{\cite{Schon2019293}}} \\ \colorbox{purple_2}{\color{white}{\cite{Chaturvedi2019264}}}\colorbox{purple_1}{\cite{Fazlic20191025}}
\end{tabular}
&
\begin{tabular}[c]{@{}l@{}}\cite{Ramrakhiyani2019102}\colorbox{purple_2}{\color{white}{\cite{Amin2019133}}} \\
\colorbox{purple_2}{\color{white}{\cite{ Sutherland2019}}}\colorbox{purple_4}{\color{white}\cite{Mao2019}} \\
\colorbox{purple_2}{\color{white}{\cite{Demeter20207634}}}
\end{tabular}
&
\begin{tabular}[c]{@{}l@{}}\cite{Ezzat2020111}\colorbox{purple_1}{\cite{Liu2021260}} \\ \cite{Gong202030885}\colorbox{purple_2}{\color{white}{\cite{Manda2020}}}
\end{tabular} \\
\cmidrule[.05pt]{2-6}
& RcNN & & \colorbox{purple_2}{\color{white}\cite{Jiang2020}} & \cite{Huang20191344} & \\
\cmidrule[0.75pt]{1-6}
{\multirow{2}{*}{{\begin{tabular}[c]{@{}l@{}}Neuro-\\ Symbolic\end{tabular}}}}
&
{LTN} &
\colorbox{purple_3}{\color{white}\cite{Bianchi2019161}}&
&
&
\\
\cmidrule[.05pt]{2-6}
&
{RNKN} &
&
\colorbox{purple_2}{\color{white}\cite{Jiang2020}} &
&
\\
\cmidrule[0.75pt]{1-6}
{\multirow{2}{*}{{\begin{tabular}[c]{@{}l@{}}Neuroevolution\end{tabular}}}}
& &
&
&
& \colorbox{purple_4}{\color{white}\cite{Skrlj2021989}}
\\
%
& & & & & \\
\bottomrule
\end{tabular}
\end{table*}
\subsection{Exploratory Data Analysis}
\label{sec:dataAnalysis}
We plot the relationships between the features extracted from the studies, and the goals from section \ref{sec:contributions} in an effort to identify any correlations between them, and ultimately to identify patterns leading to higher promise scores.
\subsubsection{Business and Technical Applications}
A collection of common NLP tasks is shown in Figure \ref{fig:problem_type}. The subset of tasks belonging to Natural Language Understanding (NLU) and Natural Language Generation (NLG) are often regarded as more difficult, and presumed to require reasoning. Given that \textit{reasoning} was one of the keywords used for search, it is not surprising that many studies report reasoning as a characteristic of their model(s). Also unsurprising is the fact that nearly half of the text classification studies (which do not belong to this subset) are not associated with any NeSy goals. The relationship between all tasks, or business applications, and NeSy goals is shown in Figure \ref{fig:buc-to-goals}.
The business application largely determines the type of model output, or technical application, as can be seen in the almost one-to-one mapping in Figure \ref{fig:biz_case-tech_app}. The exception being question answering, which has been tackled as both an inference and a classification problem. Question answering is the most frequently occurring task, and is associated mainly with reduced data, and to a much lesser degree, interpretability. On a philosophical level this seems somewhat disappointing, as one would hope that in receiving an answer, one could expect to understand why such an answer was given.
For completeness, the number of studies representing the technical applications and most frequently occurring business application is given in Figure \ref{fig:busines-tech-apps}, while Figure \ref{fig:buc-techapp-goal} combines business applications, technical applications, and goals.
\begin{figure}[htbp]
\centering
\includegraphics[scale=0.5]{figures/eps/Learning-Reasoning-diagram.eps}
\caption{Common NLP tasks}
\label{fig:problem_type}
\end{figure}
\begin{figure}[htbp]
\centering
\includegraphics[scale=0.45]{figures/eps/biz_case-goals.eps}
\caption{Relationship between Business Applications and Goals}
\label{fig:buc-to-goals}
\end{figure}
\begin{figure}[htbp]
\centering
\includegraphics[scale=0.45]{figures/eps/biz_case-tech_app.eps}
\caption{Relationship between Business Applications and Technical Applications}
\label{fig:biz_case-tech_app}
\end{figure}
\begin{figure}[htbp]
\centering
\includegraphics[scale=0.45]{figures/eps/biz_case-tech_app-goals.eps}
\caption{Relationship between Business Applications, Technical Applications, and Goals}
\label{fig:buc-techapp-goal}
\end{figure}
\begin{figure}[htbp]
\centering
\subfloat[\centering Top Business Applications (NLP tasks)]{{\includegraphics[scale=0.4]{figures/eps/biz_use_case.eps} }}%
\qquad
\subfloat[\centering Technical Applications (model output)]{{\includegraphics[scale=0.4]{figures/eps/techApplication.eps} }}%
\caption{Applications}%
\label{fig:busines-tech-apps}%
\end{figure}
\subsubsection{Type of learning}
Machine learning algorithms are classified as supervised, unsupervised, semi-supervised, curriculum or reinforcement learning, depending on the amount and type of supervision required during training \cite{kang2018machine, bonaccorso2017machine, bengio2009curriculum}. Figure \ref{fig:learning_type-tech_app-goals} demonstrates that the supervised method outweighs all other approaches.
\begin{figure}[htbp]
\centering
\includegraphics[scale=0.45]{figures/eps/learning_type-tech_app-goals.eps} %
\caption{Relationship between Learning Type, Technical Application, and Goals}%
\label{fig:learning_type-tech_app-goals}%
\end{figure}
\subsubsection{Implicit vs Explicit Reasoning}\label{sec:implicitVSexplit}
How reasoning is performed often depends on the underlying representation and what it facilitates. Sometimes the representations are obtained via explicit rules or logic, but are subsequently transformed into non-decomposable embeddings for learning. As such, we can say that any reasoning during the learning process is done implicitly. Studies utilizing Graph Neural Networks (GNNs) \cite{Lemos2020647,Zhou20212015,Huo2019159} would also be considered to be doing reasoning implicitly. The majority of the studies doing implicit reasoning leverage linguistic and/or relational structure to generate those internal representations. These studies meet 21 out of a possible 102 NeSy goals, where 102 = \#goals * \#studies, or 20.6\%. For reasoning to be considered explicit, rules or logic must be applied during or after training. Studies which implement explicit reasoning perform slightly better, meeting 21 out of 72 goals, or 29.2\% and generally require less training data. Additionally, 4 studies implement both implicit and explicit reasoning, at a NeSy promise rate of 40\%. Of particular interest in this grouping is Bianchi et al. \cite{Bianchi2019161}'s implementation of Logic Tensor Networks (LTNs), originally proposed by Serafini and Garcez in \cite{Serafini_Garcez_2016}. ``LTNs can be be used to do after-training reasoning over combinations of axioms which it was not trained on. Since LTNs are based on Neural Networks, they reach similar results while also achieving high explainability due to the fact that they ground first-order logic.\cite{Bianchi2019161}" Also in this grouping, Jiang et al. \cite{Jiang2020} propose a model where embeddings are learned by following the logic expressions encoded in huffman trees to represent deep first-order logic knowledge. Each node of the tree is a logic expression, thus hidden layers are interpretable.
Figure \ref{fig:implicit_explicit-goals} shows the relationship between implicit \& explict reasoning and goals, while the relationship between knowledge representation, type of reasoning, and goals is shown in Figure \ref{fig:implicit_explicit-kr-goals}.
\begin{figure}[htbp]
\centering
\includegraphics[scale=0.65]{figures/eps/implicit_explicit-goals.eps}
\caption{Type of Reasoning and Goals.}
\label{fig:implicit_explicit-goals}
\end{figure}
\begin{figure}[htbp]
\centering
\includegraphics[scale=0.5]{figures/eps/kr-implicit_explicit-goals.eps}
\caption{Knowledge Representation, Type of Reasoning, and Goals.}
\label{fig:implicit_explicit-kr-goals}
\end{figure}
\subsubsection{Linguistic and Relational Structure}
In the previous section we described how linguistic and relational structures can be leveraged to generate internal representations for the purpose of reasoning. Here we plot the relationships between these structures and other extracted features and their interactions - see Figure \ref{fig:leverage}. Perhaps the most telling chart is the mapping between structures and goals, where nearly half the studies leveraging linguistic structure don't meet any of the goals. This runs counter to the intuition that language is a natural fit for NeSy. However, to properly test this intuition, future work on more targeted experiments would need to be performed.
\begin{figure}[htp]
\centering
\includegraphics[width=1\textwidth]{figures/eps/leverage-all.eps}
\caption{Relationships between leveraged structures and extracted features. \textit{ Note: studies which do no leverage either structure are not shown}}
\label{fig:leverage}
\end{figure}
\subsubsection{Datasets and Benchmarks}\label{sec:datasets}
Each study in our survey is based on a unique dataset, and a variety of metrics. Given that there are nearly as many business applications, or tasks, as there are studies, this is not surprising. As such it is not possible to compare the performance of the models reviewed. However, this brings up an interesting question, and that is how one might design a benchmark for NeSy in the first place. A discussion about benchmarks at the IBM Neuro-Symbolic AI Workshop 2022\footnote{\url{https://video.ibm.com/recorded/131288165}} resulted in general agreement that the most important characteristic of a good benchmark for NeSy is in the diversity of tasks tackled. Gary Marcus pointed out that current benchmarks can be solved extensionally, meaning they can be ``gamed". With enough attempts, a model can become very good at a specific task without solving the fundamental reasoning challenge. Ostensibly, this leads to models which are not able to generalize out of distribution. In contrast, to solve a task intensionally is to demonstrate ``understanding" which is transferable to different tasks. This view is controversial with advocates of purely connectionist approaches arguing that ``understanding" is not only ill defined, but also a moving target. So instead of worrying about the semantics of ``understanding", the panelists agreed that to make the benchmarks robust to gaming is to build in enormous variance in the types of tasks they tackle. Taking this a step further, Luis Lamb proposed that instead of designing benchmarks for testing models, we should be designing challenges which encourage people to work on important real world problems.
\subsection{Taxonomies: Neural, Symbolic, \& NeSy}\label{sec:glossary}
We now describe the taxonomies introduced in Section \ref{results} as well as NeSy categories.
\subsubsection{Neural Architectures}\label{sec:nn_taxonomy}
In the main, the extracted neural terms refer to the neural architecture implemented in a given study. We group these into higher level categories such as Linear models, Early generation (which includes CNNs), Graphical models, and Sequence-to-Sequence. We also include here Neuro-Symbolic architectures such as Logic Tensor Networks (LTN) and Recursive Neural Knowledge Networks (RNKN) because they are suitable to optimization via gradient descent.
We include one study \cite{Skrlj2021989} which doesn't strictly meet the neural criteria in the sense that it does not implement gradient descent, but rather Neuroevolution (NE). Neuroevolution involves genetic algorithms for modifying neural network weights, topologies, or ensembles of networks by taking inspiration from biological nervous systems \cite{miikkulainen:encyclopedia10-ne,Lehman_Miikkulainen_2013}. Neuroevolution is often employed in the service of Reinforcement Learning (RL).
Studies which do not mention any specific models are categorised as Multilayer Perceptron (MLP).
\subsubsection{Symbolic Knowledge Representation}\label{sec:kr_taxonomy}
The definition we adopted states that NeSy is
\textit{the combination of deep learning and symbolic reasoning}. Our neural taxonomy described above reflects the \textit{deep learning} component. Since no reasoning can be done without knowledge, we use Knowledge Representation (KR) as a means of categorizing the \textit{symbolic reasoning} component. Common KR categories include: production rules, logical representation, frames, and semantic networks \cite{davis1993knowledge, bench2014knowledge, levesque1986knowledge, travis1990knowledge}.
\begin{itemize}
\item \textit{Production rules} - The production rules are a set of condition-action pairs that represent knowledge \cite{shortliffe2012computer}.
\item \textit{Logical representation} - Logical representation is the study of entailment relations—languages, truth conditions, and rules of inference \cite{levesque1986knowledge}.
\item \textit{Frames} - Frames are objects which hold entities, their properties and methods.
\item \textit{Semantic networks} - Semantic networks (frame networks) are graphs where nodes are frames, and edges are the relationships between nodes which can represent information.
\end{itemize}
Table \ref{table:nexytaxonomy} shows which studies combine which of the above neural (\ref{sec:nn_taxonomy}) and symbolic (\ref{sec:kr_taxonomy}) categories.
\subsubsection{NeSy Categories} \label{nesycategories}
NeSy systems can be categorized according to the nature of the combination of neural and symbolic techniques. At AAAI-20, Henry Kautz presented a 6 level taxonomy of Neuro-Symbolic architectures with a brief example of each \cite{Kautz}. While Kautz has not provided any additional information beyond his talk at AAAI-20, several researchers have formed their own interpretations \cite{Sarker_Zhou_Eberhart_Hitzler_2021,Garcez_Lamb_2020,Lamb_Garcez_Gori_Prates_Avelar_Vardi_2020}. We have categorized all the reviewed studies according to Kautz's taxonomy only to discover that nearly half (N=15) of the studies belong to Level 1, which arguably is not neuro-symbolic, but rather ``standard-operating-procedure" as Kautz himself put it.
Level 1 \textit{symbolic Neuro symbolic} is a special case where symbolic knowledge (such as words) is transformed into continuous vector space and thus encoded in the feature embeddings of an otherwise ``standard" ML model. We opted to include these studies if the derived input features belong to the set of symbolic knowledge representations described in Section \ref{sec:glossary}. One could still argue that this is simply a case of good old fashioned feature engineering, and not particularly special, but we want to explore the idea that deep learning can perform reasoning, albeit implicitly, if provided with a rich knowledge representation in the pre-processing phase. We classify these studies as \textit{Sequential}. Evaluating these studies as a group was particularly challenging as they have very little in common including different datasets, benchmarks and business applications. Half of the studies don't mention reasoning at all, and the ones that do are mainly executing rules on candidate solutions output by the neural models post hoc. In aggregate, only 16 out of a total of 75 (15 studies * 5 goals), or 21.3\%, possible NeSy goals were met.
Level 2 \textit{Symbolic[Neuro]} is what we describe as a \textit{Nested} architecture, where a symbolic reasoning system is the primary system with neural components driving certain internal decisions. AlphaGo is the example provided by Kautz, where the symbolic system is a Monte Carlo Tree Search with neural state estimators nominating next states. We found three studies that fit this architecture \cite{Belkebir201643, Chaturvedi2019264,Chen2021328}.
Level 3 \textit{Neuro; Symbolic} appears to hold more promise in terms of NeSy goals met. Three of the five studies with a Promise score of 3 or more belong to this category (the remaining two belong to levels 4 and 5, which we will discuss in the next subsection). There are six studies in this category, all but one of which leverage relational structure in some manner. A common theme is the use of graph representations and/or GNNs which aligns with recent research directions proposed by Garcez et al. in \cite{Lamb_Garcez_Gori_Prates_Avelar_Vardi_2020}, as well as Zhang et al. in \cite{Zhang_Chen_Zhang_Ke_Ding_2021}. We call Level 3 \textit{Cooperative}, as it is conceptually similar to Reinforcement Learning (RL). Here, a neural network focuses on one task (e.g. object detection) and interacts via input/output with a symbolic reasoner specializing in a complementary task (e.g. query answering). Unstructured input is converted into symbolic representations which can be solved by a symbolic reasoner, which in turn informs the neural component which learns from the errors of the symbolic component. The Neuro-symbolic Concept Learner \cite{Mao2019} is an example of Level 3, meeting 4 out of the 5 NeSy goals, where reasoning is performed explicitly in a ``symbolic and deterministic manner." Its ability to perform well with reduced data is particularly impressive: ``Using only 10\% of the training images, our model is able to achieve comparable results with the baselines trained on the full dataset." Similarly, \cite{Yao201842} report perfect performance on small datasets which they also attribute to the use of explicit and precise reasoning. Both studies display similar limitations, the use of synthetic datasets, and the need for handcrafted logic, a DSL (Domain Specific Language) in the case of \cite{Mao2019}, and Image Schemas in \cite{Yao201842}.
Levels 4 and 5, \textit{Neuro: Symbolic → Neuro} and \textit{Neuro\_Symbolic} respectively, were originally presented by Kautz under one heading. After his presentation, Kautz modified the slide deck separating these two levels into systems where knowledge is compiled into the weights, and where knowledge is compiled into the loss function.
Deep Learning For Mathematics \cite{lample2019deep} is an example of Level 4 where the input and output to the model are mathematical expressions. The model performs symbolic differentiation or integration, for example, given $x^2$ as input, the model outputs $2x$. The model exploits the tree structure of mathematical expressions, which are fed into a sequence-to-sequence architecture. This seems like a particularly fitting paradigm for natural language applications on the basis that structures such as parse trees can be similarly leveraged to output other meaningful structures such as for example: cause and effect relationships, argument schemes, or rhetorical devices, to name a few.
Level 5 comprises Tensor Product Representations (TPRs)\cite{Smolensky_1990}, Logic Tensor Networks (LTNs)\cite{Serafini_dAvila_Garcez_2016}, Neural Tensor Networks (NTN)\cite{Socher_Chen_Manning_Ng_2013} and more broadly is referred to as tensorization, where logic acts as a constraint. In Levels 4 and 5, reasoning can be performed both implicitly and explicitly, in that it is calculated via gradient descent, but can also be performed post hoc. We have grouped studies belonging to these two categories under the moniker of \textit{Compiled} systems, of which there are ten.
An example of a \textit{Compiled} system in our set of studies is proposed in \cite{Jiang2020} which we mentioned in Section \ref{sec:implicitVSexplit}. Here, knowledge is encoded in the form of huffman trees made of triples, and logic expressions, in order to jointly learn embeddings and model weights. The first layer of the tree consists of entities, the second layer consists of relations $(x \rightarrow y)$. Higher layers compute logic rules. The root node is the final embedding representing a document (in this case a single health record). Back propagation is used for optimization with softmax for calculating class probabilities. The model is intended for medical diagnosis decision support, where a much desirable characteristic is interpretability, and this model meets that goal.
Figure \ref{fig:NeSy_categories} illustrates our mapping from Kautz's levels to our proposed nomenclature. There are no studies in Level 6, nor in the ensemble major category, but which we include for completeness. Figure \ref{fig:studies-per-category} shows the number of studies per category, and Figure \ref{fig:cat-to-goals} illustrates the relationship between categories and goals. Table \ref{table:counts} shows the number of studies in each category per goal.
\begin{table*}[htbp]
\caption{Number of studies per goal reported as met, in each category} \label{table:counts}
\centering
\footnotesize
\begin{tabular}{lrrrr}
\toprule
{} & {Compiled} & {Cooperative} & {Nested} & {Sequential} \\
\hline
Reasoning & {\cellcolor[HTML]{8f6795}} \color[HTML]{F1F1F1} 7 & {\cellcolor[HTML]{af94b1}} \color[HTML]{F1F1F1} 4 & {\cellcolor[HTML]{d7cdd3}} \color[HTML]{000000} 2 & {\cellcolor[HTML]{73407d}} \color[HTML]{F1F1F1} 9 \\
OOD & {\cellcolor[HTML]{c0acbf}} \color[HTML]{000000} 3 & {\cellcolor[HTML]{d7cdd3}} \color[HTML]{000000} 2 & {\cellcolor[HTML]{edece6}} \color[HTML]{000000} 1 & {\cellcolor[HTML]{edece6}} \color[HTML]{000000} 1 \\
Interpretability & {\cellcolor[HTML]{c0acbf}} \color[HTML]{000000} 3 & {\cellcolor[HTML]{c0acbf}} \color[HTML]{000000} 3 & {\cellcolor[HTML]{ffffff}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{d7cdd3}} \color[HTML]{000000} 2 \\
Reduced data & {\cellcolor[HTML]{d7cdd3}} \color[HTML]{000000} 2 & {\cellcolor[HTML]{c0acbf}} \color[HTML]{000000} 3 & {\cellcolor[HTML]{edece6}} \color[HTML]{000000} 1 & {\cellcolor[HTML]{c0acbf}} \color[HTML]{000000} 3 \\
Transferability & {\cellcolor[HTML]{d7cdd3}} \color[HTML]{000000} 2 & {\cellcolor[HTML]{edece6}} \color[HTML]{000000} 1 & {\cellcolor[HTML]{edece6}} \color[HTML]{000000} 1 & {\cellcolor[HTML]{ffffff}} \color[HTML]{000000} 0 \\
\hline
{TOTAL} & {17} & {13} & {5} & {15} \\
\bottomrule
\end{tabular}
\end{table*}
\begin{figure}[htbp]
\centering
\includegraphics[scale=0.95]{figures/eps/Nesy-topology.eps}
\caption{NeSy categories. Adapted from Henry Kautz. }
\label{fig:NeSy_categories}
\end{figure}
\begin{figure}[htbp]
\centering
\subfloat[\centering NeSy category]{{\includegraphics[scale=0.4]{figures/eps/nesy_category.eps} }}%
\qquad
\subfloat[\centering Kautz category]{{\includegraphics[scale=0.4]{figures/eps/Kautz_category.eps} }}%
\caption{Number of studies per category}%
\label{fig:studies-per-category}%
\end{figure}
\begin{figure}[htbp]
\centering
\includegraphics[scale=0.5]{figures/eps/nesy_category-kautz_cat-goals.eps}
\caption{NeSy categories to NeSy Goals }
\label{fig:cat-to-goals}
\end{figure}
\begin{comment}
\begin{figure}[htbp]
\centering
\includegraphics[scale=0.65]{figures/eps/kr-neural_sup_terms.eps} %
\caption{Relationship between neural and symbolic approaches}%
\label{fig:neuraltosymbolic}%
\end{figure}
\end{comment}
\section{Discussion}\label{discussion}
All studies report performance either on par or above benchmarks, but we can't compare studies based on performance as nearly every study uses a different dataset and benchmark as discussed Section \ref{sec:datasets}. Our focus is instead on whether the goals of NeSy are being met. Our \textit{Promise Score} metric is not necessarily what the studies' authors were optimizing for or even reporting, especially studies which have not labeled themselves as NeSy per se. So we want to make it very clear that our analysis is not a judgement of the success of any particular study, but rather we seek to understand if the hypotheses about NeSy are materializing (namely, that the combination of symbolic and sub-symbolic techniques will fulfill the goals described in Section \ref{sec:contributions}. And the short answer is we're not there yet, as can be seen in Figure \ref{fig:all_promises}. For a detailed breakdown of each goal and study see Table \ref{table:promises}.
\begin{figure}[h]
\centering
\subfloat[\centering All studies]{{\includegraphics[scale=0.4]{figures/eps/all_promises.eps} }}
\qquad
\subfloat[\centering NeSy studies only]{{\includegraphics[scale=0.4]{figures/eps/nesy_promises.eps} }}
\caption{Proportion of studies which have met one or more of the 5 goals}
\label{fig:all_promises}
\end{figure}
\begin{table*}
\caption{NeSy Promises} \label{table:promises}
\centering
\begin{tabular}{lccccccc}
\toprule
\multirow{2}{*}{Reasoning} & \multicolumn{1}{p{2cm}}{\centering OOD\\Generalization} & \multirow{2}{*}{Interpretability} & \multicolumn{1}{p{1.5cm}}{\centering Reduced\\Data } & \multirow{2}{*}{Transferability} & \multirow{2}{*}{isNeSy} & \multirow{2}{*}{Score} & \multirow{2}{*}{Ref.} \\
\hline
{\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & \cite{Ramrakhiyani2019102} \\
{\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 &
\cite{Brasoveanu2019656} \\
{\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 &
\cite{Graziani2019185} \\
{\cellcolor[HTML]{450256}} \color[HTML]{F1F1F1} 1 & {\cellcolor[HTML]{450256}} \color[HTML]{F1F1F1} 1 & {\cellcolor[HTML]{450256}} \color[HTML]{F1F1F1} 1 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{450256}} \color[HTML]{F1F1F1} 1 & {\cellcolor[HTML]{805388}} \color[HTML]{F1F1F1} 3 & \cite{Bianchi2019161} \\
{\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 &
\cite{Ezzat2020111} \\
{\cellcolor[HTML]{450256}} \color[HTML]{F1F1F1} 1 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{450256}} \color[HTML]{F1F1F1} 1 & {\cellcolor[HTML]{d8ced4}} \color[HTML]{000000} 1 &
\cite{Lemos2020647} \\
{\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 &
\cite{Cui2021419} \\
{\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & \cite{Silva2011137} \\
{\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{450256}} \color[HTML]{F1F1F1} 1 & {\cellcolor[HTML]{450256}} \color[HTML]{F1F1F1} 1 & {\cellcolor[HTML]{450256}} \color[HTML]{F1F1F1} 1 & {\cellcolor[HTML]{450256}} \color[HTML]{F1F1F1} 1 & {\cellcolor[HTML]{450256}} \color[HTML]{F1F1F1} 1 & {\cellcolor[HTML]{450256}} \color[HTML]{F1F1F1} 4 &
\cite{Skrlj2021989} \\
{\cellcolor[HTML]{450256}} \color[HTML]{F1F1F1} 1 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{450256}} \color[HTML]{F1F1F1} 1 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{a687a9}} \color[HTML]{F1F1F1} 2 & \cite{Schon2019293} \\
{\cellcolor[HTML]{450256}} \color[HTML]{F1F1F1} 1 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{450256}} \color[HTML]{F1F1F1} 1 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{450256}} \color[HTML]{F1F1F1} 1 & {\cellcolor[HTML]{a687a9}} \color[HTML]{F1F1F1} 2 & \cite{Diligenti2017143} \\
{\cellcolor[HTML]{450256}} \color[HTML]{F1F1F1} 1 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{450256}} \color[HTML]{F1F1F1} 1 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{a687a9}} \color[HTML]{F1F1F1} 2 & \cite{Jiang2020} \\
{\cellcolor[HTML]{450256}} \color[HTML]{F1F1F1} 1 & {\cellcolor[HTML]{450256}} \color[HTML]{F1F1F1} 1 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{a687a9}} \color[HTML]{F1F1F1} 2 & \cite{Belkebir201643} \\
{\cellcolor[HTML]{450256}} \color[HTML]{F1F1F1} 1 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{d8ced4}} \color[HTML]{000000} 1 & \cite{Hussain20181662} \\
{\cellcolor[HTML]{450256}} \color[HTML]{F1F1F1} 1 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{d8ced4}} \color[HTML]{000000} 1 &
\cite{Liu2021260} \\
{\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 &
\cite{Mehler2012159} \\
{\cellcolor[HTML]{450256}} \color[HTML]{F1F1F1} 1 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{450256}} \color[HTML]{F1F1F1} 1 & {\cellcolor[HTML]{450256}} \color[HTML]{F1F1F1} 1 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{805388}} \color[HTML]{F1F1F1} 3 & \cite{Yao201842} \\
{\cellcolor[HTML]{450256}} \color[HTML]{F1F1F1} 1 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{450256}} \color[HTML]{F1F1F1} 1 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{a687a9}} \color[HTML]{F1F1F1} 2 &
\cite{Chaturvedi2019264} \\
{\cellcolor[HTML]{450256}} \color[HTML]{F1F1F1} 1 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{d8ced4}} \color[HTML]{000000} 1 &
\cite{Bounabi2021229} \\
{\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{450256}} \color[HTML]{F1F1F1} 1 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{450256}} \color[HTML]{F1F1F1} 1 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{a687a9}} \color[HTML]{F1F1F1} 2 & \cite{Honda2019152368} \\
{\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 &
\cite{Gong202030885} \\
{\cellcolor[HTML]{450256}} \color[HTML]{F1F1F1} 1 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{d8ced4}} \color[HTML]{000000} 1 &
\cite{Es-Sabery202117943} \\
{\cellcolor[HTML]{450256}} \color[HTML]{F1F1F1} 1 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{450256}} \color[HTML]{F1F1F1} 1 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{a687a9}} \color[HTML]{F1F1F1} 2 & \cite{Amin2019133} \\
{\cellcolor[HTML]{450256}} \color[HTML]{F1F1F1} 1 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{450256}} \color[HTML]{F1F1F1} 1 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{450256}} \color[HTML]{F1F1F1} 1 & {\cellcolor[HTML]{a687a9}} \color[HTML]{F1F1F1} 2 & \cite{Sutherland2019} \\
{\cellcolor[HTML]{450256}} \color[HTML]{F1F1F1} 1 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{d8ced4}} \color[HTML]{000000} 1 & \cite{Ayyanar2019} \\
{\cellcolor[HTML]{450256}} \color[HTML]{F1F1F1} 1 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{d8ced4}} \color[HTML]{000000} 1 &
\cite{Zhou20212015} \\
{\cellcolor[HTML]{450256}} \color[HTML]{F1F1F1} 1 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{450256}} \color[HTML]{F1F1F1} 1 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{a687a9}} \color[HTML]{F1F1F1} 2 & \cite{Manda2020} \\
{\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{450256}} \color[HTML]{F1F1F1} 1 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{450256}} \color[HTML]{F1F1F1} 1 & {\cellcolor[HTML]{d8ced4}} \color[HTML]{000000} 1 & \cite{Chen2021328} \\
{\cellcolor[HTML]{450256}} \color[HTML]{F1F1F1} 1 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{d8ced4}} \color[HTML]{000000} 1 &
\cite{Fazlic20191025} \\
{\cellcolor[HTML]{450256}} \color[HTML]{F1F1F1} 1 & {\cellcolor[HTML]{450256}} \color[HTML]{F1F1F1} 1 & {\cellcolor[HTML]{450256}} \color[HTML]{F1F1F1} 1 & {\cellcolor[HTML]{450256}} \color[HTML]{F1F1F1} 1 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{450256}} \color[HTML]{F1F1F1} 1 & {\cellcolor[HTML]{450256}} \color[HTML]{F1F1F1} 4 & \cite{Mao2019} \\
{\cellcolor[HTML]{450256}} \color[HTML]{F1F1F1} 1 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{d8ced4}} \color[HTML]{000000} 1 &
\cite{DSouza201990}\\
{\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & \cite{Huang20191344} \\
{\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 &
\cite{Tato2019623} \\
{\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{450256}} \color[HTML]{F1F1F1} 1 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{450256}} \color[HTML]{F1F1F1} 1 & {\cellcolor[HTML]{450256}} \color[HTML]{F1F1F1} 1 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{805388}} \color[HTML]{F1F1F1} 3 &
\cite{Huo2019159} \\
{\cellcolor[HTML]{450256}} \color[HTML]{F1F1F1} 1 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{450256}} \color[HTML]{F1F1F1} 1 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{450256}} \color[HTML]{F1F1F1} 1 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{805388}} \color[HTML]{F1F1F1} 3 &
\cite{Chen20201544} \\
{\cellcolor[HTML]{450256}} \color[HTML]{F1F1F1} 1 & {\cellcolor[HTML]{450256}} \color[HTML]{F1F1F1} 1 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{efeee8}} \color[HTML]{000000} 0 & {\cellcolor[HTML]{450256}} \color[HTML]{F1F1F1} 1 & {\cellcolor[HTML]{a687a9}} \color[HTML]{F1F1F1} 2 & \cite{Demeter20207634} \\
\end{tabular}
\end{table*}
In Section \ref{sec:intro_reasoning} we put forward the hypothesis that reasoning is the means by which the other goals can be achieved. This is not evidenced in the studies we reviewed. Some possible explanations for this finding are: 1) The kind of reasoning required to fulfill the other goals is not the kind being implemented; 2) The approaches are theoretically promising, but the technical solutions need further development. Next we look at each of these possibilities.
\subsection{Reasoning Challenges}\label{discussion:reasoning}
Twenty two out of the thirty four studies included for review mention reasoning as a characteristic of their solution. But there is a lot of variation in how reasoning is described and implemented. Given the overwhelming evidence of the fallibility of human reasoning, to understand language, AI researchers have sought guidance from disciplines such as psychology, cognitive linguistics, neuroscience, and philosophy. The challenge is that there are multiple competing theories of human reasoning and logic both across and within these disciplines. What we've discovered in our review, is a blurring of the lines between various types of logic, human reasoning, and mathematical reasoning, as well as counter-productive assumptions about which theory to adopt. For example, drawing inspiration from ``how people think", accepting that how people think is flawed, and subsequently attempting to build a model with a logical component, which by definition, is rooted in validity seems counter productive to us. This logical component doesn't have to be binary or even monotonic - the sheer fact that it is implemented in code necessitates a `valid' outcome, however that may be defined by the particular logic theory. Additionally, the justification of ``because that's how people think" is inconsistent. Some examples from the studies we reviewed include:
\begin{itemize}
\item \cite{Bianchi2019161} describe human reasoning in terms of a dual process of ``subsymbolic commonsense" (strongly correlated with associative learning), and ``axiomatic" knowledge (predicates and logic formulas) for structured inference.
\item In \cite{Hussain20181662} humans reason by way of analogy, and commonsense knowledge is represented in ConceptNet, a graphical representation of common concepts and their relationships.
\item For \cite{Yao201842} human reasoning can be modeled by Image Schemas (IS). Schemas are made up of logical rules on (Entity1,Relation,Entity2) tuples, such as transitivity, or inversion.
\item \cite{Es-Sabery202117943} explain their choice of fuzzy logic for ``its resemblance to human reasoning and natural language." This is a probabilistic approach which attempts to deal with uncertainty.
\item \cite{Ayyanar2019} propose that human thought constructs can be modelled as cause-effect pairs. Commonsense is often described as the ability to draw causal conclusions from basic knowledge, for example: \textit{If I drop the glass, it will break}.
\item And \cite{Chen20201544} state that ``when people perform explicit reasoning, they can typically describe the way to the conclusion step by step via relational descriptions."
\end{itemize}
But the most plausible hypothesis in our view is that of Schon et al.\cite{Schon2019293}: in order to emulate human reasoning, systems need to be flexible, be able to deal with contradicting evidence, evolving evidence, have access to enormous amounts of background knowledge, and include a combination of different techniques and logics. Most notably, no particular theory of reasoning is given. The argument put forward by Leslie Kaelbling at IBM Neuro-Symbolic AI Workshop 2022 is similarly appealing. She points to the over-reliance on the System1/System2 analogy, and advocates for a much more diverse and dynamic approach. We posit that the type of reasoning employed shouldn't be based solely on how we think people think, but on the attendant objective. This is in line with the ``goal oriented" theory from neuroscience, in that reasoning involves many sub-systems: perception, information retrieval, decision making, planning, controlling, and executing, utilizing working memory, calculation, and pragmatics. But here the irony is not lost on us, and we acknowledge that by resorting to neuroscience for inspiration, we have just committed the same mischief for which we have been decrying our peers! But if we must resort to analogies with human reasoning then it is imperative to be as rigorous as possible. In their recent book, \textit{A FORMAL THEORY OF COMMONSENSE PSYCHOLOGY How People Think People Think} \cite{gordon_hobbs_2017}, Gordon and Hobbs present a ``large-scale logical formalization of commonsense psychology in support of humanlike artificial intelligence" to act as a baseline for researchers building intelligent AI systems. Santos et al \cite{santos_kejriwal_mulvehill_forbush_mcguinness_2021} take this a step in the direction we are advocating, by testing whether there is human annotator agreement when categorizing texts into Gordon and Hobbs' theories. ``Our end-goal is to advocate for better design of commonsense benchmarks [and to] support the development of a formal logic for commonsense reasoning". It is difficult to imagine a single formal logic which would afford all of Gordon and Hobbs' 48 categories of reasoning tasks. Besold at al. \cite{besold2017neural} dedicate several pages to this topic under the heading of Neural-Symbolic Integration in and for Cognitive Science: Building Mental Models. In short, computational modelling of cognitive tasks and especially language processing is still considered a hard challenge.
Others contend that to understand language, one should approach it through the lens of argumentation; that language is a communication tool, and as such should be understood as the interaction of two or more interlocutors engaged in the exercise of persuasion. Or at a minimum there is a messenger and a receiver, each with a responsibility of encoding and decoding the information exchange. While deduction and induction have traditionally been the realm of symbolic and sub-symbolic systems respectively, abduction and non-monotonic logic are the tools of computational argumentation systems. Certain of Gordon and Hobbs' theories lend themselves well to argumentation, such as \textit{Causality}, \textit{Envisioning}, or \textit{Explanation}. None of the studies we reviewed undertook this approach. We believe this is a gap worth exploring in NeSy for NLP.
\subsection{Technical challenges}\label{discussion:technical}
There is strong agreement that a successful NeSy system will be characterized by compositionality. Compositionality allows for the construction of new meaning from learned building blocks thus enabling extrapolation beyond the training distribution. To paraphrase Garcez et al., one should be able to query the trained network using a rich description language at an adequate level of abstraction \cite{Garcez_Lamb_2020}. The challenge is to come up with dense/compact differentialble representations while preserving the ability to decompose, or unbind, the learned representations for downstream reasoning tasks.
Bianchi et al. \cite{Bianchi2019161} propose $LTN_{EE}$, an extention of Logic Tensor Networks (LTNs), in which pre-trained embeddings are fed into the LTN. They show promising results on small datasets which have the important characteristic of being capable of after-training logical inferences. However, $LTN_{EE}$ is limited by heavy computational requirements as the logic becomes more expressive, for example by the use of quantifiers.
Several other studies \cite{Mao2019,Yao201842} introduce logical inference within their solutions, but all require manually designed rules, and are limited by the domain expertise of the designer. Learning rules from data, or structure learning \cite{Embar_Sridhar_Farnadi_Getoor_2018} is an ongoing research topic as pointed out by \cite{von_Rueden_Mayer_Beckh_Georgiev_Giesselbach_Heese_Kirsch_Walczak_Pfrommer_Pick_etal_2021}. In \cite{Chaturvedi2019264} Chaturvedi et al. use fuzzy logic for emotion classification where explicit membership functions are learned. However, as stated by the authors, the classifier becomes very slow with the number of functions.
Other (\textit{compiled}) approaches involve translating logic into differentialble functions, which are either directly included as network nodes as in \cite{Jiang2020}, or added as a constraint to the loss function, as in \cite{Diligenti2017143}. To achieve this, First Order Logic (FOL) can be operationalized using t-norms for example. However, even if we can't agree on how people reason, we can probably agree how they do not, and that's with FOL. To address the many types of reasoning as discussed in the previous section, we need to be able incorporate other types of logic, such as temporal logic, modal logics, epistemic logic, non-monotonic logics and more, but which have no obvious differentiable form.
In summary, formulating logic, or more broadly reasoning, in a differentiable fashion remains challenging.
\section{Limitations \& Future Work}\label{future}
We organized our analysis according to the characteristics extracted from the studies to test whether there were any patterns leading to NeSy goals. Another approach would be to reverse this perspective, and look at each goal separately to understand the characteristics leading to its fulfillment. However, each goal is really an entire field of study in and of itself, and we don't think we could have done justice to any of them by taking this approach. We spent a lot of time looking for signal in a very noisy environment where the studies we reviewed had very little in common. More can be said about what we did not find, than what we did. Another approach might be to narrow the criteria for the type of NLP task, while expanding the technical domain. In particular, a subset of tasks from the NLU domain could be a good starting point, as these tasks are often said to require reasoning.
We tried to be comprehensive in respect to the selected studies which led to the trade-off of less space dedicated to technical details or additional context from the neuro-symbolic discussion. There are a lot of ideas and concepts which we did not cover, such as, and in no particular order, Relational Statistical Learning (RSL), Inductive Logic Programming (ILP), DeepProbLog \cite{manhaeve2018deepproblog}, Connectionist Modal Logics (CML), Extreme Learning Machines (ELM), Genetic Programming, grounding and proposinalization, Case Based Reasoning (CBR), Abstract Meaning Representation (AMR), to name but a few, some of which are covered in detail in other surveys \cite{Garcez_Gori_Lamb_Serafini_Spranger_Tran_2019,besold2017neural}.
Furthermore, we argued that we need differentiable forms of different types of logic, but we did not discuss how they might be implemented. A comprehensive point of reference such as this would be a very valuable contribution to the NeSy community, especially if the implementations were anchored in cognitive science and linguistics as discussed in \ref{discussion:reasoning}.
Finally, the need for common datasets and benchmarks cannot be overstated.
\section{Conclusion}\label{conclusion}
We analyzed recent studies implementing NeSy for NLP in order to test whether the promises of NeSy are materializing in NLP. We attempted to find a pattern in a small and widely variable set of studies, and ultimately we do not believe there are enough results to draw definitive conclusions. Only 34 studies met the criteria for our review, and many of them (in the \textit{Sequential} category) we would not consider truly integrated NeSy systems. The one thing studies which show promise \cite{Bianchi2019161, Skrlj2021989,Yao201842,Mao2019,Huo2019159} in meeting the specified goals have in common is that they all belong to the tightly integrated set of NeSy categories, \textit{Cooperative} and \textit{Compiled} which is good news for NeSy. Two out of these five report lower computational cost than baselines, and performance on par or slightly above baselines, though we must reiterate that performance comparisons are not possible as discussed in Section \ref{sec:datasets}. We've seen that studies implementing both implicit and explicit reasoning meet the most NeSy goals, but suffer from high computational cost. Furthermore, explicit reasoning still often requires hand crafted domain specific rules and logic which makes them difficult to scale or generalize to other applications. Indeed, of the five goals, transferability to new domains was the least frequently satisfied.
Our view is that the lack of consensus around theories of reasoning and appropriate benchmarks is hindering our ability to evaluate progress. Hence we advocate for the development of robust reasoning theories and formal logics as well as the development of challenging benchmarks which not only measure the performance of specific implementations, but have the potential to address real world problems. Systems capable of capturing the nuances of natural language (ie., ones that ``understand" human reasoning) while returning sound conclusions (ie., perform logical reasoning) could help combat some of the most consequential issues of our times such as mis- and dis-information, corporate propaganda such as in climate change denialism, divisive political speech, and other harmful rhetoric in the social discourse.
\begin{acks}
This publication has emanated from research supported in part by a grant from Science Foundation Ireland under Grant number 18/CRT/6183. For the purpose of Open Access, the author has applied a CC BY public copyright licence to any Author Accepted Manuscript version arising from this submission.
\end{acks}
\begin{appendix}
\section{Acronyms}
\label{sec:acronyms:appendix}
\begin{table*}[htbp]
\centering
\caption{Acronyms and Abbreviations}
\label{table:acronyms:appendix}
\renewcommand{\arraystretch}{1}
\begin{tabular}{ll}
\toprule
AI & Artificial Intelligence \\
Attn & Attention Network \\
CL & Curriculum Learning \\
CNN & Convolutional Neural Network \\
DBN & Deep Belief Network \\
DL & Deep Learning \\
GNN &Graph Neural Network \\
LLM &Large Language Models \\
LTN &Logic Tenson Network \\
ML &Machine Learning \\
NE &Neuroevolution \\
NeSy& Neuro-Symbolic AI \\
NLP &Natural Language Processing \\
NLU & Natural Language Understanding \\
NN &Neural Network \\
OOD &Out-of-distribution \\
OOP &Object-oriented programming(paradigm) \\
RcNN& Recursive Neural Network \\
RL &Reinforcement Learning \\
RNKN& Recursive Neural Knowledge Network \\
RNN &Recurrent Neural Network \\
SVM &Support Vector Machine \\
TF-IDF& Term frequency - Inverse document frequency \\
TPR &Tensor Product Representation \\
XAI &Explainable AI \\
\bottomrule
\end{tabular}
\end{table*}
\newpage
\section{Allowed Values}\label{sec:allowedvalues:appendix}
\begin{table*}[htbp]
\centering
\caption{Allowed values}
\label{table:allowedvalues:appendix}
\renewcommand{\arraystretch}{1}
\begin{tabular}{ll}
\toprule
\textbf{Feature} & \textbf{Allowed values} \\
\midrule
Business application &
{\begin{tabular}[m]{@{}l@{}}
Annotation, Constituency Parsing,
Cause-effect Identification, Dialog system,
Emotion detection, \\ Fact verification,
Image captioning, Information extraction, POS tagging, Question answering, Translation, \\
KG Completion / link prediction,
Language comprehension,
Recommendation model, Relation extraction, \\
Semantic Search, Semantic Similarity,
Sentiment analysis, Text classification,
Text games, Decision making, \\ Topic modeling / categorization,
Textual reasoning,
Decision making, Text summarization \end{tabular}}
\\
\\
Technical application &
{\begin{tabular}[]{@{}l@{}} Clustering, Generative, Regression, Inference, Classification, Dimentionality reduction, Recommendation, \\ Information extraction \end{tabular}} \\
\\
Type of learning & {\begin{tabular}[]{@{}l@{}} Supervised, Unsupervised, Semi-supervised,
Reinforcement, Curriculum \end{tabular}} \\
\\
Type of reasoning & Implicit, Explicit, Both
\\
\\
Language structure & Yes, No\\
\\
Relational structure & Yes, No\\
\\
NeSy goals & {\begin{tabular}[]{@{}l@{}}
Reasoning, OOD Generalization, Interpretability, Reduced data, Transferability
\end{tabular}}
\\ \\
Kautz category & {\begin{tabular}[]{@{}l@{}}
1. symbolic Neuro symbolic,
2. Symbolic[Neuro], 3. Neuro; Symbolic, \\
4. Neuro: Symbolic → Neuro,
5. Neuro\_Symbolic, 6. Neuro[Symbolic]
\end{tabular}} \\ \\
NeSy category & Sequential, Cooperative, Compiled, Nested \\
\bottomrule
\end{tabular}
\end{table*}
\end{appendix}
\bibliographystyle{ios1}
|
1,108,101,564,523 | arxiv | \section{Introduction}
\label{sec:introduction}
Dictionary Learning is
an important and widely used tool in Signal
Processing and Computer Vision. Its versatility
is well acknowledged and
it can be applied for denoising or for representation learning
prior to classification \cite{aharon2006rm,Mairal:2009-online-dl}.
The method consists in learning a
set of overcomplete elements (or atoms)
which are
useful for describing examples at hand.
In this context, each example is represented as a potentially sparse linear span of the atoms.
Formally, given a data matrix composed of $n$ elements of dimension $d$, $\mathbf{X} \in \mathbb{R}^{d \times n}$ and each column being an example $\mathbf{x}_i$,
the dictionary learning problem is given by:
\begin{equation}\label{eq:dl}
\min_{\mathbf{D} \in \mathbb{R}^{d \times k} , \mathbf{A} \mathbb{R}^{k \times n}} \frac{1}{2} \sum_{i=1}^n \|\mathbf{x}_i - \mathbf{D} \mathbf{a}_i\|_2^2 + \Omega_D(\mathbf{D}) + \Omega_A(\mathbf{A})
\end{equation}
where $\Omega_D$ and $\Omega_A$ represent some constraints and/or
penalties on the dictionary set $\mathbf{D}$
and the matrix coefficient $\mathbf{A} $,
each column being a linear combination coefficients $\mathbf{a}_i$
so that $ \mathbf{x}_i \approx \mathbf{D} \mathbf{a}_i$.
Typical
regularizers are sparsity-inducing penalty on $\mathbf{A}$, or unit-norm
constraint on each dictionary element although a wide variety of
penalties can be useful \cite{Tibshirani:1996-lasso,
Bach:2012-regularizers, rakotomamonjy2013applying}.
As depicted by the mathematical formulation of the problem, the learned dictionary $\mathbf{D}$ depends on training examples $\{\mathbf{x}_i\}_{i=1}^n$. However, because of the quadratic loss function in the data fitting
term, $\mathbf{D}$ is in addition, very sensitive to outlier examples. Our goal here is to address the robustness of the
approach to outliers. For this purpose, we consider
loss functions that downweight the importance of outliers in
$\mathbf{X}$ making the learned dictionary less sensitive to them.
Typical approaches in the literature, that aim at mitigating influence of outliers, use Frobenius norm or component-wise
$\ell_1$ norm as data-fitting term instead of the squared-Frobenius one
\cite{nie2010efficient,wang2016fast}.
Some works propose loss functions such as the $\ell_q$ function, with
$q \leq 1$ function or the capped function $g(u) = \min(u,\epsilon)$,
for $u > 0$ \cite{wang2013semi,jiang2015robust}.
Due to these non-smooth and non-convex loss function, the resulting dictionary
learning problem is more difficult to solve than the original one
given in Equation (\ref{eq:dl}). As such, authors have developed
algorithms based on a iterative reweighted least-square approaches
tailored to the loss function $\ell_q$ or $\min(u,\epsilon)$
\cite{wang2013semi,jiang2015robust}.
Our contribution in this paper is: (i) to introduce a generic framework for robust dictionary learning by considering as loss function the composition of the Frobenius norm and some concave loss functions (our framework encompasses previously
proposed methods while enlarging the set of applicable loss functions); (ii) to propose a generic majorization-minimization algorithm applicable to concave, smooth or non-smooth loss functions. Furthermore,
because the resulting learning problem is
non-convex, its solution is sensitive to initial conditions, hence we propose a novel heuristic for dictionary initialization that helps in detecting
outliers more efficiently during the learning process.
\section{Concave Robust Dictionary Learning}
\label{sec:rdl}
\subsection{Framework and Algorithm}
\label{ssec:our_robust_dl}
In order to robustify the dictionary learning process against outliers,
we need a learning problem that puts less emphasis on examples
that are not ``correctly'' approximated by the learned dictionary.
Hence, we propose the following generic learning problem:
\begin{equation}
\label{eq:rdl}
\min_{\mathbf{D},\mathbf{A}} \frac{1}{2} \sum_i F(\| \mathbf{x}_i - \mathbf{D} \mathbf{a}_i\|_2^2) + \Omega_D(\mathbf{D}) + \Omega_A(\mathbf{A}).
\end{equation}
where $F(\bullet)$ is a function over $\mathbb{R}_{>0}$.
Note that in
the sequel, we will not focus on the penalty and constraints over the
dictionary elements and coefficients $\mathbf{A}$. Hence, we consider
them as the classical unit-norm constraint over $\mathbf{d}_j$ and the
$\ell_1$ sparsity-inducing penalty over $\{\mathbf{a}_i\}$.
Concavity of $F$ is crucial for robustness as it helps
in down-weighting influence of large $\|\mathbf{x}_i - \mathbf{D}\mathbf{a}_i\|_2$.
For instance, if we
set $F(\bullet)= \sqrt{\bullet}$, the above problem is similar to the convex robust
dictionary learning proposed by Wang et al. \cite{wang2016fast}. In order to provide
better robustness, our goal is to introduce a generic
form of $F$ that leads to a concave loss with respect to $\|\mathbf{x}_i - \mathbf{D}\mathbf{a}_i\|_2$. instead of a linear, yet concave
one as in \cite{wang2016fast}.
In this work, we emphasize
robustness by considering $F$ as the composition of two concave
functions $F(\bullet) = g(\bullet) \circ \sqrt{\bullet} $, with $g$ a non-decreasing
concave function over $\mathbb{R}_{>0}$, such as those used for sparsity-inducing
penalties. Typically, $g(\bullet)$ can be the $q-$power, $q \leq 1$ function
$u^q$, the log function $\log(\epsilon + {u})$, the SCAD function
\cite{Fan:2001-scad}, or the capped-$\ell_1$ function
$\min(u, \epsilon)$, or the MCP function \cite{zhang2010nearly}. A key property
on $F$ is that concavity is preserved by the composition
of some specific concave functions as proved by the following lemma
which proof is omitted for space reasons.
\begin{lemma} Let $g$ be a non-decreasing concave function on
$\mathbb{R}_{>0}$ and $h$ be a concave function on a domain $\Omega$ to $\mathbb{R}_{>0}$,
then $ g \circ h$ is concave. Furthermore, if $g$ is an increasing function
then $ g \circ h$ is strictly concave.
\end{lemma}
In our framework, $h$ is the square-root function with
$\Omega = \mathbb{R}_{>0}$.
In addition, functions $g$, such as those given above, are either a concave or strictly concave functions and are all non-decreasing, hence $F = g \circ h$ is concave. Owing to concavity, for any $u_0$ and $u$ in
$\mathbb{R}_{>0}$,
$$
F(u) \leq F(u_0) + F'(u_0) (u - u_0)
$$
where $F'(u_0)$ is an element of the superdifferential of $F$ at $u_0$. As $F$ is concave, the superdifferential is always non-empty and if $F$ is smooth at $u_0$, then $F'(u_0)$ is simply the gradient of $F$ at $u_0$.
However, since $F$ is a composition of functions, in a non-smooth case, computing superdifferential is difficult unless the inner function is a linear function \cite{rockafellar2015convex}. Next lemma provides a key result showing that a supergradient of $g \circ \sqrt{\bullet}$
can be simply computed using chain rule because $\sqrt{\bullet}$ is a
bijective function on $\mathbb{R}_{>0}$ to $\mathbb{R}_{>0}$ and $g$ is non-decreasing.
\begin{lemma} Let $g$ a non-decreasing concave function on
$\mathbb{R}_{>0}$ and $h$ a bijective differentiable concave function on a domain $\mathbb{R}_{>0}$ to $\mathbb{R}_{>0}$,
then if $g_1$ is a supergradient of
$g$ at $z$ then $g_1 \cdot h'(s) $ is a supergradient of $g \circ h $ at a point $s$ so
that $z = h(s)$.
\end{lemma}
\begin{proof} As $g_1 \in \partial g(z)$, we have $\forall y, g(y) \leq
g(z) + g_1 \cdot ( y -z)$. Owing to bijectivity of $h$, define $t$ and $s$ so that $y
= h(t)$ and $z=h(s)$. In addition, concavity of $h$ gives $h(t) - h(s) \leq
h'(s) (t-s)$ and because $g$ is non-decreasing, $g_1 \geq 0$.
Combining everything, we have $g_1 \cdot ( y -z) =
g_1 \cdot (h(t) - h(s)) \leq g_1 h'(s) (t-s) $. Thus
$\forall t, g(h(t)) \leq g(h(s)) + g_1 h'(s) (t-s)$ which concludes
the proof since $g_1$ is a supergradient of $g$ at $h(s)$.
\end{proof}
\begin{algorithm}[t]
\footnotesize \caption{The proposed Robust DL method}
\label{alg:proposed_robust_dl}
\begin{algorithmic}[1] \REQUIRE Data matrix $\mathbf{X} \in \mathbb{R}^{d \times n}$, dictionary size $k$, $\lambda$, $\epsilon$, $M$.
\IF{($k > d$) and (\textit{use undercomplete initialization})}
\STATE Initialize $\mathbf{D}$ and $s$ with Algorithm~\ref{alg:undercomplete_initialization}
\ELSE
\STATE random initialization of $\mathbf{D}$, $\mathbf{A}$
\STATE $s_j = 1$ for $j=1, \ldots, n$
\ENDIF
\FOR{$i=1$ to $M$}
\REPEAT
\STATE Update $\mathbf{D}$ by solving Equation \ref{eq:dicoupdate}
\FOR{$j=1$ to $n$}
\STATE $\mathbf{a}_j \gets \frac{1}{2} ||\mathbf{x}_j - \mathbf{D} \mathbf{a}||_2^2 + \frac{\lambda}{s_j} ||\mathbf{a}||_1$
\ENDFOR
\UNTIL{convergence}
\FOR{$j=1$ to $n$}
\STATE update $s_j$ according to Equation \ref{eq:upds}
\ENDFOR
\ENDFOR
\ENSURE $\mathbf{D}$, $s$
\end{algorithmic} \end{algorithm}
Based on the above majorizing linear function property of concave
functions and because in our case $F'(u_0)$ can easily be
computed, we consider a majorization-minimization approach for
solving Problem (\ref{eq:rdl}).
Our iterative algorithm consists, at iteration $\kappa$, in
approximating the concave loss function $F$ at the current solution
$\mathbf{D}_{\kappa}$ and $\mathbf{A}_{\kappa}$ and then solve the resulting
approximate problem for $\mathbf{D}$ and $\mathbf{A}$. This yields in solving:
\begin{equation}
\label{eq:iteraterdl}
\min_{\mathbf{D},\mathbf{A}} \frac{1}{2} \sum_i s_i \| \mathbf{x}_i - \mathbf{D} \mathbf{a}_i\|_2^2 + \Omega_D(\mathbf{D}) +
\Omega_A(\mathbf{A})
\end{equation}
where $s_i = [g \circ \sqrt{\bullet}~]^\prime$ at
$\mathbf{D}_{\kappa}$ and $\mathbf{a}_{\kappa,i}$. Since, we have
$$
[g \circ \sqrt{\bullet}~]^\prime(u_0) = \frac{1}{2 \sqrt{u_0}}g^{\prime}(
\sqrt{u_0})
$$
weights $s_i$ can be defined as
\begin{equation}\label{eq:upds}
s_i = \frac{g'(\| \mathbf{x}_i -\mathbf{D}_{\kappa} \mathbf{a}_{\kappa,i}\|_2)}{2 \| \mathbf{x}_i
-\mathbf{D}_{\kappa} \mathbf{a}_{\kappa,i}\|_2}.
\end{equation}
This definition of $s_i$ can be nicely interpreted. Indeed, if
$g$ is so that
$\frac{g'(u)}{u}$ becomes small as $u$ increases, examples with
large residual values $\| \mathbf{x}_i -\mathbf{D}_{\kappa} \mathbf{a}_{\kappa,i}\|_2$ have
less importance in the learning problem (\ref{eq:iteraterdl}) because
their corresponding values $s_i$ are small.
Note how the composition $g \circ \sqrt{\bullet}$ allows us to write
the data fitting term with respect to the squared residual norm so that at
each iteration, the problem (\ref{eq:iteraterdl}) to solve
is simply a weighted
smooth dictionary learning problem, convex in each of its parameters,
that can be addressed using off-the-shelf tools.
As
such, it can be solved alternatively for $\mathbf{D}$ with fixed $\mathbf{A}$ and then
for $\mathbf{A}$ with fixed $\mathbf{D}$. For fixed $\mathbf{A}$, the optimization problem is
thus:
\begin{equation}\label{eq:dicoupdate}
\min_\mathbf{D} \frac{1}{2} \sum_i \| \tilde \mathbf{x}_i - \mathbf{D} \tilde \mathbf{a}_i\|_2^2 + \Omega_D(\mathbf{D})
\end{equation}
where $\tilde \mathbf{x}_i = \sqrt{s_i} \mathbf{x}_i$ and
$\tilde \mathbf{a}_i = \sqrt{s_i} \mathbf{a}_i$. This problem can be solved using a
proximal gradient algorithm or block-coordinate descent algorithm as
given in Mairal et al. \cite{Mairal:2009-online-dl}. For fixed $\mathbf{D}$,
the problem is separable in $\mathbf{a}_i$ and each sub-problem is equivalent
to a Lasso problem with regularization $\frac{\lambda}{s_i}$.
The above algorithm is generic in the sense that it is applicable to any continuous concave and non-decreasing function $g$, even non-smooth ones. This is in
constrast with algorithms proposed in \cite{wang2013semi,jiang2015robust} which have been tailored to
some specific functions $g$. In addition, the convergence in objective value of the algorithm is guaranteed for any of these $g$ functions, by the fact that the objective value in Equation \ref{eq:rdl} decreases at each iteration while it is obviously lower bounded.
\subsection{Heuristic for initialization}
The problem we are solving is a non-convex problem and its solution is
thus very sensitive to initialization. The presence of outliers in the
data matrix $\mathbf{X}$ magnifies this sensitivity and increases the need for
a proper initialization of $s_i$ in our iterative algorithm based
on Equation (\ref{eq:iteraterdl}). If we were able to identify
outliers before learning, then we would assign $s_i=0$ to these
samples so that they become irrelevant for the dictionary learning
problem. However, detecting outliers in a set of samples is a
difficult learning problem by itself \cite{Chandola:2007-outlier}.
Our initialization heuristic is based on the intuition that if most
examples belong to a linear subspace of $\mathbb{R}^d$ while outliers leave
outside this subspace, then these outliers can be better identified by
using an undercomplete dictionary learning than an overcomplete
one. Indeed, if the sparsity penalty is weak enough, then an
overcomplete dictionary can approximate well any example leading to a large value of $s_i$ even the
for outliers.
Hence, if the number of dictionary to learn is larger than the
dimension of the problem, we propose to initialize $\mathbf{D}$ and $s$ by
learning mini-batches of size $b<d$ of dictionary atoms using one iteration of
Alg.~\ref{alg:proposed_robust_dl} initialized with
$s_i = 1, \forall i \in [1, \ldots, n]$, a random dictionary and random
weigths $\mathbf{A}$. If
we have only a small proportions of outliers, we make the hypothesis
that the learning problem will focus on dictionary atoms that span a
subspace that better approximates non-outlier samples. Then,
as each set of learned mini-batch dictionary atoms leads to a different
error $\| \mathbf{x}_i - \mathbf{D} \mathbf{a}_i\|_2$ and thus to different $s_i$ as defined
in Equation $3$, we estimate $s_i$ as the average $s_i$ over the
number of mini-batch and we expect $s_i$ to be small if $i$-th example
is an outlier.
This initialization strategy is presented in Alg.~\ref{alg:undercomplete_initialization}.
\begin{algorithm}[t]
\footnotesize \caption{Undercomplete initialization}
\label{alg:undercomplete_initialization}
\begin{algorithmic}[1] \REQUIRE Data matrix $\mathbf X$, dictionary $\mathbf{D} \in \mathbb{R}^{d \times k}$, with $d<k$, number of atoms in each batch $b < k $, parameters $\lambda$ and $\epsilon$.
\STATE $N \gets \ceil[\big]{\frac{k}{b}}$ \COMMENT{number of batches}
\STATE $s = 0$
\STATE Initialize $\mathbf{D} = [\mathbf{d}_1, \ldots, \mathbf{d}_k]$ as a zero matrix
\FOR{$i=0$ to $(N-1)$}
\STATE I = indices related to $i$-th batch
\STATE $\hat \mathbf{D}, \hat s \gets \textrm{Algorithm~\ref{alg:proposed_robust_dl}}\big( \mathbf{X}, |I|, \lambda, \epsilon, 1 \big)$
\STATE $\mathbf{D}_I \gets \hat \mathbf{D}$ \COMMENT{assign learned dictionary to the appropriate indices}
\STATE $s \gets s + \hat s$ \COMMENT{accumulate weights}
\ENDFOR
\STATE $s \gets \frac{s}{N}$ \COMMENT{compute average}
\ENSURE $\mathbf{D}$, $s$
\end{algorithmic} \end{algorithm}
\section{Experiments}
\label{sec:experiments}
\subsection{Experiments on synthetic data}
\label{ssec:experiments_synthetic_data}
\begin{figure}[t]
\centering
~\hfill
\includegraphics[width=0.22\textwidth]{toy_2d_1.pdf}~\hfill
\includegraphics[width=0.22\textwidth]{toy_2d_2.pdf}\hfill~\\
~\hfill
\includegraphics[width=0.22\textwidth]{toy_2d_3.pdf}~\hfill~
\includegraphics[width=0.22\textwidth]{toy_2d_4.pdf}\hfill~\\
\caption{Synthetic 2D data drawn from two Gaussian distributions. The outliers are represented as the red triangles.
(top-left) original data with outliers (top-right) Clustering with K-SVD
(bottom-left) Clustering with proposed method with $g(u)= u$. (bottom-right) Clustering with proposed method using the log function.}
\label{fig:2d_experiments}
\end{figure}
\begin{figure*}[h]
\hspace{-4mm}
\begin{minipage}{0.33\linewidth}
\centering
\includegraphics[width=1.0\textwidth]{toy_multidimensional_1.pdf}\\
(a) Different dictionary sizes, with 1000 samples and 10\% are outliers.
\end{minipage}
\begin{minipage}{0.33\linewidth}
\centering
\includegraphics[width=1.0\textwidth]{toy_multidimensional_2.pdf}\\
(b) Different number of samples, where 10\% are outliers.
\end{minipage}
\begin{minipage}{0.33\linewidth}
\centering
\includegraphics[width=1.0\textwidth]{toy_multidimensional_3.pdf}\\
(c) Different outlier ratios (\%), with 1000 samples and 64 atoms.
\end{minipage}
\caption{Performance of the proposed method with multidimensional data.}
\label{fig:multidimensional_experiments}
\end{figure*}
We use synthetic generated datasets with outliers to demonstrate that our method is robust against outliers. Figure~\ref{fig:2d_experiments} presents two clusters generated from two Gaussian distributions, each containing 250 points along with 50 outliers represented as the red triangles, far away from the clusters. Figure~\ref{fig:2d_experiments} also shows the clustering results using K-SVD \cite{Aharon:2006-ksvd} as well as the proposed method when $g(u) = u$ and the $log(\epsilon + u)$ functions, respectively. Then, we compare how many of the original outliers are among the 50 highest reconstruction values. The proposed method using the log function proved to be the most robust against outliers, with 47 from the 50 true outliers detected. It is followed by the variant with the identity function, which identified 27 outliers, and finally by K-SVD, which was naturally not able to identify any of the original outliers. This example also shows that concavity of $g$ helps in better identifying outliers.
To further evaluate our proposal, we performed experiments with higher dimensional data (fixed at 32 dimensions). To generate the data, we use the approach described by Lu et al. \cite{Lu:2013-online-dl} to create synthetic data based on a dictionary and sparse coefficients. The metric adopted to compare the results is the AUC Curve (AUROC) of outlier scores $\{s_i\}$ after running Alg.~\ref{alg:proposed_robust_dl}: outliers should have scores $1/s_i$ larger than non-outliers, and each point is the average of 5 runs using newly generated data. In Fig.~\ref{fig:multidimensional_experiments}a one can observe that the behavior for both lines is the same until the number of atoms reach 32, since $k \leq d$ and the condition in the first line of Alg.~\ref{alg:undercomplete_initialization} is not met. The performance of the undercomplete initialization method also deteriorates for dictionary sizes a little bit greater than $d$, but as far as $k$ starts to increase it becomes evident that this method outperforms the default initialization. Figure~\ref{fig:multidimensional_experiments}b shows that our method stays very stable independent of the number of samples, given a constant outlier ratio, regardless of the initialization method. Finally, Fig.~\ref{fig:multidimensional_experiments}c shows the behavior of both initialization strategies in scenarios where the outlier proportion changes. It can be noticed that the AUROC values decrease slowly as long as the number of outliers in the samples increase. This is natural since when proportion of outliers is large, outliers can hardly be considered outliers anymore.
\subsection{Human attribute classification}
\label{ssec:experiments_real_data}
In order to prove that our robust dictionary learning method is really beneficial to real data contexts, we also evaluate its performance on the MORPH-II dataset \cite{Ricanek:2006-morph}, one of the largest labeled face databases available for gender and ethnicity classification, containing more than 40,000 images. Before the training and classification take place, the images are preprocessed, which consists of face detection, align the image based on the eye centers, as well as cropping and resizing. Finally, they are converted to grayscale and SIFT \cite{Lowe:1999-sift} descriptors are computed from a dense grid.
The experiments are run with the proposed method using both the default and the undercomplete initialization approaches using the log function, and then compared with state-of-the-art methods such as K-SVD and LC-KSVD \cite{Jiang:2011-lcksvd}. The classifier uses a Bag of Visual Words (BoVW) approach \cite{Csurka:2004-bag} by replacing the original K-Means algorithm with each of those methods, and then generating a image signature (histogram of frequencies) using the computed clusters, which are later fed to a SVM. This SVM uses a RBF (Radial Basis Function) kernel with tuned $\gamma$ and $C$ parameters. The number of atoms is set to 200 for all experiments.
\begin{table}[h]
\centering
\scalebox{0.70}{
\begin{tabular}{|c||c|c|}
\hline
\multirow{2}{*}{\textit{Method}} & \textit{Ethnicity} & \textit{Gender} \\
& \textit{accuracy} & \textit{accuracy} \\ \hline \hline
\textbf{Our RDL (default)} & \textbf{96.28} & \textbf{84.76} \\
\hline
\textbf{Our RDL (undercomplete)} & \textbf{96.90} & \textbf{85.79} \\
\hline
K-SVD & 96.23 & 81.88 \\
\hline
LC-KSVD1 & 96.24 & 83.91 \\
\hline
LC-KSVD2 & 95.69 & 84.69 \\
\hline
\end{tabular}
} \caption{Average classification accuracies (\%) for ethnicity and gender classification on the MORPH-II dataset.}
\label{tab:results_ethnicity_gender}
\end{table}
Each experiment is the average of 3 runs, each one using 300 selected images per class for training, and the remaining images for classification. The total number of images per class is as follows: 32,874 Africans plus 7,942 Caucasians for ethnicity classification, and 6,799 Females plus 34,017 Males for gender classification. Table~\ref{tab:results_ethnicity_gender} shows the overall accuracies. These experiments clearly demonstrate that the quality of the dictionaries computed by the proposed robust dictionary learning method is indeed superior
even to methods that uses labels for dictionary learning \cite{Jiang:2011-lcksvd}.
\section{Conclusions}
\label{sec:conclusions}
In this work, we proposed a generic dictionary learning framework which takes advantage of a composition of two concave functions to generate robust dictionaries with very little outlier interference. We also came up with a heuristic initialization which can further increase the identification of outliers, through the use of undercomplete dictionaries. Experiments on synthetic and real world datasets show that the proposed methods outperform some of the state-of-the-art methods such as K-SVD and LC-KSVD, since our approaches are able to achieve higher quality dictionaries which better generalize data.
\vfill\pagebreak
\bibliographystyle{IEEEbib}
|
1,108,101,564,524 | arxiv | \section{Introduction}
The seminal work of Onsager of 1944 on the exact solution of the planar Ising
Model \cite{Ons} has been a source of a considerable part of the subsequent
developments in the field of exactly solvable models in Statistical Mechanics
and Field Theory. One branch of these developments which originated with
Star-Triangle relation \cite{Ons,Wannier} and led to Yang-Baxter equation and,
later, to theory of Quantum Groups was particularly vigorous.
Yet the Star-Triangle relation did not play any essential role in original
Onsager's solution of 2D Ising Model. Indeed it is only mentioned in
\cite{Ons}. The crucial part was played by a certain infinite-dimensional Lie
algebra which is now called Onsager's Algebra and by what one may call
associated representation theory.
This algebra can be described by introducing the basis
$ \{ A_m , G_n \} \; , \; m = 0,\pm1,\pm2,\dots \; ; \; \\ n=1,2,\dots \;$.
Commutation relations in this basis are:
\begin{eqnarray}
[A_l,A_m] & = & 4G_{l-m} \;\; \; l \geq m \\
\mbox{} [G_l,A_m] & = & 2A_{m+l} - 2A_{m-l} \\
\mbox{} [ G_l,G_m ] & = & 0
\end{eqnarray}
Even though this algebra was at the centre of the original Onsager's solution
of 2D Ising Model, it received substantially less attention in subsequent years
then the Star-Triangle Relation. In the context of the Ising Model the
algebraic method of Onsager was superseded by simpler and more powerful
methods that rely on the equivalence of the 2D Ising Model to a
free-fermionic theory \cite{Kauf} and the dimer problem \cite{Kast}. This
caused Onsager's Algebra to remain in a shadow for quite a long time. The
situation changed in the 1980-s when several important advances related to
Onsager's Algebra took place. Some of these we briefly review below in order
to set up a background for the subsequent discussion.
Dolan and Grady \cite{Dolan} considered Hamiltonians $H$ of the form (in a
different notation) :
\begin{equation} H \; = \; A_0 + k A_1 \end{equation} where $k$ is a constant and $ A_0 , A_1 $ are
operators. They have shown that the following pair of conditions imposed upon
the operators $ A_0$ and $A_1$ - Dolan Grady relations - :
\begin{eqnarray}
\mbox{} [A_0, [A_0, [A_0 , A_1]]] & = & 16 [A_0 , A_1] \\
\mbox{} [A_1, [A_1, [A_1 , A_0]]] & = & 16 [A_1 , A_0]
\end{eqnarray}
are sufficient to guarantee that $ H $ belongs to an infinite family of
mutually commuting operators - integrals of motion for $H$. To be more precise,
only one of the relations (5,6) was considered in \cite{Dolan} - the second one
was produced as a consequence of certain duality condition imposed on the
operators $A_0 , A_1 $. The Dolan Grady relations in the form (5,6) - without
the assumption of duality - were first discussed in \cite{Davies1,Davies}.
Based on the work of Dolan and Grady, Perk \cite{Perk-Ons} and Davies
\cite{Davies1} established moreover, that the Lie Algebra generated by two
``letters'' $ A_0 , A_1$ subject to the relations (5,6) is precisely Onsager's
Algebra as it is defined in (1-3). The elements $ A_0 , A_1 $ of the basis $
\{ A_m , G_n \} $ are identified with $ A_0 , A_1 $ in (5,6) and all the rest
are expressed as commutators of $ A_0 $ and $ A_1$.
The representation of the generators $ A_0 $ and $ A_1 $ which was considered
by Onsager is:
\begin{eqnarray} A_0 \; = \; \sum_{i=1}^L \sigma_i^x &, & A_1 \; = \; \sum_{i=1}^L
\sigma_i^z \sigma_{i+1}^z \;. \end{eqnarray}
and after substitution in (4) gives the Hamiltonian of Transverse Ising Chain.
This Hamiltonian is defined on a periodic chain of length $L$ ; and $
\sigma_i^{x,z} $ are Pauli matrices representing local spins on a site with a
number $i$. It is well known that Jordan-Wigner transformation brings this
Hamiltonian to a free-fermionic form \cite{Kauf}.
An important question - whether there are other representations of the
relations (5,6) leading to nearest-neighbor spin Hamiltonians that are not
free - was answered affirmatively by von Gehlen and Rittenberg in 1985
\cite{Rittenberg} who found a family of such representations. For every integer
$ M \geq 2 $ they define:
\begin{eqnarray} A_0 \; = \; \frac{4}{M} \sum_{i=1}^L \sum_{n=1}^{M-1}\frac{X_i^n}{1-
\omega^{-n}} & , &
A_1 \; = \; \frac{4}{M} \sum_{i=1}^L \sum_{n=1}^{M-1}\frac{Z_i^n
Z_{i+1}^{M-n}}{1- \omega^{-n}}
\end{eqnarray} where $ X_i , Z_i $ are local $Z_M$ spin operators satisfying: $
[X_i,X_j]=[Z_i,Z_j]=0\;,\; Z_iX_j = \omega^{\delta_{ij}}X_jZ_i \;,\; Z_i^M =
X_i^M = I $; and $ \omega = {\rm exp}(2\pi i/M) $. When $M=2$ this
representation coincides with Onsager's original representation (7) for Ising
Model. For arbitrary integer $ M $ the spin-chain Hamiltonians of the form (1)
with $A_0 , A_1 $ given by (8) were later shown to be certain - so called {\it
Superintegrable } \cite{Albertini,Baxter2,BBP} - specializations of the
spin-chains generated by 2D Chiral Potts Model [10-14]. von Gehlen and
Rittenberg also observed numerically certain Ising-like structure of some
eigenvalues in the spectrum of these Hamiltonians. In \cite{Albertini,CPSpec}
the $ M=3 $ case was solved analytically and this structure was shown to hold
for all eigenvalues. This Ising-like form of eigenvalues was later rigorously
established by Davies \cite{Davies} to be a consequence of Onsagers's Algebra;
Davies proved that all eigenvalues of a
Hamiltonian of the form (4) defined in a finite-dimensional Hilbert space fall
into multiplets parameterized by: two real numbers $ \alpha , \beta $ ,
positive integer $ n $ , $n$ real numbers $\theta_i$ and $n$ nonnegative
integers $s_i$; eigenvalues which belong to such a multiplet being given by
the formula:
\begin{equation}
\alpha + k \beta + \sum_{j=1}^n 4m_j \sqrt{1 + k^2 + 2k cos\theta_j}
\;\;\;,\;\; m_j = -s_j, -s_j + 1, \dots , s_j .
\end{equation}
Classification of the finite-dimensional representations of Onsager's Algebra
which leads to this form of the eigenvalues was carried out by Davies
\cite{Davies} and, subsequently, by Roan \cite{Roan}.
Onsager's Algebra by itself does not define the parameters $ \{
\alpha,\beta,\theta_i,s_i \} $ entering into the eigenvalue formula (9). To
find these for a given representation of the generators $ A_0 , A_1 $ in some
Hilbert Space ${\cal H}$, one needs to find the decomposition of ${\cal H}$
into irreducible subrepresentations of Onsager's Algebra \cite{Davies}.
For the Superintegrable Chiral Potts Hamiltonians given by (8) the complete
spectrum of eigenvalues has been found in \cite{CPSpec} for the 3-state case $
M=3 $ with the aid of a certain cubic relation satisfied by the Transfer-Matrix
of Chiral Potts Model. For general $ M $ the eigenvalues of the ground-state
sector were obtained in \cite{Baxter2} by use of an inversion identity for the
same Transfer-Matrix , in \cite{Tarasov} this result was extended to other
eigenvalues. Further results related to the superintegrable Chiral Potts Model
can be found in \cite{SCP}.
Now we come to the motivation and the subject of the present paper. Since the
work of Onsager it was known that there exists an intimate relationship
between Onsager's Algebra and $ sl(2) $. This relationship was clarified by
Roan \cite{Roan} who built on the earlier work of Davies
\cite{Davies1,Davies}. Namely, Roan has shown that the Onsager's Algebra given
by (5,6) ( or equivalently by (1-3) ) is isomorphic to the fixed-point
subalgebra of $ sl(2) $-Loop Algebra $ {\cal L}(sl(2)) $ ( or ,
alternatively, of its central extension $ A_1^{(1)} $ \cite{Kac}) with respect
to the action of a certain involution. Indeed one can easily guess that there
should be a connection between Onsager's Algebra and Kac-Moody Algebra $
A_1^{(1)} $ looking at the Dolan Grady relations whose left-hand sides coincide
with the left-hand sides of the Serre relations for $ A_1^{(1)} $ \cite{Kac}.
This obviously raises the question: whether one can find a generalization of
Onsager's Algebra related in some way to $ {\cal L}(sl(N)) $ for $ N\geq 3$ and
to other Loop or Kac-Moody Algebras ? In particular: can one find some
generalizations of the Dolan Grady conditions (5,6) leading to integrability in
the sense of existence of infinite series of integrals of motion in involution?
The aim of this paper is to propose such a generalization related to
$sl(N)$-Loop Algebra for $N \geq 3$. We reserve the discussion of the
generalizations related to Loop Algebras over other classical Lie Algebras for
a future publication.
Let us summarize the results. We consider Hamiltonians of the form: \begin{equation}
H \; = \; k_1 {\rm {\bf e}}_1 + k_2 {\rm {\bf e}}_2 + \dots + k_N {\rm {\bf e}}_N \;\;\; , \;\;\;\; N \geq 3
\end{equation} where $k_i$ are some arbitrary numerical constants and $ {\rm {\bf e}}_i $ are
linear operators.
We find that if the operators $ {\rm {\bf e}}_i $ satisfy certain commutation relations (
generalized Dolan Grady relations given by the formulas (11-12) below ) then
the Hamiltonian $ H $ is a member of an infinite family of mutually commuting
integrals of motion (see formula (48)). Each of these integrals is linear in
the coupling constants $ k_i $ and is explicitly expressed in terms of $
{\rm {\bf e}}_i$. This property follows from the fact that the Lie Algebra generated by $
{\rm {\bf e}}_i $ subject to the relations (11,12) ( we call it $ sl(N) $ Onsager's
Algebra and denote it by \mbox{${\cal A}_N $}\ ) is isomorphic to the fixed-point subalgebra of
$ sl(N) $ Loop Algebra under the action of a certain involution.
The important problem is to find examples of interesting Hamiltonians of the
form (10).
The only example of a Hamiltonian satisfying the generalized Dolan Grady
relations that we have been able to find so far is the Hamiltonian of an
inhomogeneous periodic Ising chain of length $L$. To write down such a
Hamiltonian we first define a sequence of operators:
\[
g_{2k-1}\; = \; \frac{1}{2}\sigma^x_k \; \; , \;\;g_{2k}\; = \;
\frac{1}{2}\sigma^z_k \sigma^z_{k+1} \;\;, \;\;\; k = 1,2,\dots,L
\]
Then we take the operators $ {\rm {\bf e}}_i $ entering into the Hamiltonian to be given
by: $ {\rm {\bf e}}_i \:=\: g_i $ for $ i=1,2,\dots,2L $. These operators satisfy (11,12)
for the $ sl(2L) $ Onsager's Algebra. After substitution of these operators
into (10) we get the completely inhomogeneous Transverse Ising Chain with
inhomogeneities $ k_i $. If we take $ 2L = mN $ for some integer $m$, then
defining the operators $ {\rm {\bf e}}_i $ by: $ {\rm {\bf e}}_i \: = \: \sum_{s=0}^{m-1} g_{Ns+i} $
for $ i = 1,2,\dots, N $ we get a representation of $ sl(N) $ Onsager's
Algebra. The Hamiltonian corresponding to this representation is the
Hamiltonian of the Transverse Ising Chain with periodic inhomogeneities. These
Hamiltonians have been known for a long time to be free-fermionic
\cite{Book,Perk}. So the outstanding unsolved problem is to find
representations of $ sl(N) $ Onsager Algebra that give rise to models that
cannot be mapped onto free-fermions , or to prove that such representations do
not exist.
Now we outline the contents of the paper. In sec. 2 we give the definition of
the main object of our study: $sl(N)$ Onsager's Algebra which we denote by \mbox{${\cal A}_N $}\
. We specify the Algebra by giving $ N $ generators and a finite number of
defining relations that generalize the Dolan Grady conditions.
In sec. 3 we discuss an involution of the $sl(N)$ Loop Algebra and its
fixed-point subalgebra \mbox{$\tilde{\cal A}_N $}\ , as we shall see later this subalgebra is
isomorphic to \mbox{${\cal A}_N $}\ . In order to prepare the proof of this isomorphism in
subsequent sections we introduce two convenient bases in \mbox{$\tilde{\cal A}_N $}\ .
In sec. 4 we study the structure of the algebra \mbox{${\cal A}_N $}\ and prove a Proposition
which gives a set of elements that span \mbox{${\cal A}_N $}\ as a linear space. We achieve this
result by computing commutators of the generators with the elements of this
set.
In sec. 5 the isomorphism of the Lie Algebras \mbox{${\cal A}_N $}\ and \mbox{$\tilde{\cal A}_N $}\ is established.
Due to the results established in sec.3 this gives us a basis of the Algebra
\mbox{${\cal A}_N $}\ and commutation relations among the elements of this basis.
In sec. 6 the knowledge of the basis and commutation relations enables us to
find an infinite family of mutually commuting elements in the Lie Algebra \mbox{${\cal A}_N $}\.
The Hamiltonian (10) is one of these integrals of motion.
\section{Definition of \mbox{${\cal A}_N $}\ - $sl(N)$ analog of Onsager's Algebra }
In this section we introduce the Lie algebra \mbox{${\cal A}_N $}\ which is a generalization of
the original Onsager's algebra to the $sl(N)$, $ N \geq 3 $ case. The relation
of this algebra to $sl(N)$ or, more precisely, to the loop algebra of Laurent
polynomials with values in $sl(N)$ will be explained later in sec. 5.
In order to define the algebra \mbox{${\cal A}_N $}\ it is convenient to consider the Dynkin
graph of the type $A_{N-1}^{(1)}$:
\begin{center}
\begin{picture}(205,60)(5,-20)
\put(0,0){\line(4,1){100}}
\put(100,25){\line(4,-1){100}}
\put(0,0){\line(1,0){160}}
\put(180,0){\line(1,0){20}}
\put(100,25){\circle*{3.0}}
\multiput(0,0)(50,0){4}{\circle*{3.0}}
\put(200,0){\circle*{3.0}}
\multiput(165,0)(5,0){3}{\circle*{0.5}}
\put(100,29){$N$}
\put(0,-15){$1$} \put(50,-15){2} \put(100,-15){3} \put(150,-15){4}
\put(200,-15){$N-1$}
\end{picture}
\end{center}
We label the vertices of this graph by $i$ ranging from 1 to $N$ , $ i+N \mbox{$ \stackrel{\rm def}{=} $}\ i
$. To a vertex with a label $i$ we attach a letter $ {\rm {\bf e}}_i$ . Then we define \mbox{${\cal A}_N $}\
to be a complex Lie algebra generated by the letters $ {\rm {\bf e}}_i\;,\; i=1,2,\dots,N
$ subject to the following defining relations:
\begin{eqnarray}
[ {\rm {\bf e}}_i,[ {\rm {\bf e}}_i, {\rm {\bf e}}_j]] & = & {\rm {\bf e}}_j \; \; \; \;\mbox{ if $i$ and $j$ are adjacent
vertices} \\
\mbox{} [ {\rm {\bf e}}_i, {\rm {\bf e}}_j] & = & 0 \;\;\;\;\mbox{ if $i$ and $j$ are not adjacent }
\end{eqnarray}
As a linear space the algebra \mbox{${\cal A}_N $}\ is a linear span of all multiple
commutators of $ {\rm {\bf e}}_i$ between themselves taken modulo the relations (11,12).
Let us now introduce some notations. We shall often be working with multiple
commutators nested to the right, that is expressions of the form: $ [ a_{k_1},
[ a_{k_2}, [a_{k_3}, \dots [ a_{k_{m-1}}, a_{k_m}] \dots ]]] $ where $ a_{k_i}
$ are some elements in the Lie Algebra \mbox{${\cal A}_N $}\ . For such a commutator we shall
use the notation:
\[ [ a_{k_1}, [ a_{k_2}, [a_{k_3}, \dots [ a_{k_{m-1}}, a_{k_m}] \dots ]]]\;
\mbox{$ \stackrel{\rm def}{=} $}\ \;[ a_{k_1}, a_{k_2}, a_{k_3}, \dots , a_{k_{m-1}}, a_{k_m} ] \]
In multiple commutators in which the generators $ {\rm {\bf e}}_i $ appear, we shall
replace the symbol $ {\rm {\bf e}}_i $ by $ i $, for example:
\[
[ {\rm {\bf e}}_5,[ {\rm {\bf e}}_1,[ {\rm {\bf e}}_2, {\rm {\bf e}}_5]]]\; \mbox{$ \stackrel{\rm def}{=} $} \; [5,1,2,5] \; \mbox{$ \stackrel{\rm def}{=} $} \; [5,[1,2,5]] \; \mbox{$ \stackrel{\rm def}{=} $}
\; {\rm etc.} \]
Now we define elements of \mbox{${\cal A}_N $}\ which will play an important role in subsequent
discussion. These elements are denoted by $S_k(r)$ and defined as follows:
\begin{eqnarray}
S_k(r) & \mbox{$ \stackrel{\rm def}{=} $}\ & [k,k+1,k+2,\dots,k+r-1] \;\;\; k\;=\;1,2,\dots,N \;;\;
r\;=\;1,2,3,\dots
\end{eqnarray}
We shall call such an element {\it a string } of length $r$. Strings are
cyclic in their sub-indices: $ S_{k+N}(r) \; = \; S_k(r)$. Strings of length 1
are the generators of \mbox{${\cal A}_N $}\ : $ S_k(1) \; = \; {\rm {\bf e}}_k $. String of length 0 is by
convention equal to zero. As we will see in sec. 4, the whole algebra \mbox{${\cal A}_N $}\ is
spanned by strings as a linear space. Strings are linearly independent except
that the sum of all closed strings (i.e. strings whose length is divisible by
$N$ ) of a given length vanishes. This will be established in sec. 5.
The algebra \mbox{${\cal A}_N $}\ has an automorphism of order $N$ which we will use later. This
automorphism which we denote $C$ is defined by cyclic permutation of the
generators:
\begin{eqnarray}
C:\; {\rm {\bf e}}_i & \rightarrow & {\rm {\bf e}}_{i+1}
\end{eqnarray}
The automorphism $C$ is quite useful in computations of commutators, since the
action of this automorphism on strings is again cyclic permutation:
\begin{eqnarray}
C: \; S_i(r) & \rightarrow & S_{i+1}(r)
\end{eqnarray}
The obvious questions one can ask about the Lie Algebra \mbox{${\cal A}_N $}\ are: what is a
basis of this algebra and what are commutation relations among elements of this
basis. These questions are answered in the sections 4 an 5. There we shall
establish the isomorphism between \mbox{${\cal A}_N $}\ and the Lie Algebra \mbox{$\tilde{\cal A}_N $}\ which we
define and describe in the next section.
\section{The Loop Algebra ${\cal L}(sl(N))$, its involution and the fixed-point
subalgebra \mbox{$\tilde{\cal A}_N $}\ .}
As we shall see later, the Lie algebra \mbox{${\cal A}_N $}\ introduced in the previous
section is closely related to ${\cal L}(sl(N))$ - the $sl(N)$ loop algebra. In
this section we describe a certain involution $\omega$ of ${\cal L}(sl(N))$ and
the Lie subalgebra \mbox{$\tilde{\cal A}_N $}\ of ${\cal L}(sl(N))$ on which the action of $\omega$
is reduced to the identity (``fixed-point subalgebra `` of $\omega$). In
subsequent sections we shall prove that the algebras \mbox{${\cal A}_N $}\ and \mbox{$\tilde{\cal A}_N $}\ are
isomorphic and shall describe this isomorphism.
The $sl(N)$ loop algebra: ${\cal L}(sl(N)) \; = \; {\rm {\bf
C}}[t,t^{-1}]\otimes sl(N)$ has the basis $ \{ E_{ij}^{(n)} \;, \; H_k^{(n)} \}
\; \; 1\leq i \neq j \leq N \: ; \:1\leq k \leq N-1 \;\;;\;\; n = 0,\pm 1 , \pm
2, \dots \;$. In the $ N\times N $-matrix realization of $sl(N)$ the elements
of this basis have the explicit form:
\begin{eqnarray*} E_{ij}^{(n)} \;=\; t^n E_{ij}\:, & H_k^{(n)}\;=\; t^n H_k \;=\; t^n (
E_{kk}-E_{k+1k+1} ) &
\end{eqnarray*} where $ E_{kl} $ is $ N\times N $ matrix whose all entries are zero except
entry $(kl)$ which is equal to 1. The loop algebra has a linear involutive
automorphism $ \omega $ , $ \omega^2 = id $, given by:
\begin{eqnarray}
\omega : E_{ij}^{(n)} & \rightarrow & (-1)^{i+j+1+nN}E_{ji}^{(-n)} \\
\omega : H_i^{(n)} & \rightarrow & (-1)^{1+nN}H_i^{(-n)}
\end{eqnarray}
This involutive automorphism $\omega$ is a product of two involutions:
$\omega_1$ and $\omega_2$.
The first of these is an involution of the algebra of Laurent polynomials:
\[
\omega_1 : \; t^n \; \rightarrow \; (-1)^{nN}t^{-n}
\]
The second one is an involution of $sl(N)$:
\begin{eqnarray*}
\omega_2 : E_{ij} & \rightarrow & (-1)^{i+j+1}E_{ji} \\
\omega_2 : H_i & \rightarrow & -H_i
\end{eqnarray*}
It is easy to convince oneself that the subspace $ \mbox{$\tilde{\cal A}_N $}\ \; \in \; {\cal
L}(sl(N)) $ on which the involution $\omega $ acts as identity operator, is a
Lie subalgebra.
We can easily find the basis of this fixed-point subalgebra \mbox{$\tilde{\cal A}_N $}\ ; it is
formed by vectors $ \{ A_{ij}^{(n)} , G_i^{(n)}\}$ :
\begin{eqnarray}
A_{ij}^{(n)} & = & E_{ij}^{(n)} + (-1)^{i+j+1+nN}E_{ji}^{(-n)} \; \; \; \;
1\leq i<j \leq N \;,\; n=0,\pm 1,\pm 2,\dots \\
G_i^{(n)} & = & H_i^{(n)} + (-1)^{nN+1}H_i^{(-n)} \; \; \;\; 1\leq i \leq N-1
\;,\, n=1,2,\dots
\end{eqnarray}
The commutation relations of \mbox{$\tilde{\cal A}_N $}\ in the basis $ \{ A_{ij}^{(n)}\;,\;
G_i^{(n)} \} $ follow immediately from the commutation relations of ${\cal
L}(sl(N))$:
\begin{eqnarray}
[A_{ij}^{(m)},A_{kl}^{(n)}] & = & \delta_{jk}A_{il}^{(m+n)} -
\delta_{il}A_{kj}^{(m+n)} + \nonumber \\
& & \delta_{ik}(-1)^{i+j+1+mN}\theta (j<l)A_{jl}^{(n-m)} +
\delta_{ik}(-1)^{i+l+nN}\theta (l<j)A_{lj}^{(m-n)} + \nonumber \\
& & \delta_{jl}(-1)^{k+l+1+nN}\theta (i<k)A_{ik}^{(m-n)} +
\delta_{jl}(-1)^{i+l+mN}\theta (k<i)A_{ki}^{(n-m)} + \nonumber \\
& & \delta_{ik} \delta{jl} (-1)^{i+j+1+nN}\sum_{s=i}^{j-1} G_s^{(m-n)} \; \; \;
\; m\geq n \\ \mbox{}
[G_i^{(m)},A_{kl}^{(n)}] & = & ( \delta_{ik}- \delta_{ki+1} - \delta_{li} +
\delta_{li+1})(A_{kl}^{(m+n)}-(-1)^{mN}A_{kl}^{(n-m)}) \\ \mbox{}
[G_i^{(m)}, G_j^{(n)}] & = & 0
\end{eqnarray}
here $\delta_{ik}$ is the Kronecker symbol and $\theta (x)$ is the following
function: $ \theta (x) \; = \; 1(0) \; $ if $x$ is true(false).
We will also need another basis in the algebra \mbox{$\tilde{\cal A}_N $}. We shall denote the
elements of this new basis by
symbols $\mbox{$ \tilde{S} $}_i(r)$ where $ r=1,2,\dots \; $; and $ 1 \leq i \leq N $ if $r$ is
not divisible by $N$ ; and $ 1 \leq i \leq N-1 $ if $ r=Nm $ for some positive
integer $m$. The explicit expressions for the elements $\mbox{$ \tilde{S} $}_i(r)$ are as
follows:
\begin{eqnarray}
\mbox{$ \tilde{S} $}_i(k) & = & \left \{ \begin{array}{ll}
E_{ii+k} + (-1)^{k+1}E_{i+ki} & \mbox{ if $ i+k \leq N
$} \\
t E_{ii+k-N} + (-1)^{k+1}t^{-1} E_{i+k-Ni} & \mbox{ if
$ i+k \geq N+1$}
\end{array} \right.
\end{eqnarray} for $ 1\leq k\leq N-1 $;
\begin{eqnarray}
\mbox{$ \tilde{S} $}_i(Nm+k) & = & \left \{ \begin{array}{r}
(t-(-1)^Nt^{-1})(t+(-1)^Nt^{-1})^{m-1}(E_{ii+k} + (-1)^kE_{i+ki}) \\ \mbox{
if $ i+k \leq N $} \\
(t-(-1)^Nt^{-1})(t+(-1)^Nt^{-1})^{m-1}(t E_{ii+k-N} + (-1)^{k}t^{-1}
E_{i+k-Ni}) \\ \mbox{ if $ i+k \geq N+1$}
\end{array} \right.
\end{eqnarray} for $ 1\leq k\leq N-1 \; , \; m \geq 1 $;
\begin{eqnarray}
\mbox{$ \tilde{S} $}_i(Nm) & = & (t-(-1)^Nt^{-1})(t+(-1)^Nt^{-1})^{m-1}(E_{ii}-E_{i+1i+1})
\end{eqnarray} for $ m \geq 1 \; , \; 1 \leq i \leq N $.
Note that the elements defined by the last formula are linearly dependent: $
\mbox{$ \tilde{S} $}_1(Nm)+\mbox{$ \tilde{S} $}_2(Nm)+\dots+\mbox{$ \tilde{S} $}_N(Nm)\;=\;0 $.
We can express the elements of the basis $\{ A_{ij}^{(n)}\, , G_i^{(n)} \}$ in
terms of the basis $\{ \mbox{$ \tilde{S} $}_i(r) \}$ with the aid of the recursion relations:
\begin{eqnarray*}
A_{ij}^{(0)} & = & \mbox{$ \tilde{S} $}_i(j-i) \\
A_{ij}^{(-1)} & = & (-1)^{1+N+i+j} \mbox{$ \tilde{S} $}_j(N+j-i) \\
A_{ij}^{(m)} & = & (-1)^N A_{ij}^{(m-2)} + \Phi_i(Nm,j-i) -
(-1)^{i+j}\Phi_j(N(m-1),N+i-j) \\
A_{ij}^{(-m-1)} & = & (-1)^N A_{ij}^{(-m+1)} +(-1)^{mN}\Phi_i(N(m-1),j-i) \\
& & - (-1)^{(m+1)N+i+j}\Phi_j(Nm,N+i-j) \\
G_i^{(m)} & = & \Phi_i(Nm,0)
\end{eqnarray*} for $ m \geq 1 $. The vectors $ \Phi_i(Nm,s) $ are given in terms of $
\mbox{$ \tilde{S} $}_i(r) $ by the formula:
\begin{eqnarray*}
\Phi_i(Nm,s) & = & \sum_{r=1}^m c(m,r) \mbox{$ \tilde{S} $}_i(Nr+s) \; \; \; \; 0\leq s\leq N-1
\end{eqnarray*} where coefficients $ c(m,r) $ are defined by the recursion relation:
\begin{eqnarray*}
c(m+1,r) & = & (1-\delta_{1r})c(m,r-1) - (1-\delta_{mr})(1-\delta_{m+1r})(-1)^N
c(m-1,r) \\
c(1,1) & = & 1
\end{eqnarray*}
The basis of vectors $ \{ \mbox{$ \tilde{S} $}_i(r) \} $ which we described in this section will
be used in the proof of an isomorphism of the algebras \mbox{${\cal A}_N $}\ and \mbox{$\tilde{\cal A}_N $}\ .
\section{Structure of the algebra \mbox{${\cal A}_N $}\ }
In this section we study the structure of the algebra \mbox{${\cal A}_N $}\ in some detail. The
main result that we establish is formulated as the following proposition:
\begin{prop}
The Lie algebra \mbox{${\cal A}_N $}\ is spanned by the set of strings $\{ S_i(r) \}\; 1\leq
i\leq N\;,r=1,2,\dots $ as a linear space.
\end{prop}
Notice that it is not true that all strings are linearly independent.
In order to prove this proposition we shall first compute commutators of the
generators of \mbox{${\cal A}_N $}\ with all strings, that is the commutators of the form:
\begin{eqnarray*}
[ {\rm {\bf e}}_i,S_j(r)] & \mbox{$ \stackrel{\rm def}{=} $}\ & [i,S_j(r)]
\end{eqnarray*} where $ 1 \leq i,j \leq N $ and $ r=1,2,\dots $.
Due to the existence of the cyclic automorphism $C$ it is sufficient to compute
$ [1,S_j(r)] $ for all $ j $ and $r$ ; the rest of the commutators $ [i,S_j(r)]
$ is then immediately obtained by application of $C$.
The result which we get computing $ [1,S_j(r)] $ is summarized in the Lemma .
The distinctive feature of the strings which emerges from the result of the
Lemma is that a commutator of a generator with a string is again a string.
\pro Lemma
The following relations hold in \mbox{${\cal A}_N $}\ for $m \geq 0$:
\[ [1,S_k(Nm+r)] = \]
{\rm 1. If} $ 1\leq k \leq N $ , $ 1\leq r \leq N-1 $ {\rm and} $ k+r \leq N
$,
\begin{eqnarray*}
{\rm a}_m).\hspace{0.5cm} & -2S_1(Nm) & {\rm when}\ \; k=1, r=1 \\
{\rm b}_m).\hspace{0.5cm} & S_2(Nm + r-1) & {\rm when}\ \; k=1, r \geq 2 \\
{\rm c}_m).\hspace{0.5cm} & S_1(Nm + r+1) & {\rm when}\ \; k=2 \\
{\rm d}_m).\hspace{0.5cm} & 0 & {\rm when}\ \; k \geq 3 \\
\end{eqnarray*}
{\rm 2. If} $ 1\leq k \leq N $ , $ 1\leq r \leq N-1 $ {\rm and} $ k+r \geq
N+1 $,
\begin{eqnarray*}
{\rm e}_m).\hspace{0.5cm} & -S_k(Nm + r+1) & {\rm when}\ \; k+r = N+1, k \neq 2
\\
{\rm f}_m).\hspace{0.5cm} & S_1(Nm + r+1) & {\rm when}\ \; k+r = N+1, k = 2 \\
{\rm g}_m).\hspace{0.5cm} & -S_k(Nm + r-1) & {\rm when}\ \; k+r=N+2 \\
{\rm h}_m).\hspace{0.5cm} & 0 & {\rm when}\ \; k+r \geq N+3 \\
\end{eqnarray*}
{\rm 3. If} $ 1\leq k \leq N $ {\rm and} $ r = N $,
\begin{eqnarray*}
{\rm i}_m).\hspace{0.5cm} & -2S_1(Nm + N+1) & {\rm when}\ \; k = 1 \\
{\rm j}_m).\hspace{0.5cm} & S_1(Nm + N+1) & {\rm when}\ \; k = 2 \\
{\rm k}_m).\hspace{0.5cm} & S_1(Nm + N+1) & {\rm when}\ \; k = N \\
{\rm l}_m).\hspace{0.5cm} & 0 & {\rm when}\ \; 3 \leq k \leq N-1 \\
\end{eqnarray*}
{\it Proof:}
We shall prove the Lemma using induction in $m$. First we establish the base of
the induction by proving relations $ {\rm a}_0 $ through $ {\rm l}_0 $, and
then show, that relations $ {\rm a}_m $ through $ {\rm l}_m $ entail $ {\rm
a}_{m+1}$ through $ {\rm l}_{m+1} $. At each elementary step we employ either
the defining relations of \mbox{${\cal A}_N $}\ or the Jacobi identity or skew-symmetry of the
commutator. The proof given below is valid for $N \geq 4$. Proof for $N=3$
differs in some details and is omitted here.
1.) Proof of the induction base. We compute the commutator $ [1,S_k(r)] $ when
$ 1 \leq k \leq N $ and $ 1 \leq r \leq N $.
$\underline{ Case \; {\rm a_0} }$ : $ k=1 , r=1 $.
$[1,S_1(1)] \; \mbox{$ \stackrel{\rm def}{=} $}\ \; [1,1] \; = \; 0 \; \mbox{$ \stackrel{\rm def}{=} $}\ \; -2 S_1(0) $.
\vspace{0.4cm}
$\underline{ Case \; {\rm d_0} }$ : $ k \geq 3 , k+r \leq N , r \leq N-1$.
$x \; \mbox{$ \stackrel{\rm def}{=} $}\ \; [1, S_k(r)]\; \mbox{$ \stackrel{\rm def}{=} $}\ \;[1,k,k+1,\dots,k+r-1].$ Since $ k \geq 3 $
and $ k+r \leq N-1 $, 1 commutes with all $k,k+1,\dots, k+r-1$. Hence $ x=0
$.
\vspace{0.4cm}
$\underline{ Case \; {\rm b_0} }$ : $ k = 1 ,\; r \geq 2,\; k+r \leq N $.
$x \; \mbox{$ \stackrel{\rm def}{=} $}\ \; [1, S_k(r)]\; \mbox{$ \stackrel{\rm def}{=} $}\ \;[1,1,2,\dots,r].$ Since $ r \leq N-1 $, $
x\; =\; [ 1,[1,2],3,\dots,r] \;=\; [2,3,\dots,r] \; \mbox{$ \stackrel{\rm def}{=} $}\ \; S_2(r-1) $ .
\vspace{0.4cm}
$\underline{ Case \; {\rm h_0} }$ : $ k+r \geq N+3,\; r \leq N-1 $.
$x\; \mbox{$ \stackrel{\rm def}{=} $}\ \; [1,S_k(r)]\; \mbox{$ \stackrel{\rm def}{=} $} \; [1,k,k+1,\dots,N,1,2,\dots,k+r-1-N] .$ Since $
2 \leq k+r-1-N \leq N-2,$
\[
x\; = \; [k,k+1,\dots,N-1,[1,N],1,2,\dots,k+r-1-N] +
[k,k+1,\dots,N,1,1,2,\dots,k+r-1-N].
\]
Denote the first (second) summand in the right-hand side of the above formula
by $a$ ($b$). Then we find that:
\begin{eqnarray*}
a & = & -[k,k+1,\dots,N,2,\dots,k+r-1-N] + \\ & &
[k,k+1,\dots,N-1,1,[1,N],2,\dots,k+r-1-N] \\
& = & \; -[k,k+1,\dots,N,2,\dots,k+r-1-N] + \\ & &
[k,k+1,\dots,N-1,1,1,N,2,\dots,k+r-1-N] \\
& & - [k,k+1,\dots,N-1,1,N,1,2,\dots,k+r-1-N].
\end{eqnarray*}
The first two summands above and the commutator $b$ vanish since $N$ commutes
with all elements standing on its right in these expressions. Hence $ a \; = \;
-x $, because the leftmost element 1 in the third summand above commutes with
all elements standing on its left. Therefore $ x = a + b = -x $ and $ x = 0 $.
\vspace{0.4cm}
$\underline{ Case \; {\rm e_0} }$ : $ k+r = N+1,\; k \geq 3 ,\; r \leq N-1 $.
\begin{eqnarray*}
[1,S_k(r)] & \mbox{$ \stackrel{\rm def}{=} $}\ & [1,k,k+1,\dots,N-1,N]\;= \\
-[k,k+1,\dots,N-1,N,1] & \mbox{$ \stackrel{\rm def}{=} $}\ & -S_k(r+1) .
\end{eqnarray*}
$\underline{ Case \; {\rm g_0} }$ : $ k+r = N+2,\; r \leq N-1 $.
\begin{eqnarray*}
[1,S_k(r)] & \mbox{$ \stackrel{\rm def}{=} $}\ & [1,k,k+1,\dots,N-1,N,1]\;= \\
-[k,k+1,\dots,N-1,1,1,N] & = & -[k,k+1,\dots,N-1,N]\; \mbox{$ \stackrel{\rm def}{=} $} \; -S_k(r-1) .
\end{eqnarray*}
$\underline{ Case \; {\rm i_0} }$ : $ k=1\;, r = N $.
\begin{eqnarray*}
x & \mbox{$ \stackrel{\rm def}{=} $}\ & [1,S_1(N)]\; \mbox{$ \stackrel{\rm def}{=} $} \; [1,1,2,\dots,N] \\
& = & [1,[1,2],3,\dots,N] - [1,2,\dots,N,1].
\end{eqnarray*}
Denote the first(second) summand in the right-hand side of the above formula by
$a$ ($-b$). Then :
\begin{eqnarray*}
a &= &[2,3,\dots,N] + [[1,2],1,3,\dots,N] \\
&=& [2,3,\dots,N] - [1,2,3,\dots,N,1]+ [2,1,3,\dots,N,1] \\
&= & - [1,2,3,\dots,N,1] \; = \; -b
\end{eqnarray*}
Hence: $ x\; =\; a - b\; = \; -2[1,2,3,\dots,N,1]\; \mbox{$ \stackrel{\rm def}{=} $}\ \; -2S_1(N+1) $.
\vspace{0.4cm}
$\underline{ Case \; {\rm k_0} }$ : $ k=N\;, r=N $.
$x\; \mbox{$ \stackrel{\rm def}{=} $}\ \; [1,S_N(N)]\; \mbox{$ \stackrel{\rm def}{=} $}\ [1,N,1,S_2(N-2)]\; = \; a + b $. Where $ a\;=\;
[[1,N],1,S_2(N-2)] $ and $ b\;=\; [N,1,S_1(N-1)]$. Using the defining relations
of \mbox{${\cal A}_N $}\ and the Jacobi identity we find that:
\[
a\;=\;-[N,S_2(N-2)]+[1,1,N,S_2(N-2)]-[1,N,1,S_2(N-2)].
\]
Using the already proven relation $ {\rm e_0} $ and the automorphism $C$ one
finds that $ [N,S_2(N-2)]\;=\;-S_2(N-1)$. Hence $ a\;=\; S_2(N-1) - [1,S_1(N)]
- [1,S_N(N)]. $ The relation ${\rm i_0}$ then leads to $ a\;= S_2(N-1) +
2S_1(N+1) - x $. The already proven relation ${\rm b_0}$ gives: $b\;=\;
[N,S_2(N-2)]\;=\;-S_2(N-1)$. Hence we obtain $ x\; =\; a+b \;=\; 2S_1(N+1)-x $,
and $ x\;=\; S_1(N+1)$.
\vspace{0.4cm}
$\underline{ Case \; {\rm l_0} }$ : $ 3 \leq k \leq N-1 \;, r=N $.
$[1,S_k(N)]\;=\;[1,k,S_{k+1}(N-1)]\;=\;[k,1,S_{k+1}(N-1)]\;=\;0$ , since the
internal commutator in the last formula vanishes due to the already proven
relation $ {\rm h_0} $.
The rest of the cases, i.e. $ {\rm c_0}, {\rm f_0} $ and ${\rm j_0}$ are
immediate by the definition of the elements $S_i(r)$.
The induction base is proven.
2). Now we prove the induction step: the relations ${\rm a_{m+1}},\dots, {\rm
l_{m+1}}$ follow from the relations ${\rm a_{m}},\dots, {\rm l_{m}}$.
\vspace{0.4cm}
$\underline{ Case \; {\rm a_{m+1}} }$ : $ k=1 , r=1 $.
$x\;\mbox{$ \stackrel{\rm def}{=} $}\ \; [1,S_1(N(m+1)+1)]\;\mbox{$ \stackrel{\rm def}{=} $}\ \; [1,1,2,S_3(Nm+N-1)]\;=\;
[1,[1,2],S_3(Nm+N-1)]+[1,2,1,S_3(Nm+N-1)].$ Denote the first(second) summand in
the right-hand side of the last formula by $ a (b) $. Then using the defining
relations and the Jacobi identity we obtain:
\[
a\;=\; [2,S_3(Nm+N-1)]+[[1,2],1,S_3(Nm+N-1)].
\]
Denoting the second summand in the last formula by $a_2$ and using the identity
$ {\rm g_m} $ to compute the commutator $ [1,S_3(Nm+N-1)] $ , we arrive at the
equation:
\[
a_2 \;= \; -[1,2,S_3(Nm+N-2)]+[2,1,S_3(Nm_N-2)].
\]
Now we apply the identity $ {\rm e_m} $ together with the definition of a
string and find that: $a_2 \; = \; -S_1(Nm+N) - [2,S_3(Nm+N-1)]$.
Using ${\rm g_m}$ one gets $ b\;=\; -S_1(Nm+N)$. Putting expressions for $a$
and $b$ together we find:
\[
x\;=\;a+b\;=\;-2S_1(N(m+1)).
\]
$\underline{ Case \; {\rm d_{m+1}} }$ : $ k \geq 3 , k+r \leq N , r \leq N-1$.
$x \; \mbox{$ \stackrel{\rm def}{=} $}\ \; [1,S_k(N(m+1)+r)] \; \mbox{$ \stackrel{\rm def}{=} $}\ \;
[1,k,k+1,\dots,N,1,2,\dots,k-1,S_k(Nm+r)].$ The relation ${\rm d_m}$ gives: $x
\; = \;
[k,k+1,\dots,N-1,[1,N],1,2,\dots,k-1,S_k(Nm+r)]+[k,k+1,\dots,N,1,[1,2],3,\dots,k-1,S_k(Nm+r)]$. Denote the first(second) summand in $x$ by $a(b)$. Then the defining relations, the Jacobi identity and ${\rm d_m}$ give: $b\;=\; [k,k+1,\dots,N,2,3,\dots,k-1,S_k(Nm+r)]$; and $ a\;=\; -b + [k,k+1,\dots,N-1,1,[1,N],2,3,\dots,k-1,S_k(Nm+r)]$. For $x$ then we obtain:
\begin{eqnarray*}
x \;= & a+b\;= & [k,k+1,\dots,N-1,1,1,N,2,3,\dots,k-1,S_k(Nm+r)] \\
& & -[k,k+1,\dots,N-1,1,N,1,2,\dots,k-1,S_k(Nm+r)].
\end{eqnarray*}
Since in the second summand above the leftmost 1 commutes with all elements
standing on its left we can carry this generator 1 to the very left. In the
first summand above $N$ commutes with all elements standing on its right up to
$S_k(Nm+r)$. Hence we obtain:
\[
2x\;=\; [k,k+1,\dots,N-1,1,1,2,3,\dots,k-1,N,S_k(Nm+r)].
\]
Let us now compute the commutator $ [N,S_k(Nm+r)] $ standing to the very right
in the above expression. The identities ${\rm d_m}\;,\;{\rm e_m}$ together with
the application of the automorphism $C$ give: $ [N,S_k(Nm+r)]\;
=\;-\delta_{r+k,N}S_k(Nm+r+1) $. Therefore
\begin{eqnarray*}
2x &= & -\delta_{r+k,N}[k,k+1,\dots,N-1,1,1,2,\dots,k-1,S_k(Nm+r+1)] \\
&= & -\delta_{r+k,N}[k,k+1,\dots,N-1,1,S_1(Nm+r+k)] \\
&= & -\delta_{r+k,N}[k,k+1,\dots,N-2,1,N-1,S_1(Nm+N)]
\end{eqnarray*}
The relation ${\rm l_m}$ applied (together with the automorphism $C$) to the
commutator $[N-1,S_1(Nm+N)]\;$ gives $ x\;=\;0. $
\vspace{0.4cm}
$\underline{ Case \; {\rm b_{m+1}} }$ : $ k = 1 ,\; r \geq 2,\; k+r \leq N $.
$
[1,S_1(N(m+1)+r)]\;=\;[1,1,2,S_3(N(m+1)+r-2))]\;=\;[1,[1,2],S_3(N(m+1)+r-2)]+[1,2,1,S_3(N(m+1)+r-2)].$
The commutator $[1,S_3(N(m+1)+r-2)]$ is equal to zero either because of the
already proven relation ${\rm d_{m+1}}$ or, when $r=2$, because of the relation
${\rm l_m}$. Consequently $ [1,S_1(N(m+1)+r)]\;=\;
[2,S_3(N(m+1)+r-2)]\;\mbox{$ \stackrel{\rm def}{=} $}\ \; S_2(N(m+1)+r+1).$
\vspace{0.4cm}
$\underline{ Case \; {\rm h_{m+1}} }$ : $ k+r \geq N+3,\; r \leq N-1 $.
$ x \; \mbox{$ \stackrel{\rm def}{=} $}\ \; [1,S_k(N(m+1)+r)] \; = \;
[1,k,k+1,\dots,N,1,2,\dots,k-1,S_k(Nm+r)].$ By virtue of the Jacobi identity:
\begin{eqnarray*}
x & = & [k,k+1,\dots,N-1,[1,N],1,2,\dots,k-1,S_k(Nm+r)] \\
& & + \; [k,k+1,\dots,N-1,N,1,1,2,\dots,k-1,S_k(Nm+r)].
\end{eqnarray*}
Let us denote the first(second) summand in the right-hand side of the above
formula by $ a(b) $.
Due to the Jacobi identity and the defining relations:
\begin{eqnarray*}
a & = & -[k,k+1,\dots,N-1,N,2,3,\dots,k-1,S_k(Nm+r)] \\
& & +[k,k+1,\dots,N-1,1,[1,N],2,3,\dots,k-1,S_k(Nm+r)].
\end{eqnarray*}
Denoting the first(second) summand in the right-hand side of the above formula
by $a_1(a_2)$ and using the definition of a string we find: $ a_1 \; = \;
[k,k+1,\dots,N-1,N,S_2(Nm+r+k-2)].$ Then applying the already proven relation
${\rm d_{m+1}}$ and the automorphism $ C$ to compute the commutator $
[N,S_2(Nm+r+k-2)] $ one obtains $ a_1\; =\;0. $ For $ a_2 $ we have:
\begin{eqnarray*}
a_2 & = & [k,k+1,\dots,N-1,1,1,N,S_2(Nm+r+k-2)] \\
& & -[k,k+1,\dots,N-1,1,N,1,2,\dots,k-1,S_k(Nm+r)]
\end{eqnarray*}
The first term in this expression for $ a_2 $ is equal to zero because $
[N,S_2(Nm+r+k-2)]\;=\;0 $ while the second term is equal to $ -x $.
Applying the already proven relation ${\rm b_{m+1}}$ and the same reasoning as
in the computation of $a_1$
we find that $ b\; =\; 0 $. Therefore $ x\;=\; a_1 + a_2 + b \; = \; -x $ ; $
x\; = \; 0. $
\vspace{0.4cm}
$\underline{ Case \; {\rm e_{m+1}} }$ : $ k+r = N+1,\; k \geq 3 ,\; r \leq
N-1 $.
Applying the Jacobi identity we obtain:
\begin{eqnarray*}
x & \mbox{$ \stackrel{\rm def}{=} $}\ & [1,S_k(N(m+1)+r)] \\
& \mbox{$ \stackrel{\rm def}{=} $}\ & [1,k,k+1,\dots,N,1,2,\dots,k-1,S_k(Nm+r)] \\
& = & [k,k+1,\dots,N-1,[1,N],1,2,\dots,S_k(Nm+r)] \\
& & +[k,k+1,\dots,N,1,[1,2],3,\dots,k-1,S_k(Nm+r)] \\
& & + [k,k+1,\dots,N-1,N,1,2,\dots,k-1,1,S_k(Nm+r)]
\end{eqnarray*}
Let us denote the three summands standing in the right-hand side of the last
equality in the above formula by $ a\;,b\;,c\;.$
Using the defining relations, the Jacobi identity and the definition of a
string we come to the equality:
\begin{eqnarray*}
a & = & -[k,k+1,\dots,N-1,N,S_2(Nm+r+k-2)] \\
& & + [ k,k+1,\dots,N-1,1,1,N,2,3,\dots,S_k(Nm+r)]\; - \; x\; .
\end{eqnarray*}
Denoting the first(second) summand in the above expression for $ a $ by $
a_1(a_2) $, and using the relation ${\rm g_m}$ we find that $ a_1 \;=\;
[k,k+1,\dots,N-1,S_2(Nm+N-2)].$ Whereas applying the definition of a string and
the relation ${\rm b_m}$ we find that $ a_2 \; = \; -a_1 $. Therefore $ a\;=\;
-x $.
For $ b $ we obtain:
\begin{eqnarray*}
b & \mbox{$ \stackrel{\rm def}{=} $}\ & [k,k+1,\dots,N,1,[1,2],\dots,k-1,S_k(Nm+r)] \\
& = & [k,k+1,\dots,N,1,1,2,\dots,k-1,S_k(Nm+r)] \\
& & -[k,k+1,\dots,N,1,2,1,3,\dots,k-1,S_k(Nm+r)]
\end{eqnarray*}
Denote the first(second) summand in the right-hand side of the above equality
by $ b_1(-b_2) $. Then the definition of a string and ${\rm i_m }$ enable us to
write:
\begin{eqnarray*}
b_1 & = & [k,k+1,\dots,N,1,S_1(Nm+r+k-1)] \\
& = & -2[k,k+1,\dots,N,S_1(Nm+N+1)] \\
& = & -2S_k(N(m+1)+r+1)
\end{eqnarray*}
Whereas ${\rm e_m} $ applied to the commutator $ [1,S_3(Nm+r+k-3)]$ entering
$b_2$ gives $ b_2\;=\; -S_k(N(m+1) +r +1). $
The relation $ {\rm e_m} $ applied to the commutator $ [1,S_k(Nm+r)] $ entering
$c$ leads to $ c \; = b_2 $.
Finally: $ x\; = \; a+b+c \; = \; -x - 2S_k(N(m+1)+r+1)\; ; \; x \; = \;
-S_k(N(m+1)+r+1). $
\vspace{0.4cm}
$\underline{ Case \; {\rm g_{m+1}} }$ : $ k+r = N+2,\; r \leq N-1 $.
Applying the Jacobi identity we obtain:
\begin{eqnarray*}
x & \mbox{$ \stackrel{\rm def}{=} $}\ & [1,S_k(N(m+1)+r)] \\
& \mbox{$ \stackrel{\rm def}{=} $}\ & [1,k,k+1,\dots,N,1,2,\dots,k-1,S_k(Nm+r)] \\
& = & [k,k+1,\dots,N-1,[1,N],1,2,\dots,S_k(Nm+r)] \\
& & +[k,k+1,\dots,N,1,[1,2],3,\dots,k-1,S_k(Nm+r)] \\
& & + [k,k+1,\dots,N-1,N,1,2,\dots,k-1,1,S_k(Nm+r)]
\end{eqnarray*}
Let us again denote the three summands standing in the right-hand side of the
last equality in the above formula by $ a\;,b\;,c\;.$
Using the defining relations, the Jacobi identity and the definition of a
string we come to the equality:
\begin{eqnarray*}
a & = & -[k,k+1,\dots,N-1,N,S_2(Nm+r+k-2)] \\
& & + [ k,k+1,\dots,N-1,1,1,N,S_2(Nm+k+r-2)]\; - \; x\; .
\end{eqnarray*}
Using the relation ${\rm l_m}$ and the automorphism $C$ to compute the
commutator $ [N,S_2(Nm+r+k-2)] $ we arrive at $ a \;=\; -x $.
For $ b $ we obtain:
\begin{eqnarray*}
b & \mbox{$ \stackrel{\rm def}{=} $}\ & [k,k+1,\dots,N,1,[1,2],3,\dots,k-1,S_k(Nm+r)] \\
& = & [k,k+1,\dots,N,1,1,2,\dots,k-1,S_k(Nm+r)] \\
& & -[k,k+1,\dots,N,1,2,1,3,\dots,k-1,S_k(Nm+r)] \\
& = & [k,k+1,\dots,N,1,S_1(Nm+N+1)] - [k,k+1,\dots,N,1,2,1,S_3(Nm+N-1)]
\end{eqnarray*}
Now we use the already proven relation ${\rm a_{m+1}}$ to compute the
commutator $ [1,S_1(Nm+N+1)] $ inside the first bracket above and the relation
${\rm g_m}$ - to compute the commutator $ [1,S_3(Nm+N-1)] $ inside the second
one. This gives $ b\; = \; -S_k(N(m+1)+r-1) $.
Applying ${\rm g_m}$ to the commutator $ [1,S_k(Nm+r)] $ entering $c$ we get $
c \; = \; b $.
Finally: $ x\;=\; a+b+c \; =\; -x - 2S_k(N(m+1)+r-1)\;;\;
x\;=\;-S_k(N(m+1)+r-1) $.
\vspace{0.4cm}
$\underline{ Case \; {\rm i_{m+1}} }$ : $ k=1\;, r = N $.
$x \; \mbox{$ \stackrel{\rm def}{=} $}\ \; [1,S_1(N(m+1)+N] \; \mbox{$ \stackrel{\rm def}{=} $}\ \; [1,1,2,S_3(N(m+1)+N-2)] \; = \;
[1,[1,2],S_3(N(m+1)+N-2)]
+ [1,2,1,S_3(N(m+1)+N-2)]$. Denoting the first(second) summand in the
right-hand side of the last formula by $a(b)$, and using the defining relations
we obtain:
\begin{eqnarray*}
a & = & [2,S_3(N(m+1)+N-2)] + [[1,2],1,S_3(N(m+1)+N-2)]
\end{eqnarray*}
Applying the already proven relation ${\rm e_{m+1} }$ to compute the commutator
$ [1,S_3(N(m+1)+N-2)] $
one gets:
\begin{eqnarray*}
a & = & [2,S_3(N(m+1)+N-2)] - S_1(N(m+1) + N+1) + [2,1,S_3(N(m+1)+N-1)]
\end{eqnarray*}
Transforming the last summand with the aid of the already proven relation ${\rm
g_{m+1} }$ we arrive at : $ a \; = \; -S_1(N(m+1)+N+1) $.
Application of $ {\rm e_{m+1} } $ to $b$ gives $ b \; = \; a $. Hence $ x\; =
\; a+b \; = -2S_1(N(m+1)+N+1) $.
\vspace{0.4cm}
$\underline{ Case \; {\rm l_{m+1}} }$ : $ 3 \leq k \leq N-1 \;, r=N $.
$ [1,S_k(N(m+1)+N)]\;=\;[k,1,S_{k+1}(N(m+1)+N-1)]\;= 0 .$ In virtue of the
already proven relation ${\rm h_{m+1} }$.
\vspace{0.4cm}
$\underline{ Case \; {\rm k_{m+1}} }$ : $ k=N\;, r=N $.
\begin{eqnarray*}
x & \mbox{$ \stackrel{\rm def}{=} $}\ & [1,S_N(N(m+1)+N)] \;=\; [1,N,1,S_2(N(m+1)+N-2] \\
& = & [[1,N],1,S_2(N(m+1)+N-2)] + [N,1,1,S_2(N(m+1)+N-2)]
\end{eqnarray*}
Denoting the first(second) summand in the right-hand side of the last equality
by $ a(b) $ and using the defining relations we obtain:
\begin{eqnarray*}
a & = & -[N,S_2(N(m+1)+N-2]+[1,1,N,S_2(N(m+1)+N-2)]\;-\;x
\end{eqnarray*}
Transforming the second summand in the above expression for $ a $ with the aid
of the already proven relations ${\rm e_{m+1}}$ and ${\rm i_{m+1}}$ we arrive
at: $ a \;=\; -[N,S_2(N(m+1)+N-2)]+2S_2(N(m+1)+N+1) - x $.
For $b$ we have: $ b \;=\; [N,1,S_1(N(m+1)+N+1)] $. Taking into account the
already proven relation ${\rm b_{m+1}}$ one then gets: $ b \; = \;
[N,S_2(N(m+1)+N-2] $. Finally: $ x \; = \; a+b \; = -x + 2S_1(N(m+1)+N+1)\; ; x
\; = \; S_1(N(m+1) + N+ 1) $.
The remaining cases: $ {\rm c_{m+1}}\;,\; {\rm f_{m+1}}\;, \; {\rm j_{m+1}} $
are direct consequences of the definition of the strings.
Thus the induction step is proven. \begin{flushright} $ \Box $ \end{flushright}
The Lemma has obvious corollary:
\pro Corollary The elements $ x(m)\; \mbox{$ \stackrel{\rm def}{=} $}\ \; \sum_{i=1}^N S_i(Nm) $ belong to
the centre of \mbox{${\cal A}_N $}\ .
{\it Proof}
It follows at once from the statements $ {\rm i_m}\;,\;{\rm j_m}\;,\;{\rm k_m}$
and ${\rm l_m}$ of
the Lemma by application of the automorphism $C$, that $ [i,x(m)]\;=\;0\;,\;
1\leq i\leq N \;,\;m\geq 1 $
\begin{flushright} $ \Box $ \end{flushright}
Now we can proceed further and turn to the proof of the proposition 1. First of
all we notice that any multiple commutator of $ {\rm {\bf e}}_i$-s can be converted with
the aid of the Jacobi identity into a linear combination of commutators nested
to the right, i.e. commutators of the form $ [i_1,i_2,i_3,\dots,i_{m-1},i_m] $
for some set of $1 \leq i_k \leq N$. Let us compute the nested commutators in
the last expression starting from the innermost one: $[i_{m-1},i_m]$, and going
step by step outwards. At each step of this procedure we need to compute a
commutator of a generator with a string which is , according to the Lemma,
again a string. Therefore any commutator of the form: $
[i_1,i_2,i_3,\dots,i_{m-1},i_m] $ is a string (may be of zero length, then it
is equal to 0). Thus any multiple commutator of the generators is a linear
combination of strings ( with integer coefficients). This finishes the proof of
the proposition 1.
In principle now we could find commutation relations among all strings using
the Jacobi identity and the result of the Lemma. Such a computation, though, is
rather cumbersome and we did it only for $N=3$. For arbitrary $N \geq 3$ in
sec. 5 we follow another route which eventually gives a basis of the algebra
\mbox{${\cal A}_N $}\ in terms of strings and commutation relations between the elements of this
basis.
\section{Isomorphism of the algebras \mbox{${\cal A}_N $}\ and \mbox{$\tilde{\cal A}_N $}\ }
In this section we show that the algebra \mbox{${\cal A}_N $}\ defined in sec.2 and the algebra
\mbox{$\tilde{\cal A}_N $}\ defined in sec. 3 are isomorphic.
We define a linear map from \mbox{${\cal A}_N $}\ to \mbox{$\tilde{\cal A}_N $}\ which we call $\pi$. First, we define
this map on the generators of \mbox{${\cal A}_N $}\ as follows:
\begin{eqnarray} \pi( {\rm {\bf e}}_i) & \mbox{$ \stackrel{\rm def}{=} $}\ & \mbox{$ \tilde{S} $}_i(1) \;\;\; \; \; 1 \leq i \leq N
\end{eqnarray} where the vectors $\mbox{$ \tilde{S} $}_i(1)$ were defined in (23) in sec. 2.
Next, we define the map $\pi$ on the whole algebra \mbox{${\cal A}_N $}\ by the prescription:
\begin{eqnarray} \pi( [a,b] ) & \mbox{$ \stackrel{\rm def}{=} $}\ & [ \pi(a) , \pi(b) ] \end{eqnarray}
It is easy to check that the vectors $\pi( {\rm {\bf e}}_i) $ satisfy the relations
(11-12). Therefore the map $ \pi : \mbox{${\cal A}_N $}\ \; \rightarrow \; \mbox{$\tilde{\cal A}_N $}\ $ is a
homomorphism of Lie algebras. Now we wish to find out what is the image of the
algebra \mbox{${\cal A}_N $}\ under the action of $\pi$. Since \mbox{${\cal A}_N $}\ is spanned by strings, it is
sufficient to find images of all strings , i.e. vectors $ \pi( S_i(r) ) \;
\;\;1\leq i \leq N $ . Using the definition of a string, and the prescription
(27), we arrive at the recursion relation:
\begin{eqnarray} \pi(S_i(r)) & = & [ \pi( {\rm {\bf e}}_i) , \pi(S_{i+1}(r-1)) ] \end{eqnarray}
This recursion relation is supplemented by the initial conditions (26),
therefore we can solve it, the result being:
\begin{eqnarray} \pi(S_i(r)) & = & \mbox{$ \tilde{S} $}_i(r) \;\;\;\; 1 \leq i \leq N \;\;,\;\;
r=1,2,\dots \end{eqnarray} where $ \mbox{$ \tilde{S} $}_i(r) $ are defined in (23-25). Since \mbox{$\tilde{\cal A}_N $}\ is a
linear span of the vectors $ \mbox{$ \tilde{S} $}_i(r) \;\;\;\; 1\leq i\leq N \;\;,\;\;
r=1,2,\dots $, we come to the conclusion that the image of \mbox{${\cal A}_N $}\ is the whole
algebra \mbox{$\tilde{\cal A}_N $}\ : $ Im\: \pi |_{\mbox{${\cal A}_N $}\ } \; = \; \mbox{$\tilde{\cal A}_N $}\ $. Now we notice, that
all the vectors $ \mbox{$ \tilde{S} $}_i(r) \;\;\;\; (1\leq i\leq N \;\;,\;\; r=1,2,\dots) $ are
linearly independent in \mbox{$\tilde{\cal A}_N $}\, except that $ \sum_{i=1}^N \mbox{$ \tilde{S} $}_i(Nm) \; = \; 0 $
for all $ m \geq 1 $. Therefore we come to the conclusion that the kernel of
the homomorphism $\pi $ is given by:
\begin{eqnarray} Ker\: \pi & = & linear \;\; span\;\; of\;\; \{ x(m) \}_{m\geq 1}\;\; (\mbox{$ \stackrel{\rm def}{=} $}\
\; {\rm {\bf C}} \{ x(m) \}_{m\geq 1} ) \end{eqnarray} where the elements $ x(m) \; \mbox{$ \stackrel{\rm def}{=} $}\
\; \sum_{i=1}^N S_i(Nm) $ were defined in the Corollary to the Lemma. Recall
that according to this Corollary $ {\rm {\bf C}} \{ x(m) \}_{m\geq 1} $
belongs to the centre of the Lie algebra \mbox{${\cal A}_N $}\ . Hence we conclude that \mbox{${\cal A}_N $}\ is
isomorphic to a central extension of \mbox{$\tilde{\cal A}_N $}\ by $ {\rm {\bf C}} \{ x(m)
\}_{m\geq 1} $ . In order to prove that the map $\pi$ is an isomorphism we
have to show that $ Ker\: \pi = 0 $.
Now we have the following proposition:
\begin{prop}
For all $m\geq 1$ the central elements $ x(m)\; \mbox{$ \stackrel{\rm def}{=} $}\ \; \sum_{i=1}^N S_i(Nm) $
vanish.
\end{prop}
{\it Proof}
To prove this proposition we shall first compute the commutator $ C(l,m)\; \mbox{$ \stackrel{\rm def}{=} $}\
\; \\ \mbox{} [S_1(Nl),S_1(Nm)] $ for $ l,m \geq 1 $.
In virtue of the Lemma one has for $m\geq 1$:
\begin{equation}
[k,S_1(Nm+k-1)]\;=\; \left\{ \begin{array}{ll}
-2S_1(Nm+1) & \mbox{if $k=1$} \\
-S_1(Nm+k) & \mbox{if $2\leq k\leq N-1$} \\
S_N(Nm+N) & \mbox{if $k=N$ }
\end{array} \right.
\end{equation}
Applying the above equality and the Jacobi identity we then get for $l,m\geq 1$
, $ 1\leq k\leq N-1 $ :
\begin{eqnarray}
[S_k(Nl-k+1),S_1(Nm+k-1] & = & \nonumber \\
& & [k,S_{k+1}(Nl-k),S_1(Nm+k-1)] \nonumber \\
& & +(1+\delta_{k,1})[S_{k+1}(Nl-k),S_1(Nm+k)]
\end{eqnarray}
Using the last formula repeatedly we arrive at the equality:
\begin{eqnarray}
C(l,m) & = & [1,S_2(Nl-1),S_1(Nm)] \nonumber \\
& &+ 2 \sum_{k=2}^{N-1} [k,S_{k+1}(Nl-k),S_1(Nm+k-1)] \nonumber \\
& & + 2 [S_N(Nl-N+1),S_1(Nm+N-1)]
\end{eqnarray}
The equation (31) leads besides to the following equality for $ m\geq 1 $ :
\begin{equation}
[S_N(Nl-N+1),S_1(Nm+N-1)]\;=\;\left\{ \begin{array}{ll}
S_N(N(m+1)) & \mbox{if $l=1$} \\
\mbox{} [N,S_1(Nl-N),S_1(Nm+N-1)] &-
\\
-[S_1(Nl-N),S_N(Nm+N)] & \mbox{if $l\geq
2$}
\end{array} \right.
\end{equation}
Combining the relations (33) and (34) we come to:
\begin{eqnarray}
C(1,m) & = & [1,S_2(N-1),S_1(Nm)] \nonumber \\
& & + 2 \sum_{k=2}^{N-1} [k,S_{k+1}(N-k),S_1(Nm+k-1)] \nonumber \\
& & + 2S_N(N(m+1)) \;, \; \; \; m \geq 1 \\
C(l,m) & = & [1,S_2(N-1),S_1(Nm)] \nonumber \\
& & + 2 \sum_{k=2}^{N} [k,S_{k+1}(Nl-k),S_1(Nm+k-1)] \nonumber \\
& & - 2[S_1(Nl-N),S_N(Nm+N)] \;, \; \; \; m \geq 1 \;,\; l \geq 2
\end{eqnarray}
Moreover, the Lemma and the Jacobi identity also give the following equation
for $ l,m\geq 1 $:
\begin{eqnarray}
[S_1(Nl),S_1(Nm)]+2[S_1(Nl),S_N(Nm)]= & & \nonumber \\
\mbox{} [1,S_2(Nl-1),S_1(Nm)]+2[1,S_2(Nl-1),S_N(Nm)] & &
\end{eqnarray}
The next step is to compute the triple commutators of the form $
[i,S_j(p),S_k(q)] $ appearing in the equations (35-37). We shall do it as
follows. First we can compute the internal commutators $ [S_j(p),S_k(q)] $ up
to a central element using the values of commutators $ [\mbox{$ \tilde{S} $}_j(p),\mbox{$ \tilde{S} $}_k(q)] $
in the algebra \mbox{$\tilde{\cal A}_N $}. Indeed, suppose we know that:
\[
[\mbox{$ \tilde{S} $}_j(p),\mbox{$ \tilde{S} $}_k(q)] \; = \; \sum_{l,r} F_{j,p;k,q}^{l,r} \mbox{$ \tilde{S} $}_l(r)
\]
where $ F_{j,p;k,q}^{l,r} $ are known structure constants. Then since \mbox{${\cal A}_N $}\ is a
central extension of $\tilde{\mbox{${\cal A}_N $}\ }$ (by the linear span of the set $ \{ x(m)
\}_{m\geq 1} $), we have:
\[
[S_j(p),S_k(q)] \; = \; \sum_{l,r} F_{j,p;k,q}^{l,r} S_l(r)\; + \; X
\]
where $ X $ is some element in the centre of \mbox{${\cal A}_N $}\ .
Next, we compute $ [i,S_j(p),S_k(q)] $ using the result of the Lemma.
It is straightforward to find the commutators $ [\mbox{$ \tilde{S} $}_j(p),\mbox{$ \tilde{S} $}_k(q)] $ , their
computation gives the following formulas for the relevant triple commutators
for $ m \geq 1 $:
\begin{eqnarray}
[1,S_2(N-1),S_1(Nm)] & = & 2S_1(Nm+N) \nonumber \\
\mbox{}[k,S_{k+1}(N-k),S_1(Nm+k-1)] & = & S_k(Nm) \nonumber \\
\mbox{}[1,S_2(N-1),S_N(Nm)] & = & -S_1(Nm+N) \nonumber \\
\mbox{}[1,S_2(Nl-1),S_1(Nm)] & = & 2S_1(N(m+l))-8(-1)^N S_1(N(m+l-2))\;\;\;
l\geq 2 \nonumber \\
\mbox{}[k,S_{k+1}(Nl-k),S_1(Nm+k-1)] & = & S_k(N(l+m))-4(-1)^N
S_k(N(m+l-2))\;\;\; l,k \geq 2 \nonumber \\
\mbox{}[1,S_2(Nl-1),S_1(Nm)] & = & -2[1,S_2(Nl-1),S_N(Nm)]
\end{eqnarray}
Substituting these expressions into (35-37) we obtain for $ m \geq 1 $:
\begin{eqnarray}
C(1,m) & = & 2\sum_{k=1}^N S_k(N(m+1)) \; \mbox{$ \stackrel{\rm def}{=} $}\ \; 2x(m+1)\; , \nonumber \\
C(l,m) & = & C(l-1,m+1) + 2x(l+m) - 8(-1)^N x(l+m-2) \;\;\; ,\; l \geq 2
\end{eqnarray}
Now taking into account that $ C(l,m)+C(m,l)\;=\;0 $ we arrive at the following
recursion relation for $ x(m) $:\begin{equation}
mx(m)\;=\;4(m-2)(-1)^N x(m-2) \;\;\;\; m\geq 3
\end{equation}
This recursion relation is supplemented by two initial conditions: $ x(2) \; =
\; 0 $ which follows from (39) and $ x(1) \; = \; 0,$ which will be shown
shortly. Solving the recursion relation (40) with these initial conditions we
get the desired result: $ x(m)\;=\;0 $.
Now we show that $ x(1) \;=\; 0 $.
\begin{eqnarray*}
x(1) \; \mbox{$ \stackrel{\rm def}{=} $}\ \; [1,2,\dots,N]+[2,3,\dots,1]+\dots+[N,1,\dots,N-1]
\end{eqnarray*}
If $ N=3 $ , the equation $ x(1) = 0 $ is the Jacobi identity.
If $ N \geq 4 $ , using the defining relations of \mbox{${\cal A}_N $}\ we find:
\begin{eqnarray*}
[N,1,\dots,N-1] & = & [[N,1],2,3,\dots,N-1]-[1,2,\dots,N],
\end{eqnarray*}
\begin{eqnarray*}
\mbox{} [[N,1,2,\dots,k],k+1,k+2,\dots,N-1] &= & \\
\mbox{} [[N,1,2,\dots,k+1],k+2,\dots,N-1]-[k+1,\dots,k] & & k \leq N-3
\end{eqnarray*}
Applying the last equation repeatedly we arrive at:
\[
[N,1,2,\dots,N-1]\;=\;-[1,2,\dots,N]-[2,3,\dots,N]-\dots -[N-1,N,1,2,\dots,N-2]
\]
The Proposition is proven.
\begin{flushright} $\Box$ \end{flushright}
In virtue of Proposition 2 $ Ker\: \pi = 0 $ , hence the map $ \pi $ defined
in (29) is isomorphism of Lie Algebras.
\section{The Hamiltonian and the Integrals of Motion}
Now we are ready to discuss how the $sl(N)$ Onsager's algebra leads to the
existence of an infinite number of integrals of motion for a Hamiltonian which
is linear in the generators $ {\rm {\bf e}}_i$ of this algebra.
Suppose that we have a representation of the algebra \mbox{${\cal A}_N $}\ i.e. a set of $N$
linear operators $ {\rm {\bf e}}_i \;,\; 1 \leq i \leq N $ satisfying the generalized
Dolan-Grady conditions (11,12). Consider the operator $H$ - Hamiltonian:
\begin{eqnarray}
H & = & k_1 {\rm {\bf e}}_1 + k_2 {\rm {\bf e}}_2 + \dots + k_N {\rm {\bf e}}_N
\end{eqnarray} where $ k_i$ are some constants.
If we consider $ H $ as a vector in \mbox{${\cal A}_N $}\ then the image $ \pi(H) $ of $H$
under the action of the isomorphism $ \pi $ (29) is a vector in \mbox{$\tilde{\cal A}_N $}\ :
\begin{eqnarray}
\pi(H) & = & \sum_{i=1}^N k_i \mbox{$ \tilde{S} $}_i(1) \\
& = & \sum_{i=1}^{N-1}k_i( E_{ii+1} + E_{i+1i}) \; + \; k_N ( tE_{N1} +
t^{-1}E_{1N} )
\end{eqnarray}
Now let us consider the following set of vectors in \mbox{$\tilde{\cal A}_N $}\ :
\begin{eqnarray}
\tilde{I}_1 & = & \sum_{i=1}^N k_i (\mbox{$ \tilde{S} $}_i(N+1) + 2\mbox{$ \tilde{S} $}_{i+1}(N-1)) \\
& = & (t + (-1)^N t^{-1}) \pi(H) \\
\tilde{I}_{m} & = & \sum_{i=1}^N k_i (\mbox{$ \tilde{S} $}_i(Nm+1) + 2\mbox{$ \tilde{S} $}_{i+1}(Nm-1)) \\
& = & ( t - (-1)^N t^{-1})^2 ( t + (-1)^N t^{-1})^{m-2} \pi(H)
\;\;\;\; m = 2,3,\dots
\end{eqnarray}
These vectors obviously commute between themselves and with the operator $
\pi(H) $. Therefore taking the inverse of $ \pi $ we arrive to the conclusion
that the members of the following set of elements $ \{ I_m \}_{m\geq 0} $ of
the algebra \mbox{${\cal A}_N $}\ commute between themselves:
\begin{eqnarray}
I_m & = & \sum_{i=1}^N k_i (S_i(Nm+1) + 2S_{i+1}(Nm-1)) \; \; , \; m \geq 1 \\
I_0 & = & H \\
\mbox{} [ I_m , I_n ] & = & 0 \;\;,\; m \geq 0
\end{eqnarray}
Thus if the conditions (11,12) are satisfied, $ H $ is a member of the infinite
family of integrals of motion in involution. Each of these integrals is a
vector in \mbox{${\cal A}_N $}\ and is expressed in terms of the operators $ {\rm {\bf e}}_i $ according to
the definition of strings $ S_i(r) $ given in (13). Since the strings are
linearly independent, the vectors $ I_k $ are linearly independent in \mbox{${\cal A}_N $}\ as
well.
\section{Conclusion}
As we have seen, the original Dolan Grady relations that define Onsager's
Algebra admit a generalization. This generalization stands in the same
relationship to underlying $ sl(N) $ Loop Algebra for $ N \geq 3 $ as the
original Onsager's Algebra - to $ sl(2) $ Loop Algebra. The crucial property
of Dolan Grady relations - the fact that they generate an infinite series of
integrals of motion in involution - is naturally present in this
generalization.
A number of further questions present themselves. The first two of these are:
what is the analog of Onsager's Algebra for any Kac-Moody Lie Algebra and what
are the corresponding analogs of the Dolan Grady conditions? The answers are
quite straightforward and we intend to report on this in a forthcoming
publication.
Another problem is concerned with the representation theory of the $sl(N)$
Onsager's Algebra. The classification of finite-dimensional representations of
\mbox{${\cal A}_N $}\ should be carried out in order to obtain an analogue of the eigenvalue
formula (9). Finally we need to find examples of models with Hamiltonians of
the form (10) that satisfy the conditions (11,12) and cannot be mapped onto
free-fermions. Some special cases of the spin-chains associated with the
$sl(N)$ Chiral Potts Models \cite{sl(N)} seem to be natural candidates for such
models.
\vspace{1cm}
{\bf Acknowledgments} \\
We are very grateful to Professor B.M. McCoy for teaching us Onsagers's
Algebra, encouragement and useful suggestions during preparation of this
paper. We are grateful to Professor J.H.H. Perk for valuable comments and to
Professors V.E.Korepin and M.Ro\v cek for their support.
\vspace{1cm}
|
1,108,101,564,525 | arxiv | \section{Introduction}
Accreting black hole X-ray binaries (BHXBs) typically stay in one of the two
distinct spectral states \citep{tanaka95}. In the so-called high/soft
state, such a binary has a soft X-ray spectrum and a relatively high
X-ray luminosity, which is believed to be dominated by the emission
directly from the accretion disk around the black hole (BH).
Under the standard
geometrically thin and optically thick accretion disk approximation
(see Pringle 1981 and references therein),
the X-ray spectrum is an integration of the assumed blackbody-like emission
over the disk with a temperature that decreases with increasing
radius \citep{mit84}
and is thus called the multi-color disk (MCD). In the opposite
low/hard state, the spectrum is relatively flat and can often be
approximated by a power law (PL), which may extend up to several
hundred keV, whereas the luminosity is typically low.
The flat spectral shape is
usually attributed to the Comptonization of soft
disk photons by hot electrons in a surrounding corona (e.g., Zdziarski 2000).
Therefore the spectrum of a BHXB system can usually be well fitted by
a two-component model with a blackbody-like (soft) component plus
a PL-like (hard) component; a disk reflection component and
a broad iron line component are sometimes also visible in
the spectra of some systems (e.g., Cygnus X-1, Di Salvo et al.~ 2001;
GX 339--4, Ueda, Ebisawa, \& Done 1994).
Of course, there are various other effects that one needs to consider.
For the soft component,
because of the high temperature ($\sim$ 1 keV) in the inner region of the
accretion disk, disk emission is expected to be slightly Comptonized
by the free electrons in this region and the local emergent spectrum
could be approximated as a diluted blackbody spectrum rather than a blackbody
one. Furthermore, the emission from the inner accretion disk is
subject to the strong gravitational field in the vicinity of the BH,
and should also be modified by the extreme Doppler motion of the
accretion disk.
Detailed calculations suggest that the simple MCD model (diskbb in XSPEC)
could still be used to describe the distorted disk emission, but needs
to be corrected by several factors in order to infer the consistent
physical quantities: a factor $f_{color}$ accounting for the
color temperature correction of the
Comptonization in the inner disk region (e.g, Shimura \& Takahara 1995;
Merloni, Fabian, \& Ross 2000);
a factor $\eta$ indicating the difference between the apparent and
intrinsic radii of the peak temperature, and the integrated luminosity
difference between the MCD model and a more realistic accretion disk
with torque-free boundary condition
(e.g., Kubota et al.~ 1998; Gierli\'{n}ski et al.~ 1999);
factors $f_{GR}$ and $g_{GR}$ accounting for gravitational redshift and
Doppler shift and the integrated disk flux change due to the general
relativistic effects (e.g., Cunningham 1975; also see Zhang, Cui, \& Chen
1997 for more discussions).
For the hard component, the phenomenological PL model is often used. However,
the extrapolation of the PL model to the lower energy in the MCD+PL model
neglects the seed photon curvature
that should be reflected in the Comptonized spectrum
(e.g., Shrader \& Titarchuk 1998; Done, \.{Z}ycki, \& Smith 2002).
In a typical global fit of the MCD+PL model to a BHXB spectrum,
this unphysical extrapolation usually leads to
an artificial increase in the soft X-ray-absorbing
column density estimates. As a result, the absorption-corrected source
flux could be over-estimated. This problem is not very acute for
an observation with poor counting statistics and/or for a
low temperature accretion disk with the MCD emission peak located
outside the observed energy range (e.g., accreting intermediate-mass BH
candidates; Wang et al.~ 2004, Paper II). But for a stellar mass BH with an
accretion disk temperature of $\sim 1$ keV, the problem is
significant. In fact, it has been shown that the MCD+PL model fails
to give an acceptable fit to X-ray spectra of LMC X--3, when
the X-ray absorption is tightly constrained by an independent measurement
from X-ray absorption edges in dispersed X-ray spectra \citep{pag03}.
Instead of using the PL model, several groups
have developed various Comptonization models by taking care of
the radiative transfer process either numerically or analytically
(e.g., $compbb$, Nishimura, Mitsuda, \& Itoh 1986; $comptt$, Titarchuk 1994;
Hua \& Titarchuk 1995; $thcomp$, \.{Z}ycki, Done, \& Smith 1999;
$eqpair$, Coppi 1999 and references therein;
$compps$, Poutanen \& Svensson 1996; etc.).
The seed photon spectrum is assumed to be a single-temperature
blackbody in $compbb$ or its Wien approximation in $comptt$,
which clearly deviates from a multi-color black-body disk.
This deviation could be very significant, especially for a disk with a high
inner temperature, typical for a stellar mass BH; the black-body
would peak well within an X-ray band.
Whereas the $thcomp$ is a thermal Comptonization model, the $eqpair$ model
is a hybrid model of thermal and non-thermal Comptonization
(see also Gierli\'{n}ski et al.~ 1999); both models
can adopt a proper disk spectrum, but the latter is more advanced
which takes care of nearly all the
important physical processes in a disk-corona system including Compton
scattering, pair production and annihilation, bremsstrahlung, and
synchrotron radiation, etc. \citep{coppi92}. All these four models can
only produce a direction-averaged spectrum. Like $eqpair$ model,
the $compps$ model also contains most physical processes
in the disk-corona system and allows the use of a MCD spectrum, but it
can treat geometry more accurately and generate an angular dependent
spectrum in a spherical or a slab-like corona system.
It is worth noting that in the disk-corona scenario,
the Comptonization generates the PL-like component at the expense of
the MCD flux, therefore the MCD and the PL-like components are physically
related. Some Comptonization models mentioned above
(e.g., $eqpair$ and $compps$) have taken care of this relationship
whereas some have not (e.g., $thcomp$).
One then should be cautious of interpreting the physical meaning of the
MCD parameters when applying the Comptonization models.
The MCD model here describes only those un-Comptonized (un-scattered
or escaped) disk photons rather than the actually original disk emission.
In particular, this relationship does not exist in MCD+PL model, therefore
the original disk flux could be under-estimated from the fitted MCD
parameters, so are other related inferred parameters such as inner disk radius.
The under-estimate, for example, may be
responsible for the {\sl apparent}
change in the inner disk radius with the transition from one state to another,
as has been claimed for several sources such as
XTE J1550-564, GRO J1655-40 \citep{sob99a, sob99b}, and
XTE J2012+381 \citep{cam02}.
All the models mentioned above
have been widely used to model the spectra of Galactic BHXBs,
neutron star X-ray binaries, as well as the extragalactic sources
including ultraluminous X-ray sources (ULXs) and AGNs.
However, except for the works by \citet{gie99,gie01},
there has been little rigorous test of the models
against direct measurements (e.g., neutral absorption
column density, BH mass, system inclination, etc.), which are
available for several well-known
systems such as GRO~J1655--40, LMC~X--1, and LMC~X--3.
In Yao et al.~ (2005; Paper I), we have presented a Monte-Carlo method in
simulating
Comptonized multi-color disk (CMCD) spectra. The simulations used the MCD as
the source of seed photons and self-consistently accounted for the radiation
transfer in the Comptonization in a spherical or slab-like thermal plasma.
We have applied this CMCD model, implemented as a
table model in XSPEC, to a stellar mass BH
candidate XTE~J2012+381 in our Galaxy. This application
shows that the inner disk radius is {\sl not} required to change
when the source transits from the soft state to the hard state,
in contrast to the conclusion reached from the fits with the
MCD+PL model \citep{cam02}.
For a spherical corona, the toy model contains the following parameters:
the inner disk temperature ($T_{in}$), the system inclination
angle ($\theta$),
the effective thermal electron temperature ($T_c$), optical depth ($\tau$),
and radius ($R_c$) of the corona as well as the normalization
defined as
\begin{equation} \label{equ:cmcd}
K_{CMCD} = \left(\frac{R_{in}/\mathrm{km}}{D/10\mathrm{kpc}}\right)^2,
\end{equation}
where $D$ is the source distance and $R_{in}$ is the apparent inner
disk radius. For a slab-like corona, another extreme of the geometry,
we assume that the corona covers
the whole accretion disk. We find that the final emerged spectra are
insensitive to the different vertical scales,
so $R_c$ does not appear as a parameter in this geometry.
Note that $ K_{CMCD}$ differs from the normalization
for the MCD model,
\begin{equation} \label{equ:mcd}
K_{MCD} = \left(\frac{R_{in}/\mathrm{km}}{D/10\mathrm{kpc}}\right)^2\cos(\theta).
\end{equation}
More detailed discussion on the CMCD model
can be found in Paper II, in which we have applied the same model for a
spherical geometry to six ULXs
observed with {\sl XMM-Newton}. The fitted $T_{in}$
($\sim$ 0.05--0.3 keV) of these sources
are distinctly different from the values ($\sim 1$ keV)
obtained for known stellar-mass BHs, as presented in this paper and in
Paper I. Indeed, the inferred BH masses ($M_{BH}$)
of the ULXs are $\sim 10^3 M_\odot$, consistent with the intermediate-mass
BH interpretation of these sources. We have also shown that the MCD+PL model
gives an equivalent spectral description of
the ULXs, although the CMCD model provides unique constrains on the
corona properties and on the disk inclination angles, as well as on
the BH masses.
Because of the lower disk temperatures, compared to those of the
stellar mass BH systems, the nonphysical
effects of the MCD+PL model are typically not significant in the observable
photon energy range of the intermediate-mass
BH candidates.
In the present work, we conduct a critical test of the CMCD and
MCD+PL models by comparing parameters ($M_{BH}$, $\theta$, and
the equivalent neutral hydrogen absorption $N_H$)
inferred from the X-ray spectra of LMC X--1 and
X--3 with the more direct measurements based on optical and dispersed X-ray
spectra; we also compare the results from the different corona geometrical
configurations of the CMCD model. We first briefly describe these
measurements and the X-ray observations in \S 2, and then
present the spectral fitting results in \S 3. We describe the specific
comparisons in \S 4 and present the discussion and our conclusions in \S 5.
\section{Description of the Sources and Observations}
We select LMC~X--1 and X--3 for this study chiefly because of their
location in our nearest neighboring galaxy,
the Large Magellanic Cloud (LMC; $D = 50$ kpc is adopted throughout the
work). Both the well-determined distance and the relatively low foreground
soft X-ray absorption are essential to our test. These two sources are also
among the three well-known persistent BHXBs and are usually found
in the high/soft state. The low/hard state was occasionally reported
for LMC~X--3 \citep{boy00, hom00}, but never for LMC~X--1.
The remaining known persistent BHXB, Cygnus~X--1, is
in our Galaxy and stays mostly in the low/hard state
(e.g., Pottschmidt et al.~ 2003). The X-ray spectra of this source
also show a strong disk reflection component \citep{gil99, fro01} ---
a complication that is not included in the CMCD models.
Table~\ref{tab:keymeasure} summarizes the key parameters of LMC~X--1 and
X--3,
which are used for the comparison with our spectrally inferred values (\S 4).
\begin{table*}
\centering
\begin{minipage}{140mm}
\caption{Comparison of parameter measurements. \label{tab:keymeasure}}
\begin{tabular}{@{}l|ccccl@{}}
\hline
& {$\theta$} & {M$_{BH}$} & {N$_H$} & {T$_{in}$ } & \\
& {($^\circ$)} & {(M$_\odot$)}& {(10$^{20}$cm$^{-2}$)} & {(keV)} & References\\
\hline
& \multicolumn{4}{c}{\underline{LMC~X--1}}\\
Indep. est. & 24$\le\theta\le$64 & 4$\le$M$\le$12.5 &--&--& 1, 2\\
CMCD: sphere & $\lsim$43 & 4.0(3.8--4.5) & 54(52--56) & 0.93(0.91--0.94)\\
CMCD: slab & $\gsim$64 & 15.2(10.4--19.9) & 53(51--54) & 0.80(0.79--0.81)\\
MCD+PL & N/A & 3.0(2.9--3.1) & 84(79--89) & 0.93(0.92--0.95)\\
\hline
& \multicolumn{4}{c}{\underline{LMC~X--3}}\\
Indep. est. & $\theta\le$70 & M$>5.8\pm0.8$ & 3.8(3.1--4.6) &-- & 3,4,5,6\\
CMCD: sphere & $\lsim$69 & 4.19(4.17--4.21) & 4.5(4.2--4.7) & 0.98(0.97--0.99)\\
CMCD: slab & $\lsim$61 & 3.73(3.71--3.76) & 4.4(4.2--4.6) & 0.96(0.95--0.97)\\
MCD+PL & N/A & 4.2(4.1--4.3) & 7.0(6.0--8.0) & 1.02(1.01--1.03)\\
\hline
\end{tabular} \\
{The BH mass M$_{BH}$ is estimated by assuming zero spin of the BH.
See text for details.
References:
$^1$ \citet{hut83};
$^2$ \citet{gie01};
$^3$ \citet{cow83};
$^4$ \citet{pac83};
$^5$ \citet{sor01};
$^6$ \citet{pag03}.}
\end{minipage}
\end{table*}
Both LMC X--1 and X--3 were observed with {\sl Chandra}
(e.g., Cui et al.~ 2002) and {\sl XMM-Newton} (e.g., Page et al.~ 2003).
The dispersed X-ray spectra of LMC~X--3 have been used to measure the
X-ray absorption edges (mainly for Oxygen), which tightly constrains the
the absorbing matter column density $N_H$ along the line of sight
\citep{pag03}. The absorption towards LMC X--1 is, however, substantially
higher. As a result, the photon flux at the Oxygen edge is too low to
allow for a useful constraint on $N_H$ based on the existing data.
We here utilize the data from the {\sl BeppoSAX} observations, which
were carried out on 1997 October 5 for LMC X--1 and
October 11 for X--3 \citep{tre00}. The data do not have pile-up problems,
which could be present for X-ray CCD imaging observations of bright sources.
Four types of narrow-field instruments (NFIs) were on board:
Low Energy Concentrator System (LECS), Medium Energy Concentrator
Systems (MECS), High Pressure Gas Scintillation Proportional
counter (HPGSPC), and Phoswich Detector System (PDS) \citep{boella97}.
The exposure for LMC~X--1 and X--3 were about 15 ks each for the LECS and
about 40 ks each for the MECS. These two instruments were sensitive to X-rays
in the energy ranges of 0.1--10 and 1.3--10 keV respectively.
Data from the HPGSPC and the PDS, which were sensitive to photons in 4--120
keV and 15--300 keV ranges respectively, were not included because of
poor counting statistics, and also because of possible source contamination
from PSR 0540--69 (which is 25$^\prime$ away from
LMC~X--1) \citep{sew84,haa01}. We extracted the spectra from a radius
of 8$'$ and 8.4$'$ around each source
from the LECS and the MECS observations and used the energy ranges of
0.2--4 keV and 1.8--10 keV for these two instruments in this study.
The background contributions to the spectra are small and are estimated from
a blank field.
The spectra from the LECS, the MECS2, and the MECS3 were jointly fitted
for each source, using the software package XSPEC{\em 11.2.0bs}.
\section{Results}
We summarize the spectral fitting results and the
inferred source fluxes in Table~\ref{tab:fit-parameters}.
The quoted uncertainty ranges of the parameters are all at 90\%
confidence level. Fig.~\ref{fig:cmcd} shows the spectral fits with the
CMCD models. The systematic deviation of the data from the model at
low energies
($\la 1$ keV) might be due to poor calibration
of the instrument spectral response \citep{martin96}.
Fig.~\ref{fig:compton} illustrates the effects of the Comptonization in the
spherical corona systems.
\begin{table}
\caption{Spectral fit results \label{tab:fit-parameters}}
\begin{tabular}{@{}lll@{}}
\hline
{parameters} & {LMC X--1} & {LMC X--3}\\
\hline
\multicolumn{3}{c}{\underline{CMCD: sphere}} \\
$N_H$ (10$^{21}$ cm$^2$) & 5.4(5.2--5.6) & 0.45(0.42--0.47) \\
$T_{in}$ (keV) & 0.93(0.91--0.94) & 0.98(0.97--0.99) \\
$T_c$ (keV) & 19(15--23) & 20(19--23) \\
$R_c$ (R$_g$) & 11(9--19) & 10.0(9.6--16.6) \\
$\tau$ & 1.0(0.8--1.3) & 0.10(0.09--0.13) \\
$\theta$ (deg) & 28(**--43) & 59(**--69) \\
$K_{\rm CMCD}$ & 57(51--73) & 44.0(43.5--44.3) \\
$\chi^2/dof$ & 641/627 & 760/649 \\
$f_{0.2-10}$ & 8.6 & 5.7 \\
\hline
\multicolumn{3}{c}{\underline{CMCD: slab}} \\
$N_H$ (10$^{21}$ cm$^2$) & 5.3(5.1--5.4) & 0.44(0.42--0.46) \\
$T_{in}$ (keV) & 0.80(0.79--0.81) & 0.96(0.95--0.97) \\
$T_c$ (keV) & 10.0(9.0--10.6) & 22(19--24) \\
$\tau$ & 0.45(0.41--0.49) & 0.09(0.08--0.12) \\
$\theta$ (deg) & 75(64--**) & 50(**--61) \\
$K_{\rm CMCD}$ & 489(230--840) & 42.7(42.1-- 43.3)\\
$\chi^2/dof$ & 638/628 & 759/650 \\
$f_{0.2-10}$ & 8.5 & 5.7 \\
\hline
\multicolumn{3}{c}{\underline{MCD+PL}} \\
$N_H$ (10$^{21}$ cm$^2$) & 8.4(7.9--8.9) & 0.7(0.6--0.8) \\
$T_{in}$ (keV) & 0.93(0.92--0.95) & 1.02(1.01--1.03) \\
$K_{\rm MCD}$ & 28(26--30) & 23(22--24) \\
$\Gamma$ & 3.5(3.4--3.6) & 2.6(2.5--2.8) \\
$K_{PL}$ (10$^{-1}$) & 2.3(1.9--2.7) & 0.19(0.14--0.23)\\
$\chi^2/dof$ & 624/629 & 747/651 \\
$f_{0.2-10}$ & 31 & 6.4 \\
\hline
\end{tabular}\\
{The uncertainty ranges are given in parenthesis at the 90\%
confidence level; asterisks indicate that the limit is not constrained.
$R_g$ = GM/c$^2$. The $f_{0.2-10}$ is the absorption-corrected
flux in the energy range 0.2--10 keV and in unit of
$10^{-10}{\rm~erg~cm^{-2}~s^{-1}}$.}
\end{table}
\begin{figure}
\centering
\psfig{figure=f1.eps,width=0.45\textwidth}
\caption{Model fits to the {\sl BeppoSAX} spectra of LMC~X--1
({\sl panel a}) and LMC~X--3 ({\sl panel c}), and the corresponding
residuals in term of sigmas ({\sl panels b} and {\sl d}).
The {\sl solid line} in {\sl panels a} and {\sl c} show the fit of CMCD model.
The fit goodnesses from the different geometrical models (a sphere vs. a slab)
are nearly the same.
\label{fig:cmcd}}
\end{figure}
\begin{figure}
\centering
\psfig{figure=f2.eps,width=0.45\textwidth}
\caption{The effects of the Comptonization in LMC~X--1 ({\sl a}) and in
LMC~X--3 ({\sl b}). {\sl Solid line}: spherical CMCD model with the best
fit parameters; {\sl dotted line}: CMCD model with the same parameters but
$\tau$=0, which is equivalent to the MCD model.
\label{fig:compton}}
\end{figure}
Both CMCD and MCD+PL models give acceptable fits.
The model parameters are all well constrained except for the
CMCD parameter $\theta$, for which only the upper
or lower limit is constrained.
The MCD+PL model parameters we obtained here
are consistent with those reported by \citet{haa01}.
The fitted $N_H$ values from MCD+PL are systematically
higher than those from CMCDs. The same is true for the inferred
absorption-corrected fluxes, especially for LMC~X--1 which is a factor of
$\sim$ 4 higher from MCD+PL than from CMCD (Table~\ref{tab:fit-parameters}).
For LMC~X--3, $T_{in}$ from MCD+PL is slightly higher than those from CMCDs,
whereas the value of $K_{MCD}/{\rm cos}(\theta)$ is consistent with those of
$K_{CMCD}$. The best fit parameters in the two different geometric CMCDs
are nearly identical. The small value of $\tau_c$ indicates that only
a small portion of disk photons have been up-scattered to high energies.
For LMC~X--1, except for a consistent $N_H$ value, the fitted parameters
are significantly different
between the two different geometric CMCDs (Table~\ref{tab:fit-parameters}).
We will see in \S4, the results from the slab-like configuration
are inconsistent with the independent measurements.
$T_{in}$ values from the spherical CMCD and the MCD+PL
are consistent with each other, but $K_{MCD}/{\rm cos}(\theta)$ value,
assuming the best-fit CMCD $\theta$, is $\sim$ 1.5 smaller than $K_{CMCD}$.
Because a significant change in the spectral shape occurred
during the observation of LMC X--1, we have further
split the exposure into two parts, the first 30 ks and
the remaining time, in the same way as in \citet{haa01}.
To tighten the constraints on spectral parameters,
we jointly fit $N_H$ and $\theta$, which should
be the same in the two parts of the observation (Table~\ref{tab:evolution}).
From the early part to the later part, according to the
spherical CMCD model,
$T_{in}$ increased by $\sim$ 7\%
and the corona became a factor of $\sim$ 2 larger and
opaque with $\tau$ increased by a factor of $\sim$ 4. The source flux
in the 0.2-10 keV band decreased slightly, although the normalization
remained essentially the same. Because
$T_{in} \propto (\dot{M}/M_{BH})^{1/4}$, where $M_{BH}$ and $\dot{M}$ are
the BH mass and the accretion mass rate, a rising $T_{in}$ during
the observation of LMC X--1 was then caused by an increasing
accretion rate. Apparently, this change led to the thickening of the
corona. The slight decline of the
flux is likely due to a combination of the energy loss to the
corona and to the scattering of photons to energies greater than
$\ga 10$ keV. Fig.~\ref{fig:evolution} demonstrates the differences
of the Comptonization effects in the two parts of the observation.
However, it is hard to physically understand the fitted parameters of the
slab-like CMCD: From early part to the later part,
$T_{in}$ and $T_c$ need to decrease by $\sim16$\% and a factor of
4, respectively, but in the mean time both $\tau$ and $K_{CMCD}$
(hence $R_{in}$) have to increase dramatically.
We believe that the slab-like CMCD model is not suitable
for describing such a spectral state of a BHXB system when
the hard component contributes significantly to the total flux.
\begin{table}
\caption{Spectral variation of LMC X--1 \label{tab:evolution}}
\begin{tabular}{@{}lll@{}}
\hline
{parameters} & {part 1} & {part 2} \\
\hline
\multicolumn{3}{c}{\underline{CMCD: sphere}}\\
$N_H$ (10$^{21}$ cm$^2$) & 5.3(5.2--5.5) & = part 1 \\
$T_{in}$ (keV) & 0.91(0.90--0.92) & 0.96(0.93--0.99)\\
$T_c$ (keV) & 13(9--18) & 10(8--12) \\
$R_c$ (R$_g$) & 10(9--16) & 25(18--33) \\
$\tau$ & 0.55(0.5--0.7) & 2.0(1.8--2.4) \\
$\theta$ (deg) & 23(**--45) & = part 1 \\
$K_{CMCD}$ & 61(55--72) & 57(46--77) \\
$\chi^2/dof$ & \multicolumn{2}{c}{1211/1192} \\
$f_{0.2-10}$ & 8.8 & 8.3 \\
\hline
\multicolumn{3}{c}{\underline{CMCD: slab}}\\
$N_H$ (10$^{21}$ cm$^2$) & 5.3(5.2--5.4) & = part 1 \\
$T_{in}$ (keV) & 0.81(0.80--0.82) & 0.74(0.73--0.76)\\
$T_c$ (keV) & 20(19--23) & 5(**--7) \\
$\tau$ & 0.11(0.10--0.14) & 1.9(1.7--2.1) \\
$\theta$ (deg) & 75(64--**) & = part 1 \\
$K_{CMCD}$ & 348(224--495) & 1223(730--1855) \\
$\chi^2/dof$ & \multicolumn{2}{c}{1209/1194} \\
$f_{0.2-10}$ & 8.8 & 8.1 \\
\hline
\multicolumn{3}{c}{\underline{MCD+PL}} \\
$N_H$ (10$^{21}$ cm$^2$) & 8.5(8.0--9.0)& = part 1 \\
$T_{in}$ (keV) & 0.90(0.89--0.91) & 1.00(0.98--1.03) \\
$K_{MCD}$ & 52(49--55) & 23(21--26) \\
$\Gamma$ & 3.6(3.5--3.8) & 3.4(3.3--3.6) \\
$K_{PL}$ (10$^{-1}$) & 3.0(2.4--3.7) & 3.4(2.8--3.9) \\
$\chi^2/dof$ & \multicolumn{2}{c}{1180/1195} \\
$f_{0.2-10}$ & 48 & 42 \\
\hline
\end{tabular} \\
{Please refer to Table~\ref{tab:fit-parameters}.}
\end{table}
\begin{figure}
\centering
\psfig{figure=f3.eps,width=0.45\textwidth}
\caption{The spectral variation of LMC~X--1 from the first 30 ks (part 1,
{\sl solid line}) to the remaining time (part 2, {\sl dashed line}).
The models are plotted with the best fitting parameters
(Table~\ref{tab:evolution}).
\label{fig:evolution}}
\end{figure}
Now let us check the results from the MCD+PL model, which
are fairly consistent with those of \citet{haa01}. The MCD normalizations are
significantly different between the early and later
parts of the observation (Table~\ref{tab:evolution}), confirming
the analysis by \citet{haa01}, which is based on the same data and
same model. They concluded that as $T_{in}$ increased (by $\simeq$ 9\%),
$R_{in}$ decreased ($\simeq$ 38\%) from the
early to later parts of the observation.
But, as was discussed in \S 1, this apparent change in $R_{in}$
is most likely due to
the lack of the accounting for the radiative transfer between
the two components of the MCD +PL model and is therefore not physical.
Furthermore, Fig.~\ref{fig:compare} shows clearly that
for both sources, the PL component surpasses the MCD component in
contributing to the spectra at low
energies ($\lsim$ 1 keV for LMC~X--1 and $\lsim$ 0.4 keV for LMC~X--3). This
nonphysical straight extension of the PL
component to the low energy parts of the spectra is the main cause
for the required high values of $N_H$ in the MCD+PL fits, compared to
those in the CMCD fits (Tables~\ref{tab:fit-parameters} and
\ref{tab:evolution}).
\begin{figure}
\centering
\psfig{figure=f4.eps,width=0.45\textwidth}
\caption{The comparison of the spherical CMCD model and MCD+PL model with the best fit
parameters (Table~\ref{tab:fit-parameters})
for ({\sl a}) LMC~X--1 and ({\sl b}) LMC~X--3.
\label{fig:compare}}
\end{figure}
\section{Comparisons with Independent Measurements}
In addition to the above self-consistency check of the X-ray spectral models,
we compare the inferred parameter values of the BHXBs with the independent
measurements to further the test. The key results of this
comparison are included in Table~\ref{tab:keymeasure} and are discussed
in the following:
\noindent $\bullet$ {\bf X-ray Absorption}
$N_H$ may be obtained more directly and accurately via
the spectroscopy of neutral
absorption edges of Oxygen and Neon (e.g., Schulz et al.~~2002; Page et al.~~2003).
This method is nearly independent of the overall continuous spectral shape
and any other source properties. Using the dispersed spectrum of an
{\sl XMM-Newton} observation, \citet{pag03} inferred
$N_H = 0.38_{-0.07}^{+0.08}\times 10^{21}~\rm{cm}^{-2}$ for LMC~X--3,
assuming the interstellar medium abundance of \citet{wilms00}.
We have also adopted this assumption by using the
absorption model {\sl TBabs} in XSPEC.
The $N_H$ values from the CMCD models and from the X-ray absorption edge
measurement agree with each other within the quoted error
bars, whereas the value from the MCD+PL model is
significantly higher (Table~\ref{tab:keymeasure}).
\noindent $\bullet$ {\bf Disk Inclination Angle}
The upper limits on $\theta$ from the spherical CMCD model are
consistent with
the values obtained from the optical observations
(Table~\ref{tab:keymeasure}). There is a simple reason why
$\theta$ can be constrained in the spherical CMCD model: the
Comptonized flux is nearly isotropic and barely affected by the disk
inclination, whereas the observed strength of the soft disk
component is proportional to cos($\theta$). This mostly geometric
effect is strong when $\theta$ is large; e.g., no radiation come directly
from an edge-on disk. Therefore, the upper limit can be constrained
reasonably well, which is especially important for estimating the BH mass
(cf. Eqs.~\ref{equ:cmcd} and \ref{equ:mcd}). In the slab-like CMCD model,
since both of the observed soft and hard components strongly depend upon
the system inclination angle (e.g., Sunyaev \& Titarchuk 1985; see also
the Fig.~2 in Paper I), $\theta$ in principle can be well constrained in
this model. For LMC~X--3, the PL-like component only contributes
a small portion of the total flux, and the constraint of $\theta$ mainly
from the soft component, as in a spherical CMCD model.
For LMC~X--1, the PL-like component contributes
significantly to the total flux.
The slab-like corona configuration becomes problematic, giving
the inconsistent $\theta$ constraints.
\noindent $\bullet$ {\bf Black Hole Mass}
The mass of each putative BH may be estimated as
$M = c^2fR_{in}$/$G\alpha$, where
$\alpha = 6$ or 1 for a non-spin or extreme spin BH),
$f R_{in}$ is the inner disk radius, assumed to be the
same as the radius of the last marginally stable orbit around the BH, and the
factor $f$ [depending on the system inclination,
0.94 for LMC X--1 and 1.12 for LMC X--3; we adopt a new value of
$\eta=0.41$ (see \S1) derived from \citet{kub98}, compared to the old values
used $\eta=0.7$ used in \citet{zhang97} and in Paper II]
includes various corrections
related to the spectral hardening, special
and general relativity effects (e.g., Cunningham 1975;
Zhang, Cui, \& Chen 1997; Gierli\'{n}ski et al.~ 1999, 2001).
Assuming no spin for the BHs, from the spherical CMCD results,
we estimate the BH masses
as 4.0(3.8--4.5), 4.19(4.17--4.21)~$M_{\odot}$ for
LMC~X--1 and LMC~X--3, respectively. For LMC~X--1,
the value is consistent with the result from the optical study,
whereas for LMC~X--3, the derived $M_{BH}$ is slightly smaller
(Table~\ref{tab:keymeasure}) which may suggest that the LMC~X--3
is a mild spin system.
\section{Summary}
In this work, we have applied the MCD+PL model as well as our recently
constructed CMCD model to
BHXB systems LMC~X--1 and LMC~X--3, confronting the fitted parameters
with directly measured values.
We have also tested two different corona geometric configurations.
The spherical configuration passes almost all the tests:
the effective hydrogen column densities, disk
inclinations, and the BH masses of the LMC~X--1 and LMC~X--3.
This consistency suggests that the CMCD model with a spherical corona provides
a reasonably good spectral characterization of BHXBs. The model offers
useful insights into physical properties of the Comptonization coronae
and their relationship to the accretion process.
In contrast, the slab-like CMCD model is problematic in describing
the spectrum of LMC~X--1, in which the PL-like component contributes
significantly. Similarly, the MCD+PL model, though generally providing a
good fit to the spectra of BHXBs, could give misleading parameter values.
Although the tests conducted in this work are still very limited, they
have demonstrated the potential in discriminating among various models.
\section*{Acknowledgments }
We thank the anonymous referee for his/her insightful comments on our
manuscript which helps us to improve the paper greatly. Y. Yao is also
grateful to Xiaoling Zhang and Yuxin Feng for their useful discussions.
|
1,108,101,564,526 | arxiv | \section{Introduction}
In group theory, a Zappa-Sz\'{e}p product of two groups $G$ and $H$ generalizes a semi-direct product by encoding a two-way action between $G$ and $H$. In addition to a left action of $G$ on $H$, the Zappa-Sz\'{e}p product encodes an
additional right action of $H$ on $G$. An analogue of semi-direct products in operator algebra is the crossed product construction
arising from various dynamical systems. In its simplest form, a group $G$ act on a C*-algebra $\mathcal{A}$ by $*$-automorphisms, in a similar fashion as in a semi-direct product. We seek to extend the Zappa-Sz\'{e}p type construction into the field of operator algebras, which would naturally generalize the crossed product construction.
To construct a Zappa-Sz\'{e}p type structure in operator algebras, there are two key ingredients. First, a Zappa-Sz\'{e}p product of an operator algebra $\mathcal{A}$ and a group $G$ requires a left action of $G$ on $\mathcal{A}$ and a right action of $\mathcal{A}$ on $G$.
The right $\mathcal{A}$ action on $G$ requires a grading on $\mathcal{A}$. In the operator algebra literature, there are two natural ways of putting a grading: either via Fell bundles graded by groupoids,
or via product systems graded by semigroups. The Zappa-Sz\'{e}p product of a Fell bundle by a groupoid is recently studied in \cite{DL2020}. This paper aims to study Zappa-Sz\'{e}p products of product systems by groups. The second key ingredient is an appropriate replacement of the group action in a dynamical system. A Zappa-Sz\'{e}p product of a product system by a group $G$ needs to encode a two-way action between the product system and the group in the scenario. This requires the group action on the product system to be compatible with the product system action on the group. Finding a right notion of a Zappa-Sz\'{e}p product with compatible actions in the context of product systems is a key in our construction.
Let $P$ be a semigroup and $G$ a group. Suppose $G$ has a left action on $P$ and $P$ has a right action on $G$, so that one can form a Zappa-Sz\'{e}p product $P\bowtie G$. Let $X$ be a product system over $P$.
We introduce a notion of the Zappa-Sz\'{e}p action of $G$ on $X$ in Definition \ref{D:beta}. Given a Zappa-Sz\'{e}p action of $G$ on $X$, we define a Zappa-Sz\'{e}p product of $X$ by $G$.
We show in Theorem \ref{T:ZSP} that it is a product system over the Zappa-Sz\'{e}p product semigroup $P\bowtie G$. This product system is denoted by $X\bowtie G$. We then consider the scenario where the $G$-action on $P$ is homogeneous. In such a case, it turns out that we can naturally define another Zappa-Sz\'{e}p type product, denoted by $X\widetilde\bowtie G$, which is a product system over the same semigroup $P$ (Theorem \ref{T:prop.Z}).
In Section \ref{S:main}, we study covariant representations of Zappa-Sz\'{e}p actions of groups on product systems and their associated C*-algebras. A covariant representation of a Zappa-Sz\'{e}p action $(X,G,\beta)$
is a pair $(\psi, U)$ consisting of a Toeplitz representation of $X$ and a unitary representation $U$ of $G$ that satisfy a covariant relation.
First, we exhibit a one-to-one correspondence between the set of all covariant representations $(\psi, U)$ of $(X, G,\beta)$ and the set of all Toeplitz representations $\Psi$ of $X\bowtie G$
(Theorem \ref{T:Upsi}).
As an immediate consequence, we obtain a Hao-Ng isomorphism theorem: $\mathcal{T}_{X\bowtie G}\cong \mathcal{T}_X\bowtie G$ (Corollary \ref{C:HaoNgT}).
Furthermore, we show that $\psi$ is Cuntz-Pimsner covariant if
and only if so is $\Psi$, and so the Hao-Ng isomorphism also holds true for the associated Cuntz-Pimsner C*-algebras: $\mathcal{O}_{X\bowtie G}\cong \mathcal{O}_X\bowtie G$ (Theorem \ref{thm.cp} and Corollary \ref{C:HaoNgC}).
Moreover, if $(X,G,\beta)$ is homogeneous, then
$\mathcal{T}_{X\bowtie G}\cong \mathcal{T}_X\bowtie G\cong \mathcal{T}_{X\widetilde\bowtie G}$ (Theorem \ref{T:Upsi.homo} and Corollary \ref{C:HaoNgT}).
However, we do not know whether one has $\mathcal{O}_{X\widetilde\bowtie G}\cong \mathcal{O}_X\bowtie G$. Indeed, this is still unknown
even in the special case of semi-direct products. This is related to an open problem of Hao-Ng in the literature. Lastly, when the semigroup $P$ is right LCM and $X$ is compactly aligned, we show that $\psi$ is
Nica covariant if and only if so is $\Psi$ (Theorem \ref{T:Ncov}). As a result, we have the Hao-Ng isomorphism theorem for Nica-Toeplitz algebras as well: $\mathcal{N}\mathcal{T}_{X\bowtie G}\cong \mathcal{N}\mathcal{T}_X\bowtie G$ (Corollary \ref{C:HaoNgNT}).
Finally, in Section \ref{S:EX}, we present some examples of Zappa-Sz\'{e}p actions of groups on product systems and their C*-algebras.
\section{Preliminaries}
In this section, we provide some necessary background for later use.
\subsection{Zappa-Sz\'{e}p products}
Let $P$ be a (discrete) semigroup and $G$ be a (discrete) group.
By convention, in this paper, we always assume that \textsf{a semigroup $P$ has an identity}, written as $e$, unless otherwise specified.
To define a Zappa-Sz\'{e}p product semigroup of $P$ and $G$, we first need two actions between $P$ and $G$, given by
\begin{enumerate}
\item a left $G$-action on $P$: $G\times P \to P$, denoted by $(g,p)\mapsto g\cdot p$, and
\item a right $P$-action on $G$ (also called a restriction map): $P\times G\to G$, denoted by $(p,g)\mapsto g|_p$.
\end{enumerate}
Suppose that the two actions satisfy the following compatibility relations:
\begin{multicols}{2}
\begin{enumerate}
\item[(ZS1)]\label{cond.ZS1} $e\cdot p=p$;
\item[(ZS2)]\label{cond.ZS2} $(gh)\cdot p=g\cdot (h\cdot p)$;
\item[(ZS3)]\label{cond.ZS3} $g\cdot e=e$;
\item[(ZS4)]\label{cond.ZS4} $g|_{e}=g$;
\item[(ZS5)]\label{cond.ZS5} $g\cdot (pq)=(g\cdot p)(g|_p \cdot q)$;
\item[(ZS6)]\label{cond.ZS6}$g|_{pq}=(g|_p)|_q$;
\item[(ZS7)]\label{cond.ZS7} $e|_p=e$;
\item[(ZS8)]\label{cond.ZS8} $(gh)|_p=g|_{h\cdot p} h|_p$.
\end{enumerate}
\end{multicols}
\noindent
Then the Zappa-Sz\'{e}p product semigroup $P\bowtie G$ is defined by $P\bowtie G=\{(p,g): p\in P, g\in G\}$ with multiplication $(p,g)(q,h)=(p(g\cdot q), g|_q h)$. This semigroup has an identity $(e, e)$.
Recall that a left cancellative semigroup $P$ is called a right LCM semigroup if for any $p,q\in P$, either $pP\cap qP=\emptyset$ or $pP\cap qP=rP$ for some $r\in P$. In the case when $P$ is a right LCM semigroup, $P\bowtie G$ is known to be a right LCM semigroup as well \cite[Lemma 3.3]{BRRW}.
\subsection{Product systems}
We give a brief overview of product systems. One may refer to \cite{Fowler2002} for a more detailed discussion.
\begin{definition} Let $\mathcal{A}$ be a unital C*-algebra and $P$ a semigroup. A \textit{product system over $P$} with coefficient $\mathcal{A}$ is defined as $X=\bigsqcup_{p\in P} X_p$ consisting of $\mathcal{A}$-correspondences $X_p$ and an associative multiplication $\cdot:X_p \times X_q\to X_{pq}$ such that
\begin{enumerate}
\item $X_e=\mathcal{A}$ as an $\mathcal{A}$-correspondence;
\item for any $p,q\in P$, the multiplication map on $X$ extends to a unitary $M_{p,q}: X_p\otimes X_q\to X_{pq}$;
\item the left and right module multiplications by $\mathcal{A}$ on $X_p$ coincides with the multiplication maps on $X_e\times X_p\to X_p$ and $X_p\times X_e\to X_p$, respectively.
\end{enumerate}
\end{definition}
Implicit in $M_{e,q}$ being unitary, $X_p$ must be essential, that is, $\lspan\{a\cdot x: a\in X_e, x\in X_p\}$ is dense in $X_p$. This assumption is absent in Fowler's original construction as he does not require $M_{e,q}$ to be unitary. Nevertheless, when the semigroup $P$ contains non-trivial units, every $X_p$ must be essential \cite[Remark 1.3]{KL2018}. Since the semigroups $P$ and $P\bowtie G$ often contain non-trivial units, it is reasonable to
make such an assumption.
For a C*-correspondence $X$, we use $\mathcal{L}(X)$ to denote the set of all adjointable operators on $X$. It is a C*-algebra when equipped with the operator norm. For any $x,y\in X$, define the operator $\theta_{x,y}:X\to X$ by $\theta_{x,y}(z)=x\langle y,z\rangle$. It is clear that $\theta_{x,y}\in \mathcal{L}(X)$, and we use $\mathcal{K}(X)$ to denote the C*-subalgebra of $\mathcal{L}(X)$ generated by $\theta_{x,y}$. The set $\mathcal{K}(X)$ is also known as the generalized compact operators on $X$.
Suppose now that $P$ is a right LCM semigroup. The notion of compactly aligned product systems, first introduced by Fowler for product systems over quasi-lattice ordered semigroups \cite{Fowler2002}, has been
recently generalized to right LCM semigroups
in \cite{BLS2018b, KL2018}. For any $p,q\in P$, there is a $*$-homomorphism $i_p^{pq}: \mathcal{L}(X_p)\to\mathcal{L}(X_{pq})$ by setting for any $x\in X_p$ and $y\in X_q$,
\[i_p^{pq}(S)(xy)=(Sx)y.\]
\begin{definition}
\label{df.cpt.align}
We say that a product system $X$ is \emph{compactly-aligned} if for any $S\in\mathcal{K}(X_p)$ and $T\in\mathcal{K}(X_q)$ with $pP\cap qP=rP$, we have
\[i_p^r(S) i_q^r(T) \in \mathcal{K}(X_r).\]
We shall use the notion $S\vee T := i_p^r(S) i_q^r(T)$.
\end{definition}
\subsection{Representations and C*-algebras of product systems}
Product systems are one of essential tools in the study of operator algebras graded by semigroups. In its simplest form, an $\mathcal{A}$-correspondence $X$ can be viewed as a product system over $\mathbb{N}$, by setting $X_0=\mathcal{A}$ and $X_n=X^{\otimes n}$. There are two natural C*-algebras associated with such a product system: the Toeplitz algebra $\mathcal{T}_X$ and the Cuntz-Pimsner algebra $\mathcal{O}_X$ \cite{MS1998, Pimsner1997}. Fowler generalizes these to product systems over countable semigroups \cite{Fowler2002}.
In \cite{Nica1992}, Nica introduced the semigroup C*-algebra of a quasi-lattice ordered semigroup. It is the universal C*-algebra generated by isometric semigroup representations that satisfy a covariance condition, now known as the Nica-covariance condition. This extra covariance condition soon found an analogue in the C*-algebra related to product systems. In \cite{Fowler2002, FR1998, BLS2018b}, the Nica-Toeplitz algebra $\mathcal{N}\mathcal{T}_X$ is defined by imposing the extra Nica-covariance condition, thereby being a quotient of the corresponding Toeplitz algebra. Here, we give a brief overview of these three C*-algebras and their representations associated with a product system.
\begin{definition} Let $X$ be a product system over a semigroup $P$. A \textit{(Toeplitz) representation} of $X$ on a C*-algebra $\mathcal{B}$ consists of a collection of linear maps $\psi=(\psi_p)_{p\in P}$, where for each $p\in P$, $\psi_p:X_p\to \mathcal{B}$, such that
\begin{enumerate}
\item $\psi_e$ is a $*$-homomorphism of the C*-algebra $X_e$;
\item for all $p,q\in P$ and $x\in X_p, y\in X_q$, $\psi_{p}(x)\psi_q(y)=\psi_{pq}(xy)$; and
\item for all $p\in P$ and $x, y\in X_p$, $\psi_p(x)^* \psi_p(y)=\psi_e(\langle x,y\rangle)$.
\end{enumerate}
\end{definition}
Given a representation $\psi:X\to \mathcal{B}$, there is a homomorphism $\psi^{(p)}: \mathcal{K}(X_p)\to \mathcal{B}$ satisfying $\psi^{(p)}(\theta_{x,y})=\psi_p(x)\psi_p(y)^*$. The left action of $X_e$ on $X_p$ induces a $*$-homomorphism $\phi_p: X_e\to \mathcal{L}(X_p)$ by $\phi_p(a)x=a\cdot x$.
\begin{definition} A representation $\psi$ is called \textit{Cuntz-Pimsner covariant} if for all $p\in P$ and $a\in X_e$ with $\phi_p(a)\in \mathcal{K}(X_p)$, $\psi^{(p)}(\phi_p(a))=\psi_e(a)$.
\end{definition}
By \cite[Propositions 2.8 and 2.9]{Fowler2002}, there is a universal C*-algebra $\mathcal{T}_X$ (resp. $\mathcal{O}_X$) for Toeplitz (resp. Cuntz-Pimsner covariant) representations. Specifically, there is a universal Toeplitz representation $i_X$ (resp. a universal Cuntz-Pimsner covariant representation $j_X$) such that the following properties hold:
\begin{enumerate}
\item The C*-algebras $\mathcal{T}_X$ and $\mathcal{O}_X$ are generated by $i_X$ and $j_X$ respectively. That is, $\mathcal{T}_X=C^*(i_X(X))$ and $\mathcal{O}_X=C^*(j_X(X))$.
\item For every Toeplitz (resp. Cuntz-Pimsner covariant) representation $\psi$, there exists a $*$-homomorphism $\psi_*$ from $\mathcal{T}_X$ (resp. $\mathcal{O}_X$) to $C^*(\psi(X))$ such that $\psi=\psi_*\circ i_X$ (resp. $\psi=\psi_* \circ j_X$).
\end{enumerate}
The Toeplitz C*-algebra $\mathcal{T}_X$ is often quite large as a C*-algebra. For example, for the trivial product system $X$ over ${\mathbb{N}}^2$,
its Toeplitz representation is determined by a pair of commuting isometries. The Toeplitz algebra $\mathcal{T}_X$ of this product system is thus the universal C*-algebra generated by a pair of commuting isometries, which is known to be non-nuclear \cite{Murphy1996b}. This motivated Nica to study the semigroup C*-algebras of quasi-lattice ordered semigroups \cite{Nica1992} by imposing what is now known as the Nica-covariance condition. This condition is soon generalized to representations of product systems. In \cite[Definition 5.1]{Fowler2002}, Fowler defined the notion of compactly aligned product system and extended the Nica-covariance condition to such product systems. In \cite[Definition 6.4]{BLS2018b}, the Nica-covariance condition is further generalized to compactly aligned product systems over right LCM semigroups. We now give a brief overview of the Nica-covariance condition.
Recall that if $X$ is compactly aligned, then for any $S\in\mathcal{K}(X_p)$ and $T\in\mathcal{K}(X_q)$ with $pP\cap qP=rP$,
\[S\vee T:=i_p^r(S) i_q^r(T) \in \mathcal{K}(X_r).\]
\begin{definition}\label{df.NC.rep} Let $X$ be a compactly aligned product system over a right LCM semigroup $P$. A representation $\psi$ is called \emph{Nica-covariant} if for all $p,q\in P$ and $S\in\mathcal{K}(X_p)$ and $T\in \mathcal{K}(X_q)$, we have,
\[\psi^{(p)}(S) \psi^{(q)}(T) = \begin{cases}
\psi^{(r)}(S \vee T) & \text{ if } pP\cap qP=rP, \\
0 & \text{ if } pP\cap qP=0.
\end{cases}
\]
\end{definition}
One can verify that this definition does not depend on the choice of $r$ (see \cite{BLS2018b}). The Nica-Toeplitz C*-algebra $\mathcal{N}\mathcal{T}_X$ can be then defined as the universal C*-algebra
generated by the Nica-covariant Toeplitz representations of the product system $X$.
\section{Zappa-Sz\'{e}p products of Product Systems by Groups}
We first introduce the notion of Zappa-Sz\'{e}p action of a group on a product system. This allows us to define the Zappa-Sz\'{e}p product of a product system by a group, which is a product system over
the given Zappa-Sz\'{e}p product semigroup. In the special case when the Zappa-Sz\'{e}p action is homogeneous, it turns out that we can construct another Zappa-Sz\'{e}p type product system,
which is a product system over the same semigroup.
In the remaining of this section, let $P\bowtie G$ be a Zappa-Sz\'{e}p product of a semigroup $P$ and a group $G$, $\mathcal{A}$ a unital C*-algebra, and $X=\bigsqcup_{p\in P} X_p$ a product system over $P$ with coefficient $\mathcal{A}$.
\subsection{Zappa-Sz\'{e}p actions}
We first need a key notion of the Zappa-Sz\'{e}p action of $G$ on $X$. The product system $X$ naturally defines a ``$G$-restriction map" on $X$ by inheriting the $G$-restriction map on $P$. However, one has to define a $G$-action map on the product system $X$ that mimics the $G$-action on the semigroup $P$.
\begin{definition}
\label{D:beta}
(i) Let $P\bowtie G$ be a Zappa-Sz\'{e}p product of a semigroups $P$ and a group $G$, and $X$ a product system over $P$.
A \textit{Zappa-Sz\'{e}p action of $G$ on $X$} is a collection of functions $\{\beta_g\}_{g\in G}$ that satisfies the following conditions:
\begin{enumerate}
\item[(A1)]\label{cond.A1} for each $p\in P$ and $g\in G$, $\beta_g: X_p \to X_{g\cdot p}$ is a $\mathbb{C}$-linear map;
\item[(A2)]\label{cond.A2} $\beta_g\circ\beta_h=\beta_{gh}$ for all $g,h\in G$;
\item[(A3)]\label{cond.A3} the map $\beta_{e}$ is the identity map;
\item[(A4)]\label{cond.A4} for each $g\in G$, the map $\beta_g$ is a $*$-automorphism on the C*-algebra $\mathcal{A}$;
\item[(A5)]\label{cond.A5} for each $g\in G$ and $p,\, q\in P$,
\[\beta_g(xy)=\beta_g(x)\beta_{g|_p}(y) \quad\text{for all}\quad x\in X_p, \, y\in X_q;\]
\item[(A6)]\label{cond.A6} for each $g\in G$ and $p\in P$,
\[
\langle\beta_g(x), \beta_g(y)\rangle=\beta_{g|_p}(\langle x, y\rangle)\quad\text{for all}\quad x,\, y\in X_p.
\]
\end{enumerate}
(ii) If $\beta$ is a Zappa-Sz\'{e}p action of $G$ on $X$, we call the triple $(X,G,\beta)$ a \textit{Zappa-Sz\'{e}p system}.
\end{definition}
\smallskip
Some remarks are in order.
\begin{rem}
\label{R:beta}
In Definition \ref{D:beta}, to be more precise, the collection $\{\beta_g\}_{g\in G}$ should be written as $\{\beta_g^p\}_{g\in G,p\in P}$, so that $\beta_g^p: X_p\to X_{g\cdot p}$.
But it is usually clear from the context to see which $X_p$ the map $\beta_g$ acts on. So, in order to simplify our notation, we just write $\beta_g$ instead of $\beta_g^p$.
\end{rem}
\begin{rem}
In the very special case when both the $G$-action and the $G$-restriction are trivial, that is, $g\cdot p=p$ and $g|_p=g$ for all $g\in G$ and $p\in P$, Definition \ref{D:beta} corresponds to the
sense of a group action on a product system in \cite{DOK20}. For a C*-correspondence $X$ (that is a product system over ${\mathbb{N}}$), our definition coincides with the group action considered in \cite[Definition 2.1]{HaoNg2008}.
\end{rem}
\begin{rem}
It follows from \hyperref[cond.A1]{(A1)}-\hyperref[cond.A3]{(A3)} that $\beta_g^p$ is a bijection between $X_p$ and $X_{g\cdot p}$. However, $\beta_g$ is not an $\mathcal{A}$-linear map between $\mathcal{A}$-correspondences. In fact, the condition \hyperref[cond.A5]{(A5)} forces $\beta_g(xa)=\beta_g(x)\beta_{g|_p}(a)$ and $\beta_g(ax)=\beta_g(a) \beta_g(x)$ for $a\in \mathcal{A}$ and $x\in X_p$.
\end{rem}
\subsection{The Zappa-Sz\'{e}p product system $X\bowtie G$ of $(X,G,\beta)$}
Let $(X,G,\beta)$ be a Zappa-Sz\'{e}p system as defined in Definition \ref{D:beta}.
For each $p\in P$ and $g\in G$, set
\[
Y_{(p,g)}:=\{x\otimes g: x\in X_p\}.
\]
Define the left and right actions of $\mathcal{A}$ on $Y_{(p,g)}$ by
\[
a\cdot (x\otimes g)=(a\cdot x)\otimes g,\ (x\otimes g)\cdot a = x\beta_g(a) \otimes g,
\]
respectively, and an $\mathcal{A}$-valued inner product to be
\[
\langle x\otimes g, y\otimes g\rangle = \beta_{g^{-1}}(\langle x,y\rangle).
\]
\begin{proposition}
With the notation same as above, $Y_{(p,g)}$ is an $\mathcal{A}$-correspondence.
\end{proposition}
\begin{proof} We first verify that $Y_{(p,g)}$ is a right $\mathcal{A}$-module.
The inner product is $\mathcal{A}$-linear in the second component: take any $a\in \mathcal{A}$, and $x,y\in X_p$,
\begin{align*}
\langle x\otimes g, (y\otimes g) a\rangle &= \langle x\otimes g, y\beta_g(a)\otimes g\rangle \\
&= \beta_{g^{-1}}(\langle x, y\beta_g(a)\rangle) \\
&= \beta_{g^{-1}}(\langle x, y\rangle \beta_g(a)) \\
&= \beta_{g^{-1}}(\langle x, y\rangle) a \\
&= \langle x\otimes g, y\otimes g\rangle a.
\end{align*}
Since $\beta_g$ is a $*$-automorphism on $\mathcal{A}$, one has
\begin{align*}
\langle x\otimes g, y\otimes g\rangle^* &= \beta_{g^{-1}}(\langle x,y\rangle)^* \\
&= \beta_{g^{-1}}(\langle x,y\rangle^*) \\
&= \beta_{g^{-1}}(\langle y,x\rangle) \\
&= \langle y\otimes g, x\otimes g\rangle.
\end{align*}
Finally, $\beta_g$ is also a positive map on $\mathcal{A}$ because it is a $*$-automorphism. Hence
\[
\langle x\otimes g, x\otimes g\rangle = \beta_{g^{-1}}(\langle x,x\rangle)^* \geq 0.\]
Moreover, since $\beta_g$ is isometric on $\mathcal{A}$,
\[\|x\otimes g\|=\|\langle x\otimes g, x\otimes g\rangle\|^{1/2}=\|\beta_g^{-1}(\langle x, x\rangle)\|^{1/2}=\|\langle x,x\rangle\|^{1/2}=\|x\|.\]
Thus the norm on $Y_{(p,g)}$ is the same as that on $X_p$. Because $X_p$ is complete under its norm, so is $Y_{(p,g)}$. Therefore $Y_{(p,g)}$ is a Hilbert $\mathcal{A}$-module. One can clearly see that the left action of $\mathcal{A}$ induced from $X_p$ is a left action of $\mathcal{A}$ on $Y_{p,g}$. Therefore, $Y_{p,g}$ is an $\mathcal{A}$-correspondence.
\end{proof}
Let
\[
Y:=\bigsqcup_{(p,g)\in P\bowtie G} Y_{(p,g)}.
\]
For each $p,q\in P$ and $g,h\in G$, define a multiplication map $Y_{(p,g)}\times Y_{(q,h)}\to Y_{(p(g\cdot q), g|_q h)}$ by
\begin{align}
\label{E:M}
((x\otimes g), (y\otimes h))\mapsto (x\beta_g(y))\otimes (g|_q)h.
\end{align}
This extends to a map $M_{(p,g),(q,h)}:Y_{(p,g)}\otimes Y_{(q,h)}\to Y_{(p(g\cdot q), g|_q h)}$.
\begin{theorem}
\label{T:ZSP}
With the multiplication maps $M_{(p,g),(q,h)}$, $Y$ is a product system over the Zappa-Sz\'{e}p product $P\bowtie G$.
\end{theorem}
\begin{proof} First of all, the identity of $P\bowtie G$ is $(e, e)$, and one can easily check that $Y_{(e, e)}\cong \mathcal{A}$ by identifying $a\otimes e\in Y_{(e,e)}$ with $a\in \mathcal{A}$. For any $a\in \mathcal{A}$ (that is $a\otimes e\in Y_{(e,e)}$), and the left and right $\mathcal{A}$-actions on $Y_{(p,g)}$ are implemented by the multiplications:
\begin{align*}
a(x\otimes g) &= ax\otimes g = (a\otimes e)(x\otimes g), \\
(x\otimes g)a &= x\beta_g(a) \otimes g = (x\otimes g)(a\otimes e).
\end{align*}
By \condref{A1}, $\beta_g$ is a ${\mathbb{C}}$-linear isomorphism from $X_p$ to $X_{g\cdot p}$. Therefore, if $a\in X_p$ and $b\in X_q$,
\[a\beta_g(b)\in X_p X_{g\cdot q}\in X_{p(g\cdot q)}.\]
One can easily see that $M_{(p,g),(q,h)}$ is an $\mathcal{A}$-linear map. To show that it is unitary,
for any $x_p, u_p\in X_p$ and $y_q, v_q\in X_q$,
\begin{align*}
&\langle (x_p\otimes g)\otimes (y_q\otimes h), (u_p\otimes g)\otimes (v_q\otimes h)\rangle \\
=&
\langle (y_q\otimes h), \langle(x_p\otimes g), (u_p\otimes g)\rangle (v_q\otimes h)\rangle \\
=& \langle (y_q\otimes h), \beta_g^{-1}(\langle x_p,u_p\rangle) (v_q\otimes h)\rangle \\
=& \beta_h^{-1}(\langle y_q, \beta_g^{-1}(\langle x_p,u_p\rangle) v_q\rangle).
\end{align*}
On the other hand,
\begin{align*}
&\langle M_{(p,g),(q,h)}((x_p\otimes g)\otimes (y_q\otimes h)), M_{(p,g),(q,h)}((u_p\otimes g)\otimes (v_q\otimes h))\rangle \\
=&
\langle x_p\beta_g(y_q)\otimes g|_q h, u_p\beta_g(v_q)\otimes g|_p h\rangle \\
=& \beta_{h^{-1}}\beta_{(g|_p)^{-1}}(\langle x_p\beta_g(y_q), u_p\beta_g(v_q) \rangle) \\
=& \beta_{h^{-1}}\beta_{(g|_p)^{-1}}(\langle \beta_g(y_q), \langle x_p, u_p\rangle \beta_g(v_q) \rangle) \ (\text{as }M_{p,g\cdot q}\text{ is unitary}) \\
=& \beta_{h^{-1}}\beta_{(g|_p)^{-1}}(\langle \beta_g(y_q), \beta_g(\beta_g^{-1}(\langle x_p, u_p\rangle) v_q) \rangle) \\
=& \beta_h^{-1}(\langle y_q, \beta_g^{-1}(\langle x_p,u_p\rangle) v_q\rangle )\ (\text{by (A6)}).
\end{align*}
Therefore, $M_{(p,g),(q,h)}$ preserves the inner product and is thus unitary.
Finally, for $a\in X_p$, $b\in X_q$, $c\in X_r$ and $g,h,k\in G$, we compute that
\[((a\otimes g)(b\otimes h))(c\otimes k) = (a\beta_g(b)\beta_{g|_q h}(c))\otimes (g|_q h)|_r k\]
and
\[(a\otimes g)((b\otimes h)(c\otimes k)) = (a\beta_g(b\beta_h(c)))\otimes g|_{q(h\cdot r)} h|_r k.\]
From \condref{A5} one has
\[a\beta_g(b\beta_h(c)) = a\beta_g(b) \beta_{g|_q}(\beta_h(c)) = a\beta_g(b)\beta_{g|_q h}(c).\]
Also \condref{ZS8} and \condref{ZS6} yield
\[(g|_q h)|_r k = (g|_q)|_{h\cdot r} h|_r k = g|_{q(h\cdot r)} h|_r k.\]
Hence the multiplication is associative. Therefore $Y=(Y_{(p,g)})_{(p,g)\in P\bowtie G}$ is a product system over the Zappa-Sz\'{e}p semigroup $P\bowtie G$.
\end{proof}
\begin{defn}
The new product system $Y$ constructed from $(X,G,\beta)$ in Theorem \ref{T:ZSP} is called the \textit{Zappa-Sz\'{e}p product of $X$ by $G$} and denoted as $X\bowtie G$.
\end{defn}
\subsection{Another Zappa-Sz\'{e}p product $X\widetilde\bowtie G$ of a homogeneous Zappa-Sz\'{e}p system}
In this subsection, we study a special class of Zappa-Sz\'{e}p actions of groups on product systems -- homogeneous Zappa-Sz\'{e}p actions. Given such a Zappa-Sz\'{e}p action, it turns out that
\textit{another} new natural and interesting product system $X\widetilde\bowtie G$ can be constructed from the given action. Unlike $X\bowtie G$ that enlarges the grading semigroup and keeps the coefficient C*-algebra the same,
this new one, $X\widetilde\bowtie G$, enlarges the coefficient C*-algebra and keeps
the grading semigroup the same.
\label{S:homog}
\begin{definition}
Let $P\bowtie G$ be a Zappa-Sz\'{e}p product of a semigroups $P$ and a group $G$, and $X$ a product system over $P$.
A Zappa-Sz\'{e}p action $\beta$ on $X$ is called \textit{homogeneous} if $g\cdot p=p$ for any $p\in P$ and $g\in G$.
In this case, the Zappa-Sz\'{e}p system $(X,G,\beta)$ is also said to be \textit{homogeneous}.
\end{definition}
\begin{rem}
Homogeneous Zappa-Sz\'{e}p actions are a natural generalization of usual generalized gauge actions \cite{K2017}.
\end{rem}
In the case of a homogeneous Zappa-Sz\'{e}p system $(X,G,\beta)$, one can easily see that $\beta$
induces an automorphic action $\beta:G\to \operatorname{Aut}(X_p)$. This allows us to construct a new crossed product type product system over the same semigroup $P$ that encodes the Zappa-Sz\'{e}p structure.
In particular, when $p= e$, we obtain a C*-dynamical system $(\mathcal{A},G,\beta)$. Let $\mathfrak{A}=\mathcal{A}\rtimes_\beta G$ be the universal C*-crossed product of this C*-dynamical system. So $\mathfrak{A}=C^*(a, u_g: a\in \mathcal{A}, g\in G)$, where $\{a,u_g: a\in \mathfrak{A}, g\in G\}$ is the generator set of $\mathfrak{A}$.
Thus the generators satisfy the covariance condition
\[u_g a = \beta_g(a) u_g\quad\text{for all}\quad a\in \mathcal{A}\text{ and }g\in G.\]
For each $p\in P$, consider $Z_p^0=c_{00}(G,X_p)=\operatorname{span}\{x\otimes g: x\in X_p\}$. We can put an $\mathfrak{A}$-bimodule structure on $Z_p^0$: for any $a u_h\in \mathfrak{A}$ and $\xi=x_p\otimes g\in c_{00}(G,X_p)$,
\begin{align*}
(au_h)\xi = (a\beta_h(x_p))\otimes h|_p g \quad \text{ and }\quad
\xi (au_h) = (x_p\beta_g(a)) \otimes gh.
\end{align*}
Define an $\mathfrak{A}$-valued function $\langle\cdot,\cdot\rangle: Z_p^0\times Z_p^0\to \mathfrak{A}$ by setting, for any $x_p\otimes g, y_p\otimes h\in Z_p^0$,
\[\langle x_p \otimes g, y_p \otimes h\rangle=\beta_{g^{-1}}(\langle x_p, y_p\rangle) u_{g^{-1}h}.\]
By the covariance relation on $\mathfrak{A}$, one can rewrite the above identity as
\[\langle x_p \otimes g, y_p \otimes h\rangle=u_{g^{-1}} \langle x_p, y_p\rangle u_{h}.\]
\begin{proposition} The space $Z_p^0$ together with the map $\langle\cdot,\cdot\rangle$ is an inner product right $\mathfrak{A}$-module.
\end{proposition}
\begin{proof}
It is easy to see that $\langle\cdot,\cdot\rangle$ is right $\mathfrak{A}$-linear in the second variable:
take any $au_k\in \mathfrak{A}$ and $x_p\otimes g, y_p\otimes h\in Z_p^0$,
\begin{align*}
\langle x_p \otimes g, (y_p \otimes h) a u_k)\rangle &= \langle x_p \otimes g, y_p\beta_h(a)\otimes hk\rangle \\
&= \beta_{g^{-1}}(\langle x_p, y_p\beta_h(a)\rangle) u_{g^{-1} hk} \\
&= \beta_{g^{-1}}(\langle x_p, y_p\rangle) \beta_{g^{-1}h}(a) u_{g^{-1} h} u_k \\
&= \beta_{g^{-1}}(\langle x_p, y_p\rangle) u_{g^{-1} h} a u_k \\
&= \langle x_p \otimes g, y_p \otimes h\rangle a u_k
\end{align*}
Also, for any $x_p\otimes g, y_p\otimes h\in Z_p^0$,
\begin{align*}
\langle x_p \otimes g, y_p \otimes h\rangle^* &= \left(u_g^* \langle x_p, y_p\rangle u_h\right)^* \\
&= u_h^* \langle y_p, x_p\rangle u_g \\
&= \langle y_p \otimes h, x_p \otimes g\rangle.
\end{align*}
Finally, for any $x_1,\ldots,x_n\in X_p$ and $g_1,\ldots,g_n\in G$, considier $\xi=\sum_{i=1}^n x_i\otimes g_i\in Z_p^0$. We have that
\[\langle \xi, \xi\rangle = \sum_{i=1}^n \sum_{j=1}^n u_{g_i^{-1}} \langle x_i, x_j\rangle u_{g_j}. \]
Consider the $n\times n$ operator matrix $K=[\langle x_i, x_j\rangle]$. We first claim that $A\geq 0$ as an operator in $M_n(\mathcal{A})$, which is equivalent of showing \cite[Proposition 6.1]{Paschke1973} that for any $a_1,\ldots, a_n\in\mathcal{A}$,
\[\sum_{i=1}^n \sum_{j=1}^n a_i^* \langle x_i, x_j\rangle a_j \geq 0.\]
Since $X_p$ is an $\mathcal{A}$-correspondence, $a_i^* \langle x_i, x_j\rangle a_j=\langle x_i a_i, x_j a_j\rangle$. Therefore,
\[\sum_{i=1}^n \sum_{j=1}^n a_i^* \langle x_i, x_j\rangle a_j=\sum_{i=1}^n \sum_{j=1}^n\langle x_i a_i, x_j a_j\rangle=\langle \sum_{i=1}^n x_i a_i, \sum_{j=1}^n x_j a_j\rangle\geq 0.\]
This proves that the operator matrix $K=[\langle x_i, x_j\rangle]\geq 0$. Since $\mathcal{A}$ embeds injectively inside the crossed product $\mathfrak{A}=\mathcal{A}\rtimes_\beta G$, the operator matrix $K\geq 0$ as an operator in $M_n(\mathfrak{A})$. Therefore,
\[\langle \xi, \xi\rangle = \sum_{i=1}^n \sum_{j=1}^n u_{g_i^{-1}} \langle x_i, x_j\rangle u_{g_j}\geq 0. \]
Suppose that $\langle \xi,\xi\rangle=0$. We have
\[\langle \xi, \xi\rangle = \sum_{i=1}^n \sum_{j=1}^n \beta_{g_i^{-1}}( \langle x_i, x_j\rangle) u_{g_i^{-1} g_j} = 0.\]
Since there exists a contractive conditional expectation $\Phi:\mathfrak{A}\to\mathcal{A}$ by $\Phi(\sum a_g u_g)=a_e$, we have that
\[\sum_{i=1}^n \beta_{g_i^{-1}}( \langle x_i, x_i\rangle)=0.\]
Since $\beta_{g_i^{-1}}$'s are $*$-automorphisms of $\mathcal{A}$, we have that $\langle x_i, x_i\rangle =0$ for all $i$, and thus $x_i=0$ for all $i$. So we obtain that $\xi=0$.
Therefore $\langle\cdot,\cdot\rangle$ is an $\mathfrak{A}$-valued inner product on $Z_p^0$.
\end{proof}
Now let $Z_p$ be the completion of $Z_p^0$ under the norm $\|\xi\|=\|\langle \xi, \xi\rangle\|^{1/2}$. We obtain an $\mathfrak{A}$-correspondence $Z_p$.
\begin{theorem}
\label{T:prop.Z}
The collection $Z=\bigsqcup_{p\in P}Z_p$ is a product system over $P$, where the multiplication $Z_p\times Z_q\to Z_{pq}$ is given by
\[(x_p\otimes g,y_q \otimes h)\mapsto x_p \beta_g(y_q)\otimes g|_q h \quad (g,h\in G, x_p\in X_p, y_q\in X_q).\]
\end{theorem}
\begin{proof} Let $p, q\in P$. For $x_p\in X_p$ and $y_q\in X_q$, one has $\beta_g(y_q)\in X_q$ and thus $x_p \beta_g(y_q)\in X_{pq}$. Since $\beta_g$ is automorphic as mentioned above,
the multiplication is surjective. To see the multiplication induces a unitary map from $Z_p\otimes Z_q\to Z_{pq}$, take any four elementary tensors $x_p\otimes g, w_p\otimes i\in Z_p$ and $y_q\otimes h, z_q\otimes k\in Z_q$,
\begin{align*}
&\langle (x_p\otimes g)\otimes (y_q \otimes h), (w_p\otimes i)\otimes (z_q \otimes k)\rangle \\
=& \langle y_q \otimes h, \langle x_p\otimes g, w_p\otimes i\rangle (z_q \otimes k)\rangle \\
=&
\langle y_q \otimes h, \beta_{g^{-1}}(\langle x_p, w_p\rangle) u_{g^{-1}i} (z_q \otimes k)\rangle \\
=& \langle y_q \otimes h, \beta_{g^{-1}}(\langle x_p, w_p\rangle) \beta_{g^{-1}i}(z_q) \otimes (g^{-1}i)|_q k)\rangle \\
=& u_h^* \langle y_q, \beta_{g^{-1}}(\langle x_p, w_p\rangle) \beta_{g^{-1}i}(z_q)\rangle u_{ (g^{-1}i)|_q k}.
\end{align*}
On the other hand,
\begin{align*}
&\langle (x_p\otimes g) (y_q \otimes h), (w_p\otimes i) (z_q \otimes k)\rangle \\
=& \langle x_p \beta_g(y_q)\otimes g|_q h, w_p\beta_i(z_q)\otimes i|_q k\rangle \\
=& u_{g|_q h}^* \langle x_p \beta_g(y_q), w_p\beta_i(z_q)\rangle u_{i|_q k} \\
=& u_h^* u_{(g|_q)^{-1}} \langle \beta_g(y_q), \langle x_p, w_p\rangle \beta_i(z_q)\rangle u_{i|_q k} \\
=& u_h^* u_{(g|_q)^{-1}} \langle \beta_g(y_q), \beta_g(\beta_{g^{-1}}(\langle x_p, w_p\rangle) \beta_{g^{-1}i}(z_q))\rangle u_{i|_q k} \\
=& u_h^* u_{(g|_q)^{-1}} \beta_{g|_q} (\langle y_q, \beta_{g^{-1}}(\langle x_p, w_p\rangle) \beta_{g^{-1}i}(z_q)\rangle) u_{i|_q k} \\
=& u_h^* \langle y_q, \beta_{g^{-1}}(\langle x_p, w_p\rangle) \beta_{g^{-1}i}(z_q)\rangle u_{(g|_q)^{-1}} u_{i|_q k} \\
=& u_h^* \langle y_q, \beta_{g^{-1}}(\langle x_p, w_p\rangle) \beta_{g^{-1}i}(z_q)\rangle u_{g^{-1}|_{g\cdot q} i|_q k}.
\end{align*}
Because $g\cdot q=i\cdot q$,
\[u_{g^{-1}|_{g\cdot q} i|_q k}=u_{g^{-1}|_{i\cdot q} i|_q k}=u_{(g^{-1}i)|_q k}.\]
Therefore, the multiplication is indeed unitary.
The C*-algebra $Z_{e}$ can be identified as $\mathfrak{A}$ via $a\otimes g\mapsto au_g$. It is routine to verify that the left and right $\mathfrak{A}$-action on $Z_p$ are implemented by the multiplication map by $Z_{e}$. To see the associativity of the multiplication, take $x\otimes g\in Z_p$, $y\otimes h\in Z_q$, and $z\otimes k\in Z_r$. We have that
\[ \left((x\otimes g)(y\otimes h)\right)(z\otimes k) = x\beta_g(y)\beta_{g|_q h}(z)\otimes (g|_q h)|_r k\]
and
\[ (x\otimes g)\left((y\otimes h)(z\otimes k)\right) = x\beta_g(y\beta_h(z))\otimes g|_{qr} h|_r k.\]
Condition (A5) yields $\beta_g(y\beta_h(z))=\beta_g(y)\beta_{g|_q h}(z)$. From (ZS6), (ZS8), and the homogeneity assumption that $h\cdot r=r$,
we have \[(g|_q h)|_r k=(g|_q)|_{h\cdot r} h|_r k=g|_{qr} h|_r k.\]
This proves the associativity of the multiplication.
\end{proof}
\begin{definition}
The product system $Z$ obtained in Theorem \ref{T:prop.Z} is called the \textit{homogeneous Zappa-Sz\'{e}p product of $X$ by $G$} and denoted as $X\widetilde\bowtie G$.
\end{definition}
In summary, for a given \textsf{homogeneous} Zappa-Sz\'{e}p action $\beta$ of $G$ on $X$, one has two new product systems: (i) $X\bowtie G$ -- a product system over $P\bowtie G$ with coefficient C*-algebra $\mathcal{A}$,
and (ii) $X\widetilde\bowtie G$ -- a product system over $P$ with coefficient C*-algebra $\mathcal{A}\rtimes_\beta G$.
\section{C*-algebras associated to Zappa-Sz\'{e}p actions \\ and some Hao-Ng isomorphism theorems}
\label{S:main}
In this section, we study covariant representations and their associated universal C*-algebras arising from a Zappa-Sz\'{e}p system $(X,G,\beta)$.
We establish a one-to-one correspondence between the set of all covariant representations $(\psi, U)$ of $(X,G,\beta)$ and the set of all (Toeplitz) representations $\Psi$
of the Zappa-Sz\'{e}p product system $X\bowtie G$. Furthermore, it is proved that $\psi$ is Cuntz-Pimsner covariant if and only if so is $\Psi$. If $P$ is right LCM and $X$ is compactly
aligned, then $\psi$ is Nica-covariant if and only if so is $\Psi$. As a consequence, we obtain several Hao-Ng isomorphism theorems for the associated C*-algebras.
However, as we shall see, changes arise for the homogeneous Zappa-Sz\'{e}p product system $X\widetilde\bowtie G$.
\subsection{Covariant representations of $(X,G,\beta)$ and the C*-algebra ${\mathcal{T}}_X\bowtie G$}
Let $P\bowtie G$ be a Zappa-Sz\'{e}p product of a semigroup $P$ and a group $G$, ${\mathcal{A}}$ a unital C*-algebra, and $X$ a product system over $P$ with coefficient ${\mathcal{A}}$. Suppose that $\beta$ is an arbitrary Zappa-Sz\'{e}p action of $G$ on $X$.
\begin{defn}
\label{D:vrep}
Let $\psi$ be a representation of $X$ and $U$ is unitary representation of $G$ on a unital C*-algebra $\mathcal{B}$. A pair $(\psi, U)$ is called a \textit{covariant representation of a Zappa-Sz\'{e}p system $(X,G,\beta)$} if
\begin{align}
\label{E:Upsi}
U_g \psi_p(x)=\psi_{g\cdot p}(\beta_g(x)) U_{g|_p} \text{ for all } p\in P, g\in G, x\in X_p.
\end{align}
\end{defn}
\begin{theorem}
\label{T:Upsi}
There is a one-to-one correspondence $\Pi$ between the set of all representations $\Psi$ of $X\bowtie G$ on a unital C*-algebra $\mathcal{B}$
and the set of covariant representations $(\psi, U)$ of $(X,G,\beta)$.
In fact, for a given representation $\Psi$ of $X\bowtie G$, one has
\begin{align*}
\psi_p(x)&:=\Psi_{(p,e)}(x\otimes e)\quad\text{for all}\quad p\in P, x\in X_p,\\
U_g&:=\Psi_{(e, g)}(1_\mathcal{A}\otimes g)\quad\text{for all}\quad g\in G.
\end{align*}
Conversely, given a covariant representation $(\psi, U)$ of $(X,G,\beta)$, one has
\begin{align}
\label{E:DefPsi}
\Psi_{(p, g)}(x\otimes g) := \psi_p(x) U_g\quad\text{for all}\quad p\in P, g\in G, x\in X_p.
\end{align}
\end{theorem}
\begin{proof} Let $Y:=X\bowtie G$ and $\Psi$ be a representation of $Y$ on a unital C*-algebra $\mathcal{B}$.
Since $\Psi$ is a representation of $Y$, in particular we have
\begin{align*}
\Psi_{(p,g)}(x\otimes g)\Psi_{(q,h)}(y\otimes h)=\Psi_{(p(g\cdot q), g|_qh)}(x\beta_g(y)\otimes g|_qh)
\end{align*}
for all $g, h\in G$, $p,q\in P$, $x\in X_p$ and $y\in X_q$.
For $p\in P$, define $\psi_P: X_p \to \mathcal{B}$ via
\begin{align*}
\psi_p(x):=\Psi_{(p,e)}(x\otimes e)\quad\text{for all}\quad x\in X_p.
\end{align*}
Then
\begin{align*}
\psi_p(x)^*\psi_p(y)
&=\Psi_{(p,e)}(x\otimes e)^*\Psi_{(p\otimes e)}(y\otimes e)\\
&=\Psi_{(e,e)}(\langle (x\otimes e), (y\otimes e)\rangle\\
&=\Psi_{(e,e)}(\langle x, y\rangle\otimes e)\\
&=\psi_{e}(\langle x, y\rangle)\ (x, y\in X_p)
\end{align*}
and
\begin{align*}
\psi_p(x)\psi_q(y)
&=\Psi_{(p,e)}(x\otimes e)\Psi_{(q,e)}(y\otimes e)\\
&=\Psi_{(p(e\cdot q), e|_q e)}((x\otimes e)(y\otimes e))\\
&=\Psi(pq,e)(x\beta_{e}(y)\otimes e|_q e)\\
&=\Psi(pq,e)(xy\otimes e)\\
&=\psi_{pq}(xy)\ (x\in X_p, y\in X_q).
\end{align*}
So $\psi$ is a representation of $X$ on $\mathcal{B}$.
One can easily check that $U_g:=\Psi_{(e, g)}(1_\mathcal{A}\otimes g)$ is a unitary in $\mathcal{B}$ with inverse $\Psi_{(e, g^{-1})}(1_\mathcal{A}\otimes g^{-1})$, and that
\[U_g U_h=\Psi_{(e, g)}(1_\mathcal{A}\otimes g) \Psi_{(e, h)}(1_\mathcal{A}\otimes h)= \Psi_{(e, gh)}(1_\mathcal{A}\otimes gh)=U_{gh}.\] That is,
$U$ is a unitary representation of $G$ in $\mathcal{B}$.
From the definitions of $\psi_p$ and $U_g$, for any $g\in G$ and $p\in P$, we obtain
\begin{align*}
U_g \psi_p(x) &= \Psi_{(e,g)}(1_\mathcal{A}\otimes g) \Psi_{(p,e)}(x\otimes e) \\
&= \Psi_{(e,g)(p,e)}((1_\mathcal{A}\otimes g)(x\otimes e)) \\
&= \Psi_{(g\cdot p,g|_p)}(\beta_g(x)\otimes g|_p) \\
&= \Psi_{(g\cdot p,e)(e,g|_p)}((\beta_g(x)\otimes e)(1_\mathcal{A}\otimes g|_p)) \\
&= \Psi_{(g\cdot p,e)}(\beta_g(x)\otimes e) \Psi_{(e,g|_p)}(1_\mathcal{A}\otimes g|_p) \\
&= \psi_{g\cdot p}(\beta_g(x)) U_{g|_p}.
\end{align*}
Thus $(\psi, U)$ satisfies the idenity \eqref{E:Upsi}.
Conversely, given a representation $\psi$ of $X$ and a unitary representation $U$ of $G$ on a unital C*-algebra ${\mathcal{B}}$ that satisfy \eqref{E:Upsi},
define
\[
\Psi_{(p, g)}(x\otimes g) := \psi_p(x) U_g\quad\text{for all}\quad p\in P, g\in G, x\in X_p.
\]
Then one can verify that $\Psi$ is a representation of $Y$. In fact, we have
\begin{align*}
\Psi_{(p, g)}(x\otimes g) ^*\Psi_{(p, g)}(y\otimes g)
&= U_g^*\psi_p(x)^* \psi_p(y) U_g\ (\text{by the definition of } \Psi)\\
&= U_{g^{-1}} \psi_{e}(\langle x, y\rangle) U_g \ (\text{as }\psi \text{ is a representation of } X)\\
&=\psi_{e}(\beta_{g^{-1}}\langle x, y\rangle) U_{g^{-1}|_{e}} U_g \\
&=\psi_{e}(\beta_{g^{-1}}\langle x, y\rangle)\ (\text{as }g^{-1}|_{e}=g^{-1}, U_{g^{-1}|_{e}} U_g=U_{g^{-1}}U_g=I) \\
&=\Psi_{(e,e)}(\beta_{g^{-1}}(\langle x, y\rangle)\otimes e) \\
&=\Psi_{(e,e)}(\langle x\otimes g, y\otimes g\rangle)\ (\text{by the definition of }Y)
\end{align*}
for all $p\in P$, $g\in G$, $x,y\in X_p$, and
\begin{align*}
\Psi_{(p, g)}(x\otimes g) \Psi_{(q, h)}(y\otimes h)
&=\psi_p(x) U_g\psi_q(y) U_h \ (\text{by definition of } \Psi)\\
&=\psi_p(x) \psi_{g\cdot q}(\beta_g(y)) U_{g|_q} U_h\\
&=\psi_{pg\cdot q}(x\beta_g(y)) U_{g|_q h}\ (\text{as }\psi \text{ is a representation of } X)\\
&=\Psi_{(pg\cdot q, g|_qh)}(x\beta_g(y)\otimes g|_q h)\ (\text{by the definition of } \Psi)\\
&=\Psi_{(pg\cdot q, g|_qh)}((x\otimes g)(y\otimes h))
\end{align*}
for all $p,q\in P$, $g\in G$, $x\in X_p$ and $y\in X_q$.
\end{proof}
\begin{eg}
\label{Eg:Fock}
Let $X$ be a product system over a semigroup $P$. Suppose that there is a Zappa-Sz\'{e}p action $\beta$ of $G$ on $X$.
There is a natural nontrivial pair $(\psi, U)$ which satisfies all the conditions required in Theorem \ref{T:Upsi}.
Indeed, let $F(X)=\bigoplus_{s\in P} X_s$ the Fock space, and $L$ the usual Fock representation:
\begin{align*}
L_p(x)(\oplus x_s) =\oplus (x\otimes x_s)\quad (x\in X_p, \ \oplus x_s\in F(X)).
\end{align*}
Then we define an action $\tilde\beta$ of $G$ on $F(X)$ as follows:
\[
\tilde\beta_g(\oplus x_s)=\oplus (\beta_g(x_s))\quad (g\in G, \ \oplus x_s\in F(X)).
\]
Clearly, $L$ is a representation of $X$ and $\tilde\beta$ is a unitary representation of $G$. Also one can easily verify
\begin{align*}
\tilde\beta_g\circ L_p(x)=L_{g\cdot p}(\beta_g(x))\circ \tilde\beta_{g|_p}\quad (g\in G, x\in X_p).
\end{align*}
\end{eg}
We are now ready to define the Toeplitz type universal C*-algebra associated to a Zappa-Sz\'{e}p system $(X,G,\beta)$.
\begin{definition}
\label{D:TXG}
Let $\mathcal{T}_X\bowtie G$ be the universal C*-algebra generated by covariant representations of $(X,G,\beta)$.
\end{definition}
Example \ref{Eg:Fock} shows that the C*-algebra $\mathcal{T}_X\bowtie G$ is nontrivial.
\medskip
In what follows, we prove a result similar to Theorem \ref{T:Upsi} for the homogeneous Zappa-Sz\'{e}p product system $X\widetilde\bowtie G$.
\begin{theorem}
\label{T:Upsi.homo}
Suppose that a Zappa-Sz\'{e}p system $(X,G,\beta)$ is homogeneous.
Then there is a one-to-one correspondence $\widetilde\Pi$ between the set of all representations $\Psi$ of $X\widetilde\bowtie G$ on a unital C*-algebra $\mathcal{B}$
and the set of all covariant representations $(\psi, U)$ of $(X,G,\beta)$.
In fact, for a given representation $\Psi$ of $X\widetilde\bowtie G$, one has
\begin{align*}
\psi_p(x)&:=\Psi_{p}(x\otimes e)\quad\text{for all}\quad p\in P, x\in X_p,\\
U_g&:=\Psi_{e}(1_\mathcal{A}\otimes g)\quad\text{for all}\quad g\in G.
\end{align*}
Conversely, given a covariant representation $(\psi, U)$ of $X\widetilde\bowtie G$, one has
\[
\Psi_{p}(x\otimes g) := \psi_p(x) U_g\quad\text{for all}\quad p\in P, g\in G, x\in X_p.
\]
\end{theorem}
\begin{proof}
For simplicity, set $Z:=X\widetilde\bowtie G$.
Let $\Psi$ be a representation of $Z$ on $\mathcal{B}$. For all $p,q\in P$, $g,h\in G$, and $x\in X_p$, $y\in X_q$,
\[\Psi_p(x\otimes g)\Psi_q(y\otimes h)=\Psi_{pq}(x\beta_g(y)\otimes g|_q h).\]
For $p\in P$, define $\psi_p:X_p\to \mathcal{B}$ by
\[\psi_p(x):=\Psi_p(x\otimes e)\ \text{for all }x\in X_p.\]
Here, for $p=e$, we treat $x\otimes e=xu_{e}\in \mathcal{A}\rtimes_\beta G\cong Z_{e}$.
Then,
\begin{align*}
\psi_p(x)^* \psi_p(y) &= \Psi_p(x\otimes e)^* \Psi_p(y\otimes e) \\
&= \Psi_{e}(\langle x\otimes e, y\otimes e\rangle) \\
&= \Psi_{e}(\langle x,y\rangle u_{e})\\
&=\psi_{e}(\langle x,y\rangle) \\
\psi_p(x)\psi_q(y) &= \Psi_p(x\otimes e) \Psi_q(y\otimes e) \\\
&= \Psi_{pq}(x\beta_{e}(y)\otimes e|_q e) \\
&= \Psi_{pq}(xy\otimes e) \\
&=\psi_{pq}(xy).
\end{align*}
Therefore, $\psi$ is a representation of $X$ on $\mathcal{B}$.
For each $g\in G$, set $U_g:=\Psi_{e}(1_\mathcal{A}\otimes g)$. As before, one can easily check that $U$ is a unitary representation of $G$ on $\mathcal{B}$.
For any $g\in G$, $p\in P$ and $x\in X_p$,
\begin{align*}
U_g \psi_p(x) &= \Psi_{e}(1_\mathcal{A}\otimes g) \Psi_p(x\otimes e) \\
&= \Psi_p(\beta_g(x)\otimes g|_p) \\
&= \Psi_p((\beta_g(x)\otimes e)(1_\mathcal{A}\otimes g|_p)) \\
&= \Psi_p(\beta_g(x)\otimes e)\Psi_{e}(1_\mathcal{A}\otimes g|_p) \\
&= \psi_p(\beta_g(x)) U_{g|_p}.
\end{align*}
Conversely, given a representation $\psi$ of $X$ and a unitary representation $U$ of $G$ that satisfy $U_g\psi_p(x)=\psi_p(\beta_g(x)) U_{g|_p}$, define
\[
\Psi_p(x\otimes g):=\psi_p(x) U_g\quad\text{for all}\quad g\in G, p\in P, x\in X_p.
\]
For any $x,y\in X_p$ and $g,h\in G$,
\begin{align*}
\Psi_p(x\otimes g)^* \Psi_p(y\otimes h) &= U_g^* \psi_p^*(x) \psi_p(y) U_h \\
&= U_g^* \psi_{e}(\langle x,y\rangle) U_h \\
&= \psi_{e}(\beta_{g^{-1}}(\langle x,y\rangle)) U_{g^{-1}h} \\
&= \Psi_{e}(\beta_{g^{-1}}(\langle x,y\rangle)\otimes g^{-1} h) \\
&= \Psi_{e}(\langle x\otimes g, y\otimes h\rangle).
\end{align*}
For any $x\in X_p$, $y\in X_q$, and $g,h\in G$,
\begin{align*}
\Psi_p(x\otimes g) \Psi_q(y\otimes h) &= \psi_p(x) U_g \psi_q(y) U_h \\
&= \psi_p(x) \psi_q(\beta_g(y)) U_{g|_q} U_h \\
&= \psi_{pq}(x\beta_g(y)) U_{g|_q h} \\
&= \Psi_{pq}(x\beta_g(y)\otimes g|_q h) \\
&= \Psi_{pq}((x\otimes g)(y\otimes h)).
\end{align*}
Therefore, $\Psi$ is a representation of $Z$.
\end{proof}
As an immediate corollary of Theorems \ref{T:Upsi} and \ref{T:Upsi.homo}, the universal $C^*$-algebras of representations of $X\bowtie G$
and covariant representations of $(X,G,\beta)$ must coincide, and similar for $X\widetilde\bowtie G$ if $(X,G,\beta)$ is also homogeneous.
Therefore, we have the following Toeplitz type Hao-Ng isomorphism theorem.
\begin{corollary}
\label{C:HaoNgT}
Let $(X,G,\beta)$ be a Zappa-Sz\'{e}p system. Then
\begin{itemize}
\item[(i)] $\mathcal{T}_{X\bowtie G} \cong \mathcal{T}_X \bowtie G$; and
\item[(ii)]
$\mathcal{T}_{X\bowtie G} \cong \mathcal{T}_X \bowtie G \cong \mathcal{T}_{X\widetilde\bowtie G}$ provided that $(X,G,\beta)$ is homogeneous.
\end{itemize}
\end{corollary}
\subsection{Cuntz-Pimsner type representations of $(X,G,\beta)$ and the C*-algebra $\mathcal{O}_X\bowtie G$}
In this subsection, we prove that the one-to-one correspondence $\Pi$ in Theorem \ref{T:Upsi} preserves the Cuntz-Pimsner covariance: $\Psi$ is Cuntz-Pimsner covariant
if and only if so is $\psi$. Accordingly we obtain the Hao-Ng isomorphism theorem as well in this case. But unfortunately, it is unknown whether
the correspondence $\widetilde\Pi$ in Theorem \ref{T:Upsi.homo} preserves the Cuntz-Pimsner covariance.
\begin{proposition}\label{prop.cp1}
Let $(X,G,\beta)$ be a Zappa-Sz\'{e}p system and set $Y:=X\bowtie G$.
For each $p\in P$ and $g\in G$, define a map $\iota_{p,g}:\mathcal{L}(X_p)\to\mathcal{L}(Y_{(p,g)})$ by
\[\iota_{p,g}(T)(x\otimes g)=(Tx)\otimes g.\]
Then $\iota_{p,g}$ is an isometric $*$-isomorphism. Moreover, for each rank-one operator $\theta_{x,y}\in \mathcal{K}(X_p)$, $\iota_{p,g}(\theta_{x,y})$ is the rank one operator $\Theta_{x\otimes g, y\otimes g}\in\mathcal{K}(Y_{(p,g)})$, and thus $\iota_{p,g}(\mathcal{K}(X_p))=\mathcal{K}(Y_{(p,g)})$.
\end{proposition}
\begin{proof} For any $T\in\mathcal{L}(X_p)$ and $x\in X_p$, $g\in G$,
\begin{align*}
\|\iota_{p,g}(T)(x\otimes g)\|^2 &= \|\langle (Tx)\otimes g, (Tx)\otimes g\rangle\| \\
&= \|\beta_{g^{-1}}(\langle Tx, Tx\rangle\| \\
&= \|\langle Tx, Tx\rangle\|=\|Tx\|^2.
\end{align*}
Thus $\iota_{p,g}$ is isometric. It is clear that $\iota_{p,g}$ is a homomorphism. Moreover, for any $x,y\in X_p$,
\begin{align*}
\langle \iota_{p,g}(T^*)x\otimes g, y\otimes g\rangle &= \langle (T^*x)\otimes g, y\otimes g\rangle \\
&= \beta_{g^{-1}}(\langle T^* x,y\rangle)= \beta_{g^{-1}}(\langle x, Ty\rangle) \\
&= \langle x\otimes g, (Ty)\otimes g\rangle = \langle x\otimes g, \iota_{p,g}(T)y\otimes g\rangle.
\end{align*}
Hence $\iota_{p,g}(T^*)=\iota_{p,g}(T)^*$. Finally, for any $\widetilde{T}\in\mathcal{L}(Y_{(p,g)})$, take any $x\in X_p$ and define $Tx\in X_p$ such that $\widetilde{T}(x\otimes g)=Tx\otimes g$. One can check that $T$ is a $\mathcal{A}$-linear, adjointable operator in $\mathcal{L}(X_p)$, and that $\widetilde{T}=\iota_{p,g}(T)$. Therefore, $\iota_{p,g}$ is an isometric $*$-isomorphism.
Now fix a rank one operator $\theta_{x,y}\in\mathcal{K}(X_p)$. In other words, for any $z\in X_p$, $\theta_{x,y}(z)=x\langle y,z\rangle$. Now for any $g\in G$ and $z\in X_p$, we have
\begin{align*}
\iota_{p,g}(\theta_{x,y})(z\otimes g)
&= \theta_{x,y}(z)\otimes g
= (x\langle y,z\rangle)\otimes g \\
&= (x\otimes g)(\beta_{g^{-1}}(\langle y,z\rangle)\otimes e) \\
&= (x\otimes g)\langle y\otimes g, z\otimes g\rangle \\
&= \Theta_{x\otimes g, y\otimes g}(z\otimes g).
\end{align*}
This proves $\iota_{p,g}(\theta_{x,y})=\Theta_{x\otimes g, y\otimes g}$, and therefore $\iota_{p,g}(\mathcal{K}(X_p))=\mathcal{K}(Y_{(p,g)})$
\end{proof}
Let $\phi_p$ and $\Phi_{(p,q)}$ be the left action of $\mathcal{A}$ on $X_p$ and $Y_{(p,g)}$, respectively:
\[
\phi_p(a)x=ax\text{ and } \Phi_{(p,g)}(a)(x\otimes g)=ax\otimes g.
\]
\begin{theorem}\label{thm.cp}
Let $\Psi$ be a representation of $X\bowtie G$ and $(\psi, U)$ be the covariant representation of $(X,G,\beta)$ under the one-to-one correspondence $\Pi$ given in Theorem \ref{T:Upsi}. Then $\Psi$ is Cuntz-Pimsner covariant if and only if so is $\psi$.
\end{theorem}
\begin{proof}
Suppose yhat $\Psi$ is a Cuntz-Pimsner covariant representation of $Y$. Then by definition, for any $(p,g)\in P\bowtie G$,
\[\Psi^{(p,g)}(\Phi_{(p,g)}(a))=\Psi_{(e,e)}(a)\ \text{ for all }a\in\Phi_{(p,g)}^{-1}(\mathcal{K}(Y_{(p,g)})).\]
Notice that, for any $p\in P$ and $a\in {\mathcal{A}}$ with $\phi_p(a)\in \mathcal{K}(X_p)$, we have $\Phi_{(p,e)}(a)\in \mathcal{K}(Y_{(p,e)})$. Thus
\[\psi^{(p)}(\phi(a))=\Psi^{(p,e)}(\Phi(a))=\Psi_{(e,e)}(a)=\psi_{e}(a).\]
Therefore, $\psi$ is Cuntz-Pimser covariant.
Conversely, suppose that $\psi$ is Cuntz-Pimsner covariant. Take $a\in \Phi^{-1}(\mathcal{K}(Y_{(p,g)})$. Without loss of generality, we assume that
\[\Phi_{(p,g)}(a)=\Theta_{x\otimes g, y\otimes g}\ \text{for }x,y\in X_p.\]
By Proposition \ref{prop.cp1}, \[\Phi_{(p,g)}(a)=\Theta_{x\otimes g, y\otimes g}=\iota_{p,g}(\theta_{x,y})=\iota_{p,g}(\phi_p(a)).\]
We have that $\theta_{x,y}=\phi_p(a)$, and so
\begin{align*}
\Psi^{(p,g)}(\Phi_{(p,g)}(a)) &= \Psi^{(p,g)}(\Theta_{x\otimes g, y\otimes g}) \\
&= \Psi_{(p,g)}(x\otimes g)\Psi_{(p,g)}(y\otimes g)^* \\
&= \psi_p(x) U_g (\psi_p(y) U_g)^* = \psi_p(x) U_g U_g^* \psi_p(y)^* \\
&= \psi_p(x) \psi_p(y)^* = \psi^{(p)}(\theta_{x,y}) \\
&= \psi^{(p)}(\phi_p(a))
= \psi_{e}(a)=\Psi_{(e,e)}(a).
\end{align*}
Therefore, $\Psi$ is Cuntz-Pimsner covariant.
\end{proof}
\begin{definition}
\label{D:OXG}
Let $\mathcal{O}_X\bowtie G$ be the universal C*-algebra generated by the set of all covariant representations $(\psi,U)$ of $(X,G,\beta)$ with $\psi$ Cuntz-Pimsner covariant.
\end{definition}
As a corollary of Theorem \ref{thm.cp}, the universal C*-algebra of Cuntz-Pimsner covariant representations of $X\bowtie G$ and the universal C*-algebra of covariant representations $(\psi,U)$
with $\psi$ Cuntz-Pimsner covariant of $(X,G,\beta)$ must coincide. Therefore, we have the following Cuntz-Pimsner type Hao-Ng isomorphism theorem.
\begin{corollary}
\label{C:HaoNgC}
$\mathcal{O}_{X\bowtie G} \cong \mathcal{O}_X\bowtie G$.
\end{corollary}
\begin{remark}
One might notice that the above corollary has no corresponding part to Corollary \ref{C:HaoNgT} (ii) for the homogeneous case.
In fact, we do not know whether $\mathcal{O}_{X\widetilde\bowtie G}\cong \mathcal{O}_X\bowtie G$, although $\mathcal{O}_{X\widetilde\bowtie G}$ is generally a quotient of $\mathcal{O}_{X\bowtie G}$ (see Corollary \ref{C:2bowtie} below).
Even in the special situation of a homogeneous Zappa-Sz\'{e}p action of $G$ on $X$ where $g|_p=g$ for all $g\in G$ and $p\in P$,
the Zappa-Sz\'{e}p homogeneous product system $X\widetilde\bowtie G$ becomes a crossed product $X\rtimes G$. In such a case, the problem whether $\mathcal{O}_{X\rtimes G}\cong \mathcal{O}_X\rtimes G$ is known as the Hao-Ng isomorphism problem in the literature. The isomorphism is known in several special cases (see, for example, \cite{HaoNg2008} and more recent approaches from non-self-adjoint operator algebras \cite{DOK20, K2017, KR19}).
\end{remark}
\begin{corollary}
\label{C:2bowtie}
If a Zappa-Sz\'{e}p system $(X,G,\beta)$ is homogeneous, then there is a natural epimorphism from $\mathcal{O}_{X\bowtie G}$ to $\mathcal{O}_{X\widetilde\bowtie G}$.
\end{corollary}
\begin{proof}
By Corollary \ref{C:HaoNgT}, there is a natural Cuntz-Pimsner covariant representation of $X\bowtie G$ on $\O_{X\widetilde\bowtie G}$:
$X\bowtie G\hookrightarrow \mathcal{T}_{X\bowtie G}\cong \mathcal{T}_{X\widetilde\bowtie G} \twoheadrightarrow \O_{X\widetilde\bowtie G}$. By the universal property of $\O_{X\bowtie G}$, this gives a homomorphism from $\mathcal{O}_{X\bowtie G}$ onto $\mathcal{O}_{X\widetilde\bowtie G}$.
\end{proof}
\subsection{Nica-Toeplitz representations of $X\bowtie G$ and the C*-algebra $\mathcal{N}\mathcal{T}_X\bowtie G$}
Suppose that $P$ is a right LCM semigroup and that $G$ is a group. Then the Zappa-Sz\'{e}p product semigroup $P\bowtie G$ is known to be right LCM as well \cite{BRRW}. In particular, for any $(p,g), (q,h)\in P\bowtie G$, we have that
\[(p,g)P\bowtie G\cap (q,h)P\bowtie G = \begin{cases}
(r,k)P\bowtie G & \text{if } pP\cap qP=rP, k\in G,\\
\emptyset & \text{otherwise}.
\end{cases}
\]
Here, the choice of $k$ can be arbitrary since $(e,k)$ is invertible in $P\bowtie G$ and thus $(r,k)P\bowtie G=(r,e)P\bowtie G$ for all $k\in G$.
We first prove that, for a given Zappa-Sz\'{e}p system $(X,G,\beta)$ where $X$ is compactly aligned, the product system $X\bowtie G$ is compactly aligned as well.
\begin{proposition}
Let $P$ be a right LCM semigroup and $X$ a compactly aligned product system over $P$. Then, for a given Zappa-Sz\'{e}p system $(X,G,\beta)$, $X\bowtie G$ is also compactly aligned.
\end{proposition}
\begin{proof} As before, let $Y:=X\bowtie G$. For each $p\in P$ and $g\in G$, define $\iota_{p,g}:\mathcal{L}(X_p)\to\mathcal{L}(Y_{(p,g)})$ by $\iota_{p,g}(T)(x\otimes g)=(Tx)\otimes g$. By Proposition \ref{prop.cp1}, $\iota_{p,g}$ is an isometric $*$-isomorphism, and $\iota_{p,g}(\mathcal{K}(X_p))=\mathcal{K}(Y_{(p,g)})$. Fix $p,q\in P$ with $pP\cap qP=rP$. For any $g,h\in G$, we have that $(p,g)P\bowtie G\cap (q,h)P\bowtie G=(r,e)P\bowtie G$.
Let $p^{-1}r$ be the unique element in $P$ such that $p(p^{-1}r)=p$.
For any $S\in\mathcal{K}(X_p)$ and any $x\in X_p$ and $y\in X_{p^{-1}r}$, one has
\begin{align*}
\iota_{r,e}\circ i_p^r(S)(xy\otimes e)
&= (i_p^r(S)(xy))\otimes e\\
&= (Sx)y \otimes e \\
&= ((Sx)\otimes g)(\beta_{g^{-1}}(y)\otimes g_0)\ (\text{with }g_0:=(g|_{g^{-1}\cdot (p^{-1}r)})^{-1})\\
&= \left(\iota_{p,g}(S)(x\otimes g)\right)(\beta_{g^{-1}}(y)\otimes g_0)\\
&= i_{(p,g)}^{(r,e)}\circ \iota_{p,g}(S)(xy\otimes e).
\end{align*}
Similarly, we have that $\iota_{r,e}\circ i_q^r=i_{(q,h)}^{(r,e)}\circ \iota_{q,h}$. Therefore, the following diagram commutes:
\begin{figure}[h]
\centering
\begin{tikzpicture}[scale=0.9]
\node at (-3,2) {$\mathcal{K}(X_p)$};
\node at (3,2) {$\mathcal{K}(Y_{(p,g)})$};
\node at (-3,0) {$\mathcal{K}(X_r)$};
\node at (3,0) {$\mathcal{K}(Y_{(r,e)})$};
\node at (-3,-2) {$\mathcal{K}(X_q)$};
\node at (3,-2) {$\mathcal{K}(Y_{(q,h)})$};
\draw[->] (-2,2) -- (2,2);
\draw[->] (-2,0) -- (2,0);
\draw[->] (-2,-2) -- (2,-2);
\draw[->] (-3,1.5) -- (-3,0.5);
\draw[->] (-3,-1.5) -- (-3,-0.5);
\draw[->] (3,1.5) -- (3,0.5);
\draw[->] (3,-1.5) -- (3,-0.5);
\node at (-3.3, 1) {$i_p^r$};
\node at (-3.3, -1) {$i_q^r$};
\node at (0, 2.3) {$\iota_{p,g}$};
\node at (0, 0.3) {$\iota_{r,e}$};
\node at (0, -1.7) {$\iota_{q,h}$};
\node at (3.6, 1) {$i_{(p,g)}^{(r,e)}$};
\node at (3.6, -1) {$i_{(q,h)}^{(r,e)}$};
\end{tikzpicture}
\end{figure}
Now for any $S\in\mathcal{K}(Y_{(p,g)})$ and $T\in\mathcal{K}(Y_{(q,h)})$, by Propostion \ref{prop.cp1}, there exist $S_0\in\mathcal{K}(X_p)$ and $T_0\in\mathcal{K}(X_q)$ such that $\iota_{p,g}(S_0)=S$ and $\iota_{q,h}(T_0)=T$. Therefore,
\begin{align*}
i_{(p,g)}^{(r,e)}(S)i_{(q,h)}^{(r,e)}(T) &= i_{(p,g)}^{(r,e)}(\iota_{p,g}(S_0))i_{(q,h)}^{(r,e)}(\iota_{q,h}(T_0)) \\
&= \iota_{r,e}(i_p^r(S_0))\iota_{r,e}(i_q^r(T_0)) \\
&= \iota_{r,e}(i_p^r(S_0)i_q^r(T_0)).
\end{align*}
Since $X$ is compactly aligned, $i_p^r(S_0)i_q^r(T_0)\in \mathcal{K}(X_r)$. Since $\iota_{r,e}$ maps $\mathcal{K}(X_r)$ to $\mathcal{K}(Y_{(r,e)})$, we have that
\[
i_{(p,g)}^{(r,e)}(S)i_{(q,h)}^{(r,e)}(T)=\iota_{r,e}(i_p^r(S_0)i_q^r(T_0))\in\mathcal{K}(Y_{(r,e)}).
\qedhere\]
\end{proof}
\begin{theorem}
\label{T:Ncov}
Suppose that $X$ is a compactly aligned product system over a right LCM semigroup $P$.
Let $(X,G,\beta)$ be a Zappa-Sz\'{e}p system, $\Psi$ a representation of $X\bowtie G$, and $(\psi, U)$ the covariant representation of $(X,G,\beta)$ under the one-to-one
correspondence $\Pi$ given in Theorem \ref{T:Upsi}.
Then $\Psi$ is Nica-covariant if and only if so is $\psi$.
\end{theorem}
\begin{proof}
For any $p\in P$ and $g\in G$, we first show that $\Psi^{(p,g)}\circ \iota_{p,g}=\psi^{(p)}$. For any $\theta_{x,y}\in \mathcal{K}(X_p)$, $\psi^{(p)}(\theta_{x,y})=\psi_p(x) \psi_p(y)^*$. By Proposition \ref{prop.cp1}, $\iota_{p,g}(\theta_{x,y})=\Theta_{x\otimes g, y\otimes g}$, and thus
\begin{align*}
\Psi^{(p,g)}(\iota_{p,g}(\theta_{x,y})) &= \Psi_{(p,g)}(x\otimes g) \Psi_{(p,g)}(y\otimes g)^* \\
&=\psi_p(x) U_g U_g^* \psi_p(y)^* \ (\text{by }\eqref{E:DefPsi})\\
&= \psi_p(x)\psi_p(y)^* = \psi^{(p)}(\theta_{x,y}).
\end{align*}
In other words, the following diagram commutes:
\begin{figure}[h]
\centering
\begin{tikzpicture}[scale=0.9]
\node at (-3,2) {$\mathcal{K}(X_p)$};
\node at (2.5,2) {$\mathcal{B}$};
\node at (-3,0) {$\mathcal{K}(Y_{p,g})$};
\draw[->] (-3,1.5) -- (-3,0.5);
\draw[->] (-2,2) -- (2.3,2);
\draw[->] (-2,0) -- (2.3,1.8);
\node at (0, 2.3) {$\psi^{(p)}$};
\node at (0, 0.3) {$\Psi^{(p,g)}$};
\node at (-3.5, 1) {$\iota_{p,g}$};
\end{tikzpicture}
\end{figure}
Suppose that $\psi$ is Nica-covariant. For any $S\in \mathcal{K}(Y_{(p,g)})$ and $T\in \mathcal{K}(Y_{(q,h)})$, there exist $S_0\in\mathcal{K}(X_p)$ and $T_0\in\mathcal{K}(X_q)$ such that $\iota_{p,g}(S_0)=S$ and $\iota_{q,h}(T_0)=T$.
Then we have
\[ \Psi^{(p,g)}(S)\Psi^{(q,h)}(T) = \psi^{(p)}(S_0) \psi^{(q)}(T_0).\]
If $(p,g)P\bowtie G\cap (q,h)P\bowtie G=\emptyset$, then $pP\cap qP=\emptyset$ and thus $\psi^{(p)}(S_0) \psi^{(q)}(T_0)=0$ by the Nica-covariance of $\psi$. If $(p,g)P\bowtie G\cap (q,h)P\bowtie G=(r,k)P\bowtie G$, then $rP=pP\cap qP$ and thus
\begin{align*}
\Psi^{(p,g)}(S)\Psi^{(q,h)}(T) &= \psi^{(p)}(S_0) \psi^{(q)}(T_0) \\
&= \psi^{(r)}(i_p^r(S_0) i_q^r(T_0)) \\
&= \Psi^{(r,k)}(\iota_{r,k}(i_p^r(S_0) i_q^r(T_0))) \\
&= \Psi^{(r,k)}(i_{(p,g)}^{(r,k)}(\iota_{p,g}(S_0)) i_{(q,h)}^{(r,k)}(\iota_{q,h}(T_0))) \\
&= \Psi^{(r,k)}(i_{(p,g)}^{(r,k)}(S) i_{(q,h)}^{(r,k)}(T)).
\end{align*}
Therefore, $\Psi$ is also Nica-covariant. The converse is clear since $\psi$ is the restriction of a Nica-covariant representation $\Psi$ on $X\cong \bigsqcup_{p\in P} Y_{(p,e)}$.
\end{proof}
\begin{definition}
\label{D:NTXG}
Let $\mathcal{N}\mathcal{T}_X\bowtie G$ be the universal C*-algebra generated by the set of covariant representations $(\psi,U)$ of a Zappa-Sz\'{e}p system $(X,G,\beta)$ with $\psi$ Nica-covariant.
\end{definition}
It follows from \cite[Proposition 6.8]{BLS2018b} and Example \ref{Eg:Fock} that $\mathcal{N}\mathcal{T}_X\bowtie G$ is always nontrivial.
Completely similar to Corollaries \ref{C:HaoNgT} and \ref{C:HaoNgC}, we have the following Nica-Toeplitz type Hao-Ng isomorphism theorem.
\begin{corollary}
\label{C:HaoNgNT}
$\mathcal{N}\mathcal{T}_{X\bowtie G} = \mathcal{N}\mathcal{T}_X\bowtie G$.
\end{corollary}
\section{Examples}
\label{S:EX}
In this section, we provide some examples of Zappa-Sz\'{e}p actions of groups $G$ on product systems $X=\bigsqcup_{p\in P} X_p$ and the associated C*-algebras.
\begin{eg}
Let $(X,G,\beta)$ be a Zappa-Sz\'{e}p system.
\begin{itemize}
\item[(i)] If $P$ is trivial, then ${\mathcal{T}}_X \bowtie G\cong \O_X \bowtie G\cong{\mathcal{A}}\rtimes_\beta G$.
\item[(ii)] If $G$ is trivial, then ${\mathcal{T}}_X \bowtie G\cong{\mathcal{T}}_X$ and $\O_X \bowtie G\cong\O_X$. Furthermore, if $X$ is a compact aligned product system over a right LCM semigroup, then
${\mathcal{N}}{\mathcal{T}}_X \bowtie G\cong{\mathcal{N}}{\mathcal{T}}_X$.
\end{itemize}
\end{eg}
\begin{eg}
Suppose that $(X,G,\beta)$ is a homogenous Zappa-Sz\'{e}p system. If furthermore $g|_p=g$ for all $g\in G$ and $p\in P$, then
$X\bowtie G$ becomes the crossed product $X\rtimes G$, and ${\mathcal{T}}_X\bowtie G$ (resp.~$\O_X\bowtie G$) is the crossed product ${\mathcal{T}}_X\rtimes G$ (resp.~$\O_X\rtimes G$).
So it follows from Corollary \ref{C:HaoNgT} (resp.~Corollary \ref{C:HaoNgC}) that we have the Hao-Ng isomorphisms:
${\mathcal{T}}_X\rtimes G\cong T_{X\rtimes G}$ (resp. $\O_X\rtimes G\cong \O_{X\rtimes G}$).
Furthermore, if $P$ is right LCM and $X$ is compactly aligned, then ${\mathcal{N}}{\mathcal{T}}_X\bowtie G$ is the crossed product ${\mathcal{N}}{\mathcal{T}}_X\rtimes G$, and so by
the corresponding Hao-Ng isomorphism becomes
$
{\mathcal{N}}{\mathcal{T}}_{X\rtimes G}\cong {\mathcal{N}}{\mathcal{T}}_X\rtimes G
$
(cf. Corollary \ref{C:HaoNgNT}).
\end{eg}
\begin{eg}
\label{Eg:gss}
Consider the trivial product system $X_P :=\bigsqcup_{p\in P} {\mathbb{C}} v_p$ over $P$.
For $g\in G$, let
\begin{align}
\label{E:tri}
\beta_g(\lambda v_p)=\lambda v_{g\cdot p} \quad\text{for all}\quad \lambda\in{\mathbb{C}}\text{ and } p\in P.
\end{align}
It is easy to check that $\beta$ is a Zappa-Sz\'{e}p action of $G$ on $X_P$. Then we obtain the Zappa-Sz\'{e}p product system $X_P\bowtie G$ by Theorem \ref{T:ZSP}.
Suppose that $P$ is right LCM. By \cite[Theorem 4.3]{BRRW}
one has
\[
{\mathcal{N}}{\mathcal{T}}_X \bowtie G\cong \mathrm{C}^*(P\bowtie G).
\]
Thus, in this case, by the Hao-Ng isomorphism theorems in Section \ref{S:main} we have
\begin{align*}
\mathrm{C}^*(P)\bowtie G\cong {\mathcal{N}}{\mathcal{T}}_X \bowtie G\cong {\mathcal{N}}{\mathcal{T}}_{X \bowtie G}\cong \mathrm{C}^*(P\bowtie G).
\end{align*}
Let us remark that the above covers the examples studied in \cite{BRRW}.
\end{eg}
\begin{eg}
\label{Eg:resss}
Let $G$ be a group and $\Lambda$ be a row-finite $k$-graph. Let $(G,\Lambda)$ be a self-similar $k$-graph (\cite{LY-IMRN, LY-JFA})
such that for each $g\in G$
\begin{align}
\label{E:res}
d(\mu)=d(\nu)\implies g|_\mu=g|_\nu.
\end{align}
Then one can construct a Zappa-Sz\'{e}p product
${\mathbb{N}}^k\bowtie G$ as follows. For $p\in {\mathbb{N}}^k$, take $\mu \in \Lambda^p$ and define
\[
g\cdot p := p,\ g|_p:=g|_\mu.
\]
Let $X(\Lambda)=\bigsqcup_{p\in {\mathbb{N}}^k} X(\Lambda^p)$ be the $C(\Lambda^0)$-product systems over ${\mathbb{N}}^k$ associated to $\Lambda$ (see \cite{RS05} for all related details). Define
\[
\beta_g: X(\Lambda^p) \to X(\Lambda^p), \quad \beta_g(\chi_\mu)=\chi_{g\cdot\mu}
\]
for all $g\in G$ and $\mu \in \Lambda^p$.
Then $(X(\Lambda), G, \beta)$ is a Zappa-Sz\'{e}p system. Indeed, it suffices to check that it satisfies (A5) and (A6). For (A5), let $\mu\in \Lambda^p$ and $\nu\in\Lambda^q$. Then
\begin{align*}
\beta_g(\chi_\mu \chi_\nu)
&=
\delta_{s(\mu),r(\nu)}\, \chi_{g\cdot(\mu\nu)}
=\delta_{s(\mu),r(\nu)}\chi_{g\cdot\mu (g|_\mu\cdot \nu)}
=\delta_{s(g\cdot\mu),r(g_\mu\cdot \nu)}\, \chi_{g\cdot\mu} \chi_{(g|_\mu\cdot \nu)} \\
&=\beta_g(\chi_\mu)\beta_{g|_\mu}(\nu)
=\beta_g(\chi_\mu)\beta_{g|_p}(\nu).
\end{align*}
For (A6), let $\mu,\nu\in \Lambda^p$ and $v\in \Lambda^0$. We have
\begin{align*}
\langle \chi_{g\cdot \mu}, \chi_{g\cdot \nu}\rangle(v)
&= \sum_{v=r(\lambda)} \chi_{g\cdot \mu}(\lambda) \chi_{g\cdot \nu}(\lambda)
= |\{g\cdot \mu = g\cdot \nu, r(g\cdot \mu)=v\}|\\
&=|\{\mu = \nu, g\cdot r(\mu)= v\}|
=|\{\mu = \nu, g|_\mu\cdot r(\mu)= v\}|\\
&=|\{\mu = \nu, r(\mu)=(g|_\mu)^{-1}\cdot v\}|
=\langle \chi_{\mu}, \chi_{\nu}\rangle((g|_\mu)^{-1}\cdot v) \\
&=\beta_{g|_\mu}(\langle \chi_\mu, \chi_\nu\rangle)(v)
=\beta_{g|_p}(\langle \chi_\mu, \chi_\nu\rangle(v).
\end{align*}
Therefore, by Theorem \ref{T:ZSP}, we obtain a Zappa-Sz\'{e}p product $X(\Lambda)\bowtie G$ over ${\mathbb{N}}^k\bowtie G$. Also one can see that
$\O_{X(\Lambda)\bowtie G}\cong \O_{X(\Lambda)}\bowtie G \cong \O_{G, \Lambda}$,
where $\O_{G,\Lambda}$ is the self-similar $k$-graph C*-algebra associated with $(G,\Lambda)$ (\cite{LY-IMRN}).
\end{eg}
\begin{rem}
(i) One can easily see that the condition \eqref{E:res} is equivalent to $g|_\mu=g|_\nu$ for all \textit{edges} $\mu,\nu\in\Lambda$.
(ii) At first sight, the condition \eqref{E:res} seems restrictive. But it is not hard to find self-similar $k$-graphs satisfying \eqref{E:res}. For example, let
$g|_\mu=g|_\nu=:g$. Also, one can invoke \cite[Lemma 3.6]{LY-IMRN} to obtain more examples.
\end{rem}
We finish the paper by the following example, which does not really belong to Zappa-Sz\'{e}p actions considered in this paper. But it presents another natural way to construct a product system over ${\mathbb{N}}^k$ which is isomorphic to the product system $X_{G,\Lambda}$ defined in \cite[Section~4]{LY-IMRN}. This provides one of our motivations of considering homogeneous actions in Section~\ref{S:homog}, and so we decide to include
it here.
\begin{eg}
Let $(G,\Lambda)$ be a self-similar $k$-graph. Let $X(\Lambda)=\bigsqcup_{p\in {\mathbb{N}}^k} X(\Lambda^p)$ be the product systems over ${\mathbb{N}}^k$ associated to $\Lambda$ as above. Define
\[
\beta_g: X(\Lambda^p) \to X(\Lambda^p), \quad \beta_g(\chi_\mu)=\chi_{g\cdot\mu}.
\]
Then one can easily check that
\begin{enumerate}
\item[(A1)$'$] for each $p\in P$, $\beta_g: X(\Lambda^p) \to X(\Lambda^p)$ is a $\mathbb{C}$-linear bijection;
\item[(A2)$'$] for any $g,h\in G$, $\beta_g\circ\beta_h=\beta_{gh}$;
\item[(A3)$'$] the map $\beta_{e}$ is the identity map;
\item[(A4)$'$] the map $\beta_g$ is a $*$-automorphism on $X(\Lambda^0)=C(\Lambda^0)$;
\item[(A5)$'$] for $p,q\in P$, $\mu\in \Lambda_p$ and $\nu\in \Lambda_q$,
\[
\beta_g(\chi_\mu \chi_\nu)=\beta_g(\chi_\mu)\beta_{g|_\mu}(\chi_\nu);
\]
\item[(A6)$'$] for $p\in P$ and $\mu,\nu\in \Lambda_p$,
\[e
\langle\beta_g(\chi_\mu), \beta_g(\chi_\nu)\rangle=\beta_{g|_\mu}(\langle \chi_\mu, \chi_\nu\rangle).
\]
\end{enumerate}
It follows from (A1)$'$-(A4)$'$ that $(C(\Lambda^0), G)$ is a C*-dynamical system. Let
$\mathfrak{A}:=C(\Lambda^0)\rtimes_\beta G$.
In what follows, we sketch the construction of an $\mathfrak{A}$-product system $\mathcal{E}$ over ${\mathbb{N}}^k$, which is isomorphic to the product system $X_{G,\Lambda}$ defined in \cite[Section~4]{LY-IMRN}. The details are similar to those in Section~\ref{S:homog} above and left to the interested reader.
For $p\in {\mathbb{N}}^k$, we construct a C*-correspondence ${\mathcal{E}}_p$ over $\mathfrak{A}$.
For $x= \chi_v u_h$, $\xi=\chi_\mu u_g$ and $\eta=\chi_\nu u_h$ with $\xi_\mu, \eta_\nu\in X_p$,
define
\begin{align*}
\xi x := \delta_{s(\mu), g\cdot v} \chi_\mu u_{gh},\
x\xi := \delta_{v, h\cdot r(\mu)} \chi_{h\cdot \mu} u_{h|_\mu g},\
\langle \xi, \eta\rangle
:=u_{g^{-1}}\langle \chi_\mu, \chi_\nu\rangle u_h.
\end{align*}
Let $\mathcal{E}_p$ be the closure of the linear span of $\chi_\mu u_g$ with $\mu \in \Lambda^p$ and $g\in G$.
Set
\[
\mathcal{E}:=\bigsqcup_{p\in {\mathbb{N}}^k} \mathcal{E}_p.
\]
For $\xi=\sum \chi_\mu u_g\in \mathcal{E}_p$ and $\eta=\sum \chi_\nu u_h\in \mathcal{E}_q$ define
\begin{align*}
\big(\sum \chi_\mu u_g\big)\big(\sum \chi_\nu u_h\big):=\sum \chi_\mu \chi_{g\cdot \nu} u_{g|_\nu} u_h=\sum \delta_{s(\mu), r(g\cdot \nu)}\chi_{\mu g\cdot \nu} u_{g|_\nu h}.
\end{align*}
Then $\mathcal{E}$ is an $\mathfrak{A}$-product system over ${\mathbb{N}}^k$.
One can check that $\mathcal{E}$ is isomorphic to the product system $X_{G,\Lambda}$ defined in \cite[Section~4]{LY-IMRN} via the map
\[
\chi_\mu u_g\in \mathcal{E}_p\mapsto \chi_\mu j(g)\in X_{G,\Lambda,p}.
\]
\end{eg}
|
1,108,101,564,527 | arxiv | \section{Introduction}
The total cross section $\sigma_{tot}(s)$ is a fundamental quantity in collisions of strongly interacting particles. At present one of the main theoretical approaches for the
description of $\sigma_{tot}$ is the QCD-inspired formalism \cite{durand,luna001,luna009,giulia001}. In this approach the energy dependence of the total cross section
is obtained from QCD via an eikonal formulation.
The high-energy dependence of $\sigma_{tot}$ is driven by the rapid increase in gluon density at small-$x$. In this work we explore the
non-perturbative dynamics of QCD in order to describe, in both $pp$ and $\bar{p}p$ channels, the total cross sections $\sigma_{tot}^{pp,\bar{p}p}(s)$ and the ratios of the real to
imaginary part of the forward scattering amplitude, $\rho^{pp,\bar{p}p}(s)$. In our calculations we introduce an energy-dependent dipole form factor which represents the overlap
density for the partons at impact parameter $b$.
The behavior of these forward quantities is derived from the QCD parton model using standard parton-parton elementary processes and an updated set of gluon distribution,
namely the CTEQ6L1 set. In our analysis we have included recent measurements of $pp$ total cross sections at the LHC by the TOTEM Collaboration \cite{TOTEM}. This work is organized
as follows: in the next section we introduce our QCD-based model where the onset of the dominance of semihard partons is managed by the dynamical gluon mass. We investigate a form
factor which the spatial distribution of gluons changes with the energy and introduce integral dispersion relations to connect the real and imaginary parts of eikonals. In
Section III we present our results and in Section IV we draw our conclusions.
\section{The QCD-inspired eikonal model}
In our model the increase of $\sigma_{tot}^{pp,\bar{p}p}(s)$ is associated with elementary semihard processes in the hadrons. In the eikonal representation $\sigma_{tot}(s)$
and $\rho(s)$ are given by
\begin{eqnarray}
\sigma_{tot}(s) = 4\pi \int_{_{0}}^{^{\infty}} \!\! b\, db\,
[1-e^{-\chi_{_{R}}(s,b)}\cos \chi_{_{I}}(s,b)]
\label{eq01}
\end{eqnarray}
and
\begin{eqnarray}
\rho(s) = \frac{-\int_{_{0}}^{^{\infty}} \!\! b\,
db\, e^{-\chi_{_{R}}(s,b)}\sin \chi_{_{I}}(s,b)}{\int_{_{0}}^{^{\infty}} \!\! b\,
db\,[1-e^{-\chi_{_{R}}(s,b)}\cos \chi_{_{I}}(s,b)]} ,
\label{eq03}
\end{eqnarray}
respectively, where $s$ is the square of the total center-of-mass energy, $b$ is the impact parameter, and
$\chi(s,b)=\textnormal{Re}\, \chi(s,b) + i\textnormal{Im}\, \chi(s,b) \equiv\chi_{_{R}}(s,b)+i\chi_{_{I}}(s,b)$ is the (complex) eikonal function. The eikonal functions for $pp$ and
$\bar{p}p$ scatterings are the sum of the soft and semihard (SH) parton interactions in the hadron-hadron collision,
\begin{eqnarray}
\chi(s,b) = \chi_{_{soft}}(s,b) + \chi_{_{SH}}(s,b).
\end{eqnarray}
It follows from the QCD parton model that the real part of the eikonal, $\textnormal{Re}\,\chi_{_{SH}}(s,b)$, can be factored as
\begin{eqnarray}
\textnormal{Re}\,\chi_{_{SH}}(s,b) = \frac{1}{2}\, W_{\!\!_{SH}}(b)\,\sigma_{_{QCD}}(s),
\end{eqnarray}
where $W_{\!\!_{SH}}(b)$ is an overlap density for the partons at impact parameter space $b$,
\begin{eqnarray}
W_{\!\!_{SH}}(b) &=& \int d^{2}b'\, \rho_{A}(|{\bf b}-{\bf b}'|)\, \rho_{B}(b'),
\label{ref009}
\end{eqnarray}
and $\sigma_{_{QCD}}(s)$ is the usual QCD cross section
\begin{eqnarray}
\sigma_{_{QCD}}(s) &=& \sum_{ij} \frac{1}{1+\delta_{ij}} \int_{0}^{1}\!\!dx_{1}
\int_{0}^{1}\!\!dx_{2} \int_{Q^{2}_{min}}^{\infty}\!\!d|\hat{t}|
\frac{d\hat{\sigma}_{ij}}{d|\hat{t}|}(\hat{s},\hat{t}) \nonumber \\
&\times & f_{i/A}(x_{1},|\hat{t}|)f_{j/B}(x_{2},|\hat{t}|)\, \Theta \! \left( \frac{\hat{s}}{2} - |\hat{t}| \right),
\label{eq08}
\end{eqnarray}
with $|\hat{t}|\equiv Q^{2}$, $\hat{s}=x_{1}x_{2}s$ and $i,j=q,\bar{q},g$.
The eikonal functions for $pp$ and $\bar{p}p$ scatterings are written as $\chi_{pp}^{\bar{p}p}(s,b) = \chi^{+} (s,b) \pm \chi^{-} (s,b)$, with
$\chi^{+}(s,b) = \chi^{+}_{_{soft}}(s,b) + \chi^{+}_{_{SH}}(s,b)$ and $\chi^{-}(s,b) = \chi^{-}_{_{soft}}(s,b) + \chi^{-}_{_{SH}}(s,b)$. Since in the parton model
$\chi^{-}_{_{SH}}(s,b)$ decreases rapidly with increasing $s$, the difference between $pp$ and $\bar{p}p$ cross sections is due only to the different weighting of the quark-antiquark
(valence) annihilation cross sections in the two channels. Thus the crossing-odd eikonal $\chi^{-}(s,b)$ receives no contribution from semihard processes at high energies, and it
is sufficient to take $\chi_{_{SH}}(s,b)=\chi^{+}_{_{SH}}(s,b)$ and, consequently, $\chi^{-}(s,b) = \chi^{-}_{_{soft}}(s,b)$. The connection between the real and imaginary parts of
the crossing-even eikonal $\chi^{+}(s,b)$ is obtained by means of dispersion relation
\begin{eqnarray}
\textnormal{Im}\,\chi^{+}(s,b) = -\frac{2s}{\pi}\, {\cal P}\!\! \int_{0}^{\infty}ds'\,
\frac{\textnormal{Re}\,\chi^{+}(s',b)}{s^{\prime 2}-s^{2}} ,
\label{idr001}
\end{eqnarray}
valid at $s \gg 1$ GeV$^{2}$. The soft term is given by
\begin{eqnarray}
\chi^{+}_{_{soft}}(s,b) = \frac{1}{2}\, W^{+}_{\!\!_{soft}}(b;\mu^{+}_{_{soft}})\, \left[ A' +\frac{B'}{(s/s_{0})^{\gamma}}\, e^{i\pi\gamma/2}
-i C'\left[ \ln\left(\frac{s}{s_{0}}\right) -i\frac{\pi}{2} \right] \right] ,
\label{soft01}
\end{eqnarray}
where $\sqrt{s_{0}}\equiv 5$ GeV and $A'$, $B'$, $C'$, $\gamma$ and $\mu^{+}_{_{soft}}$ are fitting parameters. The odd eikonal $\chi^{-}(s,b)$ is given by
\begin{eqnarray}
\chi^{-}(s,b) &=& \frac{1}{2}\, W^{-}_{\!\!_{soft}}(b;\mu^{-}_{_{soft}})\,D'\, \frac{e^{-i\pi/4}}{\sqrt{s}},
\label{softminus}
\end{eqnarray}
where $\mu^{-}_{_{soft}}\equiv 0.5$ GeV and $D'$ is also a fitting parameter.
In the expression (\ref{eq08}) the term $d\hat{\sigma}_{ij}/d|\hat{t}|$ is the differential cross section for $ij$ scattering ($i,j=q,\bar{q},g$). We select parton-parton processes
containing at least one gluon in the initial state, i.e., we consider the processes
\begin{eqnarray}
\frac{d\hat{\sigma}}{d\hat{t}}(gg\to gg)=\frac{9\pi\bar{\alpha}^{2}_{s}}{2\hat{s}^{2}}\left(3 -\frac{\hat{t}\hat{u}}{\hat{s}^{2}}-
\frac{\hat{s}\hat{u}}{\hat{t}^{2}}-\frac{\hat{t}\hat{s}}{\hat{u}^{2}} \right) ,
\label{pp001}
\end{eqnarray}
\begin{eqnarray}
\frac{d\hat{\sigma}}{d\hat{t}}(qg\to qg)=\frac{\pi\bar{\alpha}^{2}_{s}}{\hat{s}^{2}}\, (\hat{s}^{2}+\hat{u}^{2}) \left(
\frac{1}{\hat{t}^{2}}-\frac{4}{9\hat{s}\hat{u}} \right) ,
\label{pp002}
\end{eqnarray}
\begin{eqnarray}
\frac{d\hat{\sigma}}{d\hat{t}}(gg\to \bar{q}q)=\frac{3\pi\bar{\alpha}^{2}_{s}}{8\hat{s}^{2}}\, (\hat{t}^{2}+\hat{u}^{2}) \left(
\frac{4}{9\hat{t}\hat{u}}-\frac{1}{\hat{s}^{2}} \right) ,
\label{pp003}
\end{eqnarray}
where
\begin{eqnarray}
\bar{\alpha}_{s} (Q^{2})= \frac{4\pi}{\beta_0 \ln\left[
(Q^{2} + 4M_g^2(Q^{2}) )/\Lambda^2 \right]},
\label{eq26}
\end{eqnarray}
\begin{eqnarray}
M^2_g(Q^{2}) =m_g^2 \left[\frac{ \ln
\left(\frac{Q^{2}+4{m_g}^2}{\Lambda ^2}\right) } {
\ln\left(\frac{4{m_g}^2}{\Lambda ^2}\right) }\right]^{- 12/11} ;
\label{mdyna}
\end{eqnarray}
here $\bar{\alpha}_{s} (Q^{2})$ is a non-perturbative QCD effective charge, obtained by Cornwall through the use of the pinch technique in order to derive a gauge invariant
Schwinger-Dyson equation for the gluon propagator \cite{cornwall}.
In the case of semihard gluons we consider the possibility of a ``broadening'' of the spatial distribution. Our assumption suggests an increase of the average gluon radius
when $\sqrt{s}$ increases, and can be properly implemented using the energy-dependent dipole form factor
\begin{eqnarray}
G^{(d)}_{\!\!_{SH}}(s,k_{\perp};\nu_{\!\!_{SH}})=\left( \frac{\nu_{_{SH}}^{2}}{k_{\perp}^{2}+\nu_{_{SH}}^{2}} \right)^{2},
\end{eqnarray}
where $\nu_{_{SH}}= \nu_{1}-\nu_{2}\ln ( \frac{s}{s_{0}} )$, with $\sqrt{s_{0}}\equiv 5$ GeV. Here $\nu_{1}$ and $\nu_{2}$ are constants to be fitted. From equation (\ref{ref009}) we have
\begin{eqnarray}
W_{\!\!_{SH}}(s,b;\nu_{_{SH}}) &=& \frac{1}{2\pi}\int_{0}^{\infty}dk_{\perp}\, k_{\perp}\, J_{0}(k_{\perp}b)\,[G^{(d)}_{\!\!_{SH}}(s,k_{\perp};\nu_{\!\!_{SH}}]^{2} \nonumber \\
&=& \frac{\nu^{2}_{_{SH}}}{96\pi} (\nu_{_{SH}} b)^{3} K_{3}(\nu_{_{SH}} b).
\end{eqnarray}
Since semihard interactions dominate at high energies, we consider an energy-dependence only in the case of $W_{\!\!_{SH}}(s,b;\nu_{_{SH}})$. In this way the soft overlap densities
$W^{+}_{\!\!_{soft}}(b;\mu^{+}_{_{soft}})$ and $W^{-}_{\!\!_{soft}}(b;\mu^{-}_{_{soft}})$ are static, i.e., $\mu^{+}_{_{soft}}$ and $\mu^{-}_{_{soft}}$ are not energy dependent.
\section{Results}
First, in order to determine the model parameters, we set the value of the gluon mass scale to $m_{g} = 400$ MeV. This choice for the mass scale is not only
consistent to our LO procedure, but is also the one usually obtained in other calculations of strongly interacting
processes \cite{luna001,luna009,luna010,luna2014}. Next, we carry out a global fit to high-energy forward $pp$ and
$\bar{p}p$ scattering data above $\sqrt{s} = 10$ GeV, namely the total cross section $\sigma_{tot}^{pp,\bar{p}p}$ and
the ratio of the real to imaginary part of the forward scattering amplitude $\rho^{pp,\bar{p}p}$. We use data sets
compiled and analyzed by the Particle Data Group \cite{PDG} as well as the recent data at LHC from the TOTEM
Collaboration, with the statistic and systematic errors added in quadrature. We include in the dataset the first estimate for the $\rho$ parameter made by the TOTEM
Collaboration in their $\rho$-independent measurement at $\sqrt{s}=7$ TeV, namely $\rho^{pp}=0.145\pm0.091$ \cite{TOTEM}.
The values of the fitted parameters are given in the legend of Figure 1. The result $\chi^{2}/DOF = 1.062$ for the fit was obtained
for 154 degrees of freedom. The results of the fit to $\sigma_{tot}$ and $\rho$ for both $pp$ and $\bar{p}p$ channels are
displayed in Figure 1, together with the experimental data. The curves depicted in Figure 1 were all calculated using the cutoff $Q_{min}=1.3$ GeV, the value of the CTEQ6 fixed
initial scale $Q_{0}$.
\begin{figure}[!h]
\centering
\hspace*{-0.2cm}\includegraphics[scale=0.60]{bblfigure006b.eps}
\label{fig1}
\caption{The total cross section (upper) and the ratio of the real to imaginary part of the forward scattering amplitude (lower), for $pp$ ($\bullet$) and $\bar{p}p$ ($\circ$).
The values of the fitted parameters are: $\nu_{1} = 2.770\pm0.865$ GeV, $\nu_{2} = (7.860\pm5.444)\times 10^{-2}$ GeV, $A' = 108.9\pm8.6$ GeV$^{-1}$, $B' = 30.19\pm15.78$ GeV$^{-1}$,
$C' = 1.260\pm0.437$ GeV$^{-1}$, $\gamma = 0.719\pm 0.200$, $\mu^{+}_{soft} = 0.457\pm0.209$ GeV and $D' = 21.73\pm2.26$ GeV$^{-1}$.}
\end{figure}
\section{Conclusions}
In this work we have investigated infrared effects on semihard parton-parton interactions. We have studied $pp$ and $\bar{p}p$ scattering in the LHC energy region with the
assumption that the increase of their total cross sections arises exclusively from semihard interactions.
In the calculation of $\sigma_{tot}^{pp,\bar{p}p}$ and $\rho^{pp,\bar{p}p}$ we have considered the phenomenological implication of an energy-dependent form factor for semihard
partons. We introduce integral dispersion relations to connect the real and imaginary parts of eikonals with energy-dependent form factors.
In our analysis we have included the recent data at LHC from the TOTEM Collaboration.
Our results show that very good descriptions of $\sigma_{tot}^{pp,\bar{p}p}$ and
$\rho^{pp,\bar{p}p}$ data are obtained by constraining the value of the cutoff at $Q_{min} = 1.3$ GeV.
The $\chi^{2}/DOF$ for the global fit was 1.062 for 154 degrees of freedom. This good
statistical result shows that our eikonal model, where non-perturbative effects are naturally included via a QCD
effective charge, is well suited for detailed predictions of the forward quantities to be measured at higher energies.
In the semihard sector we have considered a form factor in which the average gluon radius increases
with $\sqrt{s}$. With this assumption we have obtained another form in which the eikonal can be factored into the QCD
parton model, namely
$\textnormal{Re}\,\chi_{_{SH}}(s,b) = \frac{1}{2}\, W_{\!\!_{SH}}(s,b)\,\textnormal{Re}\,\sigma_{_{QCD}}(s)$. The imaginary
part of this {\it semi-factorizable} eikonal was obtained by means of appropriate integral dispersion relations
which take into account eikonals with energy-dependent form factors.
At the moment we are analyzing the effects of different updated sets of parton distributions on the
forward quantities, namely CTEQ6L and MSTW, investigating the uncertainty on these forward observables coming from the uncertainties associated with the dynamical
mass scale and the parton distribution functions, and exploring a new ansatz for the energy-dependent form factor.
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\smallskip
\end{document}
|
1,108,101,564,528 | arxiv | \section{Introduction}
Stimulated by the applications to the quadratic regulator problem, controllability for distributed parameter systems is usually studied with square integrable controls. Such general controls
are hardly realizable in practice and only smooth or piecewise smooth controls, like bang-bang controls, can be implemented.
Moreover, when a control is implemented numerically, via discretization, convergence estimates depend on the smoothness of the control (see~\cite{EverdZUAZlibro}).
This fact revived interest on controllability with smooth controls, and the crucial results are in~\cite{EverdZUAZArch}
(see also~\cite[Theorem~2.1]{Russel} where reachability of the wave equation under controls of class $H^s$ acting on the entire boundary is studied).
A natural guess (which is true for the wave equation but which we in part disprove for systems with memory) is that if the control is smooth then the reachable targets are ``smooth'' too, and the problem is to identify the targets which can be reached by using controls in certain smoothness classes.
In this paper we are going to examine
the following equation, which is encountered in viscoelasticity and in diffusion processes when the material has a complex molecular structure:
\begin{equation}
\label{eq:modello}
w''=\left (\Delta w+bw\right )+\int_0^t K(t-s)w(s)\;\mbox{\rm d} s \,.
\end{equation}
Here $w=w(x,t)$, the apex denotes time derivative, $w''(x,t)=w_{tt}(x,t)$, $x\in\Omega\subseteq {\rm I\hskip-2.1pt R}^d$ is a bounded region with $C^2$ boundary, $K(t)$ is a real continuous function and $\Delta=\Delta_x$ is the laplacian in the variable $x$.
Dependence on the time and expecially space variable is not explicitly indicated unless needed for clarity so that we shall write $w=w(t)=w(x,t)$ according to convenience.
We associate the following initial/boundary conditions to system~(\ref{eq:modello}):
\begin{equation}
\label{eq:iniBOUNDcond}
\begin{array}
{l}
w(0)=w_0\,,\quad w'(0)=w_1\,,\\
w(x,t)= \left\{\begin{array}{cc} f(x,t) &x\in \Gamma\\
0 & x\in\partial\Omega\setminus\Gamma
\end{array}\right.
\end{array}
\end{equation}
($\Gamma $ is a relatively open subset of $\partial\Omega$).
The function $f$ is a control, which is used to steer the pair $\left (w(t),w'(t)\right )$ to hit a prescribed target $\left (\xi,\eta\right )$ at a certain time $T$.
The spaces of the initial data and final targets and of the control $f$ will be specified below.
There is no assumption on the sign of $b$ whose presence is explained in Remark~\ref{RemaMacC}. Furtermore we note:
\begin{itemize}
\item It is known (and recalled in Sect.~\ref{subsAssumpDISCUSS})
that when $f\in L^2(0,T;L^2(\Gamma))$ and $( w_0,w_1)\in L^2(\Omega)\times H^{-1}(\Omega)$ then problem~(\ref{eq:modello})-(\ref{eq:iniBOUNDcond}) admits a unique mild solution $(w(t),w'(t))\in C([0,T];L^2(\Omega)\times H^{-1}(\Omega))$ for every $T>0$;
\item when $K=0$ (i.e. when we consider the wave equation) the solution of~(\ref{eq:modello}) is denoted $u$;
\item when we want to stress the dependence on $f$ of the solution of~(\ref{eq:modello}) we use the notation $w_f$ (the notation $u_f$ when $K=0$).~~\rule{1mm}{2mm}\par\medskip
\end{itemize}
In order to describe the result proved in~\cite{EverdZUAZArch} it is convenient to introduce the following operators $A$ and ${\mathcal A}$ in $L^2(\Omega)$:
\begin{equation}\label{eq:defiOpeA}
{\rm dom}\, A=H^2(\Omega)\cap H^1_0(\Omega)\,,\qquad A\phi=\Delta\phi+b\phi\,,\quad {\mathcal A}=(-A)^{1/2}
\end{equation}
(note that $A$ is a positive operator if $b\geq 0$ and in this case ${\mathcal A}$ is defined in a
standard way; if $b<0$ the definition of ${\mathcal A}$ is discussed in Sect.~\ref{subsAssumpDISCUSS}).
It turns out that ${\rm dom}\,{\mathcal A}=H^1_0(\Omega)$.
\begin{Definition}
Let $T>0$ and let $\mathcal{F}$ be a closed subspace of $ L^2(0,T;L^2(\Gamma))$.
We say that Eq.~(\ref{eq:modello}) is ${\rm dom}\,{\mathcal A}^{k+1}\times {\rm dom}\,{\mathcal A}^k$-controllable at time $T$ with controls $f\in \mathcal{F}$ when the following properties hold:
\begin{enumerate}
\item
If $(w_0,w_1)\in {\rm dom}\,{\mathcal A}^{k+1}\times {\rm dom}\,{\mathcal A}^k $ and $f\in \mathcal{F}$ then $(w_f(T),w_f'(T))\in {\rm dom}\,{\mathcal A}^{k+1}\times {\rm dom}\,{\mathcal A}^k $;
\item
for every $w_0$, $\xi_0$ in $ {\rm dom}\,{\mathcal A}^{k+1}$ and every $w_1$, $\eta$ in ${\rm dom}\,{\mathcal A}^k$ there exists $f\in \mathcal{F}$ such that
$(w_f(T),w_f'(T))=(\xi,\eta)$.
\end{enumerate}
\end{Definition}
The following result is proved in~\cite{PandSharp} (see also~(\cite{PandLibro,PandParma,PandESAIM}).
\begin{Theorem}\label{Theo:controLdue}
There exists a time $T_0$ and a relatively open set $\Gamma\subseteq\partial\Omega$
which have the following properties: Let $T>T_0$. For every $w_0$ and $\xi$ in $L^2(\Omega) $ and for every $w_1$ and $\eta$ in $H^{-1}(\Omega)$
there exists $f\in L^2(0,T;L^2(\partial \Omega))$ such that $\left (w_{ f}(T),w'_{ f}(T)\right )=\left (\xi,\eta\right )$.
The set $ \Gamma $ and the number $ T_0 $ do not depend on the continuous memory kernel $ K(t) $.
\end{Theorem}
Note that Theorem~\ref{Theo:controLdue} holds in particular for the wave equation (i.e. when $K=0$) and the proofs in the references above do depend on the known controllability result of the wave equation.
The result in~\cite{EverdZUAZArch} can be adapted to the case of the wave equation (without memory) as described in~\cite[Sect.~5.2]{EverdZUAZArch} (see item~\ref{item1inRemaMacC} in Remark~\ref{RemaMacC} to understand the exponents):
\begin{Theorem}\label{TeoDIevErdZUazPERiperb}
Let $T$, $T_0$ and $\Gamma$ be as in Theorem~\ref{Theo:controLdue}. System~(\ref{eq:modello}) with $K=0$ is
${\rm dom}\,{\mathcal A}^{k }\times {\rm dom}\,{\mathcal A}^{k-1}$-controllable at time $T$ with controls $f\in H^k_0(0,T;L^2(\Gamma))$.
\end{Theorem}
In the light of Theorem~\ref{Theo:controLdue} (which extends the well known controllability result of the wave equation) it is natural to guess that Theorem~\ref{TeoDIevErdZUazPERiperb} can be extended too. Instead we have the following result:
\begin{Theorem}\label{TeoINh10}
Let $T_0$, $ T $ and $\Gamma$ have the properties in Theorem~\ref{Theo:controLdue}. Then we have:
\begin{enumerate}
\item
\label{item1delTeoINh10}
System~(\ref{eq:modello}) is
${\rm dom}\,{\mathcal A} \times L^2(\Omega)$-controllable at time $T$ with controls $f\in H^1_0(0,T;L^2(\Gamma))$ (note that $L^2(\Omega)={\rm dom}\,{\mathcal A}^0$).
\item\label{item2delTeoINh10}
System~(\ref{eq:modello})
${\rm dom}\,{\mathcal A}^{2}\times {\rm dom}\,{\mathcal A} $-controllable at time $T$ with controls $f\in H^2_0(0,T;L^2(\Gamma))$.
\item
\label{item3delTeoINh10}
Let $k\geq 3$. For every $T>0$ there exist controls $f\in H^k_0(0,T;L^2(\Gamma))$ such that $(w(T),w'(T))\notin
{\rm dom}\,{\mathcal A}^{k}\times {\rm dom}\,{\mathcal A}^{k-1} $.
\end{enumerate}
\end{Theorem}
\begin{Remark}\label{RemaMacC}
{\rm
\begin{enumerate}
\item\label{item1inRemaMacC} The operator $A$ in~\cite[Sect.~5.2]{EverdZUAZArch} is defined as the laplacian with domain $H^1_0(\Omega)$ (and image $H^{-1}(\Omega)$ while we used ${\rm dom}\, A=H^2(\Omega)\cap H^1_0(\Omega)$.
\item for the sake of brevity, the properties in Theorems~\ref{Theo:controLdue} will be called ``controllability in $L^2(\Omega)\times H^{-1}(\Omega)$''.
\item In the case $\Omega=(a,b)$, item~\ref{item1delTeoINh10} of Theorem~\ref{TeoINh10}
has been proved in~\cite{PandTRIULZI}.
\item it is clear that when studying controllability we can reduce ourselves to study the system with zero initial conditions, $w_0=0$, $w_1=0$.
\item
The usual form of the system with persistent memory which is encountered in viscoelasticity is
\[
w''=\Delta w+\int_0^t M(t-s)\Delta w(s)\;\mbox{\rm d} s\,.
\]
We formally solve this equation as a Volterra integral equation in the ``unknown'' $\Delta w$. Two integrations by parts in time (followed by an exponential transformation) lead to Eq.~(\ref{eq:modello}), with $b\neq 0$ (if the initial conditions are different from zero also an affine term, which contains the initial conditions appear, but when studying controllability we
can assume $w_0=0$, $w_1=0$). For this reason we kept the addendum $bw$ in Eq.~(\ref{eq:modello}). This transformation is known as \emph{MacCamy trick} and it is detailed in~\cite{PandLibro}.~~\rule{1mm}{2mm}\par\medskip
\end{enumerate}
}
\end{Remark}
\subsection{\label{subsAssumpDISCUSS}Preliminaries}
The operator $A$ in $L^2(\Omega)$ was already defined: $A\phi=\Delta\phi+b\phi$,
${\rm dom}\, A=H^2(\Omega)\cap H^1_0(\Omega)$ (we recall that $\Omega$ is a region with $C^2$ boundary).
The operator $A$ is selfadjoint, possibly non positive since
there is no assumption on the sign of $b$. Its resolvent is compact so that the Hilbert space $L^2(\Omega)$ has an orthonormal basis $\{\phi_n\}$ of eigenvectors of the operator $A$. We denote $-\lambda_n^2$ the eigenvalue of $\phi_n$ since $ \lambda_n^2>0$ for large $n$ (it might be $ \lambda_n^2\leq 0$ if $n$ is small). The eigenvalues are repeated according to their multiplicity (which is finite).
We shall use the following known asymptotic estimate for the eigenvalues (see~\cite{Agmon}):
\begin{equation}\label{eq:stimAUTOV}
\mbox{if $\Omega\subseteq{\rm I\hskip-2.1pt R}^d$ then $\lambda_n^2\sim n^{2/d}$}\,.
\end{equation}
In particular, $\lambda_n^2$
is positive for large $n$.
Let $\gamma\in(0,1)$. If $n$ is large and $\lambda_n^2\geq 0$ then $\lambda_n^{2\gamma}$ is the nonegative determination; otherwise we fix one of the determinations. When $\gamma=1/2$, for example the one with nonnegative imaginary part. \emph{So, $\lambda_n $ denotes the chosen determination of the square root of $\lambda_n^2$, $ \lambda_n>0 $ is $ n $ is large.}
By definition,
\[
\begin{array}{l}
\displaystyle \xi=\sum _{n=1}^{+\infty} \xi_n\phi_n(x)\in {\rm dom }\,(-A)^\gamma \ \iff
\{\lambda_n^{2\gamma}\xi_n\}\in l^2 \\[2mm]
\displaystyle {\rm and}\quad
(-A)^\gamma\xi= \sum _{n=1}^{+\infty} \lambda_n^{2\gamma}\xi_n\phi_n \quad \mbox{so that in particular}\\
{\mathcal A}\xi =i\left (-A\right )^{1/2}\xi=i\left (\sum _{n=1}^{+\infty} \lambda_n\xi_n\phi_n(x)\right )
\,.
\end{array}
\]
Furthermore we define (we recall the that $\lambda_n$ is real when $n$ is large)
\begin{align*}
&R_+(t)\left ( \sum _{n=1}^{+\infty} \xi_n\phi_n(x)\right )= \sum _{n=1}^{+\infty} \left (\cos\lambda_n t\right ) \xi_n\phi_n(x)\,,
\\
&
R_-(t)\left ( \sum _{n=1}^{+\infty} \xi_n\phi_n(x)\right )= i\left (\sum _{n=1}^{+\infty} \left (\sin \lambda_n t\right ) \xi_n\phi_n(x)\right ) \,.
\end{align*}
Finally, we introduce the operator $D$: $L^2(\Gamma)\mapsto L^2(\Omega)$:
\[
u=Df\ \iff \ \left\{\begin{array}{l}
\Delta u+bu=0\ {\rm in}\ \Omega\,,\\ u= f\ {\rm on}\, \Gamma\,,\quad u=0\ {\rm on}\ \partial\Omega\setminus\Gamma\,.
\end{array}\right.
\]
It is known that ${\rm im}\, D\subseteq H^{1/2}(\Omega)\subseteq {\rm dom}\, (-A)^{1/4-\epsilon} $ for every $\epsilon>0$.
It is known (see~\cite{LasTRIEquonde}) that the mild solution of the wave equation
\[
u''=\Delta u+F\,,
\]
with initial and boundary conditions~(\ref{eq:iniBOUNDcond}) is
\begin{align}\nonumber
u(t)=&R_+(t)w_0+{\mathcal A}^{-1}R_-(t)w_1-{\mathcal A}\int_0^t R_-(t-s) D f(s)\;\mbox{\rm d} s\\
\label{eqDIu}
&+{\mathcal A}^{-1}\int_0^t R_-(t-s)F(s)\;\mbox{\rm d} s\,.
\end{align}
By definition, the mild solution of problem~(\ref{eq:modello})-(\ref{eq:iniBOUNDcond})
is the solution of the following Volterra integral equation
\begin{equation}
\label{eq:soluVisco}
w (t)=u (t)+{\mathcal A}^{-1}\int_0^t\left [\int_0^{t-s} K(r)R_-(t-s-r) w(s)\;\mbox{\rm d} r\right ] \;\mbox{\rm d} s
\end{equation}
where $u(t)$ is given by~(\ref{eqDIu}) with $F=0$.
We note that $w'(t)$ is given by
\begin{equation}
\label{eq:volteDERIVATA}
w'(t)=u'(t) +\int_0^t \left [\int_0^{t-s} K(s)R_+(t-s-r) w(s)\;\mbox{\rm d} r\right ]\;\mbox{\rm d} s
\end{equation}
where
\begin{equation}
\label{eq:diuPRIMO}
u'(t)={\mathcal A} R_-(t)w_0+R_+(t) w_1-A\int_0^t R_+(t-s) D f(s)\;\mbox{\rm d} s\,.
\end{equation}
The following result is known (see~\cite[Ch.~2]{PandLibro}):
\begin{Theorem}\label{teo:propriESoluOnde}
If $(\xi,\eta,f)\in L^2(\Omega)\times H^{-1} (\Omega)\times L^2\left (0,T;L^2(\partial\Omega)\right )$
then $\left (w_f(t),w_f'(t)\right )\in C\left ([0,T];L^2(\Omega)\right )\times C\left ([0,T];H^{-1}(\Omega)\right )$
for every $T>0$ and the linear transformation
$(\xi,\eta,f)\mapsto (w,w')$ is continuous in the indicated spaces.
\end{Theorem}
Finally, let $w_0=0$, $w_1=0$. We introduce the following operators which are continuous from $ L^2(0,T;L^2(\Gamma)) $ to $ L^2(\Omega)\times H^{-1}(\Omega) $:
\begin{align*}
&f\mapsto \Lambda_E(T) f=\left ( \begin{array}{cc}
\Lambda _{ E, 1}(T) f&\Lambda _{E, 2}(T)f
\end{array}\right)= \left ( \begin{array}{cc}
u_ f(T)&u'_f(T)
\end{array}\right)\\
&f\mapsto \Lambda_V(T) f=\left ( \begin{array}{cc}
\Lambda _{ V,1} (T)f&\Lambda _{ V,2}(T)f
\end{array}\right)= \left ( \begin{array}{cc}
w_ f(T)&w'_f(T)
\end{array}\right)\,.
\end{align*}
\emph{Our assumption is that the set $\Gamma$ and the time $T$ have been chosen so that both these operators are surjective.}
\section{Controllability with square integrable controls and moment problem}
From now on we study controllability and so we assume $w_0=0$, $w_1=0$.
We expand the solutions of Eq.~(\ref{eq:modello}) in series of $\phi_n$,
\begin{equation}\label{eq:expaWn}
w(x,t)=\sum _{n=1}^{+\infty} \phi_n(x) w_n(t)\,,\qquad w_t(x,t)=\sum _{n=1}^{+\infty} \phi_n(x) w_n'(t)\,.
\end{equation}
It is easily seen that $w_n(t)$ solves
\
w_n''(t)=-\lambda_n^2 w_n(t)+\int_0^t K(t-s)w_n(s)\;\mbox{\rm d} s-\int_\Gamma \left (\gamma_1\phi_n\right ) f(x,t)\;\mbox{\rm d} \Gamma
\]
where $\gamma_1$ is the exterior normal derivative and $\;\mbox{\rm d}\Gamma$ is the surface measure.
The initial conditions are zero since $w(0)=0$, $w'(0)=0$.
In order to represent the solution of the previous equation, we introduce $\zeta_n(t)$, the solution of
\begin{equation}
\label{eq:diZETAn}
\zeta_n''(t)=-\lambda_n^2\zeta_n(t)+\int_0^t K(t-s) \zeta_n(s)\;\mbox{\rm d} s\,,\qquad \left\{\begin{array}{l}
\zeta_n(0)=0\,,\\
\zeta_n'(0)=1\,.
\end{array}\right.
\end{equation}
Then we have
\begin{equation}\label{RApprewN}
\left\{
\begin{array}{l}
\displaystyle w_n(t)=\int_\Gamma\int_0^t \left [\zeta_n(t-s)\gamma_1\phi_n(x)\right ] f(x,s)\;\mbox{\rm d} s\,,\\[3mm]
\displaystyle w_n'(t)=\int_\Gamma\int_0^t \left [\zeta_n'(t-s)\gamma_1\phi_n(x)\right ] f(x,s)\;\mbox{\rm d} s\,.
\end{array}
\right.
\end{equation}
Let the target $(\xi,\eta)\in L^2(\Omega)\times H^{-1}(\Omega)$ have the expansion
\[
\xi(x)=\sum _{n=1}^{+\infty} \xi_n\phi_n(x)\,,\qquad \eta(x)=\sum _{n=1}^{+\infty}\left ( \eta_n\lambda_n \right )\phi_n(x)\,.
\]
Then, this target is reachable at time $T$ if and only if there exists $f\in L^2(0,T;L^2(\Gamma))$ such that
\[
\mathbb{M} _0f= c_n = \eta_n+i\xi_n
\]
where $\mathbb{M}_0$ is the \emph{moment operator}
\begin{equation}\label{eq:operMoment0}
\begin{array}{l}
\displaystyle \mathbb{M}_0 f=\int_\Gamma\int_0^T E_n^{(0)}(s)\Psi_n f(x,T-s)\;\mbox{\rm d} s\;\mbox{\rm d} \Gamma \,,\\[3mm]
\displaystyle \Psi_n=\frac{\gamma_1\phi_n}{\lambda_n}\,,\quad E_n^{(0)}(s)=\left [\zeta_n'(s)+i\lambda_n \zeta_n(s)\right ] \,.
\end{array}
\end{equation}
Note that the operator $\mathbb{M}_0$ takes values in $l^2$. So, we should write $\mathbb{M}_0 f=\{c_n\}$. The brace here is usually omitted.
The sequence $\{\Psi_n\}$ is bounded in $L^2(\partial\Omega)$ and it is almost normalized if $\Gamma$ has been chosen as in Theorem~\ref{Theo:controLdue} (see~\cite[Theorem~4.4]{PandLibro}).
It is known from~\cite{LionsLibro,PandLibro} that the operator $\mathbb{M} $ is continuous.
Our assumption is that $T$ and $\Gamma$ have been so chosen that this operator, defined on $L^2(0,T;L^2(\Gamma))$, is surjective in $l^2(\mathbb{C})$.
This implies (see~\cite[Sect.~3.3]{PandLibro}) that:
\begin{Lemma}\label{lemmaADDE}
\begin{enumerate}
\item \label{lemmaADDEitem1}
The sequence $\{e_n\}$
\[
e_n=\left [\zeta_n'(s)+i\lambda_n \zeta_n(s)\right ]\Psi_n
\]
is a Riesz sequence in $L^2(0,T;L^2(\Gamma))$, i.e. it can be transformed to an orthonormal sequence using a linear, bounded and boundedly invertible transformation.
\item
\label{lemmaADDEitem2}
The series
\[
\sum\alpha_n\Psi_n(x)\left [\zeta_n'(s)+i\lambda_n\zeta_n(s)\right ]
\]
converges in $L^2(0,T;L^2(\Gamma))$ if and only if $\{\alpha_n\}\in l^2$.
\item\label{lemmaADDEitem3}
If $K(t)=0$ then $
e_n(x,t)=\Psi_n(x) e^{i\lambda_n t}$.
\end{enumerate}
\end{Lemma}
\section{\label{sect:sisteVontroREG}The system with controls of class $H^1_0([0,T];L^2(\Gamma))$}
The property $f\in H^1_0([0,T];L^2(\Gamma))$
can be written as follows:
\begin{equation}\label{eq:RapprF}
\begin{array}{l}\displaystyle
f(x,t)=\int_0^t g(x,s)\;\mbox{\rm d} s\qquad g\in {\mathcal{N}_1}\,,\\
\displaystyle {\mathcal{N}_1}=\left \{ g\in L^2(0,T;L^2(\Gamma))\,,\quad \int_0^T g(x,s)\;\mbox{\rm d} s=0\right \}\,.
\end{array}
\end{equation}
So,
when $f\in H^1_0([0,T];L^2(\Gamma))$ we can integrate by parts the integral in~(\ref{eqDIu}) (with $F=0 $, see~\cite{PandAMO} for the rigorous justification) and we get (using that the initial conditions are zero):
\begin{align}
\nonumber u_f(t)&=-{\mathcal A}\int_0^t R_-(t-s)D\int_0^s g(r)\;\mbox{\rm d} r\,\;\mbox{\rm d} s=\\
\label{eq:ondINTepartiU}&=D\int_0^t g(r)\;\mbox{\rm d} r-\int_0^t R_+(t-s)Dg(s)\;\mbox{\rm d} s=\hat u_g(t)\,,\\
\label{eq:ondDerivINTepartiU}u'_f(t)& =-{\mathcal A}\int_0^t R_-(t-s)Dg(s)\;\mbox{\rm d} s=\tilde u_g(t)\,.
\end{align}
\begin{Remark}\label{Rema:sullaroBUSTrezzO}{\rm
These expressions show an interesting fact (compare also~\cite[Corollary~1.5]{EverdZUAZArch}). We see from Theorem~\ref{teo:propriESoluOnde} that the integrals take values respectively in $H^1_0(\Omega)$ and $L^2(\Omega)$ and
$g\mapsto \tilde u_g(t) $ is a linear continuous map from ${\mathcal{N}_1}$ to $C([0,T];L^2(\Omega))$. Instead, $g\mapsto \hat u_g(t)$ is a linear continuous map from ${\mathcal{N}_1}$ to $C([0,T]; H^{1/2}(\Omega))$ since ${\rm im}\,D\subseteq H^{1/2}(\Omega)$. We have $\hat u_g(T)\in H^1_0(\Omega)$ only
when $t=T$.
Of course these maps are continuous among the specified spaces but in every numerical computation we expect that the value of $T$ is affected by some error, and the fact that $u_f(t)\notin H^1_0(\Omega)$ if $t\neq T$ might raise some robustness issues in the numerical approximation of the steering control. This issue seemingly is still to be studied.~~\rule{1mm}{2mm}\par\medskip
}
\end{Remark}
Let us introduce
\begin{align}
\nonumber &{\mathcal{N}_1}\ni g \mapsto \Lambda_{E }^{(1)}(T)g = \left (-\hat u_g(T),-\tilde u_g(T)\right ) \\
\label{eq:defiLambda0}=&\left (\int_0^T R_+(T-s)Dg(s)\;\mbox{\rm d} s\,,\ {\mathcal A}\int_0^T R_-(T-s)Dg(s)\;\mbox{\rm d} s\right ) \, .
\end{align}
Controllability with $H^1_0$-controls $f$ of the wave equation (the case $K=0$, proved in~\cite{EverdZUAZArch}) is equivalent to surjectivity of the map $ \Lambda_{E }(T)$
from ${\mathcal{N}_1}$ to $H^1_0(\Omega)\times L^2(\Omega)$.
We introduce formulas~(\ref{eq:ondINTepartiU})-(\ref{eq:ondDerivINTepartiU}) in~(\ref{eq:soluVisco}) and~(\ref{eq:volteDERIVATA}). We get:
\begin{align}
\label{eq:ondINTepartiW}w_f(t)&=\hat u_g(t)+{\mathcal A}^{-1}\int_0^t\left [ \int_0^{t-s}K(r)R_-(t-s-r)w(s)\;\mbox{\rm d} r\right] \;\mbox{\rm d} s=\hat w_g(t)\\
\label{eq:ondDerivINTepartiW}w'_f(t)&= \tilde u_g+ \int_0^t \left [\int_0^{t-s}K(r) R_+(t-s-r) w(s)\;\mbox{\rm d} r\right ]\;\mbox{\rm d} s=\tilde w_g(t)\,.
\end{align}
We introduce the operator $\Lambda_{V}^{(1)}(T)$, to be compared with the operator $\Lambda_E^{(1)}(T)$ in~(\ref{eq:defiLambda0}):
\[
\Lambda^{(1)}_{V}(T)g=\left (-\hat w_g(T),-\tilde w_g(T)\right )\,,\qquad g\in{\mathcal{N}_1}\,.
\]
Controllability in $H^1_0(\Omega)\times L^2(\Omega)$ is equivalent to surjectivity of the map $\Lambda^{(1)}_{V}(T)$ from ${\mathcal{N}_1}$ to $H^1_0(\Omega)\times L^2(\Omega)$.
We see from here that the functions $\hat w_g(t)$ and $\tilde w_g(t)$ have the same properties as stated above for $\hat u_g(t)$ and $\tilde u_g(t)$, in particular $\hat w_g(t)\in H^{1/2}(\Omega)$ and $\hat w_g(T)\in H^1_0(\Omega)$ but there is an additional difficulty: if we want to consider $\hat u_g(T)$ we can simply ignore the contribution of $Dg(T)$. Instead, due to the Volterra stucture of Eq.~(\ref{eq:ondINTepartiW}), the term $Dg(t)$ which comes from $\hat u_g(t)$ cannot be simply ignore when looking at the function $\hat w_g$ for $t=T$.
In spite of this, using ${\rm im}\, D\subseteq H^{1/2}(\Omega)={\rm dom}\, (-A)^{1/4-\epsilon/2}$ (for every $\epsilon>0$) and solving~(\ref{eq:ondINTepartiW})-(\ref{eq:ondDerivINTepartiW})
via Picard iteration, it is simple to
prove\footnote{we use
$\left [\cdot \right]^\perp $
to denote the subspace of the annihilators in the dual space.}:
\begin{Lemma}
\label{Lemma:compa} We have:
\begin{itemize}
\item $w_f(T)\in H^{1}_0(\Omega)$, $w_f'(T)\in L^2(\Omega)$;
\item The operator $\Lambda^{(1)}_{V}(T)-\Lambda^{(1)}_{E}(T)$ is compact in $H^{1}_{0}(\Omega)\times L^2(\Omega)$ and so
${\rm Im}\,
\Lambda^{(1)}_{V}(T)$ is closed in $H^{1}_{0}(\Omega)\times L^2(\Omega)$ and
$\left [{\rm Im}\,
\Lambda^{(1)}_{V}(T)
\right ]^\perp$ is finite dimensional.
\end{itemize}
\end{Lemma}
The goal is the proof of Theorem~\ref{Theo:controLdue}, i.e. the proof that every element in $\left [{\rm Im}\, \Lambda^{(1)} _{V}(T)\right ]^{\perp}$
is equal zero. We prove this result by using the properties of the moment operator.
\subsection{Controllability with $H^1_0([0,T];L^2(\Gamma))$ controls}
Our point of departure is the expansion~(\ref{eq:expaWn}) and
the representation~(\ref{RApprewN}) of
$w_n(t)$, $w_n'(t)$.
Let
\[
K_1(t)=\int_0^t K(s)\;\mbox{\rm d} s
\]
When $f$ has the representation~(\ref{eq:RapprF}) we can manipulate~(\ref{RApprewN}) as follows:
\begin{equation}
\label{eq:RApprewNDopoINTEparti}
\begin{array}{l}
\displaystyle w_n(t)=\left (-\frac{1}{\lambda_n^2}\right )\int_\Gamma \gamma_1\phi_n \int_0^t \left (-\lambda_n^2\zeta_n(s)\right )\int_0^{t-s} g(r)\;\mbox{\rm d} r\,\;\mbox{\rm d} s\,\;\mbox{\rm d}\Gamma=\\[5mm]
\displaystyle =\left (-\frac{1}{\lambda_n^2}\right )\int_\Gamma \gamma_1\phi_n \int_0^t\left [\zeta_n''(s)-\int_0^s K(s-\nu)\zeta_n(\nu)\;\mbox{\rm d}\nu\right ]\int_0^{t-s}g(r)\;\mbox{\rm d} r\,\;\mbox{\rm d} s\,\;\mbox{\rm d}\Gamma=\\[5mm]
\displaystyle =\frac{1}{\lambda_n^2} \int_\Gamma \gamma_1\phi_n\int_0^t g(x,t-r)\;\mbox{\rm d}\Gamma\,\;\mbox{\rm d} r-\\
\displaystyle -\frac{1}{\lambda_n^2}\int_\Gamma\gamma_1\phi_n\int_0^t g(x,t-r)\left [
\zeta_n'(r)-\int_0^r K_1(r-\nu)\zeta_n(\nu)\;\mbox{\rm d}\nu
\right ] \;\mbox{\rm d} r\,\;\mbox{\rm d}\Gamma
\\[5mm]
\displaystyle w_n'(t)=\int_\Gamma \gamma_1\phi_n\int_0^t g(x,t-r)\zeta_n(r)\;\mbox{\rm d} r\,\;\mbox{\rm d} \Gamma\,.
\end{array}
\end{equation}
Let now
\[
H^1_0(\Omega)\ni \xi=\sum _{n=1}^{+\infty}\frac{\xi_n}{\lambda_n}\phi_n(x)\,,\quad
L^2(\Omega)\ni \eta=\sum _{n=1}^{+\infty}\eta_n\phi_n(x)\,,\qquad \{\xi_n\}\,,\ \{\eta_n\}\in l^2
\]
and let \[
c_n=-\xi_n+i\eta_n
\]
Of course, $\{c_n\}$ is an arbitrary element of $l^2=
l^2(\mathbb{C})$ and our goal is the proof that the following \emph{moment problem} is solvable for every $\{c_n\}\in l^2$:
\begin{align*}
&\int_\Gamma\int_0^T g(T-r)E_n^{(1)}(r)\Psi_n\;\mbox{\rm d} r\,\;\mbox{\rm d}\Gamma=c_n\,,\\
& E_n^{(1)}(r)= \left ( \zeta_n'(r)-\int_0^r K_1(r-\nu)\zeta_n(\nu)\;\mbox{\rm d}\nu \right )+i\lambda_n \zeta_n(r) \,.
\end{align*}
Here $g$ \emph{is not} an arbitrary $L^2$ function, i.e. this is not a moment problem in the space $L^2\left (0,T;L^2(\Gamma)\right )$; \emph{it is a moment problem in the Hilbert space ${\mathcal{N}_1}$.} So, it is not really $E_n^{(1)}(r)\Psi_n$ which enters this moment problem but any projection of $E_n^{(1)}(r)\Psi_n$ on the Hilbert space ${\mathcal{N}_1}$: the moment problem to be studied is
\begin{equation}\label{ProbleMOMEproiettato}
\mathbb{M}_1 g=c_n
=\int_\Gamma\int_0^T g(T-r){\mathcal P}_{{\mathcal{N}_1}}\left (E^{(1)}_n(\cdot)\Psi_n\right )\;\mbox{\rm d} r\,\;\mbox{\rm d}\Gamma
\end{equation}
where ${\mathcal P}_{{\mathcal{N}_1}}$ is \emph{any fixed projection on ${\mathcal{N}_1}$. }
the operator $\mathbb{M}_1$ is the \emph{moment operator} of our control problem.
So, the controllability problem can be reformulated as follows: \emph{to prove the existence of a suitable projection ${\mathcal P}_{{\mathcal{N}_1}}$ such that the moment problem~(\ref{ProbleMOMEproiettato}) is solvable for every $\{c_n\}\in l^2$.} In fact, surely there exist projections for which the moment problem is not solvable: the projection $Ph=0$ for every $h$ is an example.
We are going to prove that the following special projection does the job:
\begin{equation}\label{eq:Laproiez}
\left ( {\mathcal P}_{{\mathcal{N}_1}} f\right )(t)=f(t)-\frac{1}{T}\int_0^T f(s)\;\mbox{\rm d} s\,.
\end{equation}
\begin{Remark}
{\rm
The projection ${\mathcal P}_{{\mathcal{N}_1}}$ is the orthogonal projection of $L^2(0,T;L^2(\Gamma))$ onto ${{\mathcal{N}_1}}$. In fact, for every $f\in L^2(0,T;L^2(\Gamma))$ and every $g\in {{\mathcal{N}_1}}$ we have
\begin{align*}
\int_\Gamma\int_0^T\overline{g}(x,t)\left [f-{\mathcal P}_{{\mathcal{N}_1}}f\right ](x,t)\;\mbox{\rm d} t\;\mbox{\rm d}\Gamma=\\
=\frac{1}{T}\int_\Gamma
\left [\int_0^T f(x,s)\;\mbox{\rm d} s\right ]\left [\int_0^T \overline{g}(x,t)\;\mbox{\rm d} t\right ]\;\mbox{\rm d}\Gamma=0\,.\mbox{~~~~\rule{1mm}{2mm}\par\medskip}
\end{align*}
}
\end{Remark}
Let us note that the results reported in Section~\ref{sect:sisteVontroREG} in particular show that
the operator $\mathbb{M}_1$ is continuous
and the image of $\mathbb{M}_1$ is closed with finite codimension (this is Lemma~\ref{Lemma:compa}).
So, it is sufficient that we prove that if $\{\bar \alpha_n\}\perp {\rm im}\, \mathbb{M}_1$ then $\{\alpha_n\}=0$ i.e. we must prove that
\[
\bigl (\, \langle \alpha_n,\mathbb{M}_1 g\rangle=0 \qquad \forall g\in {\mathcal{N}_1}\, \bigr )\ \implies \ \{\alpha_n\}=0
\]
i.e. we prove that if the following equality holds then $\{\alpha_n\}=0$:
\begin{equation}\label{eq:LaSerConProi}
\sum _{n=1}^{+\infty} \alpha_n \mathcal{P}_{{\mathcal{N}_1}}\left (\Psi_n\left [
\zeta_n'(\cdot)-\int_0^{(\cdot)} K(\cdot-\nu)\zeta_n(\nu)\;\mbox{\rm d}\nu +i\lambda_n\zeta_n(\cdot)
\right ]\right )=0\,.
\end{equation}
We introduce explicitly the projection~(\ref{eq:Laproiez}) and we see that we must prove $\{\alpha_n\}=0$ when the following equality holds:
\begin{align}
\nonumber
&\sum _{n=1}^{+\infty} \alpha_n\Psi_n \left [
\zeta_n'(t)-\int_0^t K_1(t-\nu)\zeta_n(\nu)\;\mbox{\rm d}\nu +i\lambda_n\zeta_n(t)
\right ]=\\
\label{eq:IlprobleFINA}
&= \frac{1}{T}
\sum _{n=1}^{+\infty} \alpha_n\Psi_n \left [
\zeta_n(T)-\int_0^T\int_0^t K_1(t-\nu)\zeta_n(\nu)\;\mbox{\rm d}\nu\;\mbox{\rm d} t +i\lambda_n\int_0^T\zeta_n(s)\;\mbox{\rm d} s
\right ]
\end{align}
Note that the numerical series on the right side of~(\ref{eq:IlprobleFINA}) converges since both the series~(\ref{eq:LaSerConProi}) and the series on the left of~(\ref{eq:IlprobleFINA}) converges,
thanks to Lemma~\ref{lemmaADDE}. The series on the right side of~(\ref{eq:IlprobleFINA}) is constant.
This implies that also the sum of the series on the left is constant and so its derivative is equal zero.
We shall prove that the derivative can be computed termwise. Accepting this fact, the proof that $\{\alpha_n\}=0$ is simple: the termwise derivative is
\begin{align}
\nonumber0=\sum _{n=1}^{+\infty}\alpha_n\Psi_n\left [\zeta_n''(t)-\int_0^t K(t-\nu)\zeta_n(\nu)\;\mbox{\rm d}\nu+i\lambda_n\zeta_n'(t)\right ]=\\
\nonumber =\sum _{n=1}^{+\infty}\alpha_n\Psi_n\left [-\lambda_n^2\zeta_n(t)+i\lambda_n\zeta_n'(t)\right ]=\\
\label{eq:laserieconSECOmembCOSTA}= i \sum _{n=1}^{+\infty} \lambda_n\alpha_n\Psi_n\left [\zeta_n'(t)+i\lambda_n
\zeta_n(t)\right ]\,.
\end{align}
We noted (in Lemma~\ref{lemmaADDE} item~\ref{lemmaADDEitem1}) that $L^2(\Omega)\times H^{-1}(\Omega)$-controllability with square integrable controls of the viscoelastic system is equivalent to the fact that
$
\left \{ \Psi_n\left [\zeta_n'(t)+i\lambda_n\zeta_n(t)\right ]\right \}
$
is a Riesz sequence in $ L^2(0,T;L^2(\Gamma))$ and so $\{\lambda_n\alpha_n\}=0$ i.e. $\{\alpha_n\}=0$. Of course we implicitly used $\{\lambda_n\alpha_n\}\in l^2$, a fact we shall prove now.
In order to complete the proof we must see that it is legitimate to distribute the derivative on the series~(\ref{eq:IlprobleFINA}) and that this implies in particular $\{\lambda_n\alpha_n\}\in l^2$.
The fact that $\left \{\Psi_n[\zeta_n'+i\lambda_n\zeta_n]\right \}$ is a Riesz sequence in $L^2(0,T;L^2(\Gamma))$ shows that we can distribute the series on the left hand side of~(\ref{eq:IlprobleFINA}), which can be written as
\begin{align}
\nonumber & \sum _{n=1}^{+\infty} \alpha_n\Psi_n \left [
\zeta_n'(t) +i\lambda_n\zeta_n(t)
\right ]\\
\label{eq:IntRfGperH1DArichiaH2}
&=
\int_0^t K_1(s-\nu)\left [ \sum _{n=1}^{+\infty} \alpha_n\Psi_n\zeta_n(\nu)\right ]\;\mbox{\rm d}\nu+{\rm const} \,.
\end{align}
So, it is sufficient that we study the differentiability of the series
\[
\sum _{n=1}^{+\infty} \alpha_n\Psi_n Z_n(t)\,,\qquad Z_n(t)= \left [
\zeta_n'(t) +i\lambda_n\zeta_n(t)
\right ]
\,.
\]
Using the definition~(\ref{eq:diZETAn}) we see that
\begin{align*}
\zeta_n(t)=\frac{1}{\lambda_n} \sin\lambda_n(t)+\frac{1}{\lambda_n}\left [ \int_0^t \int_0^s K(s-\tau)\sin\lambda_n\tau\;\mbox{\rm d}\tau\right ] \zeta_n(t-s)\;\mbox{\rm d} s\,,\\
\zeta_n'(t)=\cos\lambda_n(t)+\frac{1}{\lambda_n} \int_0^t \left [\int_0^s K(s-\tau)\sin\lambda_n \tau\;\mbox{\rm d} \tau\right ] \zeta_n'(s)\;\mbox{\rm d} s
\end{align*}
and so we get the following formula for $Z_n(t)$:
\begin{equation}
\label{eqDiZetaROMANO}
\left\{\begin{array}{l}
\displaystyle Z_n=E_n+\frac{1}{\lambda_n} K*S_n*Z_n\\[2mm]
\displaystyle \mbox{where $*$ denotes the convolution and where we define}\\[2mm]
S_n(t)=\sin\lambda_n t\,,\quad C_n(t)=\cos\lambda_n t\,,\quad E_n(t)=e^{i\lambda_n t}\,.
\end{array}\right.
\end{equation}
Gronwall inequality shows that $\{Z_n(t)\}$ is bounded on bounded intervals.
We introduce the notations
\[
F^{(*1)}(t)=F(t)\,,\quad F^{(*k)}=F*F^{(*(k-1))}\,.
\]
With these notation the formula for $Z_n(t)$ shows also that
\begin{align}
\nonumber Z_n&=E_n+\frac{1}{\lambda_n}K*S_n*E_n+\frac{1}{\lambda_n^2} K^{(*2)}*S_n^{(*2)}*Z_n\,,\\
\label{asympZn}&= E_n+\sum _{r=1}^K \frac{1}{\lambda_n^r}K^{(*r)}* S_n^{(*r)}*E_n+\frac{1 }{\lambda_n^{k+1}}P_{K,n}(t)\,.
\end{align}
The functions $P_{n,K}(t)$ and $P'_{n,K}(t)$ are bounded on bounded intervals,
\[
|P_{n,K}(t)|<M_K\,,\quad |P_{n,K}(t)|<M_K
\]
where $ M_K $ does not depend on $ n $ and $ t\in [0,T] $.
So, using the fact that $\{\Psi_n\}$ is bounded in $L^2(\Gamma)$ and~(\ref{eq:stimAUTOV}) (see~\cite[Lemma~4.4]{PandLibro})
we see that
\[
\sum _{n=1}^{+\infty} \alpha_n\Psi_n\frac{1}{\lambda_n^K}M_{n,K}(t)
\]
is of class $C^1$ (and termwise differentiable) when $K$ is large enough.
We fix an index $K$ with this property and we consider the series of each one of the terms in~(\ref{asympZn}) for which $r\geq 1$:
\begin{equation}\label{eq:LeDuESerIe}
\sum _{n=1}^{+\infty}\alpha_n\Psi_n\frac{1}{\lambda_n}K*S_n*E_n\,,\qquad \sum _{n=1}^{+\infty}\alpha_n\Psi_n\frac{1}{\lambda_n^r}K^{(*r)}*S_n^{(*r)}*E_n\,.
\end{equation}
$L^2$-convergence of the series is clear. We prove that they converge to $H^1$-functions.
We consider the first series. We compute the convolution $(S_n*E_n)$ and we see that this series is equal to
\begin{align*}
&\frac{1}{2i}\sum _{n=1}^{+\infty}\alpha_n\Psi_n \frac{1}{\lambda_n}\int_0^t K(s)\left [
(t-s)e^{i\lambda_n(t-s)}-\frac{1}{\lambda_n} \sin\lambda_n(t-s)
\right ]\;\mbox{\rm d} s
\,.
\end{align*}
The series inside the integral converges in $ L^2(0,T) $ thanks to item~\ref{lemmaADDEitem3} in Lemma~\ref{lemmaADDE}.
Its termwise derivative is:
\begin{align*}
&\frac{1}{2i}\sum _{n=1}^{+\infty} \alpha_n\Psi_n\int_0^t K(s)\left [
\frac{1}{\lambda_n} e^{i\lambda_n}(t-s)+i(t-s)e^{i\lambda_n (t-s)}-\cos\lambda_n (t-s)
\right ]\;\mbox{\rm d} s=\\
&\frac{1}{2i}\int_0^t K(s)\left [
\sum _{n=1}^{+\infty} \alpha_n\Psi_n \left (
\frac{1}{\lambda_n} e^{i\lambda_n}(t-s)+i(t-s)e^{i\lambda_n (t-s)}-\cos\lambda_n (t-s)\right )
\right ]\;\mbox{\rm d} s\,.
\end{align*}
This series is $L^2$-convergent thanks to Lemma~\ref{lemmaADDE}.
A similar argument holds for every $ r\leq K $. So, $ \sum _{n=1}^{+\infty} \alpha_n\Psi_nE_n $ converges in $ H^1(0,T) $ for every $ T $, in particular $ T>T_0 $.
From~\cite[Lemma~3.4]{PandLibro} and the appendix, we see that $\alpha_n=\delta_n/\lambda_n$, $\{\delta_n\}\in l^2$, and that the derivative of the series can be computed termwise.
\emph{This ends the proof of item~\ref{item1delTeoINh10} in Theorem~\ref{TeoINh10}.}
\begin{Remark}
{\rm
This prrof holds in particular if $K=0$, and gives an alternative proof to the result in~\cite{EverdZUAZArch}. An important additional property in~\cite{EverdZUAZArch} is that a smooth steering control solves an optimization problem (and that its essential support is relatively compact in $[0,T]\times\Gamma$).~~\rule{1mm}{2mm}\par\medskip
}
\end{Remark}
\section{\label{Sect:MoreREGULARcontrols}When the control is smoother}
In this section we prove item~\ref{item2delTeoINh10} and~\ref{item3delTeoINh10} in Theorem~\ref{TeoINh10}.
We note that $f\in H^2_0(0,T;L^2(\Gamma))$ if and only if
\begin{align*}
&f(t)=\int_0^t (t-s)g(s)\;\mbox{\rm d} s\,,\\
& g\in {\mathcal{N}_2}=\left \{
g\in L^2(0,T;L^2(\Gamma))\,,\ \int_0^T g(s)\;\mbox{\rm d} s=0\,,\quad \int_0^T (T-r)g(r)\;\mbox{\rm d} r=0
\right \}\,.
\end{align*}
An analogous representation holds if $f\in H^k_0(0,T;L^2(\Gamma))$.
Using these characterizations, we integrate by parts formulas~(\ref{eqDIu}) (with $F=0$) and~(\ref{eq:diuPRIMO}) (with zero initial conditions) and when $f\in H^2_0$ we find
\begin{equation}\label{eq:LauEuprimoCONfH2}
\left\{\begin{array}{ll}
u(t)=D\int_0^t (t-r)g(r)-{\mathcal A}^{-1} \int_0^t R_-(t-s)Dg(s)\;\mbox{\rm d} s=\hat u_g(t)\,,\\
u'(t)=D\int_0^t g(r)\;\mbox{\rm d} r-\int_0^t R_+(t-s)Dg(r)\;\mbox{\rm d} r=\tilde u_g(t)\,.
\end{array}\right.
\end{equation}
This is similar to~(\ref{eq:ondINTepartiU}) and~(\ref{eq:ondDerivINTepartiU}) (and now Remark~\ref{Rema:sullaroBUSTrezzO} applies both to $u$ and $u'$).
Let
\[
L_-(t)w=\int_0^t K(r)R_-(t-r)w\;\mbox{\rm d} r\,,\qquad
L_+(t)w=\int_0^t K(r)R_+(t-r)w\;\mbox{\rm d} r\,.
\]
We have, from~(\ref{eq:soluVisco}) and~(\ref{eq:volteDERIVATA}),
\begin{equation}
\label{eq:LauEwprimoCONfH2}
\left\{\begin{array}{ll}
w_f(t)=\hat u_g(t)+{\mathcal A}^{-1}\int_0^t L_-(t-s)w(s)\;\mbox{\rm d} s\,,\\
w_f'(t)=\tilde u_g(t)+\int_0^t L_+(t-s) w(s)\;\mbox{\rm d} s\,.
\end{array}\right.
\end{equation}
The proof of item~\ref{item2delTeoINh10} in Theorem~\ref{TeoINh10} consist in
two parts: first we prove the regularity property $\left (w_f(T),w_f'(T)\right )\in {\rm dom}\,{\mathcal A}^2\times {\rm dom}\,{\mathcal A}$ and then we prove that $f\mapsto\left (w_f(T),w_f'(T)\right )\in {\rm dom}\,{\mathcal A}^2\times {\rm dom}\,{\mathcal A}$ is surjective in this space.
The proof of item~\ref{item3delTeoINh10} in Theorem~\ref{TeoINh10} is the proof that the corresponding regularity does not hold, i.e. that {due to the memory, there exist functions $f\in H^k(0,T];L^2(\Gamma))$ such that $\left (w_f(T), w'_f(T)\right )\notin {\rm dom}\,{\mathcal A}^k\times {\rm dom}\,{\mathcal A}^{k-1}$.}
We proceed as follows: we first examine the regularity issue (i.e. the positive result, when $f\in H^2_0(0,T;L^2(\Gamma))$ and the lack of regularity if $f$ is smoother) in subsection~\ref{Psubs:PiuregoMENOrego}. In the subsection~\ref{H2Controll} we prove $\left ({\rm dom}\,{\mathcal A}^2\times{\rm dom}\,{\mathcal A}\right )$-controllability when $f\in H^2_0(0,T;L^2(\Gamma))$.
\subsection{\label{Psubs:PiuregoMENOrego}Regularity and lack of regularity}
We consider $w_f(t)$ in~(\ref{eq:LauEwprimoCONfH2}). A step of Picard iteration gives
\begin{equation}\label{eq:diwinH2}
w_f(t)=\hat u_g(t)+{\mathcal A}^{-1}\int_0^t L_-(t-s)\hat u_g(s)\;\mbox{\rm d} s+A^{-1} \left (L_-^{(*2)}*w\right) (t)\,.
\end{equation}
We know from~\cite{EverdZUAZArch} that $\hat u_g(T)\in H^2_0(\Omega)={\rm dom}\,A={\rm dom}\,{\mathcal A}^2$ (also seen from~(\ref{eq:LauEuprimoCONfH2})). As we noted, $\hat u_g(t)$ has this regularity for $t=T$ but not for $t\in(0,T)$. Instead, we prove that $w_f(t)-\hat u_g(t)\in {\rm dom}\,{\mathcal A}^2$ for every $t\in [0,T]$. This is clear for the third addendum on the right side of~(\ref{eq:diwinH2}). We examine the second term, which is
\[
{\mathcal A}^{-1}\int_0^t L_-(t-s)D\int_0^s(s-r) g(r)\;\mbox{\rm d} r\,\;\mbox{\rm d} s-A^{-1} \left (L_-* R_-*g\right )(t) \,.
\]
The second addendum takes values in ${\rm dom}\, A$. We integrate by parts the first addendum and we get that it is equal to
\begin{align*}
& {\mathcal A}^{-1}\int_0^t L_-(t-s)D\int_0^s(s-r) g(r)\;\mbox{\rm d} r\,\;\mbox{\rm d} s\\
&=
{\mathcal A}^{-1}\int_0^t K(t-r)\int_0^r R_-(r-s)D\int_0^s (s-\nu) g(\nu)\;\mbox{\rm d}\nu\,\;\mbox{\rm d} s\,\;\mbox{\rm d} r\\
&= -A^{-1}\int_0^t K(t-r)\left [
D\int_0^r (r-\nu)g(\nu)\;\mbox{\rm d}\nu\right.\\
&\left.-\int_0^r R_+(r-s)Dg(s)\;\mbox{\rm d} s
\right ]\;\mbox{\rm d} r\in {\rm dom }\,A\,.
\end{align*}
The fact that $w_f'(T)\in {\rm dom}\,{\mathcal A}$ follows from the representation of $w_f'(T)$ in terms of $w_f(t)$ in the second line of~(\ref{eq:LauEwprimoCONfH2}). This fact shows that the integral even takes values in ${\rm dom}\,{\mathcal A}^2={\rm dom}\,A$ for every $t\in [0,T]$ while the first addendum $\tilde u_g(T)\in {\rm dom}\,{\mathcal A}$ from~\cite{EverdZUAZArch} (also seen from~(\ref{eq:LauEuprimoCONfH2})).
In conclusion, we proved that
\[
f\in H^2_0(0,T;L^2(\Gamma))\ \implies \left (w_f(T),w'_f(T)\right )\in {\rm dom}\,{\mathcal A}^2\times {\rm dom}\,{\mathcal A}\,.
\]
Now we prove that this result cannot be improved when $f$ is smoother. It is sufficient that we show
\[
f\in H^3(0,T;L^2(\Gamma))\centernot\implies w_f(T)\in {\rm dom}\,{\mathcal A}^3\,.
\]
Note that $f\in H^3(0,T;L^2(\Gamma))$ when
\[
f(t)=\int_0^t (t-s)^2g(s)\;\mbox{\rm d} s\,,\qquad \int_0^T (T-s)^kg(s)\;\mbox{\rm d} s=0 \quad k=0\,,\ 1\,,\ 2\,.
\]
We use this representation of $f$ and we integrate by parts the integral in~(\ref{eqDIu}) with $F=0$. We get
\begin{align*}
u_f(t)=
D\int_0^t (t-r)^2g(r)\;\mbox{\rm d} r+2A^{-1}D\int_0^t g(r)\;\mbox{\rm d} r\\
+{\mathcal A}^{-3}\left [ -2{\mathcal A}\int_0^t R_+(t-s)Dg(s)\;\mbox{\rm d} s\right ] \,.
\end{align*}
The last addendum belongs to ${\rm dom}\,{\mathcal A}^3$. In the following computation we write $G(t)$ for any term which takes values in ${\rm dom}\,{\mathcal A}^3$, not the same at every occurrence. So,
\[
u_f(t)=\int_0^t P(t-r)g(r)\;\mbox{\rm d} r+ G(t)\,,\qquad P(t)=t^2D+2A^{-1}D\,.
\]
We replace in the expression of $w_f(t)$ in~(\ref{eq:soluVisco}). Two steps of Picard iteration
gives the following representation for $w(t)$:
\begin{align*}
w(t)=\int_0^t P(t-\nu)Dg(\nu)\;\mbox{\rm d}\nu+{\mathcal A}^{-1}\int_0^t L_-(t-r)\int_0^r(r-\nu)^2Dg(\nu)\;\mbox{\rm d}\nu\,\;\mbox{\rm d} r\\
+{\mathcal A}^{-2}\int_0^t L_-(t-r)\int_0^r L_-(r-s)\int_0^s (s-\nu)^2Dg(\nu)\;\mbox{\rm d}\nu\, \;\mbox{\rm d} s\,\;\mbox{\rm d} r+G(t)\,.
\end{align*}
The integral in the second line can be integrated by parts so that the second line takes values in ${\rm dom}\,{\mathcal A}^3$. The first addendum is zero for $t=T$. So, we must study the regularity of
\begin{align*}
{\mathcal A}^{-1}\int_0^t L_-(t-r)\int_0^r(r-\nu)^2Dg(\nu)\;\mbox{\rm d}\nu\,\;\mbox{\rm d} r\\
=-{\mathcal A}^{-2}\int_0^t K(t-s)\int_0^s \frac{\;\mbox{\rm d}}{\;\mbox{\rm d} r} R_+(s-r)\int_0^r (r-\nu)^2Dg(\nu)\;\mbox{\rm d}\nu\,\;\mbox{\rm d} r\,\;\mbox{\rm d} s\\
=-{\mathcal A}^{-2} \int_0^t K(t-s)\int_0^s (s-\nu)^2Dg(\nu)\;\mbox{\rm d}\nu\,\;\mbox{\rm d} s\\
+2{\mathcal A}^{-2}\int_0^t K(t-s)\int_0^s R_+(s-r)\int_0^r (r-\nu)Dg(\nu)\;\mbox{\rm d}\nu\, \;\mbox{\rm d} r\,\;\mbox{\rm d} s\,.
\end{align*}
The last integral can be integrated by parts again and subsumed in the term $G(t)$. Instead,
\[
{\mathcal A}^{-2}D\int_0^T (T-\nu)^2\int_0^\nu K(s) g(\nu-s)\;\mbox{\rm d} s\,\;\mbox{\rm d} \nu \notin{\rm dom}\, {\mathcal A}^3
\]
since
\[
D\int_0^T (T-\nu)^2\int_0^\nu K(s) g(\nu-s)\;\mbox{\rm d} s\,\;\mbox{\rm d} \nu \notin{\rm dom}\, {\mathcal A}
\]
as it is seen for example when
$g(x,\nu)=g_0(x)g_1(\nu)$ and $g_1$ such that
\[
\int_0^T (T-\nu)^2\int_0^\nu K(s) g_1(\nu-s)\;\mbox{\rm d} s\,\;\mbox{\rm d} \nu\neq 0\,.
\]
As a specific example, when $\Omega=(0,1)$ (and $\Gamma=\{0\}$) then $Df=(1-x)f\not\in H^1_0(0,1)$ unless $f=0$.
Similar arguments hold if $f\in H^k(0,T;L^2(\Gamma))$ and $k>3$.
\subsection{\label{H2Controll}${\rm dom}\, {\mathcal A}^2\times {\rm dom}\,{\mathcal A}$-controllability when $f\in H^2_0(0,T;L^2(\Gamma))$}
This part of the proof is similar to that in the case $f\in H^1_0(0,T;L^2(\Gamma))$ and it is only sketched.
A simple examination of formulas~(\ref{eq:LauEwprimoCONfH2}) and (\ref{eq:diwinH2}) shows that the map $f\mapsto \left (w_f(T)-u_f(T),w'_f(T)-u'_f(T)\right )$ from $H^2_0(0,T;L^2(\Gamma))$ to ${\rm dom}\,{\mathcal A}^2\times {\rm dom}\,{\mathcal A} $ is compact. Hence we must prove
\[
\left [\, \left \{ (w_f(T),w'_f(T))\,,\ f\in L^2(0,T;L^2(\Gamma))\right \} \, \right ]^\perp=\{0\}
\]
(the orthogonal is respect to ${\rm dom}\,{\mathcal A}^2\times {\rm dom}\,{\mathcal A} =\left (H^2(\Omega)\cap H^1_0(\Omega)\right )\times H^1_0(\Omega)$).
We use again the formulas for $w_n(t)$ and $w_n'(t)$ in~(\ref{eq:RApprewNDopoINTEparti}) where $g(t)=f'(t)$ has now to be replaced by $\int_0^t g(s)\;\mbox{\rm d} s$. We see that
\begin{align*}
w_n(T)=\frac{1}{\lambda_n}\int_\Gamma\int_0^T g(T-r)\left [
\Psi_n\left (
\zeta_n(r)+\int_0^r K_2(r-s)\zeta_n(s)\;\mbox{\rm d} s
\right )
\right ]\;\mbox{\rm d} r\,\;\mbox{\rm d}\Gamma\,,\\
w_n'(T)=\frac{1}{\lambda_n}\int_\Gamma\int_0^T g(x,T-r)\left [\Psi_n\left (-\zeta_n'(r)+\int_0^r K_1(r-s)\zeta_n(s)\;\mbox{\rm d} s
\right )\right ]\;\mbox{\rm d} r\;\mbox{\rm d}\Gamma
\end{align*}
where
\[
K_2(t)=\int_0^t K_1(s)\;\mbox{\rm d} s=\int_0^t (t-s)K(s)\;\mbox{\rm d} s\,.
\]
We want to reach
\[
\xi(x)=\sum _{n=1}^{+\infty}\frac{\xi_n}{\lambda_n^2}\phi_n(x)\,,\quad \eta(x)=\sum _{n=1}^{+\infty} \frac{\eta_n}{\lambda_n}\phi_n(x)\,,\quad \{\xi_n\}\in l^2\,,\ \{\eta_n\}\in l^2\,.
\]
So, we must solve the moment problem in ${\mathcal{N}_2}$
\[
\int_\Gamma\int_0^T g(T-r)P_2\left (\Psi_n E^{(2)}_n\right )\;\mbox{\rm d} r\,\;\mbox{\rm d}\Gamma =c_n\,,\qquad \{c_n\}=\{-\eta_n+i\xi_n\}\in l^2
\]
where $P_2$ is the orthogonal projection on ${\mathcal{N}_2}$:
\begin{equation}\label{eq:defiPdue}
\begin{array}{l}
\left (P_2f\right )(x,t)=f(x,t)-sA_f- B_f \,,\\
A_f=\frac{12}{T^3}\int_0^T \left (s-\frac{T}{2}\right )f(x,s)\;\mbox{\rm d} s\,,\quad B_f=\frac{1}{T}\int_0^T f(s)\;\mbox{\rm d} s-\frac{1}{2}T A_f
\end{array}
\end{equation}
and
\[
E_n^{(2)}(r)=\left (
\zeta_n'(r)-\int_0^r K_1(r-s) \zeta_n(s)\;\mbox{\rm d} s
\right )+i
\left (
\lambda_n \zeta_n(r)+\lambda_n \int_0^r K_2(r-s)\zeta_n(s)\;\mbox{\rm d} s\,.
\right )
\]
Proceeding as in the case $f\in H^1_0(0,T;L^2(\Gamma))$ we see that we must prove $\{\alpha_n\}=0$ when the following equality holds:
\begin{align}
\nonumber \sum _{n=1}^{+\infty} \alpha_n\Psi_n\left( \zeta_n'(r)+i\lambda_n\zeta_n(r)\right )\\
\nonumber =\sum _{n=1}^{+\infty}\alpha_n\Psi_n\left [\int_0^r K_1(r-s)\zeta_n(s)\;\mbox{\rm d} s+i\int_0^r K_2(r-s)\lambda_n\zeta_n(s)\;\mbox{\rm d} s\right ]\\
\label{EquaProieNo2MoME}+\sum _{n=1}^{+\infty} \alpha_n\Psi_n\left [sA_n+B_n\right ]
\end{align}
where $A_n$ and $B_n$ are as in~(\ref{eq:defiPdue}) with $f$ replaced by
\[
\left (\zeta_n'-K_1*\zeta_n\right )+i\lambda_n\left ( \zeta_n+K_2*\zeta_n\right )\,.
\]
In fact, it is legitimate to distribute the series on the sum since each one of these series converge because $\left\{\Psi_n\left (\zeta_n'(r)+i\lambda_n\zeta_n(r)\right )\right\}$ is a Riesz sequence in $L^2(0,T;L^2(\Gamma))$
(see Lemma~\ref{lemmaADDE} item~\ref{lemmaADDEitem1}).
Equality~(\ref{EquaProieNo2MoME}) is similar to~(\ref{eq:IlprobleFINA}), with the right hand side of class $C^2$. So we get
\[
\alpha_n=\frac{\beta_n}{\lambda_n}\,,\qquad \{\beta_n\}\in l^2
\]
and we can compute the termwise derivative of the series. Computing the derivative and noting that $K_1'=K$, $K_2'=K_1$ we get
\[
\sum _{n=1}^{+\infty} \beta_n \Psi_n \left [-\lambda_n \zeta_n(r)+i\zeta_n'(r)\right ]= \sum _{n=1}^{+\infty} \beta_n \Psi_n\int_0^r K_1(r-s) \zeta_n(s)\;\mbox{\rm d} s-i\sum _{n=1}^{+\infty}\alpha_n\Psi_nA_n\,.
\]
This is the same as~(\ref{eq:IntRfGperH1DArichiaH2}) and so we get $\{\beta_n\}=0$, i.e. $\{\alpha_n\}=0$ as wanted.
\section*{Appendix}
In this appendix we prove the following simple result, which however is crucial in the proof of our theorem (see also~\cite[p.~323]{GohbergKrein}).
\begin{Lemma}
Let $\mathbb{J}$ be a denumerable set and let the sequence $\{e_n\}_{n\in \mathbb{J}}$ in a Hilbert space $H$ have the following properties:
\begin{enumerate}
\item\label{itemref1} if $\{\alpha_n\}\in l^2$ then $\sum \alpha_n e_n$ converges in $H$;
\item\label{itemref2} $\left \{\langle f,e_n\rangle\right \}\in l^2$ for every $f\in H$;
\item\label{itemref3} the subspace $M=\left \{ \left \{\langle f,e_n\rangle\right \}\,,\ f\in H\right \}$ is closed and its codimension is finite, equal to $k$.
\end{enumerate}
Under these conditions, there exists a set $K\subseteq \mathbb{J}$ of precisely $k$ indices
such that $\{e_n\}_{n\notin K}$ is a Riesz sequence.
\end{Lemma}
{\noindent\bf\underbar{Proof}.}\
Let $\mathbb{M}$ be the operator
\[
\mathbb{M} f= \left \{\langle f,e_n\rangle\right \}\quad H\mapsto l^2\,.
\]
It is known that $\mathbb{M}$ is a closed operator (see~\cite{AvdoIVanovbook} and~\cite[Theorem~3.1]{PandLibro}) and Assumption~\ref{itemref2} shows that its domain is closed, equal to $H$. Hence it is a continuous operator.
$M={\rm im}\,\mathbb{M}$ and Assumption~\ref{itemref3} shows that there exist $k$ (and not more) linearly independent sequences $\left \{ \alpha_n^i\right \}_{n\in\mathbb{J} }$, $i=1\,,\dots\, k$
such that
\begin{equation}\label{eq:dellaDipeLINE}
\sum _{n\in \mathbb{J}} \alpha_n^i \langle f,e_n\rangle=0\qquad \forall f\in H\,.
\end{equation}
Note that the assumptions of this lemma does not depend on the order of the elements $e_n$. If we exchange the order of two elements $e_{n_1}$ ed $e_{n_2}$ then the corresponding elements $\alpha_{n_1}^i$ and $\alpha_{n_2}^{i}$ are exchanged for every $i=1\,,\dots,k$. So we can assume $\alpha_i^{i}\neq 0$ and without restriction $\alpha_i^{i}=1$. Hence, every $e_i$, $i=1\,,\dots\,, k$ is a linear combination of the elements $e_{n}$, $ n\in\mathbb{J}\setminus \{ 1\,,\dots\,,\ k\}$.
The operator
\[
\mathbb{M}' f= \left \{\langle f,e_n\rangle_{n\in\mathbb{J}\setminus K}\right \}\quad H\mapsto l^2(\mathbb{J}\setminus K)
\]
has dense image in $l^2(\mathbb{J}\setminus K) $ otherwise we can find $\gamma$ orthogonal to its image, and adding $k$ entries equal to zero in front, we have an element orthogonal to ${\rm im}\,\mathbb{M}$, linearly independent of $\{\alpha_i\}$ and this is not possible.
Let $\mathbb{M}_0$ be the operator on $H$
\[
\mathbb{M}_0 f=\left \{\langle f,e_n\rangle \quad n\in K\right \}\,.
\]
Then, $\mathbb{M}$ is the direct sum $\mathbb{M}=\mathbb{M}_0\oplus \mathbb{M}'$ and Lemma~\ref{Lemma:compa} shows that ${\rm im}\,\mathbb{M}$ is closed. Hence,
the image of $\mathbb{M}'$ is closed too, since ${\rm im}\,\mathbb{M}$ is closed. Consequently, $\mathbb{M}'$ is
surjective from $H$ to $l^2\left (\mathbb{J}\setminus \{ 1\,,\dots\,,\ k\}\right )$ and boundedly invertible. It follows that $\left\{ e_n\right \}_{n\in\mathbb{J}\setminus K} $ is a Riesz sequence.~~\rule{1mm}{2mm}\par\medskip
|
1,108,101,564,529 | arxiv | \section{Introduction}
The Large Magellanic Cloud (LMC) has long been recognized as a fundamental
benchmark for a wide variety of astrophysical studies. As the closest bulge-less dwarf disk galaxy
\citep{b12}, it has turned out to be the ideal local
analog for the detailed study of these most common and primeval galaxies.
Ages and abundances of LMC field star populations are prime indicators of the galaxy's
chemical evolution and star formation history (SFH). This becomes even more relevant since
its formation and chemical evolution cannot be fully traced from its star cluster populations, due
to the well-known extended age gap. The LMC age-metallicity
relationship (AMR) has been the subject of a number of studies
\cite[among others]{oetal91,hetal99,cetal05,retal11}.
Among them, two perhaps best summarize our current knowledge in this field. First,
\citet[hereafter CGAH; see
also references therein]{cetal11} have examined the AMR for field star populations, based on Calcium
triplet spectroscopy of individual red giants and BVRI photometry in ten 34$\arcmin$$\times$33$\arcmin$
LMC fields. They found that:
i) the AMRs for their fields are statistically indistinguishable; ii) the disk AMR is similar
to that of the LMC star clusters and is well reproduced by closed-box models or models with a small degree of
outflow; iii) the lack of clusters with ages between 3 and 10 Gyr is not observed in the field
population; iv) the age of the youngest population observed in each field increases with galactocentric
distance; v) the rapid chemical enrichment observed in the last few Gyrs is only observed in fields
with R$<$7kpc; vi) the metallicity gradient observed in the outer disk can be explained by an increase
in the age of the youngest stars and a concomitant decrease in their metallicity; and vii) they find much
better evidence for an outside-in than inside-out formation scenario, in contradiction to generic $\Lambda CDM$ models.
Secondly, \citet[hereafter HZ09; see
also references therein)]{hz09} presented the first-ever global,
spatially-resolved reconstruction of the SFH, based on the application of their
StarFISH analysis software to the multiband photometry of twenty million stars from the Magellanic
Clouds Photometric Survey. They found that there existed a long relatively quiescent epoch (from $\sim$ 12 to 5 Gyr ago)
during which the star formation was suppressed throughout the LMC; the metallicity also remained stagnant during
this period. They
concluded that the field and cluster star formation modes have been tightly coupled throughout the LMC's history.
Although these studies represent the state-of-the-art of our knowledge of the LMC AMR, they leave
unanswered a number of outstanding questions:
What caused the general lull in SF between $\sim$ 5 and 12 Gyr ago? Are the cluster and field AMRs really tightly
coupled? Can the LMC AMR best be described by a closed-box, bursting or other chemical evolution model? What, if any,
are the radial dependences?
In addition, HZ09 did not go deep enough to derive the full SFH from information on the Main Sequence
(MS). They reached a limiting magnitude between $V$ = 20 and 21 mag, depending on the local degree of
crowding in the images, corresponding to stars younger than 3 Gyr old on the MS if the theoretical isochrones
of \citet{getal02} and a LMC distance modulus of 18.5 mag are used.
Thus, the
advantages of covering an enormous extension of the LMC is partially offset by the loss in depth of the
limiting magnitude. On the other hand, the ten fields of CGAH cover a rather small fraction of the whole
LMC. Therefore, it is desirable to obtain an overall deeper AMR for the LMC which also covers a larger area. Previous
AMRs have been founded on
theoretical isochrones, numerical models, or synthetic Color-Magnitude Diagrams (CMDs), so that an AMR
built from actual measured ages and metallicities is very valuable. A comprehensive comparison between the
field and cluster AMRs obtained using the same procedure is also lacking. All these aims demand
the availability of a huge volume of high quality data as well as a powerful technique to provide both
accurate ages and metallicities.
In this paper we address these issues for the first time. We make use of an unprecedented
database of some 5.5 million stars measured with the Washington $CT_1$ photometric system, which are
spread over a large part of the LMC main body. From this database, we produce the LMC field AMR from the birth of
the galaxy until $\sim$ 1 Gyr ago, using the $\delta$$T_1$ index and the standard giant branch isoabundance
curves to estimate ages and metallicities, respectively, of the most representative field populations.
These provide approximately independent measurements of these two quantities, minimizing the
age-metallicity degeneracy problem.
In addition, this is the
first overall LMC field star AMR obtained from Washington data; thus complementing those derived from other data sets
such as HZ09 or the AMR obtained from Washington data for LMC clusters \citep{p11a}.
Finally, we homogeneously compared the derived field star AMR to that for the LMC cluster
population with ages and metallicities put on the same scales using these two Washington datasets. This kind of
comparison has not been accomplished
before. The paper is organized as follows: Section 2 briefly describes the data handling and analysis from which
\citet{pgm12} estimated the field star ages and metallicities. Section 3 deals with the aforementioned issue of
a comprehensive AMR of the LMC field star population. In Section 4 we discuss our results and compare them with
previous studies, while Section 5 summarizes our major findings.
\section{Data handling and scope}
We obtained Washington photometric data
at the Cerro-Tololo Inter-American Observatory
(CTIO) 4 m Blanco telescope with the Mosaic \,II camera attached (36$\arcmin$$\times$36$\arcmin$ field onto a
8K$\times$8K CCD detector array) of twenty-one LMC fields, concentrated in the main body but mostly
avoiding the very crowded bar regions.
We refer the reader to \citet{pgm12} for details about the observations and
reduction and analysis of the data. Briefly, Piatti et al. analysed the $C$ and $T_1$ limiting magnitudes reached for
a 50\% completeness level from extensive artificial star tests,
produced CMDs, Hess-diagrams, MS star luminosity functions, Red Clump star $T_1$ mag histograms,
RGB distributions, etc, and presented a thorough description of the uncertainties involved and of the techniques used.
The processed data are much deeper than those used by HZ09 and generally
reach well below the MS Turnoffs (TOs) of the oldest stellar populations in the LMC
($T_{1_o}$ $\sim$ 19.9 - 21.4 mag). In addition, the
total area covered is about 2.5 times larger that that of CGAH.
We subdivided each 36$\arcmin$$\times$36$\arcmin$ field into 16 uniform 2K$\times$2K regions
(9$\arcmin$$\times$9$\arcmin$ each).
The stellar photometry was performed using the
star-finding and point-spread-function (PSF) fitting routines in the {\sc DAOPHOT/ALLSTAR} suite of programs
\citep{setal90}.
The standard Allstar - Find - Subtract procedure was repeated three times for each frame.
Finally, we combined all the independent measurements of the stars in the different filters
using the stand-alone {\sc DAOMATCH} and {\sc DAOMASTER}
programmes, kindly provided by Peter Stetson.
\citet{pgm12} used the so-called "representative" population, defined in
\citet{getal03}, to measure ages for the 21 fields in the same way as \citet{p12} did
for 11 fields of the Small Magellanic Cloud (SMC).
\citet{getal03} assumed that the observed MS in each LMC field
is the result of the superposition of MSs with different TOs (ages) and
constant luminosity functions. Thus, the intrinsic number of stars belonging to any MS interval comes from
the difference of the total number of stars in that interval and that of the adjacent intervals.
Therefore, the biggest difference is directly related to the most populated TO. This "representative" AMR differs
from those derived from modeled SFHs in the fact
that it does not include complete information on all stellar populations, but accounts for the dominant population
present in each field. Minority populations are not considered, nor dominant populations younger than $\sim$ 1 Gyr, due to our inability to age-date them.
The method has turned out to be a powerful tool for revealing the primary trends in an efficient and robust way
\citep{petal03a,petal03b,petal07}.
\citet{pgm12} clearly identified the representative star populations in the 21 studied LMC fields,
which were typically $\sim$ 25\%-50\% more frequent than the second most numerous population. They derived ages from
the $\delta$$T_1$ index, calculated by determining the difference in the $T_1$ magnitude of the red clump (RC) and
the representative MS TO \citep{getal97}. The $\delta (T_1)$ index has proven to be a powerful tool to derive
ages for star clusters older than 1 Gyr, independently of their metallicities \citep{betal98,petal02,petal09,petal11a,p11a}.
Indeed, \citet{getal97} showed that $\delta$($T_1$) is very well-correlated with $\delta$($R$) (correlation coefficient
= 0.993) and with $\delta$($V)$. We then derived ages from the
$\delta T_1$ values using equation (4) of \citet{getal97}, which was obtaining by fitting $\delta (T_1)$ values of
star clusters with well-known age estimates. This equation is only calibrated
for ages larger than 1 Gyr, in particular because the magnitude of the He-burning stage varies with age
for such massive stars, so that we are not able to produce ages for younger representative populations. Note that
this age measurement technique does not require
absolute photometry and is independent of reddening and distance as well. An additional advantage is that we do not
need to go deep enough to see the extended MS of
the representative star population but only slightly beyond its MS TO. The representative MS TO $T_1$ magnitude for
each subfield turned out to be
on average $\sim$ 0.6 mag brighter than the $T_1$ mag for the 100$\%$ completeness level of the respective subfield,
so that Piatti et al. actually
reach the TO of the representative population of each subfield.
Note that the representative stellar population is not
necessarily the oldest one reached in a subfield. Their Figs. 3 to 23 illustrate
the performance of their photometric data.
In their Table 5, they presented
the final ages and their dispersions. Such dispersions have been calculated bearing in mind the
broadness of the $T_1$ mag distributions of the representative MSTOs and RCs, and not just simply the photometric errors
at $T_1$(MSTO) and $T_1$(RC) mags, respectively. The former are clearly larger
and represent in general a satisfactory estimate of the age spread around the prevailing population ages, although
a few individual subfields have a slightly larger age spread. We refer the reader to the companion paper by \citet{pgm12}
for details concerning the methods and limitations and uncertainties involved.
The mean metallicity for each representative field population was
obtained by first entering the positions of the representative giant branch
into the [$M_{T_1}$, $(C-T_1)_o$] plane with the Standard Giant Branch (SGB) isoabundance curves traced by
\citet{gs99}. This was done to obtain, by interpolation, metal abundances
([Fe/H]) with typical errors of $\sim$ 0.20 dex. Then, they applied the appropriate age correction
to these metallicities using the age-correction procedure of \citet{getal03}, which provides
age-corrected metallicities in good general agreement with spectroscopic values \citep{petal10}.
The resulting metallicities and their dispersions are compiled in Table 6 of \citet{pgm12}.
Tables 5 and 6 of Piatti et al. 2012 are reproduced here as Tables 1 and 2 for completeness sake.
\section{The AMR}
One of the unavoidable complications in analysing measured ages and metallicities is that they have associated uncertainties.
Indeed, by considering such errors, the interpretation
of the resulting AMR can differ appreciably from that obtained using only the measured ages and metallicities without
accounting for their errors. However, the treatment of
age and metallicity errors in the AMR is not a straightforward task. Moreover, even if errors did not play an
important role, the binning of age/metallicity
ranges could also bias the results. For example, fixed age intervals have commonly been used to build cluster age
distributions using the same cluster database \citep{botal06,wetal09,p10}, with remarkably different results depending on
the details of the binning process.
These examples show that a fixed age bin size is not appropriate for yielding the intrinsic age distribution, since
the result depends on the chosen age interval and the age errors are typically a strong function of the
age. A more robust age bin is
one whose width is of the order of the age errors of the clusters in that interval.
This would lead to the selection of very narrow bins (in linear age) for young clusters and relatively broader age
bins for the older ones.
With the aim of building an age histogram that best reproduces the intrinsic open cluster
age distribution, \citet{p10} took the uncertainties in the age estimates into account in order
to define the age
intervals in the whole Galactic open cluster age range. Thus, he produced a more appropriate sampling of the open clusters
per age interval than is generated using a fixed bin size, since he included in each bin a number of clusters whose age
errors are close to the size of this bin. Indeed, the age errors for very young clusters are
a couple of Myrs,
while those for the oldest clusters are at least a few Gyrs. Therefore, smaller bins are appropriate for
young clusters,
whereas larger bins are more suitable for the old clusters. \citet{petal11a,petal11b}
have also used this precept for producing age distributions of LMC and SMC clusters, respectively.
We then searched Table 5 of \citet{pgm12} (the present Table 1) to find that
typical age errors are 0.10 $\la$ $\Delta$log($t$) $\la$ 0.15. Therefore, we produced the AMR of the LMC
field population by setting the age bin sizes according to this logarithmic law, which
traces the variation in the derived age uncertainties in terms of the measured ages. We used
intervals of $\Delta$log($t$) = 0.10. We proceeded in a similar way when binning
the metallicity range. In this case, we adopted a [Fe/H] interval of 0.25 dex. Thus, the subdivision of the
whole age and metallicity ranges was then performed on an observational-based foundation,
since the (age,[Fe/H]) dimensions are determined by the typical errors for each age/metallicity range.
However, there is still an additional issue to be
considered: even though the (age,[Fe/H]) bins are set to match the age/metallicity errors,
any individual point in the
AMR plane may fall in the respective (age,[Fe/H]) bin or in any of the eight adjacent bins. This happens
when an (age,[FeH]) point does not fall in the bin centre and, due to its errors, has
the chance to fall outside it. Note that, since we chose bin dimensions as large as the involved errors, such points
should not fall on average far beyond the adjacent bins. However, this does not necessarily happen to all 336 (age,[FeH]) points, and we should consider at the same time any other possibility.
We have taken all these effects into account to produce the AMR of the 21 studied LMC fields.
First of all, we take the AMR plane as engraved by a grid of bins as mentioned above, i.e. with logarithmic and linear
scales drawn along the age and metallicity axes. Then, if we put one of our (age,[Fe/H]) points in it, we find out that
that point with its errors covers an area which could be represented by a box of size
4$\times$$\sigma(age)$$\times$$\sigma([Fe/H])$. This (age,[Fe/H]) box may or may not fall centered on one of the AMR grid
bins, and has dimensions smaller, similar or larger than the AMR grid bin wherein it is placed. Each
of these scenarios generates a variety of possibilities, in the sense that the (age,[Fe/H]) box could cover from
one up to 25 or more AMR bins depending on its position and size. Bearing in mind all these alternatives, our strategy
consisted in weighing the contribution of each (age,[Fe/H]) box to each one of the AMR grid bins occupied by it, so that
the sum of all the weights equals unity.
The assigned weight was computed as the ratio between the area occupied by the (age,[Fe/H]) box in a AMR grid bin to the
(age,[Fe/H]) box size. When performing such a weighting process, we focused in practice on a single AMR grid bin and
calculated the weighted contribution of all the 336 (age,[Fe/H]) boxes to that AMR grid bin. We then repeated the calculation
for all the AMR grid bins. In order to know whether a portion of an (age,[Fe/H]) box falls in a AMR grid bin, we
took into account the following possibilities and combinations between them. Once an age interval is defined, we asked
whether: i) the age associated with any of the 336 (age,[Fe/H]) points is inside that age interval, ii) the age-$\sigma$(age)
value is inside that age interval; iii) the age-$\sigma$(age) value is to the left of that age interval and the age is
to the right; iv) the age+$\sigma$(age) value is inside that age interval and, v) the age+$\sigma$(age) value is to the right
of that age interval and the age is to the left. For the metallicities we proceeded in a similar way so that we finally
encompassed a total of 25 different inquiries to exactly match the positions and sizes of the 336 (age,[Fe/H]) points in the
AMR plane grid. We are confident that our analysis yields accurate morphology
and position of the main features in the derived AMRs.
Fig. 1 shows the resulting individual AMRs as labelled at the top-right margin of each panel.
It is important to keep in mind that each of the (age,[Fe/H]) points used to make each of these plots is
simply the representative, most dominant population in that subfield.
The filled boxes
represent the obtained mean values for each (age,[Fe/H]) bin; the age error bars follow the law $\sigma$log($t$) = 0.10;
and the [Fe/H]
error bars come from the full width at half-maxima (FWHMs) we derived by fitting Gaussian functions to the metallicity
distribution in each age interval.
The fit of a single Gaussian per age bin was performed using the NGAUSSFIT routine in the STSDAS/IRAF\footnote{IRAF is
distributed by the National Optical Astronomy Observatories,
which is operated by the Association of Universities for Research in Astronomy, Inc., under contract with
the National Science Foundation} package. The centre
of the Gaussian, its amplitude and its FWHM acted as variables, while the constant and the linear terms were fixed to zero,
respectively. We used Gaussian fits for simplicity. We estimated a difference from Gaussian distributions of only $\approx$
8 $\%$.
At first glance, it can be seen that the youngest and the oldest ages of each AMR vary from field to field.
The metallicity range and the shapes of the 21 AMRs are also quite variable. For example, AMRs for Fields $\#$
3, 6, and 8 do not show chemical enrichment, a feature that can be seen for example in Fields $\#$ 10, 12, 13, and
14. Moderate to intermediate chemical enrichment is seen in the remaining fields. Fields $\#$14 and 20 are
the most metal-rich and the most metal-poor fields, respectively, at any time, with a mean difference between them of $\sim$
0.8 dex.
In order to examine whether there exists any dependence of the individual AMRs with position in the LMC,
we have made use of their deprojected galactocentric distances computed by assuming that they are part of
a disk having an inclination $i$ = 35.8$\degr$ and a position angle of the line
of nodes of $\Theta$ = 145$\degr$ \citep{os02}. We refer the reader to Table 1 of \citet{ss10} which includes a summary of
orientation measurements of the LMC disk plane, as well as their analysis of the orientation and other LMC disk quantities, supporting
the present adopted values. Figs. 2 and 3 illustrate
the behaviour of the old and the young extremes and the
metal-poor and the metal-rich extremes of each AMR, respectively, as a function of the deprojected distance. Old and metal-poor
extremes are drawn with open boxes, while young and metal-rich extremes are depicted with filled boxes. The error bars for ages and
metallicities are those from Fig. 1, whereas the error bars of the deprojected distances come from the dispersion of this
quantity within the 16 subfields used in each mosaic field. As can be seen, the outer fields -defined as those with
deprojected distances $>$ 4$\degr$ \citep[][and references therein]{betal98}- contain dominant stellar populations about as old as the galaxy, while those of the inner
disk do not, with the exception of Fields $\#$9 and 18. The outer fields began at an age within our oldest age interval,
although we have represented them as a single value as a result of our binning process.
In general, the oldest dominant stellar populations in the inner disk
fields have been formed between $\sim$ 5 and 8 Gyr ago. Likewise, the main stellar formation processes in the outer disk
appears to have ceased some 5 $\pm$ 1 Gyr ago. This result confirms that of \citet{getal08} and CGAH concerning
outside-in evolution of the LMC disk as opposed to the $\Lambda CDM$ prediction for inside-out
formation.
It is interesting that the inner fields appeared to start their first strong star formation episodes at about the same time
that the outer fields were undergoing their last episode. This is certainly not what is expected
if the impulse driving the onset of star formation is some global effect like a close galactic encounter,
e.g. with the Galaxy or the SMC. The last major epoch of star formation we are sensitive to ended
about 1-2 Gyrs ago in the inner fields, with evidence for a radial age gradient.
On the metallicity side, Fig. 3 shows that for outer fields starting (open box) and ending (filled box) [Fe/H] values
are very similar, which means that they have not experienced much chemical enrichment. Taking into account
the open and filled boxes for these outer fields, we derived a mean value of [Fe/H] =
-0.90 $\pm$ 0.15 dex (note that their mean starting and ending metallicities are [Fe/H] = -0.95$\pm$0.10 dex and
-0.90$\pm$0.10 dex, respectively). This value could be considered as the representative
metallicity level for the outer disk field
during the entire life of the LMC. In the inner disk, the situation is different. Firstly, the starting metal
abundances (open boxes) are on average as metal-poor as the ending abundances for the most metal-rich outer fields. Secondly,
there exists a mean increase in the [Fe/H] values of +0.3 $\pm$ 0.1 dex, indicating significant chemical enrichment. If, in
addition, we
consider that these inner disk fields have been formed more recently than those in the outer disk, the signs of significant
recent chemical
enrichment are even more evident.
The apparent metallicity gradient exhibited in Fig. 3, in the sense that
the more distant a field from the galaxy centre, the more metal-poor it is, is
tightly coupled with the relationship shown in Fig. 2. To disentangle both dependences we fit the 336
individual metallicities (Table 2) according to the
following expression:
\begin{equation}
{\rm [Fe/H]} = C + (\partial{\rm [Fe/H]}/\partial t)\times t + (\partial{\rm [Fe/H]}/\partial a)\times a
\end{equation}
\noindent where $t$ and $a$ represent the age in Gyr and the deprojected distance in degrees.
The respective
coefficients turned out to be $C$ = -0.55 $\pm$ 0.02 dex, $\partial${\rm [Fe/H]}/$\partial$$t$ = -0.047 $\pm$
0.003 dex Gyr$^{-1}$, and $\partial${\rm [Fe/H]}/$\partial$$a$ = -0.007 $\pm$ 0.006 dex degrees$^{-1}$,
which implies a small but insignificant metallicity gradient of (-0.01 $\pm$ 0.01) dex kpc$^{-1}$, if an LMC
distance of 50 kpc is adopted
\citep{ss10}. Thus, there is no evidence for a significant metallicity gradient in the
LMC. This result agrees with that of \citet{getal06} who found that the LMC lacks any metallicity gradient.
The relatively more metal-poor stars found in the outermost regions (see
Fig. 3) are mostly a consequence of the fact that such regions are dominated by old stars which are relatively metal-poor,
whereas intermediate-age stars which are more metal-rich prevail in the innermost regions. This result
firmly confirms CGAH's findings.
We have also produced a composite AMR for the 21 LMC fields following the
same procedure used to derive the individual AMRs of Fig. 1. The result is shown in Fig. 4, where the mean points
are represented with filled boxes, while the error bars are as for Fig. 1. We have also included the individual points
of the 336 subfields plotted with gray-scale colored triangles. We used a 100 level gray-scale from black to white to
represent the most distant to the nearest star fields to the LMC centre. As can be seen, the most distant fields have been
preferentially formed at a low and relatively constant metallicity level, from the birth of the LMC until $\sim$ 6 Gyr
ago, while the inner fields have been formed later on with a steeper chemical enrichment rate. Note also that
the [Fe/H] errorbars cover a larger range than that the points represent. This is because these errorbars
do not only represent the standard dispersion of the points, but also of their measured errors (see Sect. 3).
\section{Comparison and discussion of the LMC AMR}
In Fig. 5, we have overplotted with solid lines different field star AMRs along with our presently derived composite AMR,
namely: HZ09 (yellow), \citet{retal11} (black),
\citet[hereafter PT98]{pt98} (blue), and \citet{getal98} (red).
The red line AMR is based on a closed-box model, while
the blue curve is a bursting model. We also included with red and blue filled circles the AMRs derived by
\citet{cetal08} for the LMC bar and disk, respectively. At first glance, we find that the bursting SFH modeled by
PT98 appears to be the one which best resembles the AMR derived by \citet{cetal08}, instead of
closed-box models as Carrera et al. suggested. However, such a resemblance is only apparent since PT98 constructed their
model using nearly no star formation from $\sim$ 12 up to 3 Gyr ago (see their Fig. 2). This clearly contradicts
not only Carrera et al.'s result but also ours, which show that there were many stars formed in the
LMC in that period (see Fig. 4). Indeed, we actually see no significant chemical evolution from about 12 - 6 Gyr,
even though stars were formed. In turn, the closed-box models appear to be qualitatively closer to HZ09's reconstructed AMR.
Since HZ09's AMR is based on a relatively bright limiting magnitude database and CGAH's AMRs rely on ages and metallicities
for stars distributed in ten fields (each only slightly smaller than ours), we believe that the present composite AMR has
several important advantages over these
previous ones, and possibly reconciles previous conclusions about the major
enrichment processes that have dominated the chemical evolution of the LMC from its birth until $\sim$ 1 Gyr
ago. Note that a large number of fields distributed through the galaxy are analysed here and their representative
oldest MS TOs are well measured in all fields. The composite AMR we derive results in a complex function having HZ09's AMR (or
alternatively the closed-box model) and CGAH's AMRs (or alternatively the bursting model) as lower and
approximately upper envelopes in metallicity, respectively, although the bursting model is a much better
fit. Therefore, we find evidence that the LMC has not chemically
evolved as a closed-box or bursting system, exclusively, but as a combination of both scenarios that likely have varied in
importance during
the lifetime of the galaxy, but with the bursting model dominating. The closed-box model presumably reproduces the metallicity
trend that the LMC would have had if bursting formation episodes had not taken place. However, since the LMC
would appear to have experienced such an enhanced formation event(s), important chemical enrichment has occurred from
non well-mixed gas spread through the LMC. CGAH also found that the AMRs for their ten fields are statistically
indistinguishable. We note, however, that six of their fields are aligned somewhat perpendicular to the LMC bar, reaching
quite low density outer regions, and
therefore, that their coverage represents a relatively small percentage of the whole field population. We show in Fig. 1
that, when more field stars distributed through the LMC are analyzed with age and metallicity
uncertainties robustly considered, distinct individual AMRs do arise. Indeed, Figs. 2 and 3 illustrate how different AMRs are
for inner and outer fields.
When inspecting in detail our composite field LMC AMR, the relatively quiescent epoch ($t$ $\sim$ 5 to 12 Gyr) claimed by
HZ09 and also frequently considered as a feature engraved in the cluster formation processes, i.e.
the cluster age-gap
\cite[among others]{getal97,petal02,betal04} is not observed. On the contrary, there exists a noticeable number of
fields with representative ages spanning the age gap (from $\sim$ 12 Gyr to 3 Gyr), which further strengthens the difference between cluster and field star formation during this epoch. Of course, we
do not quantitatively compare the level of SF in different epochs, we simply measure the properties of
the dominant population.
However, during this extended period, although some star formation occurred, it was
not accompanied by any significant chemical evolution until starting $\sim$6 Gyr ago. Again curiously,
there were several Gyr of star formation and chemical evolution before the cluster age gap ended. In
addition, although the ages estimated by CGAH of field stars spanning the cluster Age Gap could have uncertainties
necessarily large for individual stars, and consequently their SFH would still indicate a relatively quiescent epoch between
5 and 10 Gyr as HZ09 pointed out, we provide here evidence of the existence of
stars formed between 5 and 12 Gyr that represent the most numerous
populations in their respective regions. Note that our metallicities are generally about 0.1 - 0.2 dex lower
than CGAH for younger ages but higher for the oldest stars, indicating a smaller total chemical enrichment over the
lifetime of the galaxy compared to that found by CGAH. Our agreement with \citet{retal11} is somewhat
better. We also find that the amount of chemical evolution (as measured
by the increase in the metallicity) of the LMC fields
has varied during the lifetime of the LMC. Particularly, we find only a small range of the metal abundance
within the considered uncertainties for the outer disk fields, whereas an average increase of $\Delta$[Fe/H] = 0.3 $\pm$ 0.1 dex
appears in the inner disk fields, and this increase occurred over a relatively shorter time period. Hence, a bursting star
formation scenario turns out to
be a plausible explanation if the enhanced star formation is accompanied by a vigorous nucleosynthesis
process that takes place during the burst.
Finally, we present a homogeneous comparison between the composite field AMR with that for 81 LMC clusters with
ages ($\ga$ 1 Gyr) and metallicities derived on the same scales as here. We use the ages and metallicities compiled by
\citet{petal11a} for 45 clusters observed in the Washington system, to which we add 36 clusters with ages
estimated by
\citet{p11c} from similar data. We estimate here
their metallicities following the same procedure used for the
studied fields (see Section 2).
The resulting cluster AMR is depicted in Fig. 6 with dark-gray filled boxes superimposed onto the composite field LMC
(open boxes with error bars). As can be seen, the cluster AMR satisfactorily matches the field AMR only for the last 3 Gyr,
while it is a remarkable lower envelope of the field AMR for older ages ($t$ $>$ 11 Gyr).
The most likely explanation is a very rapid early chemical enrichment traced by the very visible globular clusters, but their coeval, low metallicity field counterparts are so rare that they are missed in our data. The origin of the 15
oldest LMC clusters still remains unexplained and constitutes one of the most intriguing enigmas in our understanding of the
LMC formation and evolution. Different studies show that they have very similar properties to
the globular clusters
in the Milky Way \citep[][among others]{betal96,mg04,vdbm04,metal09,metal10}, except for their orbits, which are within the
LMC disk instead of in an isothermal halo \citep{b07}. On the other hand, Fig. 4 show that there exist field star populations
older than 10 Gyr and about as old as the old globulars. These results go along with the curious conundrum of the absence of clusters
during the infamous Age Gap \citep{betal04}.
Since HZ09 found that
there was a relatively quiescent epoch in the field star formation from approximately 12 to 5 Gyr ago (similar to that observed for
star clusters), they also concluded that field and cluster star
formation modes are tightly coupled. Notice that the ages and metallicities used by HZ09 for the 85 clusters
are not themselves on a homogeneous scale nor on the same field age/metallicity scales.
In order to look for clues for the very low metallicities of the oldest LMC clusters, we reconstructed the cluster and
field AMRs of the SMC, also from Washington photometry obtained by us. As for the field AMR we used the ages and metallicities
derived by
\citet[his Table 4)]{p12} and applied to them the same binning and error analyses as for the composite LMC field
AMR (Fig. 4). Note that these ages and metallicities are all set on the present age/metallicity scales.
We also compiled 59 SMC clusters ($t$ $\ge$ 1 Gyr) from \citet{petal11b}, and
\citet{p11a,p11b}
with ages and metallicities tied to the same scales. Fig. 6 shows the resulting SMC AMRs depicted
with open triangles
for its field stars and with filled triangles for its star cluster population. As can be seen, cluster and field stars
apparently share similar chemical enrichment histories in the SMC, although the population of old clusters drastically
decreases beyond
$\sim$ 7 Gyr and there is only 1 older than 10 Gyr. \citet{p11b} showed, based on the statistics of
catalogued and studied clusters, that a total of only seven relatively old/old clusters remain to be studied, and an even
smaller number is obtained if the cluster spatial distribution is
considered. From this result, we conclude that the SMC cluster AMR is relatively well-known,
particularly towards its older and more metal-poor end. Therefore, it does not seem easy to connect the origin of the
oldest LMC cluster population to stripping events of ancient SMC star clusters. Moreover, the composite SMC field AMR
is on average $\sim$ 0.4 dex more metal-poor at all ages than that of the counterpart in the LMC, with
little variation, indicating that the global chemical evolution in these two galaxies was quite similar in
nature but with an offset to lower metallicity in the SMC. In particular, there was a very early and rapid
period of enrichment, followed by a long quiescent epoch with some star formation in both Clouds but
cluster formation only in the SMC and little to no metallicity increase
and finally a recent period of substantial
enrichment starting about 6Gyr ago. This is in very good agreement with the SMC AMR found by
\citet{petal10}.
The relative deficiency in heavy elements of the SMC could explain the metallicity of a few old LMC clusters, if they were captured
from the SMC \citep{betal12}, but this is an unlikely argument to explain the majority of them. In fact,
it is curious in this context that the the oldest SMC cluster is at the young and metal-rich extreme of the LMC globular cluster distributions.
\section{Summary}
In this study we present, for the first time, the AMR of the LMC field star population from ages and metallicities
derived using CCD Washington $CT_1$ photometry of some 5.5 million stars
in twenty-one 36$\arcmin$$\times$36$\arcmin$ fields distributed throughout the LMC main body
presented in \citet{pgm12}.
The analysis of the photometric data -subdivided in 336 smaller 9$\arcmin$$\times$9$\arcmin$ subfields - leads to
the following main conclusions:
i) From ages and metallicities of the representative star population in each subfield estimated by using the
$\delta$$T_1$
index and the SGB technique, respectively, we produced individual field AMRs with a robust treatment of their age and
metallicity uncertainties. These individual AMRs show some noticeable differences from field to field in several
aspects: starting and ending ages, metallicity range , shape, etc. This is contrary to CGAH, who found
very similar AMRs in their sample.
The composite AMR for the LMC fields
reveals that, while old and metal-poor field stars have been preferentially formed in the outer disk, younger and more
metal-rich stars have mostly been formed in the inner disk. This result confirms an outside-in
evolution of the galaxy, as found by \citet{cetal08}. In addition, we provide evidence of the existence
of stars formed between 6 and 12 Gyr that represent
the most numerous populations in their respective regions, although little or no chemical evolution occured during this
extended period.
ii) The resulting distribution of the ages and the metallicities as a function of the deprojected distance reveals
that there is no significant metallicity gradient in the LMC ((-0.01 $\pm$ 0.01) dex kpc$^{-1}$).
The relatively more metal-poor stars found in
the outermost regions is mainly a consequence of the fact that such regions are dominated by old stars which
are relatively
metal-poor,
whereas intermediate-age stars which are more metal-rich prevail in the innermost regions.
We also find that the range in the metallicity of the LMC fields
has varied during the lifetime of the LMC. In particular, we find only a small range of the metal abundance for the outer disk fields, whereas an average range of
$\Delta$[Fe/H] = +0.3 $\pm$ 0.1 dex is found in the inner disk fields.
iii) From the comparison of our composite AMR with theoretical ones, we conclude that the LMC has not chemically
evolved as a closed-box or bursting system, exclusively, but as a combination of both scenarios that have
had different prominence during the lifetime of the galaxy, with the bursting model generally more
dominant. Enhanced formation episodes could have possibly taken place
as a result of its interactions with the Milky Way and/or SMC.
iv) We finally accomplish a homogeneous comparison between the composite field AMR with that for LMC clusters with ages
and metallicities on the same scales. We find a satisfactory match only for the last 3 Gyr, while for older
ages ($>$ 11 Gyr) the cluster AMR results in a remarkable lower envelope of the field AMR.
The most likely explanation is a very rapid early chemical enrichment traced by the very visible globular
clusters, but their coeval, low metallicity field counterparts are so rare that they are missed in our data.
We find that such a
large difference between the metallicities of LMC field stars and clusters is not easy to explain as coming from
stripped ancient SMC clusters, although the field SMC AMR is on average $\sim$ 0.4 dex more metal-poor at all ages than
that of the LMC. The two galaxies otherwise show a very similar chemical evolution.
\acknowledgements
We greatly appreciate the comments and suggestions raised by the
reviewer which helped us to improve the manuscript.
This work was partially supported by the Argentinian institutions CONICET and
Agencia Nacional de Promoci\'on Cient\'{\i}fica y Tecnol\'ogica (ANPCyT).
D.G. gratefully acknowledges support from the Chilean
BASAL Centro de Excelencia en Astrof\'{\i}sica
y Tecnolog\'{\i}as Afines (CATA) grant PFB-06/2007.
|
1,108,101,564,530 | arxiv | \section{Introduction}
\label{sec:Introduction}
The IR luminescence of bismuth centers discovered in \mbox{\SiOiiAl:Bi} glasses
\cite{Fujimoto99, Fujimoto01} has been observed in various glasses and crystals.
Despite active studies of the bismuth-related IR luminescence (the present state
of the art is reviewed in \cite{Sun14}) and successful applications for laser
amplification and generation (see e.g. reviews \cite{Dianov09} and
\cite{Dianov13}), the origin of the luminescence centers in most systems still
remains to be established. In general, currently a belief is strengthened that
certain subvalent bismuth species are responsible for the IR luminescence (see
e.g. \cite{Sun14, Dianov10, Peng11}). In a few systems the structure of the
luminescence centers is definitively clear, namely, Bi$_5^{3+}$ subvalent
bismuth clusters in $\textrm{Bi}_5\!\left(\textrm{AlCl}_4\right)_3$ crystal,
Bi$_2^{-}$ dimers in $\left(\textrm{K-crypt}\right)_2\textrm{Bi}_2$ crystal,
$\textrm{Bi}^{+}${} ions in zeolite~Y (see review \cite{Sun14} and references within for
details). Models of subvalent bismuth centers as possible source of IR
luminescence were suggested for several systems basing on first-principle
modeling (e.g. \cite{Sun14} and references within; \cite{We13}).
Both for understanding the origin of IR luminescence centers and for possible
applications, especially in fiber optics and optical communications,
bismuthate-silicate and bismuthate-germanate systems are of interest. For many
hosts, including \mbox{$\textrm{GeO}_2$}{} and \mbox{$\textrm{SiO}_2$}, Bi doping is hindered owing to significant
ionic radius of bismuth. However in \GeOiiBi{} or \SiOiiBi{} glasses \mbox{$\textrm{Bi}_2\textrm{O}_3$}{}
appears as glass former and its content is known to vary in wide range (see e.g.
\cite{Smet90, Kargin04}). This shows promise of obtaining glasses with high
concentration of the bismuth-related luminescence centers.
In \mbox{\mbox{$\textrm{GeO}_2$}:Bi}{} and \mbox{\mbox{$\textrm{SiO}_2$}:Bi}{} glasses containing
0.03--0.05~mol.\%{} \mbox{$\textrm{Bi}_2\textrm{O}_3$}{} and no other dopants, luminescence bands around
1.67 and 1.43~\textmu{}m, respectively, were observed \cite{Bufetov11}. In \cite{We13}
we suggested models of corresponding luminescence centers based on results of
our first-principle studies. In binary \GeOiiBi{} systems, however, distinctly
different luminescence occurs. The luminescence in the 1.2--1.3~\textmu{}m{} range
excited at 0.5, 0.8 and 1.0~\textmu{}m{} was observed in \BixGeO{} glasses ($0.1 \leq
x \leq 0.4$ \cite{Su11, Su12a, Su12b, Su13} and $x \approx 0.01$
\cite{Firstov13}), in \BiMO{Ge}{12}{}{20}{} crystals quenched in N$_2$
atmosphere \cite{Yu11}, and in Mg- or Ca-doped \BiMO{Ge}{4}{3}{12}{} crystals
\cite{Yu13}. The luminescence in the 1.8--3~\textmu{}m{} range was observed in
\BixGeO{} glasses ($x \gtrsim 0.2$) \cite{Su12b}, in pure and Bi-, Mo-, or
Mg-doped \BiMO{Ge}{4}{3}{12}{} crystals, and in Mo-doped \BiMO{Ge}{12}{}{20}
crystals \cite{Su13}. Annealing glasses in oxidative atmosphere \cite{Su11,
Su12a, Su12b, Su13} or adding oxidant (CeO$_2$) in glass \cite{Wondraczek12} led
to a decrease in the luminescence intensity evidencing convincingly
oxygen-deficient character of the luminescence centers. In \GeOiiAlBi{} glasses
\cite{Peng04, Peng05}, in \GeOiiBi{} glass prepared in alumina crucible
\cite{Firstov13}, and in \mbox{\BiMO{Ge}{4}{3}{12}:Al}{} crystals \cite{Su12c}
the 1.2--1.3~\textmu{}m{} luminescence band contained a component near 1.1~\textmu{}m{}
characteristic of \SiOiiAl-based glasses \cite{Fujimoto99, Fujimoto01}.
Whilst no specific models of the luminescence centers in \GeOiiBi{} systems
were suggested in the cited papers, the authors mainly held the opinion that
such centers are formed by subvalent bismuth.
In all stable \GeOiiBi{} and \SiOiiBi{} crystals (sillenites,
\BiMO{Ge}{12}{}{20}{} and \BiMO{Si}{12}{}{20}, eulytines, \BiMO{Ge}{4}{3}{12}
and \BiMO{Si}{4}{3}{12}, benitoite, \BiMO{Ge}{2}{3}{9}) Bi atoms are known to be
threefold coordinated \cite{Kargin04}. It would be reasonable that Bi atoms
occur mainly in the same local environment in \GeOiiBi{} and \SiOiiBi{} glasses
as well. Such single threefold coordinated Bi atoms in \mbox{$\textrm{GeO}_2$}{} and \mbox{$\textrm{SiO}_2$}{}
hosts were studied in our recent work \cite{We13}. If \mbox{$\textrm{Bi}_2\textrm{O}_3$}{} content is
high enough, the groups (pairs at least) of threefold coordinated Bi atoms bound
together by bridging O atoms would occur in \GeOiiBi{} and \SiOiiBi{} as well.
Therefore one might expect that in \GeOiiBi{} and \SiOiiBi{} glasses there are
oxygen-deficient centers (\mbox{ODC}) not only typical for \mbox{$\textrm{GeO}_2$}{} and \mbox{$\textrm{SiO}_2$}{}
(namely, O vacancy and twofold coordinated Si or Ge atoms), but as well similar
\mbox{ODC}{}s containing Bi atoms (\mbox{BiODC}{}s), namely, \mbox{$=\!\textrm{Bi}\cdots\textrm{Ge}\!\equiv$}, \mbox{$=\!\textrm{Bi}\cdots\textrm{Si}\!\equiv$}, \mbox{$=\!\textrm{Bi}\cdots\textrm{Bi}\!=$}{} vacancies
and twofold coordinated Bi atoms. According to \cite{We13}, in \mbox{$\textrm{SiO}_2$}{} twofold
coordinated Bi atoms bound by bridging O atoms with Si atoms can be considered
as $\textrm{Bi}^{2+}${} centers, while in \mbox{$\textrm{GeO}_2$}{} such Bi atoms are unstable. Thus,
studying the \mbox{$=\!\textrm{Bi}\cdots\textrm{Ge}\!\equiv$}, \mbox{$=\!\textrm{Bi}\cdots\textrm{Si}\!\equiv$}, and \mbox{$=\!\textrm{Bi}\cdots\textrm{Bi}\!=$}{} vacancies as possible \mbox{BiODC}{} in
\GeOiiBi{} and \SiOiiBi{} is of interest.
\section{The modeling of bismuth-related centers}
\label{sec:Modeling}
\mbox{BiODC}{}s of O vacancy type were studied, namely, \mbox{$=\!\textrm{Bi}\cdots\textrm{Ge}\!\equiv$}, \mbox{$=\!\textrm{Bi}\cdots\textrm{Si}\!\equiv$}{} and \mbox{$=\!\textrm{Bi}\cdots\textrm{Bi}\!=$}{}
vacancies in \GeOiiBi{} and \SiOiiBi{} hosts, and \mbox{$=\!\textrm{Bi}\cdots\textrm{Ge}\!\equiv$}{} and \mbox{$=\!\textrm{Bi}\cdots\textrm{Si}\!\equiv$}{} vacancies
in \GeOiiAl{} and \SiOiiAl{} hosts. The modeling was performed using periodical
network models. $2 \times 2 \times 2$ supercells of \mbox{$\textrm{GeO}_2$}{} and \mbox{$\textrm{SiO}_2$}{}
lattice of $\alpha$ quartz structure (24 \mbox{$\textrm{GeO}_2$}{} or \mbox{$\textrm{SiO}_2$}{} groups with 72
atoms in total) were chosen as models of initial perfect network. From two to
eight \mbox{$\textrm{GeO}_2$}{} (\mbox{$\textrm{SiO}_2$}) groups in the supercell were substituted by \mbox{$\textrm{Bi}_2\textrm{O}_3$}{}
groups, from one to four. So the supercell compositions varied from
$\textrm{Bi}_2\textrm{O}_3 \cdot 22\,\textrm{GeO}_2$
($\textrm{Bi}_2\textrm{O}_3 \cdot 22\,\textrm{SiO}_2$) to
$4\,\textrm{Bi}_2\textrm{O}_3 \cdot 16\,\textrm{GeO}_2$
($4\,\textrm{Bi}_2\textrm{O}_3 \cdot 16\,\textrm{SiO}_2$), respectively.
Using ab~initio molecular dynamics (MD) the system formed by
supercells was heated to temperature as high as 1200~K (enough for both
\GeOiiBi{} and \SiOiiBi{} \cite{Kargin04}), maintained at this temperature
until the equilibrium atom velocities distribution was reached and then cooled
to 300~K. Periodical models of \GeOiiBi{} and \SiOiiBi{} networks based on final
supercell configurations were applied to study the \mbox{BiODC}{}s. Each vacancy,
\mbox{$=\!\textrm{Bi}\cdots\textrm{Ge}\!\equiv$}, \mbox{$=\!\textrm{Bi}\cdots\textrm{Si}\!\equiv$}, or \mbox{$=\!\textrm{Bi}\cdots\textrm{Bi}\!=$}, was formed by a removal of a proper O atom. When
necessary, fourfold coordinated Al center, \mbox{$\left(\textrm{AlO}_4\right)^{-}$}, was formed substituting Al
atom for Si or Ge atom and increasing the total number of electrons in the
supercell by one. Equilibrium configurations of the \mbox{BiODC}{}s were found by a
subsequent Car-Parrinello MD and complete optimization of the supercell
parameters and atomic positions by the gradient method. All these calculations
were performed using Quantum ESPRESSO{} package in the plane wave basis in generalized
gradient approximation of density functional theory using ultra-soft projector
augmented-wave pseudopotentials and Perdew–Burke–Ernzerhof functional.
Configurations of the \mbox{BiODC}{}s obtained by this means then were used to
calculate the absorption spectra. The calculations were performed with \mbox{Elk}{}
code by Bethe-Salpeter equation method based on all-electron full-potential
linearized augmented-plane wave approach in the local spin density approximation
with Perdew-Wang-Ceperley-Alder functional. Spin-orbit interaction essential for
Bi-containing systems was taken into account. Scissor correction was used to
calculate transition energies. The scissor value was calculated using modified
Becke-Johnson exchange-correlation potential. Further details and corresponding
references may be found in \cite{We13}.
\begin{figure}
\subfigure[]{%
\includegraphics[width=8.50cm, bb= 0 -10 1310 1100]
{figure1a.eps}
\label{fig:BiGe_GeO2}
}
\subfigure[]{%
\includegraphics[width=8.50cm, bb=20 0 1080 1100]
{figure1b.eps}
\label{fig:BiBi_GeO2}
}
\caption{%
\mbox{BiODC}{} in \GeOiiBi: \subref{fig:BiGe_GeO2}~\mbox{$=\!\textrm{Bi}\cdots\textrm{Ge}\!\equiv$};
\subref{fig:BiBi_GeO2}~\mbox{$=\!\textrm{Bi}\cdots\textrm{Bi}\!=$}.
}
\label{fig:Centers}
\end{figure}
On the contrary to the centers modeled in \cite{We13}, the Stokes shift
corresponding to a transition between the first excited state and the ground one
turns out to be large in all the \mbox{$=\!\textrm{Bi}\cdots\textrm{Ge}\!\equiv$}, \mbox{$=\!\textrm{Bi}\cdots\textrm{Si}\!\equiv$}, and \mbox{$=\!\textrm{Bi}\cdots\textrm{Bi}\!=$}{} centers. So in such
centers the luminescence wavelengths were estimated only roughly.
\begin{figure}
\subfigure[]{%
\includegraphics[width=2.70cm, bb=215 130 395 645]{figure2a.eps}
\label{fig:BiGe_GeO2_levels}
}
\subfigure[]{%
\includegraphics[width=2.70cm, bb=215 130 395 645]{figure2b.eps}
\label{fig:BiBi_GeO2_levels}
}
\subfigure[]{%
\includegraphics[width=2.70cm, bb=215 130 395 645]{figure2c.eps}
\label{fig:BiGe_GeO2-Al2O3_levels}
}
\\
\subfigure[]{%
\includegraphics[width=2.70cm, bb=215 130 395 645]{figure2d.eps}
\label{fig:BiSi_SiO2_levels}
}
\subfigure[]{%
\includegraphics[width=2.70cm, bb=215 130 395 645]{figure2e.eps}
\label{fig:BiBi_SiO2_levels}
}
\subfigure[]{%
\includegraphics[width=2.70cm, bb=215 130 395 645]{figure2f.eps}
\label{fig:BiSi_SiO2-Al2O3_levels}
}
\caption{%
\mbox{BiODC}{}s levels and transitions:
\subref{fig:BiGe_GeO2_levels}~\mbox{$=\!\textrm{Bi}\cdots\textrm{Ge}\!\equiv$}{} in \mbox{$\textrm{GeO}_2$};
\subref{fig:BiBi_GeO2_levels}~\mbox{$=\!\textrm{Bi}\cdots\textrm{Bi}\!=$}{} in \mbox{$\textrm{GeO}_2$};
\subref{fig:BiGe_GeO2-Al2O3_levels}~\mbox{$=\!\textrm{Bi}\cdots\textrm{Ge}\!\equiv$}{} in \GeOiiAl;
\subref{fig:BiSi_SiO2_levels}~\mbox{$=\!\textrm{Bi}\cdots\textrm{Si}\!\equiv$}{} in \mbox{$\textrm{SiO}_2$};
\subref{fig:BiBi_SiO2_levels}~\mbox{$=\!\textrm{Bi}\cdots\textrm{Bi}\!=$}{} in \mbox{$\textrm{SiO}_2$};
\subref{fig:BiSi_SiO2-Al2O3_levels}~\mbox{$=\!\textrm{Bi}\cdots\textrm{Si}\!\equiv$}{} in \SiOiiAl.
Level energies are given in $10^3$~$\textrm{cm}^{-1}$, transition wavelengths in \textmu{}m.
}
\label{fig:Levels}
\end{figure}
Calculated configurations of \mbox{$=\!\textrm{Bi}\cdots\textrm{Ge}\!\equiv$}{} and \mbox{$=\!\textrm{Bi}\cdots\textrm{Bi}\!=$}{} centers in \GeOiiBi{} are shown
in Fig.~\ref{fig:Centers}. Configurations of the corresponding centers in
\mbox{$\textrm{SiO}_2$}, \GeOiiAl, and \SiOiiAl{} are similar. \Dist{Bi}{Ge}{} distance in
\mbox{$=\!\textrm{Bi}\cdots\textrm{Ge}\!\equiv$}{} center is 3.08~\AA{} in \mbox{$\textrm{GeO}_2$}{} and 3.12~\AA{} in \GeOiiAl,
\Dist{Bi}{Si}{} distance in \mbox{$=\!\textrm{Bi}\cdots\textrm{Si}\!\equiv$}{} center is found to be 2.89~\AA{} in \mbox{$\textrm{SiO}_2$}{}
and 2.95~\AA{} in \SiOiiAl, \Dist{Bi}{Bi}{} distance in \mbox{$=\!\textrm{Bi}\cdots\textrm{Bi}\!=$}{} center in
\mbox{$\textrm{GeO}_2$}{} and \mbox{$\textrm{SiO}_2$}{} is 3.03~\AA{} and 2.94~\AA, respectively. By comparison,
calculated distance between Ge (Si) atoms in single \mbox{$\equiv\!\textrm{Ge}\!\relbar\!\textrm{Ge}\!\equiv$}{} (\mbox{$\equiv\!\textrm{Si}\!\relbar\!\textrm{Si}\!\equiv$}) vacancy in
\mbox{$\textrm{GeO}_2$}{} (\mbox{$\textrm{SiO}_2$}) is found to be 2.58~\AA{} (2.44~\AA), and in Bi$_2$ dimer the
\Dist{Bi}{Bi}{} distance is known to be 2.66~\AA{} \cite{DiatomicMolecules}. So
relatively weak covalent bond occurs between Bi and Ge (Si) atoms in \mbox{$=\!\textrm{Bi}\cdots\textrm{Ge}\!\equiv$}{}
(\mbox{$=\!\textrm{Bi}\cdots\textrm{Si}\!\equiv$}) vacancy and between two Bi atoms in \mbox{$=\!\textrm{Bi}\cdots\textrm{Bi}\!=$}{} vacancy. Regardless of the
presence of Al atom, the \Angl{O}{Bi}{O}{} angles in \mbox{$=\!\textrm{Bi}\cdots\textrm{Ge}\!\equiv$}{} and \mbox{$=\!\textrm{Bi}\cdots\textrm{Si}\!\equiv$}{}
vacancies are close to the right angle, and the \Angl{O}{Ge}{O}{} angle in
\mbox{$=\!\textrm{Bi}\cdots\textrm{Ge}\!\equiv$}{} vacancy and the \Angl{O}{Si}{O}{} one in \mbox{$=\!\textrm{Bi}\cdots\textrm{Si}\!\equiv$}{} vacancy are close to
the tetrahedral angle. The analysis of electronic density has shown Bi to be
nearly divalent in all the \mbox{BiODC}{}s under study. However the electronic
structure of these \mbox{BiODC}{}s differs essentially from that of the divalent Bi
centers (twofold coordinated Bi atoms) studied in \cite{We13}. In particular, in
the latters the excited states energies are found to exceed $19 \times
10^3$~$\textrm{cm}^{-1}${} (absorption wavelengths $\lesssim 0.55$~\textmu{}m) \cite{We13}, while
in all \mbox{$=\!\textrm{Bi}\cdots\textrm{Ge}\!\equiv$}, \mbox{$=\!\textrm{Bi}\cdots\textrm{Si}\!\equiv$}, and \mbox{$=\!\textrm{Bi}\cdots\textrm{Bi}\!=$}{} centers (Fig.~\ref{fig:Levels}) there are the
low-lying excited states with the energy of $\lesssim 9.9 \times 10^3$~$\textrm{cm}^{-1}${}
(long-wave transitions in the $\gtrsim 1.1$~\textmu{}m{} range).
The origin of states and transitions in the \mbox{$=\!\textrm{Bi}\cdots\textrm{Ge}\!\equiv$}, \mbox{$=\!\textrm{Bi}\cdots\textrm{Si}\!\equiv$}, and \mbox{$=\!\textrm{Bi}\cdots\textrm{Bi}\!=$}{} centers
may be understood in a simple model considering twofold coordinated Bi atom as
the divalent Bi center \cite{We13}. The ground state and the first excited state
of $\textrm{Bi}^{2+}${} ion are known to be \Term{2}{P}{1/2}{} and \Term{2}{P}{3/2}{}{}
(20788~$\textrm{cm}^{-1}$), respectively \cite{Moore58}. In a crystal field two sublevels,
\Term{2}{P}{3/2}{\left(1\right)}{} and \Term{2}{P}{3/2}{\left(2\right)}, of the
first excited state are formed, giving rise to the
\Term{2}{P}{1/2}{}$\,\rightarrow\,$\Term{2}{P}{3/2}{\left(1\right)}{} and
\Term{2}{P}{1/2}{}$\,\rightarrow\,$\Term{2}{P}{3/2}{\left(2\right)}{} absorption
bands and the
\Term{2}{P}{3/2}{\left(1\right)}$\,\rightarrow\,$\Term{2}{P}{1/2}{} luminescence
band. The dangling bonds of twofold coordinated Bi atom and threefold
coordinated Ge (Si) atom in \mbox{$=\!\textrm{Bi}\cdots\textrm{Ge}\!\equiv$}{} (\mbox{$=\!\textrm{Bi}\cdots\textrm{Si}\!\equiv$}) center or the dangling bonds of two
twofold coordinated Bi atoms in \mbox{$=\!\textrm{Bi}\cdots\textrm{Bi}\!=$}{} center form bonding (doubly occupied)
and anti-bonding (unoccupied) states. The corresponding levels calculated in the
tight-binding model \cite{Harrison80} without spin-orbit interaction for
geometrical parameters of the centers, obtained in our modeling, are shown in
Figs.~\ref{fig:Models}\subref{fig:model_BiGe_GeO2} and
\subref{fig:model_BiBi_GeO2} as (i) and (ii) schemes. Strong intra-atomic
spin-orbit interaction in $\textrm{Bi}^{2+}${} ion (the coupling constant is known to be $A
\approx 13860$~$\textrm{cm}^{-1}${} \cite{Moore58}) results in a splitting of both levels in
accordance with Bi atom 6p states amplitudes in the wave functions ((iii)
schemes in Figs.~\ref{fig:Models}\subref{fig:model_BiGe_GeO2} and
\subref{fig:model_BiBi_GeO2}; the values in brackets indicate total angular
momentum of the $\textrm{Bi}^{2+}${} ion states which provide Bi 6p contribution to the wave
function of the level). And finally, level splitting in a crystal field together
with Madelung’s shift result in final sets of the electronic states ((iv)
schemes in Figs.~\ref{fig:Models}\subref{fig:model_BiGe_GeO2} and
\subref{fig:model_BiBi_GeO2} according to the results of our modeling). The
luminescence owing to transition from the lowest excited state to the ground
state corresponds (regarding the 6p contributions to the wave functions) to the
\Term{2}{P}{3/2}{\left(1\right)}$\,\rightarrow\,$\Term{2}{P}{1/2}{} transition
in $\textrm{Bi}^{2+}${} ion. However the transition energy turns out to be considerably
decreased as a result of the above-described transformation of electronic
states.
\begin{figure}
\subfigure[]{%
\includegraphics[width=8.50cm, bb=25 170 580 580]{figure3a.eps}
\label{fig:model_BiGe_GeO2}
}
\subfigure[]{%
\includegraphics[width=8.50cm, bb=25 170 580 610]{figure3b.eps}
\label{fig:model_BiBi_GeO2}
}
\caption{%
On the origin of the electron states of \mbox{BiODC}{}s in \mbox{$\textrm{GeO}_2$}:
\subref{fig:model_BiGe_GeO2}~\mbox{$=\!\textrm{Bi}\cdots\textrm{Ge}\!\equiv$}; \subref{fig:model_BiBi_GeO2}~\mbox{$=\!\textrm{Bi}\cdots\textrm{Bi}\!=$}{} (look
text for details). Level energies and splittings are given in $10^3$~$\textrm{cm}^{-1}$.
}
\label{fig:Models}
\end{figure}
Both covalent (ii) and spin-orbit (iii) splittings are determined mainly by
\Dist{Bi}{Ge(Si)} (\Dist{Bi}{Bi}) distances and mutual orientation of p orbital
of Bi atom and sp$^3$ orbital of Ge (Si) atom (p orbitals of two Bi atoms).
Hence the Stokes shift of the luminescence band relative to the absorption band
corresponding to transitions between the ground and the first excited states
cannot be small, as distinct from the monovalent Bi centers \cite{We13}. Basing
on our calculations, the Stokes shift is estimated to be about 300~$\textrm{cm}^{-1}${}
($\sim\,$5\%) for \mbox{$=\!\textrm{Bi}\cdots\textrm{Ge}\!\equiv$}{} and \mbox{$=\!\textrm{Bi}\cdots\textrm{Bi}\!=$}{} centers in \mbox{$\textrm{GeO}_2$}{} and \mbox{$=\!\textrm{Bi}\cdots\textrm{Si}\!\equiv$}{} centers
in \mbox{$\textrm{SiO}_2$}, about 1200~$\textrm{cm}^{-1}${} ($\sim\,$20\%) for the \mbox{$=\!\textrm{Bi}\cdots\textrm{Bi}\!=$}{} center in \mbox{$\textrm{SiO}_2$},
and about 800~$\textrm{cm}^{-1}${} ($\sim\,$10\%) for \mbox{$=\!\textrm{Bi}\cdots\textrm{Ge}\!\equiv$}{} center in \GeOiiAl{} and
\mbox{$=\!\textrm{Bi}\cdots\textrm{Si}\!\equiv$}{} center in \SiOiiAl{} (Fig.~\ref{fig:Levels}).
If \mbox{$\left(\textrm{AlO}_4\right)^{-}$}{} center occurs in the second coordination shell of Ge (Si) atom of
the \mbox{$=\!\textrm{Bi}\cdots\textrm{Ge}\!\equiv$}{} (\mbox{$=\!\textrm{Bi}\cdots\textrm{Si}\!\equiv$}) center, the electronic density is displaced from the vacancy
towards the Al atom leading to further attenuation of interaction between Bi and
Ge (Si) atoms. As a result, \Dist{Bi}{Ge(Si)} distance increases, covalent
splittings (ii) is reduced, Bi 6p states contribution to the ground state wave
function grows, and spin-orbit splitting (iv) increases. Thus, the electronic
structure in the vicinity of Bi atom in the \mbox{$=\!\textrm{Bi}\cdots\textrm{Ge}\!\equiv$}{} (\mbox{$=\!\textrm{Bi}\cdots\textrm{Si}\!\equiv$}) center becomes more
similar to the electronic structure of twofold coordinated Bi atom. Accordingly,
the IR transition is displaced to shorter-wave range (Figs.~\ref{fig:Levels},
\subref{fig:BiGe_GeO2-Al2O3_levels} and \subref{fig:BiSi_SiO2-Al2O3_levels}).
The formation energy of \mbox{$=\!\textrm{Bi}\cdots\textrm{Si}\!\equiv$}, \mbox{$=\!\textrm{Bi}\cdots\textrm{Bi}\!=$}, \mbox{$\equiv\!\textrm{Ge}\!\relbar\!\textrm{Ge}\!\equiv$}, and \mbox{$\equiv\!\textrm{Si}\!\relbar\!\textrm{Si}\!\equiv$}{} vacancies was found to
be approximately $+0.8$, $-2.7$, $+0.9$, and $+3.1$~eV, respectively (the
formation energy of \mbox{$=\!\textrm{Bi}\cdots\textrm{Ge}\!\equiv$}{} vacancy is taken here to be zero point). Suggesting
the migration energies of O vacancy between various pairs of atoms to be
approximately in the same relations as formation energies of corresponding
vacancies, one can explain the results of \cite{Su12b} by thermally stimulated
migration of O vacancies during glass annealing. Owing to the migration,
\mbox{$=\!\textrm{Bi}\cdots\textrm{Ge}\!\equiv$}{} centers may transform into \mbox{$=\!\textrm{Bi}\cdots\textrm{Bi}\!=$}{} ones. As a result, 1.2--1.3~\textmu{}m{}
luminescence intensity decreases with 1.8--3~\textmu{}m{} luminescence increasing.
\section{Conclusion}
\label{sec:Conclusion}
In conclusion, the results of our modeling of \mbox{BiODC}{}s in \GeOiiBi{} and
\SiOiiBi{} hosts make it reasonable to suggest that the luminescence in
the 1.2--1.3~\textmu{}m{} range in \GeOiiBi{} glasses \cite{Su11, Su12a, Su12b,
Wondraczek12, Firstov13} and crystals \cite{Yu11, Yu13} is caused by \mbox{$=\!\textrm{Bi}\cdots\textrm{Ge}\!\equiv$}{}
center, an O vacancy between Bi and Ge atoms (Fig.~\ref{fig:BiGe_GeO2}). The
luminescence in the 1.8--3~\textmu{}m{} range observed in annealed \GeOiiBi{} glasses
\cite{Su12b} and in \BiMO{Ge}{4}{3}{12}{} and \BiMO{Ge}{12}{}{20}{} crystal
\cite{Su13} in the absence of the 1.2--1.3~\textmu{}m{} luminescence may be caused by
\mbox{$=\!\textrm{Bi}\cdots\textrm{Bi}\!=$}{} center, an O vacancy between two Bi atoms (Fig.~\ref{fig:BiBi_GeO2}).
The decrease in intensity of the 1.2--1.3~\textmu{}m{} luminescence may be explained
by a transformation of \mbox{$=\!\textrm{Bi}\cdots\textrm{Ge}\!\equiv$}{} centers into \mbox{$=\!\textrm{Bi}\cdots\textrm{Bi}\!=$}{} ones owing to thermally
stimulated migration of O vacancies. The luminescence near 1.1~\textmu{}m{} in
\GeOiiAlBi{} glasses \cite{Peng05, Firstov13, Su12a} and in Al-doped
\BiMO{Ge}{4}{3}{12}{} crystals \cite{Su12c} may be caused by \mbox{$=\!\textrm{Bi}\cdots\textrm{Ge}\!\equiv$}{} center
\mbox{$\left(\textrm{AlO}_4\right)^{-}$}{} center in the second coordination shell of Ge atom. Basing on our
modeling, we suppose that in Bi-doped \mbox{$\textrm{GeO}_2$}{} and \mbox{$\textrm{SiO}_2$}{} glasses containing
$\lesssim 0.1$~mol.\%{} \mbox{$\textrm{Bi}_2\textrm{O}_3$}{} the IR luminescence centers are mainly
interstitial Bi atoms forming complexes with \mbox{$\equiv\!\textrm{Ge}\!\relbar\!\textrm{Ge}\!\equiv$}{} (\mbox{$\equiv\!\textrm{Si}\!\relbar\!\textrm{Si}\!\equiv$}) vacancies
\cite{We13}, while in \GeOiiBi{} (and probably \SiOiiBi) glasses containing
$\gtrsim 10$~mol.\%{} \mbox{$\textrm{Bi}_2\textrm{O}_3$}{} the IR luminescence centers are mainly \mbox{$=\!\textrm{Bi}\cdots\textrm{Ge}\!\equiv$}{}
(\mbox{$=\!\textrm{Bi}\cdots\textrm{Si}\!\equiv$}) and \mbox{$=\!\textrm{Bi}\cdots\textrm{Bi}\!=$}{} vacancies with Bi atoms bound in the glass network.
|
1,108,101,564,531 | arxiv | \section*{Supplemental Material}
We provide details of the security calculations, the optimization of parameters and all the steps of the covert communication protocol. We also include the complete data obtained for the message transmission.
\textit{Calculation of signal and noise states.---} The security of the protocol depends on Eve's states $\rho$ when there is no communication and $\sigma$ when there is covert communication. We model the background noise originating from Raman scattering as a thermal state with mean photon number $\bar{n}_A$. In this case, the state $\rho$ is simply given in the Fock basis as
\begin{equation}
\rho = \sum_{n = 0}^{\infty} Q_{\bar{n}_A}(n) \ketbra{n}{n}
\end{equation}
where $Q_{\bar{n}_A}(n)$ is the probability mass function of thermal distribution with mean photon number $\bar{n}_A$ given by
\begin{equation}
Q_{\bar{n}_A}(n) = \frac{\bar{n}_A^n}{(1 + \bar{n}_A)^{n+1}}.
\end{equation}
On the other hand, the state when there is a communication between Alice and Bob $\sigma$ depends on the signal state $\rho_S$. The signals that Alice sends are phase-randomized coherent states with mean photon number $\mu$, which follow a Poisson distribution of photon numbers with probability mass function $P_\mu(n)$. Taking into account the constant background thermal noise, the state seen by Eve when a signal is state in that mode is given by
\begin{equation}
\rho_S = \sum_{n = 0}^{\infty} p(n) \ketbra{n}{n}
\end{equation}
where
\begin{align}
p(n) &= \sum_{r = 0}^{n} P_\mu(r) \times Q_{\bar{n}_A}(n-r)\\
&= \sum_{r = 0}^{n} \frac{e^{-\mu} \mu^r}{r!} \frac{\bar{n}_A^{n - r}}{(1 + \bar{n}_A)^{n - r + 1}}
\end{align}
The state $\sigma$ is then given by $\sigma = q \rho_S + (1 - q) \rho$ where $q$ is the probability of sending a signal in that mode.\\
The detection bias is then bounded as
\begin{equation}\label{EQ: det bias}
\epsilon\leq \sqrt{\frac{N}{8}D(\rho||\sigma)},
\end{equation}
where $2N$ is the total number of time-bins.
\textit{Decoding error probability. ---} To calculate the decoding error probability, we first calculate the probability $p_C$ that Bob's detector records a click in the correct mode where a signal is sent. Given a total transmission efficiency $\tau$ in the channel, which includes the limited detector efficiency, the probability of a correct click is given by
\begin{equation}
p_C = 1 - \frac{\exp(-\tau \mu) }{(1 + \tau\bar{n}_B)}.
\end{equation}
Similarly, the probability of obtaining a click on the wrong mode where there is only noise is given by
\begin{equation}
p_W = 1 - \frac{1}{(1 + \tau \bar{n}_B)}.
\end{equation}
From this, we can calculate the probability $p_g$ that, given there was a click, it is a correct one. Ignoring cases where we observe clicks in both modes, this is given by
\begin{equation}
p_g = \frac{p_C}{p_C + p_W}.
\end{equation}
In the protocol, we use the repetition code to protect against errors and losses, using the majority vote for decoding. Suppose that Alice repeats each bit $k$ times and Bob observes $i$ clicks; then an error occurs when at least $i/2$ of the clicks are in the wrong mode. The probability $\delta$ that Bob incorrectly decodes an individual bit is given by
\begin{align}
\delta = &\sum_{i = 0}^{k} \text{Pr[$i$ clicks]} \times \text{Pr[majority wrong]}\nonumber\\
= &\sum_{i = 0}^{k} \binom{k}{i} (p_C + p_W)^i (1 - p_C - p_W)^{k - i}\times\nonumber\\
&\sum_{j = 0}^{\lfloor i/2 \rfloor} \binom{i}{j} (p_g)^j (1 - p_g)^{i - j}. \label{eq: error}
\end{align}
To decode Alice's message reliably, Bob has to decode all the bits in the message correctly. For each bit, he can do this with probability $(1 - \delta)$. Thus, the decoding error probability for the entire message $\mathcal{E}$ is given by
\begin{equation}\label{eq: dec. error}
\mathcal{E} = 1 - (1 - \delta)^b
\end{equation}
where $b$ is the number of bits in the message.\\
\textit{Detailed covert communication protocol.---} Suppose Alice and Bob wish to covertly exchange a message of $b$ bits with a detection bias of $\epsilon$ and a decoding error probability $\mathcal{E}$. Then they perform the following protocol:
\begin{enumerate}
\item From Eqs. \eqref{eq: error} and \eqref{eq: dec. error}, calculate the smallest number of signals $d$ that is required to achieve the
desired decoding error probability. The number of repetitions in the repetition code is $k=d/b$.
\item Fix the probability $q$ of sending a signal in each slot to be $q=d/N$, where $N$ is the total number of time-bin pairs.
\item Fix the mean photon number of the signals to be $\mu$.
\item Using Eq. \eqref{EQ: det bias}, calculate the smallest $N$ such that the desired detection bias $\epsilon$ is reached.
\item Repeat these steps for different values of $\mu$ to find the optimal value that minimizes the total number of time-bins.
\item To transmit the message covertly, use a quantum random number generator to choose in which time-bin pairs to send a signal, each with
probability $q$. Let $d'$ be the actual number of signals sent.
\item Use $k'=d'/b$ as the number of repetitions of each bit and decode using a majority vote.
\end{enumerate}
\textit{Experimental results. ---} Below, we reproduce the data obtained in the transmission of each message of the covert communication protocol. The results are in Table~\ref{Tab:Result} reporting the signal probability per pulse, noise probability per bin, and error rates. We also illustrate the results in terms of bar graphs, where each bar shows the total number of ``0" and ``1" signals obtained in the transmission of each encoded bit. The message's bit value is decoded by taking the majority vote of the outcomes, resulting in a correct decoding for all messages.
For the message ``QPQI" which operates at a lower repetition rate, the smallest running time is obtained by setting larger noise levels, thus reducing the total number of time-bins while still not affecting the repetition rate. In this case, large signal detection probability compared to noise probability would lead to insecurity, so we must endure higher error rates which are compensated by a larger number of repetitions.
As a comparison, we send the same message using a continuous wave (CW) laser instead of the fiber transmitter. the parameter and the result are shown in Table~\ref{Tab:CWParams} and Table~\ref{Tab:CWResult}.
\begin{center}
\begin{table}
\begin{tabular}{ |c | c | c | c| }
\hline
Message & CQTUSTC & PRTYSAT@NINE & QPQI \\
\hline
Signal probability & $8.42\times 10^{-3}$ & $8.62\times 10^{-3}$ & $9.26\times 10^{-2}$ \\
Noise probability & $1.04\times 10^{-3}$ & $9.01\times 10^{-4}$ & $2.23\times 10^{-2}$ \\
Error rate & 13.67\% & 14.85\% & 23.62\% \\
\hline
\end{tabular}
\caption{Summary of experimental results for different covert messages. Signal probability denote the probability of detecting a click when a signal is sent in a particular mode, while the noise probability denote the probability for bins where there is only noise. The resulting error rates are sufficiently low to recover the correct bits with a majority vote decoding.}\label{Tab:Result}
\end{table}
\end{center}
\begin{figure}[hbt]
\includegraphics[width=9cm]{RecvBits_PRTYSAT_NINE.png}
\caption{(Color online) Results for the covert transmission of the message ``PRTYSAT@NINE". The message consists of 60 bits which are all decoded correctly. Each bar shows the total number of ``0" signals (dark blue) and ``1" signals (light blue). On average, $\sim$14 signals were received per bit, with an error rate of $\sim$14.9$\%$.}\label{Fig:Data2}
\end{figure}
\begin{figure}[hbt]
\includegraphics[width=9cm]{RecvBits_QPQI.png}
\caption{(Color online) Results for the covert transmission of the message ``QPQI". The message consists of 20 bits which are all decoded correctly. Each bar shows the total number of ``0" signals (dark blue) and ``1" signals (light blue). On average, $\sim$39 signals were received per bit, with an error rate of $\sim$23.6$\%$.}\label{Fig:Data3}
\end{figure}
\begin{center}
\begin{table}
\begin{tabular}{ |c | c | c | c| }
\hline
Message & CQTUSTC & PRTYSAT@NINE & QPQI \\
\hline
Bits & 35 & 60 & 20 \\
Detection bias & 0.015 & 0.069 & 0.070 \\
Repetition rate & 500 MHz & 500 MHz & 500 kHz\\
Sync pulse & Yes & Yes & No\\
Time-bins & $1.56\times 10^{12}$ & $2.17\times 10^{11}$ & $3.71\times 10^{9}$ \\
Covert signals & 68,651 & 96,919 & 8,416 \\
$\mu$ & $3.57\times 10^{-2}$ & $4.17\times 10^{-2}$ & 0.278 \\
$\bar{n}_A$ & $2.14\times 10^{-3}$ & $2.18\times 10^{-3}$ & 0.62 \\
$\bar{n}_B$ & $3.48\times 10^{-3}$ & $2.93\times 10^{-3}$ & 0.66 \\
Running time (s) & 3,120 & 434 & 7,420\\
\hline
\end{tabular}
\caption{Summary of experimental parameters for different covert messages using CW laser as noise source instead of the fiber transmitter. Here $\mu$ is the mean photon number of the signals, $\bar{n}_A$ and $\bar{n}_B$ are the mean photon number of the noise at Alice's output and Bob's input. }\label{Tab:CWParams}
\end{table}
\end{center}
\begin{center}
\begin{table}
\begin{tabular}{ |c | c | c | c| }
\hline
Message & CQTUSTC & PRTYSAT@NINE & QPQI \\
\hline
Signal probability & $7.91\times 10^{-3}$ & $9.41\times 10^{-3}$ & $9.31\times 10^{-2}$ \\
Noise probability & $8.15\times 10^{-4}$ & $9.54\times 10^{-4}$ & $2.15\times 10^{-2}$ \\
Error rate & 13.08\% & 10.40\% & 23.34\% \\
\hline
\end{tabular}
\caption{Summary of experimental results for different covert messages using CW laser as noise source instead of the fiber transmitter. Signal probability denote the probability of detecting a click when a signal is sent in a particular mode, while the noise probability denote the probability for bins where there is only noise. The resulting error rates are sufficiently low to recover the correct bits with a majority vote decoding.}\label{Tab:CWResult}
\end{table}
\end{center}
|
1,108,101,564,532 | arxiv | \section{Introduction}
\label{sec: Introduction}
One aspect of the AdS/CFT correspondence that caught significant
attention recently
is its relation to quantum information (QI). The most prominent discovery in this field is the seminal
Ryu-Takayanagi (RT) formula \cite{2006PhRvL..96r1602R},
\begin{equation}
\label{eq: RT}
S(A)
=
\frac{\text{area}(\gamma_A)}{4G_N}\,.
\end{equation}
It relates the entanglement entropy $S$ of an
entangling region $A$ on the CFT side to the area of a minimal bulk surface $\gamma_A$ in the large $N$ limit. $\gamma_A$ is referred to as RT surface and
$G_N$ is Newton's constant. Starting from the RT formula, major progress was made in understanding the QI aspects of the field theory side
by studying the bulk. Further prominent examples for gravity dual
realizations of quantities relevant for QI are
quantum error correcting codes \cite{Pastawski:2015qua}, the \textit{Fisher information metric} (FIM) \cite{Lashkari:2015hha, Banerjee:2017qti} and complexity \cite{Susskind:2014rva, Stanford:2014jda, Brown:2015bva}.
In particular, subregion complexity was proposed to be related
to the volume enclosed by RT surfaces \cite{Alishahiha:2015rta}. This
volume was recently related to a field-theory expression in \cite{Abt:2017pmf,Abt:2018ywl}.
In this paper we focus on the \textit{modular Hamiltonian} $H$ for general QFTs, which is defined by
\begin{equation}
\rho
=
\frac{e^{-H}}{\tr(e^{-H})} \label{modham}
\end{equation}
for a given state $\rho$. \footnote{We use the terms density matrix
and states interchangeably.} The modular Hamiltonian plays
an important role for QI measures such as the \textit{relative entropy}
(RE) (see e.g.~\cite{RevModPhys.74.197, Jafferis:2015del, Sarosi:2016oks, Sarosi:2016atx}) or the FIM and was studied comprehensively by many authors,
for instance in \cite{Wong:2013gua, Jafferis:2014lza, Lashkari:2015dia, Faulkner:2016mzt, Ugajin:2016opf, Arias:2016nip, Koeller:2017njr, Casini:2017roe, Sarosi:2017rsq, Arias:2017dda}. Many
interesting aspects of the modular Hamiltonian were investigated, such
as a quantum version of the Bekenstein bound \cite{Casini:2008cr,
Blanco:2013lea} or a topological condition under which the modular
Hamiltonian of a 2d CFT can be written as a local integral over the
energy momentum tensor multiplied by a local weight
\cite{Cardy:2016fqc}. However,
the modular Hamiltonian is known explicitly only for a few examples,
such as for the reduced CFT ground
state on a ball-shaped entangling region in any dimension
(see e.g.~\cite{Casini:2011kv}) or for reduced thermal states on an interval for a $1+1$ dimensional CFT (see e.g.~\cite{Lashkari:2014kda, Blanco:2017xef}).
This paper is devoted to determining further properties of the modular
Hamiltonian as given by \eqref{modham}, in particular in connection with an external variable $\lambda$ parametrizing the density matrix $\rho_\lambda$. This
parameter may be related to the energy density or the
temperature of the state, for instance, as we do in the examples considered below.
We obtain new results on the parameter dependence of
\begin{equation}
\label{eq: DeltaH def}
\Delta\corr{H_0}(A,\lambda)
=
\tr(\rho_{\lambda}^A H_0)
-
\tr(\rho_{\lambda_0}^A H_0)\,,
\end{equation}
where $\rho_{\lambda}^A=\tr_{A^c}(\rho_{\lambda})$ is a reduced state on an entangling region $A$ and $H_0$ is the modular Hamiltonian of a chosen reduced reference state $\rho_{\lambda_0}^A$, i.e.
\begin{equation}
\rho_{\lambda_0}^A
=
\frac{e^{-H_0}}{\tr(e^{-H_0})}\,.
\end{equation}
$\Delta\corr{H_0}$ plays a crucial role in the computation of the RE w.r.t.~$A$ of the one-parameter family of states $\rho_\lambda$,
\begin{equation}
\label{eq: Srel}
S_{rel}(A,\lambda)
=
\tr(\rho_\lambda^A\log \rho_\lambda^A)
-
\tr(\rho_{\lambda}^A\log \rho_{\lambda_0}^A)
=
\Delta\corr{H_0}(A,\lambda)
-
\Delta S(A,\lambda)\,,
\end{equation}
as well as the FIM
\begin{equation}
\label{eq: Fisher info}
G_{\lambda\lambda}(A,\lambda_0)
=
\partial_\lambda^2 S_{rel}(A,\lambda)|_{\lambda=\lambda_0}\,,
\end{equation}
where $\Delta S(A,\lambda)=S(A,\lambda)-S(A,\lambda_0)$ is the difference of the entanglement entropies $S(A,\lambda)$ and $S(A,\lambda_0)$ of the reduced states $\rho_\lambda^A$ and $\rho_{\lambda_0}^A$, respectively.
In particular for holographic theories, where the entanglement entropy is given by the RT formula \eqref{eq: RT}, $\Delta\corr{H_0}$ is the term that makes it difficult to
compute the RE and the FIM.
From \eqref{eq: Fisher info} we see however that $\Delta\corr{H_0}$ does not affect
$G_{\lambda\lambda}$ if it has at most linear contributions in $\lambda$. So
in these situations an explicit expression for $\Delta\corr{H_0}$ is not required to compute
$G_{\lambda\lambda}$.
We investigate the case when $\Delta\corr{H_0}$ contributes to the FIM in a non-trivial way, i.e.~when higher order $\lambda$ contributions are present in
$\Delta\corr{H_0}$. Since
\begin{equation}
\Delta\corr{H_0}(A,\lambda_0)
=
0\,,
\end{equation}
from now on we refer to higher order contributions in $\tilde{\lambda}=\lambda-\lambda_0$
instead of $\lambda$.
\\
We examine the $\tilde{\lambda}$ dependence of $\Delta\corr{H_0}$ by considering the RE,
which is a valuable quantity for studying the modular Hamiltonian \cite{Casini:2008cr, Blanco:2013joa, Blanco:2013lea, Blanco:2017akw}.
For instance, the RE is known to be non-negative and to vanish iff $\rho^A_\lambda=\rho^A_{\lambda_0}$, which implies the \textit{first law of entanglement} \cite{Blanco:2013joa},
\begin{equation}
\label{eq: 1st law}
\partial_\lambda\Delta\corr{H_0}(A,\lambda)|_{\lambda=\lambda_0}
=
\partial_\lambda\Delta S(A,\lambda)|_{\lambda=\lambda_0}\,.
\end{equation}
We see that even though the modular Hamiltonian $H_0$ is not known in general, we may use the the non-negativity of $S_{rel}$ to determine the leading order contribution of $\Delta\corr{H_0}$ in $\tilde{\lambda}$,
\begin{equation}
\label{eq: mod Ham taylor}
\Delta\corr{H_0}(A,\lambda)
=
\partial_\lambda\Delta S(A,\lambda)|_{\lambda=\lambda_0}\tilde{\lambda}
+
\mathcal{O}(\tilde{\lambda}^2)\,.
\end{equation}
For some configurations, such as thermal states dual to black string geometries with the energy density as parameter $\lambda$ and an arbitrary interval as entangling region $A$ \cite{Lashkari:2014kda, Blanco:2017xef}, the higher-order contributions in $\tilde{\lambda}$ are known to vanish\footnote{We discuss this setup in Section \ref{sec: Black strings}.}, i.e.
\begin{equation}
\label{eq: linear mod Ham}
\Delta\corr{H_0}(A,\lambda)
=
\partial_\lambda\Delta S(A,\lambda)|_{\lambda=\lambda_0}\tilde{\lambda}\,.
\end{equation}
Consequently, $\Delta\corr{H_0}$ is completely determined by entanglement entropies, and in particular only contributes trivially to the FIM, as discussed above.
However, in general higher-order contributions in $\tilde{\lambda}$ will be present.
\\
In this paper we introduce a further application of the RE that allows us to
determine under which conditions higher-order contributions in $\tilde{\lambda}$ to $\Delta\corr{H_0}$ are to be
expected for families of states that form so-called \textit{entanglement plateaux}.
The term entanglement plateau was first introduced in \cite{Hubeny:2013gta} and
refers to entangling regions $A$, $B$ that saturate the \textit{Araki-Lieb inequality} (ALI) \cite{araki1970}
\begin{equation}
\label{eq: ALI}
|S(A)-S(B)|\leq S(AB)\,.
\end{equation}
We focus on entanglement plateaux that are stable under variations of $A$ and $B$ that keep $AB$ fixed. To be more precise, we consider two families $A_\sigma$ and $B_\sigma$
of entangling regions that come with a continuous parameter $\sigma$
determining their size, where $A_{\sigma_2}\subset A_{\sigma_1}$ if
$\sigma_1<\sigma_2$ and $A_{\sigma}B_{\sigma}=\Sigma=const.$ (see Figure \ref{fig: setup}), and
saturate the ALI, i.e.
\begin{equation}
\label{eq: AL saturation}
|S(A_\sigma,\lambda)-S(B_\sigma,\lambda)|= S(\Sigma,\lambda)\,.
\end{equation}
\begin{figure}[t]
\begin{center}
\includegraphics[scale=0.1]{A_and_B}
\end{center}
\caption{The families of entangling regions $A_\sigma$ and $B_\sigma$.
We consider two families of entangling regions $A_\sigma$ (red) and $B_\sigma$ (blue)
with $A_{\sigma_2}\subset A_{\sigma_1}$ for $\sigma_1< \sigma_2$ and
$A_\sigma B_\sigma=\Sigma=const.$ In particular, this implies $B_{\sigma_1}\subset B_{\sigma_2}$.}
\label{fig: setup}
\end{figure}
We show that the only way how both $\Delta\corr{H_0}(A_\sigma,\lambda)$ and $\Delta\corr{H_0}(B_\sigma,\lambda)$ can be linear in $\tilde{\lambda}$ for all $\sigma$ in a given interval $[\xi,\eta]$ is if $\partial_\lambda^2 S(A_\sigma,\lambda)$
and $\partial_\lambda^2 S(B_\sigma,\lambda)$ are constant in $\sigma$ on $[\xi,\eta]$.
The proof of this statement is a simple application of the well-known monotonicity \cite{uhlmann1977} of the RE,
\begin{equation}
\label{eq: monotonicity of Srel}
S_{rel}(A,\lambda)\leq S_{rel}(A',\lambda)\quad\text{if}\quad A\subseteq A'\,,
\end{equation}
and holds for any quantum system, not just for those with a holographic dual.
We thus find that in the setup described above, it suffices to look at
the entanglement entropies to see when higher-order contributions of $\tilde{\lambda}$
may be expected in at least one of the $\Delta\corr{H_0}$ (i.e.~$\Delta\corr{H_0}(A_\sigma,\lambda)$ or $\Delta\corr{H_0}(B_\sigma,\lambda)$), namely if $\partial_\lambda^2S(A_\sigma,\lambda)$ or $\partial_\lambda^2S(B_\sigma,\lambda)$ is not constant in $\sigma$.
In particular if one of the $\Delta\corr{H_0}$, say $\Delta\corr{H_0}(B_\sigma,\lambda)$,
is known to be linear for all $\sigma\in[\xi,\eta]$, we learn that $\Delta\corr{H_0}(A_\sigma,\lambda)$ is not.
Consequently, it is not sufficient to work with entanglement entropies to
determine $\Delta\corr{H_0}(A_\sigma,\lambda)$ via \eqref{eq: linear mod Ham}, but more involved calculations are required.
As a result this means that the RE is not just given by entanglement entropies.
\\
Our result for entanglement plateaux has important consequences in particular for
holographic theories. There are many well-known
configurations in holography that form entanglement plateaux in the large $N$ limit. Prominent examples -- which we discuss in this paper -- are large intervals for the BTZ back hole \cite{Headrick:2007km, Blanco:2013joa, Hubeny:2013gta} and two sufficiently close intervals for black strings \cite{Headrick:2010}. For these situations, very little is known about $\Delta\corr{H_0}$, \footnote{Note that the vacuum modular Hamiltonian of two intervals is known explicitly for the 2d CFTs of the massless free fermion \cite{Casini:2009vk, Arias:2018tmw} and the chiral free scalar \cite{Arias:2018tmw}. In this paper however, we consider thermal states in strongly coupled CFTs with gravity duals.} however our result can be used to prove that non-linear $\tilde{\lambda}$ contributions play a role in the $\Delta\corr{H_0}$ occurring in these models.
For the situation of two intervals described above, this may be used to show
that the modular Hamiltonian is not an integral over the energy momentum tensor
multiplied by a local scaling, as it is the case for one interval.
\\
This paper is structured as follows. In Section \ref{sec: Black strings}
we consider the special case of black strings as a motivation and to introduce the basic arguments required to verify our result, which we prove in Section \ref{sec: Theorem} in its full generality. We then present several situations where the result can be applied in Section \ref{sec: applications}. These include
an arbitrary number of intervals for thermal states dual to black strings,
a spherical shell for states dual to black branes, a sufficiently large entangling interval for states dual to BTZ black holes and primary excitations in a CFT with large central charge, defined on a circle. Furthermore, we discuss examples where the prerequisites of our result are not satisfied in Section \ref{sec: vacuum states for CFTs}. Finally we make some concluding remarks in Section \ref{sec: conclusions}.
\section{A Simple Example: Black Strings}
\label{sec: Black strings}
Our result for modular Hamiltonians, as described in the introduction
and proved below in Section \ref{sec: Theorem}, may be applied to a vast
variety of situations. As an illustration, we begin by a simple example that introduces
the basic arguments for our result and demonstrates its
usefulness. This example involves thermal CFT states in $1+1$ dimensions of inverse temperature $\beta$ with black strings as gravity duals,
\begin{equation}
\label{eq: BS geometry}
ds_{BS}^2
=
\frac{L^2}{z^2}\Big(
-\frac{z_h^2-z^2}{z_h^2}dt^2
+
\frac{z_h^2}{z_h^2-z^2} dz^2
+
dx^2
\Big)\, ,
\end{equation}
where $z=z_h$ is the location of the black string and $L$ is the AdS radius. The asymptotic boundary, where the CFT is defined, lies at $z=0$. The energy density
\begin{equation}
\label{eq: lambda ito beta}
\lambda
=
\frac{L}{16 \pi G_N z_h^2}
=
\frac{\pi c}{6 \beta^2}\,,
\end{equation}
where $c=\frac{3L}{2 G_N}$ is the central charge of the CFT,
is chosen as the parameter for this family of states. The reference
state may be chosen to correspond to any energy density $\lambda_0$.
\\
We now demonstrate how the RE can be used to show that $\Delta\corr{H_0}$, as
defined in \eqref{eq: DeltaH def}, for a state living on two separated
intervals is in general not linear in $\tilde{\lambda}=\lambda-\lambda_0$ if the
two intervals are sufficiently close. The arguments that lead to this
conclusion will be generalized in Section \ref{sec: Theorem} below.
Consider an entangling region $A_\sigma$ that consists of two intervals $A^1_\sigma=[a_1,-\sigma]$ and $A^2_\sigma=[\sigma, a_2]$, with $\sigma>0$ and $a_1$, $a_2$ fixed (see Figure \ref{fig: BS 2 int}). The interval $B_\sigma=[-\sigma,\sigma]$ between $A^1_\sigma$ and $A^2_\sigma$ is w.l.o.g.~assumed to lie symmetric around the coordinate origin $x=0$.
\begin{figure}[t]
\begin{center}
\includegraphics[scale=0.2]{BS_1}
\end{center}
\caption{A constant time slice of the black string geometry \eqref{eq: BS geometry}. The asymptotic boundary of this geometry -- where the CFT is defined -- corresponds to the $x$-axis. The location of the black string is $z=z_h$ and depends on the energy density $\lambda$ via \eqref{eq: lambda ito beta}. If $\sigma$ is sufficiently small the RT surface $\gamma_{A_\sigma}$ of the entangling region $A_\sigma=A_\sigma^1A_\sigma^2$ (red) is the union of the RT surfaces $\gamma_\Sigma$ of $\Sigma=A_\sigma B_\sigma$ and $\gamma_{B_\sigma}$ of $B_\sigma$ (blue). This implies \eqref{eq: AL black string}.}
\label{fig: BS 2 int}
\end{figure}
If $\sigma$ is sufficiently small\footnote{For previous work regarding the modular Hamiltonian for such a situation, see e.g.~\cite{Blanco:2013joa}.},
the RT surface $\gamma_{A_\sigma}$ of $A_\sigma$ is the union of $\gamma_{B_\sigma}$ and $\gamma_\Sigma$ (see Figure \ref{fig: BS 2 int}), where $\Sigma=A_\sigma B_\sigma=[a_1, a_2]$ is the union of $A_\sigma$ and $B_\sigma$.
Consequently, the entanglement entropy of $A_\sigma$ saturates the ALI \cite{Headrick:2010}, i.e.
\begin{equation}
\label{eq: AL black string}
S(A_\sigma, \lambda)
=
S(\Sigma, \lambda)
+
S(B_\sigma, \lambda)\,,
\end{equation}
which is an immediate consequence of the RT formula \eqref{eq: RT}.
For thermal states in general CFTs defined on the real axis, the modular Hamiltonian $H_0(B_\sigma)$ of $B_\sigma$ for the reference parameter value $\lambda_0$ is given by \cite{Lashkari:2014kda, Blanco:2017xef}
\begin{equation}
\label{eq: mod Ham one interval}
H_0(B_\sigma)
=
\int_{-\sigma}^\sigma dx\, \beta_0\frac{\cosh(\frac{2\pi \sigma}{\beta_0})-\cosh(\frac{2\pi x}{\beta_0})}{\sinh(\frac{2\pi \sigma}{\beta_0})}T_{00}(x)\,,
\end{equation}
where $T_{\mu\nu}$ is the energy momentum tensor of the CFT and $\beta_0=\beta(\lambda_0)$. Thus, using \eqref{eq: DeltaH def}, we find
\begin{equation}
\label{eq: Delta mod Ham one interval}
\Delta\corr{H_0}(B_\sigma,\lambda)
=
\beta_0\Big(2\sigma\coth\Big(\frac{2\pi \sigma}{\beta_0}\Big)-\frac{\beta_0}{\pi}\Big)\tilde{\lambda}
=
\Delta S'(B_\sigma,\lambda_0)\tilde{\lambda}
\end{equation}
to be linear in $\tilde{\lambda}$.
Here, the $'$ refers to a derivative w.r.t.~$\lambda$.
The second equality is an immediate consequence of the first law of entanglement, i.e.~\eqref{eq: mod Ham taylor},
however may also be verified by a direct calculation using \cite{2006PhRvL..96r1602R, Calabrese:2004eu}
\begin{equation}
\label{eq: EE one interval}
S(B_\sigma,\lambda)
=
\frac{c}{3}\log\Big(\frac{\beta}{\pi \epsilon}\sinh\Big(\frac{2\pi \sigma}{\beta}\Big)\Big)\ ,
\end{equation}
where $\epsilon$ is a UV cutoff.
The two simple observations
\eqref{eq: AL black string} and \eqref{eq: Delta mod Ham one interval} together with the monotonicity of the RE \eqref{eq: monotonicity of Srel} are sufficient to verify that
$\Delta\corr{H_0}(A_\sigma,\lambda)$ is not linear in $\tilde{\lambda}$, except for possibly one particular $\sigma$, as we now show.
Let us assume that $\Delta\corr{H_0}(A_\sigma,\lambda)$ is linear in $\tilde{\lambda}$
for a given $\sigma$. The first law of entanglement \eqref{eq: mod Ham taylor} implies
\begin{equation}
\Delta\corr{H_0}(A_\sigma,\lambda)
=
\Delta S'(A_\sigma,\lambda_0)\tilde{\lambda}\ .
\end{equation}
Applying this result to $S_{rel}(A_\sigma,\lambda)$ and using \eqref{eq: AL black string} and \eqref{eq: Delta mod Ham one interval},
we obtain
\begin{equation}
\label{eq: Srel two intervals}
S_{rel}(A_\sigma,\lambda)
=
\Delta S'(\Sigma,\lambda_0)\tilde{\lambda}
-
\Delta S(\Sigma,\lambda)
+
S_{rel}(B_\sigma,\lambda)\ .
\end{equation}
Using \eqref{eq: lambda ito beta}, \eqref{eq: Delta mod Ham one interval} and \eqref{eq: EE one interval}, $S_{rel}(B_\sigma,\lambda)$ may be brought into the form
\begin{equation}
\label{eq: Srel for B}
S_{rel}(B_\sigma,\lambda)
=
\frac{c}{3}\Big(
\frac{1}{2}(1-b^2)(1-a\coth(a))
+
\log\Big(b\frac{\sinh(a)}{\sinh(b\,a)}\Big)
\Big)\,,
\end{equation}
where $a=2\pi\sigma/\beta_0$ and $b=\beta_0/\beta$.
For fixed $b$, $S_{rel}(B_\sigma,\lambda)$ grows with $a$ (see Figure \ref{fig: Srel plot}), which implies
that $S_{rel}(B_\sigma,\lambda)$ grows with $\sigma$ for fixed $\beta$ and $\beta_0$, or equivalently for fixed $\lambda$ and $\lambda_0$ (see \eqref{eq: lambda ito beta}).
\begin{figure}[t]
\begin{center}
\includegraphics[scale=0.1]{Srel_plot}
\end{center}
\caption{The behavior of $S_{rel}(B_\sigma, \lambda)$ \eqref{eq: Srel for B} w.r.t.~$a$ for $b=1,\dots,5$, where $a=2\pi\sigma/\beta_0$ and $b=\beta_0/\beta$. We set the global prefactor $c/3=1$ and see that $S_{rel}(B_\sigma, \lambda)$ grows with $a$ for fixed $b$. In particular, this implies that $S_{rel}(B_\sigma, \lambda)$ grows with $B_\sigma$, i.e.~$\sigma$, for fixed $\lambda$ and $\lambda_0$, which is in agreement with the monotonicity of the RE \eqref{eq: monotonicity of Srel}. For $b=1$ we find $S_{rel}(B_\sigma,\lambda)=0$, which is to be expected from \eqref{eq: Srel}, since this case corresponds to $\lambda=\lambda_0$.}
\label{fig: Srel plot}
\end{figure}
Since $S_{rel}(B_\sigma,\lambda)$ is the only $\sigma$-dependent term on the RHS of \eqref{eq: Srel two intervals}, $S_{rel}(A_\sigma,\lambda)$ grows with $\sigma$ as well.
Now assume there were two values $\xi$, $\eta$ for $\sigma$, where we set w.lo.g. $\xi<\eta$, for which $\Delta\corr{H_0}(A_\sigma,\lambda)$ is linear in $\tilde{\lambda}$.
From the above discussion we conclude
\begin{equation}
S_{rel}(A_\xi,\lambda)<S_{rel}(A_\eta,\lambda)\,.
\end{equation}
However, the monotonicity of the RE \eqref{eq: monotonicity of Srel} implies that $S_{rel}(A_\eta,\lambda)$ must be smaller than $S_{rel}(A_\xi,\lambda)$, since $A_\eta\subset A_\xi$.
So by assuming $\Delta\corr{H_0}(A_\sigma,\lambda)$ to be linear in $\tilde{\lambda}$ for more than one value of $\sigma$, we
encounter a contradiction. Consequently, $\Delta\corr{H_0}(A_\sigma,\lambda)$ may be linear in $\tilde{\lambda}$ for at most one particular $\sigma$.
\\
This simple example shows that even though the modular
Hamiltonian for two disconnected intervals is unknown, general
properties of the RE imply that the modular Hamiltonian necessarily involves contributions of higher order in $\tilde{\lambda}$. An immediate consequence of this
observation is that the modular Hamiltonian for two intervals, unlike for one interval \eqref{eq: mod Ham one interval}, can \textit{not} be of the simple form
\begin{equation}
\int_{A_\sigma} dx f^{\mu\nu}(x)T_{\mu\nu}(x)\,,
\end{equation}
where $f^{\mu\nu}$ is a local weight function, since this would lead to a $\Delta\corr{H_0}(A_\sigma,\lambda)$ that is linear in $\tilde{\lambda}$.
\\
Note that since $S_{rel}(B_\sigma,\lambda)$ is known, we are not required to consider
$\partial^2_\lambda S$, i.e.~the quantity discussed below \eqref{eq: monotonicity of Srel} in the introduction. We were able to deduce the non-linearity of $\Delta\corr{H_0}(A_\sigma,\lambda)$ directly from $S_{rel}(B_\sigma,\lambda)$ (see \eqref{eq: Srel two intervals}). In the more general cases discussed in Section \ref{sec: Theorem}, where $S_{rel}(B_\sigma,\lambda)$ is not known, this is no longer possible.
\section{Generic Entanglement Plateaux}
\label{sec: Theorem}
We now generalize the approach introduced in
Section \ref{sec: Black strings} and show how the RE determines
whether non-linear contributions to $\Delta\corr{H_0}$ in $\tilde{\lambda}$ are to be
expected. Note that we do not require $\lambda$ to be the energy density, it is
just the variable that parametrizes the family of states
$\rho_\lambda$ we consider.
The discussion in Section \ref{sec: Black strings} required the saturation of the ALI \eqref{eq: ALI}, which allowed
us to show that if $\Delta\corr{H_0}$ were linear in $\tilde{\lambda}$, the RE would increase when the size of the considered entangling region (i.e.~two intervals) decreases. However, due to the monotonicity of the RE \eqref{eq: monotonicity of Srel} this is not possible.
By looking at \eqref{eq: Srel two intervals}, we see that this contradiction does not require the explicit expressions for the (relative) entropies: If $S_{rel}(B_\sigma,\lambda)$ grows with $B_\sigma$ for fixed $\Sigma$, $S_{rel}(A_\sigma,\lambda)$ grows as well. However, this is not compatible with the monotonicity of $S_{rel}$, since $A_\sigma=\Sigma\backslash B_\sigma$ decreases if $B_\sigma$ increases. This fact allows us to generalize the arguments of Section \ref{sec: Black strings} to generic entanglement plateaux, i.e.~systems that saturate the ALI.
\subsection{Result for Generic Entanglement Plateaux}
\label{sec: gen EP}
In the general case, the prerequisites for our main statement are as follows. We consider a one-parameter family of states $\rho_\lambda$. Let $\Sigma$ be an entangling region and $A_\sigma\subseteq\Sigma$ a one-parameter family of decreasing subregions of $\Sigma$, i.e.~$A_{\sigma_2}\subset A_{\sigma_1}$ for $\sigma_1<\sigma_2$, where the parameter $\sigma$ is assumed to be continuous. Furthermore, let $B_\sigma=\Sigma\backslash A_\sigma$ be the complement of $A_\sigma$ w.r.t.~$\Sigma$ (see Figure \ref{fig: setup}).
Moreover, the ALI \eqref{eq: ALI} is assumed to be saturated for $A_\sigma$ and $B_\sigma$, i.e.
\begin{equation}
\label{eq: SA=SS+SB}
|S(A_\sigma,\lambda)-S(B_\sigma,\lambda)|
=
S(\Sigma,\lambda)
\quad
\forall\sigma,\lambda\,.
\end{equation}
Furthermore, $S(A_\sigma,\lambda)$, $S(B_\sigma,\lambda)$ and $S(\Sigma,\lambda)$ are considered to be differentiable in $\lambda$ for all $\sigma$.
Subject to these prerequisites, we now state our main result.
If both $\Delta\corr{H_0}(A_{\sigma},\lambda)$ and $\Delta\corr{H_0}(B_{\sigma},\lambda)$ are linear in $\tilde{\lambda}=\lambda-\lambda_0$ for all $\sigma$ in a given interval $[\xi,\eta]$, then
$\partial_\lambda^2S(A_\sigma,\lambda)$ and $\partial_\lambda^2S(B_\sigma,\lambda)$ are constant in $\sigma$ on $[\xi,\eta]$ for all $\lambda$.
\vspace{5mm}
\\
We prove this statement as follows.
As we discuss in the appendix, w.l.o.g.~we may restrict our arguments to the case $S(A_\sigma,\lambda)\geq S(B_\sigma,\lambda)$.
Assume that for all $\sigma\in[\xi,\eta]$, both $\Delta\corr{H_0}(A_{\sigma},\lambda)$ and $\Delta\corr{H_0}(B_{\sigma},\lambda)$ are linear in $\tilde{\lambda}$. Then, as explained in the introduction (see \eqref{eq: linear mod Ham}), we find
\begin{equation}
\label{eq: linear modhams for A and B}
\Delta\corr{H_0}(A_{\sigma},\lambda)
=
\Delta S'(A_{\sigma},\lambda_0)\tilde{\lambda}
\quad\mbox{and}\quad
\Delta\corr{H_0}(B_{\sigma},\lambda)
=
\Delta S'(B_{\sigma},\lambda_0)
\tilde{\lambda}\,,
\end{equation}
where $'$ again refers to a derivative w.r.t.~$\lambda$.
This implies together with \eqref{eq: Srel} and \eqref{eq: SA=SS+SB}
\begin{equation}
\label{eq: SrelA i.t.o. SrelB for alpha}
S_{rel}(A_\sigma,\lambda)
=
\Delta S'(\Sigma,\lambda_0)\tilde{\lambda}
-
\Delta S(\Sigma,\lambda)
+
S_{rel}(B_\sigma,\lambda)\,.
\end{equation}
Due to the monotonicity \eqref{eq: monotonicity of Srel} of $S_{rel}$ we find
\begin{equation}
S_{rel}(B_\xi,\lambda)\leq S_{rel}(B_\eta,\lambda)\,,
\end{equation}
since $B_\xi\subset B_\eta$. Using \eqref{eq: SrelA i.t.o. SrelB for alpha}, this implies
\begin{equation}
\label{eq: Srel inequality}
S_{rel}(A_\xi,\lambda)\leq S_{rel}(A_\eta,\lambda)\,.
\end{equation}
By construction we have $A_\eta\subset A_\xi$. So the only way how \eqref{eq: Srel inequality} may be compatible with the monotonicity of $S_{rel}$ is if $S_{rel}(A_\sigma,\lambda)$ is constant in $\sigma$ for $\sigma\in[\xi,\eta]$.
Thus by using \eqref{eq: Srel} and \eqref{eq: linear modhams for A and B}, we find
\begin{equation}
\label{eq: DDS=const}
-\partial^2_\lambda S_{rel}(A_\sigma,\lambda)
=
-\partial^2_\lambda(
\Delta S'(A_\sigma,\lambda_0)(\lambda-\lambda_0)
-
\Delta S(A_\sigma,\lambda)
)
=
\partial^2_\lambda S(A_\sigma,\lambda)
\end{equation}
to be constant in $\sigma$ on $[\xi,\eta]$.
Due to \eqref{eq: SrelA i.t.o. SrelB for alpha} the fact that $S_{rel}(A_\sigma,\lambda)$ is constant in $\sigma$ for $\sigma\in[\xi,\eta]$ implies that $S_{rel}(B_\sigma,\lambda)$ is as well.
In an analogous way as for $A_\sigma$, we find $\partial_\lambda^2 S(B_\sigma,\lambda)$ to be constant in $\sigma$ on $[\xi,\eta]$.
This completes the proof of the general result stated at the beginning of this section.
\subsection{Discussion for Generic Entanglement Plateaux}
\label{sec: ge EP discussion}
In Section \ref{sec: gen EP} we presented our result for a generic situation where the ALI is saturated. Some comments are in order.
First we note that even though we presented an example from holography in Section \ref{sec: Black strings} as a motivation, we did not require holography at any point during the proof. Therefore our result is true for any quantum system.
\\
Furthermore, we required $\sigma$, i.e.~the parameter of the family of entangling regions $A_\sigma$, to be continuous, as can be read off the discussion in the appendix. However, if we in addition assume the sign of $S(A_\sigma,\lambda)-S(B_\sigma,\lambda)$ to be constant in $\sigma$, we can apply the result to discrete systems, such as spin-chains, as well. The proof works analogously as in the continuous case discussed in Section \ref{sec: gen EP}.
\\
In Section \ref{sec: gen EP} we showed that $\partial_\lambda^2 S(A_\sigma,\lambda)$ and $\partial_\lambda^2 S(B_\sigma,\lambda)$ beeing constant in $\sigma$ on an interval $[\xi,\eta]$ is a necessary condition for both $\Delta\corr{H_0}(A_\sigma,\lambda)$ and $\Delta\corr{H_0}(B_\sigma,\lambda)$ to be linear in $\tilde{\lambda}$ for all $\sigma\in[\xi,\eta]$. However, this condition is not sufficient, as we now demonstrate by presenting an example where $\partial_\lambda^2 S(A_\sigma,\lambda)$ and $\partial_\lambda^2 S(B_\sigma,\lambda)$ are constant in $\sigma$ but both $\Delta\corr{H_0}(A_\sigma,\lambda)$ and $\Delta\corr{H_0}(B_\sigma,\lambda)$ are not linear in $\tilde{\lambda}$.
We consider a free massless boson CFT in two dimensions defined on a circle with radius $\ell_{CFT}$. The family of states is
chosen to consist of exited states of the form
\begin{equation}
\ket{\lambda}
=
e^{i\sqrt{2\lambda}\Phi}\ket{0}\,,
\end{equation}
where $\Phi$ is the boson field and $\ket{0}$ is the vacuum state.
We use their conformal dimension $(\lambda,0)$ to parametrize these states.
For the sake of this paper we assume the conformal dimension $\lambda$ to be
a continuous parameter\footnote{Note that the parameter $\lambda$ is assumed to be continuous in Section \ref{sec: gen EP}, since we take derivatives w.r.t.~it, e.g.~in \eqref{eq: SrelA i.t.o. SrelB for alpha}.}.
We define $A_\sigma$ to be an interval of angular size $2(\pi-\sigma)$ and
$B_\sigma=A_\sigma^c$ to be the complementary interval of angular size $2\sigma$.
Consequently, $\Sigma=A_\sigma B_\sigma$ is the entire circle and the fact that
$\ket{\lambda}$ is pure implies $S(\Sigma,\lambda)=0$ and $S(A_\sigma,\lambda)=S(B_\sigma,\lambda)$,
and therefore the saturation of the ALI \eqref{eq: ALI}.
The reference state $\ket{\lambda_0}$ can be chosen arbitrarily.
This setup was discussed in \cite{Lashkari:2015dia}, where the RE was found to be
\begin{align}
\label{eq: RE bosons A}
S_{rel}(A_\sigma,\lambda)
&
=
(1+(\pi-\sigma)\cot(\sigma))\Big(\sqrt{2\lambda}-\sqrt{2\lambda_0}\Big)^2\,,
\\
\label{eq: RE bosons B}
S_{rel}(B_\sigma,\lambda)
&
=
(1-\sigma\cot(\sigma))\Big(\sqrt{2\lambda}-\sqrt{2\lambda_0}\Big)^2\,.
\end{align}
The author of \cite{Lashkari:2015dia} states that the entanglement entropies of $A_\sigma$
and $B_\sigma$ are constant in $\lambda$. Therefore, by applying \eqref{eq: Srel}
to \eqref{eq: RE bosons A} and \eqref{eq: RE bosons B} we find
\begin{align}
\label{eq: H bosons A}
\Delta\corr{H_0}(A_\sigma,\lambda)
&
=
(1+(\pi-\sigma)\cot(\sigma))\Big(\sqrt{2\lambda}-\sqrt{2\lambda_0}\Big)^2\,,
\\
\label{eq: H bosons B}
\Delta\corr{H_0}(B_\sigma,\lambda)
&
=
(1-\sigma\cot(\sigma))\Big(\sqrt{2\lambda}-\sqrt{2\lambda_0}\Big)^2\,.
\end{align}
Obviously, both $\Delta\corr{H_0}(A_\sigma,\lambda)$ and $\Delta\corr{H_0}(B_\sigma,\lambda)$
are not linear in $\tilde{\lambda}=\lambda-\lambda_0$. However, since the entanglement entropy is constant in $\lambda$, we find $\partial_\lambda^2 S=0$
for $A_\sigma$ and $B_\sigma$, and therefore that $\partial_\lambda^2 S$ is constant in $\sigma$ for $A_\sigma$ and $B_\sigma$. Thus we see that
$\partial_\lambda^2 S$ being constant in $\sigma$ does not imply that both $\Delta\corr{H_0}(A_\sigma,\lambda)$ and $\Delta\corr{H_0}(B_\sigma,\lambda)$ are linear in $\tilde{\lambda}$.
Therefore it is a necessary but not a sufficient condition.
\\
The proof of our result presented in Section \ref{sec: gen EP} strongly relies on the first law of entanglement \eqref{eq: 1st law}. We need to emphasize that the
first law of entanglement only applies if the reference state corresponds to a parameter value $\lambda_0$ that is not a boundary point of the set of allowed parameter values $\lambda$. The fact that the first law of entanglement holds is a consequence of the non-negativity of $S_{rel}(A,\lambda)$ and $S_{rel}(A,\lambda_0)=0$. These two
properties imply that $S_{rel}$ is minimal at $\lambda=\lambda_0$ and therefore
we find
\begin{equation}
\label{eq: different 1sr law}
\partial_\lambda S_{rel}(A,\lambda)|_{\lambda=\lambda_0}
=
0\,.
\end{equation}
Using \eqref{eq: Srel} it is easy to see that \eqref{eq: different 1sr law} is equivalent to the first law of entanglement. However, if $\lambda_0$ is a boundary point of the set of allowed $\lambda$, i.e.~if it is not possible to choose $\lambda<\lambda_0$ for instance, the minimality of $S_{rel}(A,\lambda_0)$ does not necessarily imply $\partial_\lambda S_{rel}(A,\lambda)|_{\lambda=\lambda_0}$ to vanish.
The free massless boson CFT we discuss above is an example for such a situation. Here the parameter $\lambda$ is the conformal dimension of the considered states and is therefore non-negative. By choosing the reference state to be the vacuum, i.e.~$\lambda_0=0$, \eqref{eq: RE bosons A} gives
\begin{equation}
S_{rel}(A_\sigma,\lambda)
=
2(1+(\pi-\sigma)\cot(\sigma))\tilde{\lambda}\,,
\end{equation}
and therefore $\partial_\lambda S_{rel}(A_\sigma,\lambda)|_{\lambda=\lambda_0}\neq 0$.
Consequently, the first law of entanglement does not hold for this example.
Even though it has the expected properties according to our prediction, i.e.~both $\Delta\corr{H_0}(A_\sigma,\lambda)$ and $\Delta\corr{H_0}(B_\sigma,\lambda)$ are
linear in $\tilde{\lambda}$ and
$\partial_\lambda^2S(A_\sigma,\lambda)$ and $\partial_\lambda^2 S(B_\sigma,\lambda)$ are constant in $\sigma$ (see \eqref{eq: H bosons A}, \eqref{eq: H bosons B} for $\lambda_0=0$), the prerequisites of our result are not satisfied if the first law of entanglement does not hold.
\\
We only considered one-parameter families of states in Section \ref{sec: gen EP}.
However, our result can be straightforwardly generalized to an $n$-parameter family of states $\rho_\Lambda$ with $\Lambda=(\lambda^1,...,\lambda^n)$. The
reference state corresponds to $\Lambda=\Lambda_0=(\lambda^1_0,...,\lambda^n_0)$.
In an analogous way as for the one-parameter case we can show that the
only way how both $\Delta\corr{H_0}(A_\sigma,\Lambda)$ and $\Delta\corr{H_0}(B_\sigma,\Lambda)$
can be linear in $\Lambda-\Lambda_0$, i.e.~of the form\footnote{Here we use once more the first law of entanglement \eqref{eq: 1st law}.}
\begin{equation}
\begin{split}
&
\Delta\corr{H_0}(A_\sigma,\Lambda)
=
\partial_i\Delta S(A_\sigma,\Lambda)|_{\Lambda=\Lambda_0}(\lambda^i-\lambda_0^i)\,
\\
&
\Delta\corr{H_0}(B_\sigma,\Lambda)
=
\partial_i\Delta S(B_\sigma,\Lambda)|_{\Lambda=\Lambda_0}(\lambda^i-\lambda_0^i)\,,
\end{split}
\end{equation}
where $\partial_i=\partial/\partial\lambda^i$, for all $\sigma\in[\xi,\eta]$ is if $\partial_i\partial_j S(A_\sigma,\Lambda)$
and $\partial_i\partial_j S(B_\sigma,\Lambda)$ are constant in $\sigma$ on
$[\xi,\eta]$.
\\
\subsection{Alternative Formulation}
\label{sec: alternative formulation}
For the examples we discuss in Section \ref{sec: applications}, it is more convenient to use the following alternative formulation of our result:
Consider the assumptions necessary for the result to be satisfied (see Section \ref{sec: gen EP}). If $\partial_\lambda^2 S(A_\sigma,\lambda)$ or $\partial_\lambda^2 S(B_\sigma,\lambda)$ is not constant in $\sigma$ on any interval $[\xi,\eta]$, then there are only single values of $\sigma$ where both $\Delta\corr{H_0}(A_\sigma,\lambda)$ and $\Delta\corr{H_0}(B_\sigma,\lambda)$
are linear in $\tilde{\lambda}$, i.e.~there is no interval $[\xi,\eta]$ where
both $\Delta\corr{H_0}(A_\sigma,\lambda)$ and $\Delta\corr{H_0}(B_\sigma,\lambda)$ are linear in $\tilde{\lambda}$ for all $\sigma\in[\xi,\eta]$.
In the original formulation, the linearity of $\Delta\corr{H_0}(A_\sigma,\lambda)$ and $\Delta\corr{H_0}(B_\sigma,\lambda)$ in $\tilde{\lambda}$ implies that the second derivative of the entanglement entropies of $A_\sigma$ and $B_\sigma$ are constant in $\sigma$. In the alternative formulation however, non-constancy in $\sigma$ of the second derivative of one of the entanglement entropies implies that in general $\Delta\corr{H_0}$ is non-linear in $\tilde{\lambda}$ for $A_\sigma$, $B_\sigma$ or both.
In the examples of Section \ref{sec: applications}, there are non-constant second derivatives of the entanglement entropies, and therefore the alternative formulation is more appropriate.
\\
In the alternative formulation, the number of values for $\sigma$ where both $\Delta\corr{H_0}(A_\sigma,\lambda)$ and $\Delta\corr{H_0}(B_\sigma,\lambda)$ are linear in $\tilde{\lambda}$ is undetermined.
However, in Section \ref{sec: Black strings}, where we considered $A_\sigma$ to be the union of two intervals, we were able to show a stronger statement. We found that there is at most one such value for $\sigma$ and moreover, that $\Delta\corr{H_0}(A_\sigma,\lambda)$ is linear in $\tilde{\lambda}$ only for that value of $\sigma$.
The arguments of Section \ref{sec: Black strings} that lead to this conclusion
can be generalized to the case of generic entanglement plateaux if
\begin{equation}
\label{eq: Drel}
D_{rel}(B_\sigma,\lambda)
=
\Delta S'(B_\sigma,\lambda_0)\tilde{\lambda}
-
\Delta S(B_\sigma,\lambda)
\end{equation}
grows strictly monotonically with $\sigma$. In particular, if $\Delta\corr{H_0}(B_\sigma,\lambda)$ is known to be linear in $\tilde{\lambda}$, $D_{rel}(B_\sigma,\lambda)$ is the RE of $B_\sigma$, \footnote{This is an immediate consequence of the first law of entanglement \eqref{eq: 1st law}.} which is the case for the setup discussed in Section \ref{sec: Black strings}, for instance.
Just as in Section \ref{sec: gen EP}, we assume w.l.o.g.~$S(A_\sigma,\lambda)\geq S(B_\sigma,\lambda)$. Under the assumption that there are two values $\xi$, $\eta$ for $\sigma$ where $\Delta\corr{H_0}(A_\sigma,\lambda)$ is linear in $\tilde{\lambda}$, we
find, analogous to the derivation of \eqref{eq: SrelA i.t.o. SrelB for alpha},
\begin{equation}
S_{rel}(A_{\xi,\eta},\lambda)
=
\Delta S'(\Sigma,\lambda_0)\tilde{\lambda}
-
\Delta S(\Sigma,\lambda)
+
D_{rel}(B_{\xi,\eta},\lambda)\,.
\end{equation}
Since $D_{rel}(B_\sigma,\lambda)$ is assumed to grow strictly monotonically with $\sigma$, this implies for $\xi<\eta$
\begin{equation}
S_{rel}(A_\xi,\lambda)<S_{rel}(A_\eta,\lambda)\,,
\end{equation}
which is not possible due to the monotonicity of $S_{rel}$ \eqref{eq: monotonicity of Srel}, since $A_\eta\subset A_\xi$. Consequently, there can only be one value of $\sigma$
where $\Delta\corr{H_0}(A_\sigma,\lambda)$ is linear in $\tilde{\lambda}$.
\section{Applications}
\label{sec: applications}
We now apply the general result of Section \ref{sec: gen EP} to holographic states dual to black strings, black branes and BTZ black holes. Moreover, we apply the result to pure states, which we first discuss in full generality and then consider primary excitations of a CFT with large central charge as an example. In all these configurations entanglement plateaux can be constructed, i.e.~situations where the ALI is saturated \eqref{eq: SA=SS+SB}, which is the only requirement for our result.
\subsection{Black Strings Revisited}
First we consider once more, as in Section \ref{sec: Black strings}, the situation of two sufficiently close intervals
for CFTs dual to black strings \eqref{eq: BS geometry}.
The parameter $\lambda$ is chosen to be the energy density \eqref{eq: lambda ito beta}.
We can confirm the conclusion we made in Section \ref{sec: Black strings} by applying the result of Section \ref{sec: gen EP}:
Using \eqref{eq: EE one interval} is easy to see that
$\partial_\lambda^2 S(B_\sigma,\lambda)$ is not constant in $\sigma$ on any interval. So the result of Section \ref{sec: gen EP} tells us
that there is no interval $[\xi,\eta]$ where both $\Delta\corr{H_0}(A_\sigma,\lambda)$ and $\Delta\corr{H_0}(B_\sigma,\lambda)$ are linear in $\tilde{\lambda}$ for all $\sigma\in[\xi,\eta]$. We know that
$\Delta\corr{H_0}(B_\sigma,\lambda)$ is linear in $\tilde{\lambda}$ for all $\sigma$
(see \eqref{eq: Delta mod Ham one interval}), and therefore conclude that
$\Delta\corr{H_0}(A_\sigma,\lambda)$ is not, except possibly for single values of $\sigma$.
From the discussion in Section \ref{sec: alternative formulation} we are even able to conclude that there is only one such $\sigma$.
This is due to the fact that $D_{rel}(B_\sigma,\lambda)$ \eqref{eq: Drel}, which is equal to
$S_{rel}(B_\sigma,\lambda)$ here, grows strictly monotonically with $\sigma$, as pointed out in Section \ref{sec: Black strings}.
This special value of $\sigma$ corresponds to the degenerate situation
where $B_\sigma$ vanishes and $A_\sigma$ becomes a single interval, i.e.~$\sigma=0$.
The discussion of two intervals can be straightforwardly generalized to
the situation of $A_\sigma$ being the union of an arbitrary number of intervals. $B_\sigma$ is chosen to be an interval between two neighboring intervals that belong to $A_\sigma$. If the ALI is saturated, which corresponds to a situation such as the one depicted in Figure \ref{fig: BS n int},
we see in analogy to the two-interval case, that $\Delta\corr{H_0}(A_\sigma,\lambda)$ is in general not linear in $\tilde{\lambda}$.
\begin{figure}[t]
\begin{center}
\includegraphics[scale=0.2]{BS_2}
\end{center}
\caption{A constant time slice of the black string geometry \eqref{eq: BS geometry} revisited. The asymptotic boundary of this geometry -- where the CFT is defined -- corresponds to the $x$-axis. The location of the black string is $z=z_h$ and depends on the energy density $\lambda$ via \eqref{eq: lambda ito beta}. It is possible to choose a union of intervals $A_\sigma$ (red) and an interval $B_\sigma$ (blue) that lies between two intervals that belong to $A_\sigma$ in such a way that $A_\sigma$ and $B_\sigma$ saturate the ALI, i.e.~\eqref{eq: SA=SS+SB}.}
\label{fig: BS n int}
\end{figure}
\subsection{Thermal States Dual to Black Branes}
\label{sec: black branes}
Consider thermal CFT states on $d$-dimensional Minkowski space that are dual to black branes,
\begin{equation}
ds^2_{BB}
=
\frac{L^2}{z^2}\Big(
-\frac{z_h^d-z^d}{z_h^d}dt^2
+
\frac{z_h^d}{z_h^d-z^d}dz^2
+d\vec{x}_{d-1}^2\Big)\ ,
\end{equation}
where the black brane is located at $z=z_h$. Just as for black strings (see Section \ref{sec: Black strings}) the asymptotic boundary, where the CFT is defined, corresponds to $z=0$.
We choose $\Sigma$ to be a ball with radius $R$ and $B_\sigma$ another ball with radius $\sigma<R$ with the same center as $\Sigma$. Consequently, $A_\sigma=\Sigma\backslash B_\sigma$ is a spherical shell with inner radius $\sigma$ and outer radius $R$.
We choose $\lambda$ to be the energy density of the considered thermal states,
\begin{equation}
\lambda
=
\frac{(d-1)L^{d-1}}{16 \pi G_N z_h^d}\,.
\end{equation}
The reference state is chosen to be the ground state, i.e.~$\lambda_0=0$. If we only consider sufficiently small radii $\sigma$, such that the RT surface of $A_\sigma$ is given by the union of the RT surfaces of $\Sigma$ and $B_\sigma$
for all $\sigma$, we find the ALI to be saturated for this setup (see Figure \ref{fig: BS 2 int} for $d=2$).
Furthermore, we know $\Delta\corr{H_0}(B_\sigma,\lambda)$ to be linear in $\tilde{\lambda}$ for all $\sigma$ \cite{Blanco:2013joa},
\begin{equation}
\label{eq: H for BB}
\Delta\corr{H_0}(B_\sigma,\lambda)
=
\frac{2\pi \Omega_{d-2}}{d^2-1}\sigma^d\tilde{\lambda}\,,
\end{equation}
where $\Omega_{d-2}=\frac{2\pi^{(d-1)/2}}{\Gamma((d-1)/2)}$.
Moreover, $S(B_\sigma,\lambda)$ is given, via the RT formula \eqref{eq: RT}, by \cite{Blanco:2013joa}
\begin{equation}
\label{eq: EE BB}
S(B_\sigma,\lambda)
=
\frac{L^{d-1}\Omega_{d-2}}{4G_N}
\int_0^\sigma d\rho \frac{\rho^{d-2}}{z(\rho)^{d-1}}\sqrt{1+\frac{(\partial_\rho z(\rho))^2 z_h^d}{z_h^d-z(\rho)^d}}
\,,
\end{equation}
where $z(\rho)$ has to be chosen in such a way, that the integral on the RHS of
\eqref{eq: EE BB} is minimized. To our knowledge there is no analytic, integral free expression for $S(B_\sigma, \lambda)$ for generic $d$. However, in \cite{Blanco:2013joa} an expansion of $\Delta S(B_\sigma,\lambda)$ in
$\alpha\,\sigma^d\lambda$ is presented, with $\alpha=\frac{16\pi G_N}{d L^{d-1}}$, \footnote{As already pointed out in \cite{Sarosi:2016atx} there seems to be a typo in equation (3.55) of \cite{Blanco:2013joa}: The term $L^{d-1}/\ell_p^{d-1}$ needs to be inverted.}
\begin{equation}
\label{eq: Delta EE BB}
\Delta S(B_\sigma,\lambda)
=
\frac{\Omega_{d-2}L^{d-1}}{4G_N}\Big(
\frac{d\,\alpha\,\sigma^d\lambda}{2(d^2-1)}
-
\frac{d^3\sqrt{\pi}\,\Gamma(d-1)\alpha^2\sigma^{2d}\lambda^2}{
2^{d+4}(d+1)\Gamma\Big(d+\frac{3}{2}\Big)}
+
\mathcal{O}((\alpha\sigma^d\lambda)^3)\Big)\,.
\end{equation}
Due to $\partial_\lambda^2\Delta S(B_\sigma,\lambda)=\partial_\lambda^2S(B_\sigma,\lambda)$, we see that $\partial_\lambda^2S(B_\sigma,\lambda)$ is not constant in $\sigma$ on any interval. Since $\Delta\corr{H_0}(B_\sigma,\lambda)$ is linear in $\tilde{\lambda}$ \eqref{eq: H for BB} for all $\sigma$, the result of Section \ref{sec: gen EP} now tells us that $\Delta\corr{H_0}(A_\sigma,\lambda)$ may only be linear in $\tilde{\lambda}$ for single values of $\sigma$. \footnote{By applying our result to this situation we implicitly assume the first law of entanglement \eqref{eq: 1st law} to hold. However, as already pointed out in \cite{Blanco:2013joa} and Section \ref{sec: ge EP discussion}, the derivation of the first law for $\lambda_0=0$
would require to consider negative energy densities $\lambda<0$, which is unphysical. For the sake of this paper we assume the first law to be valid in the limit $\lambda_0\rightarrow 0$, since it holds for any $\lambda_0>0$.}
Just as for the black string, we can even show that there is only one such $\sigma$. From \eqref{eq: H for BB} and \eqref{eq: Delta EE BB} we conclude that $S_{rel}(B_\sigma,\lambda)$ \eqref{eq: Srel} is not constant in $\sigma$ on any interval. The monotonicity \eqref{eq: monotonicity of Srel} of the RE then implies that $S_{rel}(B_\sigma,\lambda)$ grows strictly monotonically with $\sigma$. Since $\Delta\corr{H_0}(B_\sigma,\lambda)$ is linear in $\tilde{\lambda}$ we find $D_{rel}(B_\sigma,\lambda)=S_{rel}(B_\sigma,\lambda)$ \eqref{eq: Drel} and therefore conclude that $D_{rel}(B_\sigma,\lambda)$ grows strictly monotonically with $\sigma$. The discussion in Section \ref{sec: alternative formulation} now implies that there is at most one value of $\sigma$ where $\Delta\corr{H_0}(A_\sigma,\lambda)$ is linear in $\tilde{\lambda}$. This special $\sigma$ can be found to be the degenerate case $\sigma=0$, i.e.~when $B_\sigma$ vanishes.
\subsection{BTZ Black Hole}
\label{sec: BTZ}
As a further application of the result of Section \ref{sec: gen EP} to holography we consider thermal states dual to BTZ black hole geometries,
\begin{equation}
\label{eq: BTZ geometry}
ds_{BTZ}^2
=
-\frac{r^2-r_h^2}{L^2}dt^2
+
\frac{L^2}{r^2-r_h^2}dr^2
+
r^2 d\phi^2
\,.
\end{equation}
The horizon radius $r_h$ is given -- in terms of the CFT temperature $T$ and the radius $\ell_{CFT}$ of the circle on which the CFT is defined -- by
\begin{equation}
r_h
=
\sqrt{8G_N M}L
=
2\pi L\ell_{CFT}T\,,
\end{equation}
where $M$ is the mass of the BTZ black hole.
\\
The asymptotic boundary, where the CFT is defined, corresponds to $r\rightarrow \infty$. For an interval $A_\sigma$ of sufficiently large angular size $2(\pi-\sigma)$, the RT surface consists of two disconnected parts: the horizon and the RT surface of $A^c_\sigma=B_\sigma$, as depicted in Figure \ref{fig: BTZ}.
\begin{figure}[t]
\begin{center}
\includegraphics[scale=0.18]{BTZ}
\end{center}
\caption{A constant time slice of the BTZ black hole geometry \eqref{eq: BTZ geometry}. The CFT is defined on the asymptotic boundary at $r\rightarrow \infty$. For a sufficiently large entangling region $A_\sigma$ (red) the RT surface $\gamma_{A_\sigma}$ is the union of the RT surface $\gamma_{B_\sigma}$ of its complement $B_\sigma$ (blue) and the black hole horizon. This implies \eqref{eq: SA=SS+SB}.}
\label{fig: BTZ}
\end{figure}
The entanglement entropy
is then given by \cite{Headrick:2007km, Blanco:2013joa}
\begin{equation}
\label{eq: EE BTZ}
S(A_\sigma)
=
\frac{c}{3}2\pi^2T\ell_{CFT}
+
\frac{c}{3}\log\Big(\frac{1}{\pi T \epsilon}\sinh(
2\pi\ell_{CFT}T\sigma)\Big)\,,
\end{equation}
where $\epsilon$ is a UV cutoff. The first term is the thermal entropy of the state and corresponds to the black hole horizon, while the second term is the
entanglement entropy of $B_\sigma$. We see once more that the states on $A_\sigma$ and $B_\sigma$ saturate the ALI.
As parameter $\lambda$ for this family of states we choose the square of the temperature,
\begin{equation}
\label{eq: lambda for BTZ}
\lambda
=
T^2\,,
\end{equation}
which corresponds to the mass $M$ of the black hole,
\begin{equation}
\label{eq : T^2=M}
LM
=
\frac{\pi^2\ell_{CFT}^2c}{3}\lambda\,.
\end{equation}
The reference state can be chosen to correspond to any $\lambda=\lambda_0=T_0^2$.
Using \eqref{eq: EE BTZ} it is straight forward to see that $\partial_\lambda^2 S(A_\sigma,\lambda)$ is not constant in $\sigma$ on any interval. So, even though the explicit forms of $\Delta\corr{H_0}(A_\sigma,\lambda)$ and $\Delta\corr{H_0}(B_\sigma,\lambda)$ \eqref{eq: DeltaH def} are not known, we can use the result of Section \ref{sec: gen EP} to conclude that in general at least one of $\Delta\corr{H_0}(A_\sigma,\lambda)$ or $\Delta\corr{H_0}(B_\sigma,\lambda)$ is not linear in $\tilde{\lambda}=T^2-T_0^2$.
Note that the result of Section \ref{sec: gen EP} cannot be used to determine whether $\Delta\corr{H_0}(A_\sigma, \lambda)$, $\Delta\corr{H_0}(B_\sigma, \lambda)$ or both are non-linear in $\tilde\lambda$. However, the discussion in Section \ref{sec: alternative formulation} actually allows us to show that $\Delta\corr{H_0}(A_\sigma,\lambda)$ is not linear in $\tilde\lambda$ for more than one particular $\sigma$: By applying
$S(B_\sigma,\lambda)$, i.e.~the second term in \eqref{eq: EE BTZ}, to \eqref{eq: Drel} we find
\begin{equation}
D_{rel}(B_\sigma,\lambda)
=
\frac{c}{3}\Big(
\frac{1}{2}\big(
1-\tilde{a}\coth(\tilde{a})
\big)(1-\tilde{b}^2)
+
\log\Big(\tilde{b}\frac{\sinh(\tilde{a})}{\sinh(\tilde{b}\,\tilde{a})}\Big)
\Big)\,,
\end{equation}
where $\tilde{a}=2\pi \ell_{CFT}\sqrt{\lambda_0}\sigma$ and $\tilde{b}=\sqrt{\lambda}/\sqrt{\lambda_0}$. The structure of the $\sigma$ dependence of $D_{rel}(B_\sigma,\lambda)$ is identical to the structure of the
$\sigma$ dependence of $S_{rel}(B_\sigma,\lambda)$ that was derived in Section \ref{sec: Black strings}
for two intervals (see \eqref{eq: Srel two intervals} and \eqref{eq: Srel for B}).
So in an analogous way to the discussion in Section \ref{sec: Black strings}, we find that $D_{rel}(B_\sigma,\lambda)$ grows strictly monotonically with $\sigma$. Consequently,
$\Delta\corr{H_0}(A_\sigma,\lambda)$ is not linear in $\tilde{\lambda}$ except for possibly one particular $\sigma$.
\subsection{Pure States: Primary Excitations in CFTs with Large Central Charge}
\label{sec: pure states}
It is also possible to apply the result of Section \ref{sec: gen EP} to a one-parameter family of pure states. Consider $\rho_\lambda$ to be such a family and $\Sigma$ to be the entire constant time slice, i.e.~$B_\sigma=A_\sigma^c$. Since $S(\Sigma, \lambda)=0$ and $S(A_\sigma,\lambda)=S(B_\sigma,\lambda)$, the ALI is saturated for this setup. The result of Section \ref{sec: gen EP} now tells us that if $\partial_\lambda^2S(A_\sigma,\lambda)$ is not constant in $\sigma$ on any interval $[\xi,\eta]$, it is not possible for $\Delta\corr{H_0}(A_\sigma,\lambda)$
and $\Delta\corr{H_0}(B_\sigma,\lambda)$ to be linear in $\tilde{\lambda}$ for the same $\sigma$, except for single values of $\sigma$.
\\
As an example for such a family of pure states we consider spinless primary excitations $\ket{\lambda}$ in a CFT with large central charge $c$ defined on a circle with radius $\ell_{CFT}$.
We use the conformal dimension
\begin{equation}
\label{eq: lambda for primaries}
(h_\lambda,\bar{h}_\lambda)
=
\Big(\frac{c\lambda}{24}, \frac{c\lambda}{24}\Big)
\end{equation}
to parametrize these states\footnote{We have introduced the multiplicative factor $c/24$ in the definition of $\lambda$ to simplify the formulae in this section.}
and assume $\ket{\lambda}$ to correspond to a heavy operator, i.e.~$\Delta_\lambda=h_\lambda+\bar{h}_\lambda=\mathcal{O}(c)$. Moreover, we restrict our analysis to the case $\lambda<1$ and assume the spectrum of light operators, i.e.~operators with $\Delta=h+\bar{h}\ll c$, to be sparse. The entangling regions $\Sigma$ and $B_\sigma$ are chosen to be the entire circle and an interval with angular size $2\sigma<\pi$, respectively. Consequently, $A_\sigma=B_\sigma^c$ is an interval with angular size $2(\pi-\sigma)>\pi$. The reference state corresponds to an arbitrary value $\lambda_0$ of the parameter $\lambda$.
The entanglement entropy of $B_\sigma$ for this setup was computed in \cite{Asplund:2014coa},
\begin{equation}
\label{eq: EE primaries}
S(B_\sigma,\lambda)
=
\frac{c}{3}\log\Big(\frac{2\ell_{CFT}}{\sqrt{1-\lambda}\,\epsilon}\sin\big(\sqrt{1-\lambda}\,\sigma\big)\Big)
=
S(A_\sigma,\lambda)\,,
\end{equation}
where $\epsilon$ is a UV cutoff. The second equality in \eqref{eq: EE primaries}
is a consequence of the fact that $\ket{\lambda}$ is pure\footnote{Note that the expression for $S(B_\sigma,\lambda)$ in \eqref{eq: EE primaries} is not symmetric under the transformation $\sigma\mapsto \pi-\sigma$, as one would naively expect
from the purity of $\ket{\lambda}$. The reason for that is the fact that in the derivation of $S(B_{\sigma},\lambda)$ \cite{Asplund:2014coa} $2\sigma<\pi$ was applied.} and ensures that the ALI is saturated.
It is easy to see that $\partial_\lambda^2 S(B_\sigma,\lambda)$ is not constant
in $\sigma$ on any interval. Therefore the result of Section \ref{sec: gen EP} implies that there are only single values of $\sigma$ where both $\Delta\corr{H_0}(A_\sigma,\lambda)$ and $\Delta\corr{H_0}(B_\sigma,\lambda)$ are linear in $\tilde{\lambda}=\lambda-\lambda_0$.
Analogously to the discussion regarding BTZ black holes in Section \ref{sec: BTZ}, we can actually show that $\Delta\corr{H_0}(A_\sigma,\lambda)$ is in not linear in $\tilde{\lambda}$ for any $\sigma$ with possibly one exception.
\subsection{Vacuum States for CFTs on a Circle}
\label{sec: vacuum states for CFTs}
We would like to emphasize an interesting observation regarding a family of primary states $\ket{\lambda}$ for a CFT defined on a circle with radius $\ell_{CFT}$. We define the entangling intervals $A_\sigma$ and $B_\sigma$ and the parameter $\lambda$ as in Section \ref{sec: pure states}. However, we do not require the CFT
to have large central charge. Furthermore, we do not assume any restrictions regarding the spectrum.
The reference state is chosen to be the vacuum state, i.e.~$\lambda_0=0$.
Since $\ket{\lambda}$ is a family of pure states, the ALI is saturated, as pointed out in Section \ref{sec: pure states}.
In this section we show that
our result of Section \ref{sec: gen EP} may be used to arrange the considered families of states into three categories: families where $\partial_\lambda^2S(A_\sigma,\lambda)$ and $\partial_\lambda^2S(B_\sigma,\lambda)$ are constant in $\sigma$, families where the parameter $\lambda$ is not continuous, such that the reference value $\lambda_0=0$ is separated from the other parameter values,
and finally families where the first law of entanglement \eqref{eq: 1st law} does not hold. These categories are not mutually exclusive.
For the example considered in this section, it is possible to choose these three categories since both $\Delta\corr{H_0}(A_\sigma,\lambda)$ and $\Delta\corr{H_0}(B_\sigma,\lambda)$ are linear in $\tilde{\lambda}$ for all $\sigma$, as may be seen as follows.
In general, the modular Hamiltonian $H_0(2\varsigma)$ for the
ground state of a CFT on a circle, restricted to an interval with angular size $2\varsigma$, is given by \cite{Blanco:2013joa}
\begin{equation}
\label{eq: mod Ham vac circle}
H_0(2\sigma)
=
2\pi\ell_{CFT}^2\int_0^{2\varsigma} d\phi\frac{\cos(\phi-\varsigma)-\cos(\varsigma)}{\sin(\varsigma)}T_{00}\,.
\end{equation}
Using the CFT result
\begin{equation}
\bra{\lambda}T_{00}\ket{\lambda}
-
\bra{0}T_{00}\ket{0}
=
\frac{c\tilde{\lambda}}{24\pi\ell_{CFT}^2}\,,
\end{equation}
we find from \eqref{eq: mod Ham vac circle} that
\begin{equation}
\label{eq: modHam primaries}
\Delta\corr{H_0}(A_\sigma,\lambda)
=
\frac{c}{6}\Big(1+(\pi-\sigma)\cot(\sigma)\Big)\tilde{\lambda}
\end{equation}
and
\begin{equation}
\Delta\corr{H_0}(B_\sigma,\lambda)
=
\frac{c}{6}\Big(1-\sigma\cot(\sigma)\Big)\tilde{\lambda}
\end{equation}
are linear in $\tilde{\lambda}$.
The first category of families corresponds to the case where all
prerequisites of our result of Section \ref{sec: gen EP} are satisfied. Both $\Delta\corr{H_0}(A_\sigma,\lambda)$ and $\Delta\corr{H_0}(B_\sigma,\lambda)$ are linear in $\tilde{\lambda}$ for all $\sigma$, so we conclude that $\partial_\lambda^2S$ is constant in $\sigma$ for both $A_\sigma$ and $B_\sigma$.
If $\partial_\lambda^2S(A_\sigma,\lambda)$
or $\partial_\lambda^2S(B_\sigma,\lambda)$ is not constant in $\sigma$, then at least one of the prerequisites of our result of Section \ref{sec: gen EP} is not satisfied. The examples with this property then fall into one of the other two categories introduced above.
There are two ways in which the prerequisites may be violated. One way is that the parameter $\lambda$ cannot be continuously continued to $\lambda_0=0$, which corresponds to the second category of families. In the proof of our result in Section \ref{sec: gen EP} we assume $\lambda$ to be continuous, since we take derivatives w.r.t.~$\lambda$ (see e.g.~\eqref{eq: SrelA i.t.o. SrelB for alpha}).
So if $\lambda$ has a gap at $\lambda_0$ the derivative w.r.t.~$\lambda$ is not
defined there.
The other way how the prerequisites may be violated is when the first law of entanglement does not hold. Since the conformal dimension is always non-negative, the reference value $\lambda_0=0$ is a boundary point of the set of allowed parameter values $\lambda$. As pointed out in Section \ref{sec: ge EP discussion}, the first law of entanglement may not apply in this case, since the first derivative of the RE may not vanish at $\lambda_0=0$. However, this law is an essential ingredient in the proof of Section \ref{sec: gen EP}. This situation corresponds to the third category of families.
To conclude, we note that our result of Section \ref{sec: gen EP} allows for a distinction of the three categories described in this section.
\section{Discussion}
\label{sec: conclusions}
In this paper we studied the modular Hamiltonian of a one-parameter family of reduced density matrices $\rho^{A,B}_\lambda$ on entangling regions $A$ and $B$ that form entanglement plateaux, i.e.~that saturate the ALI \eqref{eq: ALI}. These plateaux were considered to be stable under variations of $A$ and $B$ that leave $\Sigma=A B$ invariant. We parametrized these variations by introducing a continuous variable $\sigma$, i.e.~$A\rightarrow A_\sigma$, $B\rightarrow B_\sigma$, such that $A_{\sigma_2}\subset A_{\sigma_1}$ for $\sigma_1<\sigma_2$.
Our main result is that the only way how both $\Delta\corr{H_0}(A_\sigma,\lambda)$ and $\Delta\corr{H_0}(B_\sigma,\lambda)$, as defined in \eqref{eq: DeltaH def}, can be linear in $\tilde{\lambda}=\lambda-\lambda_0$ for all $\sigma$ in an interval $[\xi,\eta]$
is if $\partial_\lambda^2 S(A_\sigma, \lambda)$ and $\partial_\lambda^2 S(B_\sigma, \lambda)$ are constant in $\sigma$ on $[\xi,\eta]$. Subsequently to discussing this result for states dual to black strings as a motivation (see Section \ref{sec: Black strings}), we proved it in Section \ref{sec: gen EP} for arbitrary quantum systems using the first law of entanglement \eqref{eq: 1st law} and the monotonicity \eqref{eq: monotonicity of Srel} of the RE \eqref{eq: Srel}.
As we discussed in the introduction, if $\Delta\corr{H_0}$ is linear in $\tilde{\lambda}$
it effectively does not contribute to the FIM \eqref{eq: Fisher info}. So we see that in the
setup described above the FIM of $A_\sigma$, $B_\sigma$ or both will in general contain non-trivial contributions of $\Delta\corr{H_0}$.
Furthermore, if it is linear in $\tilde{\lambda}$, $\Delta\corr{H_0}$ is completely determined by the entanglement entropy via the first law of entanglement (see \eqref{eq: linear mod Ham}). In the setup described above however, we find that $\Delta\corr{H_0}(A_\sigma,\lambda)$, $\Delta\corr{H_0}(B_\sigma,\lambda)$ or both will in general
not have this simple form.
In Section \ref{sec: applications} we applied the result of Section \ref{sec: gen EP} to several prominent holographic examples of entanglement plateaux. By choosing $\lambda$ to be the energy density of thermal states dual to black strings, we showed that higher-order contributions in $\tilde{\lambda}$ are present in $\Delta\corr{H_0}(A_\sigma,\lambda)$ for $A_\sigma$ being the union of two sufficiently close intervals. Furthermore, we showed a similar result for thermal states dual to black branes, where $\lambda$ was again chosen to be the energy density, $\lambda_0$ was set to $0$ and $A_\sigma$ was chosen to be a spherical shell with sufficiently small inner radius $\sigma$.
In these two situations, $\Delta\corr{H_0}(B_\sigma,\lambda)$ is known to be linear in $\tilde{\lambda}$. This allowed us to determine that $\Delta\corr{H_0}(A_\sigma,\lambda)$ must be non-linear in $\tilde{\lambda}$.
Moreover, we also discussed the BTZ black hole, where we chose $A_\sigma$ to be a sufficiently large entangling interval so that $A_\sigma$ and $B_\sigma=A_\sigma^c$ saturate the ALI. For this case we were able to use our result to show that at least one $\Delta\corr{H_0}(A_\sigma,\lambda)$ or $\Delta\corr{H_0}(B_\sigma,\lambda)$ is in general non-linear in $\tilde{\lambda}=T^2-T_0^2$, where $T$ is the CFT temperature. A more detailed analysis of the entanglement entropy even allowed us to determine that $\Delta\corr{H_0}(A_\sigma,\lambda)$ will have higher order $\tilde{\lambda}$ contributions.
We showed a similar result for primary excitations in a CFT on a circle with large central charge $c$. In this case $B_\sigma$ was set to be an interval with angular size $2\sigma<\pi$ and $A_\sigma=B_\sigma^c$. The parameter $\lambda$ was chosen to
be the conformal dimension multiplied by $c/24\pi$.
\\
We emphasize that even though all these examples are very different from each
other, the fact that non-linear contributions in $\tilde{\lambda}$ are to be expected for $\Delta\corr{H_0}$, can be traced back to the same origin, namely the saturation of the ALI. This is the only property a system is required to have in order for our result to apply. Very little is known about the explicit form of the modular Hamiltonians for the holographic examples mentioned above, so it is remarkable that they share this common property.
\\
Note that for the holographic examples described above, the ALI was assumed to be saturated for all considered $\sigma$ and $\lambda$. However, whether the ALI is saturated for a given value of $\lambda$ depends on the value of $\sigma$. If $\sigma$ is chosen too large the corresponding RT surfaces undergo a phase transition \cite{Headrick:2007km, Headrick:2010, Blanco:2013joa} that causes the ALI to be no longer saturated. Consequently, our result can only be applied to make statements for $\sigma$ sufficiently small
and $\lambda$ sufficiently close to the reference value\footnote{It depends on the chosen value of $\lambda$ for which $\sigma$ the phase transition of the RT surface occurs.} $\lambda_0$.
\\
We also need to stress that the saturation of the ALI inequality for the holographic situations discussed in Section \ref{sec: applications} is a large $N$
effect. Bulk quantum corrections to the RT formula are expected to lead to additional contributions to entanglement entropies in such a way that the ALI is no longer saturated \cite{Faulkner:2013ana}. So strictly speaking our result can only be used to show that $\Delta\corr{H_0}(A_\sigma,\lambda)$ or $\Delta\corr{H_0}(B_\sigma,\lambda)$ is in general non-linear in the respective $\tilde{\lambda}$ in the large $N$ limit.
By continuity, we expect this non-linearity to hold for finite $N$ as well.
\\
We emphasize once more that even though our result was mostly applied to
examples from AdS/CFT in this paper, it is not restricted to the holographic case. We only required the monotonicity \eqref{eq: monotonicity of Srel} of the RE and the first law of entanglement \eqref{eq: 1st law} -- which is a direct implication of the non-negativity of the RE -- to prove it. Both are known to be true for any quantum system. Therefore our result is an implication of well-established properties of the
RE and holds for generic quantum systems.
\\
The RE is a valuable object for studying modular Hamiltonians \cite{Blanco:2013lea, Faulkner:2016mzt, Ugajin:2016opf, Arias:2016nip, Blanco:2017akw} and offers prominent relations between modular Hamiltonians and entanglement entropies.
Our result is a further application of the RE that reveals such a relation. Unlike the first law of entanglement, which focuses on the first order contribution of $\tilde{\lambda}$ to $\Delta\corr{H_0}$, our result makes a statement about higher-order contributions in $\tilde{\lambda}$. The fact that the entanglement entropy plays a role for the higher-order contributions in $\tilde{\lambda}$ is a non-trivial observation that deserves further analysis. Possible future projects could be devoted to investigating whether
it is possible to find more concrete relations between entanglement entropy and higher-order $\tilde{\lambda}$ contributions to $\Delta\corr{H_0}$.
This will provide further progress towards understanding the properties of the
modular Hamiltonian in general QFTs.
\begin{acknowledgments}
We would like to thank Charles Melby-Thompson, Christian Northe and Ignacio Reyes for inspiring
discussions and Ren\'e Meyer and Erik Tonni for fruitful conversations.
\end{acknowledgments}
|
1,108,101,564,533 | arxiv | \section{Introduction}
Complex networks is a new emerging branch of random graph theory.
For a long time random graphs have been mainly
studied by pure mathematics but recently
due to the availability of empirical data on real-world
networks they have attracted the attention of physics and natural
sciences (see for review
\cite{ref:ab,ref:dm,ref:n1}).
Methods of statistical physics, both empirical
and theoretical, have thus begun to play an important role in
this research area.
The empirical observations of real-networks has had
a feedback on theoretical development which now
concentrated on the understanding of the observed
features. For example fat tails in node degree distribution,
small world effect, degree-degree correlations, or high
clustering. Two complementary approaches have been developed:
diachronic, known as growing networks
\cite{ref:ab,ref:dm,ref:n1},
and synchronic being a sort of statistical mechanics
of networks \cite{ref:bck,ref:bl1,ref:dms,
ref:bjk1,ref:fdpv,ref:pn1}.
We will discuss here the latter.
This approach is a natural
extension of Erd\"os and R\'enyi ideas \cite{ref:er,ref:bb}. It is
well suited both for growing (causal) networks for which
nodes' labels reflect the causal order of nodes' attachment
to the network \cite{ref:kr,ref:bbjk} and for homogeneous
networks for which nodes' labels can be permuted freely
in an arbitrary way.
Here we shall discuss mainly homogeneous networks.
We shall shortly comment on causal networks towards the
end of the paper.
The main aim of the paper is to present
a consistent picture of statistical mechanics of networks.
Some ideas have already been introduced earlier.
They are scattered in many papers and discussed
in many different contexts. We put them together,
add some new material and introduce a guideline
to obtain a self-contained introduction
to statistical mechanics of complex networks.
The basic concept in the statistical formulation
is statistical ensemble. Statistical ensemble of networks
is defined by ascribing a statistical weight to every
graph in the given set \cite{ref:bck,ref:bjk1}. Physical quantities are
measured as weighted averages over all graphs in the ensemble.
The probability of the occurrence of a graph in
random sampling is proportional to its statistical weight.
If the statistical weight changes then also the probability of
occurrence of randomly sampled graphs will change and
in effect different random graphs will be observed.
The concept of statistical weight is crucial, since
it defines randomness in the system.
Statistical weight is built out of two ingredients:
configuration space weight and functional weight.
The configuration space weight is proportional to the
uniform probability measure on the configuration space
which tells us how to uniformly choose graphs
in the configuration space.
To illustrate the meaning of the
uniform measure consider an ensemble of Erd\"os-R\'enyi graphs
with $N$ nodes and $L$ links \cite{ref:er,ref:bb}.
The configuration space consists of $\binom{\binom{N}{2}}{L}$
graphs with labeled nodes. All those graphs are equiprobable,
and therefore the configuration space weight
is the same for each graph.
It is convenient to choose this weight to
be $1/N!$ since then it can be interpreted as a factor
which takes care of $N!$ possible permutations of nodes' labels.
This factor has the same origin as the corresponding factor in
quantum mechanics for indistinguishable particles and
it is constant for all graphs in a finite $N$-ensemble.
We can calculate the entropy of random graphs as
\begin{equation}
S = \ln \frac{1}{N!} \binom{\binom{N}{2}}{L} .
\label{S0}
\end{equation}
In the limit of large sparse graphs:
$N \rightarrow \infty$ and $\frac{2L}{N} = \alpha = \mbox{const}>2$,
the entropy is an over-extensive function of the system size:
\begin{equation}
S = \frac{\alpha - 2}{2} N \ln N + \dots ,
\label{S1}
\end{equation}
unlike in standard thermodynamics.
Let us move to weighted graphs.
The idea is to modify the Erd\"os-R\'enyi ensemble by introducing
a functional weight which explicitly depends on graph's topology.
For example, if we choose the functional weight to be a function
of the number of loops on the graph, we can suppress of favor
loops of typical graphs in the ensemble.
In a similar way we can
choose statistical weights to control the node degree distribution
to produce homogeneous scale-free graphs \cite{ref:bk}
or to introduce correlations
between degrees of neighboring nodes \cite{ref:b,ref:n3,ref:bp,ref:d2}.
Classical thermodynamics describes systems in equilibrium for which
the functional weight is given by the Gibbs measure:
$\sim \exp (-\beta E)$, where $E$ is the energy of the system.
When discussing complex networks it is convenient
to abandon the concept of energy and Gibbs measure
and consider a more general form of statistical weights because
many networks are not in equilibrium. Indeed, many networks
emerging as a result of a dynamical process like growth
are far from equilibrium \cite{ref:ab,ref:dm,ref:n1}.
It does not mean though that one cannot
introduce a statistical ensemble of growing networks.
On the contrary, one can for example consider
an ensemble of networks which result of many
independent repetitions of the growth
process terminated when the network reaches a certain size.
Such a collection of networks does not describe a thermodynamic
equilibrium. The functional weight can be deduced from the parameters
of the growth process but of course it has nothing to do with
the Gibbs measure.
In fact, many real-world networks result from a combination
of a growth process and some thermalization processes.
For example, the Internet grows but at the same time it
continuously rearranges. The latter process introduces
a sort of thermalization. Today the growth has probably still
larger influence on the topology of the underlying network
but in the future the growth may slow down due to saturation
and then equilibration processes
resulting from continuous rewirings will take over.
Similarly all evolutionary networks emerge from a
growth mixed with a sort of thermalization related to the continuous network rearrangement.
Therefore it is convenient to have a formalism which can
extrapolate between the two regimes in a flexible way.
The approach which we propose here is capable of
modeling functional properties of networks by choosing
an appropriate functional weight.
Let us return to the configuration space weight.
As we mentioned this weight is equivalent
to the uniform probability measure on the configuration
space for which all graphs are equiprobable. It is
a very crucial part of the construction of the ensemble
to carefully specify what one means by equiprobable graphs.
Consider first graphs with $N$ nodes.
There are at least two natural candidates for the uniform
measure in such a set of graphs.
Since one is interested in shape (topology) of graphs one
can define all shapes to be equiprobable. Alternatively one
can introduce labels for nodes of each graph to obtain a set
of labeled graphs and then one can define all labeled graphs
to be equiprobable. The two definitions give two different
probability measures since the number of ways
in which one can label graph nodes depends
on graph's topology and thus the probability of occurrence of
a given graph will depend on its topology too.
It turns out that the latter definition is more natural.
As we have seen above this definition leads to
Erd\"os-R\'enyi graphs. So we stick to this definition
and from here on we shall ascribe to each
labeled graph the configuration space weight $1/N!$ which
is constant in the set of graphs of size $N$.
The situation is more complex if one considers pseudographs
that is graphs which have multiple connections (more than one
link between two nodes) or self-connections (a link having the same node
at its endpoints). In this case one can also label links
and ascribe the same statistical weight to each fully labeled graph.
For this choice the statistical weight of each graph is equal
to the symmetry factor of Feynman diagrams generated in the Gaussian
perturbation field theory \cite{ref:bck}.
The paper is organized as follows. In the next section
we will recall some basic definitions. Then we will discuss
Erd\"os-R\'enyi graphs in the context of constructing
statistical ensemble and later we will generalize the
construction to weighted homogeneous graphs.
After this we will describe Monte-Carlo algorithms to
generate graphs for canonical, grand-canonical and
micro-canonical ensembles and discuss their representation in terms
of adjacency matrices. A section will be devoted to pseudographs.
In the last section we will shorty summarize the paper.
\section{Definitions}
Let us first introduce some terminology.
Graph is a set of $N$ nodes (vertices) connected by $L$ edges (links).
A graph need not be connected. It may have many disconnected
components including empty nodes (without any link).
If a graph has no multiple or self-connected links we shall
call it simple graph or graph. An example is illustrated in
Fig. \ref{fig:ex0}. Later we shall also discuss graphs with
multiple- and self-connections. To distinguish them
from simple graphs we shall call them degenerate graphs
or pseudographs.
\begin{figure}[ht]
\psfrag{1}{$1$} \psfrag{2}{$2$} \psfrag{3}{$3$} \psfrag{4}{$4$} \psfrag{5}{$5$}\psfrag{6}{$6$}
\includegraphics[width=2.5cm]{mayer_ex.eps}
\caption{An example of simple graph with $N=6,L=5$.
Vertices without links (like no. 2) are allowed.
Each vertex can have at most $N\!-\!1$ links.
Positions of vertices in the picture are meaningless.
The only information which matters is connectivity.}
\label{fig:ex0}
\end{figure}
One can consider directed or undirected graphs.
Directed graphs are built of directed links while
undirected of undirected ones. In this paper we
shall discuss undirected graphs but the discussion
can easily be generalized to directed ones as well.
Sometimes we will find it convenient to represent
an undirected link as two oriented links going
in opposite directions.
A simple graph can be represented by its adjacency matrix which
is an $N\times N$ matrix whose entries $A_{ij}$ are equal
one if there is a link between vertices $i,j$ or zero otherwise.
Since self-connections are forbidden we have $A_{ii}=0$ on the
diagonal. The adjacency matrix of an undirected graph is also symmetric
because if there is a link $i\rightarrow j$ ($A_{ij}=1$),
there must be also the opposite one $j\rightarrow i$
($A_{ji}=1$).
In this paper we want to construct statistical ensembles
of homogeneous graphs having desired properties.
We discuss three types of ensembles: ensemble of graphs
with a fixed number of nodes $N$ and varying number of links,
ensemble with a fixed number of nodes $N$ and a fixed number
of links $L$, and finally ensemble of graphs
with a given node degree sequence
$\{q_1,q_2,\dots, q_N\}$, which we shall call
grand-canonical, canonical and micro-canonical ensembles, respectively.
There are of course many other possibilities like
for instance ensembles with varying number of nodes, or
with a fixed number of loops etc, but the three
mentioned above are encountered most frequently.
To construct a statistical ensemble, for the chosen set of graphs,
we have to specify statistical weight for each graph in the
considered set.
In the next section using the Erd\"os-R\'{e}nyi graphs
and binomial graphs we will
deduce a general logical structure standing behind
the construction of ensembles of homogeneous graphs
and then we will use this structure to introduce ensembles
with an arbitrary functional weight which explicitly
depends on the node degrees.
\section{Statistical ensemble for Erd\"os-R\'{e}nyi random graphs}
For simplicity,
we start from a well-known model of Erd\"os-R\'{e}nyi's graphs
\cite{ref:er,ref:bb}. In this classical model one considers
simple graphs with $N$ labeled nodes and $L$ links \footnote{For
simple graphs it is immaterial to label links since each
link is uniquely determined by its end points.}
chosen at random out of all $\binom{N}{2}$ possibilities.
All possibilities are equiprobable and so are the corresponding
graphs -- understood as graphs whose vertices are labeled.
Usually one is interested in unlabeled graphs
that is in their shape or topology and not
in their labeled version. To explain what is meant by graph's shape
or topology, let us consider a simple graph
shown in the upper part of Fig. \ref{fig:shape}. Unlabeled
graph (topology) on the left hand side of the figure
is represented as labeled graphs on the right hand side.
\begin{figure}[h]
\includegraphics[width=10cm]{shape.eps}
\caption{Top: the graph on the left can be realized
as three different labeled graphs. A is equivalent to B, C to D
and E to F. They are equivalent
because they have the same adjacency matrix.
Bottom: triangle-shaped graph has only one realization as labeled graph.}
\label{fig:shape}
\end{figure}
There are six possible realizations, but
only three of them: A, C, E are distinct.
B is the same as A since it can be obtained from A
by a continuous deformation: one can continuously
move the vertex $2$, together with the link attached to it,
to the position of the vertex $3$,
and at the same time the vertex $3$ to the position of the vertex $2$.
Such a continuous deformation does not change graph's connectivity.
The same holds for pairs: C, D and E, F.
This can also be seen if we take into account the adjacency matrix $\mathbf{A}$.
Both A and B have identical adjacency matrices
which are different from those for C, D and E, F:
\begin{equation}
\mathbf{A}_{\mbox{\scriptsize A}} = \mathbf{A}_{\mbox{\scriptsize B}} =
\left( \begin{array}{ccc}
0 & 1 & 1 \\
1 & 0 & 0 \\
1 & 0 & 0
\end{array}
\right), \quad
\mathbf{A}_{\mbox{\scriptsize C}} = \mathbf{A}_{\mbox{\scriptsize D}} =
\left( \begin{array}{ccc}
0 & 1 & 0 \\
1 & 0 & 1 \\
0 & 1 & 0
\end{array}
\right), \quad
\mathbf{A}_{\mbox{\scriptsize E}} = \mathbf{A}_{\mbox{\scriptsize F}} =
\left( \begin{array}{ccc}
0 & 0 & 1 \\
0 & 0 & 1 \\
1 & 1 & 0
\end{array}
\right) \ .
\end{equation}
Now we remove labels from all graphs in Fig. \ref{fig:shape}
to obtain two unlabeled graphs depicted
on the left hand side. Although there are three distinct
adjacency matrices for the upper shape, all of them lead
to the same connections between vertices (unlabeled graph).
The graph in the lower line in Fig. \ref{fig:shape}.
can be labeled only in one way \footnote{
At first glance one can think that there are
six ways because one can put labels in six
different ways on a drawing of the triangle. After a while
one can see that they all are identical since they can be
transformed one into another by a transformation which
does not change connectivity. For example, if it is a drawing
of equilateral triangle one can change labels 123 into
231 by rotating it by 120$^\circ$.}
which is represented by the following adjacency matrix:
\begin{equation}
\mathbf{A}_{\mbox{\scriptsize G}} =
\left( \begin{array}{ccc}
0 & 1 & 1 \\
1 & 0 & 1 \\
1 & 1 & 0
\end{array}
\right) \ .
\end{equation}
Thus the triangular shape has only one realization as labeled graph.
Furthermore, the upper and lower
graphs are obviously distinct because none of the corresponding
labeled graphs (adjacency matrices) representing the upper graph
is equal to that for the lower graph.
In this trivial case the difference is in the number of links.
More generally, any two
graphs are distinct if the underlying labeled graphs (adjacency matrices)
cannot be converted one into another by a permutation of
nodes' labels. It is clear that for the graphs in Fig. \ref{fig:shape}
there is no such a permutation but in general case the comparison
of graphs may be a complex problem.
Let us now apply the ideas sketched above to define an ensemble
of graphs. As an example we shall consider Erd\"os-R\'{e}nyi graphs
with $N=4$, $L=3$. It consists of three distinct graphs A, B, C shown
in Fig. \ref{fig:shape2}.
Now we want to determine the statistical
weight for those graphs.
Adjacency matrices of the underlying labeled graphs
are essentially different for A, B, C since they cannot be converted
one into another by a permutation of node's labels.
\begin{figure}
\includegraphics[width=7cm]{shape2.eps}
\caption{Three possible graphs for $N=4,L=3$.
The number of ways of labeling these graphs is: $n_A=12$, $n_B=4$,
$n_C=4$.}
\label{fig:shape2}
\end{figure}
Each graph in Fig. \ref{fig:shape2} has a few possible realizations
as labeled graph. One can label four vertices in $4!=24$ ways
corresponding to permutations of $1-2-3-4$.
For the graph A, twelve of them give distinct labeled graphs.
For example, the permutation $1-2-3-4$ gives an identical
labeled graph (adjacency matrix) as $4-3-2-1$.
The same kind of symmetry applies for remaining pairs of
permutations. Therefore there are $n_A=12$ labeled graphs for $A$.
Similarly one can find that there are $n_B=4$ labeled graphs for $B$
and $n_C=4$ for $C$. Altogether, there are $n_A+n_B+n_C=20$ labeled
graphs in accordance with $n=\binom{\binom{N}{2}}{L}=20$. In the
Erd\"os-R\'{e}nyi ensemble labeled graphs are equiprobable,
so the shapes A, B, C have the following probabilities:
\begin{equation}
p_A=\frac{n_A}{n} = \frac{3}{5}, \quad
p_B=\frac{n_B}{n} = \frac{1}{5}, \quad
p_C=\frac{n_C}{n} = \frac{1}{5}.
\label{pABC}
\end{equation}
These probabilities give frequencies of the occurrence of the
shapes A, B, C in random sampling. We see that graphs (unlabeled
graphs) are not equiprobable in Erd\"os-Renyi's ensembles.
Let us denote the statistical weights for A, B, C by $w_A,w_B,w_C$
which are proportional to probabilities
of configurations in the ensemble. In our case
$w_A:w_B:w_C = p_A:p_B:p_C$. There is a
common proportionality constant in the weights.
It is convenient to choose this constant in such a way that
the weight of each labeled graph be $1/N!$ \footnote{One should
however remember that this constant is an irrelevant proportionality
factor as long as the number of nodes is fixed.}.
For this choice we have
\begin{equation}
w_A=1/2, \quad w_B=1/6, \quad w_C=1/6 ,
\end{equation}
for the graphs in Fig. \ref{fig:shape2}.
This choice compensates for the increasing factor of permutations $N!$,
when one considers ensembles with varying $N$, and intuitively
removes overcounting coming from summing over permutations
of indistinguishable node's labels.
However, one should remember that in general
the number of distinct labeled graphs of a graph
is less than $N!$ and therefore the weight
of graph is smaller than $1$. The larger is the symmetry of
a graph topology the smaller is the number of underlying
labeled graphs and thus the smaller is the statistical weight
(see for instance Fig. \ref{fig:shape}).
The partition function $Z(N,L)$ for the Erd\"os-R\'enyi
ensemble can be written in the form:
\begin{equation}
Z(N,L) = \sum_{\alpha'\in lg(N,L)} \frac{1}{N!} \;\;=
\sum_{\alpha\in g(N,L)} w(\alpha) ,
\label{eq:er}
\end{equation}
where $lg(N,L)$ is the set of all labeled graphs with
given $N,L$ and $g(N,L)$ is the corresponding set
of (unlabeled) graphs. The weight $w(\alpha) = n(\alpha)/N!$,
where $n(\alpha)$ is the number of labeled graphs of graph $\alpha$.
We are interested in quantities averaged over the ensemble.
More precisely, we are interested in quantities which depend
on topology of graph and not on node's labels. This means
that if $O(\alpha)$ is such an observable then for any two
labeled graphs $\alpha'_1$ and $\alpha'_2$
of graph $\alpha$ we have $O(\alpha'_1)=O(\alpha'_2)\equiv O(\alpha)$.
The average is defined by
\begin{equation}
\left\langle O\right\rangle \equiv
\frac{1}{Z(N,L)} \sum_{\alpha'\in lg(N,L)} O(\alpha') \frac{1}{N!} \;\;=
\frac{1}{Z(N,L)} \sum_{\alpha\in g(N,L)} w(\alpha) O(\alpha) .
\end{equation}
We shall refer to an ensemble with fixed $N,L$ as to
a \emph{canonical ensemble}.
The word ''canonical'' is used here to emphasize that the number of
links $L$ is conserved like the total number of particles in a container
with ideal gas remaining in thermal balance with a source of heat.
The partition function $Z(N,L)$ can be calculated
by pure combinatorics as we have seen in the introduction.
Now for completeness we derive it using the
adjacency matrix representation of graphs.
The adjacency matrices $\mathbf{A}$ are symmetric,
they have zeros on the diagonal and $L$ unities
above the diagonal. Thus we have
\begin{equation}
Z(N,L) = \sum_{A_{12}} \sum_{A_{13}} \dots \sum_{A_{1N}} \sum_{A_{23}} \sum_{A_{24}}
\dots \sum_{A_{N-1,N}} \delta\left[L-\sum_{p<r} A_{pr}\right] \; 1/N! ,
\end{equation}
where $\delta\left[x\right]=1$ if $x=0$ and zero elsewhere.
The sums are done over all $A_{ij}=0,1$ for all pairs $i,j: \, 1\leq i<j\leq N$.
Using an integral representation of $\delta\left[ x \right]$
and exchanging the order of summation and integration we obtain
the expected result:
\begin{eqnarray}
Z(N,L) & = & (1/N!) \frac{1}{2\pi} \int_{-\pi}^{-\pi} dk \;
e^{ikL} \left( 1+e^{-i k} \right)^{\binom{N}{2}}
= (1/N!) \frac{1}{2\pi} \int_{-\pi}^{-\pi} dk \; e^{ikL}
\sum_{m=0}^{\binom{N}{2}} \binom{\binom{N}{2}}{m} e^{-ikm} \nonumber \\
& = & (1/N!) \binom{\binom{N}{2}}{L} .
\end{eqnarray}
This method can be applied to calculate averages of various quantities.
As an example consider the
node degree distribution $\pi(q)$ which tells us what is
the probability that a randomly chosen vertex on random graph
has degree $q$:
\begin{equation}
\pi(q) = \left\langle \frac{1}{N} \sum_i \delta \left[q-q_i\right]
\right\rangle .
\label{piq}
\end{equation}
By random graph we mean that we average over graphs from
the given ensemble.
We know that for Erd\"os-R\'enyi graphs $\pi(q)$
is a Poissonian distribution in the limit
of $N\rightarrow\infty$:
\begin{equation}
\pi(q) = \frac{\bar{q}^q}{q!} \exp(-\bar{q}) ,
\label{eq:piq}
\end{equation}
where $\bar{q}=2L/N$ is the average vertex degree.
This result can be rederived using the method described above.
Let us look at the degree of a vertex labeled by one. The result
does not depend on the vertex label for homogeneous graphs since
labels have no physical meaning. One can find that
\begin{eqnarray}
\pi(q) & = & \frac{1}{Z(N,L)} \frac{1}{N!}
\sum_{A_{12}} \sum_{A_{13}} \dots
\sum_{A_{23}} \sum_{A_{24}} \dots \sum_{A_{N-1,N}}
\delta\left[L-\sum_{p<r} A_{pr}\right] \;
\delta\left[ q - \sum_{r=2}^N A_{1r} \right] \nonumber \\
& = &
\binom{\binom{N}{2}}{L-q} \binom{N-1}{q} / \binom{\binom{N}{2}}{L} .
\label{eq:exact_p}
\end{eqnarray}
which in the limit $\bar{q}=\mbox{const},\,N\rightarrow\infty$ gives
Eq. (\ref{eq:piq}) as expected.
So far we have discussed the canonical ensemble of Erd\"os-R\'{e}nyi
graphs with $N,L$ fixed. Erd\"{o}s and R\'{e}nyi
introduced also a related model called {\it binomial model}
where the number of nodes $N$ is fixed but the number of links $L$
is not fixed {\em a priori}. One starts from $N$ empty vertices and connects
every pair of vertices with a probability $p$.
In this statistical ensemble the probability of obtaining
a labeled graph with given $L$ is $P(L)=p^L (1-p)^{\binom{N}{2}-L}$.
Thus the partition function is
\begin{eqnarray}
Z(N,\mu) & = & \sum_L \sum_{\alpha\in lg(N,L)} \frac{1}{N!} P(L(\alpha))
= (1-p)^{\binom{N}{2}} \sum_L \left(\frac{p}{1-p}\right)^L
\sum_{\alpha\in lg(N,L)} \frac{1}{N!} \nonumber \\
& \propto & \sum_L \exp (-\mu L) \; Z(N,L)
\propto \sum_L \exp (-\mu L + S(N,L)) ,
\label{grand}
\end{eqnarray}
where $\frac{p}{1-p}\equiv \exp(-\mu)$ or $\mu = \ln \frac{1-p}{p}$,
and the entropy $S(N,L)$ is given by Eq. (\ref{S0}).
We skipped an $L$-independent factor
in front of the sum in the second line substituting equality
by proportionality sign.
The weight of labeled graphs is $w(\alpha)=1/N! \; \exp(-\mu L(\alpha))$,
where $\mu$ is a constant which can be interpreted as chemical potential
for links in the \emph{grand-canonical} ensemble (\ref{grand}).
One can calculate the average number of links or
its variance as derivatives of the grand-canonical partition
function with respect to $\mu$:
$\langle L \rangle = -\partial_\mu \ln Z(N,\mu)$
and $\langle L^2 \rangle -\langle L \rangle^2
= \partial^2_\mu \ln Z(N,\mu)$.
The sum of states can be done exactly:
\begin{equation}
Z(N,\mu) = \sum_{L=0}^{\binom{N}{2}} e^{-\mu L} \frac{1}{N!} \binom{\binom{N}{2}}{L} =
\frac{1}{N!} (1+e^{-\mu})^{\binom{N}{2}} .
\end{equation}
It is easy to see that for fixed finite $\mu$ the average
number of links behaves as $N^2$ or more precisely as
\begin{equation}
\langle L \rangle = p \frac{N(N-1)}{2} =
\frac{1}{1 + e^\mu} \frac{N(N-1)}{2} .
\label{Lp}
\end{equation}
Thus for $N\rightarrow\infty$ the graphs become dense.
The mean value of node degree
$\langle \bar{q} \rangle =2\langle L \rangle /N$ increases to infinity.
The situation changes when $\mu$ goes to infinity with increasing $N$.
This happens in particular if the probability $p$
scales as $p \sim 1/N$ since then $\mu$ behaves as
$\mu \sim \ln N$. In this case
$L$ is proportional to $N$ (\ref{Lp}) and
both the terms $\mu L$ and $S$
in the exponent of Eq. (\ref{grand}) behave as $N\ln N$
and compensate each other. The corresponding graphs
become sparse and the mean node degree
$\langle \bar{q} \rangle =2\langle L \rangle /N$ is now finite.
The situation in which $\mu$ scales as $\ln N$
is very different from the situation known from classical
statistical physics, where such quantities like chemical potential
$\mu$ are intensive and do not depend on system size $N$
in the thermodynamic limit $N\rightarrow\infty$.
The difference between canonical and grand-canonical ensembles
gradually disappears in the large $N$ limit \cite{ref:bck,ref:dms}.
For canonical ensemble or sparse graphs the node degree
$\bar{q} = 2L/N = \alpha$ is kept constant
when $N\rightarrow \infty$ while in grand-canonical one it may fluctuate
around the average $\langle \bar{q} \rangle =
2\langle L\rangle /N = \alpha$. However, the magnitude of
fluctuations around the average disappears in the large $N$ limit
since
\begin{equation}
\langle L^2 \rangle - \langle L\rangle^2 =
\binom{N}{2} \frac{e^{-\mu}}{(1+e^{-\mu})^2} ,
\end{equation}
and $\Delta q =
\sqrt{\langle L^2 \rangle - \langle L\rangle^2} / \langle L \rangle
\sim 1/N \rightarrow 0$,
so effectively the system selects graphs with $\bar{q} = \alpha$.
Sometimes one also considers an ensemble of graphs
with a predefined node degree sequence $\{q_1,q_2,\dots,q_N\}$.
We shall call it \emph{micro-canonical}.
Again, in the simplest case one assumes that all
labeled graphs are equiprobable in this ensemble. Properties
of random graphs in such an ensemble
strongly depend on the degree sequence.
\section{Weighted homogeneous graphs}
In the previous section we described
ensembles for which all labeled graphs
had the same statistical weight. Random graphs
in such ensembles have well known
properties. It turns out, however, that most
of these properties do not correspond to those
observed for real world networks. One needs a more
general set-up to define an ensemble of complex random
networks. Such a set-up can be introduced as follows.
One considers the same set of graphs as in Erd\"os-R\'enyi model
but one ascribes to each graph a different statistical weight.
In other words, one chooses a probability measure
on the set of graphs which differs from the uniform measure.
In the generalized ensemble, each graph
in addition to the configuration space weight $1/N!$
has a functional weight $W(\alpha)$.
For homogeneous random graphs this weight
is assumed to depend only on graph topology. This means
that the weight does not depend on nodes' labels:
if $\alpha'_1$ and $\alpha'_2$ are labeled graphs
of $\alpha$ then $W(\alpha'_1)=W(\alpha'_2)\equiv W(\alpha)$.
The partition function for a weighted canonical ensemble
reads
\begin{equation}
Z(N,L) = \sum_{\alpha'\in lg(N,L)} (1/N!)\, W(\alpha') \;\;=
\sum_{\alpha\in g(N,L)} w(\alpha) W(\alpha) ,
\label{eq:can_g}
\end{equation}
where $w(\alpha)$ is the same factor $w(\alpha) = n(\alpha)/N!$
as before (\ref{eq:er}), being just the ratio of the number of
labeled graphs $\alpha'$ of $\alpha$ (obtained by
permutations of nodes' labels giving distinct adjacency matrices)
and the number of all nodes' labels permutations $N!$.
For Erd\"os-R\'enyi graphs the functional weight is $W(\alpha)=1$.
The simplest non-trivial example is
a family of product weights $W$:
\begin{equation}
W(\alpha) = \prod_{i=1}^N p(q_i) ,
\label{eq:prod}
\end{equation}
where $p(q)$ is a positive function depending on
one node degree $q$.
This functional weight does not introduce
correlations between node degrees. We shall refer to
random graphs generated by this partition function as
\emph{uncorrelated networks}. One should however remember
that the total weight does not entirely factorize because the
configuration space weight $w(\alpha) = n(\alpha)/N!$ written
as a function of node degrees $w(q_1,q_2,\dots,q_N)$ does
not factorize. There is also another factor which prevents
the model from the full factorization and independence
of node degrees, namely this is the total number of links
$2L = q_1+q_2+\dots + q_N$ which for given $L$ and $N$ introduces
correlations between $q_i$'s. For example, if one of $q_i$'s is
large, say of order $2L$, then the remaining ones have to be small
in order not to violate the constraint on the sum.
For a wide class of weights $p(q)$ one can however show
that in the large $N$ limit the probability that a randomly
chosen graph has degrees $q_1,\dots,q_N$ approximately factorizes:
\begin{equation}
\pi(q_1,\dots,q_N) \sim \prod_{i=1}^N \pi(q_i) .
\label{fact}
\end{equation}
For large $N$, the node degree distribution $\pi(q)$ (\ref{piq}),
that is the probability that a random node on random graph has
degree $q$ \cite{ref:bck,ref:bjk1,ref:bl1}, can be approximated by
\begin{equation}
\pi(q) = \frac{p(q)}{q!} \exp(-Aq-B) ,
\label{piqAB}
\end{equation}
where parameters $A,B$ are determined from the conditions
for the normalization $\sum_q \pi(q)=1$ and for the average
$\sum_q q \pi(q) = \bar{q} \equiv 2L/N$.
For example, for $p(q)=1$ which corresponds to
Erd\"os-R\'{e}nyi graphs one finds $A = -\ln \bar{q} = \ln 2L/N$
and $B = \bar{q} = 2L/N$, therefore $\pi(q)$ is given by the Poissonian
from Eq. (\ref{eq:piq}).
Since the node degree distribution $\pi(q)$
for weighted graphs (\ref{eq:prod}) depends on $p(q)$, one can
choose the latter to obtain a desired form of
the node degree distribution $\pi(q)$. Let $\pi(q)$
be a desired node degree distribution such that
\begin{equation}
\sum_q \pi(q)=1 \ , \quad \bar{q} = \sum q \pi(q) .
\end{equation}
If we choose the weight (\ref{eq:prod}) with
\begin{equation}
p(q) = q! \pi(q)
\end{equation}
in canonical ensemble with $N$ nodes
and $L$ links, in the limit of $N\rightarrow \infty$ and $2L/N = \bar{q}$
we obtain homogeneous random graphs
with this node degree distribution.
In this case the constants $A$, $B$ from Eq. (\ref{piqAB}) vanish
automatically: $A=B=0$.
In particular by an appropriate choice of $p(q)$ we
can generate scale free graphs with the node degree Barab\'{a}si - Albert distribution \cite{ref:bamodel}:
$\pi(q) = \frac{4}{q(q+1)(q+2)}$ for $q=1,2,\dots$ and $\pi(0)=0$
as an ensemble of graphs $L=N$, $\bar{q} = 2$
with $p(q) = q! \frac{4}{q(q+1)(q+2)}$ for $q=1,2,\dots$ and $p(0)=0$.
However, for finite $N$ the node degree distribution
$\pi(q)$ deviates from the limiting shape due to finite size corrections,
which are particularly strong for fat tailed distributions $\pi(q) \sim q^{-\gamma}$.
The maximal node degree
scales as $q_{max}\sim N^{1/(\gamma-1)}$ for $\gamma\ge 3$ and
as $q_{max} \sim N^{1/2}$ for very fat tails: $2 < \gamma < 3$
\cite{ref:bk,ref:bpv} as a result of structural constraints
which also lead to the occurrence of correlations between node degrees.
One can define more complicated weights than those given by
Eq. (\ref{eq:prod}). A natural candidate for networks
with degree-degree correlations is the following
weight \cite{ref:bl1,ref:b}:
\begin{equation}
W(\alpha) = \prod_{l=1}^L p(q_{a_l},q_{b_l}) ,
\end{equation}
where the product runs over all links of the graph,
and the weight $p(q_a,q_b)$ is a symmetric function
of degrees of nodes at the end points of the link.
One can choose this function to favor assertive
or disassertive behavior \cite{ref:bl1,ref:b,ref:n3,ref:bp,ref:d2}.
In a similar way one can
introduce probability measures on the set of graphs
which mimic some other functional properties of real
networks, like for example higher clustering
\cite{ref:n2,ref:bjk2,ref:bjk3,ref:d1,ref:pn2}.
One can do this in micro-canonical, canonical, grand-canonical
or any other ensemble. This is just the most general
set-up to handle homogeneous networks.
\section{Monte-Carlo generator of homogeneous networks}
Erd\"os-R\'enyi graphs are exceptional in the sense
that one can calculate for them almost all quantities of interest
analytically. This is not the case for weighted networks.
Various methods have been proposed for generating random graphs
\cite{ref:k}.
In this section we will describe a Monte-Carlo method
which allows one to study a wide class of random weighted
graphs experimentally by a sort of numerical experiments.
The basic idea behind this type of experiments is to sample
the configuration space of graphs with the probability
proportional to the statistical weight or in other
words to generate graphs with a desired probability.
Again, the Erd\"os-R\'enyi graphs are exceptional
because one can generate them one by one independently of
each other. This is just because they are equiprobable.
For weighted graphs the situation is not that easy since
there are no efficient algorithms to pick up
an element from a large set with the given probability.
The naive algorithm which relies on picking up an element
uniformly and then accepting it with the given
probability has a very low acceptance rate.
Therefore one has to use another idea. We will
describe below how to generate graphs using
dynamical Monte-Carlo technique.
The idea is to run a random walk process
in the set of graphs which visits configurations with
a frequency proportional to their
statistical weight. Mathematically, this means
that one has to invent a stationary
Markov chain (process) for which the stationary distribution
is proportional to the statistical weights of graphs:
$\sim W(\alpha)/Z$.
The Markov chain is defined by
transition probabilities $P(\alpha\rightarrow \beta)$
that the random walker will go in one step
from a configuration (graph) $\alpha$ to $\beta$.
The probabilities are stored in a transition matrix $\mathbf{P}$:
$P_{\alpha\beta} \equiv P(\alpha\rightarrow \beta)$
which is also called Markov's matrix. For a stationary process,
the transition matrix $\mathbf{P}$
is constant during the random walk.
Random walk is initiated from a certain graph
$\alpha_0$ and then elementary steps are
repeated producing a sequence (chain) of graphs
$\alpha_0 \rightarrow \alpha_1 \rightarrow \alpha_2 \rightarrow \dots \ $.
The probability $p_\beta(t+1)$ that
a graph $\beta$ will be generated in the $(t+1)$-th step
of the Markov process can be calculated as
\begin{equation}
p_\beta(t+1) = \sum_\alpha p_\alpha(t) P_{\alpha \beta} .
\end{equation}
The last equation can be written as
\begin{equation}
\mathbf{p} (t+1) = \mathbf{P}^{\tau} \mathbf{p} (t) ,
\label{interp}
\end{equation}
where $\tau$ denotes transposition, and $\mathbf{p}$ is a
vector of elements $p_\alpha$. One should note that the stationary
state: $\mathbf{p}(t+1) = \mathbf{p}(t)$ corresponds to a left eigenvector
of $\mathbf{P}$ to the eigenvalue \footnote{One can show that all eigenvalues
of a Markov transition matrix lie inside or on the unit
circle $|\lambda| \le 1$.}
$\lambda=1$.
If the process is ergodic, which means that
any configuration can be reached by a sequence of transitions
starting from any initial configuration,
and if the transition matrix fulfills the detailed balance
condition:
\begin{equation}
W_\alpha P_{\alpha \beta} =
W_\beta P_{\beta \alpha} \quad \forall \alpha,\beta ,
\label{eq:db}
\end{equation}
then the stationary state can be shown to approach the
desired distribution:
$p_\alpha(t) \rightarrow W_\alpha/Z$ for $t \rightarrow \infty$.
We used a short-hand notation $W_\alpha$ for $W(\alpha)$.
In other words, when the length of the Markov chain becomes
infinite the probability of occurrence of graphs
in the Markov chain becomes proportional to their
statistical weights and becomes independent of the initial
configuration. Therefore the average over graphs
generated in this Markovian random walk is a good
estimator of the average over the weighted ensemble.
The price to pay for generating graphs in this way
is that the consecutive graphs in the Markov chain may
be correlated with each other.
Therefore one has to find a minimal number of steps for which
one can treat measurements on such graphs as independent.
One should note that the only
characteristics of the Markov process which
has a physical meaning from the point of view of the
simulated ensemble is the stationary distribution.
All other dynamical properties of the random walk
which are encoded in the form of transition matrix
$P(\alpha\rightarrow\beta)$
are irrelevant. Many different
transition matrices $\mathbf{P}$
may have the same stationary distribution.
Indeed, many of them fulfill the detailed balance condition for
given weights $W_\alpha$ (\ref{eq:db}). The best known
choice of $\mathbf{P}$ is
\begin{equation}
P_{\alpha\beta} =
\min\left\{1,\frac{W_\beta}{W_\alpha}\right\} .
\label{metrop}
\end{equation}
This choice is quite general and can be used in many
different contexts. It is called Metropolis
algorithm.
For the current configuration $\alpha$ one proposes a change
to a new configuration $\beta$ which differs slightly from
$\alpha$ and then one accepts it with
the Metropolis probability (\ref{metrop}). One
repeats this many times producing a chain of configurations.
The proposed modifications should not be too large
since then the acceptance rate would be small. Therefore
the algorithm makes only small steps (moves) in the configuration
space which form a sort of weighted random walk.
\section{Monte-Carlo generator of canonical ensemble}
Now, we want to apply this method to generate Erd\"os-R\'{e}nyi
graphs. Let us begin with the canonical ensemble with $N,L$ fixed.
A good candidate for elementary transformation of graph
is rewiring of a link as shown in Fig. \ref{fig:tmove},
because it does not change $N$ and $L$.
As mentioned before it is convenient to introduce
a representation in which each undirected link is represented
by two directed links.
The rewiring is done in two steps \cite{ref:bck}. First we
choose a directed link $ij$ and a vertex $k$ at random.
Then we rewire the link $ij$ to $ik$. If there is already
a link between $i$ and $k$ or if the vertex $k$ coincides with $i$,
we reject the rewiring since it would
otherwise lead to a multiple- or self-connection.
One should note that the result of rewiring $ij$
is not the same as of rewiring $ji$.
\begin{figure}[h]
\psfrag{i}{$i$} \psfrag{j}{$j$} \psfrag{k}{$k$}
\includegraphics[width=13cm]{tmove.eps}
\caption{The idea of rewiring: a random link (dotted line)
is rewired from vertex $j$ to a random vertex $k$ (left hand side).
Alternatively (right hand side) a random, oriented link
(dotted line) is rewired from vertex of its end $j$
to a random vertex $k$. The opposite link
$j\rightarrow i$ is simultaneously rewired.}
\label{fig:tmove}
\end{figure}
The move is accepted with the Metropolis probability.
For the canonical ensemble of Erd\"os-R\'enyi
graph this probability is equal to one
since functional weights are $W_\alpha=W_\beta=1$ in Eq. (\ref{metrop}).
Let us see how rewiring transformations work in practice.
Consider as an example the set of graphs shown in Fig. \ref{fig:shape2}.
If we pick up the link $3-2$ in the graph A
and rewire it to $3-1$, we will obtain the graph B.
If we rewire the link $2-3$ to $2-4$, we will get the graph C.
So using the procedure of rewiring showed in Fig. \ref{fig:tmove}
we can obtain every graph in the ensemble.
The rewiring transformation is ergodic in this set of graphs.
To summarize, our procedure of generating graphs in this training
ensemble looks as follows. We construct an arbitrary graph having
$N=4$ nodes and $L=3$ links to initiate the procedure
and then we repeat iteratively rewirings for
randomly chosen links and vertices. The only restriction is
that the rewirings do not produce self- or multiple-connected
links. We keep on repeating until we obtain ''thermalized graphs''.
Then we can begin measuring quantities on the generated sequence
of random graphs.
Let us check that the described Monte-Carlo procedure indeed
generates graphs with the expected probabilities (\ref{pABC}).
Let us calculate the Markovian matrix $\mathbf{P}$ for the
rewiring procedure in this ensemble. First we calculate the
transition probability from A to B. The graph
A can be converted into B in one step if
we rewire the link ''b'' in Fig. \ref{fig:tmove_ex} to the vertex 2,
or alternatively the link ''e'' to the vertex 1. We see that for this
change we can choose two out of six links and one of four vertices
to obtain the desired change. Thus the probability of choosing
links is $2/6$ and of choosing correct vertex is $1/4$, so
the total probability is $P(A\rightarrow B)=2/6 \cdot 1/4 = 2/24$.
Let us now calculate $P(A\rightarrow C)$.
To obtain $C$ from $A$ we have to rewire
''a'' to 3 or ''f'' to 4. Thus $P(A\rightarrow C)=2/6 \cdot 1/4 = 2/24$.
We can find $P(A\rightarrow A)$ from the condition:
$P(A\rightarrow A)+P(A\rightarrow B)+P(A\rightarrow C)=1$.
This yields $P(A\rightarrow A)=20/24$.
\begin{figure}[ht]
\includegraphics[width=3cm]{tmove_ex.eps}
\caption{The representation of graph A in Fig. \ref{fig:shape2}
as directed graph. }
\label{fig:tmove_ex}
\end{figure}
Repeating the calculations for the remaining cases we find:
$P(B\rightarrow A)=6/24$,
$P(B\rightarrow B)=18/24$, $P(B\rightarrow C)=0$ and
$P(C\rightarrow A)=6/24$, $P(C\rightarrow B)=0$,
$P(C\rightarrow C)=18/24$. The results can be collected
in a transition matrix:
\begin{equation}
\mathbf{P} = \frac{1}{24} \left[\begin{array}{ccc}
20 & 2 & 2 \\
6 & 18 & 0 \\
6 & 0 & 18
\end{array}\right].
\end{equation}
We can now determine the stationary probability distribution
of the Markov process as a left eigenvector to the eigenvalue
one of the transition matrix $\mathbf{P}$. We obtain
$p_A:p_B:p_C = 3:1:1$ in agreement with Eq. (\ref{pABC}).
This is not surprising since
$\mathbf{P}$ satisfies the detailed balance condition (\ref{eq:db})
and the corresponding changes are ergodic.
We have checked above by explicit calculation
that the algorithm gives correct weights of
Erd\"os-R\'enyi graphs for $N=3,L=4$.
One can give a general argument that for any $N,L$
the algorithm generates labeled graphs which are
equiprobable. Suppose that we have a certain labeled
graph $\alpha$ and want to get $\beta$ by rewiring $ij$ to $ik$.
(see Fig. \ref{fig:tmove}). The total probability
$P(\alpha\rightarrow\beta)$ can be written
as a product of two factors: the probability $P_c$ of choosing
a particular candidate for a new configuration and
the probability $P_a$ of accepting it.
Because we choose a link $i\rightarrow j$ from $2L$ possible
directed links and a vertex $k$ from $N$ vertices we
have $P_c = 1/(2LN)$. Inserting
$P(\alpha'\rightarrow\beta') = 1/(2LN) \,
P_a(\alpha'\rightarrow\beta')$ in the Eq. (\ref{eq:db})
and similarly for $\alpha'\leftrightarrow \beta'$ we get
\begin{equation}
w_{\alpha'} P_a(\alpha'\rightarrow\beta') =
w_{\beta'} P_a(\beta'\rightarrow\alpha') .
\label{eq:can1}
\end{equation}
But $w_{\alpha'}=1/N!$ for all labeled graphs,
thus $P_a(\alpha'\rightarrow\beta')=P_a(\beta'\rightarrow\alpha')$.
This means that every move should be accepted unless it
violates the multiple- or self-connections constraints.
The rejection does not change the frequency of the occurrence of
simple graphs but only restricts the space of sampled graphs
to what we need. The weights of (unlabeled) graphs $\alpha$
are in this case $w(\alpha)= n(\alpha)/N!$ where $n(\alpha)$
is the number of distinct graphs of $\alpha$.
\begin{table}[h]
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
Graphs & \multicolumn{6}{|c|}{\includegraphics[width=7cm]{pent_ex.eps}} \\
\hline
$w_\alpha$ & 1/2 & 1/2 & 1/12 & 1/2 & 1/8 & 1/24 \\
$p_\alpha$ & 0.286 & 0.286 & 0.048 & 0.286 & 0.071 & 0.024 \\
\hline
\multicolumn{7}{|c|}{Simulations} \\
\hline
R & \multicolumn{6}{|c|}{} \\
\hline
10 & 0.285(1) & 0.286(1) & 0.048(1) & 0.286(1) & 0.071(1) & 0.024(1) \\
5 & 0.286(1) & 0.285(1) & 0.047(1) & 0.286(1) & 0.071(1) & 0.024(1) \\
2 & 0.285(1) & 0.285(1) & 0.047(1) & 0.286(1) & 0.072(1) & 0.024(1) \\
1 & 0.284(1) & 0.286(1) & 0.048(1) & 0.284(1) & 0.072(1) & 0.025(1) \\
\hline
\end{tabular}
\caption{Theoretically calculated weights $w_\alpha$
of graphs in the canonical ensemble $N=5,L=4$ are
normalized to ensure probabilistic interpretation:
$p_\alpha = w_\alpha/\sum_\beta w_\beta$,
and compared with the experimental frequencies in the Markov chain
using algorithm of rewiring. The results differ by the number
$R$ of rewirings between consecutive measurements. }
\label{tab1}
\end{center}
\end{table}
Let us numerically test the algorithm.
In table \ref{tab1} we compare the weights calculated analytically
and computed from Monte-Carlo generated graphs
for $N=5,L=4$. There are six different graphs in this ensemble.
We generated $10^6$ graphs. Each of
them was obtained from the previous one by $R$
rewirings, or more precisely by $R$ attempts of rewiring \footnote{
Even if a rewiring is rejected we count it in since it
corresponds to the transition $P(\alpha \rightarrow \alpha)$
from graph to the same graph.}.
As we can see in table \ref{tab1},
the frequency of occurrence of each graph is in
an excellent agreement with the expected weights.
The results do not depend on the separation $R$
between the measurements.
In the chain of $10^6$ graphs, each graph in this ensemble
is produced many times. For larger ensembles
the algorithm would not be able to visit all graphs since
the number of graphs is very large (\ref{S1}). In this
case the algorithm would
choose only those graphs which are most representative. To make sure
that the algorithm has reached the stationary distribution
one should start a couple of random walks from different corners of the
configuration space and run the algorithm so long as the
statistical properties of graphs generated in all the random
walks become identical.
Generalization of the algorithm to a weighted ensemble
is straightforward. We insert
the statistical weights $W_\alpha$ of this ensemble
into the Metropolis formula (\ref{metrop}).
Consider in particular a product weight (\ref{eq:prod}).
We choose a link $ij$ and a vertex $k$
on the current configuration $\alpha$ at random
and attempt to rewire the link to $ik$ to obtain
a new configuration $\beta$.
The change is accepted with the probability
\begin{equation}
P_a(\alpha\rightarrow\beta) =
\min\left\{1,\frac{W_\beta}{W_\alpha}\right\} =
\min\left\{1,\frac{p(q_j-1)p(q_k+1)}{p(q_j)p(q_k)}\right\} .
\end{equation}
The degrees $q_j,q_k$ are taken from $\alpha$.
Clearly the rewiring changes the degrees
$q_j\rightarrow q_j-1$, $q_k\rightarrow q_k+1$,
and leaves the degrees of remaining nodes intact.
The ratio $W_\beta/W_\alpha$ can be calculated for
any form of statistical weights, so the algorithm is
general.
\section{Monte-Carlo generator of grand-canonical ensemble}
The rewiring procedure described in the previous section
does not change $N$ and $L$. If we want to simulate graphs
from a grand-canonical ensemble for which $L$ is variable,
we have to supplement the set of elementary transformations
in the algorithm by transformations which change the number of links.
We can introduce two mutually
reciprocal transformations: adding and deleting a link.
Both they preserve the number of nodes $N$ but change the
number of links: $L \rightarrow L\pm 1$.
The two transformations must be carefully balanced.
On a given graph $\alpha$ we have to choose one of them.
Let the link addition be selected with the probability
$p_+$ and the removal with $p_-$. Once the move is
selected we have to choose a link-candidate for which the
move is to be applied. It is convenient to split the total
transition probability into three factors:
\begin{equation}
P(\alpha\rightarrow \beta) = p_\pm P_c(\alpha\rightarrow \beta)
P_a(\alpha\rightarrow\beta) ,
\end{equation}
where $p_\pm$ stands for one of $\{p_-,p_+\}$,
$P_c(\alpha\rightarrow \beta)$ for the probability of choosing
a candidate configuration for the change
and $P_a(\alpha\rightarrow \beta)$ for the probability
of accepting the move. Let $\alpha$ and $\beta$ be two graphs
which differ by a link which is present on $\beta$
but absent on $\alpha$: $L(\alpha)=L(\beta)-1$.
The transition probability for adding
a link to $\alpha$ has to be balanced with the probability of
removing the link from $\beta$.
In order to add a link to $\alpha$ we have to choose two vertices
to which the addition of a link is attempted.
The probability of choosing a given pair of vertices,
if we choose two vertices independently, is
$P_c(\alpha\rightarrow\beta)=2/N^2$. Thus the total
probability of this move is
\begin{equation}
P_{\alpha\beta}=
P(\alpha\rightarrow\beta) = p_+ \, \frac{2}{N^2} \,
P_a(\alpha\rightarrow\beta) .
\end{equation}
In the reciprocal transformation we have to choose this
link among all links. The probability of choosing one among $L$ links
is $P_c(\beta\rightarrow\alpha)=1/L_\beta=1/(L_\alpha+1)$.
Thus the total probability of this move is
\begin{equation}
P_{\beta\alpha}= P(\beta\rightarrow\alpha) = p_- \, \frac{1}{L_\beta} \,
P_a(\beta\rightarrow\alpha) .
\end{equation}
Now we have to insert the last two equations to the
detailed balance condition which for the grand-canonical
ensemble additionally includes the factor $e^{-\mu L}$:
\begin{equation}
W_\alpha e^{-\mu L_\alpha} P_{\alpha \beta} =
W_\beta e^{-\mu L_\beta} P_{\beta \alpha} \ .
\label{eq:db_gc}
\end{equation}
Using this we can calculate the ratio
\begin{equation}
\frac{P_a(\alpha\rightarrow\beta)}{P_a(\beta\rightarrow\alpha)} =
\exp(-\mu) \, \frac{p_-}{p_+} \,
\frac{N^2}{2L_\beta} \, \frac{W_\beta}{W_\alpha} .
\end{equation}
If one chooses the same number of attempts for adding
and removing a link: $p_+=p_-$, then the ratio
$p_+/p_-=1$ will disappear from the last equation
and the acceptance probabilities for
adding or removing a link in the Metropolis algorithm
will read
\begin{equation}
P_a(\alpha\rightarrow\beta) =
\min\left\{1, \exp(-\mu) \,
\frac{N^2}{2(L_\alpha+1)} \, \frac{W_\beta}{W_\alpha} \right\} ,
\label{eq:gcan_rown}
\end{equation}
and
\begin{equation}
P_a(\beta\rightarrow\alpha) =
\min\left\{1, \exp(+\mu) \,
\frac{2L_\beta}{N^2} \, \frac{W_\alpha}{W_\beta}\right\} ,
\label{eq:gcan_rown2}
\end{equation}
respectively.
As before if we want to produce only simple graphs we must
have an additional condition which eliminates moves leading
to self- or multiple connections.
The algorithm is complete. One should note
that there is no reason to do additional rewirings because
a rewiring of a link
$ij$ to a link $ik$ is equivalent to removing
the link $ij$ and adding $ik$.
In principle one could propose other algorithms. For example,
one could consider a modified algorithm in which
the move removing a link is done in a different way.
Instead of picking up a link as a candidate,
one could pick up two vertices at random, and then remove
a link if there is any between them.
The probability of choosing a pair of vertices would
be $2/N^2$ and it would cancel with the identical factor
for the probability of choosing candidates in the move
adding a link. The fractions
$N^2/2L$ and $2L/N^2$ would in this case disappear from equations
(\ref{eq:gcan_rown}) and (\ref{eq:gcan_rown2}).
The two algorithms of course generate
the same ensemble. However, the modified algorithm would have
much worse acceptance rate for sparse networks since
the chance that there is a link between
two randomly chosen vertices on a sparse graph
is very small. Most of the chosen pairs of vertices
are not connected by a link
and therefore the algorithm will do nothing since
there is no link to remove.
This problem is absent for the
algorithm which we described previously since in that case
only existing links are chosen as candidates for removal.
One can easily estimate that the probability of accepting
a link removal (\ref{eq:gcan_rown2}) is not very small.
Indeed, even for sparse graph the factor $e^{\mu} 2L/N^2$
in Eq. (\ref{eq:gcan_rown2}) is of order unity. In this
case both $\exp(\mu)$ and $L$ for large $N$ grow proportionally
to $N$ and their product balances the factor
$N^2$ in the denominator. The
algorithm has a finite acceptance rate which does not
vanish when the system size grows.
As an exercise, let us consider an example of unweighted
($W_\alpha=1$) graphs with $N=3$. This ensemble consists
of four graphs shown in table \ref{tab2}.
Their statistical weights can be easily found to be
$1/3!, \, 3e^{-\mu}/3!, \, 3e^{-2\mu}/3!, \, e^{-3\mu}/3!$,
so we expect that the frequency of occurrence in random
sampling should be $1:3e^{-\mu}:3e^{-2\mu}:e^{-3\mu}$.
As we see in table \ref{tab2},
the results of Monte-Carlo simulations
are in perfect agreement with this expectation.
\begin{table}[h]
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|}
\hline
& Graphs & \multicolumn{4}{|c|}{\includegraphics[width=5cm]{gr_3.eps}} \\
\hline
$\mu = 0$ & Theor. & \ \ 0.125 \ \ & \ \ 0.375 \ \ & \ \ 0.375 \ \ & \ \ 0.125 \ \ \\
& Exp. & 0.125(1) & 0.374(1) & 0.374(1) & 0.126(1) \\
\hline
$\mu = 0.2$ & Theor. & 0.166 & 0.408 & 0.334 & 0.091 \\
& Exp. & 0.166(1) & 0.408(1) & 0.334(1) & 0.091(1) \\
\hline
$\mu = 0.5$ & Theor. & 0.241 & 0.439 & 0.266 & 0.054 \\
& Exp. & 0.241(1) & 0.439(1) & 0.266(1) & 0.054(1) \\
\hline
\end{tabular}
\caption{Comparison of the probability distribution
of graphs in a grand-canonical ensemble with $N=3$
nodes: calculated theoretically
and computed in a run of Monte-Carlo simulation in which
$10^6$ graphs were generated.}
\label{tab2}
\end{center}
\end{table}
One can easily apply this technique to any form of statistical
weights. In particular we can consider the product weights (\ref{eq:prod}).
The probability of accepting a new configuration by adding
or removing a link between $ij$ reads
\begin{eqnarray}
\mbox{min} \left\{ 1, \frac{N^2}{2(L+1)}
\exp(-\mu) \frac{p(q_i+1)p(q_j+1)}{p(q_i)p(q_j)} \right\} &
\mbox{for adding a link,} \nonumber \\
\mbox{min} \left\{ 1, \frac{2L}{N^2}
\exp(+\mu) \frac{p(q_i-1)p(q_j-1)}{p(q_i)p(q_j)} \right\} &
\mbox{for deleting a link,} \nonumber
\end{eqnarray}
where $L$ and $q_i,q_j$ refer to the current configuration.
\section{Monte-Carlo generator of micro-canonical ensemble}
Another frequently encountered ensemble is an ensemble of graphs
which have a given node degree sequence $\{q_1,q_2,\dots,q_N\}$.
The partition function $Z$ has the form:
\begin{equation}
Z(N,\left\{ q_i \right\}) = \sum_{\alpha'\in lg(N,L)}
\left( \prod_{i=1}^N \delta \left[ q_i(\alpha') - q_i\right] \right)
\, 1/N! \, W(\alpha') ,
\end{equation}
where the product of delta functions allows one to include only
those graphs which have a prescribed degree distribution $q_i$.
As before the factor $1/N!$ is fixed in this ensemble
and could in principle be skipped.
The canonical partition function $Z(N,L)$ is related to
the micro-canonical ones:
\begin{equation}
Z(N,L) = \sum_{q_1=0}^\infty \dots \sum_{q_N=0}^\infty
Z(N,\left\{ q_i \right\}) \, \delta \left[ q_1+q_2+\dots +q_N - 2L \right] .
\end{equation}
To generate graphs from micro-canonical ensemble one has to
have a Markov process preserving node degrees.
The main idea is to combine
simultaneous rewirings \cite{ref:msz}
as shown in Fig. \ref{fig:xmove}.
We shall call this combination ''X-move''.
\begin{figure}[h]
\psfrag{i}{$i$} \psfrag{j}{$j$} \psfrag{k}{$k$} \psfrag{l}{$l$}
\includegraphics[width=5cm]{xmove.eps}
\caption{The idea of ''X-move'': two oriented links (dotted lines)
$ij$ and $kl$ chosen in a random way are rewired, exchanging their endpoints.
Then the opposite links (solid lines) are also rewired. }
\label{fig:xmove}
\end{figure}
At each step one picks up two random links: $ij$ and $kl$,
and rewires them to $il$ and $kj$.
This procedure is ergodic, i.e. it explores the whole
configuration space. Such a transformation
was discussed in \cite{ref:msz} where it was used to
randomize graphs with a given nodes' degree sequence.
In that case the functional weight was $W_\alpha=1$ and
rewirings were done with probability equal to one.
In general case if one considers non-trivial
$W_\alpha$, one has to accept the change with
a corresponding Metropolis probability (\ref{metrop}).
In this way one can for example generate graphs
whose statistical weights depend on the number
of triangles. In a sense one can perform
a weighted randomization of networks with the given
node degree sequence. Introducing a weight
into randomization may be very important in the
construction of scoring functions in problems
of motif searching \cite{ref:m,ref:bl2,ref:ia}.
If one tries to determine relations between structural motifs
and the functionality of network, it is very important
to properly construct scoring function which may
clearly account for the existence of a particular
subgraph on a network and its function. Scoring functions
are usually measured as a sort of distance between
a network which displays a certain function and
a random network which does not.
An important problem in such studies is how to construct those
networks which should serve as the background reference.
The simplest idea is to use networks obtained
by uniform randomization. This may however introduce some bias
and may be misleading. Imagine
for example that a motif which is responsible for a certain network
function is built out of a couple of triangular loops and
that triangular loops alone have no function.
It is clear that one would like to control the abundance
of triangular loops to distinguish between specific motifs and
motifs which are more frequent by pure chance just because of higher
abundance of triangles. Therefore it may be important to control
the number of triangles in the randomized reference networks
used in the scoring function. It was just an example, but in
general case it might be useful to perform a weighted randomization
taking into account some desired features of reference networks.
\section{Graph generator and adjacency matrices}
All the elementary transformations: rewiring, adding or removing a link,
and the X-move have a simple representation in terms of adjacency
matrices. Rewiring relies on picking up at random an element
$A_{ij} = 1$ of the adjacency matrix and flipping it with
an element $A_{ik}=0$ so that after the move
$A_{ij} = 0$ and $A_{ik}=1$. For undirected links
adjacency matrices are symmetric and therefore at the same time
one has to flip $A_{ji} = 1$ and $A_{ki}=0$.
To add a vertex one chooses at random $A_{ij}$ and if
$A_{ij}=0$ and $i\ne j$, one changes it into $A_{ij}=1$
(and for $A_{ji}$ correspondingly).
To remove a link one picks up a non-vanishing element
$A_{ij}=1$ and substitutes it with $A_{ij}=0$. To perform
X-move one picks up two non-vanishing elements of $\mathbf{A}$ at random,
say $A_{ij}=1$ and $A_{kl}=1$, and flips them simultaneously with
$A_{il}=0$ and $A_{kj}=0$ to:
$A_{ij}=0$, $A_{kl}=0$, $A_{il}=1$ and $A_{kj}=1$.
Of course one also flips their four symmetric
counterparts. In practice, when simulating sparse graphs
one does not use the matrix
representation since it would require $N^2$ storage capacity.
For sparse matrices the number of non-vanishing
matrix elements is proportional to $N$ and one can
use a linear storage structure. It directly corresponds to the underlying
graph structure. Using linear storage one can code graphs having of
order $10^6$ nodes or even more on a PC.
\section{Degenerated graphs (pseudographs)}
In previous sections we described ensembles of simple
graphs. Let us now discuss pseudographs that is graphs
which may have multiple- and self-connections.
A degenerate undirected pseudograph can be represented
by a symmetric adjacency matrix $\mathbf{A}$ whose off diagonal entries
$A_{ij}$ count the number of links between vertices $i$ and $j$,
and the diagonal ones $A_{ii}$ count twice the number of
self-connecting links attached to vertex $i$.
For example, the graph depicted in Fig. \ref{fig:pseudo}
has the following adjacency matrix:
\begin{equation}
\mathbf{A} = \left( \begin{array}{cccccc}
0 & 0 & 1 & 0 & 0 & 1 \\
0 & 2 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 2 \\
1 & 0 & 0 & 0 & 2 & 0
\end{array} \right).
\label{eq:pseudo}
\end{equation}
\begin{figure}[h]
\psfrag{1}{$1$} \psfrag{2}{$2$} \psfrag{3}{$3$} \psfrag{4}{$4$} \psfrag{5}{$5$} \psfrag{6}{$6$}
\includegraphics[width=7cm]{pseudo.eps}
\caption{Left hand side: the example of pseudograph with
$N=6,L=5$. Right hand side: its representation as directed graph.}
\label{fig:pseudo}
\end{figure}
In the representation where each undirected link is
represented as two opposite directed links
all matrix elements including the diagonal ones
count the number of directed links emerging
from the vertex.
As before let us first consider labeled pseudographs.
However, in order to have a unique representation of a graph
one has to label links as well. We did not have to do
this for simple graphs since in that case each link was uniquely
determined by its endpoints. It is not anymore the case for
degenerate graphs since there may be more than one link
between two nodes. A pseudograph
with $N$ nodes and $L$ links can be fully labeled
by $N$ node labels and $2L$ labels of directed links.
Each fully labeled graph has
thus the configuration space weight equal to $1/(N!(2L)!)$.
Let us work out the consequences of this choice.
Denote $\alpha$ a graph, $\alpha'$ a labeled
graph of $\alpha$ with labeled nodes only, and $\alpha''$
a fully labeled graph of $\alpha$ with labeled nodes and
labeled links. From here on labeled graph means a graph which has only
labels on nodes while \emph{fully labeled graph} a graph which
has additionally labels on directed links.
The configuration space weight of $\alpha$ can be calculated
as a sum over all fully labeled graphs $\alpha''$ as follows:
\begin{equation}
w_\alpha = \sum_{\alpha'' \in flg(\alpha)} \frac{1}{N!(2L)!} =
\sum_{\alpha' \in lg(\alpha)} \frac{1}{N!}
\left( \prod_{i} \frac{1}{2^{A_{ii}/2} \left( A_{ii}/2\right)!} \right)
\prod_{i>j} \frac{1}{A_{ij}!} ,
\label{wp}
\end{equation}
where $flg(\alpha)$ denotes the set of fully
labeled graphs of graph $\alpha$, $lg(\alpha)$ the set of
labeled graphs (labeled nodes only) of graph $\alpha$ .
The expression $A_{ii}/2$ counts the number of
self-connecting links attached to vertex $i$,
and $A_{ij}$ is the multiplicity of links connecting $i$ and $j$.
If there are no self-connections ($A_{ii}=0$)
and no multiple connections ($A_{ij}\le 1$), the configuration
space weight (\ref{wp}) reproduces the weight of simple graphs.
One can easily understand the appearance of the
combinatorial factors in general case. Suppose that
we permute links' labels of a fully labeled graph
leaving nodes' labels intact. Among all $(2L)!$ permutations
not all are distinct. If we have $A_{ij}$ links
between vertex $i$ and $j$ and we will permute their labels, then
all $A_{ij}!$ permutations will give the same labeled
graph (if we simultaneously permute labels of the directed partners).
Similarly if we have vertex with a self-connection and
we exchange labels of the two directed links emerging from
this vertex, the fully labeled graph will not change. Thus
for each self-connection two permutations lead to the same
fully labeled graph. To summarize, the number of distinct
permutations of link labels is reduced from $(2L)!$ by dividing
out the factor $2$ for each self-connection and $k!$ for each $k$-link
multiple connection which just gives Eq. (\ref{wp}).
It turns out that these weights
are identical to the combinatorial factors of
Feynman diagrams which appear in perturbative
series of a mini-field
theory \cite{ref:bck}. One can thus interpret random
pseudographs as Feynman diagrams and use perturbation theory
to enumerate them.
Let us consider as an example a canonical ensemble of pseudographs
with $N=3$ and $L=3$. There are $14$ graphs in this ensemble.
They are shown in Fig. \ref{fig:pseudo_3}.
\begin{figure}[h]
\includegraphics[width=12cm]{pseudo_3.eps}
\caption{All pseudographs in the canonical ensemble with $N=3,L=3$.}
\label{fig:pseudo_3}
\end{figure}
In table \ref{tab3} we compare the
theoretically calculated probability distribution
of graphs:
\begin{equation}
p_\alpha = \frac{w_\alpha}{\sum_\beta w_\beta} ,
\end{equation}
using the weights calculated by the formula (\ref{wp})
with the probability distribution obtained experimentally
from the frequency histogram
of graphs produced by the Monte-Carlo generator.
Now the generator works exactly as before except that
it does not reject moves leading to a self- or multiple-connections.
The results are in perfect accordance.
\begin{table}[h]
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline
Graphs & A & B & C & D & E & F & G \\
\hline
Weights & 1/6 & 1/2 & 1/12 & 1/4 & 1/2 & 1/8 & 1/4 \\
\hline
Theor. & 0.066 & 0.197 & 0.033 & 0.099 & 0.197 & 0.049 & 0.099 \\
\hline
Exp. & 0.066(1) & 0.197(1) & 0.033(1) & 0.099(1) & 0.197(1) & 0.049(1) & 0.099(1) \\
\hline
\hline
Graphs & H & I & J & K & L & M & N \\
\hline
Weights & 1/16 & 1/4 & 1/8 & 1/8 & 1/48 & 1/16 & 1/96 \\
\hline
Theor. & 0.0247 & 0.099 & 0.049 & 0.049 & 0.0082 & 0.0247 & 0.0041 \\
\hline
Exp. & 0.0246(1) & 0.099(1) & 0.049(1) & 0.049(1) & 0.0082(1) & 0.0247(1) & 0.0041(1) \\
\hline
\end{tabular}
\caption{Comparison of theoretical and experimental (Monte-Carlo) computations of
frequencies of graphs' occurrence in the ensemble with $N=3,L=3$ \label{tab3}.}
\end{center}
\end{table}
As an example let us calculate the weight of graph M in Fig. \ref{fig:pseudo_3}.
The weight of each labeled graph of graph M,
according to the formula (\ref{wp}), is equal
\begin{equation}
w_{M'} = \frac{1}{3!} \cdot \frac{1}{2^3} \frac{1}{2!} = \frac{1}{96} ,
\end{equation}
where the first factor comes from $1/N!$, the
second from the three self-connections, and the third
from the fact that the two self-connections are attached
to the same vertex and thus can be permuted without changing
graph's connectivity. There are six distinct labeled graphs M'
of graph M and thus
\begin{equation}
w_{M} = \sum_{M'} \frac{1}{96} = \frac{6}{96} = \frac{1}{16} .
\end{equation}
One should note that the number of distinct labeled graphs
varies from graph to graph. For example, for graph L there is only
one labeled graph. In this case $w_{L'} = 1/3! \cdot 1/2^3 = 1/48$
and $w_L = 1/48$. The calculation can be easily repeated
for each graph in Fig. \ref{fig:pseudo_3}
yielding the weights $w_\alpha$ listed
in table \ref{tab3}.
As follows from Eq. (\ref{wp}),
the partition function for the canonical ensemble of
pseudographs can be written in three different ways:
\begin{equation}
Z(N,L) = \sum_{\alpha'' \in flg(N,L)} \frac{1}{N! (2L)!} \ =
\sum_{\alpha' \in lg(N,L)} \frac{1}{N!}
\left( \prod_{i} \frac{1}{2^{A_{ii}/2} \left( A_{ii}/2\right)!} \right)
\prod_{i>j} \frac{1}{A_{ij}!} \ =
\sum_{\alpha \in g(N,L)} w_\alpha .
\label{zw}
\end{equation}
The first sum runs over fully labeled graphs and has the
simplest form since all fully labeled graphs have the same
weight. The weight of labeled graphs in the second sum
varies. We note that labeled graphs are isomorphic with
adjacency matrices that is each labeled graph is
uniquely represented by a generalized adjacency
matrix \footnote{The elements $A_{ij}$ are indexed by
nodes' labels, but the information about links' labels
is lost in the adjacency matrix representation.}
like Eq. (\ref{eq:pseudo}). The
sum over $\alpha' \in lg(N,L)$ can be thus interpreted
as a sum over all generalized adjacency
$N \times N$ symmetric matrices $\mathbf{A}$
such that $\sum_{ij} A_{ij} = 2L$. We see that
not all adjacency matrices have the same statistical weight
unlike for simple graphs since the weights depend on
the number of self- and multiple-connections.
A weighted ensemble of pseudographs is constructed as before
by introducing an additional functional weight $W(\alpha)$
under the sum defining the partition function (\ref{zw}):
\begin{equation}
Z(N,L) = \sum_{\alpha'' \in flg(N,L)} \frac{1}{N! (2L)!} W(\alpha'') \ =
\sum_{\alpha \in g(N,L)} w_\alpha W(\alpha) .
\end{equation}
As before the functional weight $W(\alpha'')$ does not depend
on graph's labeling but only on graph's topology.
In other words
if $\alpha''_1$ and $\alpha''_2$ are two different fully
labeled graphs of graph $\alpha$ then
$W(\alpha''_1)=W(\alpha''_2) \equiv W(\alpha)$.
We can now consider various weights: for example a
product weight as in Eq. (\ref{eq:prod}) to mimic
graphs with uncorrelated node degrees.
But even in this case the total weight does not factorize since
the configuration space weight $w(\alpha)$ written
as a function of node degrees $w(q_1,q_2,\dots, q_N)$
does not factorize. Due to the absence of the structural
constraints the approximation given by equations (\ref{fact}) and
(\ref{piqAB}) has now much weaker finite size corrections.
A grand-canonical ensemble for pseudographs
with arbitrary product weights (\ref{eq:prod})
has the following partition function:
\begin{equation}
Z(N,\mu) = \sum_L \exp (-\mu L)
\sum_{\alpha \in g(N,L)} w_\alpha \prod_{i=1}^N p(q_i(\alpha)) .
\label{eq:grps}
\end{equation}
This means that all pseudographs with fixed nodes' degrees $\{q_i\}$
have the same functional weight $\sim p(q_1)\cdots p(q_N)$,
which seems to be similar to
that generated by the Molloy-Reed
construction of pseudographs \cite{ref:mr}.
Let us comment on this.
In the Molloy-Reed construction
one generates a sequence of non-negative integers $\{q_1,q_2,\dots,q_N\}$
for example as independent identically distributed
numbers with the distribution $p(q)$. One interprets $q_i$'s as node degrees.
The only requirement is that the sum $q_1+q_2+\dots +q_N=2L$ is even.
In the first step of the construction
each integer $q_i$ is represented as a hub built out of
a vertex and $q_i$ outgoing branches which can be viewed
as directed links emerging from this vertex.
In the second step the directed links are paired randomly
in couples of links in opposite direction to
form undirected links connecting vertices.
This procedure generates
the same subset of pseudographs as
the partition function $Z(N,\mu)$ (\ref{eq:grps}).
However, statistical weights are different.
To see this, let us consider a subset of Molloy-Reed graphs obtained for
a given set $\{q_i\}$.
There are $N$ labeled
vertices and $2L=\sum_i q_i$ labeled directed links. All permutations of
labels of links and nodes are equiprobable
exactly as it was before for fully labeled pseudographs (\ref{wp}).
If one calculates corresponding symmetry factors
for node-labeled graphs the same combinatorial factors arise
as in Eq. (\ref{wp}): if one pairs two directed links $a$ and $b$
which belong to the same vertex one obtains a self-connecting link.
The pair $ab$ is identical as $ba$
since both the links begin and end at the same vertex.
This reduces the number of distinct permutations by factor of $2$
as in Eq. (\ref{wp}). Similarly for $k$ pairs of directed links between two
vertices one can exchange the order of pairing in $k!$ ways
each time obtaining the same multiple connection, so the corresponding
factor is $1/k!$ again as in Eq. (\ref{wp}).
The conditional probability of choosing a particular graph
$\alpha$ under the condition that in the first step of the construction
the set of $\{q_1,q_2,\dots,q_N\}$ has been selected, is
\begin{equation}
w_{M-R}(\{q_i\}|\alpha) =
\frac{w_\alpha}{\sum_{\beta \in g\{q_1,\dots ,q_N\}} w_{\beta}} ,
\end{equation}
where the sum is done over all (unlabeled)
pseudographs $\beta$ from the micro-canonical set
of fixed degrees $\{q_1,\dots ,q_N\}$.
The total probability is thus
\begin{equation}
w_{M-R}(\alpha) = P(\{q_i\}) \, w_{M-R}(\{q_i\}|\alpha) ,
\end{equation}
where $P(\{q_i\})$ is the probability that
in the first step of the construction the set
$\{q_1,\dots ,q_N\}$ is selected.
This probability is proportional to the product
of $p(q_i)$'s multiplied by the number of permutations of
$\{q_1,\dots ,q_N\}$ which give the same set.
We denote this number by $\mbox{Perm}(q_1,\dots, q_N)$.
The order of $q_i$'s does not matter since we consider unlabeled
graphs. For example, the following
permutations (sequences): $(q_1,q_2,q_3)=(3,3,2)$,
$(3,2,3)$ and $(2,3,3)$ give the same set $\{3,3,2\}$,
so in this case we have $\mbox{Perm}(3,3,2)=3$.
In general, the number is given by
\begin{equation}
\mbox{Perm}(q_1,\dots, q_N) = \frac{N!}{n_0! n_1!\cdots} ,
\end{equation}
where $n_0,n_1,\dots$ are degree's multiplicities:
$n_q=\sum_i \delta\left[ q_i-q\right]$. Thus
\begin{equation}
P(\{q_i\}) \propto
\left( \prod_{i=1}^N p(q_i(\alpha)) \right) \mbox{Perm}(\{q_i\}) .
\end{equation}
\begin{figure}[h]
\includegraphics[width=12cm]{grps.eps}
\caption{A set of 10 pseudographs for $N=3$ and $p(q)=1/3$ for
$q=0,1,2$. Top: three hubs for $\{q_i\}=\{2,2,2\}$
are generated and then directed links are paired randomly giving three
pseudographs A1,A2,A3. Bottom: the rest of pseudographs from
this ensemble.}
\label{fig:grps}
\end{figure}
\begin{table}[h]
\renewcommand*{\arraystretch}{1.3}
\begin{center}
\begin{tabular}{|c|c|c|c|c||c|c|}
\hline
L & $\{q_i\}$ & $\mbox{Perm}(q_1,q_2,q_3)$ & Graphs & $w_\alpha$ weights (\ref{wp}) & $w_{M-R}$ weights (\ref{eq:wmr}) & G-C weights from Eq. (\ref{eq:grps}) \\
\hline \hline
3 & 2,2,2 & 1 & A1,A2,A3 & $(\frac{1}{6}:\frac{1}{8}:\frac{1}{48}) = \frac{1}{15} (8:6:1)$ &
$\frac{1}{14} (\frac{8}{15},\frac{6}{15},\frac{1}{15})$ & $e^{-3\mu}(\frac{1}{6},\frac{1}{8},\frac{1}{48})$ \\
\hline \hline
2 & 2,2,0 & 3 & B1,B2 & $\frac{1}{8}:\frac{1}{4}=\frac{1}{3}(1:2)$ &
$\frac{3}{14} (\frac{1}{3},\frac{2}{3})$ & $e^{-2\mu}(\frac{1}{8},\frac{1}{4})$ \\
\hline
& 2,1,1 & 3 & C1,C2 & $\frac{1}{2}:\frac{1}{4}=\frac{1}{3}(2:1)$ &
$\frac{3}{14} (\frac{2}{3},\frac{1}{3})$ & $e^{-2\mu}(\frac{1}{2},\frac{1}{4})$ \\
\hline \hline
1 & 2,0,0 & 3 & D & $\frac{1}{4}$ & $\frac{3}{14}$ & $e^{-\mu}\frac{1}{4}$ \\
\hline
& 1,1,0 & 3 & E & $\frac{1}{2}$ & $\frac{3}{14}$ & $e^{-\mu}\frac{1}{2}$ \\
\hline \hline
0 & 0,0,0 & 1 & F & $\frac{1}{6}$ & $\frac{1}{14}$ & $\frac{1}{6}$ \\
\hline
\end{tabular}
\caption{Weights calculated for the Molloy-Reed construction
and for the corresponding grand-canonical ensemble with $N=3$.
For each sequence $q_1,q_2,q_3$ we give the combinatorial
number $\mbox{Perm}(q_1,q_2,q_3)$ of permutations leading to
the same graph. Altogether, there are $14$ different sequences
of length three, with $q_i=0,1$ or $2$ as can be seen in the
third column of the table.}
\label{tab:grps}
\end{center}
\end{table}
Collecting all the factors together and normalizing
to have the probabilistic interpretation we obtain
the following expression for
the total weight (probability) for Molloy-Reed's
pseudographs:
\begin{equation}
w_{M-R}(\alpha) = \left( \prod_{i=1}^N p(q_i(\alpha)) \right)
\frac{\mbox{Perm}(q_1(\alpha),\dots, q_N(\alpha))}
{\sum_{k_1,\dots, k_N} \mbox{Perm}(k_1,\dots, k_N)} \,
\frac{w_\alpha}{\sum_{\beta \in g\{q_1,\dots ,q_N\}} w_{\beta}} .
\label{eq:wmr}
\end{equation}
The first factor comes from picking $N$ numbers $q_i$ at random,
the second counts permutations
and the third includes the weight generated by pairing directed links.
As we see, despite many similarities the Molloy-Reed ensemble
and the grand-canonical lead to different weights.
As an example, in Fig. \ref{fig:grps} we show an ensemble of 10 pseudographs
with $N=3$ generated by Molloy-Reed algorithm for $p(q)=1/3$ for
$q=0,1,2$ and zero elsewhere. We compare statistical weights
of the generated graphs with the corresponding ones in the
grand-canonical ensemble. As we can see in table \ref{tab:grps},
the weights are different in the two ensembles.
\section*{Summary}
We have discussed a statistical approach
to homogeneous random graphs. This framework is
a natural extension of the Erd\"os-R\'enyi theory
to the case of weighted graphs: one considers the
same set of graphs but with modified statistical weights.
The statistical weights of homogeneous graphs depend
only on graphs' topology. In other words, if one assigns
some labels to its nodes, they will
have no physical meaning similarly as the numbers
of indistinguishable particles in quantum mechanics.
One can permute them and the graph and its statistical
weight will stay intact. The only information which matters
is the number (entropy) of distinct permutations of nodes' labels.
All permutations of node's labels
are equivalent, unlike for growing
networks where those permutations have to preserve the
causal order corresponding to the order of node's attachment to the graph.
The statistical weight of a homogeneous graph is proportional
to the number of all labeled graphs of this graph while
of a growing network to the number of causally labeled graphs.
This leads to a difference between homogeneous and growing networks.
For example, a typical homogeneous graph has a larger diameter than
the corresponding growing network with the same node degree
distribution (\ref{eq:exact_p}). Generally, geometrical properties of homogeneous
graphs are different from those of growing networks for which
correlations between the time of node's
attachment and its degree induce
node-node correlations of a specific type \cite{ref:kr,ref:bbjk}.
Such correlations are absent for homogeneous graphs.
Various functional properties of homogeneous networks can
be modeled by an appropriate choice of functional weight.
One can easily produce networks with an assertive mixing,
higher clustering or any desired property which can
reflect any real-data observation.
Homogeneous networks can be simulated numerically.
We have also described a Monte-Carlo algorithm to
generate canonical, grand-canonical and micro-canonical
ensembles which performs a sort of weighted random walk
(Markov chain) in the configuration space with a desired
stationary distribution. We advocated the
importance of the possibility of generating random
networks with desired statistical properties
for advanced motif searching \cite{ref:m,ref:bl2,ref:ia}.
Many real networks have resulted from hybrid processes
of growth mixed with some thermalization. The framework discussed
in this paper can flexibly extrapolate between the two regimes.
It allows one to directly investigate the relation between
structural and functional properties of complex networks.
\bigskip
\noindent
{\bf Acknowledgments}
\medskip
\noindent
We thank Piotr Bialas, Jerzy Jurkiewicz and Andrzej Krzywicki
for stimulating discussions. This work was partially supported by
the Polish State Committee for Scientific Research (KBN) grant
2P03B-08225 (2003-2006) and Marie Curie Host Fellowship
HPMD-CT-2001-00108 and by EU IST Center of Excellence "COPIRA".
|
1,108,101,564,534 | arxiv |
\subsection{Locally Computed Gradient}
Consider the link $l\sim (i,j)$, since $w_l=w_{ij}=w_{ji}$ and $w_{ii}=1-\sum_{s\in N_i}w_{is}$, we have:
\begin{equation}\label{diff}
\frac{\text{d}w_{st}}{\text{d}w_l}=
\begin{cases}
+1 &\text{if } s=i \text{ and } t=j\\
+1 &\text{if } s=j \text{ and } t=i\\
-1 &\text{if } s=i \text{ and } t=i\\
-1 &\text{if } s=j \text{ and } t=j\\
0 &\text{else.}
\end{cases}
\end{equation}
The gradient $g_l$ of the function $h(\mathbf{w})$ for $l\sim (i,j)$ can be calculated as follows:
\begin{align}
g_l&=\frac{\text{d}\hfill h(\mathbf{w})}{\text{d}w_l}\nonumber \\
&=\frac{\text{d}\hfill f(W)|_{W=I-\mathcal{I}\times \text {diag}(\mathbf{w})\times \mathcal{I}^T}}{\text{d}w_l}\nonumber \\
&=\sum_{s,t}\frac{\partial f}{\partial w_{st}}\frac{\text{d}w_{st}}{\text{d}w_l}\nonumber \\
&=\frac{\partial f}{\partial w_{ij}}\frac{\text{d}w_{ij}}{\text{d}w_l}+\frac{\partial f}{\partial w_{ji}}\frac{\text{d}w_{ji}}{\text{d}w_l}+\frac{\partial f}{\partial w_{ii}}\frac{\text{d}w_{ii}}{\text{d}w_l}+\frac{\partial f}{\partial w_{jj}}\frac{\text{d}w_{jj}}{\text{d}w_l}\nonumber\\
&=\frac{\partial f}{\partial w_{ij}}+\frac{\partial f}{\partial w_{ji}}-\frac{\partial f}{\partial w_{ii}}-\frac{\partial f}{\partial w_{jj}}\nonumber\\
&=p\big ((W^{p-1})_{ji}+(W^{p-1})_{ij}-(W^{p-1})_{ii}-(W^{p-1})_{jj}\big ).\label{line4}
\end{align}
In the last equality we used equation \eqref{derivativef}.
It is well know from graph theory that if we consider $W$ to be the adjacency matrix of a weighted graph $G$, then $(W^s)_{ij}$ is a function of the weights on the edges of the $i-j$ walks (i.e.~the walks from $i$ to $j$) of length exactly $s$ (in particular if the graph is unweighted $(W^s)_{ij}$ is the number of distinct $i-j$ $s$-walks \cite{West2000}). Since for a given $p$ the gradient $g_l$, $l\sim (i,j)$, depends on the $\{ii, jj, ij, ji\}$ terms of the matrix $W^{p-1}$, $g_l$ can be calculated locally by using only the weights of links and nodes at most $\frac{p}{2}$ hops away from $i$ or $j$\footnote{
If a link or a node is more than $p/2$ hops away both from node $i$ and node $j$, then it cannot belong to a $i-j$ walk of length $p$.
}
. Practically speaking, at each step, nodes $i$ and $j$ need to contact all the nodes up to $p/2$ hops away in order to retrieve the current values of the weights on the links of these nodes and the values of weights on the nodes themselves. For example, when $p=2$, then the minimization is the same as the minimization of the Frobenius norm of $W$ since $Tr(W^2)=\sum _{i,j}w_{ij}^2=||W||_F^2$, and the gradient $g_l$ can be calculated as $g_l=2\times (2W_{ij}-W_{ii}-W_{jj})$ which depends only on the weights of the vertices incident to that link and the weight of the link itself.
An advantage of our approach is that it provides a trade-off between locality and optimality. In fact, the larger the parameter $p$, the better the solution of problem \eqref{tracemin} approximates the solution of problem \eqref{minim}, but at the same time the larger is the neighborhood from which each node needs to retrieve the information. When $p=2$, then $g_l$ where $l\sim (i,j)$ only depends on the weights of subgraph induced by the two nodes $i$ and $j$. For $p=4$, the gradient $g_l$ depends only on the weights found on the subgraph induced by the set of vertices $N_i\cup N_j$, then it is sufficient that nodes $i$ and $j$ exchange the weights of all their incident links.
\subsection{Choice of Stepsize and Projection set}
The global convergence of gradient methods (i.e.~for any initial condition) has been proved under a variety of different hypotheses on the function $h$ to minimize and on the step size sequence $\gamma^{(k)}$.
In many cases the step size has to be adaptively selected on the basis of the value of the function or of the module of its gradient at the current estimate, but this cannot be done in a distributed way for the function $h(\mathbf{w})$. This leads us to look for convergence results where the step size sequence can be fixed ahead of time. Moreover the usual conditions, like Lipschitzianity or boundness of the gradient, are not satisfied by the function $h(.)$ over all the feasible set.
For this reason we add another constraint to our original problem~\eqref{unconsTM} by considering that the solution has to belong to a given convex and compact set $X$. Before further specifying how we choose the set $X$, we state our convergence result.
\begin{prop}\label{prop3}
Given the following problem
\begin{eqnarray}\label{unconsTM2}
\text{minimize} & &h(\mathbf{w})=Tr \left((I-\mathcal{I}\times \text {diag}(\mathbf{w})\times \mathcal{I}^T)^p\right),\nonumber\\
\text{subject to} & &\mathbf{w} \in X
\end{eqnarray}
where $X \subseteq \mathbb{R}^m $ is a convex and compact set, if $\sum_k \gamma^{(k)}=\infty$ and $\sum_k \left(\gamma^{(k)}\right)^2< \infty$, then the following iterative procedure converges to the minimum of $h$ in $X$:
\begin{equation}\label{e:grad_proj_vect}
\mathbf{w}^{(k+1)}= P_X\left(\mathbf{w}^{(k)} - \gamma^{(k)} \mathbf{g}^{(k)}\right),
\end{equation}
where $P_X(.)$ is the projection operator on the set $X$ and $\mathbf{g}^{(k)}$ is the gradient of $h$ evaluated in $\mathbf{w}^{(k)}$.
\end{prop}
\begin{proof}
The function $h$ is continuous on a compact set $X$, so it has a point of minimum. Moreover also the gradient $\mathbf{g}$ is continuous and then bounded on $X$. The result then follows from Proposition~$8.2.6$ in \cite[pp. 480]{bertsekas03}.
\end{proof}
For example, $\gamma ^{(k)}=a/(b+k)$ where $a>0$ and $b\geq 0$ satisfies the step size condition in Proposition~\ref{prop3}.
While the convergence is guaranteed for any set $X$ convex and compact, we have two other requirements. First, it should be possible to calculate the projection $P_X$ in a distributed way. Second, the set $X$ should contain the solution of the optimization problem~\eqref{unconsTM}.
About the first issue, we observe that if $X$ is the cartesian product of real intervals, i.e. if $X=[a_1,b_1]\times[a_2,b_2]\times \dots [a_m,b_m]$, then we have that the $l$-th component of the projection on $X$ of a vector $\mathbf{y}$ is simply the projection of the $l$-th component of the vector on the interval $[a_l,b_l]$, i.e.:
\begin{equation}
\left[P_X(\mathbf{y})\right]_l=P_{[a_l,b_l]}(y_l)=
\begin{cases}
a_l&\text{if } y_l<a_l,\\
y_l&\text{if } a_l \leq y_l\leq b_l,\\
b_l&\text{if } b_l<y_l.
\end{cases}
\end{equation}
Then in this case Eq.~\eqref{e:grad_proj_vect} can be written component-wise as
$$w_l^{(k+1)}= P_{[a_l,b_l]}(w_l^{(k)} - \gamma^{(k)} g_l^{(k)}).$$
We have shown in the previous section that $g_l$ can be calculated in a distributed way, then the iterative procedure can be distributed.
About the second issue, we choose $X$ in such a way that we include in the feasibility set all the weight matrices with spectral radius at most $1$.
The following lemma indicates how to choose $X$.
\begin{lemma} \label{l:proj}
Let $W$ be a real and symmetric matrix where each row (and column) sums to $1$, then the following holds,$$\rho (W)=1 \ \Longrightarrow \ \max _{i,j}|w_{ij}|\leq 1.$$
\end{lemma}
\begin{proof}
Since $W$ is real and symmetric, then we can write $W$ as follows
$$W=S\Lambda S^T,$$
where $S$ is an orthonormal matrix ($S^TS=SS^T=I$), and $\Lambda$ is a diagonal matrix having $\Lambda _{kk}=\lambda _k$ and $\lambda _k$ is the $k$-th largest eigenvalue of $W$. Let $\mathbf{r}_k$ and $\mathbf{c}_k$ be the rows and columns of $S$ respectively and $r_k^{(i)}$ be the $i$-th element of this vector. So, $$W=\sum _k\lambda _k\mathbf{c}_k\mathbf{c}_k^T,$$ and
\begin{align}
|w_{ij}|&=|\sum _k\lambda _kc_k^{(i)}c_k^{(j)}|\label{Teq1}\\
&\leq \sum _k|c_k^{(i)}||c_k^{(j)}|\label{Teq2}\\
&= \sum _k|r_i^{(k)}||r_j^{(k)}|\label{Teq3}\\
&\leq ||\mathbf{r}_i||_2||\mathbf{r}_j||_2\label{Teq4}\\
&=1.\label{Teq5}
\end{align}
The transition from \eqref{Teq1} to \eqref{Teq2} is due to the fact $\rho (W)=1$, the transition from \eqref{Teq3} to \eqref{Teq4} is due to Cauchy--Schwarz inequality. The transition from \eqref{Teq4} to \eqref{Teq5} is due to the fact that $S$ is an orthonormal matrix.
\end{proof}
A consequence of Lemma~\ref{l:proj} is that if we choose $X=[-1,1]^m$ the weight vector of the matrix solution of problem~\eqref{minim} necessarily belongs to $X$ (the weight matrix satisfies the convergence conditions). The same is true for the solution of problem~\eqref{unconsTM} for $p$ large enough because of Proposition~\ref{problem_equivalence}. The following proposition summarizes our results.
\begin{prop}
If the graph of the network is strongly connected, then the following distributed algorithm converges to the solution of the Schatten norm minimization problem for $p$ large enough:
\begin{equation}
w_l^{(k+1)}= P_{[-1,1]}(w_l^{(k)} - \gamma^{(k)} g_l^{(k)}), \;\; \forall l =1, \dots, m,
\end{equation}
where $\sum_k \gamma^{(k)}=\infty$ and $\sum_k \left(\gamma^{(k)}\right)^2< \infty$.
\end{prop}
\section{Introduction}\label{sec:intro}
A network is formed of nodes (or agents) and communication links that allow these nodes to share information and resources. Algorithms for efficient routing and efficient use of resources are proposed to save energy and speed up the processing. For small networks, it is possible for a central unit to be aware of all the components of the network and decide how to optimally use a resource on a global view basis. As networks expand, the central unit needs to handle a larger amount of data, and centralized optimization may become unfeasible especially when the network is dynamic. In fact, the optimal configuration needs to be computed whenever a link fails or there is any change in the network. Moreover, nodes may have some processing capabilities that are not used in the centralized optimization. With these points in mind, it becomes more convenient to perform distributed optimization relying on local computation at each node and local information exchange between neighbors. Such distributed approach is intrinsically able to adapt to local network changes.
A significant amount of research on distributed optimization in networks has recently been carried out. New faster techniques (\cite{Wei11a,Wei11b,Ghadimi11}) have been proposed for the traditional dual decomposition approach for separable problems that is well known in the network community since Kelly's seminal work on TCP (\cite{Kelly98}). A completely different approach has been recently proposed in~\cite{Nedic09}: it combines a consensus protocol, that is used to distribute the computations among the agents, and a subgradient method for the minimization of a local objective. Convergence results hold in the presence of constraints (\cite{Nedic10}), errors due to quantization (\cite{Nedic09b}) or to some stochastic noise (\cite{Ram10}) and in dynamic settings (\cite{Nedic09a,Lobel11,Masiero11}).
Finally a third approach relies on some intelligent random exploration of the possible solution space, e.g.~using genetic algorithms (\cite{Alouf10}) or the annealed Gibbs sampler (\cite{Kauffmann07}).
In this report we study distributed techniques to optimally select the weights of average consensus protocols (also referred to as ave-consensus protocols or algorithms). These protocols allow nodes in a network, each having a certain measurement or state value, to calculate the average across all the values in the network by communicating only with their neighbors. Consensus algorithms are used in various applications such as environmental monitoring of wireless sensor networks and cooperative control of a team of machines working together to accomplish some predefined goal. For example, a group of vehicles moving in formation to the same target must reach consensus on the speed and direction of their motion to prevent collisions. Although the average seems to be a simple function to compute, it is an essential element for performing much more complex tasks including optimization, source localization, compression, and subspace tracking (\cite{Rabideau:1996, Ren:2007, Boyd06}).
In the ave-consensus protocol, each node first selects weights for each of its neighbors, then at each iteration the estimates are exchanged between neighbors and each node updates its own estimate by performing a weighted sum of the values received (\cite{Saber07, Ren:2005}). Under quite general conditions the estimates asymptotically converge to the average across all the original values.
The weights play an essential role to determine the speed of convergence of the ave-consensus.
For this reason in this report we study how to select the weights in a given network in order to have fast convergence independently from initial nodes' estimates.
In \cite{Xiao04}, the authors refer to this problem as the Fastest Distributed Linear Averaging (FDLA) weight selection problem. They show FDLA problem is equivalent to maximize the spectral gap of the weight matrix $W$ and that, with the additional requirement of $W$ being symmetric, is a convex \emph{non-smooth} optimization problem. Then, the (symmetric) FDLA problem can be solved offline by a centralized node using interior point methods, but, as we discussed above, this approach may not be convenient for large scale networks and/or when the topology changes over time.
In this work, we propose to select the consensus weights as the values that minimize the Schatten $p$-norm of the weight matrix $W$ under some constraints (due also to the network topology). The Schatten $p$-norm of a matrix is the $p$-norm of its singular values as we will see later. We show that this new optimization problem can be considered an approximation of the original problem in \cite{Xiao04} (the FDLA) and we reformulate our problem as an equivalent unconstrained, convex and \emph{smooth} minimization that can be solved by the gradient method. More importantly, we show that in this case the gradient method can be efficiently distributed among the nodes.
We describe a distributed gradient procedure to minimize the Schatten $p$-norm for an even integer $p$ that requires each node to recover information from nodes that are up to $\frac{p}{2}$-hop distant. Then the order $p$ of the Schatten norm is a tuning parameter that allows us to trade off the quality of the solution for the amount of communication/computation needed. In fact the larger $p$ the more precise the approximation, but also the larger the amount of information nodes need to exchange and process. The simulations are done on real networks (such as Enron's internal email graph and the dolphins social network) and on random networks (such as Erdos Renyi and Random Geometric Graphs). Our simulation results show that our algorithm provides very good performance in comparison to other distributed weighted selection algorithms already for $p=2$, i.e. when each node needs to collect information only from its direct neighbors.
Finally, we show that nodes do not need to run our weight optimization algorithm \emph{before} being able to start the consensus protocol to calculate the average value, but the two can run in parallel.
The report is organized as follows:
In Section~\ref{PF} we formulate the problem we are considering and we give the notation used across the report.
Section~\ref{relatedwork} presents the related work for the weight selection problem for average consensus.
In Section \ref{TM} we propose Schatten $p$-norm minimization as an approximation of the original problem and in section~\ref{COTM} we show how its solution can be computed in a distributed way and evaluate its computation and communication costs.
Section \ref{PE} compares the performance of our algorithm and that of other known weight selection algorithms on different graph topologies (real and random graphs). We also investigate the case when the weight optimization algorithm and the consensus protocol runs simultaneously, and then the weight matrix changes at every time slot. Section~\ref{EXT} discusses some methods to deal with potential instability problems and with misbehaving nodes.
Section \ref{Conc} summarizes the report.
\section{Problem Formulation} \label{PF}
Consider a network of $n$ nodes that can exchange messages between each other through communication links. Every node in this network has a certain value (e.g.~a measurement of temperature in a sensor network or a target speed in a unmanned vehicle team), and each of them calculate the average of these values through distributed linear iterations. The network of nodes can be modeled as a graph $G=(V,E)$ where $V$ is the set of vertices, labeled from $1$ to $n$, and $E$ is the set of edges, then $(i,j)\in E$ if nodes $i$ and $j$ are connected and can communicate (they are neighbors) and $|E|=m$. We label the edges from $1$ to $m$. If link $(i,j)$ has label $l$, we write $l \sim (i,j)$. Let also $N_i$ be the neighborhood set of node $i$. All graphs in this report are considered to be \emph{connected} and \emph{undirected}.
Let $x_i(0)\in \mathbb{R}$ be the initial value at node $i$. We are interested in computing the average
$$x_{ave}=(1/n)\sum _{i=1}^nx_i(0),$$
in a decentralized manner with nodes only communicating with their neighbors.
The network is supposed to operate synchronously: when a global clock ticks, all nodes in the system perform the iteration of the averaging protocol. At iteration $k+1$, node $i$ updates its state value $x_i$ as follows:
\begin{equation}\label{wsum}
x_i(k+1)= w_{ii}x_i(k)+\sum _{j\in N_i}w_{ij}x_j(k),
\end{equation}
where $w_{ij}$ is the weight selected by node $i$ for the value sent by its neighbor $j$ and $w_{ii}$ is the weight selected by node $i$ for it own value.
As it is commonly assumed, in this report we consider that two neighbors select the same weight for each other, i.e.~$w_{ij}=w_{ji}$.
The matrix form equation is:
\begin{equation}\label{Mwsum}
\mathbf{x}(k+1)=W\mathbf{x}(k),
\end{equation}
where $\mathbf{x}(k)$ is the state vector of the system and $W$ is the weight matrix. The main problem we are considering in this report is how a node $i$ can choose the weights $w_{ij}$ for its neighbors so that the state vector $\mathbf{x}$ of the system converges fast to consensus. There are centralized and distributed algorithms for the selection of $W$, but in order to explain them, we need first to provide some more notation.
We denote by $\mathbf{w}$ the vector of dimensions $m\times 1$, whose $l$-th element $w_l$ is the weight associated to link $l$, then if $l\sim(i,j)$ it holds $w_l=w_{ij}=w_{ji}$. $A$ is the adjacency matrix of graph $G$, i.e. $a_{ij}=1$ if $(i,j) \in E$ and $a_{ij}=0$ otherwise. $\mathcal{C}_G$ is the set of all real $n\times n$ matrices $M$ following graph $G$, i.e. $m_{ij}=0$ if $(i,j) \notin E$. $D$ is a diagonal matrix where $d_{ii}$ (or simply $d_i$) is the degree of node $i$ in the graph $G$. $\mathcal{I}$ is the $n\times m$ incidence matrix of the graph, such that for each $l\sim(i,j)\in E\;$ $\mathcal{I}_{il}=+1$ and $\mathcal{I}_{jl}=-1$ and the rest of the elements of the matrix are null. $L$ is the laplacian matrix of the graph, so $L=D-A$. It can also be seen that $L=\mathcal{I}\mathcal{I}^T$.
The $n\times n$ identity matrix is denoted by $I$.
Given that $W$ is real and symmetric, it has real eigenvalues (and then they can be ordered). We denote by $\lambda _i$ the $i$-th largest eigenvalue of $W$, and by $\mu$ the largest eigenvalue in module non considering $\lambda_1$, i.e.~$\mu=\max\{\lambda_2,-\lambda_n\}$. $\sigma _i$ is the $i$-th largest singular value of a matrix. $\textrm{Tr}(X)$ is the trace of the matrix $X$ and $\rho(X)$ is its spectral radius. $||X||_{\sigma p}$ denotes the Schatten $p$-norm of matrix $X$, i.e. $||X||_{\sigma p}=(\sum _i\sigma _i^p)^{1/p}$. Finally we use the symbol $\frac{\text{d}\hfill }{\text{d}X}f(X)$, where $f$ is a differentiable scalar-valued function $f(X)$ with matrix argument $X\in \mathbb{R}^{m\times n}$, to denote the $n\times m$ matrix whose $(i,j)$ entry is $\frac{\partial f(X)}{\partial x_{ji}}$.
Table \ref{tab:notation} summarizes the notation used in this report.
\begin{table}[h]
\caption{Notion }
\centering
\scalebox{1}{
\begin{tabular}{lll}
\hline\hline
Symbol & Description & Dimension\\
\hline
$G$ & network of nodes and links & -\\
$V$ & set of nodes/vertices & $|V|=n$\\
$E$ & set of links/edges & $|E|=m$\\
$\mathbf{x}(k)$ & state vector of the system at iteration $k$ & $n\times 1$\\
$W$ & weight matrix & $n\times n$\\
$\mathbf{W}_i$ & vector of weights selected by node $i$ to its neighbors & $d_i\times 1$\\
$\mathbf{w}$ & vector of weights on links & $m\times 1$\\
$\text{diag}(\mathbf{v})$ & diagonal matrix having the elements of the $n\times 1$ vector $\mathbf{v}$ & $n\times n$ \\
$\mathcal{C}_G$ & set of $n\times n$ real matrices following $G$& - \\
$D$ & degree diagonal matrix & $n\times n$\\
$A$ & adjacency matrix of a graph & $n\times n$\\
$\mathcal{I}$ & incidence matrix of a graph & $n\times m$\\
$L$ & laplacian matrix $L=D-A=\mathcal{I}\mathcal{I}^T$ & $n\times n$\\
$\lambda _i$ & $i$th largest eigenvalue of $W$ & scalar\\
$\Lambda$ & eigenvalue diagonal matrix $\Lambda _{ii}=\lambda _i$ & $n\times n$\\
$\sigma _i$ & $i$th largest singular value & scalar\\
$\mu$ & second largest eigenvalue in magnitude of $W$ & scalar\\
$\rho (X)$ & spectral radius of matrix $X$ & scalar\\
$\textrm{Tr}(X)$ & trace of the matrix $X$ & scalar\\
$||X||_{\sigma p}$ & Schatten $p$-norm of a matrix $X$ & scalar\\
$\frac{\text{d}\hfill}{\text{d}X}f(X)$ & Derivative of $f(X)$, $X\in \mathbb{R}^{m\times n}$, $f(X)\in \mathbb{R}$ & $n\times m$\\
$P_S(.)$ & Projection on a set $S\subset\mathbb{R}^m$ & $m\times 1$\\
\hline
\end{tabular}}
\label{tab:notation}
\end{table}
\subsection{Convergence Conditions}
In \cite{Xiao04} the following set of conditions is proven to be necessary and sufficient to guarantee convergence to consensus for any initial condition:
\begin{eqnarray}
& & \mathbf{1}^T W = \mathbf{1}^T, \label{e:stoch}\\
& & W \mathbf{1}=\mathbf{1},\label{e:stoch2}\\
& & \rho(W - \frac{1}{n}\mathbf{1}\mathbf{1}^T)<1, \label{e:spectral}
\end{eqnarray}
where $\mathbf{1}$ is the vector of all ones. We observe that the weights are not required to be non-negative.
Since we consider $W$ to be symmetric in this report, then the first two conditions are equivalent to each other and equivalent to the possibility to write the weight matrix as follows: $W=I-\mathcal{I}\times \text{diag}(\mathbf{w})\times \mathcal{I}^T$, where $I$ is the identity matrix and $\mathbf{w}\in \mathbb{R}^m$ is the vector of all the weights on links $w_l$, $l=1...m$.
\subsection{Fastest Consensus}
\label{s:fastest}
The speed of convergence of the system given in \eqref{Mwsum} is governed by how fast $W^k$ converges. Since $W$ is real symmetric, it has real eigenvalues and it is diagonalizable. We can write $W^k$ as follows (\cite{Meyer:2000}):
\begin{equation}\label{Wprojdec}
W^k=\sum _i\lambda _i^k G_i,
\end{equation}
where the matrices $G_i$'s have the following properties: $G_i$ is the projector onto the null-space of $W-\lambda_i I$ along the range of $W-\lambda_i I$, $\sum_i G_i=I$ and $G_i G_j=0^{n\times n} \ \ \forall i\neq j$.
Conditions~(\ref{e:stoch}) and~(\ref{e:spectral}) imply that $1$ is the largest eigenvalue of $W$ in module and is simple. Then $\lambda_1=1$, $G_1=1/n \mathbf{1}\mathbf{1}^T$ and $|\lambda_i|<1$ for $i>1$.
From the above representation of $W^k$, we can deduce two important facts:
\begin{enumerate}
\item First we can check that $W^k$ actually converges, in fact we have $\lim _{k\rightarrow \infty}\mathbf{x}(k)=$ $\lim_{k\rightarrow \infty} W^k \mathbf{x}(0)=\frac{1}{n}\mathbf{1}\mathbf{1}^T x(0)= x_{ave}\mathbf{1}$ as expected.
\item Second, the speed of convergence of $W^k$ is governed by the second largest eigenvalue in module, i.e.~on $\mu = \max \{\lambda _2,-\lambda _n\}=\rho\left(W-G_1 \right)$. For obtaining the fastest convergence, nodes have to select weights that minimizes $\mu$, or equivalently maximize the spectral gap\footnote{
The spectral gap is the difference between the largest eigenvalue in module and the second largest one in module. In this case it is equal to $1-\mu$.
} of $W$.
\end{enumerate}
Then the problem of finding the weight matrix that guarantees the fastest convergence can be formalized as follows:
\begin{equation}\label{minim}
\begin{aligned}
& \underset{W}{\text{Argmin}}
& & \mu (W)\\
& \text{subject to}
& & W=W^T, \\
&&& W\mathbf{1}=\mathbf{1},\\
&&& W\in \mathcal{C} _G,
\end{aligned}
\end{equation}
where the last constraint on the matrix $W$ derives from the assumption that nodes can only communicate with their neighbors and then necessarily $w_{ij}=0$ if $(i,j) \not \in E$.
Problem~\ref{minim} is called in~\cite{Xiao04} the ``symmetric FDLA problem".
The above minimization problem is a convex one and the function $\mu (W)$ is non-smooth convex function. It is convex since when $W$ is a symmetric matrix, we have $\mu (W)=\rho(W-G_1)=||W-G_1||_2$ which is a composition between an affine function and the matrix L-2 norm, and all matrix norms are convex functions. The function $\mu (W)=\rho(W-G_1)$ is non-smooth since the spectral radius of a matrix is not differentiable at points where the eigenvalues coalesce \cite{Fan:1995}. The process of minimization itself in \eqref{minim} tends to make them coalesce at the solution. Therefore, smooth optimization methods cannot be applied to \eqref{minim}. Moreover, the weight matrix solution of the optimization problem is not unique. For example it can be checked that for the network in Fig.~\ref{conject}, there are infinite weight values that can be assigned to the link $(2,3)$ and solve the optimization problem \eqref{minim}, including $w_{23}=0$. Additionally, this shows that adding an extra link in a graph (e.g.~link $(2,3)$ in the Fig.~\ref{conject}), does not necessarily reduce the second largest eigenvalue of the optimal weight matrix.
\begin{figure}[h]
\begin{center}
\setlength{\unitlength}{0.6cm}
\begin{picture}(6,5)
\thicklines
\put(1,2){\line(2,1){2}}
\put(1,2){\line(2,-1){2}}
\put(0.7,2.1){$1$}
\multiput(3,1.1)(0,0.4){5}
{\line(0,1){0.2}}
\put(3,1){\line(1,0){2}}
\put(3,0.4){$2$}
\put(3,3){\line(1,0){2}}
\put(3,3.1){$3$}
\put(5,1){\line(0,1){2}}
\put(5,0.4){$4$}
\put(5,3.1){$5$}
\put(7,2){\line(-2,1){2}}
\put(7,2){\line(-2,-1){2}}
\put(7,2.1){$6$}
\end{picture}
\caption{Network of 6 nodes.}
\label{conject}
\end{center}
\end{figure}
We address in this report a novel approach for the weight selection problem in the average consensus protocol by allowing nodes to optimize a global objective in a totally distributed way. The problem \eqref{minim} is in practice difficult to implement in a distributed way because of the non-smoothness of the function $\mu$. We present in this report a differentiable approximation of the problem, and we show how the new optimization problem can be implemented in a fully decentralized manner using gradient techniques. We then compare the approximated solution with the optimal one and other distributed weight selection algorithms such as the metropolis or the max degree ones.
\section{Related Work} \label{relatedwork}
Xiao and Boyd in \cite{Xiao04} have shown that the symmetric FDLA problem \eqref{minim} can be formulated as a Semi-Definite Program (SDP) that can be solved by a centralized unit using interior point methods.
The limit of such centralized approach to weight selection is shown by the fact that a popular solver as \texttt{CVX}, matlab software for disciplined convex programming~\cite{cvx:2011}, can only find the solution of~\eqref{minim} for networks with at most tens of thousands of links. The optimal solution in larger networks can be found iteratively using a centralized subgradient method. A possible approach to distribute the subgradient method and let each node compute its own weights is also proposed in~\cite{Xiao04}, but it requires at each time slot an iterative sub-procedure to calculate an approximation of some eigenvalues and eigenvectors of the matrix $W$ (global information not local to nodes in a network).
Kim \emph{et al.~} in \cite{Kim:2009} approximate the general FDLA using the $q$th-order spectral norm (2-norm) minimization ($q$-SNM). They showed that if a symmetric weight matrix is considered, then the solution of the $q$-SNM is equivalent to that of the symmetric FDLA problem. Their algorithm's complexity is even more expensive than the SDP. Therefore, solving the problem \eqref{minim} in a distributed way is still an open problem.
Some heuristics for the weight selection problem that guarantee convergence of the average protocol and attract some interest in the literature either due to their distributed nature or to the easy implementation are the following (see \cite{Xiao05distributedaverage, Xiao04}):
\begin{itemize}
\item max degree weights (MD): \\$w_l=\frac{1}{\Delta +1} \ \ \forall l=1...m$.
\item local degree (metropolis) weights (LD): \\$w_l = \frac{1}{\text{max}\{d_i,d_j\}+1} \ l\sim (i,j) \ \ \forall l=1,2,\dots m$.
\item optimal constant weights (OC): \\$w_l=\frac{2}{\lambda _1(L)+\lambda _{n-1}(L)} \ \ \forall l=1...m.$
\end{itemize}
where $\Delta=\max _i\{d_i\}$ is the maximum degree in the network and $L$ is the Laplacian of the graph. The weight matrix can be then deduced from $\mathbf{w}$: \[W=I-\mathcal{I}\times \text{diag}(\mathbf{w})\times \mathcal{I}^T.\]
\begin{comment}
\section{Existing Algorithms for Fast Consensus}\label{OCI}
The function to minimize in the optimization~\eqref{minim} is a non-smooth convex function, so distributed implementation techniques are challenging. In this section, we will present two approaches proposed by Xiao and Boyd in~\cite{Xiao04} to solve the optimization problem, a centralized approach and a distributed one and a similar approach proposed in~\cite{Boyd06}.
\subsection{Centralized Solution}
Xiao and Boyd in \cite{Xiao04} have shown that problem \eqref{minim} can be formulated as a Semi-Definite Program (SDP) that can be solved using interior point methods running in polynomial time (see \cite{Nesterov:1994}). The semi-definite program is the following:
\begin{equation} \label{SDP}
\begin{aligned}
& \text{minimize}
& & s\\
& \text{subject to}
& & -sI\preceq I-\mathcal{I}\times \text {diag}(\mathbf{w})\times \mathcal{I}^T-G_1\\
& & & I-\mathcal{I}\times \text {diag}(\mathbf{w})\times \mathcal{I}^T-G_1\preceq sI, \\
\end{aligned}
\end{equation}
where $s$ is an auxiliary real variable and $A\preceq B$ if and only if $B-A$ is positive semi-definite, and $G_1$ is the projection matrix as in \ref{Wprojdec}.
The output of this program is the optimal weight vector $\mathbf{w}$ such that $w_l , l=1...m$ is the weight selected for link $l$. The limit of such centralized approach to weight selection is also shown by the fact that a popular solver as \texttt{CVX}, matlab software for disciplined convex programming~\cite{cvx:2011}, can only find the solution of~\eqref{SDP} for networks with at most tens of thousands of links.
Moreover any topology change will require to be notified to the central unity that should solve again the problem.
\subsection{Distributed Solution} \label{BoydGradient}
Contrary to the centralized approach, in a distributed solution all the nodes in the network contribute to calculate the solution of the optimization problem. The whole network then benefits from nodes' processing capabilities.
The authors of \cite{Xiao04} present a sub-gradient method for selecting weights on links in a network by minimizing the unconstrained problem:
\[ \text{minimize } r(\mathbf{w})=\rho (I-\mathcal{I}\times \text {diag}(\mathbf{w})\times \mathcal{I}^T-G_1).\]
Each link weight is updated according to the following sub-gradient iteration:
\begin{equation}\label{gradientiter}
w_l^{(k+1)}=w_l^{(k)}-\gamma ^{(k)}g_l^{(k)}/||\mathbf{g}^{(k)}||,
\end{equation}
where $w_l^{(k)}$ is the weight on link $l$ at iteration $k$, $g_l^{(k)}$ is the \mbox{$l$-th} component of a subgradient $\mathbf{g}^{(k)}$ of the function to minimize calculated in $\mathbf{w}^{(k)}$ and $\gamma ^{(k)}$ is the stepsize satisfying the following sufficient conditions for convergence, $ \lim _{k\rightarrow \infty}\gamma ^{(k)}=0 \text{ and } \sum _{k=1}^\infty \gamma ^{(k)}=\infty$.
The components of the sub-gradient can be calculated as follows:
\begin{itemize}
\item if $r(\mathbf{w})=\lambda _2(W)$, then
\[ g_l=-(u_i-u_j)^2 , \textrm{ if } l\sim(i,j), \ l=1,...,m\]
where $u_i$ is the $i$-th component of a unit eigenvector of the weight matrix $W(k)$ relative to the eigenvalue $\lambda_2$.
\item if $r(\mathbf{w}) =-\lambda _n(W)$, then
\[ g_l=(u_i-u_j)^2 , \textrm{ if } l\sim(i,j), \ l=1,...,m\]
where $u_i$ is the $i$-th component of a unit eigenvector of the weight matrix $W(k)$ relative to the eigenvalue $\lambda_n$.
\end{itemize}
We observe that the stepsize used in~\eqref{gradientiter}
is normalized by $||\mathbf{g}^{(k)}||$ which cannot be locally computed by each node.
While this problem can probably be circumvented by a different choice of the stepsize (without loosing the convergence properties of~\eqref{gradientiter}
), there are other aspects that make problematic this distributed implementation.
In fact this iterative procedure requires at every step to calculate $\lambda_2(W(k))$ and $\lambda_n(W(k))$, and determine one eigenvector for the one of these two eigenvalues that is the largest in module.
For the solution to be really distributed also these quantities have to be calculated in a distributed way. This is not an easy task. There are some distributed iterative techniques (\cite{Kempe:2004,Franceschelli:2009,Yang:2008}) that converge asymptotically to the correct eigenvalue-eigenvector pair. Then each step of the optimization procedure requires itself the convergence of an iterative sub-procedure to calculate the two eigenvalue and the eigenvector with significant computation and communication costs. We remark in particular that at each step the sub-procedure has to run long enough to guarantee that the estimations are accurate enough to not jeopardize the convergence of the optimization procedure. Deciding when to terminate the sub-procedure at each step may require itself another distributed mechanisms or the use of worst-case bounds on the errors.
A similar optimization problem but with some additional constraints is to find the fastest converging algorithm for randomized gossiping, and it has been studied in~\cite{Boyd06}. The authors provide a subgradient method that projects the variables violating the constraints back onto the feasible set. The projection can be done in a distributed way and the stepsize sequence can be calculated at each node. Nevertheless, the gradient of the cost function depends also in this case on eigenvalues and eigenvectors of the underlying graph, so its calculation incurs the same problems exposed above.
The main contribution of this report is to consider a different cost function that approximates that in the original \mbox{problem (\ref{minim})} (i.e.~the module of the second largest eigenvalue) and to propose a distributed gradient method to obtain the optimal solution. The main difference in comparison to the approach described in this section---beside the obvious difference to deal with differentiable function rather than with non-smooth one--- is that each component of the gradient can be expressed in a closed form and calculated by each node only on the basis of some local information, without the need to perform an iterative sub-procedure at each step.
\end{comment}
\section{Schatten Norm Minimization} \label{TM}
We change the original minimization problem in~\eqref{minim} by considering a different cost function that is a monotonic function of the Schatten Norm.
The minimization problem we propose is the following one:
\begin{equation}\label{tracemin}
\begin{aligned}
& \underset{W}{\text{Argmin}}
& & f(W)=||W||_{\sigma p}^p \\
& \text{subject to}
& & W=W^T, \\
&&& W\mathbf{1}=\mathbf{1},\\
&&& W\in \mathcal{C} _G,
\end{aligned}
\end{equation}
where $p$ is an even positive integer. The following result establishes that~\eqref{tracemin} is a smooth convex optimization problem and also it provides an alternative expression of the cost function in terms of the trace of $W^p$. For this reason we refer to our problem also as \emph{Trace Minimization} (TM).
\begin{prop}\label{prop1}
$f(W)=||W||_{\sigma p}^p=\textrm{Tr}(W^p)$ is a scalar-valued smooth convex function on its feasible domain when $p$ is an even positive integer.
\end{prop}
\begin{proof}
We have $\textrm{Tr}(W^p)=\sum _{i=1}^n\lambda _i^p$. Since $W$ is symmetric, its non-zero singular values are the absolute values of its non-zero eigenvalues~(\cite{Meyer:2000}). Given that $p$ is even, then $\sum_{i=1}^{n}\lambda _i^p=\sum_{i=1}^n \sigma _i^p$. Therefore, $\textrm{Tr}(W^p)=||W||_{\sigma p}^p$.
The Schatten norm $||W||_{\sigma p}$ is a nonnegative convex function, then $f$ is convex because it is the composition of a non-decreasing convex function---function $x^p$ where $x$ is non-negative---and a convex function (see \cite{BoydBook:2004}).
The function is also differentiable and we have
\begin{equation}\label{derivativef}
\frac{\text{d}\hfill}{\text{d}W}\textrm{Tr}(W^p)=pW^{p-1},
\end{equation}
(see \cite[p.~411]{Bernstein:2009}).
\end{proof}
We now illustrate the relation between \eqref{tracemin} and the optimization \eqref{minim}. The following lemmas will prepare the result:
\begin{lemma}
\label{l:asympt}
For any symmetric weight matrix $W$ whose rows (and columns) sum to $1$ and with eigevalues $\lambda_1(W)\ge \lambda_2(W)\ge \dots \ge \lambda_n(W)$, there exist two integers $K_1\in \{1,2,\dots n-1\},K_2 \in \{0,1,2,\dots n-1\}$ and a positive constant $\alpha<1$ such that for any positive integers $p$ and $q$ where $p=2q$ we have:
\begin{equation}
\label{e:asympt}
1+ \tau(W)^p K_1\leq \textrm{Tr}(W^p) \leq 1+\tau(W)^p (K_1+K_2\alpha^p),
\end{equation}
where
\begin{equation}
\tau(W) =
\begin{cases}
\rho(W)=\max \{\lambda_1(W),-\lambda_n(W)\} & \text{if } \rho(W) > 1,
\\
\mu(W)=\max\{\lambda_2(W),-\lambda_n(W)\} & \text{if } \rho(W) \leq 1.
\end{cases}
\end{equation}
\end{lemma}
\begin{proof}
Let us consider the matrix $W^2$ and denote by $\nu_1,\nu_2,\dots \nu_r$ its distinct eigenvalues ordered by the largest to the smallest and by $m_1,m_2, \dots m_r$ their respective multiplicities. We observe that they are all non-negative and then they are also different in module. For convenience we consider $\nu_{s}=m_{s}=0$ for $s>r$.
We can then write:
$$\textrm{Tr}(W^{p})=\sum_{i=1}^n \lambda_i^p=\sum_{i=1}^r m_i \nu_i^q.$$
The matrix $W^2$ has $1$ as an eigenvalue. Let us denote by $j$ its position in the ordered sequence of distinct eigenvalues, i.e.~$\nu_j=1$. Then it holds:
$$\textrm{Tr}(W^p)=1+(m_j-1)+ \sum_{i \neq j} m_i \nu_i^q. $$
If $\rho(W)=1$ (i.e.~$1$ is the largest eigenvalue in module of $W$), then $1$ is also the largest eigenvalue of $W^2$ ($\nu_1=1$). If $m_1>1$, then it has to be either $\lambda_2(W)=1$ (the multiplicity of the eigenvalue $1$ for $W$ is larger than $1$) or $\lambda_n(W)=-1$. In both cases $\tau(W)=\mu(W)=1$,
$$\textrm{Tr}(W^p)=1+(m_1-1)+ \sum_{i > 1} m_i \nu_i^q $$
and the result holds with $K_1=m_1-1$, $K_2=\sum_{i>1} m_i$ and $\alpha=\sqrt{\nu_2}$. If $m_1=1$, then $\nu_2=\lambda_2^2$. We can write:
$$\textrm{Tr}(W^p)=1+\nu_2^q \left(m_2+\sum_{i>2} m_i \left(\frac{\nu_i}{\nu_2}\right)^q \right)$$
and the result holds with $K_1=m_2$, $K_2=\sum_{i>2} m_i$, and $\alpha=\sqrt{\nu_3/\nu_2}$.
If $\rho(W) > 1$, then $\nu_1=\rho(W)^2>1$ and we can write:
$$\textrm{Tr}(W^p)=1+\nu_1^q \left(m_1+\sum_{\underset{i\neq j}{i>1}} m_i (\frac{\nu_i}{\nu_1})^q + (m_j-1) (\frac{1}{\nu_1})^q \right). $$
Then the result holds with $\tau(W)=\sqrt{\nu_1}=\rho(W)$, $K_1=m_1$, $K_2=\sum_{i>1} m_i$, and $\alpha= \sqrt{\nu_2/\nu_1}$.
\end{proof}
\begin{lemma}
\label{l:tau}
Let us denote by $W_{(p)}$ the solution of the minimization problem \eqref{tracemin}. If the graph of the network is strongly connected then $\tau\left(W_{(p)}\right) < 1$ for $p$ sufficiently large.
\end{lemma}
\begin{proof}
If the graph is strongly connected then there are multiple ways to assign the weights such that the convergence conditions~\eqref{e:stoch}-\eqref{e:spectral} are satisfied.
In particular the local degree method described in~Sec.~\ref{relatedwork} is one of them. Let us denote by $W_{(LD)}$ its weight matrix. A consequence of the convergence conditions is that $1$ is a simple eigenvalue of $W_{(LD)}$, and that all other eigenvalues are strictly less than one
in magnitude (see~\cite{Xiao04}). It follows that $\tau\left(W_{(LD)}\right)$ in Lemma~\ref{l:asympt} is strictly smaller than one and that $\lim_{p \to \infty} \textrm{Tr}\left(W_{(LD)}^p\right)=1$. Then there exists a value $p_0$ such that for each $p>p_0$
$$\textrm{Tr}\left(W_{(LD)}^p\right)<2.$$
Let us consider the minimization problem~\eqref{tracemin} for a value $p>p_0$. $W_{(LD)}$ is a feasible solution for the problem, then
$$\textrm{Tr}(W_{(p)}^p)\le \textrm{Tr}(W_{(LD)}^p)<2.$$
Using this inequality and Lemma \ref{l:asympt}, we have:
$$1+\tau\left(W_{(p)}\right)^p \le 1+\tau\left(W_{(p)}\right)^p K_1\leq \textrm{Tr}(W_{(p)}^p) < 2,$$
from which the thesis follows immediately.
\end{proof}
We are now ready to state our main results by the following two propositions:
\begin{prop}\label{problem_equivalence}
If the graph of the network is strongly connected, then the solution of the Schatten Norm minimization problem~\eqref{tracemin} satisfies the consensus protocol convergence conditions for $p$ sufficiently large. Moreover as $p$ approaches $\infty$, this minimization problem is equivalent to the minimization problem~\eqref{minim} (i.e.~to minimize the second largest eigenvalue $\mu(W)$).
\end{prop}
\begin{proof}
The solution of problem~\eqref{tracemin}, $W_{(p)}$ is necessarily symmetric and its rows sum to $1$.
From Lemma~\ref{l:tau} it follows that for $p$ sufficiently large $\tau\left(W_{(p)}\right)<1$ then by the definition of $\tau(.)$ it has to be $\rho(W_{(p)})=1$ and $\mu(W_{(p)})<1$. Therefore $W_{(p)}$ satisfies all the three convergence conditions~\eqref{e:stoch}-\eqref{e:spectral} and then the consensus protocol converges.
Now we observe that with respect to the variable weight matrix $W$, minimizing $\textrm{Tr}(W^p)$ is equivalent to minimizing $(\textrm{Tr}(W^p)-1)^{1/p}$. From Eq.~\eqref{e:asympt}, it follows:
$$\tau(W)K_1^{\frac{1}{p}}\le (\textrm{Tr}(W^p)-1)^{\frac{1}{p}}\le \tau(W)(K_1+K_2\alpha^p)^\frac{1}{p}.$$
$K_1$ is bounded between $1$ and $n-1$ and $K_2$ is bounded between $0$ and $n-1$, and $\alpha<1$,then it holds:
$$ \tau(W)K_1^{\frac{1}{p}}\le (\textrm{Tr}(W^p)-1)^{\frac{1}{p}}\le \tau(W) K^\frac{1}{p},$$
with $K=2(n-1)$.
For $p$ large enough $\tau\left(W_{(p)}\right)=\mu(W_{(p)})$, then
$$ \left|(\textrm{Tr}(W_{(p)}^p)-1)^{\frac{1}{p}} - \mu(W_{(p)}) \right| \le \mu(W_{(p)}) \left( K^\frac{1}{p} -1\right)\le K^\frac{1}{p} -1.$$
Then the difference of the two cost functions converges to zero as $p$ approaches infinity.
\end{proof}
\begin{prop}
The Schatten Norm minimization~\eqref{tracemin} is an approximation for the original problem~\eqref{minim} with a guaranteed error bound,
$$|\mu (W_{(SDP)})-\mu (W_{(p)})|\leq \mu (W_{(SDP)})\times \epsilon (p),$$
where $\epsilon (p)=(n-1)^{1/p}-1$ and where $W_{(SDP)}$ and $W_{(p)}$ are the solutions of \eqref{minim} and \eqref{tracemin} respectively.
\end{prop}
\begin{proof}
Let $S$ be the feasibility set of the problem \eqref{minim} (and \eqref{tracemin}), we have $\mu (W)=\max \{\lambda _2(W), -\lambda _n(W)\}$ and let $g(W)=\left(\sum _{i\geq 2}\lambda _i^p(W)\right)^{\frac{1}{p}}$ . Since $W_{(SDP)}$ is a solution of \eqref{minim}, then
\begin{equation}\label{eq1}
\mu (W_{(SDP)})\leq \mu (W), \ \ \forall W\in S.
\end{equation}
Note that the minimization of $g(W)$ is equivalent to the minimization of $Tr(W^p)$ when $W\in S$ (i.e. $\underset{W\in S}{\text{Argmin}} \ g(W)=\underset{W\in S}{\text{Argmin}} \ Tr(W^p)$), then
\begin{equation}\label{eq2}
g(W_{(p)})\leq g(W), \ \ \forall W\in S.
\end{equation}
Finally for a vector $\mathbf{v} \in \mathbb{R}^m$ all norms are equivalent and in particular $||\mathbf{v}||_\infty \leq ||\mathbf{v}||_p\leq m^{1/p}||\mathbf{v}||_\infty$ for all $p\geq 1$. By applying this inequality to the vector whose elements are the $n-1$ eigenvalues different from $1$ of the matrix $W$, we can write
\begin{equation}\label{eq3}
\mu (W)\leq g(W)\leq (n-1)^{1/p}\mu (W), \ \ \forall W\in S.
\end{equation}
Using these three inequalities we can derive the desired bound:
\begin{align}
\mu (W_{(SDP)})\overset{\eqref{eq1}}{\leq} \mu (W_{(p)}) \overset{\eqref{eq3}}{\leq} g(W_{(p)}) \overset{\eqref{eq2}}{\leq} g(W_{(SDP)}) \nonumber\\
\leq (n-1)^{1/p}\mu (W_{(SDP)}),
\end{align}
where the number above the inequalities shows the equation used in deriving the bound.
Therefore $\mu (W_{(SDP)})\leq \mu (W_{(p)}) \leq (n-1)^{1/p}\mu (W_{(SDP)})$ and the proposition directly follows.
\end{proof}
{\bf Remark:} Comparing the results of Schatten Norm minimization \eqref{tracemin} with the original problem \eqref{minim}, we observe that on some graphs the solution of problem~\eqref{tracemin} already for $p=2$ gives the optimal solution of the main problem \eqref{minim}; this is for example the case for complete graphs\footnote{
This can be easily checked.
In fact, for any the matrix that guarantees the convergence of average consensus protocols it holds $\mu(W) \ge 0$ and $\textrm{Tr}(W^2)\ge 1$ (because $1$ is an eigenvalue of $W$).
The matrix $\hat W=1/n \mathbf{1} \mathbf{1}^T$ (corresponding to each link having the same weight $1/n$) has eigenvalues $1$ and $0$ with multiplicity $1$ and $n-1$ respectively. Then $\mu(\hat W)=0$ and $\textrm{Tr}(\hat W^2)=1$. It follows that $\hat W$ minimizes both the cost function of problem~\eqref{minim} and~\eqref{tracemin}.
}. However, on some other graphs, it may give a weight matrix that does not guarantee the convergence of the consensus protocol because the second largest eigenvalue is larger than or equal to $1$ (the other convergence conditions are intrinsically satisfied). We have built a toy example, shown in Fig.~\ref{ToyN}, where this happens. The solution of \eqref{tracemin} assigns weight $0$ to the link $(i,j)$; $w_{ij}=0$ separates the network into two disconnected subgraphs, so $\mu (W)=1$ in this case. We know by Lemma~\ref{l:tau} that this problem cannot occur for $p$ large enough. In particular for the toy example the matrix solution for $p=4$ already guarantees convergence. We discuss how to guarantee convergence for any value of $p$ in Section \ref{EXT}.
\begin{figure}
\begin{center}
\includegraphics[scale=0.35]{figures/graph.eps}
\caption{For this network the matrix solution of Schatten Norm minimization~\eqref{tracemin} with $p=2$ does not guarantee convergence of average consensus to the true average because $w_{ij}=0$ which separates the network into two parts, each of which can converge to a totally different value (but not to the average of initial values).}
\label{ToyN}
\end{center}
\end{figure}
Given that problem~\eqref{tracemin} is smooth and convex, it can be solved by interior point methods which would be a centralized solution. In the next section we are going to show a distributed algorithm to solve problem~\eqref{tracemin}.
\section{A Distributed Algorithm for Schatten Norm minimization}\label{COTM}
\input{distributedalgTM.tex}
\input{localgradient.tex}
\input{stepsize.tex}
\subsection{Complexity of the Algorithm}
Our distributed algorithm for Schatten Norm minimization requires to calculate at every iteration, the stepsize $\gamma ^{(k)}$, the gradient $g_l^{(k)}$ for every link, and a projection on the feasible set $X$. Its complexity is determined by the calculation of link gradient $g_l$, while the cost of the other operations is negligible.
In what follows, we detail the computational costs (in terms of number of operations and memory requirements) and communication costs (in terms of volume of information to transmit) incurred by each node for the optimization with the two values $p=2$ and $p=4$.
\subsubsection{Complexity for $p=2$}
For $p=2$, $g_l=2\times (2W_{ij}-W_{ii}-W_{jj})$, so taking into consideration that nodes are aware of their own weights ($W_{ii}$) and of the weights of the links they are incident to ($W_{ij}$), the only missing parameter in the equation is their neighbors self weight ($W_{jj}$). So at every iteration of the subgradient method, nodes must broadcast their self weight to their neighbors. We can say that the computational complexity for $p=2$ is negligible and the communication complexity is $1$ message carrying a single real value ($w_{ii}$) per link, per node and per iteration.
\subsubsection{Complexity for $p=4$}
For $p=4$, the node must collect information from a larger neighborhood. The gradient at link $l\sim (i,j)$ is given by $g_l=4\big ( (W^3)_{ij}+(W^3)_{ji}-(W^3)_{ii}-(W^3)_{jj}\big )$.
From the equation of $g_l$ it seems like the node must be aware of all the weight matrix in order to calculate the 4 terms in the equation, however this is not true. As discussed in the previous section, each of the 4 terms can be calculated only locally from the weights within 2-hops from $i$ or $j$. In fact, $(W^3)_{ij}$ depends only on the weights of links covered by a walk with 3 jumps: Starting from $i$ the first jump reaches a neighbor of $i$, the second one a neighbor of $j$ and finally the third jump finishes at $j$, then we cannot move farther than 2 hops from $i$. Then this term can be calculated at node $i$ as follows: Every node $s$ in $N_i$, sends its weight vector $\mathbf{W}_s$ to $i$ ($\mathbf{W}_s$ is a vector that contains all weights selected by node $s$ to its neighbors). The same is true for the addend $(W^3)_{ji}$. The term $(W^3)_{ii}$ depends on the walks of length 3 starting and finishing in $i$, then node $i$ can calculate it once it knows $\mathbf{W}_s$ for each $s$ in $N_i$. Finally, the calculation of the term $(W^3)_{jj}$ at node $i$ requires $i$ to know more information about the links existing among the neighbors of node $j$. Instead of the transmission of this detailed information, we observe that node $j$ can calculate the value $(W^3)_{jj}$ (as node $i$ can calculate $(W^3)_{ii}$) and then can transmit directly the result of the calculation to node $i$. Therefore, the calculation of $g_l$ by node $i$ for every link $l$ incident to $i$ can be done in three steps:
\begin{enumerate}
\item Create the subgraph $H_i$ containing the neighbors of $i$ and the neighbors of its neighbors by sending ($\mathbf{W}_i$) and receiving the weight vectors ($\mathbf{W}_s$) from every neighbor $s$.
\item Calculate $(W^3)_{ii}$ and broadcast it to the neighbors (and receive $(W^3)_{ss}$ from every neighbor $s$).
\item Calculate $g_l$.
\end{enumerate}
We evaluate now both the computational and the communication complexity.
\begin{itemize}
\item Computation Complexity: Each node $i$ must store the subgraph $H_i$ of its neighborhood. The number of nodes of $H_i$ is $n_H\leq \Delta ^2 +1$, the number of links of $H_i$ is $m_H\leq \Delta ^2$ where $\Delta$ is the maximum degree in the network. Due to sparsity of matrix $W$, the calculation of the value $(W^3)_{ii}$ requires $O(\Delta ^3)$ multiplication operation without the use of any accelerating technique in matrix multiplication which ---we believe--- could further reduce the cost. So the total cost for calculating $g_l$ is in the worst case $O(\Delta ^3)$. Notice that the complexity for solving the SDP for \eqref{minim} is of order $O(m^3)$ where $m$ is the number of links in the network. Therefore, on networks where $\Delta << m$, the gradient method would be computationally more efficient.
\item Communication Complexity: Two packets are transmitted by each node on each link at steps $1$ and $2$. So the complexity would be two messages per link per node and per iteration. The first message carries at most $\Delta$ values (the weight vector $\mathbf{W}_i$) and the second message carries one real value ($(W^3)_{ii}$).
\end{itemize}
\section{Performance Evaluation}\label{PE}
In this section we evaluate the speed of convergence of consensus protocols when the weight matrix $W$ is selected according to our algorithm. As we have discussed in Section~\ref{s:fastest}, this speed is asymptotically determined by the second largest eigenvalue in module ($\mu(W)$), that will be one of two performance metrics considered here. The other will be defined later. The simulations are done on random graphs (Erd\"os-Renyi (ER) graphs and Random Geometric Graphs (RGG)) and on two real networks (the Enron company internal email exchange network \cite{enron} and the dolphin social network \cite{dolphin}). The random graphs are generated as following :
\begin{itemize}
\item For the ER random graphs, we start from $n$ nodes fully connected graph, and then every link is removed from the graph by a probability $1-Pr$ and is left there with a probability $Pr$. We have tested the performance for different probabilities $Pr$.
\item For the RGG random graphs, $n$ nodes are thrown uniformly at random on a unit square area, and any two nodes within a connectivity radius $r$ are connected by a link. We have tested the performance for different connectivity radii. It is known that for a small connectivity radius, the nodes tend to form clusters.
\end{itemize}
The real networks are described as following:
\begin{itemize}
\item The Enron company has 151 employees where an edge in the graph refers to an exchange of emails between two employees (only internal emails within the company are considered where at least 3 emails are exchanged between two nodes in this graph).
\item The dolphin social network is an undirected social network of frequent associations between 62 dolphins in a community living off Doubtful Sound, New Zealand.
\end{itemize}
\subsection{Comparison with the optimal solution}
We first compare $\mu\left(W_{(p)}\right)$ for the solution $W_{(p)}$ of the Schatten p-norm (or Trace) minimization problem~\eqref{tracemin} with its minimum value obtained solving the symmetric FDLA problem~\eqref{minim}. To this purpose we used the \texttt{CVX} solver (see section~\ref{relatedwork}). This allows us also to evaluate how well problem~\eqref{tracemin} approximates problem~\eqref{minim} for finite values of the parameter $p$.
The results in Fig.~\ref{ERfdla} have been averaged over $100$ random graphs with $20$ nodes generated according to the Erdos-Renyi (ER) model, where each link is included with probability $Pr \in \{0.2,0.3,0.4,0.5\}$. We see from the results that as we solve the trace minimization for larger $p$, the asymptotic convergence speed of our approach converges to the optimal one as proven in Proposition \ref{problem_equivalence}.
\begin{figure}
\begin{center}
\includegraphics[scale=0.35]{figures/ERfdlaNEW.eps}
\caption{Performance comparison between the optimal solution of the FDLA problem (labeled FDLA) and the approximated solutions obtained solving the Schatten Norm minimization for different values of $p$ (labeled TM).}
\label{ERfdla}
\end{center}
\end{figure}
\subsection{Other distributed approaches: Asymptotic Convergence Rate}
We compare now our algorithm for $p=2$ and $p=4$ with other distributed weight selection approaches described in section~\ref{relatedwork}.
Fig.~\ref{ERtrace4} shows the results on connected Erd\"os-Renyi (ER) graphs and Random Geometric Graphs (RGG) with $100$ nodes for different values respectively of the probability $Pr$ and of the connectivity radius $r$.
We provide 95\% confidence intervals by averaging each metric over $100$ different samples.
\begin{figure}
\begin{center}
\includegraphics[scale=0.35]{figures/ERtrace4.eps}
\includegraphics[scale=0.35]{figures/RGGtrace4.eps}
\caption{Performance comparison between Schatten Norm minimization (TM) for $p=2$ and $p=4$ with other weight selection algorithms on ER and RGG graphs.}
\label{RGGtrace4}
\label{ERtrace4}
\end{center}
\end{figure}
We see in Fig.~\ref{RGGtrace4} that TM for $p=2$ and $p=4$ outperforms other weight selection algorithms on ER by giving lower $\mu$. Similarly on RGG the TM algorithm reaches faster convergence than the other known algorithms even when the graph is well connected (large connectivity radius). However, the larger the degrees of nodes, the higher the complexity of our algorithm. Interestingly even performing trace minimization for the smallest value $p=2$ nodes are able to achieve faster speed of convergence than a centralized solution like the OC algorithm.
Apart from random networks, we performed simulations on two real world networks: the Enron company internal email exchange network \cite{enron} and the dolphin social network \cite{dolphin}.
The table below compares the second largest eigenvalue $\mu$ for the different weight selection algorithms on these networks:
\begin{table}[h]
\centering
\begin{tabular}{|l|c|c|c|c|c|}
\hline
~ & MD & OC & LD & TM p=2 & TM p=4 \\ \hline
Enron $\mu$ & 0.9880 & 0.9764 & 0.9862 & 0.9576 & 0.9246 \\ \hline
Dolphin $\mu$ & 0.9867 & 0.9749 & 0.9796 & 0.9751 & 0.9712 \\
\hline
\end{tabular}
\end{table}
The results show that for Enron network, our totally distributed proposed algorithm TM for p=4 has the best performance ($\mu=0.9246$) among the studied weight selection algorithms followed by TM for p=2 ($\mu=0.9576$) because they have the smallest $\mu$. On the Dolphin's network, again TM for p=4 had the smallest $\mu$ ($\mu =0.9712$) but OC had the second best performance ($\mu =0.9749$) where TM for p=2 ($\mu =0.9751$) was close to the OC performance.
\subsection{Communication Overhead for Local Algorithms }
Until now we evaluated only the asymptotic speed of convergence, independent from the initial values $x_i(0)$, by considering the second largest eigenvalue $\mu(W)$.
We want to study now the transient performance.
For this reason, we consider a random initial distribution of nodes' values and we define the convergence time to be the number of iterations needed for the error (the distance between the estimates and the actual average) to become smaller than a given threshold. More precisely, we define the normalized error $e(k)$ as
\begin{equation}\label{consensuserror}
e(k)=\frac{{||\mathbf{x}(k)-\bar{\mathbf{x}}||}_2}{{||\mathbf{x}(0)-\bar{\mathbf{x}}||}_2},
\end{equation}
where $\bar{\mathbf{x}}=x_{ave}\mathbf{1}$, and the convergence time is the minimum number of iterations after which $e(k)<0.001$ (note that $e(k)$ is non increasing).
As the Schatten norm minimization problem itself may take a long time to converge, whereas other heuristics can be obtained instantaneously, the complexity of the optimization algorithm can affect the overall procedure. If we consider a fixed network (without changes in the topology), the weight optimization procedure is done before the start of the consensus cycles,\footnote{For example, the cycle of the daily average temperature in a network of wireless environmental monitoring sensors is one day because every day a new averaging consensus algorithm should be run.} and then the same weights are used for further average consensus cycles. Therefore, the more stable the network, the more one is ready to invest for the optimization at the beginning of consensus. The communication overhead of the local algorithms is plotted in Fig.~\ref{Complexity}. For each algorithm we consider the following criteria to define its communication overhead. First we consider the number of messages that should be exchanged in the network for the weight optimization algorithm to converge. For example, in our networking settings (RGG with $100$ nodes and connectivity radius $0.1517$) the initialization complexity of MD algorithm is 30 messages per link because the maximum degree can be obtained by running a maximum consensus algorithm that converges after a number of iterations equal to the diameter (the average diameter for the graphs was 15 hops), while by LD the nodes only need to send their degrees to their neighbors which makes its complexity for establishing weights only 2 messages per link which is the least complexity among other algorithms. The trace minimization algorithm complexity is defined by the number of iterations needed for the gradient method to converge, multiplied by the number of messages needed per iteration as mentioned in the complexity section. In our networking setting, the $TM$ for $p=2$ took on average $66.22$ messages per link to converge while the $TM$ for $p=4$ took $1388.28$ messages.\footnote{The step size $\gamma _k$ is calculated with values $a=10/p$ and $b=100$, and convergence is obtained when $||g||$ drops below the value $0.02$.} Notice that OC depends on global values (eigenvalues of the laplacian of the graph) and is not included here because it is not a local algorithm and cannot be calculated with simple iterative local methods.
In addition to the initialization complexity, we add the communication complexity for the consensus cycles. We consider that the convergence of the consensus is reached when the consensus error of Eq.~\eqref{consensuserror} drops below $0.1\%$. The results of Fig.~\ref{Complexity} show that if the network is used for 1 or 2 cycles the best algorithm is to use $TM$ for $p=2$, followed by $LD$, followed by $MD$, and the worst overhead is for $TM$ for $p=4$. If the network is used between 3 and 5 cycles, then $TM$ where $p=4$ becomes better that $MD$ but still worst than the other two. Further more, the $TM$ where $p=4$ becomes better than $LD$ for the 6th and 7th cycles. And finally, if the network is stable for more than 7 cycles, the $TM$ for $p=4$ becomes the best as the asymptotic study shows.
\begin{figure}
\begin{center}
\includegraphics[scale=0.35]{figures/Complexity.eps}
\caption{Communication overhead of local algorithms.
}
\label{Complexity}
\end{center}
\end{figure}
\subsection{Joint Consensus-Optimization (JCO) Procedure}\label{JCO}
In the following experiments we address also another practical concern. It may seem our approach requires to wait for the convergence of the iterative weight selection algorithm before being able to run the consensus protocol.
This may be unacceptable in some applications specially if the network is dynamic and the weights need to be calculated multiple times. In reality, at each slot the output of the distributed Schatten norm minimization is a new feasible weight matrix, that can be used by the consensus protocol, and (secondarily) should also have faster convergence properties than the one at the previous step. It is then possible to interleave the weight optimization steps and the consensus averaging ones: at a given slot each node will improve its own weight according to \eqref{line4} and use the current weight values to perform the averaging \eqref{wsum}. We refer to this algorithm as the joint consensus--optimization (JCO) procedure. Weights can be initially set according to one of the other existing algorithms like LD or MD. The convergence time of JCO depends also on the choice of the stepsize, that is chosen to be $\gamma ^{(k)}=\frac{1}{p(1+k)}$.
\begin{figure}
\begin{center}
\includegraphics[scale=0.35]{figures/ERtrace4_sim.eps}
\includegraphics[scale=0.35]{figures/RGGtrace4_sim.eps}
\caption{Convergence time of different weight selection algorithms on ER and RGG graphs. TM-JCO-LD $p=4$ is the joint consensus-optimization algorithm initialized with the LD algorithm's weight matrix and the same for TM-JCO-MD $p=4$ but initialized with the MD algorithm's one.
}
\label{RGGtrace4_sim}
\label{ERtrace4_sim}
\end{center}
\end{figure}
The simulations show that our weight selection algorithm outperforms the other algorithms also in this case. In particular, Fig.~\ref{RGGtrace4_sim} shows the convergence time for various weight selection criteria on ER and RGG graphs. For each of the network topology selected, we averaged the data in the simulation over 100 generated graphs, and for each of these graphs we averaged the convergence time of the different algorithms over 20 random initial conditions (the initial conditions were the same for all algorithms). Notice that running at the same time the optimization with consensus gave good results in comparison to LD, MD, and even OC algorithms. We also notice, that the initial selection of the weights does not seem to have an important role for the TM-JCO approach. In fact, despite the LD weight matrix leads itself a significantly faster convergence than the MD weight matrix, the difference between TM-JCO-MD and TM-JCO-LD is minor, suggesting that the weight optimization algorithm moves fast away from the initial condition.
\section{Stability and Misbehaving Nodes}
\label{EXT}
In this section we first explain how the convergence of the consensus protocol can be guaranteed also for ``small" $p$ values (see the remark in section~\ref{TM}) and then we discuss how to deal with some forms of nodes' misbehavior.
\subsection{Guaranteeing Convergence of Trace Minimization}
The conditions \eqref{e:stoch}-\eqref{e:spectral} guarantee that the consensus protocol converges to the correct average independently from the initial estimates.
In this section, for the sake of conciseness, we call a weight matrix that satisfies these set of conditions a \emph{convergent matrix}. A convergent matrix is any matrix that guarantees the convergence of average consensus protocols. We showed in Proposition~\ref{problem_equivalence} that for $p$ large enough, the solution $W_{(p)}$ of \eqref{tracemin} is a convergent matrix. However, for ``small"~$p$ values, it may happen that $\mu(W_{(p)})\ge 1$ (the other conditions are intrinsically satisfied) and then the consensus protocol does not converge for all the possible initial conditions. We observe that if all the link weights and the self weights in $W_{(p)}$ are strictly positive then $W_{(p)}$ is a convergent matrix. In fact from Perron-Frobenius theorem for nonnegative matrices \cite{seneta2006non} it follows that a stochastic weight matrix $W$ for a strongly connected graph where $w_{ij}>0$ if and only if $(i,j)\in E$ satisfies \eqref{e:spectral} (i.e. $\mu (W)< 1$).
Then, the matrix may not be convergent only if one of the weights is negative.
Still in such a case nodes can calculate in a distributed way a convergent weight matrix that is ``close" to the matrix $W_{(p)}$. In this section we show how it is possible and then we discuss a practical approach to guarantee convergence while not sacrificing the speed of convergence of $W_{(p)}$ (when it converges).
We obtain a convergent matrix from $W_{(p)}$ in two steps. First, we project $W_{(p)}$ on a suitable set of matrices that satisfy conditions \eqref{e:stoch} and \eqref{e:spectral}, but not necessarily \eqref{e:stoch2}, then we generate a symmetric convergent matrix from the projection.
Let $\hat{W}=W_{(p)}$ be the matrix to project, the solution of the following projection is guaranteed to satisfy~\eqref{e:stoch} and~\eqref{e:spectral}:
\begin{equation}\label{projTM}
\begin{aligned}
& \underset{W}{\text{Argmin}}
& & ||W-\hat{W}||_F^2\\
& \text{subject to}
& & W\mathbf{1}=\mathbf{1},\\
&&& W\in \mathcal{C} _G^\prime,
\end{aligned}
\end{equation}
where $\mathcal{C} _G^\prime$ is the set of non-negative matrices such that $w_{ij}\ge \delta> 0$ if $(i,j)\in E$, $w_{ij}=0$ if $(i,j) \notin E$, and $||.||_F$ is the Frobenius matrix norm.
The constant $\delta>0$ is a parameter that is required to guarantee that the feasible set is closed.
Now, we show how it is possible to project a matrix $\hat W$ according to \eqref{projTM} in a distributed way.
We observe that this approach is feasible because we do not require the projected matrix to be symmetric (and then satisfy \eqref{e:stoch2}).
The key element for the distributed projection is that the Frobenius norm is separable in terms of the variables $\mathbf{W}_i$ (the $d_i\times 1$ vector of weights selected by node $i$ for its neighbors), so that problem~\eqref{projTM} is equivalent to:
\begin{equation}\label{localprojTM}
\begin{aligned}
& \underset{\mathbf{W}_1,...,\mathbf{W}_n}{\text{Argmin}}
& & \sum _{i=1}^nr(\mathbf{W}_i)\\
& \text{subject to}
& & \mathbf{W}_i^T\mathbf{1}_{d_i}\leq 1 \ \ \forall i,\\
&&& \mathbf{W}_i \ge \delta >0 \ \ \forall i,
\end{aligned}
\end{equation}
where $\mathbf{1}_{d_i}$ is the $d_i\times 1$ vector of all ones,
and $r(\mathbf{W}_i)$ is defined as follows:
\begin{align}
r(\mathbf{W}_i)&=(w_{ii}-\hat{w}_{ii})^2+\sum _{j\in N_i}(w_{ij}-\hat{w}_{ij})^2\\
&=(\mathbf{W}_i-\mathbf{\hat{W}}_i)^T(\mathbf{W}_i-\mathbf{\hat{W}}_i)+\left((\mathbf{W}_i-\mathbf{\hat{W}}_i)^T\mathbf{1}_{d_i}\right)^2\\
&=(\mathbf{W}_i-\mathbf{\hat{W}}_i)^T\left(I_{d_i}+\mathbf{1}_{d_i}\mathbf{1}^T_{d_i}\right)(\mathbf{W}_i-\mathbf{\hat{W}}_i),
\end{align}
where $I_{d_i}$ is $d_i$-identity matrix.
Since the variables in \eqref{localprojTM} are separable in $\mathbf{W}_1, ...,\mathbf{W}_n$, then each node $i$ can find the global solution for its projected vector $\mathbf{W}_i^{(proj)}$ by locally minimizing the function $r(\mathbf{W}_i)$ subject to its constraints.
Once the weight vectors $\mathbf{W}_i^{(proj)}$ are obtained, the projection of $W_{(p)}$ on the set $\mathcal{C} _G^\prime$ is uniquely identified. We denote it $W^{(proj)}$. We can then obtain a convergent weight matrix $W^{(conv)}$ by modifying $W^{(proj)}$ as follows.
For every link $l\sim (i,j)$, we set:
$$w^{(conv)}_l=\min\left\{\left(\mathbf{W}^{(proj)}_i\right)_{\alpha(j)},\left(\mathbf{W}^{(proj)}_j\right)_{\alpha(i)}\right\},$$
where $\alpha(j)$ (similarly $\alpha(i)$) is the index of the node $j$ (similarly $i$) in the corresponding vector. Then we calculate the convergent weight matrix: $$W^{(conv)}=I-\mathcal{I}\times \text{diag}(\mathbf{w^{(conv)}})\times \mathcal{I}^T.$$
While the matrix $W^{(conv)}$ is convergent, its speed of convergence may be slower than the matrix $W_{(p)}$, assuming this converges too. Then the algorithm described above should be ideally limited to the cases where $W_{(p)}$ is known to not be convergent. Unfortunately in many network scenarios this may not be known a priori.
We discuss a possible practical approach in such cases.
Nodes first compute $W_{(p)}$. If all the link-weights and self-weights are positive then the matrix $W_{(p)}$ can be used in the consensus protocol without any risk. If one node has calculated a non-positive weight, then it can invoke the procedure described above to calculate $W^{(conv)}$. Nodes can then run the consensus protocol using only the matrix $W^{(conv)}$ at the price of a slower convergence or they can run the two consensus protocols in parallel averaging the initial values both with $W^{(conv)}$ and $W_{(p)}$. If the estimates obtained using $W_{(p)}$ appear to be converging to the same value of the estimates obtained using $W^{(conv)}$, then the matrix $W_{(p)}$ is likely to be convergent and the corresponding estimates should be closer to the actual average\footnote{
Note that if $\mu(W_{(p)})>1$ the estimates calculated using $W_{(p)}$ diverge in general, then it should be easy to detect that the two consensus protocols are not converging to the same value.
}.
\subsection{Networks with Misbehaving Nodes}
The convergence of the average consensus relies on all the nodes correctly performing the algorithm.
If one node transmits an incorrect value, the estimates of all the nodes can be affected.
In this section we address this particular misbehavior. In particular, let $x_i(k)$ be the estimate of node $i$ at iteration $k$, if $x_i(k)\neq w_{ii}(k-1)x_i(k-1)+\sum _{j\in N_i}w_{ij}(k-1)x_j(k-1)$, then we call $i$ a \emph{misbehaving node}. \emph{Stubborn} nodes are a special class of misbehaving nodes that keep sending the same estimate at every iteration (i.e. a node $i$ is a stubborn node when at every iteration $k$ we have $x_i(k)=x_i(k-1)\neq w_{ii}(k-1)x_i(k-1)+\sum _{j\in N_i}w_{ij}(k-1)x_j(k-1)$). The authors of \cite{Acemoglu2011} and \cite{Ben-Ameur2012} showed that networks with stubborn nodes fail to converge to consensus. In \cite{Ben-Ameur2012}, they proposed a robust average consensus algorithm that can be applied on networks having one stubborn node and converges to consensus. To the best of our knowledge, dealing with multiple stubborn nodes is still an open issue. It turns out that with a minor modification of our JCO algorithm, the nodes can detect an unbounded number of misbehaving nodes under the following assumptions:
\begin{itemize}
\item {\bf Assumption 1:} There is no collusion between misbehaving nodes (every node that detects a misbehaving neighbor declares it).
\item {\bf Assumption 2:} At each iteration a misbehaving node sends the same (potentially wrong) estimate to all its neighbors.
\end{itemize}
The second assumption can be automatically satisfied in the case of a broadcast medium.
In the JCO procedure in section \ref{JCO}, nodes perform one weight optimization step and one average consensus step at every iteration. Consider an iteration $k$, weight optimization requires nodes to receive the weight vectors used by their neighbors (in particular, node $i$ will receive $\mathbf{W}_j^{(k-1)}$ from every neighbor $j\in N_i$), and the averaging protocol requires them to receive their neighbors estimates (in particular, node $i$ will receive $x_j(k)$ from every neighbor $j\in N_i$). We also require that nodes send the estimates of their neighbors, e.g.~node $i$ will receive together with the vector $\mathbf{W}_j^{(k-1)}$ another vector $\mathbf{X}_j(k-1)$ from every neighbor $j\in N_i$ where $\mathbf{X}_j(k-1)$ is the vector of the estimates of the neighbors of node $j$.
With such additional information, the following simple algorithm allows nodes to detect a misbehaving neighbor:
\begin{quote}
\underline{Misbehaving Neighbor Detection Algorithm - Node $i$}\\
$\{x_j(k),\mathbf{X}_j(k-1),\mathbf{W}_j^{(k-1)} \}$: the message received from a neighbor $j$ at iteration $k$\\
$\alpha(i)$: index of a node $i$ in the corresponding vector\\
{\bf for all} $j\in N_i$\\
\hspace*{1cm} $C=w_{jj}(k-1)x_j(k-1)+\mathbf{X}_j^T(k-1)\mathbf{W}_j^{(k-1)}$\\
\hspace*{1cm} {\bf if} $\left(x_j(k)\neq C \right)$ or $\left(x_i(k-1)\neq \left(\mathbf{X}_j(k-1)\right)_{\alpha(i)}\right)$ or $\left(w_{ij}(k-1)\neq \left(\mathbf{W}_{j}^{(k-1)}\right)_{\alpha(i)}\right)$\\
\hspace*{1.5cm} Declare $j$ as misbehaving node.\\
\hspace*{1cm} {\bf end if}\\
{\bf end for}
\end{quote}
The first condition ($x_j(k)\neq w_{jj}(k-1)x_j(k-1)+ \mathbf{X}_j^T(k-1)\mathbf{W}_j^{(k-1)}$) corresponds to the definition of a misbehaving node and allows neighbors to detect a node sending a wrong estimate. The second and third conditions ($x_i(k-1)\neq \left(\mathbf{X}_j(k-1)\right)_{\alpha(i)}$) or ($w_{ij}(k-1)\neq \left(\mathbf{W}_{j}^{(k-1)}\right)_{\alpha(i)}$) detect if node $j$ is modifying the content of any element in the vectors $\mathbf{X}_j(k-1)$ and $\mathbf{W}_j^{(k-1)}$ before sending them to its neighbors. More precisely, because of Assumption 2, if a node changes any element in the previously mentioned vectors, then this message will reach all neighbors including the neighbors concerned by this modification. These neighbors will remark this modification by checking the second and the third condition, and, due to Assumption~1, they will declare the node as misbehaving.
Once a node is declared a misbehaving node, the others can ignore it by simply assigning a null weight to its links in the following iterations.
\section{Conclusion}\label{Conc}
We have proposed in this report an approximated solution for the Fastest Distributed Linear Averaging (FDLA) problem by minimizing the Schatten $p$-norm of the weight matrix. Our approximated algorithm converges to the solution of the FDLA problem as $p$ approaches $\infty$, and in comparison to it has the advantage to be suitable for a distributed implementation. Moreover, simulations on random and real networks show that the algorithm outperforms other common distributed algorithms for weight selection.
\begin{comment}
|
1,108,101,564,535 | arxiv | \section{Introduction}
The second part of Hilbert's sixteenth problem
asks for a study of the number and relative positions of the limit cycles
of a planar polynomial system of ordinary differential equations.
The problem was included in Smale's list of problems for the next century \cite{Smale1998},
where he stated that
``except for the Riemann hypothesis,
this seems to be the most elusive of Hilbert's problems.''
Although far from solved, this problem has attracted a great deal of attention and influenced several developments within the field of dynamical systems.
See \cite{Ilyashenko}, \cite{Li},
\cite{UribeHossein}, \cite{Roussarie}, \cite{Gaiko}, and \cite{Caubergh} for surveys.
The finiteness of the number of limit cycles for any polynomial system was shown by Ilyashenko \cite{Ilya} and independently \'Ecalle \cite{Ecalle}.
The problem of proving the existence of an upper bound depending only on the degree of the polynomial system remains open;
even in the case of quadratic vector fields no uniform upper bound is known.
Several steps of progress
\cite{Chen},
\cite{Khovanski}, \cite{Varnchenko},
\cite{Petrov}, \cite{IlyaYakoInvent}, \cite{Gavrilov},
\cite{BinNovYak},
\cite{BinDor}
have been made on establishing uniform quantitative bounds for a special ``infinitesimal'' Hilbert's sixteenth problem that was proposed by Arnold \cite{ArnoldProb}
and which is restricted to near Hamiltonian systems.
Concerning lower bounds, in \cite{ChLl} it is shown that there are degree-$d$ polynomial systems whose number of limit cycles grows as $d^2 \log d$ with $d$ (cf. \cite{HanLi}).
A number of papers consider limit cycle enumeration problems for particular classes of interest, such as Li\'enard systems \cite{DumPanRou}, \cite{Llibre},
quadratic systems \cite{Bautin}, \cite{Gavrilov}, systems on a cyclinder \cite{LN} which are
related to systems with homogeneous nonlinearities \cite{CalRuf},
and systems arising in control theory
\cite{LeoKuz}.
The current work concerns the
following probabilistic perspective on
enumeration of limit cycles of planar systems associated to polynomial vector fields.
\begin{prob}
Study the number
and distribution in the plane (including their relative positions) of the limit cycles of a vector field whose component functions are random polynomials.
\end{prob}
This probabilistic point of view was introduced by A. Brudnyi in \cite{Brudnyi1}
with attention toward a special local problem of estimating the number of small amplitude limit cycles
near a randomly perturbed center focus.
We will revisit that setting below where we prove a limit law for the number of limit cycles (see Section \ref{sec:Brudnyi}).
But first we direct attention toward a non-perturbative problem concerning limit cycles of a random vector field with components sampled from the Kostlan-Shub-Smale ensemble.
The \emph{first} part of Hilbert's sixteenth problem concerns the topology of real algebraic manifolds (see the above mentioned survey \cite{Li} that includes discussion of both the first and second part of Hilbert's sixteenth problem).
This occupies a separate setting from the second part, namely, real algebraic geometry as opposed to dynamical systems. Yet, the current work builds on some of the insights from recent studies on the topology of random real algebraic hypersurfaces
\cite{NazarovSodin1} \cite{GaWe0}, \cite{sarnak}, \cite{GaWe2}, \cite{LLstatistics}, \cite{GaWe3}, \cite{NazarovSodin2}, \cite{SarnakWigman}, \cite{FLL}.
\subsection{Limit cycles for the Kostlan-Shub-Smale ensemble}\label{sec:KSS}
Our first concern will be in estimating the global number of limit cycles when the vector field components are random polynomials
sampled from the so-called Kostlan-Shub-Smale ensemble.
\begin{equation}\label{eq:affine}
p(x,y) = \sum_{0 \leq j + k \leq d }
a_{j,k}
\sqrt{\frac{d!}{(d-j - k)! j! k!}} x^{j} y^{k}, \quad a_{j,k} \sim N(0,1), \text{ i.i.d.}
\end{equation}
Among the Gaussian models of random polynomials, the Kostlan model is distinguished as the unique model that uses the monomials as a basis and is invariant under change of coordinates by orthogonal transformations of projective space $\mathbb{R} \mathbb{P}^2$ (there are other models with this invariance but their description requires Legendre polynomials). Moreover, the complex analog of the Kostlan model, obtained by taking $a_{j,k} \sim N_{\mathbb{C}}(0,1)$, is the only unitarily invariant Gaussian model of complex random polynomials. For these reasons, the Kostlan-Shub-Smale model has become a model of choice in studies of random multivariate polynomials.
Kostlan \cite{kostlan:93} adapted Kac's univariate method \cite{kac43} to the study of zero sets of multivariate polynomials,
and Shub and Smale \cite{Bez2} further showed that the
average number of real solutions to a random system
of $n$ equations in $n$ unknowns where the polynomials have degrees
$d_1,...,d_n$ equals $\sqrt{d_1 \cdots d_n}$,
which is the square root
of the maximum possible number of zeros as determined by Bezout's theorem
In particular, a planar vector field with random polynomial components of degree $d$
has $\sqrt{d^2}=d$ many equilibria on average.
We show that the average number of limit cycles as well grows (at least) linearly in the degree.
\begin{thm}[Lower bound for average number of limit cycles]
\label{thm:main}
Let $p,q$ be random polynomials of degree $d$ sampled independently
from the Kostlan ensemble.
Let $N_d$ denote the number of limit cycles of the vector field
$$F(x,y) = \binom{p(x,y)}{q(x,y)}.$$
There exists a constant $c_0>0$ such that
\begin{equation}\label{eq:LB}
\mathbb{E} N_d \geq c_0 \cdot d ,
\end{equation}
for all $d$.
\end{thm}
\begin{remark}
Addressing relative positions of limit cycles, it follows from the method of the proof of Theorem \ref{thm:main}, which localizes the problem to small disjoint (and unnested) annuli,
that the same lower bound holds while restricting to the number $\hat{N}_d$ of \emph{empty} limit cycles (we refer to a limit cycle as ``empty'' if it does not surround any other limit cycle).
Together with the above-mentioned result of Kostlan-Shub-Smale
this determines the growth rate of $\mathbb{E} \hat{N}_d$ to be linear in $d$; for all sufficiently large $d$ we have
\begin{equation}\label{eq:empty}
c_0 d \leq \mathbb{E} \hat{N}_d \leq d .
\end{equation}
Indeed, each empty limit cycle contains an equilibrium point in its interior and distinct empty limit cycles have disjoint interiors.
\end{remark}
The estimates \eqref{eq:empty}
suggest the existence of a constant $c>0$ such that
$\mathbb{E} \hat{N}_d \sim c \cdot d $ as $d \rightarrow \infty$.
We prove an analogous asymptotic result for a related model where the vector field components are random real analytic functions sampled from
the Gaussian ensemble induced by the Bargmann-Fock inner product.
This model arises as a rescaling limit of the Kostlan-Shub-Smale model \cite{BeGa}.
\begin{thm}[asymptotic for number of empty limit cycles]\label{thm:law}
Let $f,g$ be random real-analytic functions sampled independently
from the Gaussian space induced by the Bargmann-Fock inner product.
Let $\hat{N}_R$ denote the number of empty limit cycles situated within the of the disk of radius $R$ of the vector field
$$F(x,y) = \binom{f(x,y)}{g(x,y)}.$$
There exists a constant $c>0$ such that
\begin{equation}\label{eq:law}
\mathbb{E} \hat{N}_R \sim c \cdot R^2, \quad \text{as } R \rightarrow \infty.
\end{equation}
\end{thm}
The proof of Theorem \ref{thm:main}
uses transverse annuli and an adaptation of the ``barrier construction'' originally developed by Nazarov and Sodin for the study of nodal sets of random eigenfunctions \cite{NazarovSodin1}.
The proof of Theorem \ref{thm:law}
is based on yet another tool from the study of random nodal sets, the integral geometry sandwich from \cite{NazarovSodin2}, which reduces the current problem to controlling long limit cycles.
It is of interest, but seems very difficult in the above non-perturbative settings, to obtain asymptotic results or even upper bounds for the average total number of limit cycles (without restricting to empty limit cycles).
In the next sections we discuss this direction in a special setting.
\subsection{Limit cycles surrounding a perturbed center focus}\label{sec:Brudnyi}
In \cite{Brudnyi1}, Brudnyi considered
the limit cycles situated in the disk $\mathbb{D}_{1/2}$ of radius $1/2$ centered at the origin for the random vector field
\begin{equation}\label{eq:Brudnyi}
F(x,y) = \binom{y + \varepsilon p(x,y)}{-x + \varepsilon q(x,y)}
\end{equation}
where
\begin{equation}\label{eq:p}
p(x,y) = \sum_{1 \leq j+k \leq d} a_{j,k} x^j y^k
\end{equation}
and
\begin{equation}\label{eq:q}
q(x,y) = \sum_{1 \leq j+k \leq d} b_{j,k} x^j y^k
\end{equation}
are random polynomials with the vector of coefficients sampled uniformly from the $d(d+3)$-dimensional Euclidean unit ball $\displaystyle \left\{ \sum_{1 \leq j+k \leq d} (a_{j,k})^2 + (b_{j,k})^2 \leq 1 \right\}$,
and where $\varepsilon=\varepsilon(d) =\frac{1}{40\pi\sqrt{d}}$.
Using complexification and pluripotential theory, Brudnyi showed \cite{Brudnyi1} that, with $F$ as in \eqref{eq:Brudnyi}, the average number of limit cycles of the system $\binom{\dot{x}}{\dot{y}} = F(x,y)$ residing within the disk $\mathbb{D}_{1/2}$ is $O((\log d)^2)$,
which was later improved to $O(\log d)$ in \cite{Brudnyi2}.
We suspect that this can be further improved to $O(1)$ based on the result below
showing almost sure convergence of the limit cycle counting statistic for a slightly modified model,
where the vector of coefficients is sampled uniformly from the cube $[-1,1]^{d(d+3)}$ rather than from the unit ball of dimension $d(d+3)$.
A natural reason to prefer the cube is that it represents independent sampling of coefficients.
Note that we relax the smallness of the perturbation allowing $\varepsilon = \varepsilon(d) \rightarrow 0$ at an arbitrary rate as $d \rightarrow \infty$.
\begin{thm}[limit law for the perturbed center focus]\label{thm:as}
Let $p,q$ be random polynomials of degree $d$ with coefficients sampled uniformly and independently from $[-1,1]$, and suppose $\rho < 1$.
Suppose $\varepsilon = \varepsilon(d) \rightarrow 0$ as $d \rightarrow \infty$.
Let $N_d(\rho)$ denote the number of limit cycles situated within the disk $\mathbb{D}_{\rho}$ of the vector field
\begin{equation}\label{eq:perturbedcenter}
F(x,y) = \binom{y + \varepsilon p(x,y)}{-x + \varepsilon q(x,y)}.
\end{equation}
Then $N_d(\rho)$
converges almost surely (as $d \rightarrow \infty$) to
a non-negative random variable $X(\rho)$
with $\mathbb{E} X(\rho) < \infty$.
\end{thm}
The proof of this result shows that the original limit can be replaced with the iterated limit, taking $d \rightarrow \infty$ before $\varepsilon \rightarrow 0$. This ultimately reduces the problem to studying an \emph{infinitesimal} perturbation involving random bivariate power series.
As we will detail in the proof, the limiting random variable $X(\rho)$ counts the number of zeros of a certain random univariate power series (see \eqref{eq:Ainfinity} below). It is of interest to study the behavior of $X(\rho)$ as the radius $\rho$ approaches unity.
We conjecture that the precise asymptotic behavior of $\mathbb{E} X(\rho)$
is given by $$\displaystyle \mathbb{E} X(\rho) \sim \frac{1}{\pi} \sqrt{ - \log (1-\sqrt{\rho}) }, \quad \text{as } \rho \rightarrow 1^-.$$
This conjecture is supported by the following heuristic which rests on an assumption that Tao and Vu's ``replacement principle'' \cite{TaoVu} can be applied in the current setting: If the random series (see the function $\mathscr{A}_\infty$ defined in the proof of Theorem \ref{thm:as} below) whose zeros are counted by $X(\rho)$ is replaced by an analogous Gaussian series (replacing each coefficient by a Gaussian random variable of the same variance)
then the resulting Gaussian series is closely related to the one considered in \cite[Thm. 3.4]{Flasche} where a precise asymptotic has been derived.
The validity of replacing a random series with a Gaussian one as $\rho \rightarrow 1^-$ seems plausible considering universality principles in related settings \cite{TaoVu}, \cite{DoVu}, \cite{Kabluchko}.
\begin{remark}
A normal distribution law was established in \cite[Thm. 1.7]{Brudnyi1} for the number of zeros of certain related families of analytic functions, but it follows from Theorem \ref{thm:as}, with some attention to the details of its proof,
that a normal distribution limit law does not hold for $N_d(\rho)$ when $\rho<1$.
Indeed, $N_d(\rho)$ converges, without rescaling,
to a nondegenerate random variable $X(\rho)$ with support $\{0,1,2,...\}$.
\end{remark}
The underlying reason for the almost sure convergence of the number of limit cycles is that, within this particular model of random polynomials, $p,q$ tend toward random bivariate power series convergent in the unit bidisk $\mathbb{D} \times \mathbb{D} \subset \mathbb{C}^2$, in particular, convergent in the real disk $\mathbb{D}_\rho$ for any $\rho<1$.
It is desirable to count limit cycles beyond this domain of convergence, i.e., in a disk with radius $\rho>1$.
Under an assumption that $\varepsilon(d)$ shrinks sufficiently fast, the methods of \cite{Brudnyi1} can be utilized to obtain the estimate $O((\log d)^2)$. For convenience, to state this result we return to the context of Brudnyi's model,
sampling the vector of coefficients from the unit ball instead of a cube.
\begin{thm}\label{thm:addendum}
Let $p,q$ be random polynomials with the vector of coefficients sampled uniformly from the Euclidean unit ball, and suppose $\rho > 1$.
Suppose $0<\varepsilon = \varepsilon(d) \leq \frac{(2 \rho)^{-d(d+3)}}{40\pi \sqrt{d}} $.
The expectation of the number of limit cycles of the vector field
\begin{equation}
F(x,y) = \binom{y + \varepsilon p(x,y)}{-x + \varepsilon q(x,y)}
\end{equation}
situated within the disk $\mathbb{D}_{\rho}$
is bounded from above by a constant times $(\log d)^2$.
\end{thm}
\subsection{Infinitesimal perturbations}
Finally, let us consider
enumeration of limit cycles for a random system of the form \eqref{eq:Brudnyi}
while first taking the limit $\varepsilon \rightarrow 0$ and then $d \rightarrow \infty$.
This is a more tractable problem of enumerating the limit cycles that arise when the perturbation is \emph{infinitesimal}.
For generic $p,q$, this problem reduces to studying the zeros of the first Melnikov function (also known as the Poincare-Pontryagin-Melnikov function).
We first consider the case when $p,q$ are
sampled from the Kostlan-Shub-Smale model,
where we obtain the following precise asymptotic valid for any radius $\rho>0$.
\begin{thm}[square root law for Kostlan perturbations]\label{thm:sqrt}
Let $p,q$ be random polynomials of degree $d$ sampled independently from the Kostlan ensemble.
For $\rho>0$ let $N_{d,\varepsilon}(\rho)$ denote the number of limit cycles situated in the disk $\mathbb{D}_\rho$ of the vector field
\begin{equation}
F(x,y) = \binom{y+\varepsilon p(x,y)}{-x + \varepsilon q(x,y)}.
\end{equation}
Then as $\varepsilon \rightarrow 0$, $N_{d,\varepsilon}(\rho)$ converges almost surely to a random variable $N_d(\rho)$ whose expectation satisfies the following asymptotic in $d$.
\begin{equation}\label{eq:arctan}
\mathbb{E} N_d(\rho) \sim \frac{\arctan \rho}{\pi}\sqrt{d} , \quad \text{as } d \rightarrow \infty.
\end{equation}
\end{thm}
The proof of this result identifies the $\varepsilon \rightarrow 0$ limit as the number of zeros of a Poincar\'e-Pontryagin-Melnikov integral (the first Melnikov function associated to the perturbation) which is a random Gaussian function in this setting. We apply the Kac-Rice formula to determine the average number of zeros, and we use Laplace's method in the asymptotic analysis as $d \rightarrow \infty$. This type of application of the Kac-Rice formula is novel, and the technique can be adapted more generally to the study of randomly perturbed Hamiltonian systems, see Lemmas \ref{lemma:Nd}, \ref{lemma:EK}, \ref{lemma:twopt}.
We next consider a family of models where $p$ and $q$ have independent coefficients with mean zero and variances that have a power law relationship to the degree of the associated monomial. We collect the variances as deterministic weights, $c_m$ depending on the degree $m$. Namely,
\begin{equation}\label{eq:pweight}
p(x,y) = \sum_{m=1}^d \sum_{j+k = m} c_m a_{j,k} x^j y^k,
\end{equation}
and
\begin{equation}\label{eq:qweight}
q(x,y) = \sum_{m=1}^d \sum_{j+k = m} c_m b_{j,k} x^j y^k,
\end{equation}
with
\begin{equation}\label{eq:powerlawvar}
c_m^2 \sim m^\gamma, \quad \text{as } m \rightarrow \infty, \quad \gamma > 0
\end{equation}
and $a_{j,k}$, $b_{j,k}$ are i.i.d. with mean zero, unit variance, and finite moments.
\begin{thm}[random coefficients with power law variance]\label{thm:powerlaw}
Let $p,q$ be random polynomials of degree $d$
as in \eqref{eq:pweight}, \eqref{eq:qweight} with independent coefficients having power law variance as in \eqref{eq:powerlawvar} with $\gamma>0$.
Let $N_{d,\varepsilon}$ denote the number of limit cycles
(throughout the entire plane) of the vector field
\begin{equation}
F(x,y) = \binom{y+\varepsilon p(x,y)}{-x + \varepsilon q(x,y)}.
\end{equation}
Then as $\varepsilon \rightarrow 0$, $N_{d,\varepsilon}$ converges almost surely to a random variable $N_d$ whose expectation satisfies the following asymptotic in $d$.
\begin{equation}
\mathbb{E} N_d \sim \frac{1+\sqrt{\gamma}}{2\pi} \log d, \quad \text{as } d \rightarrow \infty.
\end{equation}
\end{thm}
The proof uses Do, Nguyen, and Vu's results \cite{DoVu},
which concern real zeros of univariate random polynomials with coefficients of polynomial growth. Unfortunately, this method does not extend to the case where $p,q$ have i.i.d. coefficients, i.e., when $\gamma=0$. We leave this case as an open problem that motivates advancing the theory of univariate random polynomials in a particular direction.
\subsection*{Outline of the paper}
In Section \ref{sec:prelim},
we review some preliminaries:
the construction and basic properties of the Kostlan-Shub-Smale ensemble, the notion of a transverse annulus from dynamical systems, the Poincar\'e-Pontryagin-Melnikov integral from perturbation theory of Hamiltonian systems, and the Kac-Rice formula for real zeros of random functions.
In Section \ref{sec:main} we present the proofs of Theorems \ref{thm:main} and \ref{thm:law}.
In Section \ref{sec:Brudnyi}
we present the proofs of Theorems \ref{thm:as}
and \ref{thm:addendum}.
In Section \ref{sec:infinitesimal}
we present the proofs of Theorems
and \ref{thm:sqrt} and \ref{thm:powerlaw}. We conclude with some discussion of future directions and open problems in Section \ref{sec:concl}.
\subsection*{Acknowledgements}
The author thanks
Zakhar Kabluchko for
directing his attention toward
the above mentioned result from
Hendrik Flasche's dissertation \cite{Flasche}.
\section{Preliminaries}\label{sec:prelim}
In this section we review
some basics of Gaussian random polynomials
and dynamical systems
that will be needed.
\subsection{The Kostlan-Shub-Smale ensemble of random polynomials}
Let $\mathcal{P}_d$ denote the space of polynomials
of degree at most $d$ in two variables,
and recall that there is a natural
isomorphism (through homogenization) between $\mathcal{P}_d$ and the space $\mathcal{H}_d$ of homogeneous polynomials of degree $d$ in three variables.
A Gaussian ensemble can be specified by choosing a scalar product on
$\mathcal{H}_d$ in which case $f$ is sampled according to:
$$\text{Probability} (f\in A) = \frac{1}{v_{n,d}}\int_A e^{-\frac{\|f \|^2}{2}} dV(f), $$
where $v_{n,d}$ is the normalizing constant that makes this a probability density function,
and $dV$ is the volume form
induced by the scalar product.
The \emph{Kostlan-Shub-Smale ensemble}
(which we may simply refer to as the ``Kostlan ensemble''),
results from choosing as a scalar product
the \emph{Fischer product}\footnote{This scalar product also goes by many other names,
such as the ``Bombieri product'' \cite[p. 122]{KhLu}.} defined as:
$$\langle f, g \rangle_F = \frac{1}{d! \pi^{3}}\int_{\mathbb{C}^{3}} f(x,y,z) \overline{g(x,y,z)} e^{-|(x,y,z)|^2} dx dy dz.$$
The monomials are orthogonal with respect to the Fischer product, and the weighted monomials $\binom{d}{\alpha}^{1/2}x^{\alpha_1}y^{\alpha_2}z^{\alpha_3}$ form an orthonormal basis.
Consequently, the following expression relates the Fischer norm of $f=\sum_{|\alpha|=d}f_\alpha x^{\alpha_1}y^{\alpha_2}z^{\alpha_3}$ with its coefficients in the monomial basis (see \cite[Equation (10)]{NewmanShapiro}):
\begin{equation}
\|f\|_F=\left(\sum_{|\alpha|=d}|f_\alpha|^2 \binom{d}{\alpha}^{-1} \right)^{\frac{1}{2}}.\end{equation}
Here we are using multi-index notation $\alpha=(\alpha_1,\alpha_2,\alpha_3)$,
with $|\alpha|:=\alpha_1+\alpha_2+\alpha_3$,
and $\binom{d}{\alpha} := \frac{d!}{\alpha_1! \alpha_2! \alpha_3!}$.
Having chosen a scalar product,
we can build the random polynomial
$f$ as a linear combination, with independent Gaussian coefficients,
using an orthonormal basis for the associated scalar product.
The weighted monomials $\sqrt{\binom{d}{\alpha}} x^{\alpha_1}y^{\alpha_2}z^{\alpha_3} $ form an orthonormal basis for the Fischer product.
Thus, sampling $f$ from the Kostlan model (here again $\alpha=(\alpha_1,\alpha_2,\alpha_3)$ is a multi-index),
we have
$$f(x,y,z)=\sum_{|\alpha|=d}\xi_\alpha \sqrt{\binom{d}{\alpha}}x^{\alpha_1}y^{\alpha_2}z^{\alpha_3}, \quad \xi_\alpha\sim N\left(0,1 \right), \text{ i.i.d.}$$
Restricting to the affine plane $z=1$,
and setting $p(x,y,1)=f(x,y,z)$,
we arrive at \eqref{eq:affine}.
\begin{remark}\label{rmk:basis}
The monomial basis is the natural basis for this model, but we are free to write the expansion with i.i.d. coefficients in front of another basis as long as it is orthonormal with respect to the Fischer product. This fact will be used in the proof of Theorem \ref{thm:main}.
\end{remark}
Among Gaussian models built using the monomials as an orthogonal basis, the Kostlan ensemble has the distinguished property of being the unique Gaussian ensemble that is rotationally invariant
(i.e., invariant under any orthogonal transformation of projective space).
The two-point correlation function (or covariance kernel) of $p$, defined as $K(v_1,v_2) := \mathbb{E} p(v_1)p(v_2)$, where
$v_1=(x_1,y_1), v_2=(x_2,y_2)$,
satisfies
\begin{equation}\label{eq:2ptK}
K(v_1,v_2) = (1+x_1x_2 + y_1 y_2)^d,
\end{equation}
which can be seen by applying the multinomial formula after using $\mathbb{E} \xi_\alpha \xi_\beta = \delta_{\alpha,\beta}$ to obtain
\begin{equation}
\mathbb{E} p(v_1) p(v_2) = \sum_{|\alpha|=d} \binom{d}{\alpha} (x_1 x_2)^{\alpha_1}(y_1 y_2)^{\alpha_2}.
\end{equation}
The homogeneous random polynomial $f$ is distributed the same as if composed with
an orthogonal transformation $T$ of projective space.
The random polynomial $p$ is not invariant under the induced action $\psi \circ T \circ \psi^{-1}$ on affine space, but if we multiply it by the factor
$(1+x^2 + y^2)^{-d/2}$ then it is. We state this as a remark that will be used later.
\begin{remark}\label{rmk:invariant}
For $p \in \mathcal{P}_d$ a random Kostlan polynomial of degree $d$,
the random function $(1+x^2 + y^2)^{-d/2} p(x,y)$ is invariant under the action $\psi \circ T \circ \psi^{-1}$ on affine space induced by
an orthogonal transformation $T$ of projective space.
Indeed, point evaluation of this function at $(x,y)$ is equivalent to homogenization of $p$ followed by point evaluation at $\psi^{-1}(x,y)$.
\end{remark}
\subsection{Transverse annuli}
In the study of dynamical systems,
a fundamental and widely used notion is that of a ``trapping region''.
In particular, a so-called \emph{transverse annulus} is useful in implementations of the Poincare-Bendixson Theorem.
\begin{definition}
Suppose $F$ is a planar vector field.
An annular region $A$ with $C^1$-smooth boundary
is called a \emph{transverse annulus} for $F$ if
\begin{itemize}
\item[1.] $F$ is transverse to the boundary of $A$ with
$F$ pointing inward on both boundary components
or outward on both boundary components, and
\item[2.] $F$ has no equilibria in $A$.
\end{itemize}
\end{definition}
The next proposition follows from the Poincare-Bendixson Theorem \cite{Gu}.
\begin{prop}\label{prop:transann}
Any transverse annulus for a planar vector field $F$ contains at least one periodic orbit of $F$.
\end{prop}
\begin{remark}\label{rmk:generic}
In the setting where $F$ is a random vector field (sampled from the Kostlan-Shub-Smale ensemble), periodic orbits of $F$ are almost surely limit cycles; this follows from the fact that generic polynomial vector fields
contain no periodic orbits other than limit cycles along with the fact that the Kostlan measure is absolutely continuous with respect to the Lebesgue measure on the parameter space (the space of coefficients).
\end{remark}
\subsection{The Poincar\'e-Pontryagin-Melnikov integral}
Let us consider the perturbed Hamiltonian system
\begin{equation}\label{eq:Hpert}
\begin{cases}
\dot{x} = \partial_y H(x,y) + \varepsilon p(x,y) \\
\dot{y} = -\partial_x H(x,y) + \varepsilon q(x,y)
\end{cases},
\end{equation}
with $H,p,q$ polynomials.
For $\varepsilon=0$
the trajectories of \eqref{eq:Hpert}
follow (connected components of) the level curves of $H$.
For $\varepsilon>0$ small, the limit cycles of the system can be studied (in non-degenerate cases)
using the Poincar\'e-Pontryagin-Melnikov integral (first Melnikov function) given by
\begin{equation}\label{eq:Abel}
\mathscr{A}(h) = \int_{C_h} p \, dy - q \, dx,
\end{equation}
where $C_h$ denotes a connected component of the level set $\{ (x,y): H(x,y) = h \}$.
The following classical result is fundamental in the study of limit cycles of perturbed Hamiltonian systems, see \cite[Sec. 2.1 of Part II]{ChLi} and \cite[Sec. 26]{IlyashenkoBook}.
\begin{thm}\label{thm:PPM}
Let $\mathscr{P}_{\varepsilon}(h)$ be the Poincar\'e
first return map of the system \eqref{eq:Hpert} defined on some
segment transversal to the level curves of $H$, where $h \in (a,b)$ marks the value taken by $H$.
Then, $\mathscr{P}(h) = h + \varepsilon \mathscr{A}(h) + \varepsilon^2 E(h, \varepsilon)$, as $\varepsilon \rightarrow 0$,
where $\mathscr{A}$ denotes the integral \eqref{eq:Abel} and where $E(h,\varepsilon)$ is analytic and uniformly bounded for $\varepsilon$ sufficiently small and for $h$ in a compact neighbourhood of $(a,b)$.
\end{thm}
In particular, if the zeros of
$\mathscr{A}$ are isolated and non-degenerate,
then each zero of $\mathscr{A}$ corresponds to a limit cycle of the system \eqref{eq:Hpert}.
\subsection{The Kac-Rice formula}
The following result,
in its various forms, is fundamental in the study of real zeros of random analytic functions.
We follow \cite[Thm. 3.2]{AzaisWscheborbook},
which will suit our needs,
but we mention in passing that the Kac-Rice formula also holds in multi-dimensional and non-Gaussian settings.
\begin{thm}[Kac-Rice formula]\label{thm:KR}
Let $f:I \rightarrow \mathbb{R}$ be a random function with $I \subset \mathbb{R}$ an interval.
Suppose the following conditions are satisfied
\begin{itemize}
\item[(i)] $f$ is Gaussian,
\item[(ii)] $f$ is almost surely of class $C^1$,
\item[(iii)] For each $t \in I$, $f(t)$ has a nondegenerate distribution,
\item[(iv)] Almost surely $f$ has no degenerate zeros.
\end{itemize}
Then
\begin{equation}\label{eq:KRformula}
\mathbb{E} |\{t \in I:f(t) =0\} | = \int_I \int_\mathbb{R} |B| \rho_t(0,B) dB dt,
\end{equation}
where $ \rho_t(A,B)$ denotes the
joint probability density function
of $A=f(t)$ and $B=f'(t)$.
\end{thm}
As observed by Edelman and Kostlan \cite{EdelmanKostlan95}, We can express \eqref{eq:KRformula} in terms of the two-point correlation function $K(r,t):=\mathbb{E} f(r) f(t)$, namely,
\begin{equation}\label{eq:EKform}
\mathbb{E} |\{t \in I:f(t) =0\} | = \frac{1}{\pi} \int_I \sqrt{\frac{\partial^2}{\partial t \partial r} \log K(r,t) \rvert_{r=t=\tau}} d\tau.
\end{equation}
This identity is based on the fact that the
joint density $\rho_t(A,B)$ of the Gaussian pair $A=f(t), B=f'(t)$ can be expressed in terms of the covariance matrix of $f(t)$ and $f'(t)$ which in turn can be computed from evaluation along the diagonal $r=t$ of appropriate partial derivatives of $K(r,t)$.
\section{Proofs of Theorems \ref{thm:main} and \ref{thm:law}}\label{sec:main}
\begin{proof}[Proof of Theorem \ref{thm:main}]
It is enough to prove a lower bound of the form \eqref{eq:LB}
while restricting our attention to just those limit cycles that are contained in the unit disk $\mathbb{D} = \{ (x,y) \in \mathbb{R}^2 : x^2 + y^2 < 1 \}$.
Let $A_d = A_d(0,0) = \{ (x,y) \in \mathbb{R}^2 : d^{-1/2} < |(x,y)| < 2d^{-1/2} \}$ denote the annulus centered at $(0,0)$ with inner radius $d^{-1/2}$ and outer radius $2d^{-1/2}$.
For $0<r<1$ and $0 \leq \theta < 2\pi$ let $A_d(r,\theta)$ denote the image of $A_d$ under the mapping $\psi \circ T_{r,\theta} \circ \psi^{-1} : \mathbb{R}^2 \rightarrow \mathbb{R}^2$,
where $\psi:S^2 \rightarrow \mathbb{R}^2$
denotes central projection
and $T_{r,\theta} : S^2 \rightarrow S^2$
denotes the rotation of $S^2$
such that $\psi \circ T_{r,\theta} \circ \psi^{-1} : \mathbb{R}^2 \rightarrow \mathbb{R}^2$
fixes the line through the origin with angle $\theta$ and maps the origin to $(r\cos \theta,r \sin \theta)$.
From elementary geometric considerations, we notice that $A_d(r,\theta)$ is an annulus with elliptical boundary components, each having angle of alignment $\theta$.
The two elliptical boundary components are asymptotically concentric and homothetic.
As $d \rightarrow \infty$
the center is located at
$(r \cos \theta, r \sin \theta) + O(d^{-1})$.
The major semi-axis of the smaller ellipse equals $a d^{-1/2} + O(d^{-1})$,
and the major semi-axis of the larger ellipse is $2a d^{-1/2} + O(d^{-1})$, with $a = \sqrt{1+r^2}$. The minor semi-axes of the inner and outer boundary components are $d^{-1/2}$ and $2d^{-1/2}$ respectively.
For some $k>0$ independent of $d$,
we can fit at least $k \cdot d$
many disjoint annuli $A_d(r_i,\theta_i)$ in the unit disk $\mathbb{D}$.
Letting $I_j$ denote the indicator random variable
for the event $\mathcal{E}_j$ that the $j$th annulus is transverse for $F$,
we note that if $\mathcal{E}_j$ occurs then Proposition
\ref{prop:transann} and Remark \ref{rmk:generic} guarantee that there is at least one limit cycle contained in the annulus associated to $\mathcal{E}_j$.
Since the annuli are disjoint,
the limit cycles considered above are distinct (even though the events $I_j$ are dependent),
and this gives the following lower bound for the expectation of $N_d$
\begin{align}\label{eq:linearity}
\mathbb{E} N_d &\geq \mathbb{E} \sum I_j \\
&= \sum \mathbb{E} I_j \\
&= \sum \mathbb{P} \mathcal{E}_j,
\end{align}
where $j$ ranges over an index set of size at least $k \cdot d$,
and we have used linearity of expectation
going from the first line to the second.
\begin{prop}\label{prop:LB}
There exists a constant $c_1>0$, independent of $d$, such that for any $0<r<1$ and $0 \leq \theta < 2\pi$,
the probability that $A_d(r,\theta)$
is a transverse annulus for $F$ is at least $c_1$.
\end{prop}
We defer the proof of the proposition in favor of first seeing how it is used to finish the proof of the theorem.
We can apply Proposition \ref{prop:LB} to each of the events $\mathcal{E}_j$ to get
$\mathbb{P} \mathcal{E}_j \geq c_1$ for all $j$.
Combining this with \eqref{eq:linearity},
we obtain
$$\mathbb{E} N_d \geq c_1 \cdot k \cdot d.$$
Since $k>0$ and $c_1>0$ are each independent of $d$,
this proves Theorem \ref{thm:main}.
\end{proof}
It remains to prove Proposition \ref{prop:LB}.
We will need the following lemmas.
\begin{lemma}\label{lemma:supest}
Let $F = \binom{p}{q}$ be a random vector field with polynomial components $p,q$ sampled from the Kostlan ensemble of degree $d$.
Let $D_d = \mathbb{D}_{3d^{-1/2}}$ denote the disk
of radius $3d^{-1/2}$ centered at the origin.
There exists a constant $C_0 > 0$ such that
$$ \mathbb{P} \left\{ \sup_{(x,y) \in D_d} | F(x,y)| \geq C_0 \right\} < \frac{1}{3}.$$
\end{lemma}
\begin{proof}[Proof of Lemma \ref{lemma:supest}]
Define $\hat{p}(\hat{x},\hat{y}) = p(d^{-1/2}\hat{x},d^{-1/2}\hat{y})$, and
$\hat{q}(\hat{x},\hat{y}) = q(d^{-1/2}\hat{x},d^{-1/2}\hat{y})$,
and note that $(x,y) \in D_d$
corresponds to $(\hat{x},\hat{y}) \in \mathbb{D}_2(0)$.
It suffices to show that
\begin{equation}\label{eq:twoevents}
\mathbb{P} \left\{ \| \hat{p} \|_{\infty,\mathbb{D}_2(0)} > C_0/2 \right\} < \frac{1}{6}, \quad \text{and} \quad \mathbb{P} \left\{ \| \hat{q} \|_{\infty,\mathbb{D}_2(0)} > C_0/2 \right\} < \frac{1}{6},
\end{equation}
where we used the notation $\displaystyle \| \hat{p} \|_{\infty,\mathbb{D}_2(0)} := \sup_{(\hat{x},\hat{y}) \in \mathbb{D}_2(0)} \hat{p}(\hat{x},\hat{y}).$
Indeed, the event that $\displaystyle \| F\|_{\infty,D_d} > C_0$
is contained in the union of the
two events considered in \eqref{eq:twoevents}.
We only need to prove the first statement in \eqref{eq:twoevents} since $\hat{p}$ and $\hat{q}$ are identically distributed.
For $(\hat{x},\hat{y}) \in \mathbb{D}_2(0)$ we have
\begin{align}
|\hat{p} (\hat{x},\hat{y})| &= \left| \sum_{ |\beta| \leq d }
a_{\beta}
\sqrt{\frac{d!}{(d-|\beta|)!\beta_1!\beta_2 !}} \frac{\hat{x}^{\beta_1} \hat{y}^{\beta_2}}{d^{|\beta|/2}} \right| \\
&\leq \sum_{ |\beta| \leq d }
|a_{\beta}|
\frac{2^{|\beta|}}{\sqrt{\beta_1!\beta_2 !}}.
\end{align}
This implies
\begin{align}
\mathbb{E} \|\hat{p} \|_{\infty,\mathbb{D}_2(0)}
&\leq \sum_{ |\beta| \geq 0 }
\sqrt{\frac{2}{\pi}}
\frac{2^{|\beta|}}{\sqrt{\beta_1!\beta_2 !}} \\
&\leq \sqrt{\frac{2}{\pi}} \sum_{ k \geq 0 }
\frac{k \, 2^{k}}{\sqrt{\lfloor k/2 \rfloor}!},
\end{align}
which is a convergent series.
So, we have shown that
$$\mathbb{E} \|\hat{p} \|_{\infty,\mathbb{D}_2(0)} \leq M<\infty,$$
with $M>0$ independent of $d$.
Applying Markov's inequality we have
$$\mathbb{P} \{ \|\hat{p} \|_{\infty,\mathbb{D}_2(0)} > C_0/2 \} \leq \frac{2 M}{C_0},$$
and \eqref{eq:twoevents} follows
when $C_0$ is chosen larger than $12M$.
\end{proof}
For $0 \leq r \leq 1$, write $a=\sqrt{1+r^2}$,
and consider the random vector field
\begin{equation}\label{eq:defBr}
B_r(x,y) = \binom{ \xi_0 b_1(x,y)}{ \eta_0 b_2(x,y) } ,
\end{equation}
where $\xi_0,\eta_0$
are independent standard normal random variables, and
$$b_1(x,y) = \frac{1}{1+a} \left( -d^{1/2}ay + \frac{d^{1/2}}{V_d}x(3-d^2 x^2-d^2y^2 )\right)$$
$$b_2(x,y) = \frac{1}{1+a} \left( d^{1/2}x + \frac{d^{1/2}}{V_d} a y (3 - d^{2} x^2 - d^{2} y^2 )\right),$$
where
\begin{equation}\label{eq:Vd}
V_d = 3+\sqrt{\frac{2}{(1-1/d)(1-2/d)}} + \sqrt{\frac{6}{(1-1/d)(1-2/d)}}.
\end{equation}
\begin{lemma}\label{lemma:normone}
Let $B_r$ be the vector field defined in \eqref{eq:defBr}, and let $h_1(x,y,z)$ and $h_2(x,y,z)$ denote the homogeneous polynomials of degree $d$ that coincide with the components $b_1$ and $b_2$ of $B_r$ on the affine plane $z=1$.
Then $h_1$ and $h_2$ each have unit Fischer norm.
\end{lemma}
\begin{proof}[Proof of Lemma \ref{lemma:normone}]
We first recall that the monomials are orthogonal with respect to the Fischer inner product,
and next we recall the norms of each of the monomials appearing in $B_r(x,y)$:
$$\| x z^{d-1} \|_F = \| y z^{d-1} \|_F = d^{-1/2}$$
$$\| x^3 z^{d-3} \|_F = \| y^3 z^{d-3} \|_F = \sqrt{\frac{6}{d(d-1)(d-2)}}$$
$$\| x^2y z^{d-3} \|_F = \| y^2x z^{d-3} \|_F = \sqrt{\frac{2}{d(d-1)(d-2)}}$$
Then letting $h_1(x,y,z)$ and $h_2(x,y,z)$ denote the homogeneous polynomials of degree $d$ that coincide with $b_1$ and $b_2$ (respectively) on the affine plane $z=1$, we find
$$ \|h_1(x,y,z) \|_F = \frac{1}{1+a} \left( a+ \frac{1}{V_d}\left[3+ \frac{\sqrt{6}+\sqrt{2}}{\sqrt{(1-1/d)(1-2/d)}}\right]\right),$$
which is unity by the choice of $V_d$,
and similarly
$$\| h_2(x,y,z) \|_F = \frac{1}{1+a} \left( 1 + \frac{a}{V_d} \left[ 3+ \frac{\sqrt{6}+\sqrt{2}}{\sqrt{(1-1/d)(1-2/d)}}\right]\right) = 1.$$
\end{proof}
Let $T_r = \psi \circ R_r \circ \psi^{-1}$
where $\psi : S^2 \rightarrow \mathbb{R}^2$ denotes central projection,
and $R_r$ denotes the rotation of $S^2$ such that $\psi \circ R_r \circ \psi^{-1}$ fixes the $x$-axis and maps $(r,0)$ to the origin.
Let $\hat{n}_r$
denote the inward-pointing unit normal vector
on $\partial A(r,0)$,
and let
$\hat{N}_r = \hat{n}_r \circ T_r$ denote
its pullback
by the transformation $T_r$.
More explicitly, we can write
\begin{equation}\label{eq:defNr}
\hat{N}_r(x,y) = \pm \frac{1}{\sqrt{x^2/a^2 + y^2}} \binom{x/a}{y} + O(d^{-1/2}),
\end{equation}
where the sign is ``$+$'' for the inner boundary component and ``$-$'' for the outer component.
\begin{lemma}\label{lemma:barrier}
Fix $C_0>0$.
For $0 \leq r \leq 1$,
let $B_r$ and $\hat{N}_r$ be as defined above.
There exists a constant $c_2>0$ independent of both $d$ and $r$, such that there is probability at least $c_2$ of both of the following being satisfied.
\begin{itemize}
\item[(i)] Everywhere along $\partial A_d$, the scalar product $\langle B_r(x,y), \hat{N}_r(x,y) \rangle$ of $B_r$ and $\hat{N}_r$ satisfies $$\langle B_r(x,y), \hat{N}_r(x,y) \rangle \geq 2C_0 .$$
\item[(ii)] We have $$\inf_{(x,y) \in A_d} \| B_r(x,y) \| \geq 2C_0.$$
\end{itemize}
\end{lemma}
\begin{remark}\label{rmk:perturb}
Note that, properties (i) and (ii)
imply that, for any vector field $F_0$ satisfying $|F_0(x,y)| \leq C_0$ for all $(x,y) \in A_d$, the perturbation
$B_r+F_0$ of $B_r$ satisfies
$\langle B_r + F_0 , \hat{N}_r \rangle > 0$ on $\partial A_d$, and we also have that $B_r + F_0$ does not vanish in $A_d$, i.e., $A_d$ is a transverse annulus for $B_r+F_0$.
\end{remark}
\begin{proof}[Proof of Lemma \ref{lemma:barrier}]
Fix $0\leq r \leq 1$.
Let us decompose $B_r(x,y)$ defined in \eqref{eq:defBr} as
\begin{equation}\label{eq:Bdecomp}
B_r(x,y) = \frac{1}{1+a} \left( B_t(x,y) + \frac{B_n(x,y)}{V_d} \right),
\end{equation}
where
$$B_t(x,y) = d^{1/2} \binom{-\xi_0 a y}{\eta_0 x} ,$$
and
$$B_n(x,y) = d^{1/2} \left( 3 - d(x^2 + y^2) \right) \binom{\xi_0 x}{\eta_0 a y}.$$
For $C_1 \gg 1 \gg \varepsilon > 0$ fixed,
consider the event $\mathcal{E}$ that
\begin{equation}\label{eq:interval}
\xi_0,\eta_0 \in (C_1 - \varepsilon, C_1 + \varepsilon).
\end{equation}
The probability $c_2$ of $\mathcal{E}$ is positive and independent of $d$ and $r$.
We will show that $\mathcal{E}$
implies each of the properties (i) and (ii).
First we rewrite $B_t$ and $B_n$ using the
condition \eqref{eq:interval} defining the event $\mathcal{E}$:
\begin{equation}\label{eq:Bcdecomp}
B_t(x,y) = C_1 d^{1/2} \binom{-a y}{x} + d^{1/2} \binom{- \gamma_1 a y}{ \gamma_2 x} ,
\end{equation}
and
\begin{equation}\label{eq:Bgdecomp}
B_n(x,y) = C_1d^{1/2} \left( 3 - d(x^2 + y^2) \right) \binom{x}{a y} + d^{1/2} \left( 3 - d(x^2 + y^2) \right) \binom{\gamma_1 x}{ \gamma_2 a y},
\end{equation}
with $-\varepsilon < \gamma_i < \varepsilon$ for $i=1,2$.
Note that $B_t$ is approximately
orthogonal to the vector field $\hat{N}_r$ along $\partial A_d$, and $B_n$ is approximately parallel to $\hat{N}_r$ along $\partial A_d$.
This is the basic idea used in the estimates that follow.
We recall from \eqref{eq:defNr}
that $\hat{N}_r(x,y) = \pm \frac{1}{\sqrt{x^2/a^2 + y^2}} \binom{x/a}{y}+O(d^{-1/2})$ on $\partial A_d$.
Since $\binom{x/a}{y}$ is orthogonal to $\binom{-a y}{x}$,
we have, using \eqref{eq:Bcdecomp} and \eqref{eq:defNr},
\begin{equation}\label{eq:BtNr}
\left| \langle B_t(x,y) , \hat{N}_r(x,y) \rangle \right| = d^{1/2} \left| \left(\gamma_2 - \gamma_1 \right)\frac{xy}{\sqrt{x^2/a^2+y^2}}\right| + O(d^{-1/2}), \quad \text{on } \partial A_d.
\end{equation}
Applying the estimates
$$ d^{1/2} \frac{|xy|}{\sqrt{x^2/a^2+y^2}} \leq 2a < 4 , \quad (x,y) \in \partial A_d,$$
$$ |\gamma_2- \gamma_1| < 2 \varepsilon,$$
in \eqref{eq:BtNr} we obtain, for all $d$ sufficiently large,
\begin{equation}\label{eq:Bt}
\left| \langle B_t(x,y) , \hat{N}_r(x,y) \rangle \right| < 8\varepsilon, \quad \text{on } \partial A_d.
\end{equation}
Using \eqref{eq:Bgdecomp}, we obtain for $(x,y) \in \partial A_d$
\begin{align}\label{eq:alignBnNr}
\langle B_n(x,y) , \hat{N}_r(x,y) \rangle &= \pm a \frac{d^{1/2} \left( 3 - d(x^2 + y^2)\right)}{\sqrt{x^2/a^2+y^2}} \left( C_1(x^2/a^2+y^2) + \gamma_1 x^2/a^2 + \gamma_2 y^2 \right) \\
&\geq \pm a \frac{d^{1/2} \left( 3 - d(x^2 + y^2)\right)}{\sqrt{x^2/a^2+y^2}} (x^2/a^2+y^2)
\left( C_1 -2\varepsilon \right) \\
&\geq \pm d^{1/2} \left( 3 - d(x^2 + y^2)\right)\sqrt{x^2+y^2}
\left( C_1 -2\varepsilon \right),
\end{align}
where the choice of $\pm$ sign is determined according to the component of $\partial A_d$ as in \eqref{eq:defNr}.
We have
$$\pm d^{1/2} \left( 3 - d(x^2 + y^2)\right)\sqrt{x^2+y^2} = 2, \quad \text{on } \partial A_d, $$
which holds on both components of $\partial A_d$.
This gives
\begin{equation}\label{eq:Bn}
\langle B_n(x,y) , \hat{N}_r(x,y) \rangle \geq 2C_1 - 4 \varepsilon, \quad \text{on } \partial A_d.
\end{equation}
The two estimates \eqref{eq:Bt} and \eqref{eq:Bn} together give
$$\langle B_r(x,y) , \hat{N}_r(x,y) \rangle \geq \frac{1}{1+a} \left( \frac{2C_1 - 4 \varepsilon}{V_d} -8\varepsilon \right),\quad \text{on } \partial A_d.$$
Since $1+a<3$ and $V_d$ converges to a postive constant $3 + \sqrt{6}+\sqrt{2}$ as $d \rightarrow \infty$,
for an appropriate choice of $C_1 \gg \varepsilon > 0$ we have
$\frac{1}{1+a} \left( \frac{1}{V_d} (2C_1 - 4\varepsilon) -8\varepsilon \right) > 2C_0$
for all $d$ sufficiently large.
This completes the verification that property (i)
holds.
Next we check property (ii).
Throughout the annulus $A_d$ we have
\begin{align}\label{eq:normlowerbound}
(1+a)^2 \| B_r \|^2 &= \| B_t \|^2 + \frac{1}{V_d^2}\|B_n \|^2 + \frac{2}{V_d} \langle B_t , B_n \rangle \\
&\geq \| B_t \|^2 + \frac{2}{V_d} \langle B_t , B_n \rangle \\
&\geq \| B_t \|^2 - | \langle B_t , B_n \rangle |,
\end{align}
where we have used in the last line that $V_d$ which is defined in \eqref{eq:Vd} is at least $3$. Next we estimate $| \langle B_t , B_n \rangle |$.
\begin{align}\label{eq:normlowerboundcont}
| \langle B_t , B_n \rangle | &= 4 a d C_1 |(\gamma_2-\gamma_1)xy|
+2ad |(\gamma_2^2-\gamma_1^2)xy| \\
&\leq 8 a C_1 |\gamma_1-\gamma_2|
+4a |\gamma_2^2-\gamma_1^2|\\
&\leq 16 a C_1 \varepsilon + 4a \varepsilon^2 \\
&\leq 32 C_1 \varepsilon + 8 \varepsilon^2,
\end{align}
Thus,
\begin{align}
(1+a)^2 \| B_r \|^2 &\geq \| B_t \|^2 - | \langle B_t , B_n \rangle | \\
&\geq \| B_t \|^2 - 32 C_1 \varepsilon - 8 \varepsilon^2 \\
&\geq (C_1-\varepsilon)^2 - 32 C_1 \varepsilon - 8 \varepsilon^2,
\end{align}
which is larger than $(6C_0)^2$ for an appropriate choice of $C_1 \gg \varepsilon >0$.
So we have
$$ (1+a)^2\| B_r \|^2 \geq (6C_0)^2,$$
which shows that property (ii) is satisfied
since $1+a < 3$.
\end{proof}
\begin{proof}[Proof of Proposition \ref{prop:LB}]
Fix $0<r<1$ and $0\leq \theta < 2\pi$.
Applying a rotation of the $xy$-plane about the origin (and using the rotation invariance of the Kostlan ensemble), we may assume that $\theta=0$.
Let $\mathcal{E}$ denote the event that
$A_d(r,0)$ is a transverse annulus for $F$,
i.e., $\mathcal{E}$ is the event that the following two conditions both hold:
\begin{itemize}
\item The vector field $F$ points into $A_d(r,0)$
on both boundary components, i.e.,
$\langle F , \hat{n} \rangle > 0$ at each point
on $\partial A_d(r,0)$ where $\hat{n}$
denotes the inward pointing normal vector.
\item $F$ has no equilibria in $A_d(r,0)$.
\end{itemize}
Equivalently, $\mathcal{E}$ is the event that
$A_d(r,0)$ is a transverse annulus for $G$ defined as
$$G(x,y) = (1+x^2+y^2)^{-d/2} F(x,y).$$
By Remark \ref{rmk:invariant},
the component functions of $G$ are invariant (as random functions) under the group of transformations induced by
orthogonal transformations of projective space.
Let $T_r = \psi \circ R_r \circ \psi^{-1}$,
where $\psi$ denotes central projection, and $R_r$ is the rotation such that $T_r$ fixes the $x$-axis and maps the origin to $(r,0)$.
As defined above, let $\hat{N}_r = \hat{n}_r \circ T_r$ denote the pullback to $\partial A_d$ of the inward-pointing unit normal vector on $\partial A_d(r,0)$ by the map $T_r$, where recall $A_d = A_d(0,0)$.
For each $(x,y) \in \partial A_d$,
we recall from \eqref{eq:defNr}
that
$\hat{N}_r(x,y) = \pm \frac{1}{\sqrt{x^2/a^2 + y^2}}\binom{x/a}{y} + O(d^{-1/2})$.
The event $\mathcal{E}$ occurs if
$\langle G \circ T_r, \hat{N}_r \rangle > 0$
at each point on $\partial A_d$
and $G \circ T_r$ has no equilibria in $A_d$.
We may replace $G \circ T_r$ by $G$
without changing the probability of the event
by Remark \ref{rmk:invariant}.
Since $G$ is a nonvanishing scalar multiple of $F$,
we may then replace $G$ by $F$.
This leads us to consider the event $\mathcal{E}_0$ that both of the following are satisfied
\begin{itemize}
\item $\langle F , \hat{N}_r \rangle > 0$
on $\partial A_d$.
\item $F$ has no equilibria in $A_d$.
\end{itemize}
This event has the same probability as $\mathcal{E}$.
As stated in Remark \ref{rmk:basis}, in the description of the Kostlan polynomial as a random linear combination with Gaussian coefficients,
one is free to choose the basis (as long as it is orthonormal with respect to the Fischer product).
Since the degree-$d$ homogenizations of the components of $B_r$ each have unit Fischer norm (by Lemma \ref{lemma:normone}), each can be used as elements in an orthonormal basis (orthonormal with respect to the Fischer product)
while expanding the components of the random vector field $F$ as a linear combination with i.i.d. Gaussian coefficients.
We write this in an abbreviated form as
\begin{align}
F(x,y) &= \binom{p(x,y)}{q(x,y)} \\
&= B_r(x,y) + F_r^\perp(x,y)\\
&= \binom{\xi_0 b_1(x,y)}{\eta_0 b_2(x,y)} + \binom{f_1^\perp(x,y)}{f_2^\perp(x,y)},
\end{align}
where in the first component all the terms involving the basis elements besides those in $b_1$ are collected in $f_1^\perp$, and in the second component all the terms involving the basis elements besides those in $b_2$ are collected in $f_2^\perp$.
Let
$$\tilde{B}_r(x,y) = \binom{\xi_1 b_1(x,y)}{\eta_1 b_2(x,y)}$$
be an independent copy of $B_r(x,y)$,
i.e., $\xi_1$ and $\eta_1$ are standard normal random variables independent of eachother and of $\xi_0$ and $\eta_0$.
Define
$$F_r^{\pm} = F_r^\perp \pm \tilde{B}_r.$$
Then $F_r^{\pm}$ are each distributed as $F$,
and we can write
$$ F(x,y) = B_r(x,y) + \frac{1}{2}(F_r^+ + F_r^-) .$$
Define $\mathcal{E}_1$ to be the event
described in Lemma \ref{lemma:barrier}
concerning $B_r(x,y)$.
Define $\mathcal{E}_2$ to be the event that
$\|F_r^+(x,y)\|_\infty \leq C_0$,
and define $\mathcal{E}_3$ to be the event that
$\|F_r^-(x,y)\|_\infty \leq C_0$,
If $\mathcal{E}_1$, $\mathcal{E}_2$, and $\mathcal{E}_3$ all occur
then Lemma \ref{lemma:barrier} and Remark \ref{rmk:perturb} imply that $\mathcal{E}_0$ occurs. Hence, we have
\begin{equation} \label{eq:intersection}
\mathbb{P} \{ \mathcal{E}_0 \} \geq \mathbb{P} \{ \mathcal{E}_1 \cap \mathcal{E}_2 \cap \mathcal{E}_3 \} .
\end{equation}
By Lemma \ref{lemma:supest},
the complementary events $\mathcal{E}_2^c$ and $\mathcal{E}_3^c$
each have probability less than $1/3$.
Notice that $\mathcal{E}_1$ is independent of $\mathcal{E}_2$ and $\mathcal{E}_3$, but $\mathcal{E}_2$ and $\mathcal{E}_3$ are not independent of each other.
Thus, we use a union bound with \eqref{eq:intersection} to estimate the probability of $\mathcal{E}_0$
\begin{align}
\mathbb{P} \{ \mathcal{E}_0 \} &\geq \mathbb{P} \{ \mathcal{E}_1 \cap \mathcal{E}_2 \cap \mathcal{E}_3 \} \\
&= \mathbb{P} \mathcal{E}_1 \mathbb{P} \{ \mathcal{E}_2 \cap \mathcal{E}_3 \} \\
&\geq \mathbb{P} \mathcal{E}_1 \left( 1 - \mathbb{P} \mathcal{E}_2^c - \mathbb{P} \mathcal{E}_3^c \right)\\
&\geq (1/3)\mathbb{P} \mathcal{E}_1 > 0,
\end{align}
which proves the proposition
since Lemma \ref{lemma:barrier}
provides that the probability of $\mathcal{E}_1$
is positive and independent of $d$.
This concludes the proof of the proposition.
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm:law}]
Let $f,g$ be random real-analytic functions sampled independently
from the Gaussian space induced by the Bargmann-Fock inner product
and normalized by the deterministic scalar $\exp\left\{ -(x^2+y^2)/2 \right\}$.
Explicitly,
$$f(x,y) = \exp\left\{ -(x^2+y^2)/2 \right\} \sum_{j,k \geq 0} a_{j,k} \frac{x^j y^k}{\sqrt{j! k!}} , \quad a_{j,k} \sim N(0,1).$$
Including the factor $\exp\left\{ -(x^2+y^2)/2 \right\}$ ensures that $f,g$ are invariant under translations \cite{BeGa}, and multiplication of a vector field by a non-vanishing scalar function does not affect its trajectories, in particular, its limit cycles.
Let $\hat{N}_R$ denote the number of empty limit cycles situated within the disk of radius $R$ of the vector field
$$F(x,y) = \binom{f(x,y)}{g(x,y)}.$$
We will use the integral geometry sandwich \cite{NazarovSodin1}
\begin{equation}\label{eq:IGS}
\int_{\mathbb{D}_{R-r}} \frac{\mathcal{N}(x, r)}{|\mathbb{D}_r |} dx
\leq \mathcal{N}(0,R) \leq \int_{\mathbb{D}_{R+r}} \frac{\mathcal{N}^*(x, r)}{| \mathbb{D}_r |} dx,
\end{equation}
where
$\mathcal{N}(x,r)$ denotes the number of empty limit cycles completely contained in the disk $\mathbb{D}_r(x)$ of radius $r$ centered at $x$, $\mathcal{N}^*(x,r)$ denotes the number of empty limit cycles that intersect $\overline{\mathbb{D}_r(x)}$, and $|\mathbb{D}_r|=\pi r^2$ denotes the area of $\mathbb{D}_r$.
The statement of the integral geometry sandwich in \cite{NazarovSodin2} is for connected components of a nodal set, but as indicated in the proof it is an abstract result that holds in more generality (cf. \cite{SarnakWigman}, \cite{ALL}) that includes the case at hand.
Dividing by $|\mathbb{D}_R|$, we rewrite \eqref{eq:IGS} as
\begin{equation}
\left(1-\frac{r}{R}\right)^2\frac{1}{|\mathbb{D}_{R-r}|}\int_{\mathbb{D}_{R-r}} \frac{\mathcal{N}(x, r)}{|\mathbb{D}_r |} dx
\leq \frac{\mathcal{N}(0,R)}{|\mathbb{D}_R|} \leq \left(1+\frac{r}{R}\right)^2\frac{1}{|\mathbb{D}_{R+r}|}\int_{\mathbb{D}_{R+r}} \frac{\mathcal{N}^*(x, r)}{| \mathbb{D}_r |} dx.
\end{equation}
Taking expectation
and using translation invariance to conclude that $\mathbb{E} \mathcal{N}(x,r)$ is independent of $x$, we obtain
\begin{equation}\label{eq:sandexp}
\left(1-\frac{r}{R}\right)^2 \frac{\mathbb{E} \mathcal{N}(0, r)}{|\mathbb{D}_r |}
\leq \frac{\mathbb{E} \mathcal{N}(0,R)}{|\mathbb{D}_R|} \leq \left(1+\frac{r}{R}\right)^2 \frac{\mathbb{E} \mathcal{N}^*(0, r)}{| \mathbb{D}_r |} .
\end{equation}
Next we assert that
\begin{equation}\label{eq:tangencies}
\mathcal{N}^*(0,r) \leq \mathcal{N}(0,r) + \mathscr{T}(r),
\end{equation}
where
$$\mathscr{T}(r) = \# \left\{ (x,y) \in \partial \mathbb{D}_r : \langle F(x,y),\binom{x}{y} \rangle = 0 \right\}$$ denotes the number of tangencies of $F$ with the circle of radius $r$.
Indeed, each limit cycle that intersects but is not completely contained in $\mathbb{D}_r$
must have at least one entry and exit point along $\partial \mathbb{D}_r$. By considering the intersection of $\partial \mathbb{D}_r$ with the interior of the limit cycle and selecting one of its connected components, we may choose such an entry-exit pair to be the endpoints of a circular arc of $\partial \mathbb{D}_r$ that is completely contained in the interior of the limit cycle. At these entry and exit points, $F$ is directed inward and outward, respectively, and by the intermediate value theorem applied to $\langle F, \binom{x}{y}\rangle$ there is an intermediate point along $\partial \mathbb{D}_r$ where $F$ is tangent to $\partial \mathbb{D}_r$. By our choice of the entry-exit pair, the point of tangency is in the interior of the limit cycle. Thus, empty limit cycles correspond to distinct such points of tangency, and this verifies \eqref{eq:tangencies}.
Next we use the Kac-Rice formula to prove the following lemma concerning the expectation of the number $\mathscr{T}_r$ of such tangencies along $\partial \mathbb{D}_r$.
\begin{lemma}\label{lemma:tang}
The expectation of $\mathscr{T}_r$ satisfies
\begin{equation}\label{eq:lineartang}
\mathbb{E} \mathscr{T}_r = 2 \sqrt{1+r^2}.
\end{equation}
\end{lemma}
\begin{proof}[Proof of Lemma \ref{lemma:tang}]
Let $h_1(r,\theta)$ denote
$\frac{x}{r}f(x,y)+\frac{y}{r}g(x,y)$
evaluated at $x=r\cos (\theta), y=r \sin (\theta)$.
For fixed $r$ the zeros of $h_1$ correspond to the tangencies counted by $\mathscr{T}_r$.
Hence, applying Theorem \ref{thm:KR}
(it is easy to check that the conditions (i)-(iv) are satisfied)
gives
\begin{equation}\label{eq:KRtang}
\mathbb{E} \mathscr{T}_r = \int_{0}^{2\pi} \int_\mathbb{R} |B| \rho_\theta(0,B) dB d\theta,
\end{equation}
where $\rho_{\theta}(A,B)$
denotes the joint probability density
of $A=h_1(r,\theta)$ and $B=\partial_\theta h_2(r,\theta)$.
The vector field $\binom{x/r}{y/r}$ is invariant with respect to rotations about the origin,
and $f,g$ are also invariant (meaning the distribution of probability on the space of these random functions is invariant) with respect to rotations about the origin.
This implies that the inside integral in \eqref{eq:KRtang} is independent of $\theta$,
and this gives
\begin{equation}\label{eq:KRtang2}
\mathbb{E} \mathscr{T}_r = 2\pi \int_\mathbb{R} |B| \rho_0(0,B) dB .
\end{equation}
Computation of $h_2(r,0)$ gives
$$ h_2 (r,0) = r f_y(r,0) + g(r,0) .$$
Since $f,g$ are independent of each other, and evaluation of $f$ is independent of $f_y$ evaluated at the same point, we have that
$rf_y(r,0) + g(r,0)$ is independent of $f(r,0)$.
So the joint density $\rho_0$ is the product of densities
of $rf_y(r,0)+g(r,0) \sim N(0,1+r^2)$
and $f(r,0) \sim N(0,1)$
which gives
$$\rho_0(A,B) = \frac{1}{2\pi\sqrt{1+r^2}} \exp\left\{-\frac{A^2}{2} \right\} \exp\left\{-\frac{B^2}{2(1+r^2)} \right\} ,$$
and in particular
\begin{equation}
\rho_0(0,B) = \frac{1}{2\pi\sqrt{1+r^2}} \exp\left\{-\frac{B^2}{2(1+r^2)} \right\} .
\end{equation}
Then \eqref{eq:KRtang2} becomes
\begin{equation}
\mathbb{E} \mathscr{T}_r = \frac{2\pi}{\sqrt{2\pi}} \int_\mathbb{R} |B|
\frac{1}{\sqrt{2\pi(1+r^2)}} \exp\left\{-\frac{B^2}{2(1+r^2)} \right\} B,
\end{equation}
where we have separated the constants so that the integral gives the absolute moment of a Gaussian of mean zero and variance $1+r^2$.
From this we conclude
\begin{equation}
\mathbb{E} \mathscr{T}_r = 2\sqrt{1+r^2},
\end{equation}
which gives \eqref{eq:lineartang} as desired and concludes the proof of the lemma.
\end{proof}
Applying \eqref{eq:tangencies} in
\eqref{eq:sandexp} and returning to the abbreviated notation $\hat{N}_R = \mathcal{N}(0,R)$, we have
\begin{equation}\label{eq:sandexp2}
\left(1-\frac{r}{R}\right)^2 \frac{\mathbb{E} \hat{N}_r}{|\mathbb{D}_r |}
\leq \frac{\mathbb{E} \hat{N}_R}{|\mathbb{D}_R|} \leq \left(1+\frac{r}{R}\right)^2 \frac{\mathbb{E} \hat{N}_r+ \mathbb{E} \mathscr{T}(r)}{| \mathbb{D}_r |} .
\end{equation}
Let $\varepsilon>0$ be arbitrary.
We will show that there exists $r$ such that for all $R \gg r$ sufficiently large we have
\begin{equation}\label{eq:goalrR}
\left| \frac{\mathbb{E} \hat{N}_R}{|\mathbb{D}_R|} - \frac{\hat{N}_r}{| \mathbb{D}_r |} \right| < \varepsilon .
\end{equation}
From \eqref{eq:lineartang}
we have
$$ \frac{\mathbb{E} \mathscr{T}(r) }{ | \mathbb{D}_r|} = O(r^{-1}),$$
and using this we choose $r$ large enough so that
\begin{equation}\label{eq:smalltang}
\frac{\mathbb{E} \mathscr{T}(r) }{ | \mathbb{D}_r|} < \frac{\varepsilon}{8}.
\end{equation}
We have
$$\left(1-\frac{r}{R} \right)^2 \frac{\mathbb{E} \hat{N}_r }{ | \mathbb{D}_r|} > \frac{\mathbb{E} \hat{N}_r }{ | \mathbb{D}_r|} - 2\frac{r}{R} \frac{\mathbb{E} \hat{N}_r }{ | \mathbb{D}_r|} ,$$
and we may choose $R>r$ sufficiently large (with the above choice of $r$ now fixed) so that
\begin{equation}\label{eq:lower}
\left(1-\frac{r}{R} \right)^2 \frac{\mathbb{E} \hat{N}_r }{ | \mathbb{D}_r|} > \frac{\mathbb{E} \hat{N}_r }{ | \mathbb{D}_r|} - \varepsilon .
\end{equation}
Choosing $R$ larger if necessary we also have
\begin{equation}\label{eq:upper}
\left(1+\frac{r}{R} \right)^2 \frac{\mathbb{E} \hat{N}_r }{ | \mathbb{D}_r|} < \frac{\mathbb{E} \hat{N}_r }{ | \mathbb{D}_r|} + \frac{\varepsilon}{2} .
\end{equation}
Then \eqref{eq:smalltang}, \eqref{eq:lower}, and \eqref{eq:upper} imply
\begin{equation}
\frac{\mathbb{E} \hat{N}_r}{|\mathbb{D}_r |} - \varepsilon
< \frac{\mathbb{E} \hat{N}_R}{|\mathbb{D}_R|} < \frac{\mathbb{E} \hat{N}_r}{| \mathbb{D}_r |} + \varepsilon ,
\end{equation}
which implies \eqref{eq:goalrR}.
It follows that
$\frac{\mathbb{E} \hat{N}_R}{|\mathbb{D}_R|}$ converges as $R \rightarrow \infty$, i.e., there exists a constant $c$ such that
\begin{equation}
\mathbb{E} \hat{N}_R \sim c \cdot R^2, \quad \text{as } R \rightarrow \infty.
\end{equation}
Positivity of the constant $c$ follows from a simple adaptation of the proof of Theorem \ref{thm:main}; instead of considering shrinking elliptical annuli one can take a collection of circular annuli of fixed radius. One can fit
$\geq k\cdot R^2$ many disjoint such annuli in $\mathbb{D}_R$. The subsequent constructions simplify as well since the distortion factor $a$ is absent.
We omit further details since no new complications arise.
\end{proof}
\section{Limit cycles surrounding a perturbed center focus: Proofs of Theorems \ref{thm:as} and \ref{thm:addendum}}\label{sec:smallpert}
\begin{proof}[Proof of Theorem \ref{thm:as}]
Recall from the statement of the theorem that $N_d(\rho)$ denotes the number of limit cycles in the disk $\mathbb{D}_\rho$ of the vector field $$F(x,y)=\binom{y+\varepsilon(d)p_d(x,y)}{-x + \varepsilon(d) q_d(x,y)}, $$
where
\begin{equation}\label{eq:seriesp}
p_d(x,y) = \sum_{1 \leq j+k \leq d} a_{j,k} x^j y^k,
\end{equation}
\begin{equation}\label{eq:seriesq}
q_d(x,y) = \sum_{1 \leq j+k \leq d} b_{j,k} x^j y^k.
\end{equation}
Let $p_\infty$ and $q_\infty$
denote the random bivariate power series
obtained by letting $d\rightarrow \infty$ in \eqref{eq:seriesp}, \eqref{eq:seriesq}.
Fix $R$ satisfying $\rho < R < 1$.
Since $|a_{j,k}|, |b_{j,k}| \leq 1$
the series \eqref{eq:seriesp}, \eqref{eq:seriesq} are each majorized by
\begin{equation}\label{eq:majorization}
\sum_{j,k \geq 1} |x|^j |y|^k = \sum_{j\geq 1} |x|^j \sum_{k\geq 1} |y|^k,
\end{equation}
which converges uniformly in $\{(x,y)\in \mathbb{C}^2 :|x|\leq R, |y| \leq R \}$.
In particular, $p_\infty$ and $q_\infty$ converge absolutely and uniformly in the bidisk $\mathbb{D}_R \times \mathbb{D}_R$.
Since our goal is to prove almost sure convergence of $N_d(\rho)$, it is important to note that $p_d$ and $p_\infty$
are naturally coupled
in a single probability space;
the $d$th-order truncation of $p_\infty$
is distributed as $p_d$.
Thus, in order to prove the desired almost sure convergence, we sample $p_\infty$, $q_\infty$ and then show that,
almost surely, the sequence $N_d(\rho)$ associated with the truncations $p_d$, $q_d$
converges as $d \rightarrow \infty$.
We take note of the following error estimates based on the tail of the majorization \eqref{eq:majorization},
\begin{equation}\label{eq:majortail}
\sup_{\mathbb{D}_R \times \mathbb{D}_R} |p_\infty - p_d| < R^{2d}/(1-R)^2, \quad \sup_{\mathbb{D}_R \times \mathbb{D}_R} |q_\infty - q_d| < R^{2d}/(1-R)^2.
\end{equation}
Consider the vector field
\begin{equation}\label{eq:infinity}
F_\infty(x,y) = \binom{y + \varepsilon p_\infty(x,y)}{-x + \varepsilon q_\infty(x,y)},
\end{equation}
and let $\mathscr{P}_{\infty,\varepsilon}:\mathbb{R}_+\rightarrow \mathbb{R}_+$ denote the corresponding Poincar\'e first return map along the positive $x$-axis.
Then by Theorem \ref{thm:PPM}, we have the following perturbation expansion
\begin{equation}\label{eq:pertexp}
\mathscr{P}_{\infty,\varepsilon}(r) = r + \varepsilon \mathscr{A}_\infty(r) + \varepsilon^2 E(r,\varepsilon) ,
\end{equation}
where $\mathscr{A}_\infty(r)$ is the first Melnikov function
given by the integral
\begin{equation}\label{eq:A1}
\mathscr{A}_\infty(r) = \int_{x^2+y^2=r^2} p_\infty dy - q_\infty dx.
\end{equation}
Let $X(\rho)$ denote the number of zeros of $\mathscr{A}_\infty$ in $(0,\rho)$.
$X(\rho)$ is our candidate limit for the sequence $N_d(\rho)$.
Before proving this convergence, let us first verify the second part of the conclusion in the theorem, namely, $\mathbb{E} X(\rho)< \infty$.
Note that $\mathscr{A}_\infty$ is analytic in the unit disk, and by the non-accumulation of zeros for analytic functions,
$X(\rho)$ is finite-valued.
On the other hand, the finiteness of its expectation $\mathbb{E} X(\rho)$ requires additional estimates shown in the following lemma.
\begin{lemma}\label{lemma:Finite}
The expectation $\mathbb{E} X(\rho)$ of the number of zeros of $\mathscr{A}_\infty$ in $(0,\rho)$ satisfies
\begin{equation}
\mathbb{E} X(\rho) < \infty.
\end{equation}
\end{lemma}
\begin{proof}[Proof of Lemma \ref{lemma:Finite}]
As above, fix $R$ satisfying $\rho < R < 1$.
Writing the integral \eqref{eq:A1} in polar coordinates, we have
\begin{equation}\label{eq:Aint}
\mathscr{A}_\infty(r) = \int_{0}^{2\pi} p_\infty(r \cos(\theta),r \sin(\theta)) r\cos(\theta) d\theta + q_\infty(r \cos(\theta),r \sin(\theta)) r\sin(\theta) d\theta.
\end{equation}
This defines an analytic function valid for complex values of $r$, and for $r\in \mathbb{D}_R$ with $\theta \in [0,2\pi]$ we have $(r\cos(\theta),r\sin(\theta)) \in \mathbb{D}_R \times \mathbb{D}_R$.
From the majorization \eqref{eq:majorization},
we have $|p_\infty|$ and $|q_\infty|$ are uniformly bounded by $1/(1-R)^2$ in $\mathbb{D}_R \times \mathbb{D}_R$.
Using this to estimate the above integral we obtain
\begin{equation}\label{eq:Asupest}
\sup_{r \in \mathbb{D}_R} |\mathscr{A}_\infty(r)| \leq 2\pi \frac{2R}{(1-R)^2}.
\end{equation}
Integrating term by term in \eqref{eq:Aint},
we can write $\mathscr{A}_\infty(r)$ as a series.
\begin{equation}\label{eq:Ainfinity}
\mathscr{A}_\infty(r) = \sum_{m=0}^\infty \zeta_m r^{2m+2},
\end{equation}
where
\begin{equation}\label{eq:zetam}
\zeta_m = \int_{0}^{2\pi} \sum_{j+k=2m+1} (a_{j,k} \cos(\theta) + b_{j,k} \sin(\theta) ) (\cos(\theta))^j(\sin(\theta))^k d\theta.
\end{equation}
(The odd powers of $r$ do not survive in the series due to symmetry.)
Let $f(r) = \sum_{m=0}^\infty \zeta_m r^{2m} = A_\infty(r)/r^2$.
In order to see that
$\mathbb{E} X(\rho) < \infty$
we consider the number $Y(\rho)$ of complex zeros of $f$ in the disk $\mathbb{D}_\rho$, and we use the following estimate based on Jensen's formula \cite[Ch. 5, Sec. 3.1]{Ahlfors}
$$Y(\rho) \leq \frac{1}{\log(\rho/R)} \log\left(\frac{M}{ |\zeta_1|}\right),$$
where
$$M:=\sup_{|r|=R} |f(r)|.$$
This gives
$$\mathbb{E} Y(\rho) \leq \frac{1}{\log(\rho/R)} \left( \mathbb{E} \log M - \mathbb{E} \log |\zeta_1| \right) .$$
From \eqref{eq:Asupest} we have
\begin{equation}
\sup_{|r|=R} |f(r)| \leq \frac{4\pi}{R(1-R)^2},
\end{equation}
and hence it suffices to show that
\begin{equation}\label{eq:finite}
- \mathbb{E} \log |\zeta_1| < \infty.
\end{equation}
We obtain, after computing the integrals in \eqref{eq:zetam},
\begin{equation}\label{eq:zeta1}
\zeta_1= 2\pi ( a_{3,0} + b_{2,1}/4 + a_{1,2}/4 + b_{0,3}).
\end{equation}
Then the desired estimate \eqref{eq:finite} follows from the triangle inequality and finiteness of the integral $\displaystyle \int_{-1}^1 \log |u| du.$
This completes the proof of the lemma.
\end{proof}
We have that $\mathscr{A}_\infty$ is smooth on $(0,\rho)$
(in fact $\mathscr{A}_\infty$ is analytic in the unit disk), and the probability density of its point evaluations are bounded.
Hence, we can apply Bulinskaya's Lemma \cite[Prop. 1.20]{AzaisWscheborbook} to conclude that almost surely the zeros of $\mathscr{A}_\infty$ are all non-degenerate, i.e.,
the derivative $\mathscr{A}_\infty'$ does not vanish at any zero.
Non-degeneracy implies the following lemma.
\begin{lemma}\label{lemma:C1}
Almost surely, the zero set of $\mathscr{A}_\infty$ is stable with respect to $C^1$-small perturbations.
\end{lemma}
Let us use the following notation for the $C^1$-norm.
$$\left\| \mathscr{F} \right\|_{C^1} := \sup_{0<r<\rho} |\mathscr{F}(r)| + \sup_{0<r<\rho} |\mathscr{F}'(r)| .$$
Let $\mathscr{P}_d:\mathbb{R}_+ \rightarrow \mathbb{R}_+$ denote the Poincar\'e first return map along the positive $x$-axis of the vector field
\begin{equation}
F(x,y) = \binom{y + \varepsilon p_d(x,y)}{-x + \varepsilon q_d(x,y)}.
\end{equation}
In order to show that $N_d(\rho)$ converges to $X(\rho)$, by Lemma \ref{lemma:C1} it is enough to show that $(\mathscr{P}_d(r) - r)/\varepsilon(d)$ approaches $\mathscr{A}_\infty$ in the $C^1$-norm as $d \rightarrow \infty$, i.e.,
\begin{equation}\label{eq:C1close2}
\lim_{d \rightarrow \infty}\left\| \frac{\mathscr{P}_{d}(r) - r}{\varepsilon(d)} - \mathscr{A}_\infty(r) \right\|_{C^1}=0.
\end{equation}
Since Theorem \ref{thm:PPM} provides that the error term $E(r,\varepsilon)$ in the perturbation expansion \eqref{eq:pertexp} is uniformly bounded, we have that
$(\mathscr{P}_{\infty,\varepsilon}(r)-r)/\varepsilon$ approaches $\mathscr{A}_\infty$ in the $C^1$-norm as $\varepsilon \rightarrow 0$.
In particular, since $\varepsilon(d) = o(d)$, we have
\begin{equation}\label{eq:C1close}
\lim_{d \rightarrow \infty}\left\| \frac{\mathscr{P}_{\infty,\varepsilon(d)}(r) - r}{\varepsilon(d)} - \mathscr{A}_\infty(r) \right\|_{C^1}=0.
\end{equation}
We have
\begin{equation}\label{eq:reduced}
\left\| \frac{\mathscr{P}_d(r) - r}{\varepsilon(d)} - \mathscr{A}_\infty(r) \right\|_{C^1} \leq \left\| \frac{\mathscr{P}_{\infty,\varepsilon(d)} - \mathscr{P}_d}{\varepsilon(d)} \right\|_{C^1} + \left\| \frac{\mathscr{P}_{\infty,\varepsilon(d)}(r) - r}{\varepsilon(d)} - \mathscr{A}_\infty(r) \right\|_{C^1},\end{equation}
and, as we have noted in \eqref{eq:C1close}, the second term on the right hand side of \eqref{eq:reduced} converges to zero as $d\rightarrow \infty$.
Hence, our immediate goal \eqref{eq:C1close2} is reduced to proving the following lemma.
\begin{lemma}\label{lemma:delta}
Let $\delta>0$ be arbitrary.
For all $d$ sufficiently large, we have
\begin{equation}\label{eq:suff} \left\| \frac{\mathscr{P}_{\infty,\varepsilon(d)} - \mathscr{P}_d}{\varepsilon(d)} \right\|_{C^1} < \delta. \end{equation}
\end{lemma}
\begin{proof}[Proof of Lemma \ref{lemma:delta}]
In order to show this, we first change to polar coordinates $x=r \cos \theta, y = r \sin \theta$ and write our systems $\binom{\dot{x}}{\dot{y}} = \binom{y+\varepsilon(d) p_d(x,y)}{-x + \varepsilon(d) q_d(x,y)}$ and $\binom{\dot{x}}{\dot{y}} = \binom{y+\varepsilon p_\infty(x,y)}{-x + \varepsilon q_\infty(x,y)}$ as
\begin{equation}\label{eq:dH_d}
\frac{d r}{d \theta} = \varepsilon(d)H_d(r,\theta),
\end{equation}
and
\begin{equation}\label{eq:dH_inf}
\frac{d r}{d \theta} = \varepsilon H_{\infty,\varepsilon}(r,\theta),
\end{equation}
where
$$H_d(r,\theta) = \frac{(x p_d(x,y)+y q_d(x,y))/r}{1+\varepsilon(d)(x q_d(x,y)-y p_d(x,y))/r^2},$$
and
$$H_{\infty,\varepsilon}(r,\theta) = \frac{(x p_\infty(x,y)+y q_\infty(x,y))/r}{1+\varepsilon (x q_\infty(x,y)-y p_\infty(x,y))/r^2}.$$
Fix an initial condition $r(0) = r_0$. We will use $r_d(\theta)$ and $r_\infty(\theta)$ to denote the solutions to the systems \eqref{eq:dH_d}
and \eqref{eq:dH_inf}, respectively, with the same initial condition $r(0)=r_0$.
The estimates \eqref{eq:majorization}, \eqref{eq:majortail} imply that
given any $\delta_0$, we have for all $d$ sufficiently large
\begin{equation} \label{eq:delta0}
\sup_{(r,\theta) \in \mathbb{D}_R \times [0,2\pi]} \left| H_{\infty,\varepsilon(d)}(r,\theta) - H_d(r,\theta) \right| < \delta_0.
\end{equation}
We also have from the estimates \eqref{eq:majortail} that there exists $M>0$ such that
\begin{equation}\label{eq:M}
\sup_{(r,\theta) \in \mathbb{D}_R \times [0,2\pi]} |H_d(r,\theta)| \leq M , \quad \sup_{(r,\theta) \in \mathbb{D}_R \times [0,2\pi]} |H_\infty(r,\theta)| \leq M.
\end{equation}
We have
\begin{equation}
\frac{d}{d \theta} \left( r_\infty(\theta)-r_d(\theta) \right)
= \varepsilon \left[ H_\infty(r_\infty(\theta),\theta)- H_d(r_d(\theta),\theta) \right],
\end{equation}
which implies
\begin{equation} \label{eq:intEst}
|r_\infty(\theta)-r_d(\theta)| \leq \varepsilon \int_{0}^{\theta} \left| H_\infty(r_\infty(\theta),\theta)- H_d(r_d(\theta),\theta) \right| d\theta.
\end{equation}
Applying \eqref{eq:M} to estimate \eqref{eq:intEst} we obtain
\begin{equation} \label{eq:firstEst}
|r_d(\theta) - r_\infty(\theta)| < 2 M \varepsilon(d).
\end{equation}
We have $\mathscr{P}_{\infty,\varepsilon(d)}(r_0) = r_\infty(2\pi)$, $\mathscr{P}_d(r_0) = r_d(2\pi)$,
where recall $r_\infty, r_d$ are the solutions to the systems \eqref{eq:dH_inf}, \eqref{eq:dH_d} with initial condition $r_\infty(0) = r_d(0)=r_0$.
Using this while setting $\theta = 2\pi$ in \eqref{eq:intEst2pi} we have
\begin{equation} \label{eq:intEst2pi}
|\mathscr{P}_{\infty,\varepsilon(d)}(r_0)-\mathscr{P}_d(r_0)| \leq \varepsilon \int_{0}^{2\pi} \underbrace{\left| H_\infty(r_\infty(\theta),\theta)- H_d(r_d(\theta),\theta) \right|}_{(*)} d\theta.
\end{equation}
The above integrand $(*)$ satisfies
\begin{equation}\label{eq:*}
(*) \leq \left| H_d(r_d(\theta),\theta) - H_\infty(r_d(\theta),\theta) \right| + \left| H_\infty(r_d(\theta),\theta)-H_\infty(r_\infty(\theta),\theta)\right|.
\end{equation}
We have from \eqref{eq:delta0}
\begin{equation}\label{eq:*part1}
| H_d(r_d(\theta),\theta) - H_\infty(r_d(\theta),\theta) | < \delta_0, \quad \theta \in [0,2\pi],
\end{equation}
and we also have for all $d$ sufficiently large
\begin{equation}\label{eq:*part2}
|H_\infty(r_d(\theta),\theta)-H_\infty(r_\infty(\theta),\theta)| < \delta_0, \quad \theta \in [0,2\pi].
\end{equation}
The latter follows from the
uniform equicontinuity of the sequence $H_{\infty,\varepsilon(d)}$
since we have as stated in \eqref{eq:firstEst}
that $|r_\infty(\theta)-r_d(\theta)|$
is arbitrarily small for $d$ sufficiently large.
Using \eqref{eq:*}, \eqref{eq:*part1}, \eqref{eq:*part2} to estimate \eqref{eq:intEst2pi}, we obtain
\begin{equation}
|\mathscr{P}_{\infty,\varepsilon(d)}(r_0)-\mathscr{P}_d(r_0)| \leq \varepsilon(d) 4\pi \delta_0.
\end{equation}
This gives
\begin{equation}\label{eq:supPoincare}
\sup_{r\in \mathbb{D}_R} \left| \frac{\mathscr{P}_{\infty,\varepsilon(d)}(r)-\mathscr{P}_d(r)}{\varepsilon(d)} \right| < 4\pi \delta_0,
\end{equation}
and using Cauchy estimates \cite[Ch. 4, Sec. 2.3]{Ahlfors} for the derivatives
we obtain
\begin{equation}
\sup_{r\in \mathbb{D}_\rho} \left| \frac{\mathscr{P}'_{\infty,\varepsilon(d)}(r)-\mathscr{P}'_d(r)}{\varepsilon(d)} \right| < \frac{4\pi}{R-\rho} \delta_0,
\end{equation}
and together with \eqref{eq:supPoincare} this gives
\begin{equation}
\left\| \frac{\mathscr{P}_{\infty,\varepsilon(d)}(r)-\mathscr{P}_d(r)}{\varepsilon(d)} \right\|_{C^1} < 4\pi \left( 1+ \frac{1}{R-\rho}\right) \delta_0.
\end{equation}
Since $\delta_0$ was arbitrary, this verifies
\eqref{eq:suff} and concludes the proof of the lemma.
\end{proof}
Lemma \ref{lemma:delta} along with
\eqref{eq:reduced} and \eqref{eq:C1close}
implies \eqref{eq:C1close2}.
Hence, almost surely, we have for all $d$ sufficiently large, $(\mathscr{P}_d(r)-r)/\varepsilon(d)$
lies within the $C^1$-neighborhood of stability of $\mathscr{A}_\infty$ provided by Lemma \ref{lemma:C1}, and we conclude that the fixed points of $\mathscr{P}_d$ in the interval $(0,\rho)$ are in one-to-one correspondence with the zeros of $\mathscr{A}_\infty$.
This verifies the desired almost sure convergence of $N_d(\rho)$ to $X(\rho)$.
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm:addendum}]
We shall reduce this problem to one where we can apply the results from \cite{Brudnyi1}. Some lines below closely follow the proof of \cite[Thm. A]{Brudnyi1}.
We have
\begin{equation}
p(x,y) = \sum_{1 \leq j+k \leq d} a_{j,k} x^j y^k, \quad
q(x,y) = \sum_{1 \leq j+k \leq d} b_{j,k} x^j y^k,
\end{equation}
with the vector of all coefficients
sampled uniformly from the $d(d+3)$-dimensional unit ball $\displaystyle \left\{ \sum_{1 \leq j+k \leq d} (a_{j,k})^2 + (b_{j,k})^2 \leq 1 \right\}$.
We change variables by a scaling
$u=x/(2\rho)$, $v=y/(2\rho)$.
In these coordinates, the system
\begin{equation}\label{eq:xy}
\binom{\dot{x}}{\dot{y}} = \binom{-y + \varepsilon p(x,y)}{x + \varepsilon q(x,y)}
\end{equation}
becomes
\begin{equation}\label{eq:uv}
\binom{\dot{u}}{\dot{v}} = \binom{- v + \varepsilon \hat{p}(u,v)}{ u + \varepsilon \hat{q}(u,v)},
\end{equation}
where
$\hat{p}(u,v) = 2 \rho \cdot p(2\rho u, 2 \rho v)$,
and $\hat{q}(u,v) = 2 \rho \cdot q(2\rho u, 2 \rho v)$.
We are concerned with limit cycles
situated in the disk $\mathbb{D}_{1/2}$ of the $(u,v)$-plane.
Changing to polar coordinates $u=r \cos \phi, v = r \sin \phi$, we obtain
\begin{equation}\label{eq:dH}
\frac{\partial r}{\partial \phi} = H(r,\phi) r, \quad H(r,\phi) = \frac{\varepsilon(u\hat{p}(u,v)+v \hat{q}(u,v))/r^2}{1+\varepsilon(u\hat{q}(u,v)-v\hat{p}(u,v))/r^2}.
\end{equation}
We complexify the radial coordinate $r$,
and let $U = \mathbb{D} \times [0,2\pi]$.
Then a simple modification of the estimates in \cite[p. 236]{Brudnyi1} gives
$$ \sup_{U} \left| \frac{\hat{p}(u,v)}{r} \right| \leq (2\rho)^d \sqrt{\sum (a_{j,k})^2} \sqrt{d} \leq (2\rho)^d \sqrt{d} ,$$
$$ \sup_{U} \left| \frac{\hat{q}(u,v)}{r} \right| \leq (2\rho)^d \sqrt{\sum (b_{j,k})^2} \sqrt{d}
\leq (2\rho)^d \sqrt{d},$$
where we used $\sum (a_{j,k})^2 + (b_{j,k})^2 \leq 1$.
These estimates imply
\begin{align*}
\sup_{U} | H(r,\phi)| &\leq \frac{\varepsilon (2\rho)^d\sqrt{d}}{1-\varepsilon (2\rho)^d \sqrt{d}} \\
&\leq 3 \sqrt{d} (2\rho)^d \varepsilon.
\end{align*}
Since $\varepsilon \leq \frac{(2\rho)^{-d(d+3)}}{40\pi \sqrt{d}}$,
we have
$$ \sup_{U} | H(r,\phi)| \leq \frac{3}{40\pi},$$
so that the hypothesis of \cite[Prop. 3.1]{Brudnyi1} is satisfied
and we conclude that
the Poincar\'e map $p_\mu$ as in \cite[p. 237]{Brudnyi1}
along with the functions
$$g_\mu(r) = \frac{p_\mu(r)}{r} - 1, \quad h_\mu(r) = \frac{40 \sqrt{d}}{\sqrt{2}}g_\mu(r) $$
are analytic in $\mathbb{D}_{3/4}$
and depend anaytically on the vector $\mu$ of coefficients throughout the ball
in $\mathbb{C}^{d(d+3)}$ of radius $2N$, with $N = \frac{1}{40\pi \sqrt{d}}$.
The rest of the proof, consisting of an application of \cite[Thm. 2.3]{Brudnyi1} now
follows exactly as in \cite[p. 237]{Brudnyi1}.
\end{proof}
\section{Infinitesimal perturbations: Proofs of Theorems \ref{thm:sqrt} and \ref{thm:powerlaw}}\label{sec:infinitesimal}
Before proving Theorem \ref{thm:sqrt}, we state several lemmas that hold more generally for perturbed Hamiltonian systems.
Consider the system
\begin{equation}\label{eq:systemKostpertHam}
\begin{cases}
\dot{x} = \partial_x H(x,y) + \varepsilon p(x,y) \\
\dot{y} = -\partial_y H(x,y) + \varepsilon q(x,y)
\end{cases},
\end{equation}
with $p,q$ Kostlan random polynomials of degree $d$.
Let $A$ be a period annulus of the unperturbed Hamiltonian system ($\varepsilon=0$), and let $N_{d,\varepsilon}(A)$ denote the number of limit cycles in $A$.
\begin{lemma}\label{lemma:Nd}
Fix a period annulus $A$ of the Hamiltonian system. As $\varepsilon \rightarrow 0^+$ the number $N_{d,\varepsilon}(A)$ of limit cycles in $A$ converges almost surely to the random variable $N_d(A)$ that counts the number of zeros of the first Melnikov function for the system \eqref{eq:systemKostpert} given by
\begin{equation}\label{eq:firstMelnikov}
\mathscr{A}(t) = \int_{C_t} p dy - q dx,
\end{equation}
where $C_t = \{H(x,y)=t\}$.
\end{lemma}
\begin{proof}[Proof of Lemma \ref{lemma:Nd}]
Since $p,q$ are Gaussian random polynomials
and the curves $C_t$ provide a smooth foliation of $A$, the random function $\mathscr{A}$ is a smooth (in fact analytic) non-degenerate Gaussian random function, and the probability density of its point evaluations $\mathscr{A}(t)$ are uniformly bounded.
Bulinskaya's Lemma \cite[Prop. 1.20]{AzaisWscheborbook} then implies
that almost surely the zeros of $\mathscr{A}$ are all non-degenerate;
the lemma now follows from an application of Theorem \ref{thm:PPM}.
\end{proof}
\begin{lemma}\label{lemma:EK}
The expectation $\mathbb{E} N_d(A) $ of the number of zeros of $\mathscr{A}$ in $A$ is given by
\begin{equation}\label{eq:EK}
\mathbb{E} N_d(\rho) = \frac{1}{\pi} \int_a^b \sqrt{ \frac{\partial^2}{\partial r \partial t} \log(\mathscr{K}(r,t)) \rvert_{r=t=\tau} } \, d\tau,
\end{equation}
where $a$ and $b$ are chosen so that $A$ is a component of the set $\{ a < H(x,y) < b \}$ and where
\begin{equation}\label{eq:2pt}
\mathscr{K}(r,t) = \mathbb{E} \mathscr{A}(r) \mathscr{A}(t)
\end{equation}
denotes the two-point correlation function of $\mathscr{A}$.
\end{lemma}
\begin{proof}[Proof of Lemma \ref{lemma:EK}]
Since $\mathscr{A}$ satisfies the conditions of Theorem \ref{thm:KR}, this follows from an application of Edelman and Kostlan's formulation \eqref{eq:EKform} of the Kac-Rice formula.
\end{proof}
\begin{lemma}\label{lemma:twopt}
The two-point correlation function of $\mathscr{A}$ defined in \eqref{eq:2pt} satisfies
\begin{equation} \label{eq:kernel}
\mathscr{K}(r,t)= \int_{C_r} \int_{C_t} (1+x_1 x_2 +y_1 y_2)^d \left[ dy_1 dy_2 + dx_1 dx_2 \right].
\end{equation}
\end{lemma}
\begin{proof}[Proof of Lemma \ref{lemma:twopt}]
Let us write
\begin{equation}\label{eq:ArAt}
\mathscr{A}(r) = \int_{C_r} p(v_1) dy_1 - q(v_1)dx_1, \quad \mathscr{A}(t) = \int_{C_r} p(v_2) dy_2 - q(v_2)dx_2,
\end{equation}
with $v_1=(x_1,y_1), v_2=(x_2,y_2)$.
Using \eqref{eq:ArAt}
and linearity of expectation,
the expectation in
\eqref{eq:2pt} can be expressed as
\begin{equation}
\int_{C_r} \int_{C_t} \mathbb{E} \left[ p(v_1) p(v_2) dy_1 dy_2 - p(v_1)q(v_2)dy_1 dx_2 - q(v_1)p(v_2)dx_1 dy_2 + q(v_1)q(v_2) dx_1 dx_2 \right],
\end{equation}
and since the pointwise evaluations
of $p, q$ are centered Gaussians independent of eachother,
we have, for each $v_1=(x_1,y_1), v_2=(x_2,y_2)$,
$$\mathbb{E} p(v_1)q(v_2) = \mathbb{E} q(v_1)p(v_2) = 0,$$
which leads to the following simplified expression for the two-point correlation.
\begin{equation}
\mathscr{K}(r,t) = \int_{C_r} \int_{C_t} \left[ \mathbb{E} p(v_1) p(v_2) dy_1 dy_2 + \mathbb{E} q(v_1)q(v_2) dx_1 dx_2 \right].
\end{equation}
We write this as
\begin{equation}\label{eq:covar}
\mathscr{K}(r,t) = \int_{C_r} \int_{C_t} \left[ K(v_1, v_2) dy_1 dy_2 + K(v_1,v_2) dx_1 dx_2 \right],
\end{equation}
where $K(v_1,v_2) = \mathbb{E} p(v_1) p(v_2)$ denotes the covariance kernel for the Kostlan ensemble.
Finally, substitute in \eqref{eq:covar} the known expression $$ K(v_1,v_2) = (1+x_1 x_2 + y_1 y_2)^d$$
from \eqref{eq:2ptK}
for the covariance kernel of the Kostlan ensemble.
This gives \eqref{eq:kernel} and concludes the proof of the lemma.
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm:sqrt}]
As in the statement of the theorem, $N_{d,\varepsilon}(\rho)$ denotes the number of limit cycles in $\mathbb{D}_\rho$ of the system
\begin{equation}\label{eq:systemKostpert}
\begin{cases}
\dot{x} = y + \varepsilon p(x,y) \\
\dot{y} = -x + \varepsilon q(x,y)
\end{cases},
\end{equation}
with $p,q$ Kostlan random polynomials of degree $d$.
By Lemma \ref{lemma:Nd}, as $\varepsilon \rightarrow 0^+$ $N_{d,\varepsilon}(\rho)$ converges almost surely to the number $N_d(\rho)$ of zeros of $\mathscr{A}$ in $(0,\rho)$ with
$$\mathscr{A}(r) = \int_{C_r} p dx - q dy, \quad C_r = \{ x^2 + y^2 = r^2 \} .$$
Let $\mathscr{K}(r,t) = \mathbb{E} \mathscr{A}(r) \mathscr{A}(t)$ denote the two-point correlation function of $\mathscr{A}$ as in \eqref{eq:2pt}.
We parameterize $C_r = \{ x^2 + y^2 = r^2 \}$ by $(x_1,y_1) = (r \cos \theta_1, r \sin \theta_1)$ and
$C_t$ by $(x_2,y_2) = (t \cos \theta_2, t \sin \theta_2)$.
In terms of this parameterization we can write the integral \eqref{eq:kernel} as
$$ \mathscr{K}(r,t) = \int_{0}^{2\pi} \int_{0}^{2\pi} (1+rt\cos(\theta_1-\theta_2))^d r t \cos(\theta_1-\theta_2) d\theta_1 d\theta_2.
$$
Changing variables in the inside integral with $u=\theta_1-\theta_2$, $du=d\theta_1$ leads to an integrand independent of $\theta_2$, and we obtain
\begin{equation}\label{eq:du}
\mathscr{K}(r,t) = 2 \pi \int_{0}^{2\pi} (1+rt\cos(u))^d r t \cos(u) du.
\end{equation}
Applying Laplace's method \cite[Sec. 4.2]{deBruijn} for asymptotic evaluation of the integral \eqref{eq:du}, we find
\begin{equation}\label{eq:Laplace}
\mathscr{K}(r,t) = 2\pi (1+rt)^d \sqrt{\frac{2\pi rt (rt+1)}{d}}(1+E_d(r,t)), \quad \text{as } d \rightarrow \infty,
\end{equation}
where $E_d(r,t) = O(d^{-1})$.
From \eqref{eq:Laplace} we obtain
\begin{equation}\label{eq:1/dasymp}
\frac{\log \mathscr{K}(r,t)}{d} = \log(1+rt) + \frac{1}{d} \log 2\pi \sqrt{\frac{2\pi rt(1+rt)}{d}} + \frac{1}{d}\log(1+E_d(r,t)).
\end{equation}
The functions $\frac{\log \mathscr{K}(r,t)}{d}$
are analytic in a complex neighborhood $U$ of $(r,t) \in [0,\infty) \times [0,\infty)$
and converge to $\log(1+rt)$ as $d \rightarrow \infty$.
The convergence is uniform on compact subsets of $U$. This justifies, by way of Cauchy estimates, differentiation of the asymptotic \eqref{eq:1/dasymp} to obtain
\begin{equation} \label{eq:asymp}
\lim_{d \rightarrow \infty} \frac{1}{d}\frac{\partial^2}{\partial r \partial t}\log \mathscr{K}(r,t) \rvert_{r=t=\tau} = \frac{1}{(1+\tau^2)^2},
\end{equation}
where the convergence is uniform for $\tau \in [0,\rho]$.
Applying this to \eqref{eq:EK} gives
\begin{equation}\label{eq:sqrtLaw}
\mathbb{E} N_d(\rho) \sim \frac{\sqrt{d}}{\pi} \int_0^\rho \frac{1}{(1+\tau^2)} \, d\tau, \quad \text{as } d \rightarrow \infty,
\end{equation}
and computing the integral in \eqref{eq:sqrtLaw} we find
\begin{equation}
\mathbb{E} N_d(\rho) \sim \frac{\sqrt{d}}{\pi} \arctan(\rho), \quad \text{as } d \rightarrow \infty,
\end{equation}
as desired. This concludes the proof of the theorem.
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm:powerlaw}]
The initial step follows the above proof of Theorem \ref{thm:sqrt}. Let $\mathscr{A}$ denote the first Melnikov function of the system $\binom{\dot{x}}{\dot{y}} = F(x,y)$, where recall $\displaystyle F(x,y) = \binom{y+\varepsilon p(x,y)}{-x + \varepsilon q(x,y)} $ with
\begin{equation}
p(x,y) = \sum_{m=1}^d \sum_{j+k = m} c_m a_{j,k} x^j y^k, \quad
q(x,y) = \sum_{m=1}^d \sum_{j+k = m} c_m b_{j,k} x^j y^k,
\end{equation}
where the deterministic weights $c_m$ satisfy \eqref{eq:powerlawvar}.
From another application of
Lemma \ref{lemma:Nd} we have that $N_{d,\varepsilon}$
converges almost surely to $N_d$ as $\varepsilon \rightarrow 0^+$, where $N_d$ denotes the number of zeros of the first Melnikov function.
Using polar coordinates and integrating term by term, we obtain
\begin{align*}
\mathscr{A}(r) &= \int_{x^2+y^2=r^2} p dy - q dx \\
&= \int_{0}^{2\pi} p(r \cos(\theta),r \sin(\theta)) r\cos(\theta) d\theta + q(r \cos(\theta),r \sin(\theta)) r\sin(\theta) d\theta \\
&= \sum_{m=0}^{\lfloor (d-1)/2 \rfloor} c_{2m+1} \zeta_m r^{2m+2},
\end{align*}
where
\begin{align}
\zeta_m &= \int_{0}^{2\pi} \sum_{j+k=2m+1} (a_{j,k} \cos(\theta) + b_{j,k} \sin(\theta) ) (\cos(\theta))^j(\sin(\theta))^k d\theta \\
&= \sum_{j+k=2m+1} a_{j,k}\int_{0}^{2\pi} (\cos(\theta))^{j+1}(\sin(\theta))^k d\theta + b_{j,k}\int_{0}^{2\pi} (\cos(\theta))^j(\sin(\theta))^{k+1} d\theta.
\end{align}
The variance $\sigma_m^2 = \mathbb{E} \zeta_m^2 $ of $\zeta_m$ is given by
\begin{equation}
\label{eq:sigma_m}
\begin{split}
\sigma_m^2 = \sum_{k=0}^{2m+1} \mathbb{E} a_{2m+1-k,k}^2 & \left( \int_{0}^{2\pi} (\cos(\theta))^{2m+2-k}(\sin(\theta))^k d\theta\right)^2 \\
+ \mathbb{E} & b_{2m+1-k,k}^2\left( \int_{0}^{2\pi} (\cos(\theta))^{2m+1-k}(\sin(\theta))^{k+1} d\theta\right)^2 .
\end{split}
\end{equation}
We use the identities \cite{Grad}
$$ \int_{0}^{2\pi} (\cos(\theta))^{2m+2-k}(\sin(\theta))^k d\theta =
\begin{cases}
2\pi \frac{(2m-k+1)!!(k-1)!!}{(2m+2)!!} &, \quad k \text{ even} \\
0 &, \quad k \text{ odd}
\end{cases},$$
$$ \int_{0}^{2\pi} (\cos(\theta))^{2m+1-k}(\sin(\theta))^{k+1} d\theta =
\begin{cases}
0 &, \quad k \text{ even} \\
2\pi \frac{(2m-k)!!(k)!!}{(2m+2)!!} &, \quad k \text{ odd}
\end{cases},$$
in order to obtain
\begin{equation}\sigma_m^2 = 4\pi^2 \sum_{\ell=0}^m
\mathbb{E} a_{j,k}^2 \left(\frac{(2m-2\ell+1)!!(2\ell-1)!!}{(2m+2)!!}\right)^2 +
\mathbb{E} b_{j,k}^2 \left(\frac{(2m-2\ell-1)!!(2\ell+1)!!}{(2m+2)!!}\right)^2 .
\end{equation}
The asymptotic behavior as $m\rightarrow \infty$
is dominated by just two terms (one for $\ell=0$ and one for $\ell=m$), and
we have
$$\sigma_m^2 \sim 8\pi^2 \left( \frac{(2m+1)!!}{(2m+2)!!}\right)^2 \sim 8\pi m^{-1}.$$
Let us write $\zeta_m = \sigma_m \hat{\zeta}_m$, where $\hat{\zeta}_m$ has mean zero, unit variance, and uniformly bounded moments, and
\begin{equation}
\mathscr{A}(r) = \sum_{m=0}^{\lfloor (d-1)/2 \rfloor} c_{2m+1} \sigma_m \hat{\zeta}_m r^{2m+2}.
\end{equation}
Letting
\begin{equation}
f(s) = \sum_{m=0}^{\lfloor (d-1)/2 \rfloor} c_{2m+1} \sigma_m \hat{\zeta}_m s^{m},
\end{equation}
we have $\mathscr{A}(r) = r^2 f(r^2)$.
Note that $\mathscr{A}$ and $f$ have the same number of zeros in $(0,\infty)$.
Since $c_{2m+1}^2 \sigma_m^2 \sim 8\pi 2^\gamma m^{\gamma-1}$ as $m \rightarrow \infty$, we can apply \cite[Corollary 1.6]{DoVu} to conclude
\begin{equation}
\mathbb{E} N_d \sim \frac{1+\sqrt{\gamma}}{2 \pi} \log d, \quad \text{as } d \rightarrow \infty,
\end{equation}
as desired.
\end{proof}
\section{Future Directions and Open Problems}\label{sec:concl}
\subsection{The Real Fubini-Study Ensemble}
The Kostlan model is sometimes referred to as the Complex Fubini-Study model, since it arises
from the inner product associated to integration with respect to the complex Fubini-Study metric.
The Gaussian model induced by the inner product alternatively associated with integration with respect to the real Fubini-Study metric has been referred to as the ``Real Fubini-Study model'' \cite{sarnak}.
While an explicit description of the Real Fubini-Study model is more complicated than that of the Complex Fubini-Study model and requires expansions in terms of Legendre polynomials (or more generally Gegenbauer polynomials in higher-dimensions), it has the attractive feature of exhibiting more extreme behavior.
For instance, the average number of equilibria grows quadratically, and probabilistic studies on the first part of Hilbert's sixteenth problem \cite{LLstatistics}, \cite{NazarovSodin2}, \cite{SarnakWigman} show that the number of ovals in a random curve given by the zero set of a random polynomial sampled from the Real Fubini-Study model grows quadratically.
It seems likely that the proof of Theorem \ref{thm:main} can be extended to prove a quadratic lower bound on the average number of limit cycles of a random vector field with components sampled from the Real Fubini-Study model.
However, the lower bounds shown in \cite{ChLl} grow faster than quadratically, having an additional factor of $\log d$.
Does the average (over the Real Fubini-Study ensemble) grow faster than quadratically?
\subsection{Zeros of Random Abelian Integrals}\label{sec:abelian}
As indicated in the lemmas of Section \ref{sec:infinitesimal}, some of the methods in the proof of Theorem \ref{thm:sqrt} apply more generally to the study of perturbed Hamiltonian systems
\begin{equation}
\begin{cases}
\dot{x} = \partial_y H(x,y) + \varepsilon p(x,y) \\
\dot{y} = -\partial_x H(x,y) + \varepsilon q(x,y)
\end{cases},
\end{equation}
where $H$ is a fixed generic Hamiltonian and $p,q$ are Kostlan random polynomials of degree $d$.
The associated first Melnikov function
\begin{equation}
\mathscr{A}(t) = \int_{C_t} p \, dy - q \, dx
\end{equation}
for this problem is an Abelian integral.
Here recall $C_t$ is a connected component of the level set $\{ (x,y) \in \mathbb{R}^2: H(x,y) = t \}$.
Since the level sets of $H$ can have multiple connected components, the function $\mathscr{A}$ is multi-valued, and we are interested in the zeros of its real branches.
We can apply Lemma \ref{lemma:EK} along each branch and collect the results to obtain an exact formula for the expected number of real zeros of $\mathscr{A}$.
Asymptotic analysis, on the other hand, is delicate
and will be carried out in a forthcoming work.
\subsection{Random Li\'enard Systems}
Smale posed the problem \cite{Smale1998} of estimating the number of limit cycles for the special class of vector fields
\begin{equation}\label{eq:Lienard}
F(x,y) = \binom{y - f(x)}{-x},
\end{equation}
where $f(x)$ is a real polynomial of odd degree $2k+1$ and satisfying $f(0)=0$.
Without requiring a smallness assumption on $f$, the trajectories of \eqref{eq:Lienard} retain the same topological structure as the perturbed center focus; there is a single equilibrium at the origin and the trajectories wind around this point. In particular, the Poincar\'e map is always globally defined along the entire positive $x$-axis.
For this reason, the following probabilistic version of Smale's problem seems to provide a non-perturbative problem that is more approachable (perhaps using methods of \cite{Brudnyi1}, \cite{Brudnyi2}) than the one mentioned at the end of Section \ref{sec:KSS}.
\begin{prob}
Determine an upper bound on the expected global number of limit cycles of the vector field \eqref{eq:Lienard}
where $f$ is a random univariate polynomial.
\end{prob}
The outcome will depend on the choice of model from which $f$ is sampled.
\subsection{Limit cycles on a cylinder}
A problem of Pugh,
with a revision suggested by Lins-Neto \cite{LN}, asks to study the number of solutions of the one-dimensional differential equation with boundary condition
\begin{equation}\label{eq:cylinder}
\frac{dx}{dt} = f(t,x), \quad x(0)=x(1) ,\end{equation}
with $f$ a polynomial in $x$ whose coefficients are analytic $1$-periodic functions in $t$.
Pugh's original problem asked for an upper bound depending only on the degree in $x$. After presenting counter-examples, Lins-Neto proposed to take coefficients that are trigonometric polynomials in $t$ and asked for an upper bound depending on both the degrees in $x$ and $t$.
Solutions of \eqref{eq:cylinder} can be viewed as limit cycles on a cylinder.
We pose a randomized version of the problem where we take
\begin{equation}
f(t,x)=\sum_{\lambda_i \leq \Lambda} a_i \phi_i(t,x)
\end{equation}
to be a Gaussian band-limited function, i.e., a truncated eigenfunction expansion. The basis $\{\phi_i\}$ consists of Laplace eigenfunctions of the cylinder $\phi_i$ with eigenvalue $\lambda_i$, and we consider the frequency cut-off $\Lambda$ in place of degree as the large parameter in this model.
The random function $f$ is periodic in $t$ and translation-invariant in $x$.
\begin{prob}
Let $f(t,x)$ be a Gaussian band-limited function with frequency cut-off $\Lambda$.
Study the average number $\mathbb{E} N_\Lambda([a,b])$ of solutions of \eqref{eq:cylinder} over a finite interval $x(0) \in [a.b]$.
\end{prob}
Assuming finiteness of this average number, the translation-invariance in $x$ implies that the ``first intensity'' of limit cycles is constant, i.e., we have
\begin{equation}
\mathbb{E} N_\Lambda([a,b]) = (b-a) F(\Lambda).
\end{equation}
Based on the natural scale provided by the wavelength $1/\Lambda$, a na\"ive guess is that $F(\Lambda) \sim C\cdot\Lambda$ as $\Lambda \rightarrow \infty$, for some constant $C>0$.
\subsection{High-dimensional Systems and the May-Wigner Instability}
Studies in population ecology indicate that an ecological system with a large number of species
is unlikely to admit stable equilibria.
This phenomenon is referred to as the \emph{May-Wigner instability} and was first observed by May
\cite{MAY1972} in linear settings to be a consequence of Wigner's semi-circle law from random matrix theory. An updated and more refined treatment comparing several random matrix models is provided in \cite{Allesina2012}, and
a non-linear version of the May-Wigner instability (yet still based on random matrix theory) is presented in
\cite{Fyodorov}.
All of the results in this area concern the local stability of equilibria. However, as pointed out in \cite{Allesina2015}, a lack of equilibria does not preclude the persistence of a system; the coexistence of populations can be achieved through the existence of stable limit cycles and other
stable invariant sets.
This naturally leads to the following problem.
\begin{prob}
Study the existence of stable limit cycles and other stable structures
in high-dimensional random vector fields.
Find an asymptotic lower bound on the probability that at least one stable forward-invariant set exists as the dimension becomes large.
\end{prob}
It may be useful to adapt elements from the proof of Theorem \ref{thm:main},
while replacing the transverse annulus with an appropriate trapping region.
However, additional tools will be required to address the high-dimensional nature of the problem.
\bibliographystyle{abbrv}
|
1,108,101,564,536 | arxiv | \section*{Introduction}
We start by explaining how subfactor theory, conformal field theory, and the groups of Richard Thompson got connected. Second, we outline a general program: what do we wish and what are our motivations and aims. Third, we explain our general formalism based on the manipulation of diagrams and explain how it relates to existing constructions in the literature of group theory. We end by describing the content of this present article and briefly mentioning some works in progress.
{\bf Subfactor, planar algebra, conformal field theory, and braid.}
In the 1980's Vaughan Jones initiated subfactor theory which rapidly became a major field in operator algebras \cite{Jones83}.
Connections with mathematical physics naturally appeared (see \cite{Evans-Kawahigashi92, Kawahigashi18-ICM,Evans22}) but also with seemingly completely unrelated subjects such as knot theory with the introduction of the Jones polynomial \cite{Jones85,Jones87}.
Jones worked very hard in finding new point of views and powerful formalisms to study objects.
This led to the introduction of Jones' planar algebras: a description of the standard invariant of subfactors using diagrams similar to string diagrams in tensor categories \cite{Jones99}.
{\bf Richard Thompson's groups $F,T,$ and $V$.}
On a rather different part of mathematics lives the three Richard Thompson group $F\subset T\subset V$ \cite{Cannon-Floyd-Parry96}.
These groups naturally appear in various parts of mathematics such as logic, topology, dynamics, complexity theory to name a few.
They follow rather unusual properties and display new phenomena in group theory.
For instance, the groups $T$ and $V$ were the first examples found of finitely presented simple infinite groups.
Moreover, Brown and Geoghegan proved that $F$ satisfied the topological finiteness property $F_\infty$ and was of infinite geometric dimension, providing the first torsion-free group of this kind, see Section \ref{sec:def-finiteness} for definitions \cite{Brown-Geoghegan84}.
These groups seem completely unrelated to subfactors although they share one common feature: elements of Thompson's groups can be described by planar diagrams (a pair of rooted trees) as shown by Brown \cite{Brown87}.
{\bf From subfactors to almost CFT, and Thompson's group $T$.}
In the 2010's Jones discovered an unexpected connection between subfactors and the Thompson groups \cite{Jones17}, see also the survey \cite{Brothier20-survey}.
The story goes as follows.
Subfactors and conformal field theory (in short CFT) in the formalism of Doplicher-Haag-Roberts (DHR), have been closely linked since the late 1980's when Longo quickly realised that the Jones index is equal to the square of the statistical dimension in DHR theory \cite{Doplicher-Haag-Roberts71, Longo89}.
Moreover, in the mid 1990's Longo and Rehren prove that any CFT produces a subfactor \cite{Longo-Rehren95}.
However, the converse remains mysterious and is one of the most important question in the field, see \cite{Evans-Gannon11, Bischoff17, Xu18}.
Jones famous question was: "Does every subfactor has something to do with conformal field theory?"
There are some case by case reconstruction results but the most exotic and interesting subfactors are not known to provide CFT at the moment.
Using planar algebras Jones constructed a field theory associated to {\it each} subfactor that is not quite conformal but has a rather discrete group of symmetry.
This symmetry group is nothing else than Thompson's group $T$.
{\bf New connections and Jones' technology.}
From there, new field theories were introduced but also connections between subfactors, Thompson's groups, and braid groups \cite{Jones17,Jones16}.
For the last connection we invite the reader to consult the survey of Jones \cite{Jones19-survey}.
Moreover, Jones found an efficient technology for constructing actions of Thompson's group $F$ using diagrams.
Originally, this was done by transforming tree-diagrams of Thompson group into string diagrams in a planar algebra or a nice tensor category.
Jones quickly realised that it could be greatly generalised founding {\it Jones' technology}.
It works as follows.
Consider a monoid or a category $\mathcal C$ admitting a calculus of fractions.
This allows to formally inverting elements (or more precisely morphisms) of $\mathcal C$ obtaining a fraction groupoid $\Frac(\mathcal C)$ and fraction groups $\Frac(\mathcal C,e)$ (that are isotropy groups at object $e$).
Now, the new part of Jones is that any functor $\Phi:\mathcal C\to \mathcal D$ starting from $\mathcal C$ and ending in any category can produce an action of $\Frac(\mathcal C)$ and thus of $\Frac(\mathcal C,e)$.
Hence, if a complicated group $G$ can be expressed as $\Frac(\mathcal C,e)$ from a somewhat simpler category $\mathcal C$, then we can produce many actions of $G$ using the simpler structure $\mathcal C$.
For instance, any isometry $H\to H\otimes H$ between Hilbert spaces provides a unitary representation of $F$ (which in fact extends to $V$).
This is done by applying Jones' technology to the categorical description of the Thompson groups.
Moreover, Jones' technology is very explicit giving practical algorithms for extracting information of the action constructed such as matrix coefficients in the case of unitary representations.
Applications has been given in various fields such as: knot theory, mathematical physics, general group theory, study of Thompson's groups, operator algebras, noncommutative probability theory \cite{Jones19-survey,Aiello-Brothier-Conti21, Jones18-Hamiltonian, Brothier-Stottmeister19,Brothier-Stottmeister-P, Brothier19WP, Brothier-Jones19,Brothier-Jones19bis, Kostler-Krishnan-Wills20, Kostler-Krishnan22}.
\subsection*{Program: wishes, motivations, and aims.}
Our program is to consider groups $G=\Frac(\mathcal C,e)$ constructed from categories $\mathcal C$ of diagrams that are well-suited for approximating CFT, but also for applying Jones' technology, and are interesting groups on their own.
More specifically:
\begin{itemize}
\item (Calculus of fractions)
We wish to be able to decide easily if a category of diagrams $\mathcal C$ admits a cancellative {\it calculus of fractions} requiring $\mathcal C$ to be left-cancellative and satisfying Ore's property (i.e.~if $t,s\in\mathcal C$ have same target, then there exists $f,g\in\mathcal C$ satisfying $tf=sg$). This permits to construct groups.
\item (From subfactors to CFT) Given any subfactor, Jones constructed from its planar algebra $\mathcal P$ a field theory where Thompson's group $T$ took the role of the space-time diffeomorphism group.
With $\mathcal P$ fixed we wish to perform a similar construction but having a more sophisticate and specific symmetry group $\Frac(\mathcal C,e)$.
\item (Exceptional group properties) Forgetting about CFT and subfactors we wish to construct fraction groups $\Frac(\mathcal C,e)$ that are interesting on their own: groups satisfying some of the exceptional properties of the Thompson groups. This would provide new examples of rare phenomena in group theory but also would help understanding why the Thompson groups are so special.
\item (Applying Jones' technology)
We wish to apply efficiently Jones' technology to the fraction groups $\Frac(\mathcal C,e)$ for constructing actions.
This would occur if $\mathcal C$ admits a concrete presentation with few generators as a category or better describe by explicit skein relations of diagrams (see below).
\item (Operator algebras) The {\it Pythagorean algebra} $P$ was a C*-algebra that naturally appeared when applying Jones' technology to the Thompson groups \cite{Brothier-Jones19bis}.
It provides a powerful method for constructing representations of the Cuntz algebra using only {\it finite dimensional} operators, new class of representations of the Thompson groups, and moreover exhibits a rare example of a non-nuclear C*-algebra satisfying the lifting property \cite{Cuntz77,Brothier-Jones19bis,Brothier-Wijesena22,Brothier-Wijesena22bis,Courtney21}.
We wish to extend these connections and applications in a larger setting.
\end{itemize}
\subsection*{Our class of diagrams and structures.}
{\bf Between forests and string diagrams.}
Here is our choice that we believe fit into all the constraints and wishes enunciated above.
This class is inspired by the tree diagrams used by Brown to describe elements of Thompson's group $F$ and string diagrams appearing in Jones' planar algebras.
Recall that an element of $F$ is described by a class of pairs of trees $(t,s)$ that have the same number of leaves.
Here, {\it tree} means a finite ordered rooted binary tree. Two such pairs defined the same element of $F$ when they are equal up to removing or adding corresponding carets on each tree.
We often describe the element associated to $(t,s)$ by a diagram in the plane with the roots of $t$ on the bottom and its leaves on top that are connected to the leaves of $s$ where $s$ is placed upside down on top of $t$.
Now removing a pair of corresponding carets is the operation of removing a diamond as described in the following diagrams:
\[\includegraphics{ts.pdf}\]
More generally, we can consider pairs of {\it forests} rather than {\it trees} and obtaining the Thompson groupoid.
Elements of Jones' planar algebras (or other similar structures like Etingof-Nikshych-Ostrik's fusion categories or rigid C*-tensor categories) are described by linear combinations of {\it string diagrams} \cite{Jones99,Etingof-Nikshych-Ostrik05}.
String diagrams are planar diagrams made of a large outer disc, some inner labelled discs, and some non-intersecting strings that start and end at certain boundary points of discs and perhaps some closed loops.
Here is an example with three inner discs having 4,2,4 boundary points (when read from top to bottom and left to right) and one closed loop:
\[\includegraphics{tangle.pdf}\]
A class of such diagrams, as originally defined by Jones, is generated by a collection of labelled $n$-valent vertices and satisfies relations expressed by linear combinations of string diagrams constructed from the generators.
We call the later {\it skein relations} (as in the context of knot theory and Conway tangles) that are identities closed under taking concatenation of diagrams.
Explicit examples of skein relations of planar algebras can be found for instance in \cite{Peters10} or \cite[Section 3]{Morrison-Peters-Snyder10} or in the PhD thesis of Liu \cite{Liu-thesis}.
Having such planar presentations for Jones' planar algebras is powerful as it permits to express in few symbols very rich algebraic structures.
We define classes of certain planar diagrams called {\it forest categories} using forest diagrams whose vertices are labelled and satisfy certain skein relations, although removing any linear structures. The set of skein relations completely describes the forest category and provides a {\it forest presentation}.
{\bf Forest categories.}
We consider all (finite ordered rooted binary) forests that we represent as planar diagrams with roots on the bottom and leaves on top that we order from left to right.
We fix a set $S$ and colour each interior vertex of a forest with an element of $S$ obtaining a {\it coloured forest}.
Here is an example of coloured forest with three roots, eleven leaves, and colour set $S=\{a,b\}:$
\[\includegraphics{forest.pdf}\]
Note, we do not colour the leaves just like boundary points of the exterior disc of a string diagram is not coloured in Jones' framework.
This produces a small category (where composition is given by horizontal stacking) that we call the {\it free forest category over $S$}.
This category is always left-cancellative.
Now, if $S$ is a single colour, then we recover the usual category of {\it monochromatic} forests whose fraction group is Thompson's group $F$.
If $S$ has more than two colours, then this category never satisfies Ore's property and thus does not admit any calculus of fractions nor produces fraction groups.
The idea for obtaining Ore's property is to mod out by some relations similar to skein relations.
These relations are expressed by a pair of coloured trees $(u,u')$ that have the same number of leaves.
Now, two forests $f,f'$ are equivalent if by substituting {\it subtrees} of $f$ isomorphic to $u$ by $u'$ (or the opposite) we can transform $f$ into $f'$.
The quotient is a small category $\mathcal F$ of equivalence classes of diagrams that we call a {\it forest category}.
Such a $\mathcal F$ is expressed in a compact way by a {\it forest presentation} $(S,R)$ where $S$ is the set of colours and $R$ is a set of pairs of trees $(u,u')$.
A forest presentation allows to apply Jones' technology efficiently and to study in an effective manner associated structures like fraction groups.
Here is an example of a forest category $\mathcal F$ presented by $(S,R)$ with two colours $S=\{a,b\}$ and one relation $R=\{(u,u')\}$ so that
\[\includegraphics{uuprime.pdf}\]
For instance, the following two forests are equivalent:
\[\includegraphics{ffprime.pdf}\]
{\bf Forest monoids.}
A forest presentation $(S,R)$ defines a monoid $\mathcal F_\infty:=\FM\langle S|R\rangle$ that we call a {\it forest monoid}.
It is obtained by considering infinite forests with roots and leaves indexed by $\mathbf{N}_{>0}$ that have finitely many nontrivial trees.
It is indeed a monoid with composition being vertical stacking of diagrams.
Moreover, note that this monoid can be obtained from the forest category by performing a certain inductive limit.
It is an auxiliary structure that is useful for studying $\mathcal F$.
{\bf Forest groups.}
In some occasion $\mathcal F$ is a Ore category (i.e.~it is left-cancellative and satisfies Ore's property) and thus we may formally invert forests obtaining a {\it fraction groupoid} $\Frac(\mathcal F)$.
Now, by considering any $r\geq 1$ we have an isotropy group $\Frac(\mathcal F,r)$ corresponding to all $f\circ g^{-1}$ where $f,g$ have the same number of leaves and have $r$ roots.
We choose a favourite group $G:=\Frac(\mathcal F,1)$ (produced with trees) which we call {\it the} fraction group of $\mathcal F$ and call all these groups {\it forest groups}.
Note that $\Frac(\mathcal F,r)\simeq G$ for any $r$ since we work with binary forests (implying that the fraction groupoid $\Frac(\mathcal F)$ is {\it path-connected} and thus all its isotropy groups are isomorphic).
Similarly, the forest monoid $\mathcal F_\infty$ provides a fraction group $H:=\Frac(\mathcal F_\infty)$ that can be obtained as an inductive limit of the $\Frac(\mathcal F,r)$ (letting $r$ tending to infinity).
All these groups embed in each other but are in general not isomorphic in an obvious way (unlike the classical Thompson group case).
Jones' technology is more effective for studying $G$ rather than $H$ and we thus focus more on $G$.
{\bf Previous constructions in the literature.}
Examples of groups constructed from diagrams of forests with more than one caret have been previously considered.
The first appeared in the work of Stein by allowing carets of various degrees, then in the work of Brin and the description of partition of hypercubes with several binary carets (one caret per dimension), and more recently in the work of Burillo, Nucinkis, and Reeves in the description of the Cleary irrational-slope Thompson group using two different binary carets \cite{Stein92,Brin-dV1, Burillo-Nucinkis-Reeves21}.
They all act by transformation of the unit interval or higher dimensional cube that are piecewise affine.
We will come back to these examples but before let's say that the last example cited fits exactly into the framework of this article, the first fits in an extension of it, and the second not quite as we will explain later.
{\bf Generalisation and extension of the formalism.}
In this first article we present the general theory of forest categories and their forest groups for coloured binary forests as briefly explained above.
This produces forest groups similar to Thompson's group $F$.
We list below some classical tree-diagrams that have been considered to construct Thompson-like groups by various authors.
Each of these extension of Thompson's group $F$ can be applied and combined in our framework of forest categories such as:
\begin{itemize}
\item add permutations of leaves (obtaining Thompson's groups $T$ and $V$ see \cite{Brown87});
\item replace binary forests by $n$-ary forests for any $n\geq 2$ (by Higman and Brown obtaining Higman-Thompon's groups $V_{n,r}$ and the Brown subgroups $F_{n,r}, T_{n,r}$ \cite{Higman74,Brown87});
\item allow forests whose vertices have various degrees (by Stein \cite{Stein92});
\item label leaves with elements of an auxiliary group (independently by Tanushevski, Witzel-Zaremsky, the author \cite{Tanushevski16,Witzel-Zaremsky18, Brothier22, Brothier21});
\item add braids on top of leaves (independently by Brin and Dehornoy obtaining the braided Thompson group $BV$ \cite{Brin-BV1,Brin-BV2,Dehornoy06});
\item perform more sophisticate Brin-Zappa-Sz\'ep products between forests and an auxiliary category (by Witzel-Zaremsky via general cloning systems of groups \cite{Witzel-Zaremsky18}).
\end{itemize}
All these extensions are interesting and rich in applications.
We can adapt these constructions in the more general framework of coloured forests with no more technical problems than in the classical case of the Thompson groups.
In this first article we stick to the binary case for simplicity and clarity of the exposition.
Although, we do explain how to incorporate permutations and braids using the approaches of Brown and Brin \cite{Brown87,Brin-BV1}.
This produces {\it $X$-forest categories} denoted $\mathcal F^X$ (for $X=F,T,V,BV$) and {\it $X$-forest groups} similar to the three Thompson groups $F,T,V$ and the braided Thompson group $BV$.
{\bf Forest groups among Thompson-like groups.}
Here, we want to locate our class of forest groups among various Thompson-like groups existing in the literature.
We have listed above a number of ways to generalise the notion of forest groups but here we discuss about the smaller class of forest groups obtained from forest categories of coloured {\it binary} forests without any decoration on leaves, so no permutations, braids, nor auxiliary group, (so manifestly constructed like $F$).
We start by giving properties of forest groups we have observed, then list some known groups that happen to be forest groups, and finally discuss about few families of Thompson-like groups that are related or not to our class of groups.
{\it Properties and observations on forest groups.}
A forest group contains a copy of $F$. In particular, they inherent various properties of $F$ such as not being elementary amenable, having exponential growth, and having infinite geometric dimension.
Although, they do not share all properties of $F$.
Indeed, most of the examples we have encounter have torsion in their abelianisation and have sometimes torsion themselves.
There are some forest group that contains a copy of the free group of rank two.
Some forest groups are nontrivial extension of $F$ and so are their derived group (preventing them to be simple).
There exist forest groups that decompose as nontrivial direct products and can have some finite conjugacy classes or even a nontrivial center.
Every forest group admits a nice simplicial complex on which it acts.
Although, this complex does not have any obvious cubical structure and we don't know any CAT(0) cubical complex on which a generic forest group acts.
{\it Some forest groups, see Section \ref{sec:example}}
{\it Thompson's group.}
Thompson's group $F$ is a forest group. It is the unique forest group whose underlying forest category is monochromatic.
Its (standard) forest presentation consists on one colour and no relations.
{\it Higman-Thompson group.}
The group $F_{3,1}$ obtained from ternary (monochromatic) trees is isomorphic to the forest group arising from the skein relation given by
\[\includegraphics{F31.pdf}\]
By anticipating definitions and symbols given in Section \ref{sec:formalism} we can write its underlying forest category by
$$\FC\langle a,b| b_1 a_1 = a_1 b_2\rangle.$$
The author was surprised to find an isomorphism and did not anticipated it from the diagrammatic description. The advantage of having a bicoloured description of $F_{3,1}$ is to have several embeddings of $F$ and $F_{3,1}$ inside it and to produce a priori new groups that are the $T,V,$ and $BV$-versions of it.
{\it Cleary's irrational-slope Thompson's group.}
The irrational-slope Thompson group of Cleary is a group of homeomorphism of the unit interval like $F$ but with elements having slopes powers of the golden ratio rather than powers of two \cite{Cleary00}.
Burillo, Nucinkis, and Reeves have provided a tree-diagrammatic description of it in \cite{Burillo-Nucinkis-Reeves21}, see also the follow up article with the $T$ and $V$-versions of this group \cite{Burillo-Nucinkis-Reeves22}.
It is easy to derive from this tree-like description that Cleary's group is a forest group obtained from a forest category with two colours and one relation
$$\FC\langle a,b| a_1a_2=b_1b_1\rangle.$$
We recommend the article of Cannon, Floyd, and Parry and the two of Burillo, Nucinkis, and Reeves as they provide very detailed and well-illustrated descriptions specific groups that can naturally be interpreted as forest groups \cite{Cannon-Floyd-Parry96,Burillo-Nucinkis-Reeves21,Burillo-Nucinkis-Reeves22}.
{\it Brin's higher-dimensional Thompson groups and Belk-Zaremsky's extensions of it.}
Brin has constructed higher-dimensional Thompson groups $dV$ for each dimension $d\geq 1$ \cite{Brin-dV1,Brin-dV2}.
Informally, this is done by considering piecewise affine bijections between standard dyadic partitions of hypercubes of dimension $d$.
An element of $dV$ is described by two such partitions with same number of pieces and one correspondence saying which piece is sent to which.
A partition is described by a binary tree having carets coloured by $\{1,\cdots, d\}$ (one colour per dimension) and a labelling of the leaves corresponding to a labelling of the pieces of the partition.
Refining a partition corresponds to grow the associated tree. Unfortunately, they seem to not exists any meaningful way for labelling the leaves of the grown tree that is compatible with the labelling of the elements of the partitions, see the original articles of Brin but also an article of Burillo and Cleary for more details \cite{Burillo-Cleary10}.
Hence, we may define a forest group admitting similar diagrammatic description but it will not describe transformation of hypercubes and is thus not a priori isomorphic to Brin group $dV$. For $d=2$, such a forest group admits the following forest presentation:
$$\FC\langle a,b| a_1 b_1b_3=b_1a_1a_3\rangle$$
which translates the two way to cut a square into four equal pieces.
Note that Brin's construction extends to any nonempty set $S$ (including infinite sets) giving the group $SV$ of certain transformations of a hypercube where the axis are labelled by $S$.
Belk and Zaremsky have extended Brin's construction by adding in the data a group action $G\curvearrowright S$ \cite{Belk-Zaremsky22}, see also the expository \cite{Zaremsky22}.
This provides {\it twisted Brin-Thompson groups} $SV_G$ that are particularly interesting when $S$ is infinite.
Elements of $SV_G$ admit similar descriptions using forests with colour set $S$ but as in the original case of Brin this diagrammatic description must be taken with precaution.
{\it Auxiliary group and morphism.}
Given a group $\Gamma$ and group morphism $\phi:\Gamma\to\Gamma\oplus\Gamma$ one can construct a Thompson-like group $G(\Gamma,\phi)$ obtained by taking binary (monochromatic) trees, decorating their leaves with elements of $\Gamma$, and using $\phi$ for growing trees.
This was independently discovered by Tanushevski, Witzel-Zaremsky, and the author \cite{Tanushevski16, Witzel-Zaremsky18, Brothier22}.
See the appendix of \cite{Brothier22} for details and a precise comparison of the three approaches.
We have discovered with surprise that this class of groups intersects nontrivially our class of forest groups.
The intersection we found between the two classes is exactly the following.
All groups $G(\Gamma,\phi)$ where $\phi(g)=(g,e)$ ($e$ being the neutral element of $\Gamma$) and $\Gamma=\ker(H\to \mathbf{Z})$ where $H$ is the fraction group of a Ore monoid admitting a homogeneous presentation and where $H\to\mathbf{Z}$ is the word-length map of the presentation.
We will present in details this connection in the immediate successor of this article \cite{Brothier22-HPM}.
{\it Guba-Sapir's diagram groups.}
It is natural to wonder if forest groups are related to Guba-Sapir diagram groups \cite{Guba-Sapir97}.
Indeed, both classes are constructed by planar diagrams similar to coloured forests (see in particular their dual diagrammatic representations appearing in \cite[Section 4]{Guba-Sapir97}), are related to the Thompson groups, and admit obvious $T$ and $V$-versions.
However, their resemblance seems misleading.
Indeed, all diagram groups (the $F$-version) are torsion-free with torsion-free abelianisations and act freely properly and isometrically on a CAT(0) cubical complex in contrast with our examples of forest groups.
{\it Hughes' FSS groups.}
Note that picture groups (i.e.~braided diagram groups in Guba-Sapir terminology corresponding to Thompson-like group similar to $V$) forms the same class than Hughes' finite similarity structure groups (in short FSS groups) by \cite{Hughes09, Farley-Hughes17}.
Hence, from our observations of above, it seems that FSS groups and forest groups don't have a large intersection.
{\it Piecewise affine maps or action on trees.}
Many Thompson-like groups act as piecewise affine maps on an interval or a similar structure.
This is the case for the Higman-Thompson groups, Stein groups, Brin's $dV$, and many other.
Similarly, they typically act on totally disconnected spaces like the Cantor space of binary sequences (by swapping finite prefixes) or more generally as almost automorphisms of trees. The family of groups just mentioned act in that way and other like R{o}ver-Nekrashevych groups, Brin's $dV$ groups and its generalisations like Belk-Zaremsky's twisted Brin-Thompson groups \cite{Rover99,Nekrashevych04,Belk-Zaremsky22}.
So far we have not find any such actions for forest groups. Although, they all admit a canonical action on a totally ordered space playing the role of the dyadic rationals modulo one for $F,T,V$, see Section \ref{sec:Q-space}.
{\it Thumann's operad groups.}
In his PhD thesis Thumann defines {\it operad groups} which play a special role in this article \cite{Thumann17}.
A nice operad defines three operad groups: a planar, symmetric, and braided one corresponding to $F,V$, and $BV$.
The class of operad groups is huge and as far as the author observed all Thompson-like groups similar to $F,V,BV$ can be naturally identified with operad groups including our class of forest groups.
Examples that seem to not admit an obvious description as operad groups are the one constructed using Brin-Zappa-Sz\'ep products as in the theory of cloning systems of Witzel-Zaremsky such as Thompson group $T$ constructed using cyclic permutations or $V^{mock}$ constructed from mock-symmetric permutations that seat in between $F$ and $V$, see \cite[Section 9]{Witzel-Zaremsky18}. Although, all these examples are probably isomorphic to operad groups anyway by encoding the Brin-Zappa-Sz\'ep product in the operad (like every group is the fraction group of itself).
Thumann provides a powerful abstract and very general framework.
Indeed, using some of Thumann's result we deduce that many forest groups are of type $F_\infty$ (for any $n\geq 1$ the group admits a classifying space with finite $n$-skeleton, see Section \ref{sec:def-finiteness}).
We could present all the theory of forest categories and groups using Thumann's operad approach.
Although, we found many advantages in developing a more specific formalism that enables us to easily and explicitly construct new classes of examples, studying them with specific diagrammatic tools, and capture key features that we want all our structures to share in our program.
\subsection*{Content of this present article.}
In this present article we introduce the formalism of forest categories and their associated fraction groups called forest groups.
In particular, we define {\it forest presentations} which will play a key role in applying Jones' technology in future articles.
We provide general criteria to decide that a forest category admits a calculus of fractions.
This is a key strength of our formalism: we are able to decide for many forest presentations if they can produce groups or not.
We present two canonical actions of forest groups: one on a simplicial complex giving a classifying space and another on a totally ordered space which mimics the classical action of the Thompson groups on the set of dyadic rationals of the unit torus or interval.
We deduce explicit presentations of forest groups.
We do not apply Jones' technology nor explore connections with CFT in this article.
Instead, we consider an exceptional property that the Thompson groups $F,T,V$ share.
Indeed, we prove that a large class of forest groups satisfy the topological finiteness property of being of type $F_\infty$.
It is a rare property for (infinite) groups that has pushed the study of $F,T,V$: Thompson-like groups are the main source of infinite simple groups with good finiteness properties such as $F_\infty$, see Section \ref{sec:def-finiteness} for definitions and references.
We end by providing a huge class of explicit forest categories that are all Ore categories and whose fraction groups are of type $F_\infty$.
As previously mentioned, the framework chosen has been inspired by Jones' planar algebra, Jones' reconstruction program of CFT, and Brown's diagrammatic description of Thompson's groups.
The presentation does not use much categorical language but is rather set theoretical and constructive in the vein of the styles of Jones, Brin, and Dehornoy which inspired the author.
On the technical side we use previous work of Dehornoy on monoids and Thumann on operad groups.
The first permits to prove the existence of certain forest groups and the second to establish a finiteness property.
{\bf Detailed plan and main results.}
In Section \ref{sec:formalism} we define forest categories and monoids using presentations (generators and skein relations).
Forest categories are monoidal small categories but we treat them as classical algebraic structures: sets equipped with two binary operations.
We define morphisms between them and in particular forest subcategories and quotients.
Presented forest categories can be alternatively defined as the solution of a universal problem.
This permits us to announce how Jones' technology will be used.
We provide explicit {\it category} presentations of forest categories and monoids.
Section \ref{sec:Dehornoy} is about left-cancellativity and Ore's property. These are the two conditions we want for constructing groups.
We start by giving definition and some obvious but useful reformulations of them.
Then we present some more advance techniques due to Dehornoy that we apply to our specific class of forest categories and monoids.
These techniques provide powerful characterisations and criteria for checking left-cancellativity of forest categories and in fewer cases Ore's property.
In practice we prove Ore's property by explicitly constructing a cofinal sequence of tree.
Sections \ref{sec:forest-groups} and \ref{sec:presentation} are about forest groups and presentations of them.
Given a presented Ore forest category $\mathcal F=\FC\langle S|R\rangle$ we define the forest groups $G:=\Frac(\mathcal F,1)$ and $H:=\Frac(\mathcal F_\infty).$
Using obvious diagrammatic maps we prove that $G$ and $H$ embed in each other and moreover contain Thompson group $F$.
The monoid presentation of $\mathcal F_\infty$ obtained in Section \ref{sec:formalism} provides a group presentation of $H$.
For $G$ we define a complex $E\mathcal F$ canonically constructed from $\mathcal F$ on which $G$ acts.
The complex $E\mathcal F$ is obtained by considering the two actions
$$G\curvearrowright \Frac(\mathcal F)\curvearrowleft \mathcal F$$
given by restricting the composition of the groupoid $\Frac(\mathcal F)$.
From there we can define a $G$-poset and its associated order complex which is $E\mathcal F$.
This latter is a free $G$-simplical complex and Ore's property of $\mathcal F$ implies that $E\mathcal F$ is contractible.
Hence, the quotient $B\mathcal F:=G\backslash E\mathcal F$ is a classifying space of $G$ and in particular $G$ is isomorphic to its Poincar\'e group (taken at any point).
We deduce an infinite presentation of $G$ determined by a forest presentation $(S,R)$ and the choice of a colour $a\in S$.
Using a similar reduction performed for the classical Thompson groups we deduce presentations with less generators and relations that are sometime finite and obtain the following.
\begin{lettertheo}
Let $\mathcal F=\FC\langle S|R\rangle$ be a presented Ore forest category with forest groups $G=\Frac(\mathcal F,1)$ and $H=\Frac(\mathcal F_\infty)$.
The groups $G,H$ admit explicit group presentations in terms of $S,R$, see Theorem \ref{theo:groupG-presentation}.
Moreover, if $S$ is finite (resp.~$S$ and $R$ are finite), then $G$ and $H$ are finitely generated (resp.~finitely presented).
\end{lettertheo}
Note that the converse is false: there exists a Ore forest category that does not admit any finite forest presentation but its forest group is finitely presented and in fact of type $F_\infty$, see Section \ref{sec:example}.
This theorem provides explicit finite presentations for many forest groups and in particular recovers results of the literature like the presentation of Cleary irrational-slope Thompson group given in \cite{Burillo-Nucinkis-Reeves21}.
Note, in this article we only cover the case of $F$-forest groups but strongly believe that group presentations can be obtained (with some substantial work) to the other $T,V,BV$ cases by combining our presentations of $\Frac(\mathcal F,1)$ with the presentations of $T,V,$ and $BV$ given by Cannon-Floyd-Parry and by Brin \cite{Cannon-Floyd-Parry96,Brin-BV2}.
In Section \ref{sec:Q-space} we construct a canonical action $G^V\curvearrowright Q_{\mathcal F}$ for all $V$-forest groups $G^V:=\Frac(\mathcal F^V,1)$.
This extends the classical action of Thompson's group $V$ on the set of dyadic rationals $\mathbf{Z}[1/2]/\mathbf{Z}$ in the unit torus.
We provide three equivalent descriptions of it when restricted to $G^T$.
One is simply the homogeneous space $G^T/G$, the second is obtained by quotienting a piece of the fraction groupoid $\Frac(\mathcal F)$, and the third is constructed using Jones' technology.
The later approach permits to extend {\it canonically} the action to $G^V$.
The poset structure of $Q_\mathcal F$ permits to characterise $G^F$ (resp.~$G^T$) inside $G^V$ as the subgroups of order-preserving (resp.~order preserving up to cyclic permutations) transformations.
We prove some strong transitivity statements analogous to the classical Thompson group case.
Using these results and adapting an elegant argument due to Brown and Geoghegan we deduce the following theorem on finiteness properties, see \cite[Section 4B, Remark 2]{Brown87} and \cite[Theorem 9.4.2]{Geoghegan-book}.
\begin{lettertheo}
Let $G$ be a forest group and let $G^T$ be its $T$-version.
If $G$ satisfies the topological finiteness property $F_n$ for a certain $n\geq 1$, then so does $G^T$.
\end{lettertheo}
In most examples, a finiteness property holding for the $F$-version of a Thompson-like group generally holds for its $T$-version. Although, it is usually proved by following the whole argument of the $F$-case and adapting it step-by-step to the $T$-case.
Our theorem allows us to prove a permanence property that holds for all forest groups regardless of the strategy adopted to prove that $G$ satisfies a given finiteness property.
There is a (obvious) homological version of this theorem but we have not found any application of it.
However, the topological version of above have application as we are about to see.
Section \ref{sec:Thumann} is about Thumann's theory of operad groups and his finiteness theorem \cite{Thumann17}.
Thumann formalism and theorem are impressively general.
He covers and adapts in a single categorical approach a number of key techniques pioneered by Brown that have been refined and extended (in a nontrivial way and using new ideas) to study many more groups like Stein groups, Brin's higher dimensional $dV$, braided Thompson group $BV$, Guba-Sapir diagram groups and Hughes' finite similarity structure groups, to cite a few \cite{Brown87,Stein92,Fluch-Marschler-Witzel-Zaremsky13, Bux-Fluch-Marschler-Witzel-Zaremsky16,Farley-Hughes15}.
We refer the reader to the articles of Zaremsky and Witzel for a first read on these techniques and strategy \cite{Zaremsky21,Witzel19}.
We prove that all forest groups are operad groups and establish a dictionary between the two theories.
This allows us to precisely translate and specialise the theorem of Thumann to forest groups.
We define the {\it spine} of a forest category $\mathcal F$ that is roughly speaking the mcm-closure of the set of trees with two leaves (where mcm stands for ``minimal common multiples'') and deduce the following.
\begin{lettertheo}\label{ltheo:Finfty}
If $\mathcal F$ is a Ore forest category with a finite spine, then the $X$-forest groups $\Frac(\mathcal F^X,1)$ is of type $F_\infty$ for $X=F,T,V,BV.$
\end{lettertheo}
Note that Thumann's result takes care of the case $F,V,$ and $BV$ using three parallel technical arguments corresponding to planar, symmetric, and braided operad groups, respectively.
Using our previous theorem we add the missing $T$-case.
Moreover, note that this theorem covers a class of groups that is closed under taking braiding.
This is remarkable but possible thanks to the difficult proof on the braided Thomson group $BV$ \cite{Bux-Fluch-Marschler-Witzel-Zaremsky16} that was extended very cleverly by Thumann and recovered by us in this context of forest groups.
Using Dehornoy's criteria on left-cancellativity and specialising to the two colours case we deduce the following result.
\begin{lettercor}
If $\mathcal F$ is a forest category with two colours and one relation (a pair of trees with roots of different colours), then if it satisfies Ore's property then it is a Ore category (hence is left-cancellative) and the $X$-forest groups $\Frac(\mathcal F^X,1)$ is of type $F_\infty$ for $X=F,T,V,BV.$
\end{lettercor}
These results can be generalised in the spirit of Higman, Brown, and Stein by considering forests with vertices of various valencies and by considering groups formed by pairs of {\it forests} rather than trees, see Remark \ref{rem:Stein-Finfty}.
Although, these finiteness theorems have their limits.
We present the seemingly very elementary forest category $\mathcal F:=\FC\langle a,b| a_1b_1=b_1a_1, a_1a_1=b_1b_1\rangle$ with two colours, two relations (of length two) in Section \ref{sec:rebel}.
It produces a group $G$ of type $F_\infty$ but this fact is surprisingly difficult to prove and resists to Thumann's theorem and a second approach due to Witzel \cite{Witzel19}. We will prove $F_\infty$ in a future article using a third approach inspired from the work of Tanushevski and Witzel-Zaremsky \cite{Tanushevski16,Witzel-Zaremsky18,Brothier22-HPM}.
We end this first article by providing a huge class of forest presentations $(S,R)$ for which the associated forest category $\mathcal F$ is automatically a Ore category and whose fraction group is of type $F_\infty$ (and so are its $T,V,$ and $BV$-versions).
Given any nonempty family $\tau=(\tau_a:\ a\in S)$ of monochromatic trees with the same number of leaves we define a forest category $\mathcal F_\tau$, see Section \ref{sec:class-example}.
\begin{lettertheo}
The forest category $\mathcal F_\tau$ is a Ore category whose spine injects in the index set $S$ union a point.
In particular, $\mathcal F_\tau$ admits forest groups $G^X_\tau:=\Frac(\mathcal F^X_\tau,1)$ for $X=F,T,V,BV.$
If $S$ is finite of order $n$, then $G_\tau^X$ is of type $F_\infty$.
Moreover, the $F$-group $G_\tau$ admits an explicit presentation with no more than $4n-2$ generators and $8n^2-4$ relations.
\end{lettertheo}
This theorem permits to construct easily and explicitly a plethora of interesting examples of groups.
In particular, if $S=\{a,b\}$, then each pair of $(t,s)$ of monochromatic trees with the same number of leaves provides a group $G_{(t,s)}$ of type $F_\infty$ and we can provide an explicit group presentation of $G_{(t,s)}$ with 6 generators and 28 relations.
Though, the forest presentation of $\mathcal F_\tau$ is very small with two colours and one relation. This makes very easy to construct actions of $G_{(t,s)}$ using Jones' technology.
It would be interesting to know when they are pairwise isomorphic and which further properties they satisfy other than being of type $F_\infty$.
{\bf Some future work.}
One of the main strength of our approach is to be able to easily construct new examples of groups.
In order to keep reasonable the length of this first article we have postponed a number of detailed study of specific examples and classes of examples to future articles.
The immediate successor to this paper is about a specific class of forest categories \cite{Brothier22-HPM}.
They are built from homogenenously presented monoids and can be interpreted as forest categories with {\it one-dimensional} skein relations.
They produce very specific forest groups that are split extensions of the Thompson groups.
We classify them and relate precisely certain properties of the monoids and the associated forest groups.
After this second article we will continue working on the program described earlier with an emphasise on constructing and studying specific examples or classes of examples.
Choice of examples may be dictate by specific Jones' planar algebra and CFT or in view of establishing group properties or to construct interesting operator algebras.
This is a vast program that will demand time to develop and grow.
The author is happy to help and answer questions to anybody wishing to work on it.
We are also certainly not against suggestions, advices, and collaborations!
\begin{center}\begin{large}{\bf Acknowledgement}\end{large}\end{center}
All figures have been created with Tikzit. We thank the programmers for making freely available such a great tool.
\section{A class of categories built from coloured forests}\label{sec:formalism}
In this section we define what is a {\it forest category}. This is an algebraic structure similar than a monoid but with several units (a small category). It is roughly equal to a collection of forest that can be concatenated and whose vertices have various colours.
We start by fixing the terminology and introducing what we call {\it coloured trees and forests}.
From there we introduce {\it free forest categories} that are analogues of free monoids but obeying nontrivial relations induced by the diagrammatic structure.
Finally, we introduce relations and quotients of these free forest categories. This leads to the notion of {\it forest presentations}, {\it forest categories}, and morphisms between them.
\subsection{Coloured forests and trees}
\subsubsection{Monochromatic forests and trees}
We start by defining what we intend by {\it monochromatic} trees and forests (hence without colours).
Consider the infinite regular rooted binary monochromatic tree $\mathcal T_2$.
The vertex set of $\mathcal T_2$ is the set of all finite binary sequences of $0$ and $1$ (also called words) including the trivial one that is the {\it root} of $\mathcal T_2$.
The edge set is equal to all sets $\{w, w0\}$ and $\{w,w1\}$ for $w$ a binary sequence.
We direct the tree keeping only the pairs $(w,w0)$ and $(w,w1)$ corresponding in having all edges going away from the root.
We call them {\it left and right edges}, respectively.
We say that $w0$ and $w1$ are the {\it children} of $w$ and $w$ is their {\it parent}.
If $u=wv$ for some words $w,v$ with $v$ nontrivial, then $u$ is a {\it descendent} of $w$ and $w$ is an {\it ancestor} of $u$.
The union of $(w,w0)$ and $(w,w1)$ is called a {\it caret}.
We identify $\mathcal T_2$ with a graph embedded in the plane with its root at the bottom and where a left edge is going to the top left and a right edge to the top right such as
\[\includegraphics{T2.pdf}\]
A {\it monochromatic tree} $t$ is a finite rooted subtree of $\mathcal T_2$ so that each of its vertex has either no children or two.
A vertex of $t$ with no children is called a {\it leaf} and otherwise an {\it interior vertex}.
We may drop the terms monochromatic if the context is clear.
We order from left to right the leaves of $t$ using the embedding of $\mathcal T_2$ in the plane.
The {\it trivial tree}, denoted $I$, is a tree with a single vertex.
This vertex is equal to its root and is at the same time its unique leaf.
For pictorial reason we represent $I$ by a vertical bar rather than a point.
There is a unique tree with two leaves that we denote $Y$.
Note that $Y$ is equal to a single caret and more generally any tree with $n+1$ leaves is equal to the union of $n$ carets.
A monochromatic {\it forest} $f$ is a finite list of trees $(f_1,\cdots, f_n)$.
We order the roots of $f$ from left to right.
Note that every forest can be obtained from two trees.
Indeed, if $s$ is a rooted subtree of $t$, then the complement of $s$ inside $t$ is a forest $f$ and all forest arise in that way.
We define composition of forests so that $t=s\circ f$. That is: $t$ is obtained by stacking vertically $f$ on top of $s$ and lining up leaves of $s$ with roots of $f$.
More generally we can compose any two forests as long as we match number of leaves and roots.
We write $\Leaf(f), \Root(f),\Ver(f)$ for the set of leaves, roots, and interior vertices, respectively, of a forest $f$.
Note that the vertex set of $f$ is equal to the disjoint union of $\Ver(f)$ and $\Leaf(f)$.
Moreover, $\Ver(f)$ is in bijection with the carets of $f$.
Here is the composition of two forests $f,g$.
The forest $f$ has two roots, three leaves, one interior vertex while $g$ has three roots, five leaves, and two interior vertices.
The composition $f\circ g$ is a forest with two roots, fives leaves, and three interior vertices.
\[\includegraphics{fguncoloured.pdf}\]
\subsubsection{Coloured forests and trees}
We now add colours.
Let $S$ be a {\it nonempty} set.
A \textit{coloured forest (over $S$)} is a pair $(f,c)$ where $f$ is a monochromatic forest (as defined above) and a {\it colouring map} $c:\Ver(f)\to S$.
A coloured tree is a coloured forest with one root.
We call $S$ the {\it set of colours} and its element a {\it colour}.
If the context is clear we may drop the term colour and write $f$ for $(f,c)$.
A caret coloured by $a$ or a {\it $a$-caret} is a caret whose origin is a vertex coloured by $a$.
We may interpret a monochromatic forest as a forest coloured by a single colour.
{\bf Notation.}
Typically we write $a,b,c$ or $x,y,z$ for colours, $t,s,u,v$ for trees, $f,g,h,k$ for forests.
Recall that $I$ designates the trivial tree.
For each colour $a\in S$ there is a unique tree with two leaves of colour $a$ which we denote by $Y_a, Y(a).$
\subsection{The collection of all forests, operations, quotient}
Let $S$ be a set of colour and write $\mathcal{UF}=\mathcal{UF}(S)=\FC\langle S\rangle$ for the set of all coloured forests over $S$.
\subsubsection{Compositions and tensor products of forests}
We define two binary operations on $\mathcal{UF}$.
{\bf Composition.}
The first one is called \textit{composition} and is only partially defined.
Consider $n\geq 1$, $(f,c_f),(g,c_g)\in\mathcal{UF}$ so that $f$ has $n$ leaves and $g$ has $n$ roots.
Let $f\circ g$ or simply $fg$ be the forest obtained by concatenating vertically $g$ on top of $f$ where the leaves of $f$ are lined up with the roots of $g$: the $j$th leaf of $f$ gets attached to the $j$th root of $g$ for $1\leq j\leq n.$
We have the following obvious identifications:
$$\Ver(f\circ g)=\Ver(f)\sqcup \Ver(g), \ \Root(f\circ g) = \Root(f), \text{ and }\Leaf(f\circ g)=\Leaf(g).$$
Define the colouring map $c_{f\circ g}:\Ver(f\circ g)\to S$ so that $c_{f\circ g}(v)=c_f(v)$ if $v\in \Ver(f)$ and $c_g(v)$ otherwise.
The composition of $(f,c_f)$ with $(g,c_g)$ denoted by $(f,c_f)\circ (g,c_g)$ is equal to $(f\circ g, c_{f\circ g}).$
We will often omit the colouring maps.
Diagrammatically we express as follows the composition of two coloured forests where the colours are $a,b,c$:
\[\includegraphics{fg.pdf}\]
{\bf Convention.}
Given $f,g\in\mathcal{UF}$, if we write $f\circ g$ we implicitly assume that this composition makes sense meaning that $f$ has its number of leaves equal to the number of roots of $g$.
In that case we may say that the pair $(f,g)$ is composable.
Similarly, we may consider $k$-tuples that are composable.
{\bf Tensor product.}
We define a second binary operation called the \textit{tensor product} that is defined everywhere.
Consider two forests $(f,c_f),(g,c_g)\in\mathcal{UF}$.
Write $f\otimes g$ for the forest obtained by concatenating horizontally $f$ with $g$ where $f$ is placed to the left of $g$.
We have the following obvious identifications
$$\Ver(f\otimes g) = \Ver(f)\sqcup \Ver(g),\ \Root(f\otimes g) = \Root(f)\sqcup \Root(g),$$
and
$$\Leaf(f\otimes g) = \Leaf(f)\sqcup \Leaf(g).$$
Define the colouring map $c_{f\otimes g}:\Ver(f\otimes g)\to S$ so that $c_{f\otimes g}(v)=c_f(v)$ if $v\in \Ver(f)$ and $c_g(v)$ otherwise.
The tensor product of $(f,c_f)$ with $(g,c_g)$ denoted $(f,c_f)\otimes (g,c_g)$ is equal to $(f\otimes g, c_{f\otimes g}).$
We will often omit the colouring maps.
Here is an example:
\[\includegraphics{ftensorg.pdf}\]
Note that if we consider forests as lists of trees, then the tensor product is the concatenation of lists:
$$(f_1,\cdots,f_n)\otimes (g_1,\cdots,g_m):=(f_1,\cdots,f_n, g_1,\cdots, g_m).$$
{\bf Subtrees and subforests.}
If $t$ is a tree, then $s$ is a \textit{rooted subtree} of $t$ if $s$ is a tree and there exists a forest $f\in\mathcal{UF}$ satisfying $t=s\circ f$.
A \textit{subtree} $u$ of a tree $t$ is a tree satisfying that there exist a rooted subtree $s$ of $t$ and some forests $f,g,h$ satisfying that
$t= s\circ (f \otimes u \otimes g)\circ h$.
Similarly, we define rooted subforests and subforests.
{\bf Partial order.}
We define a partial order $\leq$ on $\mathcal{UF}$.
If $f,g\in\mathcal{UF}$, then we write $f\leq g$ if and only if there exists $h\in\mathcal F$ satisfying $f\leq g\circ h$.
Hence, $f\leq g$ if and only if $f$ is a rooted subforest of $g$.
This is indeed a partial order.
Moreover, if $f\leq g$, then $h\otimes f\otimes k\leq h\otimes g\otimes k$ and $r\circ f\leq r\circ g$ for all forests $f,g,h,k,r$.
If $f\leq g$, then we may say that $g$ is obtained by {\it growing} the forest $f$.
\begin{remark}
We have adopted the diagrammatic convention that the roots of a forest are at the bottom and the leaves on top. Moreover, $f\circ g$ stands for the forest with $f$ on the bottom and $g$ on top.
Hence, the composition can be perceived as concatenation of forests when reading the mathematical symbols from left to right corresponds in reading the diagram from bottom to top.
In the literature there exist all possible conventions; each of them having their advantages and drawbacks.
We warn the reader that even the author have been using opposite conventions in previous articles.
Hence, covariant functors appearing in \cite{Brothier21} would become contravariant functors for the conventions of this present article.
\end{remark}
\subsubsection{Equivalence relations on the set of forests}\label{sec:ER}
{\bf Relations and equivalence relations.}
Consider $S,\mathcal{UF}$ as above and write $\mathcal T$ for the set of trees of $\mathcal{UF}$.
A \textit{forest relation in $\mathcal{UF}$} or simply a \textit{relation} is a pair of trees $(t,t')$ in $\mathcal T$ so that $t$ and $t'$ have the same number of leaves.
If $R$ is a set of relations in $\mathcal{UF}$, then we consider $\overline R\subset \mathcal{UF}\times\mathcal{UF}$ the smallest equivalence relation in $\mathcal{UF}$ that contains $R$ and is closed under taking compositions and tensor products.
Hence, if $(f,f')\in \overline R$ and $g,h,k,p\in \mathcal{UF}$, then $(g\circ f\circ h,g\circ f'\circ h)\in \overline R$ and $(k\otimes f \otimes p, k\otimes f'\otimes p)\in \overline R$.
Given $f,f'\in \mathcal{UF}$ we write $f\sim_R f'$ or simply $f\sim f'$ for indicating that $(f,f')\in\overline R$.
{\bf Quotient.}
Let $R$ be a set of relations in $\mathcal{UF}$ with associated equivalence relation $\overline R$.
We write
$$\mathcal F:=\FC\langle S| R\rangle$$
for the quotient of $\mathcal{UF}$ with respect to equivalence relation $\overline R$.
If $f\in\mathcal{UF}$, then we write $[f]$ for its class in $\mathcal F$.
In order to keep light notations we may identify the equivalence class $[f]$ with a representative $f$.
{\bf Binary operations.}
By definition of $\overline R$ we have that if $f,f',g,g',h,h'\in\mathcal{UF}$ satisfy $f\sim f', g\sim g', h\sim h'$, then $f\circ g\sim f'\circ g'$ and $f\otimes h\sim f'\otimes h'.$
This allows us to define on $\mathcal F$ a composition and a tensor product as follows:
$$[f]\circ [g] := [f\circ g] \text{ and } [f]\otimes [h]:=[f\otimes h],$$
for $f,g,h\in\mathcal{UF}.$
We will assume that $\mathcal F$ is equipped with these two binary operations.
{\bf Partial order.}
The partial order $\leq$ of $\mathcal{UF}$ provides a partial order on $\mathcal F$ as follows.
Define the relation: $[f]\leq [g]$ if and only if $[g] = [f]\circ [h]$ for some $[h]\in\mathcal F$ where $[f],[g]\in\mathcal F$.
Note that $[f]\leq [g]$ if and only if there exists $f',g'\in\mathcal{UF}$ in the classes of $[f],[g]$, respectively, satisfying $f'\leq g'.$
{\bf Elementary forests.}
If $a\in S$, $1\leq j\leq n$, then we write $a_{j,n}$ or simply $a_j$ for the forest having $n$ roots, its $j$th tree is $Y_a$, and all the other trees are trivial.
Hence,
$$a_{j,n} = I^{\otimes j-1} \otimes Y_a \otimes I^{\otimes n-j}.$$
For instance:
\[\includegraphics{a24.pdf}\]
Here we use the notation $I^{\otimes m}$ for the forest equal to $m$ trivial trees place next to each other taking the convention that $I^{\otimes m}$ is the empty diagram if $m\leq 0$.
We say that $a_{j,n}$ (or its class $[a_{j,n}]$) is an \textit{elementary forest} and write $\mathcal E(S)$ for the set of all elementary forests coloured by $S$.
{\bf Forest with at most one nontrivial tree.}
By extending the notation of elementary forests we write $t_{j,n}$ for the forest with $n$ roots having its $j$th tree equal to $t$ and all other trivial.
For instance:
\[\includegraphics{t24.pdf}\]
{\bf Generators.}
It is not hard to see that any forest $f$ of $\mathcal{UF}$ (resp. $[f]$ of $\mathcal F$) is a finite composition of elementary forests.
Hence, the elementary forests are \textit{generators} of $\mathcal{UF}$ and $\mathcal F$ for the composition.
Note that the smallest subset of $\mathcal{UF}$ (resp. $\mathcal F$) containing $\{Y_a:\ a\in S\}$ (resp. $\{[Y_a]:\ a\in S\}$) that is closed under taking composition and tensor products with the trivial tree is equal to $\mathcal{UF}$ (resp. $\mathcal F$).
\begin{remark}
Instead of taking pairs of {\it trees} for relations we could take pairs of {\it forests} $(f,f')$ providing they have the same number of leaves but also {\it roots}. This seems more general but in fact it is not.
Indeed, a relation of the form $(f,f')$ can be obtained from the set of all relations $(tf,tf')$ where $t$ runs over all trees composable with $f$.
Most of the examples investigate will have $S$ and $R$ finite. In fact, many interesting examples arise when $S$ has two elements $a,b$ and $R$ has one or two relations.
\end{remark}
\subsection{Forest category}
\subsubsection{Algebraic structure}
We now analyse the algebraic structure of the set $\mathcal F$ equipped with the two operations $\circ, \otimes.$
The triple $(\mathcal F,\circ,\otimes)$ has an obvious structure of a monoidal small category, i.e.~a category whose collections of objects and morphisms are sets and equipped with a monoidal product (or tensor product).
We interpret $\mathcal F$ as a category for the convenience of the terminology.
Although, we think of $\mathcal F$ as a classical algebraic structure: a set equipped with two composition laws.
{\bf Categorical structure.}
The set of object $\ob(\mathcal F)$ of $\mathcal F$ is the set of natural number $\mathbf{N}:=\{0,1,2,\cdots\}$ (taking the convention that $0$ is a natural number).
If $m,n\in\mathbf{N}$, then we write $\mathcal F(m,n)$ for the set of morphism from $m$ to $n$ equal to all forests having $m$ leaves and $n$ roots.
Hence, a morphism is a forest with source its number of leaves and target its number of roots.
Note that this is the opposite convention of \cite{Brothier21}.
Observe that $\mathcal F(m,n)$ is empty if $m<n$ and $\mathcal F(m,m)$ contains one element: the trivial forest $I^{\otimes m}$ with $m$ roots.
This is the identity automorphism of $\mathcal F(m,m)$.
Moreover, note that $\mathcal F(m,0)$ is empty unless $m=0$. (Here we identify the empty forest with the trivial forest with $0$ roots.)
The composition is associative: it is clear in the free case since it corresponds in concatenating labelled graphs. The general case follows since it is a free forest category mod out by an equivalence relation closed under composition.
All together we deduce that $(\mathcal F,\circ)$ is a small category.
{\bf Monoidal structure.}
The monoidal structure is given by the binary operation $\otimes.$
On objects it is defined as $n\otimes m:=n+m$ for $n,m\in\mathbf{N}.$
For morphisms (i.e.~forests) it is defined as above by horizontal concatenations of forests.
The tensor unit is the empty forest.
A similar reasoning than above shows that all the axioms of a monoidal category are satisfied.
Hence, $(\mathcal F,\circ,\otimes)$ is a monoidal small category.
We will continue to write $f\in\mathcal F$ for a forest $f$ and will say that $f$ is an element of $\mathcal F$ (rather than calling $f$ a morphism of the category $\mathcal F$).
Note that the object $0$ and the empty diagram are rather useless. We only added them in order to have a tensor unit and to fulfil precisely the axioms of a monoidal category.
\begin{definition}
\begin{enumerate}
\item The monoidal category $(\mathcal F,\circ,\otimes)$ is called a \textit{forest category}.
\item Let $S$ be a nonempty set and $R$ a set of pairs of coloured trees over $S$ that have the same number of leaves.
Consider $\mathcal F:=\FC\langle S| R\rangle$ as defined above and the binary operations of composition $\circ$ and tensor product $\otimes$.
We say that $(S,R)$ is a {\it forest presentation}, $\FC\langle S| R\rangle$ a {\it presented forest category}, $S$ the {\it set of colours}, and $R$ the {\it set of forest relations}.
We may drop the word ``forest'' if the context is clear.
\item When the set of relations $R$ is empty we write $\FC\langle S\rangle$ or $\mathcal{UF}\langle S\rangle$ rather than $\FC\langle S| \ \emptyset \rangle$ and call it the \textit{universal forest category} or the \textit{free forest category} over the set $S$ where the symbol $\mathcal U$ stands for ``universal''.
\end{enumerate}
\end{definition}
It is sometime convenient to discuss about {\it abstract} forest categories where a forest presentation has not been specified. Similarly, we may want to discuss about colours in the absence of a presentation.
\begin{definition}
\begin{enumerate}
\item An {\it abstract forest category} is a monoidal small category $(\mathcal C,\circ,\otimes)$ with set of object $\mathbf{N}$ so that there exist a presented forest category $\FC\langle S| R\rangle$ and a monoidal functor $\phi:\FC\langle S| R\rangle\to \mathcal C$ that is the identity on object and is bijective on morphisms.
In that case we say that $(S,R)$ is \textit{a forest presentation} of the abstract forest category $\mathcal C$.
\item An abstract forest category is called \textit{free} or \textit{universal} if it admits a presentation $(S',R')$ where $R'=\emptyset$.
\item An {\it abstract colour} of an abstract forest category $\mathcal F$ is a tree with two leaves.
\end{enumerate}
\end{definition}
\begin{remark}\label{rem:colour}
We may drop the term {\it abstract} if the context is clear.
Note that if $\mathcal F=\FC\langle S|R\rangle$ is a presented forest category, then $S\to\mathcal F(2,1), a\mapsto Y_a$ realises a surjection from the set of colours $S$ to the set of abstract colours $\mathcal F(2,1)$.
This justifies the last definition of abstract colours.
\end{remark}
\subsubsection{Forest monoids}
{\bf Infinite forests.}
An \textit{infinite forest} $f$ (over $\mathcal F$) is a sequence of trees $(f_n:\ n\geq 1)$.
Pictorially, we imagine $f$ has an infinite planar diagram contained in a horizontal strip of the plane with $f_{n}$ a tree placed to the left of $f_{n+1}$.
We define and order the leaves of $f$ from left to right just as we did for forests.
We compose two infinite forests using vertical concatenations.
This provides a well-defined associative binary operation that is defined for {\it all} pairs of infinite forests.
We are interested in a smaller set of infinite forests.
{\bf Monoid of finitely supported forests.}
Let $f$ be an infinite forest over $\mathcal F$.
The {\it support} of $f$ is the set of $n\geq 1$ satisfying that $f_n$ is not the trivial tree (the tree with one leaf).
Define $\mathcal F_\infty$ to be the set of all {\it finitely supported} infinite forests over $\mathcal F$.
Note that the composition of two finitely supported infinite forests is still finitely supported.
Moreover, the trivial infinite forest is obviously in $\mathcal F_\infty$ giving a unit for the composition.
We deduce that $(\mathcal F_\infty,\circ)$ is a monoid.
We call $\mathcal F_\infty$ the \textit{forest monoid} associated to $\mathcal F$.
If $P=(S,R)$ is a forest presentation of $\mathcal F$, then we write
$$\mathcal F_\infty:=\FM\langle S| R\rangle$$
and say that $P$ is a forest presentation of the monoid $\mathcal F_\infty$.
We may also call $\mathcal F_\infty$ the \textit{forest monoid over the presentation $P$}.
{\bf Notation.}
Note the distinction of symbols: $\mathcal F=\FC\langle S| R\rangle$ and $\mathcal F_\infty=\FM\langle S| R\rangle$ where $\FC$ and $\FM$ stand for forest category and forest monoid, respectively, and the symbol $\infty$ stands for forests with infinitely many trees.
We may write $I^{\otimes\infty}$ for the unit of $\mathcal F_\infty$.
If $f\in\mathcal F_\infty$ is supported on $\{1,\cdots,n\}$, then we may write $f_1\otimes f_2\cdots \otimes f_n\otimes I^{\otimes \infty}$ for $f$ where $f_j$ is the $j$th tree of $f$.
{\bf Elementary infinite forests and generators.}
If $a\in S$ and $j\geq 1$, then we write $a_j\in \mathcal F_\infty$ for the infinite forest $I^{\otimes j-1}\otimes Y_a\otimes I^{\otimes \infty}$ having for $j$th tree $Y_a$ and all other trees trivial.
We say that $a_j$ is an \textit{elementary forest} of $\mathcal F_\infty$ and write $\mathcal E(S)$ or $\mathcal E(S)_\infty$ or $S_\infty$ the set of all of them.
Note that we have a bijection
$$S\times \mathbf{N}_{>0}\to S_\infty, \ (a,j)\mapsto a_j$$
and $S_\infty$ generates the monoid $\mathcal F_\infty$.
We extend the notation $a_j$ to all tree $t$ and index $j$ by writing $t_j$ for the infinite forest $I^{\otimes j-1}\otimes t\otimes I^{\otimes \infty}$ having a its $j$th tree equal to $t$ and all other trivial.
{\bf Partial order.}
If $\mathcal F_\infty$ is a forest monoid, then we can define a partial order $\leq$ such as:
$f\leq f'$ if there exists $p\in \mathcal F_\infty$ satisfying $f'=f\circ p$.
This is indeed a partial order analogous to the partial order defined on forest categories.
\begin{remark}\label{rk:forest-monoid}
(Forest monoids obtained from directed systems)
Define the map
$$\phi:\mathcal F\to \mathcal F_\infty,\ (f_1,\cdots,f_n)\mapsto (f_1,\cdots,f_n, I, I,\cdots)$$ where $(f_1,\cdots,f_n)$ is a finite list of trees.
Intuitively, $\phi(f)$ is obtained by adding infinitely many trivial trees to the right of $f$ and thus we may simply write $\phi(f)=f\otimes I^{\otimes \infty}.$
Observe that $\phi$ is preserving the composition: $\phi(f\circ g)=\phi(f)\circ \phi(g)$ for $f,g\in\mathcal F$.
Hence, $\phi$ is a covariant functor from $\mathcal F$ to $\mathcal F_\infty$.
Moreover, $\phi$ preserves the partial orders: if $f\leq f'$, then $\phi(f)\leq \phi(f')$.
It is easy to see that, as a function, $\phi$ is surjective but not injective.
Indeed, $\phi(f\otimes I) = \phi(f)$ for all $f\in\mathcal F$.
Although, if we consider the system of inclusion maps
$$(\iota_{m,n}^k:\mathcal F(m,n)\to \mathcal F(m+k,n+k), f\mapsto f\otimes I^{\otimes k} \text{ for } k\geq 0 \text{ and } 1\leq n\leq m),$$
then the inductive limit of the directed system $(\mathcal F(m,n), \iota_{m,n}^k:\ k\geq 0 \text{ and } 1\leq n\leq m)$ can be identified in an obvious manner with $\mathcal F_\infty.$
(Exterior law)
Unlike the composition and the order we cannot extend the tensor product of $\mathcal F$ to $\mathcal F_\infty$.
However, one can tensor an element of $\mathcal F$ with an element of $\mathcal F_\infty$ obtaining a map:
$$\mathcal F\times \mathcal F_\infty\to \mathcal F_\infty,\ (f, g)\mapsto f\otimes g.$$
This justifies the notation $f\mapsto f\otimes I^{\otimes\infty}$ defining the map $\phi$ of above.
\end{remark}
\subsection{Usual presentations, universal properties, and Jones' technology}\label{sec:universal-property}
\subsubsection{Category presentation}\label{sec:cat-pres}
Consider a forest presentation $(S,R)$ with associated categories
$$\mathcal{UF}=\FC\langle S\rangle, \mathcal{UF}_\infty=\FM\langle S\rangle, \mathcal F=\FC\langle S|R\rangle, \mathcal F_\infty=\FM\langle S|R\rangle.$$
Note that a {\it forest} presentation $(S,R)$ is not a presentation in the usual sense for $\mathcal F$ or $\mathcal F_\infty$
Our plan is to deduce from $(S,R)$ a category presentation of $\mathcal F$ and a monoid presentation of $\mathcal F_\infty$.
From there we will deduce universal properties of $\mathcal F$ (and $\mathcal F_\infty$) with respect to \textit{any} category (not only a forest category).
{\bf This universal property combined with Jones' technology will permit us to construct actions of forest groups. It is one of the main motivations of our work which will be extensively studied in future articles.}
In order to not create confusions we systematically say in this section that $S,R$ are, respectively, sets of \textit{forest generators}, \textit{forest relations}, and $(S,R)$ is a \textit{forest presentation} of $\mathcal F$.
For a (classical) category or monoid $\mathcal C$ we use the terminology \textit{category generators} or {\it monoid generators}, etc., for the usual notions as defined in \cite[Section 1.4]{Dehornoy-book}.
{\bf Thompson-like relations.}
Observe that the diagrammatic structures provide the following category relations that are inherent to all forest categories and forest monoids coloured by $S$:
\begin{align}\label{eq:Thompson-relations}
\TR(S):=&\{(b_{q,n}\circ a_{j,n+1}\ , \ a_{j,n}\circ b_{q+1,n+1}):\ a,b\in S \text{ and } 1\leq j <q\leq n\};\\
\TR(S)_\infty:=&\{(b_{q}\circ a_{j}\ , \ a_{j}\circ b_{q+1}):\ a,b\in S \text{ and } 1\leq j <q\}.
\end{align}
We call them \textit{Thompson-like relations} (over the set $S$).
The first appears in $\mathcal F$ while the second in $\mathcal F_\infty$.
{\bf Forest relations give category relations.}
Consider a forest relation $(u,u')\in R$ which is a pair of trees with the same number of leaves.
The tree $u$ can be written (uniquely up to the Thompson-like relations) as a word of elementary forests so that its $k$th letter is of the form $y_{i_k,k}$ where $y\in S$ and $1\leq i_k\leq k$.
Similarly, write $u'$ as a word with letters in $\mathcal E(S)$.
This provides a relation that we write $R(u,u',1,1)$.
Now, by shifting the $k$th letter $y_{i_k,k}$ to $y_{i_k+j-1, k+n-1}$ we obtain a new category relation $R(u,u',j,n)$ corresponding to the pair of forests $(u_{j,n},u'_{j,n})$ (here $u_{j,n}=I^{\otimes j-1} \otimes u\otimes I^{\otimes n-j}$).
Similarly, we obtain a one parameter family of monoid relations $R(u,u',j)$ by erasing the second index $n$.
The important point is that there are no additional category relations in $\mathcal F$ and $\mathcal F_\infty$ as stated in the following proposition.
\begin{proposition}\label{prop:universal-category}
If $\mathcal F:=\FC\langle S| R\rangle$ is a presented forest category, then
\begin{align*}
&(\mathcal E(S), \{ R(u,u',j,n):\ (u,u')\in R, 1\leq j\leq n\}\cup \TR(S)) \text{ and }\\
&(S_\infty, \{ R(u,u',j):\ (u,u')\in R, 1\leq j\}\cup \TR(S)_\infty)
\end{align*}
are category presentation of $\mathcal F$ and monoid presentation of $\mathcal F_\infty$, respectively.
\end{proposition}
\begin{proof}
First, consider the free forest monoid case with $\mathcal{UF}_\infty=\mathcal F\langle S\rangle_\infty$.
An easy adaptation of the monochromatic case (which is a well-known fact, see for instance \cite[Proposition 2.4.5]{Belk-PhD}) gives that $(S_\infty,\TR(S)_\infty)$ is a monoid presentation of $\mathcal{UF}_\infty$.
From there we deduce the category presentation $(\mathcal E(S),\TR(S))$ for the free forest category $\mathcal{UF}$.
Now, going from $\mathcal{UF}$ to $\mathcal F$ (resp.~$\mathcal{UF}_\infty$ to $\mathcal F_\infty$) is obvious by the definition of $\mathcal F$ (resp.~$\mathcal F_\infty$) which is the quotient of $\mathcal{UF}$ (resp.~$\mathcal{UF}_\infty$) by the equivalence relation $\overline R$ equal to the closure of $R$ by compositions and tensor products.
\end{proof}
\begin{example}A category presentation of the free forest category $\mathcal{UF}:=\FC\langle S\rangle$ is $(\mathcal E(S), \TR(S)).$
The Thompson-like relation $(b_{3,3}\circ a_{2,4}, a_{2,3}\circ b_{4,4})$ is expressed diagrammatically as follows:
\[\includegraphics{b33a24.pdf}\]
Consider $\mathcal F:=\FC\langle a,b| a_1a_2=b_1b_1\rangle.$
A category presentation of $\mathcal F$ is given by
the category generator set
$$\mathcal E(a,b):=\{ a_{j,n}, b_{j,n}:\ 1\leq j\leq n\}$$
and the category relations
$$(a_{j,n}a_{j+1,n+1} \ , \ b_{j,n} b_{j,n+1}) \text{ for all } 1\leq j\leq n$$
and
$$(x_{q,n}\circ y_{j,n+1}\ , \ y_{j,n}\circ x_{q+1,n+1}):\ x,y\in \{a,b\} \text{ and } 1\leq j <q\leq n.$$
Here is a diagrammatic description;
\[\includegraphics{Ruuprime.pdf}\]
A monoid presentation of $\mathcal F_\infty$ is obtained by removing the second index of the generators, e.g.~$a_{j,n}\leftrightarrow a_j$.
\end{example}
\subsubsection{Universal property and Jones' technology}
The next corollary is a direct consequence of the previous proposition and the definition of category presentation.
\begin{corollary}
A presented forest category $\mathcal F:=\FC\langle S| R\rangle$ satisfies the following universal property.
Consider a category $\mathcal C$ containing a set of morphisms $X$.
Assume that there is a map $\phi:\mathcal E(S)\to X$.
Extend $\phi$ to finite paths of morphisms of $\mathcal E(S)$ as
$$f_1\circ \cdots\circ f_n \mapsto \phi(f_1)\circ \cdots \circ \phi(f_n).$$
Assume that, if $(f,f')$ is a category relation of $\mathcal F$, then $\phi(f)=\phi(f')$.
Then there exists a unique covariant functor $\Phi:\mathcal F\to \mathcal C$
whose restriction to $\mathcal E(S)$ is $\phi$.
\end{corollary}
\begin{remark}
In practice, we will be considering {\it contravariant} functors for constructing unitary representation or actions on groups via Jones' technology. Although, the covariant case is interesting as well. For instance, it produces actions of Thompson's groups on inverse limits such as totally disconnect compact spaces, profinite groups, and other.
\end{remark}
We will be mainly considering \textit{monoidal} functors and the following corollary.
In that case we only need to consider a {\it forest} presentation rather than a {\it category} presentation.
So far, it has been the most common way for producing explicit actions of groups via Jones' technology.
\begin{corollary}
Let $\mathcal F:=\FC\langle S| R\rangle$ be a presented forest category and $(\mathcal C,\circ,\otimes)$ a monoidal category.
Assume there exists an object $e$ in the collection of objects of $\mathcal C$ and some morphism $\phi(a):e\to e \otimes e$ for all $a\in S$.
Define $\phi$ on elementary forests such that
$$\phi(a_{j,n}):= \id_e^{\otimes j-1}\otimes \phi(a)\otimes \id_e^{\otimes n-j} \text{ for all } a\in S, 1\leq j\leq n,$$
where $\id_e$ denotes the identity automorphism of the object $e$.
There exists a unique contravariant monoidal functor $\Phi: \FC\langle S\rangle\to C$ satisfying that $\Phi$ restricts to $\phi$ on the set of elementary forests $\mathcal E(S)$.
If moreover, for each forest relation $(u,u')\in R$ we have that $\Phi(u)=\Phi(u')$, then $\Phi$ factorises uniquely into a contravariant monoidal functor $\overline\Phi:\mathcal F\to \mathcal C$.
\end{corollary}
\begin{example}
Consider the presented forest category $\mathcal F:=\FC \langle a,b | a_1a_{2}=b_1b_{1}\rangle.$
Let $\Hilb$ be the category of (complex) Hilbert spaces having isometries for morphisms and monoidal structure given by the classical tensor product of Hilbert spaces.
For any Hilbert space $H$ and isometries $A,B:H\to H\otimes H$ satisfying
\begin{equation}\label{eq:ABone}(\id_H\otimes A)\circ A = (B\otimes \id_H)\circ B.\end{equation}
there exists a unique contravariant monoidal functor $\Phi:\mathcal F\to \Hilb$ satisfying that $\Phi(Y_a)=A, \Phi(Y_b)=B.$
We will see in Section \ref{sec:example} that $\mathcal F$ is a Ore category whose fraction group $G=\Frac(\mathcal F,1)$ isomorphic to the Cleary irrational-slope Thompson's group \cite{Cleary00}.
Using Jones' technology (which we don't develop in this article) we can build explicitly from $\Phi$ a unitary representation of $G^V$ (the $V$-version of $G$ defined in \cite{Burillo-Nucinkis-Reeves22}).
Hence, any pairs of isometries $(A,B)$ satisfying Equation \ref{eq:ABone} provides a unitary representation of the three irrational-slope Thompson groups $G,G^T,$ and $G^V$.
\end{example}
\subsection{The collection of all forest categories}
Our main object of study is a forest category considered as an algebraic structure like a monoid or a group.
We will now define what are the right notion of maps between them.
This leads us to introduce the category of forest categories. This is a category that is not captured by classical set theory and we treat it in that way.
The notion of morphism between forest categories may seem restrictive at a first glance. However, it is the correct notion assuring that range of morphisms are forest categories (leading to the notions of quotient and embedding).
We could spend more time defining various operations and constructions in this category (like free product, direct product, etc.) but leave it to future articles when we will derive applications of these operations.
\subsubsection{The category of forest categories}
{\bf Collection of objects.}
Let $\Forest$ be the collection of all forest categories.
This is not a set since the class of all nonempyt sets embeds in it via $S\mapsto \mathcal{UF}\langle S\rangle.$
{\bf Morphisms.}
Consider two forest categories $\mathcal F,\tilde\mathcal F$.
A {\it morphism} $\phi$ from $\mathcal F$ to $\tilde\mathcal F$ is a covariant monoidal functor from $\mathcal F$ to $\tilde\mathcal F$ satisfying that $\phi(1)=1$ for the object $1$.
In other terms, $\phi$ is a map from $\mathcal F$ to $\tilde\mathcal F$ such that if $f$ is a forest with $n$ roots and $m$ leaves, then $\phi(f)$ is a forest of $\tilde\mathcal F$ with $n$ roots and $m$ leaves.
Moreover, $\phi(f\circ g) = \phi(f)\circ \phi(g)$ and $\phi(f\otimes h)=\phi(f)\otimes \phi(h)$ for all $f,g,h\in\mathcal F$.
We write $\Hom(\mathcal F,\tilde\mathcal F)$ for the class of morphisms form $\mathcal F$ to $\tilde\mathcal F$.
Elements of $\Hom(\mathcal F,\tilde\mathcal F)$ are typically written using the symbols $\phi,\varphi,\psi,\chi.$
Note, since there is only one tree with one leaf in each forest category we have $\phi(I)=I$ and thus $\phi$ sends trivial forests to trivial forests.
Similarly, it maps abstract colours to abstract colours.
\begin{proposition}
The collection of all forest categories together with the collection of morphisms defines a category that we call the \textit{category of forest categories} denoted $\Forest$.
\end{proposition}
\begin{proof}
A composition of covariant monoidal functor is a covariant monoidal functor implying that a composition of morphisms is a morphism.
Since morphisms are functions it is obvious that the composition is associative.
For each forest category we can consider the identity map which defines an identity functor.
All together we deduce that $\Forest$ is a category.
\end{proof}
{\bf Terminology.}
We will adopt a set-theoretical terminology for qualifying properties of morphisms.
Hence, we will say that $\phi:\mathcal F\to\tilde\mathcal F$ is injective or surjective rather than $\phi$ is a monomorphism or an epimorphism, respectively.
It is not hard to see that if a morphism $\phi:\mathcal F\to\tilde\mathcal F$ is bijective, then the inverse map is a morphism.
Hence, an isomorphism from $\mathcal F$ to $\tilde\mathcal F$ is a bijective morphism $\phi:\mathcal F\to\tilde\mathcal F$ with inverse written $\phi^{-1}.$
Similarly, an automorphism of $\mathcal F$ is a bijective endomorphism of $\mathcal F$.
\begin{proposition}\label{prop:colour-morphism}
Consider two presented forest categories $\mathcal F=\FC\langle S| R \rangle$ and $\tilde\mathcal F=\FC\langle \tilde S| \tilde R\rangle$.
A morphism $\phi:\mathcal F\to\tilde\mathcal F$ is characterised by its restriction to $\mathcal F(2,1)$ (the set of trees of $\mathcal F$ with 2 leaves).
In particular, $\Hom(\mathcal F,\tilde\mathcal F)$ is a set which is finite when $S$ is and there is an injective map:
$$r:\Hom(\mathcal F,\tilde\mathcal F)\to \{S\to \tilde S\}, \ \phi\mapsto r(\phi)$$ satisfying
$$\phi(Y_a) = Y_{r(\phi)(a)} \text{ for all } \phi\in \Hom(\mathcal F,\tilde\mathcal F), a\in S.$$
\end{proposition}
\begin{proof}
This is a direct consequence of the universal property satisfied by a forest category among monoidal categories, see Proposition \ref{prop:universal-category}.
\end{proof}
Here is a diagrammatic example. Consider a morphism $\phi:\mathcal F\to\tilde\mathcal F$ and write $r$ the map $r(\phi)$ from the colour set of $\mathcal F$ to the one of $\tilde\mathcal F$.
We have:
\[\includegraphics{phi.pdf}\]
\begin{remark}
Note that there exist covariant monoidal functors between forest categories that are not {\it morphisms} in our sense.
This means that there exists maps $\phi:\mathcal F\to\mathcal G$ that preserve compositions and tensor products but not the number of roots and leaves.
It rarely happens but it does. We give below an example where $\mathcal F$ and $\mathcal G$ are rather pathological. Note that in many cases a covariant and monoidal functor between two specific forest categories will be automatically a morphism in our sense.
Consider the monochromatic forest category $\mathcal F$ that has only one tree $t_n$ with $n$ leaves for each $n\geq 1.$
This can be obtained using the forest presentation $(S,R)$ where $S$ is a singleton and where $R$ is the set of {\it all} pairs of trees $(t,s)$ with the same number of leaves.
If $\mathcal T\subset \mathcal F$, is the subset of trees, then we set:
$$\phi:\mathcal T\to\mathcal F, t_n\mapsto t_{2n-1}\otimes I.$$
Now, extend $\phi$ on $\mathcal F$ as follows:
$$\phi:\mathcal F\to\mathcal F, f_1\otimes\cdots\otimes f_r\mapsto \phi(f_1)\otimes \cdots \otimes \phi(f_r)$$
for all forest $f=f_1\otimes\cdots \otimes f_r$ where each $f_j$ is a tree.
The map $\phi$ preserves composition implying that it is a covariant functor. Moreover, it is monoidal by construction.
However, it is not a morphism of forest categories because it doubles the number of roots and leaves.
\end{remark}
\subsubsection{Kernel, quotient, range, factorisation, and forest subcategory}\label{sec:forest-sub}
Consider two forest categories $\mathcal F,\mathcal G$ and a morphism $\phi:\mathcal F\to\mathcal G$.
{\bf Kernel.}
The \textit{kernel} $\ker(\phi)$ of $\phi$ is the set of pairs $(f,f')\in\mathcal F\times \mathcal F$ satisfying $\phi(f)=\phi(f').$
Note that $\ker(\phi)$ is closed under taking composition, tensor product, rooted subforests, and is the graph of an equivalence relation denoted $\sim_\phi.$
If $R_\phi\subset\ker(f)$ is the subset of pairs of {\it trees}, then it is a set of forest relations that generates $\ker(f)$ (i.e.~the closure of this set for composition and tensor product is equal to $\ker(f)$).
{\bf Quotient and factorisation.}
Let $\chi_\phi:\mathcal F\to \mathcal F/\sim_\phi$ be the canonical quotient map.
Observe that $\mathcal F/\sim_\phi$ has an obvious structure of forest category.
Equip with this structure $\chi_\phi$ is a surjective morphism of forest categories.
Moreover, if $(S,R)$ is a forest presentation of $\mathcal F$, then $(S,R\cup R_\phi)$ is a forest presentation of $\mathcal F/\sim_\phi$.
Finally, the morphism $\phi:\mathcal F\to\mathcal G$ factorises uniquely into an injective morphism $\overline\phi:\mathcal F/\sim_\phi\hookrightarrow\mathcal G$.
{\bf Forest subcategory.}
The range $\phi(\mathcal F)$ of $\phi$ has an obvious structure of forest category and is isomorphic to $\mathcal F/\sim_\phi$ via $\overline\phi$.
We say that $\phi(\mathcal F)$ is a {\it forest subcategory} of $\mathcal F$.
Note that $\phi(\mathcal F)$ is generated by $\phi(\mathcal F)\cap \mathcal G(2,1)$ (its trees with two leaves) where {\it generated} means closed under composition, tensor product, and by adding the trivial tree.
Moreover, this characterises forest subcategories: $\{\mathcal G(X):\ \emptyset\neq X\subset\mathcal G(2,1)\}$ is the set of all forest subcategories of $\mathcal G$.
Note, if $a$ is a colour of $\mathcal G$, then $\mathcal G(\{Y_a\})=\mathcal G(a)$ is the forest subcategory of $\mathcal G$ equals to all monochromatic forests coloured by $a$.
{\bf Quasi-forest subcategories.}
We slightly extend the definition of above by saying that $\mathcal G(X)$ is a {\it forest quasi-subcategory} of $\mathcal G$ where $X$ is a nonempty subset of {\it trees} of $\mathcal G$ (rather than a nonempty subset of trees with exactly two leaves).
In particular, if $t\in\mathcal G$ is a nontrivial tree, then it defines a forest quasi-subcategory $\mathcal G(\{t\})=\mathcal G(t)$ of trees made exclusively with the ``caret'' $t$.
All these definitions and facts can be adapted to forest monoids in the obvious manner.
\subsection{Generalisations of forest categories}
We briefly generalise our setting of forest categories adding permutations and braids. This will be used to produce groups similar to Thompson's groups $T,V$, and $BV$.
These constructions are obvious extensions of the classical cases of Thompson's groups $T,V$ exposed in Cannon-Floyd-Parry and by Brin \cite{Cannon-Floyd-Parry96,Brin-BV1}, see also the PhD thesis of Belk \cite[Section 7.4]{Belk-PhD}.
Moreover, they appear as particular cases of the constructions of Thumann of symmetric and braided operads corresponding to the $V$ and $BV$ cases \cite[Section 3.1]{Thumann17}.
\subsubsection{Diagrams for groups}
We start by describing the symmetric, braid, and cyclic groups using diagrams.
{\it Symmetric group.}
Let $\Sigma_n$ be the symmetric group of order $n$: the group of bijections of $\{1,\cdots,n\}$.
Given $\sigma\in \Sigma_n$ we consider $[\sigma]$ to be the isotopy class of the diagram drawn in $\mathbf{R}^2$ equal to $\cup_{1\leq k\leq n} [ (k,1) , (\sigma(k),0)]$ where $[ (k,1) , (\sigma(k),0)]$ is the segment going from $(k,1)$ to $(\sigma(k),0)$ for $1\leq k\leq n.$
We interpret and call the points $(j,0)$ and $(k,1)$ as the $j$th root and $k$th leaf of $[\sigma]$.
Write $[\sigma]\circ[\tau]$ for the (isotopy class of the) horizontal concatenation of $\tau$ on top of $\sigma$ and note that $[\sigma]\circ[\tau]=[\sigma\circ \tau].$
Here is an example of two permutations of $\Sigma_4$ and their composition:
\[\includegraphics{sigmatau.pdf}\]
{\it Braid group.}
By considering two types of crossing, over-crossing and under-crossing, we obtain the classical graphical description given by Artin of the braid groups $B_n$ over $n$ strands \cite{Artin47}.
Note, the usual group morphism $B_n\twoheadrightarrow \Sigma_n$ corresponds in forgetting the distinction between over and under-crossings.
{\it Cyclic group.}
Consider the cyclic permutation $g:j\mapsto j+1$ of $\Sigma_n$ with $j$ written modulo $n$.
This provides an embedding $\mathbf{Z}/n\mathbf{Z}\to \Sigma_n, 1\mapsto g$.
We identify $\mathbf{Z}/n\mathbf{Z}$ with this image in $\Sigma_n$ and consider the diagrammatic description of $\mathbf{Z}/n\mathbf{Z}$ deduced from the one of $\Sigma_n$.
\subsubsection{Free $X$-forest categories}
{\it The $V$-case.}
Consider a free forest category $\mathcal{UF}$ over a set of colours $S$.
If $f\in\mathcal{UF}$ is a forest with $n$ leaves and $\sigma\in \Sigma_n$, then we write $f\circ \sigma$ for the horizontal concatenation of $f$ with $[\sigma]$ on top of it where the $j$th leaf of $f$ is lined up with the $j$th root of $\sigma.$
Similarly, if $f$ has $r$ roots and $\tau\in \Sigma_r$, then $\tau\circ f$ stands for the diagram obtained by concatenating $f$ on top of $[\tau]$.
Now, $\tau\circ f = f^\tau\circ \tau^f$ for uniquely determined forest $f^\tau$ and permutation $\tau^f$ as explained in \cite[Section 7.4]{Belk-PhD} (up to adding colours).
Here is an example with $\tau\in \Sigma_3$ and $f$ a forest with three roots:
\[\includegraphics{BZS.pdf}\]
This defines a composition and a small category $\mathcal{UF}^V$ generated by a copy of $\mathcal{UF}$ and a copy of $\Sigma_n$ for each $n$.
(We may interpret $\mathcal{UF}^V$ as a Brin-Zappa-Sz\'ep product of the category $\mathcal{UF}$ and the groupoid of all finite permutations.)
All elements of $\mathcal{UF}^V$ (except the empty diagram) can uniquely be written in the form $f\circ \sigma$.
{\it The $T$ and $BV$-cases.}
We observe that if $\tau$ is cyclic and $f$ is a forest, then $\tau^f$ is again a cyclic permutation.
This allows us to define $\mathcal{UF}^T$ as the subcategory of $\mathcal{UF}^V$ generated by $\mathcal{UF}$ and all the cyclic groups $\mathbf{Z}/n\mathbf{Z}.$
Similarly, using the graphical composition explained in \cite{Brin-BV1} we define $\mathcal{UF}^{BV}$ where now $\Sigma_n$ is replaced by $B_n$.
We say that $\mathcal{UF}^X$ is the free $X$-forest category over the set of colours $S$ for $X=F,T,V,BV$ where $\mathcal{UF}^F:=\mathcal{UF}$.
\subsubsection{$X$-forest categories}
Consider a presented forest category $\mathcal F=\FC\langle S|R\rangle$ and write $\chi:\mathcal{UF}\to\mathcal F$ for the canonical morphism.
Observe that if $\chi(f)=\chi(f')$, then $f$ and $f'$ have the same number of roots and leaves.
Moreover, if the $j$th tree of $f$ has $n_j$ leaves, then so does the $j$th tree of $f'$.
This implies that $\tau\mapsto \tau^f$ only depends on the class of $f$ inside $\mathcal F$ where $\tau$ is either a permutation or a braid.
Moreover, $\chi(f)=\chi(f')$ implies that $\chi(f^\tau)=\chi(f'^\tau).$
This permits to define $\mathcal F^X$ from $\mathcal{UF}^X$ in the obvious manner.
We call $\mathcal F^X$ a {\it $X$-forest category} or the {\it $X$-version} of the forest category $\mathcal F$.
\subsubsection{$Y$-forest monoids}
A similar construction can be applied to a forest monoid $\mathcal F_\infty$.
Consider the group $\Sigma_{(\mathbf{N})}$ of {\it finitely supported} permutations of $\mathbf{N}_{>0}.$
Elements of $\Sigma_{(\mathbf{N})}$ can be represented as diagrams similar to above and together with $\mathcal F_\infty$ forms a monoid $\mathcal F_\infty^V$.
Replacing permutations by braids (that are finitely supported) we obtain a monoid $\mathcal F_\infty^{BV}$.
Now, this construction does not apply to the $T$-case since there are no directed structures for the sequence of finite cyclic groups.
If $Y=F,V,BV$ (hence missing $T$), then we call $\mathcal F_\infty^Y$ the {\it $Y$-forest monoid} or the {\it $Y$-version} of the monoid $\mathcal F_\infty$ setting by convention $\mathcal F_\infty^F=\mathcal F_\infty$.
\section{Ore forest categories and monoids}\label{sec:Dehornoy}
Our main objective is to construct groups from forest categories using a left-cancellative {\it calculus of fractions}.
This is equivalent for a category to be left-cancellative and to satisfy a property due to Ore, see \cite[Chapter I]{Gabriel-Zisman67} or \cite[Chapter 3]{Dehornoy-book} for details.
A category with these two properties will be called a \textit{Ore category}.
In this section we investigate left-cancellativity and Ore's property.
First, we define and rephrase them via obvious but useful characterisations.
Second, we provide more advanced techniques to decide if a forest category is left-cancellative and in some fewer cases satisfies Ore's property.
This second analysis is an adaptation of the work of Dehornoy on complete presentations of monoids \cite{Dehornoy03}.
It will provide very useful and powerful criteria that can be, in some cases, directly read from the forest presentation.
\subsection{Definitions and obvious characterisations}
\subsubsection{Cancellative}
A category $\mathcal C$ is {\it left-cancellative} if $f\circ g=f\circ h$ implies $g=h$.
Right-cancellativity is defined analogously and {\it cancellative} means left and right cancellative.
By choice we will only consider the property of being {\it left}-cancellative.
Here are some straightforward but useful observations.
\begin{observation}\label{obs:LC}
Consider a forest category $\mathcal F$ and its associated forest monoid $\mathcal F_\infty$.
For each $f\in\mathcal F$ we consider the (partially defined) multiplication map:
$L_f: g\mapsto f\circ g.$
The following assertions are equivalent.
\begin{enumerate}
\item The forest category $\mathcal F$ is left-cancellative;
\item The forest monoid $\mathcal F_\infty$ is left-cancellative;
\item The map $L_t$ is injective for all tree $t$ of $\mathcal F$;
\item The map $L_s$ is injective for all tree $s$ of $\mathcal F$ with two leaves;
\item Any forest subcategory of $\mathcal F$ (including $\mathcal F$ itself) is left-cancellative.
\end{enumerate}
\end{observation}
\begin{example}\label{ex:LC}
\begin{enumerate}
\item A free forest category is cancellative.
\item
The forest category $\FC\langle a,b| a_1b_1=b_1a_1, \ a_1a_2=b_1b_2\rangle$ is not left-cancellative. Indeed, $Y_a(Y_a\otimes Y_b)=Y_a(Y_b\otimes Y_a)$ while $Y_a\otimes Y_b\neq Y_b\otimes Y_a.$
\item A monochromatic forest category is left-cancellative if and only if it is free.
Here is a proof.
Let $\chi:\mathcal{UF}\to\mathcal F$ be the quotient morphism of the monochromatic free forest category onto a monochromatic forest category.
Assume $\chi$ has a nontrivial kernel. Consider a pair of trees $(t,s)$ such that $t\neq s$ and $\chi(t)=\chi(s)$.
Moreover, choose this pair $(t,s)$ so that the number of leaves of $t$ is minimal.
Such a pair exists by assumption.
Since there is only one monochromatic tree $Y$ with two leaves we necessarily have $Y\leq t, Y\leq s$ and they both decomposes as follows:
$$t=Y\circ(h\otimes k) \text{ and } s=Y\circ(h'\otimes k')$$
with $h,k,h',k'$ some monochromatic trees not all trivial.
Since $\chi(t)=\chi(s)$ and $\chi$ is monoidal we deduce that $\chi(h)=\chi(h')$ and $\chi(k)=\chi(k')$.
Therefore, $(h,h'),(k,k')$ are elements of the kernel of $\chi$ and are trees with stritcly less leaves than $t$.
This implies that $h=h'$ and $k=k'$ implying $t=s$, a contradiction.
\item The forest category $\FC\langle a,b| a_1^n=b_1b_2\cdots b_n\rangle$ is left-cancellative for any $n\geq 1$.
In fact, we will see that all presented forest categories with two colours and one relations of the form $a_1\cdots=b_1\cdots$ are left-cancellative.
\end{enumerate}
\end{example}
\subsubsection{Ore's property}
A category $\mathcal C$ satisfies {\it Ore's property} if for all $f,g\in\mathcal C$ with same target there exists $p,q\in\mathcal C$ satisfying $f\circ p=g\circ q.$
\begin{remark}
Recall that a forest category $\mathcal F$ is equipped with a partial order $\leq$ defined as: $f\leq f'$ if there exists $p\in\mathcal F$ satisfying $f'=f\circ p.$
Observe that $\mathcal F$ satisfies Ore's property if and only if given $f,g\in\mathcal F$ with the same number of roots there exists $h\in\mathcal F$ satisfying that $f\leq h$ and $g\leq h$.
We may witness this property by only looking at the set $\mathcal T$ of trees. We deduce that the poset of $(\mathcal T,\leq)$ of trees is {\it directed} if and only if $\mathcal F$ satisfies Ore's property.
\end{remark}
\begin{definition}
We say that a sequence of trees $(t_n)_{n\geq 1}$ is \textit{cofinal} if it is increasing (i.e.~$n\leq m$ implies $t_n\leq t_m$) and if for any $t\in\mathcal T$ there exists $k\geq 1$ so that $t\leq t_k$.
More generally, we consider cofinal {\it nets} of trees.
\end{definition}
Here are useful observations and characterisations of Ore's property.
\begin{observation}\label{obs:Ore}
Let $\mathcal F$ be a forest category, $\mathcal F_\infty$ its associated forest monoid, and $\mathcal T\subset\mathcal F$ the subset of trees equipped with the usual partial order $\leq$.
The following assertions are equivalent.
\begin{enumerate}
\item The forest category $\mathcal F$ satisfies Ore's property;
\item The forest monoid $\mathcal F_\infty$ satisfies Ore's property;
\item The poset $(\mathcal T,\leq)$ is directed;
\item The poset $(\mathcal T,\leq)$ admits a cofinal net;
\item Every quotient of $\mathcal F$ (including $\mathcal F$ itself) satisfies Ore's property.
\end{enumerate}
\end{observation}
If $\mathcal F(2,1)$ is countable (which implies that $\mathcal F$ is countable), then the fourth item can be replaced by ``$(\mathcal T,\leq)$ admits a cofinal {\it sequence}".
\begin{example}
\begin{enumerate}
\item Consider a nonempty set $S$ and the free forest category $\mathcal{UF}$ over $S$.
We have that $\mathcal{UF}$ does not satisfies Ore's property unless $S$ has only one point.
Indeed, if $a,b\in S, a\neq b$, then there are no tree dominating both $Y_a$ and $Y_b$ inside $\mathcal{UF}$.
\item The forest category $\mathcal F=\FC\langle a,b| a_1b_1=b_1a_1\rangle$ satisfies Ore's property and thus so does its quotient $\FC\langle a,b| a_1b_1=b_1a_1 , \ a_1a_2=b_1b_2\rangle$.
Note that the first is left-cancellative while the second is not.
\item The forest category $\FC\langle a,b| a_1^n=b_1b_2\cdots b_n\rangle$ satisfies Ore's property for any $n\geq 1$.
\end{enumerate}
\end{example}
\begin{definition}
A small category $\mathcal C$ that is left-cancellative and satisfies Ore's property is called a \textit{Ore category}.
In that case we say that $\mathcal C$ admits a (left-cancellative) {\it calculus of fractions}.
\end{definition}
We will see in Section \ref{sec:forest-groups} how to construct (fraction) groupoids and groups from Ore forest categories.
{\bf Warning.} In the literature, several different meanings are given to the term {\it Ore category}. It usually translates the existence of a calculus of fractions but requiring weaker or stronger conditions than ours.
\subsection{Dehornoy's criteria}
We present criteria due to Dehornoy which given a presentation provides tools to decide if the associated algebraic structure is left-cancellative or satisfies Ore's property.
We adapt and specialise these criteria to our situation of forest presentations.
Most details about these notions can be found in \cite{Dehornoy03}. Although, we will need several statements from other articles of Dehornoy that we will precisely cite.
{\bf Notations.} In all this section $\sigma$ is a nonempty set of \textit{letters} or \textit{generators}, $\word(\sigma)$ denotes the finite words in $\sigma$ including the trivial one denoted by $e$.
The \textit{length} $|u|=\length(u)$ of a word $u$ is the number of letters that composed $u$ taking the convention $|e|=0.$
We write $\rho$ for a set of pairs $(u,v)\in \word(\sigma)^2$ satisfying $|u|=|v|$.
Elements of $\rho$ are called \textit{relations}.
We consider the pair $\pi:=(\sigma,\rho)$ that we interpret as the presentation of a monoid $M$ that we denote by $\Mon\langle \sigma|\rho\rangle$ where $\Mon$ stands for {\it monoid}.
We continue to write $u\in\word(\sigma)$ for its image in the monoid $M$ and may write $u=_\rho v$ to express that two words $u,v\in \word(\sigma)$ are equal in the monoid $M$, i.e.~$(u,v)$ belongs to the smallest equivalence relation generated by $\rho.$
{\bf Terminology.}
We say that $\pi$ is a {\it homogeneous monoid presentation}.
This means that all relations of $\rho$ are pairs of words $(u,u')$ in the letter set $\sigma$ such that $|u|=|u'|$.
In particular, $u$ does not contain any inverse of letters of $\sigma$ (it is a monoid presentation).
This last property is also called {\it positive}.
As usual we write $\sigma^{-1}$ for the set of formal inverses of elements of $\sigma$.
We consider the group $G=\Gr\langle \sigma|\rho\rangle$ with presentation $(\sigma,\rho)$ where $\Gr$ stands for {\it group}.
\subsubsection{Complete presentation}
{\bf Reversibility \cite[Definition 1.1]{Dehornoy03}.}
Consider words $w,w'\in \word(\sigma\cup \sigma^{-1})$ (hence composed of letters and formal inverse of letters) and say that $w\curvearrowright^0 w'$ is true if $w'$ is obtained from $w$
\begin{enumerate}
\item by deleting an occurrence of $u^{-1}u$ for $u \in \word(\sigma)$ or;
\item by replacing an occurrence of $u^{-1} v$ by $v'u'^{-1}$ such that $(uv',vu')\in \rho$ and $u,v,u',v'\in \word(\sigma)$.
\end{enumerate}
We say that $w$ is (right-)reversible to $w''$ (or $w$ \textit{reverses} to $w''$) if we can go from $w$ to $w''$ with finitely many transformations as above.
We then write $w\curvearrowright w''$ removing the superscript 0.
When there are several presentations involved we may add ``w.r.t.~the presentation $\pi$'' to express which kind of relations we are allowed to use in the reversing process.
{\bf Dehornoy's strong cube condition \cite[Definition 3.1]{Dehornoy03}.}
The presentation $(\sigma,\rho)$ satisfies the strong cube condition (in short SCC) at $(u,v,w)\in \word(\sigma)^3$ if the following holds:
$$[u^{-1} w w^{-1} v \curvearrowright v' u'^{-1} \text{ for some } u',v'\in \word(\sigma)]\Rightarrow (uv')^{-1}(vu')\curvearrowright e.$$
Given $X\subset \word(\sigma)$ we say that $(\sigma,\rho)$ satisfies the SCC on $X$ if the cube condition holds for all $(u,v,w)\in X^3.$
Note that the order of $u,v,w$ matters this is why we use triples rather than sets.
\begin{observation}
If $(\sigma,\rho)$ is a presentation and $u\in \word(\sigma)$, then the SCC is satisfied at $(u,u,u).$
Moreover, note that if $u^{-1} w w^{-1} v \curvearrowright v' u'^{-1}$, then necessarily $uv'=_\rho vu'$ (see \cite[Lemma 1.10]{Dehornoy03}).
\end{observation}
{\bf Complete presentation.}
Dehornoy defined the very useful concept of \textit{completeness} for presentations.
We only consider homogeneous monoid presentations. For those, there is a way to characterise completeness using the notion of SCC.
We use this characterisation as our definition.
\begin{definition}\cite[Proposition 4.4]{Dehornoy03}
A homogeneous monoid presentation $(\sigma,\rho)$ is called \textit{complete} if it satisfies the SCC at $\sigma$.
\end{definition}
The following proposition is the main strength of the concept of completeness.
\begin{proposition}\label{prop:DehornoyRCF}\cite[Propositions 6.1 and 6.7]{Dehornoy03}
Let $(\sigma,\rho)$ be a complete presentation with associated monoid $M$.
The monoid $M$
\begin{enumerate}
\item is left-cancellative if and only if $u^{-1}v\curvearrowright e$ for any relation $(au,av)\in \rho$ with $a\in \sigma, u,v\in \word(\sigma)$;
\item has Ore's property if and only if there exists $\sigma\subset \sigma'\subset \word(\sigma)$ so that for any $u,v\in \sigma'$ there exists $u',v'\in \sigma'$ satisfying $(uv')^{-1} (vu')\curvearrowright e.$
\end{enumerate}
\end{proposition}
\subsubsection{The special case of complemented presentation}
We consider the specific case of {\it complemented} presentations introduced in \cite{Dehornoy97}.
Complemented presentations are easy to work with as there is at most one reversing process per pairs of words.
{\bf Warning.} We warn the reader that {\it complete} and {\it complemented} are not the same notions.
The first was defined above and stands for presentations that have no "hidden" relations in an intuitive sense while the second stands for presentations that have "few" relations as we are about to define.
\begin{definition}
A presentation $(\sigma,\rho)$ is {\it complemented} if for each $a,b\in \sigma$ there is at most one relation of the form $a\cdots=b\cdots$ in $\rho$.
When they exists we write $(a\backslash b)$ and $(b\backslash a)$ the unique elements of $\word(\sigma)$ satisfying $(a(a\backslash b) \ , \ b(b\backslash a))\in \rho.$
\end{definition}
If $(\sigma,\rho)$ is complemented, then for any pair of \textit{words} (not only letters) $(u,v)$ there is at most one pair of words $(u',v')$ satisfying $u^{-1} v\curvearrowright v'u'^{-1}.$
Similarly, we write $u'=(v\backslash u)$ and $v'=(u\backslash v)$ when they exist so that $$u^{-1} v \curvearrowright (u\backslash v) (v\backslash u)^{-1} \text{ and thus } u (u\backslash v) =_\rho v(v\backslash u).$$
The SCC translates as follows in this special case.
\begin{observation}
Let $(\sigma,\rho)$ be a complemented presentation.
The SCC at $(u,v,w)\in \word(\sigma)^3$ is satisfied if and only if either one of the $(x\backslash y)$ is undefined for $x,y\in\{u,v,w\}$ or
$$[(u\backslash v)\backslash (u\backslash w)] \backslash [(v\backslash u) \backslash (v\backslash w)]=e.$$
\end{observation}
We obtain the following characterisation of completeness for complemented presentations.
\begin{proposition}
A complemented presentation $(\sigma,\rho)$ is complete if and only if $$[(a\backslash b)\backslash (a\backslash c)]\backslash [(b\backslash a)\backslash (b\backslash c)] \text{ is either undefined or equal to } e \text{ for each } a,b,c\in \sigma.$$
\end{proposition}
Here is a way to slightly reduce the number of SCC to check.
\begin{observation}\label{obs:complemented}
If $(\sigma,\rho)$ is a complemented presentation and $(u,v,w)\in\word(\sigma)^3$ are not all distinct, then the SCC is satisfied at $(u,v,w).$
\end{observation}
Note that the observation of above is no longer true in general for non-complemented presentations.
The monoid presentation $\Mon\langle a,b| a^2=b^2, a^2=ba\rangle$ gives such an example at $(a,a,b)$.
\subsection{Applications of Dehornoy's techniques to forest categories}
We now come back to our class of forest categories. Our aim is to use Dehornoy's result for proving that certain forest categories are Ore categories. We start by introducing and recalling notations and then outline the general strategy.
{\bf Notation.}
From now on $(S,R)$ is the forest presentation of a forest category $\mathcal F$ (implying by convention that $S\neq\emptyset$).
To $\mathcal F$ is associated a monoid $\mathcal F_\infty$ having the monoid presentation $P_\infty:=(S_\infty,R_\infty)$.
Recall from Section \ref{sec:cat-pres} that
$$S_\infty=\mathcal E(S)_\infty=\{b_j:\ b\in S, j\geq 1\}$$ is the set of elementary forests of $\mathcal F_\infty$
and
\begin{align*}
& R_\infty := \cup_{n\in\mathbf{N}} R_n \cup \TR(S) \\
& R_n:=\{ (u_n,v_n):\ n\geq 1, (u,v)\in R\}\\
& \TR(S)=\TR(S)_\infty:= \{ y_q x_j = x_j y_{q+1}:\ x,y\in S, 1\leq j<q\}.
\end{align*}
By inspection we deduce the following.
\begin{observation}
The presentation $P_\infty$ is homogeneous and positive, i.e.~$R_\infty$ is a set of pairs of words (the letter set being $S_\infty$) of same length.
\end{observation}
Our general strategy for deducing properties of $\mathcal F$ from Dehornoy's results on monoids is the following.
\begin{enumerate}
\item Check that $P_\infty$ is complete using the SCC at $S_\infty$;
\item Deduce properties of the monoid $\mathcal F_\infty$ using Dehornoy's results;
\item Deduce properties for the category $\mathcal F$ using Observations \ref{obs:LC} and \ref{obs:Ore}.
\end{enumerate}
We will often consider the case where $P_\infty$ is a {\it complemented} presentation. From there it is easy to check if $P_\infty$ is complete and left-cancellative. In some cases we can even establish Ore's property but this later property will be more likely proved by constructing a cofinal sequence of trees in $\mathcal F$.
The first item is checking the SCC at each triple $(x_i,y_j,z_k)$ where $x,y,z\in S$ are colours and $i,j,k\geq 1$ indices.
It is not hard to see that the SCC is satisfied at such a triple if and only if it is satisfied at $(x_{i+1}, y_{j+1},z_{i+1})$. Indeed, if $(u,v)$ is a relation, then so is its shift ($I\otimes u,I\otimes v)$ and this shift corresponds in shifting by 1 all indices.
Therefore, when checking the SCC at $(x_i,y_j,z_k)$ we can always assume that $\min(i,j,k)=1$.
We start by considering an easier case where there are few relations.
\subsection{Complemented forest presentations}
We define complemented forest presentations and derive criteria of completeness for their associated monoid presentations.
\subsubsection{Definition and first observations}
\begin{definition}\label{def:complemented}
Let $P=(S,R)$ be a forest presentation.
We say that $P$ is {\it complemented} if for any two colours $a,b\in S$ there is at most one relation of $R$ of the form $a_1\cdots=b_1\cdots$ and moreover if $a=b$ then there are no such relations.
\end{definition}
Note that here we ask that there are no relations of $R$ where both words start by the same letter. This will give shorter statements and will cover all the cases we wish to study.
We have the following unsurprising fact.
\begin{proposition}
If $(S,R)$ is a complemented forest presentation, then the associated monoid presentation $(S_\infty,R_\infty)$ is complemented.
\end{proposition}
\begin{proof}
Consider a complemented forest presentation $(S,R)$ and two generators $x_i,y_j\in S_\infty$ where $i,j\geq 1$ and $x,y\in S$.
If $i\neq j$, then there is a unique Thompson-like relation of the form $x_iy_j=y_j x_{i+1}$ if $i>j$ and $y_jx_i=x_i y_{i+1}$ if $i<j$.
Moreover, there are no other relations in $R_\infty$ starting from these letters.
If $i=j$, then by assumption there is at most one relation of the form $x_i\cdots=y_i\cdots$ which belongs to $R_i$.
We conclude that $(S_\infty,R_\infty)$ is complemented.
\end{proof}
{\bf A few identities for complemented presentations.}
We recall and establish few elementary computations for complemented presentations in the special case of forest categories.
These computations will allow us to easily check a number of SCC.
Let $(S,R)$ be a complemented forest presentation and $(S_\infty,R_\infty)$ the associated monoid presentation.
This later presentation is complemented in the usual sense and thus we have a partially defined map
$$\word(S_\infty)^2\to \word(S_\infty), (u,v)\mapsto (u\backslash v)$$
so that $$u^{-1} v\curvearrowright (u\backslash v)\cdot (v\backslash u)^{-1}.$$
By plugging $e$ for $u$ we deduce
$$(e\backslash v) = v \text{ and } (v\backslash e) = e \text{ for all } v\in \word(S_\infty).$$
{\it Relations coming from the forest presentation.}
Assume that $(a_1u_1, b_1v_1)\in R$.
This produces a one parameter family of relations $((a_ju_j,b_jv_j) :\ j\geq 1)$ in $R_\infty$.
Here we have
$$u_j=(a_j\backslash b_j), v_j=(b_j\backslash a_j) \text{ so that } (a_j\cdot (a_j\backslash b_j), b_j\cdot (b_j\backslash a_j))\in R_\infty.$$
Observe that both $u_j,v_j$ have same length $n$ (which does not depend on $j\geq 1$) and moreover $u_j$ is a word whose $k$th letter is a certain $c_m$ with $1\leq m\leq k+1$ for all $1\leq k\leq n.$
{\it Thompson-like relations.}
If $1\leq j<q$ and $a,b\in S$, then $a_q b_j = b_j a_{q+1}$ is a relation of $R_\infty$ implying that
$$(a_q\backslash b_j) = b_j \text{ and } (b_j\backslash a_q)=a_{q+1} \text{ for all } 1\leq j<q, a,b\in S.$$
More generally, consider $w_1\in \word(S_\infty)$ which corresponds to a tree $t$ rooted at the first node of a certain length $n$ and write $w_j$ for the word corresponding to $I^{\otimes j-1}\otimes t\otimes I^{\otimes \infty}$ obtained by shifting all indices by $j-1$.
We have the following identities:
\begin{align*}
& (a_q\backslash w_j) = w_j \text{ and } (w_j\backslash a_q) = a_{q+n}\\
& (a_j\backslash w_q) = w_{q+1} \text{ and } (w_q\backslash a_j) = a_j \text{ for all } a\in S, 1\leq j<q.
\end{align*}
This is obtained by observing that the first letter of $w_j$ has for index $j$, the second $j$ or $j+1$, etc., and by applying $n$ Thompson-like relations.
We obtain a similar statement for forests described by words $w$ rather than trees.
{\it Mix of both types of relations.}
Consider $x\in S$, some indices $1\leq j<q$, and a forest relation $(a_1 u_1,b_1 v_1)$.
This provides a relation $(a_ju_j,b_jv_j)$ where $u_j=(a_j\backslash b_j)$ and $v_j=(b_j\backslash a_j)$.
Now, $u_j$ is a word of a certain length $n\geq 0$ with $k$th letter of the form $y_{i_k}$ with $y\in S$ and $j\leq i_k\leq j+k+1.$
By applying $n$ Thompson-like relations we deduce:
$$x_{q+1} u_j = u_j x_{q+n+1}$$ giving the identity
$$[x_{q+1}\backslash ( a_j\backslash b_j)] =(a_j\backslash b_j) \text{ and } [(a_j\backslash b_j)\backslash x_{q+1}] = x_{q+n+1}.$$
Now, if we exchange the indices we obtain that:
$$x_j u_{q} = u_{q+1} x_j$$ giving:
$$[x_j\backslash (a_q\backslash b_q)] = (a_{q+1}\backslash b_{q+1}) \text{ and } [(a_q\backslash b_q)\backslash x_j] = x_j.$$
\subsubsection{Characterisation of completeness}
Here is a very useful characterisation of completeness which automatically provides left-cancellativity.
\begin{proposition}\label{prop:LC-complemented}
Let $(S,R)$ be a complemented forest presentation.
The associated monoid presentation $P_\infty=(S_\infty,R_\infty)$ is complete if and only if
for all triple of {\it distinct} colours $(x,y,z)$ of $S$ we have that
\begin{equation*}[(x_1\backslash y_1)\backslash (x_1\backslash z_1)]\backslash [(y_1\backslash x_1)\backslash (y_1\backslash z_1)]\end{equation*}
is either undefined or equal to $e$.
In that case the forest category $\mathcal F=\FC\langle S|R\rangle$ is left-cancellative.
\end{proposition}
Note that if $S$ has one or two colours and $(S,R)$ is complemented, then automatically $P_\infty$ is complete and $\mathcal F$ is left-cancellative.
\begin{proof}
Since $(S,R)$ is complemented, then so is $(S_\infty,R_\infty)$ and the function $(u,v)\mapsto (u\backslash v)$ is well-defined on a certain domain of $\word(S_\infty)^2.$
Now, $P_\infty$ is complete if and only if it satisfies the SCC at each triple of generators $(x_i,y_j,z_k)$ with $x,y,z\in S$ and $i,j,k\geq 1$.
Since the presentation is complemented the SCC at $(x_i,j_j,z_k)$ is satisfied if and only if the expression
\begin{equation}\label{eq:SCCprop}E:=[(x_i\backslash y_j)\backslash (x_i\backslash z_k)]\backslash [(y_j\backslash x_i)\backslash (y_j\backslash z_k)]\end{equation}
is either undefined or equal to $e$.
Fix three colours $x,y,z\in S$ and some indices $i,j,k\geq 1$.
{\bf Claim: If $i,j,k$ are not all equal, then \eqref{eq:SCCprop} is undefined or equal to $e$.}
We inspect several SCC using identities established before the proposition.
\begin{enumerate}
\item Case 1: $i>j,k$.
We obtain
$$E=[ y_j\backslash z_k]\backslash [x_{i+1} \backslash (y_j\backslash z_k)].$$
Assume $(y_j\backslash z_k)$ is defined since if not we are done.
Now, $i+1$ is strictly larger than the index of the first letter of the word $(y_j\backslash z_k)$.
We deduce that $[x_{i+1} \backslash (y_j\backslash z_k)]= (y_j\backslash z_k)$ and thus $E=e$.
\item Case 2: $i<j,k$.
We have
$$E=[y_{j+1}\backslash z_{k+1}]\backslash [ x_i \backslash (y_j\backslash z_k)].$$
Assume $(y_j\backslash z_k)$ exists.
Since all the letters of $(y_j\backslash z_k)$ have indices strictly larger than $i$ we deduce that $x_i \backslash (y_j\backslash z_k)= (y_{j+1}\backslash z_{k+1})$ implying that $E=e$.
\item Case 3: $k>i,j$.
We have
$$E=[(x_i\backslash y_j)\backslash z_{k+1}]\backslash [(y_j\backslash x_i) \backslash z_{k+1}].$$
If $(x_i\backslash y_j)$ is well-defined then so is $(y_j\backslash x_i).$
Moreover, these two words have same length, say $m$.
We deduce that $(x_i\backslash y_j)\backslash z_{k+1}=z_{k+m+1}$ and $(y_j\backslash x_i) \backslash z_{k+1}=z_{k+m+1}$ implying $E=e$.
\end{enumerate}
The remaining three cases can be treated likewise using similar identities and are left to the reader. This establishes the claim.
Consider now that $i=j=k$. By symmetry of the problem we may assume $i=1.$
Moreover, by Observation \ref{obs:complemented} the SCC is satisfied whenever $x,y,z$ are not all distinct.
By characterisation of completeness we deduce that $P_\infty$ is complete if and only if the SCC is satisfied at $(x_1,y_1,z_1)$ for each triple $(x,y,z)$ of distinct colours.
If this is the case, we have that $\mathcal F_\infty$ is left-cancellative if there are no relations in $R_\infty$ with both words starting with the same letter.
This is indeed the case by the definitions of $R_\infty$ and of complemented forest presentation.
This implies that $\mathcal F$ is left-cancellative since $\mathcal F_\infty$ is.
\end{proof}
We deduce a criteria for having Ore forest categories when there are few relations of small length.
\begin{corollary}\label{cor:Ore-FC}
Let $(S,R)$ be a forest presentation, $\mathcal F$ its associated forest category, and assume that all relations of $R$ are of the form $a_1x=b_1y$ where $a,b\in S$, $a\neq b$, and $x,y$ are letters. Moreover, if $a,b\in S, a\neq b$, then there exists one relation $a_1x=b_1y$.
If $$[(x_1\backslash y_1)\backslash (x_1\backslash z_1)]\backslash [(y_1\backslash x_1)\backslash (y_1\backslash z_1)]=e$$ for all triple $(x,y,z)$ of distinct colours, then $\mathcal F$ is a Ore forest category.
\end{corollary}
\begin{proof}
Consider $S,R,\mathcal F$ as above.
By assumption $(S,R)$ is complemented.
The associated monoid presentation $P_\infty=(S_\infty,R_\infty)$ is complete by the last proposition.
Moreover, there are no relations in $R_\infty$ starting by the same letter implying that $\mathcal F_\infty$ is left-cancellative by Proposition \ref{prop:DehornoyRCF}.
Finally, all relations of $R_\infty$ are pairs of words of length two. This implies that the set of letters $S_\infty$ is closed under reversing and thus satisfies the second item of Proposition \ref{prop:DehornoyRCF}.
Therefore, $\mathcal F_\infty$ is a Ore monoid and thus $\mathcal F$ is a Ore category.
\end{proof}
We deduce that forest presentations with zero or one relation are complemented and complete.
\begin{corollary}\label{cor:presentation-free}
Consider the forest presentation $(S,R)$ where $R$ is either empty or equal to a single relation of the form $a_1\cdots=b_1\cdots$ with $a\neq b$.
Then the associated monoid presentation $(S_\infty,R_\infty)$ is complemented, complete, and the forest category $\FC\langle S|R\rangle$ is left-cancellative.
\end{corollary}
\begin{proof}
By definition $(S,R)$ is complemented and so is $(S_\infty,R_\infty)$
We can thus use the last proposition to check if $(S_\infty,R_\infty)$ is complete which will automatically imply that $\FC\langle S|R\rangle$ left-cancellative.
Consider a triple of distinct letters $(x,y,z)$ and let us prove that the SCC is satisfied at $(x_1,y_1,z_1)$.
This is the case when
$$E:=[(x_1\backslash y_1)\backslash (x_1\backslash z_1)]\backslash [(y_1\backslash x_1)\backslash (y_1\backslash z_1)]$$
is either undefined or equal to $e$.
If $E$ is well-defined, then there is one relation of the form $x_1\cdots=y_1\cdots$ and another of the form $x_1\cdots=z_1\cdots$ contradicting our assumption.
\end{proof}
\subsection{Beyond the complemented case}
We may consider forest presentations $(S,R)$ that are not necessarily complemented (and thus may have more relations than unordered pairs of generators).
We make the assumption that all the relations of $R$ are of the form $a_{1}\cdots=b_{1}\cdots$ where $a,b\in S$ are {\it distinct} colours.
We provide a simplified criterion for checking completeness by removing a number of SCC cases.
\begin{proposition}
Consider a forest presentation $(S,R)$ of a forest category $\mathcal F$ with no relations starting from the same letter.
As usual we write $\mathcal F_\infty$ for the associated forest monoid with associated monoid presentation $P_\infty=(S_\infty,R_\infty)$.
The presentation $P_\infty$ is complete if and only if the SCC is satisfied at $(x_1,y_1,z_1)$ for all triple of colours $(x,y,z)$ where $x\neq z\neq y$.
In that case, $\mathcal F$ is left-cancellative.
\end{proposition}
\begin{proof}
Fix a forest presentation $(S,R)$ as above.
We have that $P_\infty$ is complete if and only if the SCC is satisfied at each triple $(x_i,y_j,z_k)$.
We are going to show that many such SCC are automatically satisfied whatever $(S,R)$ is.
Note, that when $P_\infty$ is complete, then there are no relations in $R_\infty$ starting from the same letter.
Hence, we deduce that $\mathcal F_\infty$ (and thus $\mathcal F$) is left-cancellative.
Therefore, we only need to prove the first statement.
The strategy of the proof is to use completeness of certain sub-presentations of $P_\infty$ of the form $(S_\infty, Q_\infty)$ with $Q_\infty\subset R_\infty$ satisfying the assumption of Corollary \ref{cor:presentation-free}.
More specifically, $Q_\infty$ will be either equal to $\TR(S)$ or to $\TR(S)\cup \bigcup_n Q_n$ that is deduced from a {\it single} forest relation of the type $a_{1}u=b_{1} v$ (so that $Q_n=(a_nu_n,b_nv_n)$).
We will then use that the SCC is satisfied at any triple of generators for this sub-presentation deducing SCC inside the larger presentation $P_\infty$ in some cases.
Consider a triple of generators $(x_i,y_j,z_k)$.
{\bf Claim 1: The SCC is satisfied at $(x_i,y_j,z_k)$ when $i,j,k$ are all distinct.}
Assume that $i,j,k$ are all distinct.
Now, if $$x_i^{-1} z_k z_k^{-1} y_j\curvearrowright fg^{-1}$$ with respect to the presentation $P_\infty$, then observe that only Thompson-like relations have been used for performing this reversing.
Indeed, $x_i^{-1}z_k\curvearrowright z_{k'} x_{i'}^{-1}$ and $z_k^{-1} y_j\curvearrowright y_{j'} z_{k'}^{-1}$ where the dash indices are equal to the original one plus 0 or 1, i.e.~$j'\in\{j,j+1\}.$
Moreover, no other reversings are allowed since there is only one relation of the form $x_i\cdots=z_k\cdots$ when $i\neq k$.
Hence, $x_i^{-1} z_k z_k^{-1} y_j\curvearrowright z_{k'} x_{i'}^{-1} y_{j'} z_{k'}^{-1}.$
By inspection of the various cases ($i>j>k$ or $j>k>i$, etc) we observe that $i'\neq j'$.
Hence, $x_{i'}^{-1} y_{j'}\curvearrowright y_{j''} x_{i''}^{-1}$.
We have only used relations of $\TR(S)$ for performing the reversings.
Since the presentation $(S_\infty,\TR(S))$ is complete (by Corollary \ref{cor:presentation-free}) the SCC is satisfied at $(x_i,y_j,z_k)$ and thus $(x_i g)^{-1} (y_j f)$ reverses to $e$ with respect to the presentation $(S_\infty,\TR(S)).$
Hence, $(x_i g)^{-1} (y_j f)$ reverses to $e$ for the larger presentation $P_\infty$ and thus the SCC is satisfied at $(x_i,y_j,z_k)$ for $P_\infty$ proving the claim.
{\bf Claim 2: The SCC is satisfied at $(x_i,y_j,z_k)$ when $i,j,k$ are not all equal.}
Assume now that $|\{i,j,k\}|=2$ and $x_i^{-1} z_k z_k^{-1} y_j\curvearrowright fg^{-1}$ w.r.t.~$P_\infty$.
We proceed as in the previous claim but possibly allowing one extra relation not in $\TR(S)$.
Indeed, at most one forest relation is used in the reversing process.
For instance, if $i=k<j$, then
$$x_i^{-1} z_i z_i^{-1} y_j\curvearrowright x_i^{-1} z_i y_{j+1} z_i^{-1}.$$
Now, $x_i^{-1}z_i$ may reverse to $u_iv_i^{-1}$ using a relation $(x_iu_i,z_iv_i)$ where $u_i,v_i$ are words with first letter having index $i$ or $i+1$.
Hence, $x_i^{-1} z_i z_i^{-1} y_j\curvearrowright u_iv_i^{-1}y_{j+1} z_i^{-1}.$
Now, $j+1$ is strictly larger than the index of the first letter of $v_i$.
Hence, we only have relations from $\TR(S)$ for the remaining reversing operations.
We deduce that $$x_i^{-1} z_i z_i^{-1} y_j\curvearrowright u_iy_{j+1+m} v_i^{-1} z_i^{-1}=:fg^{-1}$$
where $m$ is the length of the word $v_i$.
Observe that we only used one single relation $(x_iu_i,z_iv_i)$ that was not in $\TR(S)$.
Now, the sub-presentation
$$(S_\infty, \TR(S)\cup \{ (x_nu_n,z_nv_n):\ n\geq 1\})$$
of $P_\infty$ is complemented and complete by Corollary \ref{cor:presentation-free}.
Therefore, the SCC is satisfied at $(x_i,y_i,z_k)$ for this presentation and thus $(x_ig)^{-1}(y_i f)$ reverses to $e$ for this presentation.
This implies that $(x_ig)^{-1}(y_i f)$ reverses to $e$ for $P_\infty$ and thus the SCC is satisfied at $(x_i,y_i,z_k)$.
The other cases where the three indices are not all equal can be treated similarly.
We have shown that the SCC is satisfied for all triple $(x_i,y_j,z_k)$ where $i,j,k$ are not all equal.
Assume now that $i=j=k$. From our previous observation we know that the SCC is satisfied at the triple $(x_i,y_i,z_i)$ if and only if it is satisfied for the index $i=1$ so we may assume $i=1$.
To conclude the proof it is sufficient to prove that the SCC is satisfied at $(x_1,y_1,x_1)$ and at $(x_1,y_1,y_1)$.
For the first note that:
$$x_1^{-1} x_1 x_1^{-1}y_1 \curvearrowright x_1^{-1} y_1$$
uniquely and from there we use one single relation that is not in $\TR(S)$.
We can then conclude as in the previous claims.
Similarly, in the second case we have
$$x_1^{-1}y_1 y_1^{-1}y_1\curvearrowright x_1^{-1} y_1$$
uniquely and can conclude similarly as above.
\end{proof}
\section{Forest groups}\label{sec:forest-groups}
Fix a Ore forest category $\mathcal F$.
We will present the \textit{calculus of fractions} of $\mathcal F$.
This will allow us to define a groupoid $\Frac(\mathcal F)$ and in particular its isotropy groups $\Frac(\mathcal F,n)$ that are all (equivalence classes of) pairs of forests with the same number of leaves and a fixed number of roots $n$.
We will be particularly interested in $\Frac(\mathcal F,1)$ defined from pairs of {\it trees}.
\subsection{Fraction groupoids of forest categories}
We recall the construction of the \textit{fraction groupoid} of $\mathcal F$.
Extensive details can be found in \cite{Gabriel-Zisman67}. All statements below are standard and easy exercises.
{\bf Pairs of forests.}
Let $\mathcal P_\mathcal F$ be the set of pairs $(f,g)\in\mathcal F\times \mathcal F$ where $f$ and $g$ have the same number of leaves.
Consider the smallest equivalence relation $\sim$ of $\mathcal P_\mathcal F$ generated by
$$(f,g)\sim (f\circ h, g\circ h) \text{ for all } (f,g)\in \mathcal P_\mathcal F , \ h\in\mathcal F$$ and write $\Frac(\mathcal F)$ for the quotient $\mathcal P_\mathcal F/\sim$.
We write $[f,g]$ for the equivalence class of $(f,g)$ inside $\Frac(\mathcal F)$.
{\bf A binary operation: fraction groupoids.}
We define a partially define binary operation on $\Frac(\mathcal F)$.
Consider $(f,g), (f',g')\in \mathcal P_\mathcal F$ such that $|\Root(g)|=|\Root(f')|$.
By Ore's property there exists $p,p'\in\mathcal F$ satisfying $gp=f'p'$.
We define a binary operation as follows:
$$[f,g]\circ [f',g']:= [fp, g'p'].$$
We recall well-known facts on calculus of fractions that we specialize to our context.
\begin{proposition}
The following assertions are true.
\begin{enumerate}
\item The binary operation $\circ$ of above is indeed well-defined and associative (it does not depend on the choices of $p,p'$ nor on the representatives $(f,g), (f',g')$);
\item The algebraic structure $(\Frac(\mathcal F),\circ)$ is a groupoid;
\item The formula $f\mapsto [f,I^{\otimes n}]$, where $n=|\Leaf(f)|$, defines a covariant functor from $\mathcal F$ to $\Frac(\mathcal F)$ (it preserves compositions).
\item We have the identities $[f,g]\circ [g,h] = [f,h]$ and $[f,g]^{-1}=[g,f]$ for all $[f,g],[g,h]\in\Frac(\mathcal F).$
\end{enumerate}
\end{proposition}
\begin{definition}
We call $\Frac(\mathcal F)$ the \textit{fraction groupoid} of $\mathcal F$ and its elements {\it fractions}.
A {\it forest groupoid} is a groupoid $\mathcal G$ for which there exists a forest category $\mathcal F$ satisfying that $\mathcal G\simeq \Frac(\mathcal F)$.
\end{definition}
\begin{remark}
Note that the fraction groupoid $\Frac(\mathcal F)$ is obtained by formally inverting elements of $\mathcal F$ and is sometimes denoted by $\mathcal F[\mathcal F^{-1}].$
Formally, $[f,g]$ corresponds to $f \circ g^{-1}$ and we shall often use this notation which makes item 3 and 4 obvious.
It is common to denote $[f,g]$ as a fraction $\frac{f}{g}$ by analogy with fractions of real numbers or more generally elements of localized rings. This explains the terminology.
Every small category admits an enveloping groupoid made of signed-path of morphisms. What is exceptional here is that being a Ore category assures that every element (or morphism) of this enveloping groupoid (which is the fraction groupoid) is a word of at most length two: $f\circ g^{-1}.$
Note that the map $\mathcal F\to \Frac(\mathcal F),f\mapsto [f,I^{\otimes n}]$ where $n=|\Leaf(f)|$ is not injective in general. Indeed, for a Ore forest category it is injective if and only if $\mathcal F$ is also right-cancellative (that is $\mathcal F$ is left and right-cancellatives not only left-cancellative), see Lemma 3.8 and Proposition 3.11 in \cite{Dehornoy-book}.
Even weaker conditions can be considered for having a right-calculus of fractions (for instance by requiring a weaker form of left-cancellativity) but we will not consider them in our study, see \cite{Gabriel-Zisman67}.
\end{remark}
{\bf Diagrammatic description of elements of $\Frac(\mathcal F)$.}
We may extend the diagrammatic description of $\mathcal F$ to its fraction groupoid $\Frac(\mathcal F)$.
Elements of $\mathcal F$ view inside $\Frac(\mathcal F)$ are represented by the same diagrams than before.
Now, an element of the form $[I^{\otimes n},g]=g^{-1}$, where $n= |\Leaf(g)|$, is represented by the diagram of $g$ but drawn upside down with now leaves on the bottom and roots on top.
The multiplication is still the vertical stacking.
Hence, we represent an element $[f,g]=f\circ g^{-1}$ as the diagram of $f$ and stack on top of it the upside down diagram of $g$ as in the following example:
\[\includegraphics{fginverse.pdf}\]
Now, we know how to diagrammatically multiply $f$ with $g^{-1}$ but also $f$ with $f'$ and $g^{-1}$ with $(g')^{-1}$.
It remains to understand the diagrammatic multiplication of $g^{-1}$ with a forest $h$.
Choose some forests $p,q$ satisfying $gp=hq$.
We obtain that $$g^{-1}\circ h = p\circ (gp)^{-1}\circ hq\circ q^{-1} = p\circ q^{-1}.$$
This can be expressed diagrammatically by stacking $h$ on top of $g^{-1}$ and performing various {\it universal} skein relations of the form
\[\includegraphics{YmY.pdf}\]
inherent to all Ore forest categories combined with specific forest relations until obtaining the diagram of $p\circ q^{-1}$.
We will not use substantially diagrammatic multiplication in $\Frac(\mathcal F)$.
We thus skip a detailed description of it that can be derived without much trouble from the classical monochromatic case of the Thompson groups.
For this later we refer the reader to the Chapter 7 of the PhD thesis of Belk and the article of Jones where many such (monochromatic) diagrammatic computations are performed \cite{Belk-PhD,Jones21}.
\subsection{Fraction groups associated to forest categories.}
We are interested in constructing and studying groups rather than groupoids.
Given $\mathcal F$ as above and its fraction groupoid $\Frac(\mathcal F)$ we can consider for each $r\geq 1$ the set of pairs $(f,g)$ where $f,g$ have $r$ roots and have the same number of leaves.
The set of such classes $[f,g]$ is a group that we write $\Frac(\mathcal F,r)$.
If we consider $\Frac(\mathcal F)$ using categorical terminology, then $\Frac(\mathcal F,r)$ is nothing else than the automorphism group of the object $r$ inside the groupoid $\Frac(\mathcal F)$ also called the \textit{isotropy group} of the object $r$.
\begin{definition}
Let $\mathcal F$ be a Ore forest category.
The \textit{fraction group} of $\mathcal F$ is $\Frac(\mathcal F,1)$.
It is the set of fractions $[f,g]=f\circ g^{-1}$ where $f,g$ are \textit{trees} with the same number of leaves equipped with the restriction of the multiplication of $\Frac(\mathcal F)$.
A {\it forest group} is a group $G$ isomorphic to the fraction group of a forest category. We may also say that $\Frac(\mathcal F,1)$ is {\it the} forest group associated to $\mathcal F$.
\end{definition}
\begin{example}
Let $\mathcal F$ be the monochromatic free forest category.
It is left-cancellative and satisfies Ore's property.
Its fraction group $\Frac(\mathcal F,1)$ is (isomorphic to) Thompson's group $F$ via the Brown diagrammatic description of \cite{Brown87}.
Moreover, $\Frac(\mathcal F,r)$ for $r\geq 1$ is the Higman-Thompson's group usually denoted by $F_{2,r}$ in the literature. Note that $F_{2,r}$ is the ``$F$-version'' of the Higman-Thompson's group $V_{2,r}$ that was first defined by Brown in \cite{Brown87}.
More examples can be found in Section \ref{sec:example} and Section \ref{sec:class-example}.
\end{example}
\begin{remark}
It is worth noting that if $\mathcal F$ is a Ore forest category, then $\Frac(\mathcal F,r)$ is isomorphic to $\Frac(\mathcal F,1)$ for all $r\geq 1$.
Indeed, since we are considering \textit{binary} forests there exists a tree $t$ with $r$ leaves.
Now, the conjugation by $t$ provides a group isomorphism from $\Frac(\mathcal F,r)$ to $\Frac(\mathcal F,1)$.
In categorical language, the groupoid $\Frac(\mathcal F)$ is path-connected implying that all its isotropy groups are isomorphic.
(If we add the empty diagram and the object $0$ to $\mathcal F$ in the purpose of having a tensor unit and fulfilling scrupulously all the axioms of a monoidal category, we obtain one additional isolated path-connected component associated to $0$ giving the trivial isotropy group $\Frac(\mathcal F,0)$.)
If we were considering {\it ternary} forests rather than {\it binary}, then there would be no morphisms between $1$ and $2$ for instance. Although, for the classical case of ternary forests with one colour and no quotients the Higman-Thompson's groups $F_{3,r},r\geq 1$ are all isomorphic but for a different reason, see \cite[Proposition 4.1]{Brown87}.
\end{remark}
\subsection{Functoriality}
The collection of all Ore forest category together with morphisms between them forms a subcategory of $\Forest$ that we write $\Forest_{Ore}$ and call the \textit{category of Ore forest categories}.
We collect straightforward facts concerning morphisms between forest categories, fraction groupoids, and fraction groups. It shows that all our constructions are functorial.
They are all easy to prove and well-known in greater generality.
\begin{proposition}\label{prop:morphisms}
Let $\mathcal F,\mathcal G$ be Ore forest categories.
The following assertions are true.
\begin{enumerate}
\item If $\theta\in\Hom(\mathcal F,\mathcal G)$, then $\theta$ uniquely extends into groupoid and group morphisms
$$\Frac(\theta):\Frac(\mathcal F)\to\Frac(\mathcal G) \text{ and } \Frac(\theta,1):\Frac(\mathcal F,1)\to\Frac(\mathcal G,1)$$
via the formula
$$[f,g]\mapsto [\theta(f),\theta(g)].$$
We often write $\theta$ for $\Frac(\theta)$ or $\Frac(\theta,1)$ if the context is clear.
\item The assignments $\mathcal F\mapsto \Frac(\mathcal F), \theta\mapsto\Frac(\theta)$ and $\mathcal F\mapsto \Frac(\mathcal F,1), \theta\mapsto\Frac(\theta,1)$ define covariant functors that we write:
$$\Frac:\Forest_{Ore}\to \Groupoid \text{ and } \Frac(-,1):\Forest_{Ore}\to \Gr$$
where $\Groupoid$ is the category of groupoids and $\Gr$ the category of groups.
\item If a morphism $\theta:\mathcal F\to\mathcal G$ is injective (resp.~surjective), then $\Frac(\theta)$ and $\Frac(\theta,1)$ are both injective (resp.~surjective).
\end{enumerate}
\end{proposition}
\begin{remark}
The converse of item 3 of the last proposition is false in general: there exist morphisms of Ore categories that are not injective or surjective but extend to injective or surjective morphisms between the corresponding groupoids.
Such examples appear for Ore categories that are not right-cancellative.
Consider the presented monoid
$$M:=\Mon\langle a,b| aa=ba\rangle$$
and the morphisms
$$M\to \mathbf{N}, a,b\mapsto 1 ; \ \mathbf{N}\to M, 1\mapsto a.$$
The first is not injective and the second not surjective. However, they extend into isomorphism of groups between $\Frac(M)$ and $\Frac(\mathbf{N})=\mathbf{Z}.$
We can adapt these examples into the context of forest categories by considering the the presented forest category $$\mathcal F:=\FC\langle a,b| a_1a_1=b_1a_1\rangle$$ and free monochromatic forest category $\mathcal G$. They are both Ore categories admitting fraction groupoids and groups.
The morphisms of above define maps between the colour sets of $\mathcal F$ and $\mathcal G$ which define morphisms of forest categories.
These two morphisms are not isomorphisms of forest categories but extend into isomorphisms of groupoids and groups.
\end{remark}
From the observations of above we deduce an interesting corollary.
\begin{corollary}\label{cor:Finside}
Any forest group $G$ contains a copy of Thompson's group $F$.
More precisely, if $G$ is the fraction group of $\mathcal G=\FC\langle S|R\rangle$ and $\mathcal{UF}$ is the monochromatic free forest category, then each colour $a\in S$ defines an injective morphism $C_a:\mathcal{UF}\hookrightarrow\mathcal G,Y\mapsto Y_a$ called the colouring map that extends into an injective group morphism $F\hookrightarrow G$.
\end{corollary}
\begin{proof}
Let $G$ be a forest group.
There exists a presented Ore forest category $\mathcal G=\FC\langle S|R\rangle$ satisfying that $G$ is isomorphic to $\Frac(\mathcal G,1)$.
Fix a colour $a\in S$ and define the colouring map
$$C_a:\mathcal{UF}\to\mathcal G, Y\mapsto Y_a$$
from the monochromatic free forest category $\mathcal{UF}$ to $\mathcal G$.
Such a morphism exists by universal property of $\mathcal{UF}$.
Let us show that $C_a$ is injective. We assume the contrary and follow a similar argument given in Example \ref{ex:LC} item 3.
Assume $C_a$ has a nontrivial kernel and consider a pair of trees $(t,s)\in\ker(C_a)$ satisfying $t\neq s$ with minimal number of leaves.
Decompose $t$ and $s$ as $$t=Y \circ (h\otimes k),\ s= Y \circ (h'\otimes k')$$
where $h,k,h',k'$ are trees.
Since $C_a$ preserves composition and tensor product we have that $(h,h')$ and $(k,k')$ are in $\ker(C_a)$.
By minimality of the number of leaves we conclude that $h=h', k=k'$ implying $t=s$, a contradiction.
Therefore, $C_a$ is injective and extends into an injective {\it group} morphism $\Frac(C_a,1): \Frac(\mathcal{UF},1)\to \Frac(\mathcal G,1)$ by the last proposition.
\end{proof}
\begin{remark}
Note that the proof of the last proposition shows that if $\mathcal G$ is a left-cancellative forest category, then it contains a copy of the monochromatic free forest category.
\end{remark}
We deduce that all forest groups share common properties since they all contain the group $F$. We list few of those in the corollary below. Although, this list is far from being exhaustive.
\begin{corollary}\label{cor:geometric-dimension}
A forest group is infinite, contains free Abelian group of arbitrary rank, has infinite geometric dimension (see Section \ref{sec:def-finiteness}), is not elementary amenable, and has exponential growth.
\end{corollary}
\begin{remark}
A number of properties of $F$ are not shared by all $F$-forest groups.
For instance, there exist forest groups with torsion, with their abelianisation with torsion, some forest groups contain a copy of the free group of rank two, some forest groups decompose as a nontrivial direct product, can have nontrivial centre, etc.
\end{remark}
\subsection{Fraction groups of forest monoids and the CGP}\label{sec:CGP}
Let $\mathcal F$ be a Ore forest category which implies that its forest monoid $\mathcal F_\infty$ is a Ore monoid.
We can define a calculus of fractions on $\mathcal F_\infty$ just as we did for $\mathcal F$.
We briefly recall the construction that is rather identical to the one performed for $\mathcal F$.
Although, the construction is slightly simpler since we do not need to care about numbers of leaves and roots.
Consider the product $\mathcal F_\infty\times\mathcal F_\infty$ and define the equivalence relation $\sim$ on it that is generated by $(f,g)\sim (f\circ h, g\circ h)$ with $f,g,h\in\mathcal F_\infty.$
Write $\Frac(\mathcal F_\infty)$ for the quotient of $\mathcal F_\infty\times\mathcal F_\infty$ by $\sim$ and write $[f,g]=\frac{f}{g}=f\circ g^{-1}$ for the equivalence class of $(f,g)$ inside $\Frac(\mathcal F_\infty).$
Given $[f,g],[f',g']\in \Frac(\mathcal F_\infty)$ we define
$$[f,g]\circ [f',g']:=[fp,g'p'] \text{ for some } p,p' \text{ satisfying } gp=f'p'.$$
This is a well-defined associative binary operation on $\Frac(\mathcal F_\infty)$.
\begin{definition}
The algebraic structure $(\Frac(\mathcal F_\infty),\circ)$ is a group called the \textit{fraction group of the forest monoid $\mathcal F_\infty$}.
\end{definition}
We prove that the two groups $\Frac(\mathcal F,1)$ and $\Frac(\mathcal F_\infty)$ embed in each other.
\begin{proposition}\label{prop:GHinclusion}
Let $\mathcal F$ be a Ore forest category and consider the associated groups $G:=\Frac(\mathcal F,1)$ and $H:=\Frac(\mathcal F_\infty).$
For each colour $a$ of $\mathcal F$ there exists an injective group morphism $\gamma_a:H\to G$.
For each $k\geq 1$ there exists an injective group morphism $\eta_k:G\to H$.
In particular, any property of groups that is closed under taking subgroup is either satisfied for $G$ and $H$ or for none of them. This applies for instance for analytic properties like Haagerup property, Cowling-Haagerup weak amenability, or growth properties, etc.
\end{proposition}
\begin{proof}
Consider $\mathcal F,G,H$ as above.
Fix a colour $a$ of $\mathcal F$ (that is an element of $S$ if $(S,R)$ is a forest presentation of $\mathcal F$ or an tree with two leaves) and a natural number $k\geq 1.$
The map $$\mathcal F\to\mathcal F_\infty, f\mapsto I^{\otimes k-1} \otimes f \otimes I^{\otimes \infty}$$ is a functor from the category $\mathcal F$ to the monoid $\mathcal F_\infty$.
It extends into a groupoid morphism from $\Frac(\mathcal F)$ to $\Frac(\mathcal F_\infty)$ and restricts into a group morphism $\eta_k:\Frac(\mathcal F,1)\to \Frac(\mathcal F_\infty)$ that is injective since the original map $\mathcal F\to\mathcal F_\infty$ is.
The range of $\eta_k$ corresponds to pairs of trees with roots at the $k$th spot.
Here is an example of the mapping $\eta_3$:
\[\includegraphics{eta_3.pdf}\]
Define the following sequence of trees of $\mathcal F$:
$$t_{n+1}:= a_{1,1}a_{2,2}\cdots a_{n,n} \text{ for all } n\geq 1$$
so that $t_{n+1}$ is a monochromatic tree of colour $a$ with $n+1$ leaves and a long right-branch.
We say that $t_{n+1}$ is a {\it right $a$-vine} (with $n$ interior vertices).
Here is the diagram of $t_3$:
\[\includegraphics{t_3.pdf}\]
Observe that for any forest $f\in\mathcal F_\infty$ the sequence
$$n\mapsto [t_n\circ f|_n, t_{n+m}] = t_n \circ f|_n \circ t_{n+m}^{-1}$$ is eventually constant where $f|_n$ is obtained by truncating the infinite forest $f$ to its $n$th first trees and $m$ is the number of carets of $f|_n$ (which is eventually constant too).
This defines a map from $\mathcal F_\infty$ to $G$ which extends into a group embedding $\gamma_a:H\to G$.
\end{proof}
\begin{remark}
We observed in Remark \ref{rk:forest-monoid} that $\mathcal F_\infty$ is obtained as a direct limit of set of forests with finitely many roots.
Similarly, $\Frac(\mathcal F_\infty)$ is the direct limit of the system of groups $(\Frac(\mathcal F,r):\ r\geq 1)$ for the (injective) connecting morphisms
$$\iota_r^{r+n}:\Frac(\mathcal F,r)\to \Frac(\mathcal F,r+n), \ [f,g]\mapsto [f\otimes I^{\otimes n}, g\otimes I^{\otimes n}].$$
This explains the notation where $\Frac(\mathcal F_\infty)$ is interpreted as $\varinjlim_{r} \Frac(\mathcal F,r)$: the limit of the fraction groups $\Frac(\mathcal F,r)$ for $r$ tending to infinity.
Now, the morphism $\eta_1:\Frac(\mathcal F,1)\to \Frac(\mathcal F_\infty)$ of the last proposition corresponds to the limit morphism $\iota_1^\infty:=\varinjlim_{n}\iota_1^{1+n}$.
\\
The group $\Frac(\mathcal F,1)$ is often interpreted as a subgroup of $\Frac(\mathcal F_\infty)$ in the literature. Although, we do the opposite and consider $\Frac(\mathcal F_\infty)$ as a subgroup of $\Frac(\mathcal F,1)$ using the morphism $\gamma_a$ of the last proposition.
With this viewpoint we will provide in Section \ref{sec:presentation} presentations of $\Frac(\mathcal F,1)$ and $\Frac(\mathcal F_\infty)$ where one is the truncation of the other.
\end{remark}
\begin{definition}\label{def:CGP}
We write $G(a)$ for the range of $\gamma_a$ which is a subgroup of $G$ isomorphic to $H$.
If $G=G(a)$, then we say that $G$ has the {\it colouring generating property} at $a$ (in short the CGP at $a$ or simply the CGP).
In that case, $\gamma_a$ defines an isomorphism between $H$ and $G$.
\end{definition}
Note that elements of $G(a)$ consists of elements $g=t\circ s^{-1}$ which admits representatives $t,s$ so that all vertices between the root and the right-most leaf of $t$ and $s$ are coloured by $a$.
We will see that in some cases $G(a)=G$. This happens when $\mathcal F$ is the monochromatic free forest category and $G=F$ as pointed out by Brown but there are many other examples, see Section \ref{sec:example}.
\begin{remark}
The isomorphism $$\gamma_a:H\to G(a),f\mapsto [t_n\circ f|_n, t_{n+m}]$$ consists in taking a finitely supported forest $f$ and to add on top and bottom a right $a$-vine.
This cannot be generalised to right-vines that are not {\it monochromatic}.
Indeed, let $(a^{(n)},n\geq 1)$ be a sequence of colours and put $\tilde t_{n+1}:=a^{(1)}_{1,1}\circ\cdots\circ a^{(n)}_{n,n}$ the right-vine with $j$th vertex coloured by $a^{(j)}$ for $1\leq j\leq n.$
Now, given $f\in\mathcal F_\infty$ we consider
$$\tilde\gamma(f,n):=[\tilde t_{n}\circ f|_{n}, \tilde t_{n+m}]= \tilde t_n\circ f|_{n} \circ \tilde t_{n+m}^{-1}$$ where $m$ is the number of vertices of $f|_{n}$.
In order to have a monoid morphism we need to have that this fraction is eventually constant in $n$.
But $\tilde\gamma(f,n+1)$ corresponds to $\tilde\gamma(f,n)$ to which we add a $a^{(n+1)}$-vertex and a $a^{(n+p+1)}$-vertex.
Hence, $\tilde\gamma(f,n)=\tilde\gamma(f,n+1)$ implies that $a^{(n+1)}=a^{(n+p+1)}.$
\end{remark}
\subsection{Forest groups similar to $T,V,BV$.}
Consider a Ore forest category $\mathcal F$ and let $\mathcal F^X, \mathcal F^Y_\infty$ be the associated $X$-forest category and $Y$-forest monoid, respectively, where $X=F,T,V,BV$ and $Y=F,V,BV$.
It is rather obvious that if $\mathcal F$ is a Ore category, then so are $\mathcal F^X$ and $\mathcal F^Y_\infty$.
Hence, we can define the fraction groupoids $\Frac(\mathcal F^X)$ and the fraction groups $\Frac(\mathcal F^X,1)$, $\Frac(\mathcal F_\infty^Y).$
We often denote the groups as follows:
$$G^X=\Frac(\mathcal F^X,1) \text{ and } H^Y=\Frac(\mathcal F_\infty^Y)$$
removing the superscript when $X=F$ or $Y=F$.
A group isomorphic to $G^X$ is called a {\it $X$-forest group}.
We also say that $G^X$ and $H^Y$ are the {\it $X$-version} of $G$ and {\it $Y$-version} of $H$, respectively.
An element of $G$ is of the form $t\circ s^{-1}$ with $t,s$ trees.
Now, an element of $\mathcal F^V$ is of the form $f\circ \pi$ where $f$ is a forest and $\pi$ a permutation.
Hence, an element of $G^V$ is of the form $t\circ \pi \circ \sigma^{-1}\circ s$ with $t,s$ trees and $\pi,\sigma$ permutations.
Since $\tau:=\pi\circ\sigma^{-1}$ is itself a permutation we deduce that all elements of $G^V$ are described by {\it triples} rather than {\it quadruples}.
We may write $[t,\tau,s]$ or $[t\circ \tau,s]$ for $t\circ \tau\circ s^{-1}$.
Similarly, elements of $G^T$ and $G^{BV}$ can be described by an equivalence class of a triple $(t,\tau,s)$ with $\tau$ a cyclic permutation or a braid, respectively.
An easy adaptation of the proofs of Corollary \ref{cor:Finside} and Propostion \ref{prop:GHinclusion} provides the following.
\begin{corollary}
Consider $X\in\{F,T,V,BV\}, Y\in\{F,V,BV\},$ a Ore forest category $\mathcal F$ and the associated groups $G^X,H^Y$.
The group $G^X$ contains a copy of the group $X$.
Moreover, $G^Y$ and $H^Y$ embed in each other.
\end{corollary}
\subsection{Examples of forest groups}\label{sec:example}
We provide examples of forest presentations that define Ore forest category and thus forest groups.
We anticipate definitions and results of next sections and articles.
In particular, explicit presentations of forest groups given in Theorem \ref{theo:groupG-presentation}. We use often the notion of CGP introduced in Section \ref{sec:CGP}.
If $\mathcal F$ is a Ore forest category, then we use the notations $G^X:=\Frac(\mathcal F^X,1)$ and $H^Y:=\Frac(\mathcal F_\infty^Y)$ for $X=F,T,V,BV, Y=F,V,BV$.
We write $G^{ab}$ for the abelianisation $G/[G,G]=G/G'$ of the group $G$.
\subsubsection{Monochromatic forest categories}
Consider a monochromatic forest category $\mathcal F$. It admits a forest presentation of the form $(S,R)$ where $S=\{x\}$ is a singleton and $R$ is a set of pairs of trees.
Now, $\mathcal F$ is left-cancellative if and only if $R$ is empty.
In that case $\mathcal F$ is the monochromatic free forest category that is a Ore category.
We obtain that $G\simeq H\simeq F$ and moreover $G^X\simeq X$ for $X=T,V,BV$.
In particular, $G^{BV}$ is Brin's braided Thompson group $BV$ and $H^{BV}$ corresponds to the group $\widehat{BV}$ considered by Brin in \cite{Brin-BV1}.
\subsubsection{Ternary Thompson group $F_{3,1}$}
Consider the presented forest category
$$\mathcal F=\FC\langle a,b| a_1b_2=b_1a_1\rangle.$$
Graphically we may use the following diagrams for the $a$-caret and $b$-caret which provides an intuitive forest relation:
\[\includegraphics{F3presentation.pdf}\]
Since there are two colour and a single relation with words of length two we automatically obtain that it is left-cancellative by Corollary \ref{cor:presentation-free} and satisfies Ore's property by Corollary \ref{cor:Ore-FC}. Moreover, one can show it satisfies the CGP.
Hence, $G$ and $H$ exist and are isomorphic.
Moreover, they admit the following infinite group presentation with generators:
$$\{a_j,b_j:\ j\geq 1\}$$
and relations
\begin{itemize}
\item $x_q y_j = y_j x_{q+1}$ for all $x,y=a,b$ and $1\leq j<q$;
\item $a_j b_{j+1} = b_j a_j$ for all $j\geq 1.$
\end{itemize}
We are going to show that $H$ and $G$ are in fact isomorphic to the ternary Higman-Thompson group $F_3=F_{3,1}$ obtained from the monochromatic free {\it ternary} forest category $\mathcal F_3$.
We do it by renaming the generators $a_j,b_j$ of $H$ and observing that they satisfy the relation of a well-known presentation of $F_{3,\infty}:=\Frac((\mathcal F_3)_\infty)$ itself isomorphic to $F_3:=\Frac(\mathcal F_3,1)$.
Indeed, define
$$\begin{cases} z_{2n}=b_n\\
z_{2n-1} = a_n
\end{cases} \text{ for all } n\geq 1.$$
Now, substitute $z$ in the presentation of $H$.
For instance, the Thompson-like relation $a_q b_j=b_j a_{q+1}$ becomes $z_{2q-1} z_{2j} = z_{2j} z_{2q+1}$ and the translated forest relation $b_j a_j=a_j b_{j+1}$ becomes $z_{2j} z_{2j-1}=z_{2j-1} z_{2j+2}$ for $1\leq j<q.$
We deduce the following new presentation of $H$:
$$\Gr\langle z_n, n\geq 1 | z_q z_j = z_j z_{q+2}, \ 1\leq j<q\rangle.$$
Consider now the forest monoid $F_{3,\infty}^+:=(\mathcal F_3)_\infty$ of ternary forests.
Observe that if $t_j$ is the elementary forest of $F_{3,\infty}^+$ that has a single ternary-caret at the $j$th root, then we have the relations $t_qt_j=t_j t_{q+2}$ for all $1\leq j<q$ and in fact this provides a monoid presentation of $F_{3,\infty}^+$ and thus a group presentation of its fraction group denoted $F_{3,\infty}$ by Brown.
Now, sending $z_j$ to $t_j$ provides a monoid isomorphism from $\mathcal F_\infty$ to $F_{3,\infty}^+$ inducing a group isomorphism from $H=\Frac(\mathcal F_\infty)$ to $F_{3,\infty}.$
Note that $G\simeq H\simeq F_{3}$ admits two embedding of $F$ given by the two colours $a$ and $b$, see Corollary \ref{cor:Finside}.
Moreover, by sending a monochromatic ternary caret to $a_1b_2$ we obtain an embedding of $F_3$ in itself which is not the identity.
Finally, the $T,V,$ and $BV$-versions of $G$ are not (or at least not isomorphic in an obvious way) the usual Higman-Thompson groups $T_{3,1}, V_{3,1}$ nor the ternary Brin braided Thompson group $BV_{3}$.
\subsubsection{Cleary irrational-slope Thompson group}
Consider the presented forest category
$$\mathcal F:=\FC\langle a,b| a_1a_1=b_1b_2\rangle$$
with two generators $a,b$ and one relation with words of length two.
Using Corollaries \ref{cor:presentation-free} and \ref{cor:Ore-FC} or Theorem \ref{theo:class-example} we have that $\mathcal F$ is a Ore category.
Moreover, it satisfies the CGP (see Section \ref{sec:CGP}).
Hence, $G\simeq H$ and is isomorphic to the Cleary irrational-slope Thompson group further studied by Burillo, Nucinkis, and Reeves \cite{Cleary00,Burillo-Nucinkis-Reeves21}.
It is the group of piecewise affine homeomorphisms of the unit interval with finitely many breakpoints and all slopes powers of the golden number $(\sqrt 5 -1 )/2.$
By Theorem \ref{theo:groupG-presentation} it admits the infinite group presentation
$$\Gr\langle a_j,b_j, j\geq 1| x_qy_j=y_j x_{q+1}, \ a_ia_i = b_ib_{i+1}, 1\leq j<q, i\geq 1\rangle$$
and the finite group presentation with generators
$$\{a_1,b_1,a_2,b_2\}$$
and relations
\begin{itemize}
\item $[a_1^{-1}x_i, a_1^{-j} y_2 a_1^j]=e$ for $x,y\in\{a,b\}, i,j\in\{1,2\}$ with $(x,i)\neq (a,1)$;
\item $a_1a_1=b_1b_2$;
\item $a_2a_2=b_2a_1^{-1}b_2a_1$.
\end{itemize}
From there we deduce that $G^{ab}\simeq \mathbf{Z}\oplus\mathbf{Z}\oplus\mathbf{Z}/2\mathbf{Z}.$
Similarly, the larger groups $G^T$ and $G^V$ are the $T$ and $V$ versions of Cleary's group introduced by Burillo-Nucinkis-Reeves in \cite{Burillo-Nucinkis-Reeves22}.
Cleary proved that $G$ is of type $F_\infty$ and his proof can be extended without troubles to prove that $G^T$ and $G^V$ are of type $F_\infty$.
The forest category $\mathcal F$ is an example of a Ore forest category with a finite spine equal to $\{a_1,b_1, a_1a_1\}$, see Definition \ref{def:spine}.
Hence, by Theorem \ref{theo:Finfty} the group $G$ is of type $F_\infty$ and so are $G^T,G^V$ but also $G^{BV}.$
Hence, we obtain the result that the braided version of $G$ is of type $F_\infty$ as well.
The nice thing of our formalism is that we can easily generalise these results to similar forest groups.
For instance, consider
$$\mathcal F_n=\FC\langle a,b| a_1^n=b_1b_2\cdots b_n\rangle \text{ for } n\geq 1$$
obtained by considering a left $a$-vine and right $b$-vine both having $n$ interior vertices.
Here is the forest relation of $\mathcal F_3$:
\[\includegraphics{a_13b_13.pdf}\]
By Theorems \ref{theo:class-example} this is a Ore forest category producing the forest group $G_n:=\Frac(\mathcal F_n,1)$ that is of type $F_\infty$ (the spine is equal to $\{a_1,b_1, a_1^n\}$).
Moreover, it satisfies the CGP and thus $G_n\simeq H_n:=\Frac((\mathcal F_n)_\infty).$
Similar results hold for its $T,V,BV$ versions.
The group $G_n$ admits the finite group presentation with generators
$$\{a_1,b_1,a_2,b_2\}$$
and relations
\begin{itemize}
\item $[a_1^{-1}x_i, a_1^{-j} y_2 a_1^j]=e$ for $x,y\in\{a,b\}, i,j\in\{1,2\}$ with $(x,i)\neq (a,1)$;
\item $a_1^n=b_1b_2(a_1^{-1}b_2a_1) (a_1^{-2} b_2 a_1^2)\cdots (a_1^{3-n} b_2 a_1^{n-3})$;
\item $a_2^n=b_2(a_1^{-1}b_2a_1) (a_1^{-2} b_2 a_1^2)\cdots (a_1^{2-n} b_2 a_1^{n-2})$.
\end{itemize}
Moreover, $$G_n^{ab}\simeq \mathbf{Z}\oplus\mathbf{Z}\oplus\mathbf{Z}/n\mathbf{Z}$$ (with generators the images of $a_1,b_1, a_2b_2^{-1}$)
implying in particular that the $G_n$ are pairwise non-isomorphic.
One can generalise in many ways this construction.
Using again Theorem \ref{theo:class-example} we can replace the two vines $a_1^n$ and $b_1\cdots b_n$ by {\it any} pair of {\it monochromatic} trees obtaining the group $G_{(t,s)}$ considered in Section \ref{sec:two-colours}.
Furthermore, we can consider any number of colours and even infinitely many.
\subsubsection{Brin's higher dimensional Thompson groups}
Brin's groups $dV$ with $d\geq 1$ are higher dimensional generalisation of Thompson's group $V$ so that $1V=V$ and $dV$ acts by piecewise affine maps on the hypercube $[0,1]^d$.
An element of $dV$ is characterised by two standard dyadic partitions of $[0,1]^d$: $P=\{R_1,\cdots,R_k\}$ and $P'=\{R_1',\cdots,R_k'\}$ having the same number of pieces and one bijection $\beta$ between the elements of $P$ and $P'$.
The pieces $R_j$ are higher dimensional rectangle obtained by iteratively cutting the unit hypercube into two equal pieces along one axis.
We now describe a description of elements of $dV$ using trees given by Brin and further exploit by Burillo and Cleary \cite{Brin-dV1,Burillo-Cleary10}.
We can encode the data of $P,P'$ as a pair of trees with $d$ possible colours of carets, each colour corresponding to one axis.
Now, if $a,b$ are two colours, then the two operations $Y_a(Y_b\otimes Y_b)$ and $Y_b(Y_a\otimes Y_a)$ provide the same result of cutting the hypercube into four using two different axis.
We define the forest category $\mathcal F$ with $d$ colours and relations $a_1(b_1b_3)=b_1(a_1a_3)$ for all pairs of colours $a,b$.
We obtain a Ore forest category. Although, its fraction group is not Brin's group $dV$ but share similar metric properties.
What makes it different is that the forest relations are not compatible with bijections $\beta$, see \cite[Section 1]{Burillo-Cleary10} for details.
Note that the {\it decolouring map} $$D:\mathcal F\to \mathcal{UF}, Y_a\mapsto Y$$ consisting in removing all colours is a well-defined and surjective morphism from $\mathcal F$ to the monochromatic free forest category $\mathcal{UF}$.
It induces a surjective group morphism $G\twoheadrightarrow F$ which admits a section.
Hence, the group $G$ is a nontrivial split extension of $F$ when $d\geq 2$ (and similarly for the $T,V,BV$-versions).
\subsubsection{An example not satisfying the CGP}
Consider the presented forest category
$$\mathcal F=\FC\langle a,b| b_1a_1b_3=a_1a_2a_3\rangle.$$
The forest presentation has two colours and one relation implying it is complemented with associated monoid presentation complete.
In particular, $\mathcal F$ is left-cancellative.
Moreover, one can prove it satisfies Ore's property. It is proved by showing that every tree can be grown into a monochromatic tree of colour $a$.
Hence, $\mathcal F$ is a Ore category producing the groups $G$ and $H$.
The group $G$ admits the following infinite presentation: generating set $\{a_j,b_j,\hat b_j:\ j\geq 1\}$ and relations:
$$\begin{cases}
x_{q} y_{j} = y_{j} x_{q+1} \\
\hat b_{q} y_{j} = y_{j} \hat b_{q+1} \\
b_j b_{j+1} a_j = a_j a_{j+1} a_{j+2}\\
\hat b_j \hat b_{j+1} a_j = e
\end{cases}$$
for all $1\leq j<q, x,y\in\{a,b\}$.
Note that this presentation is not homogeneous.
A presentation of $H$ is deduced by removing all generators with a hat and relations containing them.
We obtain the presentation of $H$ with generating set
$\{a_j,b_j:\ j\geq 1\}$ and relations:
$$\begin{cases}
x_{q} y_{j} = y_{j} x_{q+1} \text{ for all } 1\leq j<q , x,y\in\{a,b\} \\
b_j b_{j+1} a_j = a_j a_{j+1} a_{j+2}
\end{cases}.$$
This forest category does not satisfies the CGP and thus the embedding $\gamma_a:H\to G$ of Proposition \ref{prop:GHinclusion} is not surjective.
\subsubsection{Monoids give forest groups}
In the future article \cite{Brothier22-HPM} we will explain how certain monoid gives forest groups and study them.
We provide here two examples of those that have interesting finiteness properties.
Consider
$$\mathcal F=\FC\langle a,b| a_1a_1=b_1b_1, a_1b_1=b_1a_1\rangle.$$
This is a Ore forest category that is an extension of $F$.
It has infinite spine but nevertheless is of type $F_\infty$.
Moreover, one can prove that $G$ is isomorphic to the wreath product $\mathbf{Z}/2\mathbf{Z}\wr_Q F=\oplus_{Q}\mathbf{Z}/2\mathbf{Z}\rtimes F$ associated to the action $F\curvearrowright Q$ that is the classical action of $F$ on the set of dyadic rationals of the unit torus.
This group and its abelianisation have torsion and moreover have a nontrivial center isomorphic to $\mathbf{Z}/2\mathbf{Z}.$
Similarly, $G^T$ and $G^V$ are isomorphic to $\mathbf{Z}/2\mathbf{Z}\wr_Q T$ and $\mathbf{Z}/2\mathbf{Z}\wr_Q V$ that are not simple nor perfect.
Consider the forest presentation with colours $\{x_i:\ i\geq 1\}$ and relations
$$(x_k)_j (x_i)_j = (x_i)_j(x_{k+1})_j \text{ for all } 1\leq i<k \text{ and } i\geq 1.$$
It is constructed from the Thompson monoid.
Its associated forest category $\mathcal F$ is a Ore category. Since $\mathcal F(2,1)$ is infinite we have that $\mathcal F$ does not admit any forest presentation with finitely many colours.
Nevertheless, its forest group is finitely presented and in fact of type $F_\infty$ as we will prove it in \cite{Brothier22-HPM}.
\section{Presentations of forest groups}\label{sec:presentation}
In all this section we consider $P:=(S,R)$ a forest presentation of a forest category $\mathcal F$.
Let $\mathcal F_\infty$ be the associated forest monoid.
Assume that $\mathcal F$ is left-cancellative and satisfies Ore's property.
This implies that both $\mathcal F$ and $\mathcal F_\infty$ admit a right-calculus of fractions and embed in their fraction groupoids $\Frac(\mathcal F)$ and $\Frac(\mathcal F_\infty)$.
Note that this latter is a group.
Consider the fraction groups $G:=\Frac(\mathcal F,1)$ and $H:=\Frac(\mathcal F_\infty)$.
In this section we introduce contractible simplicial complex $E\mathcal F$ admitting a free simplicial action $G\curvearrowright E\mathcal F$.
In particular, the space of orbits $B\mathcal F:=G\backslash E\mathcal F$ is a classifying space of $G$.
From $B\mathcal F$ we deduce an explicit infinite group presentation of $G$ in terms $P$.
We recall an explicit embedding of $H$ inside $G$.
We provide a reduced group presentation $\Gr\langle \sigma|\rho\rangle$ of $G$.
A presentation of $H$ is given by $\Gr\langle \sigma_0|\rho_0\rangle$ for some subsets $\sigma_0\subset \sigma$ and $\rho_0\subset \rho.$
We prove that when $\mathcal F$ is finitely forest generated (resp.~forest presented), then the groups $G$ and $H$ are finitely generated (resp.~presented) and we give upper bounds for the number of their generators and relations.
\subsection{A complex associated to each forest category}\label{sec:complex}
{\bf Associated structures.}
Recall that $\mathcal F$ is a category with object $\mathbf{N}$ and a forest $f$ is a morphism with origin $\omega(f)=|\Leaf(f)|$ and target $\tau(f)=|\Root(f)|$.
We extend these origin and target maps to the groupoid $\Frac(\mathcal F)$.
Hence, $[f,g]=f\circ g^{-1}$ has for origin the number of roots of $g$ and target the number of roots of $f$.
{\bf A $G$-space.}
Since $G\subset \Frac(\mathcal F)$ we can consider the restriction of the multiplication of $\Frac(\mathcal F)$ to composable pairs of $G\times \Frac(\mathcal F)$.
This provides an action
$$G\times Q\to Q, (g,x)\mapsto g\circ x$$
where $$Q=\{[t,f]:\ t \text{ tree } f \text{ forest }, \ |\Leaf(t)|=|\Leaf(f)|\} = \{ x\in \Frac(\mathcal F):\ \tau(x)=1\}.$$
{\bf A $G$-directed set.}
Now, an element of $Q$ can be multiplied to the right by an element of $\Frac(\mathcal F)$ but if we want to preserve $Q$ we must multiply by elements of $\mathcal F$.
Hence, we consider
$$\Frac(\mathcal F,1) \curvearrowright Q \curvearrowleft \mathcal F.$$
The right action $Q\curvearrowleft \mathcal F$ defines a partial order:
$$x\leq x\circ f,\ x\in Q, f\in\mathcal F.$$
We obtain a partially ordered set (in short poset) $(Q,\leq)$.
Since the left and right actions of any subsets of $\Frac(\mathcal F)$ mutually commute we have that $\leq$ is invariant under the action of $G$.
Moreover, the poset $(Q,\leq)$ is directed (i.e.~two elements of $Q$ admits an upper bound) since $\mathcal F$ satisfies Ore's property.
{\bf A contractible $G$-free simplicial complex.}
Let $\Delta(Q)$ be the ordered complex deduced from $(Q,\leq)$.
It is the abstract simplicial complex with vertices $Q$ and $k$-simplices (for $k\geq 1$) the strict chain $x_0<\cdots<x_k$ whose faces are the sub-chains.
Write $E\mathcal F:=|\Delta(Q)|$ for the geometric realisation of $\Delta(Q)$ which is a simplicial complex with $k$-simplices $|x_0<\cdots<x_k|$ that we may identify with $\Delta(Q)$ if the context is clear.
Since $G\curvearrowright Q$ is free and order preserving it induces a free simplicial action $G\curvearrowright E\mathcal F$.
Moreover, since $Q$ is directed we deduce that $E\mathcal F$ is contractible.
Therefore, the quotient $B\mathcal F:= G\backslash E\mathcal F$ is a classifying space for $G$, i.e.~$B\mathcal F$ is a path-connected CW-complex, $\pi_1(B\mathcal F,p)\simeq G$, and $\pi_n(B\mathcal F,p)=\{e\}$ for all $n\geq 2$ where $p$ is a point of $B\mathcal F$.
The projection map $E\mathcal F\to B\mathcal F$ is a fibre bundle and since $E\mathcal F$ is simply connected (since it is contractible) $E\mathcal F$ is a universal cover of $B\mathcal F$.
A $k$-cell of $B\mathcal F$ is an orbit of the form $G\cdot |x_0<\cdots<x_k|.$
\begin{remark}
Note that the constructions of $E\mathcal F$ and $B\mathcal F$ are canonical and thus do not depend on any choice of forest presentation of $\mathcal F$.
Moreover, these constructions are functorial.
\end{remark}
\subsection{An infinite group presentation}
Since $G\curvearrowright E\mathcal F$ is a free action on a contractible space we have that the Poincar\'e group $\pi_1(B\mathcal F,p)$ at any point of $B\mathcal F:=G\backslash E\mathcal F$ is isomorphic to $G$.
Moreover, $G\simeq \pi_1(B\mathcal F,p)=\pi_1(B\mathcal F^{(2)},p)=\pi_1(G\backslash E\mathcal F^{(2)},p)$ where $B\mathcal F^{(2)}$ stands for the 2-skeleton of $B\mathcal F$ (the subcomplex of $B\mathcal F$ obtained by removing all $k$-cells for $k\geq 3$).
From there we are able to extract informations on group presentations of $G$ in terms of the simplicial structure of $E\mathcal F^{(2)}.$
We start by providing a group presentation of $G$ in terms of the forest presentation $P=(S,R)$ of $\mathcal F$.
{\bf Orientation.}
Equip the simplicial complex $E\mathcal F=|\Delta(Q)|$ with the orientation deduced from the partial order $\leq$ of $Q$.
Hence, a 1-simplex $|x<y|$ of $E\mathcal F$ is now interpreted as a directed edge starting at $x$ and ending at $y$.
{\bf A subtree of $E\mathcal F$.}
We now construct a subtree of $E\mathcal F$ whose image in $B\mathcal F$ is a maximal subtree.
Here, a tree in $E\mathcal F$ means a path-connect subcomplex contained in the 1-skeleton of $E\mathcal F$ that does not contain any cycle.
The reader should not confuse such a tree in $E\mathcal F$ and a tree of the forest category $\mathcal F$.
{\it Vertices.}
Fix a colour $a\in S$ and define by induction a sequence of trees $t_n\in\mathcal F$ with $n$ leaves and monochromatic in $a$ such that
$$t_1=I \text{ and } t_{n+1}= t_n\circ a_{n,n} \text{ for } n\geq 1$$
so that $t_{n+1}=a_{1,1}a_{2,2}\cdots a_{n,n}.$
The tree $t_n$ corresponds graphically to a long right branch that we call a {\it right vine of colour $a$} or a {\it right $a$-vine} following the terminology of Belk \cite[Section 1.3]{Belk-PhD}.
They previously appear in the proof of Proposition \ref{prop:GHinclusion}.
Note that $t_n$ is an element of $Q$ and thus a vertex of the complex $E\mathcal F:=|\Delta(Q)|.$
{\it Edges.}
For each $n\geq 1$ note that $t_n<t_{n+1}$ and thus we have a directed edge $|t_n<t_{n+1}|$ of $E\mathcal F$ going from $t_n$ to $t_{n+1}$.
Let $T_a$ be the directed graph inside $E\mathcal F$ with vertex set
$$V(T_a):=\{t_n:\ n\geq 1\}$$ and edge set $$E(T_a)=\{ |t_n<t_{n+1}|:\ n\geq 1\}.$$
Graphically $T_a$ is a single infinite ray.
{\bf A maximal subtree of $B\mathcal F$.}
Consider the quotient map $$q:E\mathcal F\to B\mathcal F,\ \sigma\mapsto G\cdot\sigma.$$
Recall that we have the origin-map
$$\omega:\Frac(\mathcal F)\to\mathbf{N}, \ [f,g]\mapsto |\Root(g)|$$
and observe that $\omega$ is $G$-invariant.
Moreover, if $x,y\in Q$ and $\omega(x)=\omega(y)$, then there exists $g\in G$ satisfying $y=g\cdot x.$
Indeed, one can simply take $g:= y\circ x^{-1}$ which is in $\Frac(\mathcal F,1)=G$.
This two facts implies that $\omega$ factorises into a bijection:
$$\omega:G\backslash P\to \mathbf{N}_{>0}.$$
Since $\omega(t_n)=n$ for all $n\geq 1$ we deduce that $\{t_n:\ n\geq 1\}$ is a set of representatives of the vertices of $B\mathcal F$, i.e.~the quotient map $E\mathcal F\to B\mathcal F$ restricts into a bijection from $V(T_a)$ to $B\mathcal F^{(0)}$.
This implies that $q(T_a)$ is a maximal subtree of $B\mathcal F$ containing all vertices of $B\mathcal F$.
\begin{remark}
Note that $(t_n:\ n\geq 1)$ is an increasing sequence of trees such that $t_n$ has $n$ leaves.
We could make the same construction with any such sequence which will provide a different presentation of the group $G$.
However, the specific choice of $t_{n+1}=a_{1,1}\cdots a_{n,n}$ seems to be the most practical to work with.
\end{remark}
{\bf Group presentation of $G$.}
We have found a classifying space $B\mathcal F$ of $G$ and thus its Poincar\'e group is isomorphic to $G$.
Moreover, $q(T_a)$ is a maximal subtree of $B\mathcal F$.
By classical theory (see for instance \cite[Chapter 4]{Stillwell-book} or \cite[Theorem 3.1.16]{Geoghegan-book}) a presentation of $G$ is obtained by taking the free group over the oriented edges of $B\mathcal F$ that we mod out by the boundary of each 2-cell of $B\mathcal F$ and by the edges of $q(T_a)$.
Let us describe precisely this presentation in our context.
{\bf Naive description.}
Consider a nontrivial forest $f\in\mathcal F$ and write $\ell,r$ for its number of leaves and roots, respectively.
We have an edge $|t_r<t_rf|$ in $E\mathcal F$ whose image by the quotient map $q:E\mathcal F\to B\mathcal F$ connects $q(t_r)$ and $q(t_\ell)$.
Therefore, any nontrivial forest $f$ defines an edge in $B\mathcal F$ and conversely any edge of $B\mathcal F$ is of this form.
We obtain a surjective map
$$\mathcal F^*\twoheadrightarrow \text{Edge}(B\mathcal F),\ f\mapsto q(|t_r<t_rf|)$$
where $\mathcal F^*$ is the set of nontrivial forests.
Let $\Gr\langle\mathcal F\rangle$ be the free group over the set of forests $\mathcal F$ writing $\bar f$ the generator of $\Gr\langle \mathcal F\rangle$ associated to $f\in\mathcal F$.
Let $\pi$ be the quotient of $\Gr\langle\mathcal F\rangle$ by the relations:
\begin{enumerate}
\item $\bar f =e$ if $f$ is a trivial forest (i.e.~$f$ has the same number of leaves and roots);\\
\item $\bar f =e$ if the edge $q(|t_r<t_rf|)$ associated to $f$ is in $q(T_a)$;\\
\item $\bar f \cdot \bar f'=\bar f''$ if $f,f',f''$ correspond to the edges equal to the boundary of a 2-cell of $B\mathcal F$ so that $\bar f,\bar f''$ have same start and $\bar f',\bar f''$ have same end.
\end{enumerate}
From the discussion of above we obtain that $\pi\simeq \pi_1(B\mathcal F,p)\simeq G$.
{\bf More refined description.}
Let us better interpret the relations of above using the structure of $\mathcal F$.
The second item is equivalent to $G\cdot |t_r\leq t_r f| = G\cdot | t_r\leq t_\ell |$.
Which means $f=t_r^{-1}\circ t_\ell$ and is a product of some $a_{m,m}$.
Item two is thus equivalent to
$$\bar a_{n,n}=e \text{ for all } n\geq 1.$$
For the third item: consider a 2-cell of $B\mathcal F$. It is of the form $G\cdot |t_r<t_r f<t_rff'|$ for some $r\geq 1$ and forests $f,f'.$
Now, $G\cdot |t_r<t_rf|$ corresponds to $\bar f$ and $G\cdot |t_rf< t_rff'|$ to $\bar{f'}.$
The third item is equivalent to
$$\bar f \cdot \bar f' = \overline{f\circ f'} \text{ for $(f,f')$ a pair of composable nontrivial forests.}$$
Finally, if we allow $f$ or $f'$ to be the trivial forests in the expression of above we deduce the first item.
We deduce a group presentation of $G$ and relate it with the morphisms $\gamma_a:H\to G(a)\subset G$ of Proposition \ref{prop:GHinclusion}.
\begin{proposition}\label{prop:presentation-G}
Let $\mathcal F$ be a Ore forest category, $a$ in the colour set of $\mathcal F$, $G:=\Frac(\mathcal F,1)$ the fraction group of $\mathcal F$, $B\mathcal F$ the classifying space of $\mathcal F$ constructed as above, and $p$ a point of $B\mathcal F$.
We have the following isomorphisms:
\begin{equation}\label{eq:presentation-pi}G\simeq \pi:=\pi_1(B\mathcal F,p)\simeq \Gr\langle \mathcal F | \ \bar{a_{n,n}}=e,\ \bar f \cdot \bar f' = \overline{f\circ f'}\rangle\end{equation}
where $f\mapsto \bar f$ denote the canonical embedding from $\mathcal F$ into the free group over the set $\mathcal F$, and where the relations of above stand for each $n\geq 1$, and $(f,f')$ composable pairs of forests.
The group morphism
$$\theta_0:\Gr\langle\mathcal F\rangle\to G,\ \overline f\mapsto t_r \circ f \circ t_\ell^{-1} \text{ where } f\in\mathcal F(r,\ell)$$
factorises into a group isomorphism $\theta$ from the presented group of above and $G$.
Moreover, the set $\theta(\mathcal F\otimes I)$ of all $\theta(\bar f)$ with $f$ a forest with last tree trivial generates the group $G(a)$.
\end{proposition}
\subsection{Practical infinite and reduced group presentations}
We now fix a forest presentation $(S,R)$ of $\mathcal F$ and express presentations of $G$ and $H$ using the elementary forests.
Note that we freely identify $G$ and $\pi$ via the isomorphism $\theta$ of the last proposition.
Using Propositions \ref{prop:universal-category} and \ref{prop:presentation-G} we deduce that $G$ admits the presentation with generating set
$$\{ \overline{b_{j,n}}:\ b\in S, 1\leq j\leq n\}$$
and set of relations:
\begin{enumerate}
\item $\overline{a_{n,n}}=e$ for all $n\geq 1$;
\item $\overline{b_{q,n}}\circ \overline{a_{j,n+1}} = \overline{a_{j,n}}\circ \overline{b_{q+1,n+1}}$ for all $a,b\in S$ and $1\leq j <q\leq n$;
\item $\overline{U_{j,n}}=\overline{U'_{j,n}}$ for all forest relations $(u,u')\in R$ and $1\leq j\leq n$
\end{enumerate}
where $U_{j,n}$ is a word in the elementary forest expressing the forest $u_{j,n}:= I^{\otimes j-1}\otimes u\otimes I^{\otimes n-j}$ and where $\overline{U_{j,n}}$ denotes its image in the free group $\Gr\langle \mathcal F\rangle$.
The third kind of relation is written $R(u,u',j,n)$.
First, observe that $\overline{b_{j,n}} = \overline{b_{j,j+1}}$ for all $n>j$.
Hence, the generator set of $G$ can be written as
$$\{\overline{b_{j,j+1}}, \ \overline{b_{j,j}} :\ j\geq 1\}.$$
Second, the generators of the first kind generates the subgroup $G(a)$ which is isomorphic to the fraction group $\Frac(\mathcal F_\infty).$
An isomorphism is given by $$\overline{b_{j,j+1}}=[t_n \circ b_{j,n}, t_{n+1}]\mapsto b_j \text{ for all } b\in S, 1\leq j <n.$$
Third, the conjugation by $\overline{a_{1,2}}$ corresponds in shifting both indices, i.e.~
$$\overline{a_{1,2}}^{-1} \circ \overline{b_{k,r}} \circ \overline{a_{1,2}} = \overline{b_{k+1,r+1}}.$$
This implies that we only need to retain the relations $R(u,u',1,1)$, $R(u,u',1,2)$, $R(u,u',2,2)$, and
$R(u,u',2,3)$.
Indeed, $R(u,u',j,n)=R(u,u',j,j+1)$ for all $j<n$ by the first observation and $\ad(\overline{a_{1,2}}^{-1})(R(u,u',i,m))=R(u,u',i+1,m+1)$ by the third for all $i\leq m$.
Similarly, we may only retain the indices $j=1,2$ for the generating set.
Moreover, using the relation $\overline{a_{j,j}}=e$ for all $j$ we may remove the generators $\overline{a_{1,1}},\overline{a_{2,2}}.$
We deduce the following.
\begin{observation}
The set
$$\sigma:=\{ \overline{b_{1,2}}, \overline{b_{2,3}}, \overline{c_{1,1}}, \overline{c_{2,2}}:\ b\in S, c\in S\setminus\{a\}\}$$ generates the group $G.$
The subset $\sigma_0:=\{\overline{b_{1,2}}, \overline{b_{2,3}}: \ b\in S\}$ generates the subgroup $G(a)$ isomorphic to $H$.
In particular, if $\mathcal F$ has finitely many colours, then $G$ and $H$ are finitely generated.
\end{observation}
If $U_{j,n}$ has letters $\overline{b_{k,r}}$ with $k$ larger than $2$, then we replace it by $\overline{a_{1,2}}^{2-k} \circ \overline{b_{2,r-k+2}}\circ \overline{a_{1,2}}^{k-2}$ so that the relations $R(u,u',j,n)$ are all expressed using the letter of the smaller generating set $\sigma$.
An easy adaptation of the proof of \cite[Section 3]{Cannon-Floyd-Parry96} permits to reduce the Thompson-like relations to a few commutators equal to the identity.
To have more compact notation we give a presentation with the following correspondence of symbols:
$$b_j\leftrightarrow \overline{b_{j,j+1}},\ \widehat b_j\leftrightarrow \overline{b_{j,j}}
, R(u,u',j)\leftrightarrow (\overline{U_{j,j+1} },\overline{U'_{j,j+1} }), \text{ and } \widehat R(u,u',j)\leftrightarrow (\overline{U_{j,j} },\overline{U'_{j,j} }).$$
\begin{theorem}\label{theo:groupG-presentation}
Let $\mathcal F=\FC\langle S|R\rangle$ be a presented Ore forest category with associated forest group $G:=\Frac(\mathcal F,1)$.
Fix a colour $a\in S$.
The group $G$ admits the group presentation with generating set
$$\{ \widehat b_j, b_j:\ b\in S, j\in\{1,2\}\}$$
and set of relations
\begin{enumerate}
\item $[ a_1^{-1} x_i \ ,\ a_1^{-j}\widehat y_2a_1^j]=e$ for all $i,j\in\{1,2\}$ and $x,y\in S$ with $(x,i)\neq (a,1)$;
\item $[ a_1^{-1} x_i \ ,\ a_1^{-j}y_2a_1^j]=e$ for all $i,j\in\{1,2\}$ and $x,y\in S$ with $(x,i)\neq (a,1)$;
\item $\widehat a_1= \widehat a_2 =e;$
\item $\widehat{R}(u,u',i)=R(u,u',i)=e \text{ for all } (u,u')\in R, i=1,2.$
\end{enumerate}
In particular, if $S$ is finite, then $G$ is finitely generated admitting a generating set of cardinal $4|S|-2$ and if both $S,R$ are finite, then $G$ is finitely presented admitting a presentation with $4|R|+8|S|^2-4|S|+2$ relations.
Moreover, the subgroup of $G$ generated by $\{b_j:\ b\in S, j\in\{1,2\}\}$ is isomorphic to $H:=\Frac(\mathcal F_\infty).$
A presentation of $H$ is obtained by considering the relations (2) and (4) excluding $\widehat R(u,u',i)$.
An infinite presentation of $G$ is given by the generating set
$$\{ \widehat b_j, b_j:\ b\in S, j\geq 1\}$$
and set of relations
\begin{enumerate}
\item $x_q\circ y_j = y_j \circ x_{q+1} \text{ for all } 1\leq j <q \leq m, x,y\in S;$
\item $\widehat x_q\circ y_j = y_j \circ \widehat x_{q+1} \text{ for all } 1\leq j <q \leq m, x,y\in S;$
\item $\widehat a_n=e$ for all $n\geq 1$;
\item $\widehat{R}(u,u',i)=R(u,u',i)=e \text{ for all } (u,u')\in R, i=1,2.$
\end{enumerate}
An infinite presentation of $H$ is obtained by considering the generating set $\{b_j:\ b\in S, j\geq 1 \}$ and the relations (2), (4) excluding $\widehat R(u,u',i)$.
\end{theorem}
\begin{remark}
\begin{enumerate}
\item The last theorem proved that if a Ore forest category is finitely generated (resp.~presented) as a forest category, then its fraction group is finitely generated (resp.~presented) as a group.
However, the converse is not true. A counterexample is given in Section \ref{sec:example}.
\item Let $\mathcal F$ be a Ore forest category.
We have constructed for the forest monoid $\mathcal F_\infty$ a homogeneous monoid presentation providing a positive homogeneous group presentation of $\Frac(\mathcal F_\infty)$.
In particular, if $\mathcal F$ satisfies the CGP at a certain colour, then $\Frac(\mathcal F,1)$ admits a positive homogeneous group presentation.
Although, the presentation of $\Frac(\mathcal F,1)$ provided in the last theorem is not homogeneous in general, see Section \ref{sec:example}.
\end{enumerate}
\end{remark}
\section{A replacement of the dyadic rationals for forest groups}\label{sec:Q-space}
Consider a forest category $\mathcal F$ and let $G^X=\Frac(\mathcal F^X,1)$ be its $X$-forest group for $X=F,T,V$.
The classical Thompson groups $F,T,V$ act on the set of dyadic rationals $\mathbf{Q}_2:=\mathbf{Z}[1/2]/\mathbf{Z}$ of the torus in an obvious manner by restricting the classical action on the unit torus.
This section is about constructing an analogous space $Q$ for the forest groups $G,G^T,G^V$.
We start by building the action $G^T\curvearrowright Q$ that we describe in three different ways.
The last of it uses Jones' technology and permit to be extended to $G^V$ in a canonical way.
We derive key transitive properties of it.
From there we deduce a finiteness theorem by adapting an elegant argument of Brown and Geoghegan.
\subsection{Three descriptions of a $G^T$-space}
{\it First description.} We consider the homogeneous space action $G^T\curvearrowright G^T/G$ given by $$g\cdot hG:=ghG,\ g\in G^T, hG\in G^T/G.$$
{\it Second description.}
Consider the two commuting actions $$G^T\curvearrowright \Frac(\mathcal F^T)\curvearrowleft \mathcal F$$ on the fraction groupoid $\Frac(\mathcal F^T)$ obtained by restriction of the composition.
Note, this is similar to the way we constructed a classifying space of $G$ in Section \ref{sec:complex}.
This gives $G^T\curvearrowright Q^T$ where $$Q^T=\{ t\circ \pi\circ f^{-1}:\ t, \pi ,f \},$$
where $t$ is a tree with $n$ leaves, $\pi\in\mathbf{Z}/n\mathbf{Z}$ a cyclic permutation, and $f$ is a forest with $n$ leaves.
In other words, $Q^T$ is the subset of $\Frac(\mathcal F^T)$ of elements of target $1$.
Consider now the equivalence relation generated by the right action of $\mathcal F$ on $Q^T$ that is
$$x\sim y \text{ if } x\mathcal F\cap y\mathcal F\neq\emptyset$$
and denote by $Q^T/\mathcal F$ the space $Q^T$ quotiented by this equivalence relation.
The action $G^T\curvearrowright Q^T$ factorises into an action $G^T\curvearrowright Q^T/\mathcal F$.
Note that any element of $Q^T/\mathcal F$ is obtained as a class of the form $t\pi\mathcal F$ with $t$ a tree and $\pi$ a cyclic permutation.
In fact a set of representatives of $Q^T/\mathcal F$ is given by all the $t\pi$ satisfying the property:
$$[t\pi\mathcal F\cap t'\pi'\mathcal F\neq\emptyset] \Rightarrow [t\pi\mathcal F\supset t'\pi'\mathcal F].$$
In that case, $t'\pi'=t\pi h = (th^\pi) \pi^h$ for a certain forest $h$ and where $(\pi,h)\mapsto (\pi^h, h^\pi)$ denotes the Brin-Zappa-Sz\'ep product (or bicrossed product) of the groupoid of cyclic permutations and the forests category $\mathcal F$.
Here is a diagrammatic example when on the left we have a product $\pi h$ and on the right a product $h^\pi \pi^h$.
\[\includegraphics{BZS-cyclic.pdf}\]
Note that $\pi$ and $\pi^h$ are both cyclic and $h, h^\pi$ have the same number of roots and leaves.
The second action is conjugate to the first via the map
$$Q^T/\mathcal F\to G^T/G, \ t \pi\mathcal F\mapsto t\pi t^{-1}G.$$
To see that it is well-defined note that if $f$ is a forest then $t\pi \cdot f = t f^\pi \pi^f$.
Moreover, a key point is that $f$ and $f^\pi$ have the same number of leaves (and roots) implying that $t f^\pi (tf)^{-1}\in G$.
We deduce:
$$t f^\pi \pi^f (tf^\pi)^{-1} G = tf^\pi \pi^f (tf)^{-1} G = t\pi t^{-1} G.$$
The $G^T$-equivariance of the map is also obvious.
{\it Third description.}
We adapt to {\it any} Ore forest category $\mathcal F$ an example of actions constructed in \cite{Brothier-Jones19bis} just before Exercise 2.2.
This is done using Jones' technology.
Let $(\Set,\sqcup)$ be the monoidal category of sets equipped with disjoint union for the monoidal structure.
Let $\Phi:\mathcal F\to \Set$ be the unique monoidal contravariant functor satisfying $\Phi(n)=\{1,\cdots,n\}$ and $\Phi(t)(1)=1$ for all tree $t$.
Hence, if $f=(f_1,\cdots,f_r)$ is a forest with $r$ roots and $n$ leaves, then $\Phi(f)$ is a map from $\{1,\cdots,r\}$ to $\{1,\cdots,n\}$ sending $j$ to
$$j^f:=|\Leaf(f_1,\cdots,f_{j-1})|+1.$$
The number $j^f$ corresponds to the first leaf of the $j$th tree of $f$.
It is remarkable that $\Phi$ indeed exists whatever the forest category $\mathcal F$ is.
It is a consequence of the universal property of a forest category among categories given in Section \ref{sec:universal-property}.
Following Jones' technology we define the set $X_0$ equal to all pairs $(t,j)$ where $t$ is a tree and $j$ a leaf of $t$ (identified with an element of $\{1,\cdots,r\}$ where $r=|\Leaf(t)|$).
The space $X$ is obtained by quotienting $X_0$ by the equivalence relation generated by
\begin{equation}\label{eq:bicross}(t,j)\sim (tf, j^f) \text{ where } j^f = |\Leaf(f_1,\cdots,f_{j-1})|+1.\end{equation}
Here is an example of two elements of $X_0$ in the same class where the distinguish leaf is a black dot:
\[\includegraphics{Xleaf.pdf}\]
We write $[t,j]$ for the class of $(t,j)$ inside $X:=X_0/\sim$.
Hence, elements of $X$ can be thought as a tree $t$ with a distinguish leaf $\ell$ of it.
Now, we can grow this tree into $tf$ getting for distinguish leaf $\ell^f$: the unique leaf of $tf$ which is a descendant of $\ell$ satisfying that the geodesic from $\ell$ to $\ell^f$ only contains left edges (i.e.~is a left vine).
A key point is that Jones' technology provides an action of the larger group $G^V$.
We have the Jones action
$$G^V\curvearrowright X,\ g \cdot [t,j]:= [sp' , \pi(j^p)]$$
where $G^V\ni g=s\circ \pi \circ t'^{-1}$ with $s,t',t$ trees, $\pi$ a permutation, and $p,p'$ some forests satisfying $t'p'=tp$.
In particular, $$s\circ \pi\circ t^{-1} \cdot [t,j]=[s, \pi(j)]$$
when we have equality of $t$ with $t'$.
The restriction of this action to $G^T$ is conjugate to the two actions described above via the map
$$Q^T/\mathcal F\to X,\ t\pi \mathcal F \mapsto [t, \pi^{-1}(1)].$$
{\bf Classical case.}
When $\mathcal F$ is the monochromatic free forest category, then $G=F, G^T=T,$ and $G^V=V$.
Moreover, if $\mathbf{Q}_2=\mathbf{Z}[1/2]/\mathbf{Z}$ is the set of dyadic rationals of the unit torus, then the evaluation map
$$T\to \mathbf{Q}_2, g\mapsto g(0)$$
factorises into a bijection
$$T/F\to\mathbf{Q}_2.$$
The usual action $T\curvearrowright \mathbf{Q}_2$ and the homogeneous action $T\curvearrowright T/F$ are conjugated by this bijection.
The extension to an action of $V$ is obtained by restricting the classical action of $V$ on infinite binary sequences and by identifying $\mathbf{Q}_2$ with finitely supported ones.
\subsection{Total order}
We now mostly work with the third description of the action that we extend to $G^V$.
This is the Jones action $G^V\curvearrowright X$ where $X$ is the set of classes $[t,j]$ where $t$ is a tree of $\mathcal F$ and $j$ a leaf of $t$.
Moreover, if $a_k$ is the elementary forest with one $a$-caret at the $k$th root, then $[t,j]=[ta_k,j']$ where $j'=j+1$ if $k<j$ and $j'=j$ if $k\geq j.$
We define a total order $\preceq$ on $X$ (a partial order for which any two elements are comparable).
Consider two elements $[t,j]$ and $[t',j']$ of $X$.
Here, $t,t'$ are trees with $n$ and $n'$ leaves, respectively, and $1\leq j\leq n, 1\leq j'\leq n'$.
Since $\mathcal F$ satisfies Ore's property there exists some forests $f,f'$ so that $tf=t'f'=:z$.
Hence, $[t,j]=[tf,j^f]=[z,j^f]$ and $[t',j']=[t'f',j'^{f'}]=[z,j'^{f'}]$.
We define the order $\preceq$ by setting
$$[t,j]\preceq [t',j'] \text{ if } j^f\leq j'^{f'}.$$
We claim that the formula of above defines a partial order.
\begin{proof}[Proof of the claim]
Consider $[t,j],[t',j']\in X$ and assume there exists a pair of forests $(f,f')$ satisfying
$$tf=t'f' \text{ and } j^f\leq j'^{f'}.$$
The claim resides in proving that if $(h,h')$ is another pair of forests satisfying $th=t'h'$, then we must have $j^h\leq j'^{h'}.$
First, note that given any forest $h$ with $n$ roots and $m$ leaves we have a map
$$\{1,\cdots,n\}\to \{1,\cdots,m\}, \ j\mapsto j^h$$
(defined in \eqref{eq:bicross}) that is increasing in $j$.
Second, observe that $j^{hk}=(j^h)^k\geq j^h$ and in particular $h \mapsto j^h$ is increasing in $h$ for the usual partial order on the set of forests.
From these observations it is easy to deduce that $\preceq$ is a well-defined partial order.
Indeed, choose $(h,h')$ so that $th=t'h'$.
By Ore's property there exists a pair of forests $(u,v)$ so that $tfu=thv=:z.$
Now,
\begin{align*}
& [t,j]=[tf,j^f]=[z,j^{fu}] =[z,j^{hv}];\\
& [t',j']=[t'f',j'^{f'}]= [tf, j'^{f'}] = [tfu,j'^{f'u}] = [z, j'^{f'u}] \text{ and similarly } \\
& [t',j'] = [t'h',j'^{h'}]= [th, j'^{h'}]= [thv, j'^{h'v}] = [z,j'^{h'v}].
\end{align*}
We obtain
$$j^{hv}=j^{fu} = (j^{f})^u \leq (j'^{f'})^u = j'^{f'u} = j'^{h'v}.$$
Hence, $(j^h)^v\leq (j'^{h'})^v$ implying that $j^h\leq j'^{h'}.$
\end{proof}
We have proven that $\preceq$ is indeed a partial order.
Moreover, Ore's property assures that any two elements of $X$ are comparable.
Therefore, $\preceq$ is a total order on $X$.
We write $\prec$ for the strict order associated to $\preceq$.
The following proposition proves that $G$ and $G^T$ can be defined using $G^V\curvearrowright X$ and the partial order $\preceq$ just like the classical Thompson groups action acting on $\mathbf{Q}_2$.
\begin{proposition}\label{prop:order-preserving}
Consider the action $\alpha:G^V\curvearrowright X$ and the total order $\preceq$ on $X$.
We have that $G$ is equal to the set of elements of $G^V$ that is preserving the order $\preceq$, i.e.~
$$G=\{ g\in G^V:\ \alpha_g(x)\preceq \alpha_g(y) \text{ for all } x,y\in X, x\preceq y\}.$$
The group $G^T$ is the subset of $G^V$ of elements preserving the order $\preceq$ up to cyclic permutation.
That is $G^T$ is the set of $g\in G^V$ satisfying that for all chain $x_1\preceq x_2\preceq \cdots \preceq x_k$ in $X$ there exists a cyclic permutation $\pi\in\mathbf{Z}/k\mathbf{Z}$ so that
$$\alpha_g(x_{\pi(1)}) \preceq \alpha_g(x_{\pi(2)})\preceq \cdots \preceq \alpha_g(x_{\pi(k)}).$$
\end{proposition}
\begin{proof}
Consider $k\geq 1$ and a chain $x_1\preceq \cdots\preceq x_k$ in $X$.
We have that $x_i=[t_i,j_i]$ for some tree $t_i$ and natural number $j_i$.
Up to using Ore's property and taking larger tree representatives we may assume that $t=t_1=\cdots=t_k$.
Hence, $x_i=[t,j_i]$ with $t$ a tree with $n$ leaves and $i\mapsto j_i$ is increasing from $\{1,\cdots,k\}$ to $\{1,\cdots,n\}.$
Every chain of $X$ is thus of this form for suitable $t$ and $j_i$.
Consider $g=[s\circ \pi ,t']\in G^V$.
Up to growing $t$ and $t'$ we may assume $t=t'$.
We obtain that $g\cdot x_i= [s, \pi(j_i)]$ for all $i$.
It is now clear that elements of $G$ preserves the order and elements of $G^T$ preserves the order up to cyclic permutations.
Conversely, if $\pi$ is nontrivial, then $g\notin G$ and there exists $i<j$ such that $\pi(j)<\pi(i)$. This implies that $\alpha_g([t,j])=[s,\pi(j)]\prec [s,\pi(i)]=\alpha_g([t,i])$ while $[t,i]\prec [t,j]$.
Hence, $g$ does not preserve the order $\preceq.$
Similarly, we prove that if $\pi$ is not cyclic, then $g$ does not preserve at least one chain up to cyclic permutations.
\end{proof}
{\bf Classical case.}
Assume that $\mathcal F$ is the monochromatic free forest category and thus $G,G^T,G^V$ are equal to $F,T,V$.
We have a bijection between $X$ and finitely supported sequences $\{0,1\}^{(\mathbf{N})}$ but also with the dyadic rationals $[0,1)\cap \mathbf{Z}[1/2]$ of the half-open interval $[0,1)$.
A bijection from the second to the third space is given by the map
$$\{0,1\}^{(\mathbf{N})}\to [0,1], x\mapsto \sum_{n=1}^\infty \frac{x_n}{2^n}.$$
Equip $\{0,1\}^{(\mathbf{N})}$ with the lexicographical order and $[0,1)\cap \mathbf{Z}[1/2]$ with the usual order of the real line.
We obtain that $(X,\preceq)$ is isomorphic to these two totally ordered sets. Moreover, the bijections considered are isomorphisms of posets.
We recover that $F$ (resp.~$T$) corresponds to order-preserving (resp.~up to cyclic permutations) transformations.
\subsection{Transitivity properties}
\begin{proposition}\label{prop:Q-space}
Let $\mathcal F,G$ as above and write $\alpha:G^T\curvearrowright X$ the Jones action described above.
Define $X_k$ to be all subsets of $X$ of cardinal $k$ and $\alpha_k:G^T\curvearrowright X_k$ the action deduced from $\alpha$ for $k\geq 1.$
The following assertions are true.
\begin{enumerate}
\item For each $A\in X_k$ there exists a tree $t$ with $n$ leaves and some distinct natural numbers $1\leq j_1,\cdots,j_k\leq n$ satisfying $$A=\{ [t,j_i]:\ 1\leq i\leq k\}.$$
\item The action $\alpha_k$ is transitive for all $k\geq 1.$
\item If $A\in X_k$, then the fixed-point subgroup
$$\Fix(A):=\{g\in G^T:\ \alpha_g(x)=x, \text{ for all } x\in A\}$$
is isomorphic to $G^k$ and the stabiliser subgroup
$$\Stab(A)=\{g\in G^T:\ \alpha_g(A)=A\}$$
is isomorphic to $G^k\rtimes \mathbf{Z}/k\mathbf{Z}$ where $\mathbf{Z}/k\mathbf{Z}$ shifts indices modulo $k$.
\end{enumerate}
\end{proposition}
\begin{proof}
The first statement was shown in the proof of Proposition \ref{prop:order-preserving}.
Proof of (2).
Consider $k\geq 1$ and $A,B\in X_k$.
From the first item of the proposition we may write $A$ and $B$ as sets of the form $\{[t,j_1],\cdots,[t,j_k]\}$ and $\{[s,i_1],\cdots, [s,i_k]\}$ where both $t$ and $s$ are trees with the same number of leaves, say $n$.
Moreover, we may assume $1\leq j_1<j_2<\cdots<j_k\leq n$, $1\leq i_1<i_2<\cdots<i_k\leq n$ up to re-indexing the elements of $A$ and $B$.
Up to applying $g=t\circ \pi \circ t^{-1}\in G^T$ to $A$ where $\pi(j_1)=1$ is a cyclic permutation we may assume $j_1=1$ and similarly we may assume $i_1=1$.
We claim that there exists trees $t'$ and $s'$ with $n'$ leaves and a sequence $1=l_1<l_2<\cdots<l_k \leq n'$ satisfying that
$$(t,j_p)\sim (t', l_p) \text{ and } (s,i_p)\sim (s',l_p) \text{ for all } 1\leq p\leq k.$$
We proceed by induction on $k$.
If $k=1$, then we have nothing to do since $j_1=i_1=1.$
Assume $k\geq 2.$
Consider $j_2$ and $i_2$ and choose two trees $f_1$ and $h_1$ satisfying
$$|\Leaf(f_1)| + j_2 = |\Leaf(h_1)|+i_2=:l_2.$$
Now, choose again two trees $f_n$ and $h_n$ satisfying
$$|\Leaf(f_1)|+|\Leaf(f_n)|=|\Leaf(h_1)| + |\Leaf(h_n)|.$$
Finally, define the forests with $n$ roots $f=(f_1,\cdots,f_n)$ and $h=(h_1,\cdots,h_n)$ such that $f_j=h_j$ is trivial if $1\neq j\neq n$.
Observe that
$$(t,j_2)\sim (tf, l_2) \text{ and } (s,i_2)\sim (sf,l_2).$$
Moreover, $(t,j_1)\sim (tf,j_1)$ and $(s,i_1)\sim (s,l_1)$ since $j_1=i_1=1$.
Finally, $tf$ and $sh$ are two trees with the same number of leaves.
We continue this process in adding trees on top of $t$ and $s$ until all the indices match.
Using the claim we may assume that
$$A=\{ [t',l_p]:\ 1\leq p\leq k\} \text{ and } B=\{[s',l_p]:\ 1\leq p\leq k\}$$
where $t',s'$ are trees with the same number of leaves.
Taking $g:=s'\circ t'^{-1}\in G\subset G^T$ we deduce that $\alpha_k(g)(A)=B.$
We have proven that $\alpha_k$ is transitive.
Proof of (3).
Consider $k\geq 1$ and $A\in X_k$.
By (2) we may assume $A=\{ [t, p]:\ 1\leq p\leq k\}$ where $t$ is a tree with $k$ leaves.
Consider $g\in G^T$ and observe that it can always be decomposed as
$$g= tf \circ \pi \circ (th)^{-1}$$
with $f,h$ forests with $k$ roots and $\pi$ a cyclic permutation.
Assume that $g$ fixes each point of $A$.
We are going to show that the $j$th tree of $f$ and $h$ have the same number of leaves for each $j$.
Note that
$$g\cdot [t,p] = g\cdot [th,p^h] = [tf, \pi(p^h)],$$
see \eqref{eq:bicross} for the notation $p^h$.
Since $g\cdot [t,p]=[t,p]$ and $[t,p]=[tf,p^f]$ we deduce that
$$\pi(p^h)=p^f \text{ for all } 1\leq p\leq k.$$
Note that $1^h=1=1^f$ implying that $\pi(1)=1$ and thus $\pi$ is the trivial permutation (since it is cyclic and fixes one point).
Hence, $p^h=p^f$ for all $1\leq p\leq k$.
This implies that $|\Leaf(f_j)|=|\Leaf(h_j)|$ for all $1\leq j\leq p$.
Conversely, this condition assures that $tf\circ (th)^{-1}$ fixes each point of $A$.
We deduce
$$\Fix(A)=\{ tf\circ (th)^{-1}:f,h\}$$
where $f=(f_1,\cdots,f_k), h=(h_1,\cdots,h_k)$ and $|\Leaf(f_j)|=|\Leaf(h_j)|$ for all $1\leq j\leq k.$
We obtain the first statement of (3) by observing that the map
$$G^k\to \Fix(A), (f_j\circ h_j^{-1},\ 1\leq j\leq k)\mapsto tf\circ (th)^{-1}$$
realises a group isomorphism.
Now $\Stab(A)$ is generated by $\Fix(A)$ and a group $C$ of permutations of the elements of $A$.
Moreover, $\Fix(A)$ and $C$ forms a semidirect product $\Fix(A)\rtimes C$ where the action is deduced from $C\curvearrowright A$.
The only possible permutations are necessarily of the form $[t,p]\mapsto [t,p+n]$ for a fixed $n$ and where the index $p$ is considered modulo $k$.
These permutation are realised by $t\circ \pi\circ t^{-1}$ where $\pi$ a cyclic permutation of the leaves.
After identification of $\Stab(A)$ with $G^k$ we deduce the last statement.
\end{proof}
\subsection{A finiteness result}\label{sec:def-finiteness}
We recall few definitions of topological finiteness properties of group introduced by Wall \cite{Wall65}. There exist homological analogous that we don't present here as we did not find any applications with them. We refer the reader to the book of Geoghegan \cite[Sections 7 and 8]{Geoghegan-book} for details.
{\bf Classifying space and universal cover.}
Let $G$ be a group.
A classifying space $BG$ of $G$ (or $K(G,1)$-complex) is a (pointed) path-connected CW complex such that all its homotopy groups are trivial except the first one $\pi_1(BG,p)$ (the Poincar\'e group) which is isomorphic to $G$.
A universal cover $EG$ of $BG$ is necessarily contractible admitting a free (cellular) action $G\curvearrowright EG$ whose quotient $G\backslash EG$ is isomorphic to $BG$.
Conversely, every classifying space arise in that way.
All classifying spaces of a fixed group $G$ are pairwise homotopically equivalent but they may not be isomorphic as {\it complexes}.
Hence, we may define invariants of the group using properties of the family of its classifying spaces.
{\bf Topological finiteness properties.}
If $G$ is a group and $n\geq 0$, then we say that $G$ satisfies the {\it topological finiteness property} $F_n$ (or simply is of {\it type} $F_n$) if there exists a classifying space $BG$ with finite $n$-skeleton (i.e.~$BG$ has finitely many $k$-cells for all $k\leq n$).
Equivalently, $G$ is of type $F_n$ if there exists a contractible CW complex $EG$ on which $G$ acts freely and such that there are only finitely many $k$-cells modulo $G$ for each $k\leq n$.
The group $G$ is of type $F_\infty$ if it is of type $F_n$ for all $n\geq 1.$
Note that all groups are of type $F_0$ admitting a classifying space with one vertex and $F_1$ (resp.~$F_2$) is equivalent to be finitely generated (resp.~finitely presented).
{\bf Geometric dimension.}
The {\it geometric dimension} $\dim_g(G)$ of $G$ is the minimal dimension (as a complex) of a classifying space of $G$ which is either a natural number or infinity $\infty$.
It is increasing with respect to inclusion of groups and satisfies that $\dim_g(\mathbf{Z}^r)=r$ and $\dim_g(\mathbf{Z}/k\mathbf{Z})=\infty$ for any $k\geq 2.$
{\bf Exceptional properties of Thompson-like groups.}
The group $F$ was the first example of a torsion-free group with infinite geometric dimension and of type $F_\infty$ as proved by Brown and Geoghegan \cite{Brown-Geoghegan84}.
Since the result of Brown-Geoghegan there have been many groups similar to $F$ that have been proved to be of type $F_\infty$.
Here is a list that is very far from being exhaustive which includes the family of groups: $T,V$, Higman-Thompson's groups, Stein's groups, Brin's higher dimensional Thompson's groups $dV$, Brin and Dehornoy's braided Thompson's group $BV$, and large subclasses of Hughes' FSS groups, Guba-Sapir's diagram groups, Rover-Nekrashevych's groups, cloning system of groups, operad groups, and a number of other generalisations \cite{Brown87, Stein92,Kochloukova-Martinez-Perez-Nucinkis13, Fluch-Marschler-Witzel-Zaremsky13, Bux-Fluch-Marschler-Witzel-Zaremsky16,Farley-Hughes15,Witzel-Zaremsky18, Thumann17}.
We refer to a recent preprint of Skipper and Wu for additional examples and references \cite{Skipper-Wu21}.
For all these families of groups the scheme of the proof is based on two technical results: Brown's criterion and Bestvina-Brady's discrete Morse theory \cite{Brown87,Bestvina-Brady97}.
It relies on finding a suitable filtration of a CW complex (often a simplicial one) and computing connectivity of links rather than finding a CW complex with finite skeleton (which is rare to find in practice).
We refer the reader to the article of Zaremsky for a very clear and pedagogical explanation on this strategy \cite{Zaremsky21}.
Even though many families of Thompson-like groups have been proved to be either not finitely presented or of type $F_\infty$ there exist groups satisfying properties in between.
Indeed, for any $n\geq 1$ there exists a Thompson-like groups that is of type $F_n$ but not of type $F_{n+1}$ using a construction and result of Tanushevski \cite{Tanushevski16}.
This is obtained by considering diagrams of monochromatic binary forests and by decorating their leaves with elements of a group with the desirable finiteness property.
Although, all these examples are nontrivial split extension of Thompson group ($F,T$, or $V$) and are thus not simple.
Skipper, Witzel, and Zaremsky have constructed a sequence of Rover-Nekrashevych groups that are of type $F_n$ but not of type $F_{n+1}$ and moreover are simple (giving the first family of this kind), see \cite{Skipper-Witzel-Zaremsky19} and also the family of groups constructed by Belk and Zaremsky \cite{Belk-Zaremsky22}.
Note that all forest groups have infinite geometric dimension since they contain a copy of $F$, see Corollary \ref{cor:Finside}.
Although, it is a delicate question to decide if they have finiteness properties.
Indeed, finiteness properties (topological or homological) fail to be close under most permanence properties.
{\bf A result.}
We adapt a clever strategy due to Brown and Geoghegan explained in \cite[Section 4B, Remark 2]{Brown87} and in the book \cite[Theorem 9.4.2]{Geoghegan-book} to deduce that $G^T$ is of type $F_n$ when $G$ is.
To do this we use a criteria stated by Brown \cite[Proposition 1.1]{Brown87} on homological finiteness property which we simplify and translate to our need.
\begin{proposition}\label{prop:stab-F-infty}
Consider $n\geq 1$, a cellular action $G\curvearrowright C$ on a contractible CW-complex assuming that the stabiliser of each cell is a group of type $F_n$ and $C$ has finitely many $k$-cells modulo $G$ for each $k\geq 0.$ Then $G$ is a group of type $F_n$.
\end{proposition}
\begin{theorem}\label{theo:T-F-infty}
Consider $n\geq 1$, a Ore forest category $\mathcal F$, and $G:=\Frac(\mathcal F,1), G^T:=\Frac(\mathcal F^T,1)$ the associated $F$ and $T$-forest groups, respectively.
If $G$ is of type $F_n$, then so is $G^T.$
\end{theorem}
\begin{proof}
Consider $n,G,G^T$ as above and assume $G$ is of type $F_n$.
Let $\ell^2(G^T/G)$ be the real Hilbert space with standard orthonormal basis $B$ indexed by the set $G^T/G$.
Define $C\subset \ell^2(G^T/G)$ to be the simplicial complex whose $k$-simplices are the convex hull of $k+1$ distinct elements of the basis $B$.
The complex $C$ is contractible and moreover the action $G^T\curvearrowright G^T/G$ induces a simplicial action $G^T\curvearrowright C$.
Using item 2 of Proposition \ref{prop:Q-space}, we deduce that $C$ has one $k$-simplex modulo $G^T$ for each $k\geq 0$.
Moreover, the stabiliser of a $k$-simplex is isomorphic to $G^{k+1}\rtimes \mathbf{Z}/(k+1)\mathbf{Z}$ by item 3 of the same proposition.
This later group is of type $F_n$ since $G$ is.
We deduce from Proposition \ref{prop:stab-F-infty} that $G^T$ is of type $F_n.$
\end{proof}
\begin{remark}\label{rem:FPm}
An {\it homological} version of this last theorem can be given by replacing $F_n$ by $FP_n$ where $FP_n$ stands for having a projective $\mathbf{Z} G$-resolution of $\mathbf{Z}$ that is finitely generated in dimension smaller than $n$, see \cite[Section 8]{Geoghegan-book} for details.
Note that $F_n$ always implies $FP_n$ for a group $G$ (take the augmented cellular chain complex of the universal cover of a classifying space of $G$ which is a {\it free} $\mathbf{Z} G$-resolution of $\mathbf{Z}$).
The converse holds for $n=1$ and for all $n\geq 2$ providing $G$ is finitely presented.
Hence, new applications in the homological setting would only happen for forest groups $G$ that are not finitely presented but are of type $FP_n$ for a certain $n\geq 2$ (thus necessarily finitely generated).
We have not witnessed any forest groups with these properties so far.
\end{remark}
\section{Thumann's operad groups and a finiteness theorem}\label{sec:Thumann}
In his PhD thesis Thumann introduced the class of {\it operad groups} and proved that a large collection of them are of type $F_\infty$.
Thumann's theorem recovers at once a number of result of the literature by adapting and extending their techniques of proof in a unified categorical framework \cite{Thumann17}.
\subsection{Operad groups and forest groups}
In this section we present Thumann's operad groups and explain that forest groups are particular case of those.
We do it for the ordinary $F$-case and briefly explain why the $V$ and $BV$ versions are operad groups as well.
We use Thumann's notations and terminology and rephrase them in our formalism.
This allows the reader to easily find additional details in Thumann's article and to present notions and results in the two languages (which are significantly different).
Consider a forest category $\mathcal F$ which we recall is a monoidal category $(\mathcal F,\circ,\otimes)$ with set of objects $\mathbf{N}$ and morphisms the forests that we compose by vertical stacking.
A forest with $\ell$ leaves and $r$ roots is a morphism from $\ell$ to $r$ or an {\it arrow} $\ell\to r$.
\subsubsection{Operad, colours, and operations.}
In Thumann's framework \textit{colour} means something else than colours for vertices of forests in our framework.
Hence, we write \textit{Op-colour} when we refer to the colour in the operad group context.
An {\it operad} $\mathcal O$ is a set of \textit{Op-colours} together with sets of \textit{operations} $\mathcal O(a_1,\cdots,a_n;b)$ with $a_i,b$ Op-colours so that $a_i$ are the input and $b$ the output Op-colours.
Such an operation is represented (like string diagrams in Jones' planar algebras or in tensor categories) by a triangle with $n$ horizontal lines on the left and one on the right corresponding to the $n$ inputs and the one output.
Here is an example when $n=3$:
\[\includegraphics{operad.pdf}\]
They can be composed associatively by concatenating diagrams horizontally.
Moreover, there is an identity for each Op-colour $1_a\in \mathcal O(a;a).$
{\bf Forest case.}
Define $\mathcal O_\mathcal F$ to be the operad having a single Op-colour $*$ (i.e.~it is {\it Op-monochromatic}) so that the set of operations $\mathcal O_\mathcal F(a_1,\cdots,a_n;b)$ with $a_1=\cdots=a_n=b=*$ is equal to the set of trees with $n$ leaves.
Our composition of forests correspond to the composition of the operads.
The identity $1_*\in\mathcal O_\mathcal F(*,*)$ is the trivial tree $I$.
Therefore, $\mathcal O_\mathcal F$ corresponds to the set of trees of $\mathcal F$ equipped with the composition of trees with elementary forests.
\subsubsection{Colour-tame.}
Thumann requires that operads are {\it colour-tame} in order to apply his machinery.
He notes that this may not be necessary but for technical reasons we must assume it.
We will not define it but simply notice that if an operad is Op-monochromatic), then it is automatically colour-tame.
In particular, the operad $\mathcal O_\mathcal F$ associated to any forest category is colour-tame.
\subsubsection{Monoidal category associated to an operad}
Given an operad $\mathcal O$ we consider $\mathcal{S}(\mathcal O)$ the monoidal category with object finite lists of Op-colours and morphisms finite lists of operations. The tensor product is the juxtaposition.
{\bf Forest case.}
A finite list of Op-colours in $\{*\}$ is a natural number and a finite list of trees is a forest. Juxtaposition corresponds to the tensor product of forests.
We deduce that $\mathcal{S}(\mathcal O_\mathcal F)$ and $\mathcal F$ are isomorphic as monoidal categories.
\subsubsection{Adding permutations or braids}
Thumann defines three different types of operads: {\it planar, symmetric, and braided} operads.
The first is the one we just defined, the second consists in adding permutation, and the third braids.
Graphically, we now consider the diagram of a braid drawn horizontally followed to the right by an operation where the strands of the braids are connected to the inputs of the operation.
For permutations, we do the same except that under and over crossing are replace by a single type of crossing (corresponding to the usual maps $B_n\twoheadrightarrow \Sigma_n$).
Formally, this can be achieved using a Brin-Zappa-Sz\'ep product of the category $\mathcal{S}(\mathcal O)$ with the groupoid of symmetries or braids.
Hence, any planar operad can produce three different categories: planar, symmetric, and braided.
{\bf Forest case.}
For forest categories we have proceed in a very similar way. Given a forest category $\mathcal F$ we can add symmetries or braids obtaining $V$ and $BV$-forest categories.
Hence, $F,V,BV$-forest categories correspond to planar, symmetric, and braided operads.
Note that Thumann did not consider the $T$-case corresponding in incorporating only cyclic permutations.
We are mainly focusing on the $F$-case in this article. Although, we will translate results for the $F,V,$ and $BV$-cases since it is easy to do from Thumann's formalism.
Moreover, using Theorem \ref{theo:T-F-infty} we will deduce results for the remaining $T$-case.
\subsubsection{Operad groups}
Given a list of Op-colours $Y$ we define the {\it operad group}
$$\pi_1(\mathcal O,Y):=\pi_1(\mathcal{S}(\mathcal O),Y)),$$
equal to all loops in $\mathcal{S}(\mathcal O)$ starting (and thus ending) at $Y$ that we equip with the composition of path.
This means formal compositions of morphisms and their inverses in $\mathcal{S}(\mathcal O)$.
If $\mathcal{S}(\mathcal O)$ admits a {\it cancellative calculus of fractions}, then elements of $\pi_1(\mathcal{S}(\mathcal O),Y)$ are nothing else than $[f,g]=f\circ g^{-1}$ with $f,g$ list of operations ending at $Y$.
The multiplication and inverse are the obvious ones: $[f,g]\circ [g,h]=[f,h]$ and $[f,g]^{-1}=[g,f].$
{\bf Forest case.}
Cancellative calculus of fractions is the same notion of being a Ore category in our sense (i.e.~left-cancellative and satisfying Ore's property).
Assume we are in this case.
A list of Op-colours $Y$ is a natural number $n$ and the operad group $\pi_1(\mathcal O_\mathcal F,Y)$ corresponds to the group of fractions $[f,g]$ of forests $f,g$ having both $n$ roots.
We deduce that $\pi_1(\mathcal O_\mathcal F,Y)$ is the forest group group $\Frac(\mathcal F,n)$.
If we consider the symmetric or braided version of $\mathcal O_\mathcal F$ we obtain the $V$ and $BV$-forest groups, respectively.
\subsubsection{Degree and transformation}
{\bf Degree}
An operation of {\it degree} $d$ is an operation with $d$ inputs.
The category $\mathcal I(\mathcal O)$ with object the Op-colours and morphism all the degree 1 operations is considered.
For technical reasons it is assumed that $\mathcal I(\mathcal O)$ is a groupoid and later we will ask that it is of type $F_\infty^+$ (a groupoid where any subgroup of any of its isotropy group is of type $F_\infty$).
{\bf Transformation.}
A {\it transformation} of $\mathcal{S}=\mathcal{S}(\mathcal O)$ is a list of morphisms of $\mathcal I=\mathcal I(\mathcal O)$ forming a subcategory $\mathcal T=\mathcal T(\mathcal O)$ of $\mathcal{S}$.
Since $\mathcal I$ is assumed to be a groupoid we deduce that $\mathcal T$ is a groupoid as well.
We consider $\mathcal T\mathcal C=\mathcal T\mathcal C(\mathcal O)$ equal to the set of all operations of $\mathcal O$ mod out by $\mathcal T.$
Elements of $\mathcal T\mathcal C$ are thus classes of operations modulo the transformations.
Compositions defines a partial order for operations that passes through the quotient by transformations.
Moreover, the degree map $\deg$ defined as the number of inputs of an operation passes through the quotient too.
We obtain that $\deg:(\mathcal T\mathcal C,\leq)\to (\mathbf{N},\leq)$ is strictly increasing, i.e.~$x<y$ implies $\deg(x)<\deg(y).$
We say that $(\mathcal T\mathcal C,\leq,\deg)$ is a {\it graded poset}.
{\bf Forest case.}
Now, a degree $d$ operation of $\mathcal O_\mathcal F$ is a tree with $d$ leaves.
Hence, $\mathcal I=\mathcal I(\mathcal O_\mathcal F)$ is the category with one object $*$ and one morphism: the trivial tree $I$. It is the trivial group and in particular is a groupoid of type $F_\infty^+$ whatever $\mathcal F$ is and whatever we work with the $F,V$, or $BV$-version.
The groupoid $\mathcal T$ is then equal to the collection of trivial forests $\{I^{\otimes n}, n\geq 1\}$ that is the groupoid having for set of object $\mathbf{N}$, no morphisms between distinct natural numbers, and one single endomorphism for each $n\in \mathbf{N}$.
A transformation is thus a trivial forest.
Once again this does not change for the other versions $V$ and $BV$ (hence permutations and braids are {\it not} in $\mathcal T$).
An operation of $\mathcal O_\mathcal F$ is a tree. Multiplying by a trivial forest (to the right for our conventions and to the left for the conventions of Thumann) does not change it.
Hence, $\mathcal T\mathcal C$ is in fact equal to the set of operations of $\mathcal O_\mathcal F$ that is the set of trees.
The partial order is the usual one: $t\leq t\circ f$ that we have considered.
The degree map is the initial map or number of leaves map also considered as the origin map $\omega$:
$$\deg(t):=|\Leaf(t)|=\omega(t).$$
The graded poset $(\mathcal T\mathcal C,\leq,\deg)$ is the set of trees equipped with the same partial order $t\leq t\circ f$ we have defined and the number of leaves map $t\mapsto |\Leaf(t)|.$
For the other cases $V,BV$ one should interpret tree by a tree with possibly a permutation or a braid on top.
\subsubsection{Spine, elementary transformation classes, finitely generated, and finite type}
{\bf Spine of a graded poset.}
Consider a graded poset $(P,\leq,\deg)$ and its subset $M$ of minimal elements.
We say that a subset $Sp\subset P$ is the {\it spine} of $P$ if it is the smallest subset $Sp\subset P$ satisfying that $M\subset Sp$ and for all $v\in P$ the set $\{s\in Sp:\ s<v\}$ contains a greatest element, i.e.~comparable and larger with all other element in the set of above.
Note that the spine is unique if it exists.
Thumann proved that every graded poset admits a spine, see \cite[Section 3.4.1]{Thumann17}.
To prove it he constructs explicitly the spine providing a convenient equivalent definition.
The construction is as follows: let $Sp_0$ be the minimal elements of $P$.
Given $x,y\in P$ distinct define the set $M_0(x,y)$ of $z\in P$ larger than both $x$ and $y$.
Now, define $M(x,y)$ to be the subset of all minimal elements of $M_0(x,y)$.
Note that $M(x,y)$ could be empty if the poset is not directed.
Let $Sp_1:= \cup_{(x,y)} M(x,y)$ be the union of all $M(x,y)$ where $x,y$ are in $Sp_0$ and $x\neq y$.
Define inductively $Sp_{n+1}:=\cup_{(x,y)} M(x,y)$ for $n\geq 1$ where the union is over the pairs $(x,y)\in Sp_n\times Sp_n$ so that $x\neq y$.
Finally, the spine of $P$ is equal to the union $\cup_{n\geq 0} Sp_n.$
{\bf Spine of the graded poset associated to an operad.}
Define $\mathcal T\mathcal C^*(\mathcal O)$ to be full subposet of $\mathcal T\mathcal C(\mathcal O)$ spanned by the degree classes of at least 2.
Minimal elements of $(\mathcal T\mathcal C^*(\mathcal O),\leq)$ form the collection of {\it very elementary transformation classes} denoted $VE$.
The spine of the poset $(\mathcal T\mathcal C^*(\mathcal O), \leq,\deg)$ forms the collection of {\it elementary transformation classes} denoted $E$.
{\bf Finitely generated operads and operads of finite type.}
The operad is \textit{finitely generated} if $VE$ is finite and of \textit{finite type} if $E$ is finite.
{\bf Forest case.}
{\it Spine.}
Consider again a forest category $\mathcal F$ and the graded poset $(\mathcal T\mathcal C(\mathcal O_\mathcal F),\leq,\deg)$ of above.
We have that $\mathcal T\mathcal C^*(\mathcal O_\mathcal F)$ is the set of all nontrivial trees: the set of trees minus $I$.
Moreover, observe that if $x,y$ are trees, then $M(x,y)$ corresponds to the minimal common right-multiples of $x$ and $y$.
The subset of minimal elements of $\mathcal T\mathcal C^*(\mathcal O_\mathcal F)$ is the set $\mathcal F(2,1)$ of all trees with two leaves that we call the set of abstract colours of $\mathcal F$.
Hence, the spine of the graded poset is obtained by taking minimial common right-multiples of trees starting from $\mathcal F(2,1)$.
We rephrase this construction using composition rather than the poset structure and call {\it spine of $\mathcal F$} the set equal to the spine of the graded poset $(\mathcal T\mathcal C(\mathcal O_\mathcal F),\leq,\deg)$.
Write $\cm(x,y)$ for the common (right-)multiples of both $x$ and $y$ and $\mcm(x,y)$ for the minimal elements of $\cm(x,y)$.
\begin{definition}\label{def:spine}
Let $\mathcal F$ be a forest category.
Define the following sequence of sets:
$$\Sp_0(\mathcal F) =\mathcal F(2,1) ,\ \Sp_{n+1}(\mathcal F) := \bigcup_{f,g\in \Sp_n(\mathcal F), f\neq g}\mcm(f,g) \text{ for all } n\geq 0.$$
The \textit{spine} $\Sp(\mathcal F)$ of $\mathcal F$ is the union
$$\Sp(\mathcal F):=\bigcup_{n\geq 0} \Sp_n(\mathcal F).$$
\end{definition}
{\it Elementary transformation.}
The very elementary transformations are the set of trees of two leaves: the abstract colour set.
The elementary transformations are the elements of $\Sp(\mathcal F)$.
{\it Finitely generated.}
The operad $\mathcal O_\mathcal F$ is finitely generated if and only if $\mathcal F$ has finitely many abstract colours.
{\it Finite type.}
The operad $\mathcal O_\mathcal F$ is of finite type if and only if the spine of $\mathcal F$ is finite.
This last property will be crucial in applying Thumann's theorem.
\subsection{Topological finiteness properties}
We state Thumann's theorem regarding finiteness properties of operad groups that we translate and specialise to forest categories.
We explain the limit of this theorem by exhibiting one striking example.
\subsubsection{Thumann's theorem}
Here is Thumann's main theorem of \cite{Thumann17}.
\begin{theorem}\label{theo:Thumann}
Consider an operad $\mathcal O$ (that is either planar, symmetric, or braided).
If the following five conditions are satisfied:
\begin{enumerate}
\item it has finitely many Op-colours;
\item it is colour-tame;
\item it admits a cancellative calculus of fractions;
\item it is of finite type;
\item the category $\mathcal I(\mathcal O)$ is a groupoid of type $F_\infty^+$,
\end{enumerate}
then for any object $Y$ of $\mathcal{S}(\mathcal O)$ the operad group $\pi_1(\mathcal O,Y)$ is of type $F_\infty$.
\end{theorem}
Consider now a Ore forest category $\mathcal F$ and its $X$-forest group $\Frac(\mathcal F^X,1)$ with $X$ being either $F,V,$ or $BV$.
Its associated operad $\mathcal O_\mathcal F$ is by definition Op-monochromatic and thus satisfies the first two items.
By assumption it is a Ore category and thus satisfies the third item.
The category $\mathcal I(\mathcal O_\mathcal F)$ is equal to the trivial group and thus satisfies the last item.
Note that the forest group $\Frac(\mathcal F^X,1)$ is isomorphic to $\pi_1(\mathcal O_\mathcal F, Y)$ where $Y$ is the list with one element which is the unique Op-colour of $\mathcal O_\mathcal F$ and $X=F,V,BV$ when $\mathcal O_\mathcal F$ is planar, symmetric, or braided, respectively.
Finally, recall that $\mathcal O_\mathcal F$ is of finite type if and only if the spine of $\mathcal F$ is finite.
We obtain the following corollary for forest groups.
\begin{corollary}\label{theo:Finfty-withoutT}
Let $\mathcal F$ be a Ore forest category.
If the spine of $\mathcal F$ is finite, then the forest group $\Frac(\mathcal F^X,1)$ is of type $F_\infty$ for $X\in\{F,V,BV\}.$
\end{corollary}
Using Theorem \ref{theo:T-F-infty} we deduce the following result encompassing the $T$-case.
\begin{theorem}\label{theo:Finfty}
Let $\mathcal F$ be a Ore forest category and let $X$ be $F,T,V,$ or $BV$.
If the spine of $\mathcal F$ is finite, then the forest group $\Frac(\mathcal F^X,1)$ is of type $F_\infty$.
\end{theorem}
We now deduce that a forest presentation with two colours and one relation provides a forest group (when it exists) of type $F_\infty$. This produces a very large class of examples for which the theorem above applies. However, it has the weakness to not prove {\it existence} of the forest group.
In the next section we will present a class of forest presentations for which forest groups always exist.
\begin{corollary}
Consider a presented forest category
$$\mathcal F=\FC\langle a,b| t=s\rangle$$
with two colours and one relation $t=s$ so that $t$ and $s$ have their roots coloured by $a$ and $b$, respectively.
If $\mathcal F$ satisfies Ore's property, then it is a Ore category whose fraction group $\Frac(\mathcal F,1)$ is of type $F_\infty$ and so are its $T,V$ and $BV$-versions.
\end{corollary}
\begin{proof}
This is a combination of Corollary \ref{cor:presentation-free} and Theorem \ref{theo:Finfty}.
Consider $\mathcal F$ as above and moreover assume it satisfies Ore's property.
Having $t$ and $s$ with roots of different colours implies that the forest presentation $(a,b|t=s)$ is complemented and complete. We deduce that $\mathcal F$ is left-cancellative.
Therefore, $\mathcal F$ is a Ore category and thus the fraction group $\Frac(\mathcal F,1)$ can be constructed.
Now, we show that the spine of $\mathcal F$ has at most three elements.
Indeed, by definition $\Sp_0(\mathcal F)=\{Y_a,Y_b\}$.
Now, $\Sp_1(\mathcal F)=\mcm(Y_a,Y_b)=\{t\}$ is a singleton.
Since $\Sp_1(\mathcal F)$ is a single point we obtain that $\Sp_2(\mathcal F)$ is empty and thus so are all the $\Sp_n(\mathcal F)$ with $n\geq 3.$
Therefore, $\Sp(\mathcal F)=\{Y_a,Y_b,t\}$ which is finite.
We can now apply the last theorem which implies that $\Frac(\mathcal F,1)$ is of type $F_\infty$ and so are its $T,V$, and $BV$-versions.
\end{proof}
\begin{remark}\label{rem:Stein-Finfty}
\begin{enumerate}
\item Since we work over binary forests there is always a morphism between $n$ and $m$ inside the fraction groupoid $\Frac(\mathcal F)$.
This implies that all the forest groups $(\Frac(\mathcal F,n),\ n\geq 1)$ are pairwise isomorphic.
This is why we state the corollary only for the specific forest group $\Frac(\mathcal F,1)$ with $n=1$.
\item
Observe that if $\mathcal F=\FC\langle a,b| t=s\rangle$ with $t,s$ trees with {\it three} leaves and roots of different colours, then automatically $\mathcal F$ is a Ore category.
By Corollary \ref{cor:presentation-free} it is left-cancellative and by Corollary \ref{cor:Ore-FC} it satisfies Ore's property.
This only provides very few examples (in fact four of them with two of them isomorphic to $F$) but has the merit to cover the Cleary irrational-slope Thompson group and the {\it bicoloured} description of the ternary (Brown-)Higman-Thompson $F_{3,1}$, see Section \ref{sec:example}.
\item
The discussion of this section can be generalised to forest categories $\mathcal F$ with interior vertices having various valencies (corresponding to the extension of Stein of the groups of Higman and Brown).
The only difference resides in the definition of the spine of $\mathcal F$.
Indeed, for this more general case consider $\Sp_0(\mathcal F)$ to be the set of all {\it minimal trees with at least} two leaves rather than trees with {\it exactly} two leaves.
Then $\Sp_{n+1}(\mathcal F)$ is defined as before as the union of the $\mcm(x,y)$ with $x\neq y$ in $\Sp_n(\mathcal F)$ and $\Sp(\mathcal F)=\cup_{n\geq 1} \Sp_n(\mathcal F).$
If this spine is finite and $\mathcal F$ is a Ore category, then $\Frac(\mathcal F^X,n)$ is a group of type $F_\infty$ for $X=F,T,V,BV$ and any $n\geq 1$.
In this case it does make sense to consider various $n$ in contrary to the binary case as the groups indexed by $n$ may produce several non-isomorphic groups.
\end{enumerate}
\end{remark}
\subsubsection{Strategy and limits of Thumann's theorem}\label{sec:rebel}
{\bf Strategy of Thumann.}
The strategy of Thumann resides in considering the exact same complex than our: $E\mathcal F$ the geometric realisation of the ordered complex of the directed poset $(Q,\leq)$ where $Q$ is the set of $[t,f]$ with $t$ a tree and $f$ a forest.
He considers the origin map $\omega:Q\to\mathbf{N}, [t,f]\mapsto |\Root(f)|$ that is extended as a Morse function $\omega:E\mathcal F\to\mathbf{R}_+$ by making it affine on each simplex.
This map provides a filtration $E\mathcal F_j:=\omega^{-1}([0,j])$ that is compatible with the (left) action of $G:=\Frac(\mathcal F,1)$.
Now, $G\backslash E\mathcal F_j$ is a finite complex and thus by Brown's criterion we have that $G$ is of type $F_\infty$ if the connectivity of the pair $(E\mathcal F_{j+1},E\mathcal F_j)$ tends to infinity in $j$.
Using Bestvina-Brady discrete Morse theory it is then sufficient to prove that the descending links $\link_{\downarrow}(v)$ in $E\mathcal F$ of vertices $v\in E\mathcal F^{(0)}=Q$ has its connectivity tending to infinity when $\omega(v)$ tends to infinity.
This last part is the technical core of the proof.
One can show that $\link_{\downarrow}(v)$ only depends on $\omega(v)=j$ and is isomorphic to some complex $L_j$.
Now, one can decompose $L_j$ with respect to subcomplexes related to $L_k$ for $k<j$.
The Hurewicz theorem and the Mayer-Vietoris sequence give some lower bound for the connectivity of $L_j$ with respect to the connectivity of some $L_k$ with $k<j$.
When good hypothesis on $\mathcal F$ are added (in this case: its spine is finite) one can deduce that the connectivity of $L_j$ tends to infinity and thus $G$ is of type $F_\infty$.
Note, this proof is performed separately for the three cases of planar, symmetric, and braided operad groups corresponding to $\mathcal F$ and its $V$ and $BV$ versions, respectively.
{\bf Garside family and Witzel's theorem.}
There is another important strategy that can be used for this kind of groups.
The idea is to find a smaller subcomplex of $X\subset E\mathcal F$ that is stabilised by $G$ and still contractible (so a universal cover of a classifying space of $G$).
We then apply a similar strategy to $X$ using Brown's criterion and Bestvina-Brady discrete Morse theory.
If $X$ is significantly smaller than $E\mathcal F$, then proving that the connectivity of descending links are getting large may be easier to do.
Dehornoy defined Garside family for monoids and more generally for certain small categories \cite{Dehornoy-book}.
Witzel proved (in greater generality) the key result that if we keep the same vertex set $Q$ but allow only simplices of the form $x<xf_1<\cdots<xf_n$ where all $f_j$ are in the Garside family (or equivalently $f_n$ is in the Garside family), then we obtain a subcomplex $X\subset E\mathcal F$ that is $G$-closed and still {\it contractible} \cite{Witzel19}.
One can then apply the strategy of above to $X$ to deduce finiteness properties of $\mathcal F$.
Of course, the smaller the Garside family is, the smaller $X$ is, and the better is the theorem of Witzel for doing computations.
Using Dehornoy's work we have that every Ore forest category $\mathcal F$ admits a {\it smaller} Garside family $\Gar(\mathcal F)$.
It is the closure of the trees with two leaves under the operation of taking minimal common multiples and right-divisors.
In particular, $\Gar(\mathcal F)$ contains the spine $\Sp(\mathcal F)$ of $\mathcal F$.
Hence, if the spine is infinite, then we cannot apply Thumann's result but we can still use Witzel strategy by taking the subcomplex $X$.
The complex $X$ will be still quite large since $\Gar(\mathcal F)$ is infinite but we may hope to simplify a number of computations.
{\bf One rebel example.}
We present one example of forest category that has a small forest presentation with two colours and two relations of length two, provides a forest group very similar to $F$, but does not fulfil the assumption of Thumann's theorem and for which Witzel's theorem does not help.
Consider the forest category $$\mathcal F=\FC\langle a,b| a_{1,1}b_{1,2} = b_{1,1} a_{1,2}, \ a_{1,1}a_{1,2}=b_{1,1}b_{1,2}\rangle.$$
This example is inspired by a thin monoid provided by Dehornoy in \cite[Example 4.4]{Dehornoy02}.
It is not hard to prove that $\mathcal F$ is left-cancellative and satisfies Ore's property.
A quick computation shows that $\Sp_n(\mathcal F)$ is never empty for every $n$ implying that $\Sp(\mathcal F)$ is infinite.
Hence, we cannot use Thumann's theorem.
Moreover, following an argument of Dehornoy we have that $\Gar(\mathcal F)$ is as large as possible and equal to the whole category $\mathcal F$. Hence, the subcomplex $X$ is equal to $E\mathcal F$ in this case.
However, we will see in a future article that its forest group $G:=\Frac(\mathcal F,1)$ is of type $F_\infty$ using a different complex that is inspired by the work of Tanushevski and Witzel-Zaremsky \cite{Tanushevski16,Witzel-Zaremsky18}.
Moreover, we will show that $G$ decomposes as a wreath-product of the form $\mathbf{Z}_2\wr_X F$ where $F\curvearrowright X$ is the usual action of $F$ on the dyadic rationals of $[0,1)$.
The group $G$ can be described as the usual Thompson group $F$ using pairs of trees (with a single type of caret) but where leaves are labelled by $0$ or $1$ (the elements of $\mathbf{Z}/2\mathbf{Z}$).
From this point of view it is somehow the most elementary forest group strictly larger than $F$ that we could think of.
However, it has infinite spine and its Garside family $\Gar(\mathcal F)$ si equal to $\mathcal F$ preventing the use of two very general theorems on finiteness properties.
\section{A large class of forest groups}\label{sec:class-example}
As stressed before, given a monoid or more generally a category it is a difficult task to check if it is left-cancellative or satisfies Ore's property.
In this section, we will provide a large class of examples of forest categories that are all Ore categories and thus produce forest groups.
These categories have forest presentations of a particular kind built from a family of {\it monochromatic} trees.
The monochromaticity makes possible to prove at once that they all satisfy the property of Ore.
Moreover, using a general criteria of Dehornoy we prove they are left-cancellative.
Finally, their spine is as small as it could be: equal to the set of trees with two leaves and a single additional tree.
We start by giving the general construction and then specialise to the two colours case.
\subsection{General construction}
Recall that $\mathcal{UF}\langle S\rangle$ denotes the free forest category over a set $S$ and $\mathcal{UF}\langle *\rangle$ the {\it monochromatic} free forest category where $*$ denotes its unique colour.
\begin{observation}
Let $t\in\mathcal{UF}\langle*\rangle$ be a monochromatic tree and let $\mathcal{UF}\langle*\rangle(t)\subset \mathcal{UF}\langle*\rangle$ be the quasi-forest subcategory generated by $t$ as defined in Section \ref{sec:forest-sub} (i.e.~all forests made of $I$ and $t$).
Define inductively the following sequence of monochromatic trees $(t_n)_{n\geq 1}$ so that $t_1=t$ and $t_{n+1}=t_n \circ f_n$ where $f_n$ is the forest composable with $t_n$ whose each tree is equal to $t$ (hence $t_{n+1}$ is obtained from $t_n$ by attaching to each of its leaf a copy of $t$).
We have that for any tree $s\in\mathcal{UF}\langle*\rangle$, there exists a tree $z\in\mathcal{UF}\langle*\rangle(t)$ satisfying $s\leq z$.
In particular, the sequence $(t_n)_{n\geq 1}$ is cofinal in the set of trees of $\mathcal{UF}\langle*\rangle$ (i.e.~for all tree $s\in \mathcal{UF}\langle*\rangle$ there exists $n\geq 1$ so that $s\leq t_n$ and $t_k\leq t_l$ for all $1\leq k\leq l$).
\end{observation}
\begin{proof}
Consider two monochromatic trees $t,s\in\mathcal{UF}\langle*\rangle$ assuming that $t$ is nontrivial.
We prove the observation by induction on the number of leaves of $s$.
If $s$ has only one caret, then it is smaller than $t$ since $t$ is nontrivial.
Note, this obvious fact uses in a crucial way that $\mathcal{UF}\langle*\rangle$ is monochromatic.
If not, write $s$ as $s_0\circ f$ where $f$ is an elementary forest with one caret say at the $j$th root of $f$.
By the induction assumption we have that $s_0\leq z$ for a certain $z\in\mathcal{UF}\langle*\rangle(t)$.
We can decompose $z$ as $s_0\circ h$.
Now, if the $j$th tree of $z$ is nontrivial, then $z\geq s$ since $h\geq f$.
If not, attach to the $j$th root of $h$ the tree $t$.
We obtain a larger tree in $\mathcal{UF}\langle*\rangle(t)$ which is larger than $s$.
\end{proof}
We now define our class of examples.
Fix a nonempty index set $S$ and let $(\tau_a:\ a\in S)$ be a family of nontrivial monochromatic trees of $\mathcal{UF}\langle *\rangle$ such that all of them have the same number of leaves.
Let $$C_a:\mathcal{UF}\langle*\rangle\to \mathcal{UF}\langle S\rangle, Y\mapsto Y_{a}$$
be the colouring map consisting in colouring all vertices of a monochromatic forest of $\mathcal{UF}\langle*\rangle$ with the colour $a$.
Define $\mathcal F_\tau=\mathcal F_{(\tau_a:\ a\in S)}$ to be the forest category with colour set $S$ and forest relations $(C_a(\tau_a), C_{b}(\tau_{b}))$ for all $a\neq b$ in $S$.
We may write $\mathcal F=\mathcal F_\tau$ if the context is clear.
\begin{remark}
Note that we do not require $a\in S\mapsto \tau_a$ to be injective.
Hence, even if there are only finitely many monochromatic trees with a fixed number of leaves we have no limitation on the cardinal of $S$ since we may choose to have $a\mapsto \tau_a$ constant.
\end{remark}
Here is a very satisfactory theorem that provides a huge family of examples and whose proof is rather short. It was discovered after proving that various one parameter families of forest categories (typically with two colours and one relations) were Ore categories.
\begin{theorem}\label{theo:class-example}
Let $S$ be a nonempty set and $\tau:S\to\mathcal{UF}\langle*\rangle$ a map from $S$ into the set of monochromatic trees.
Assume that all the trees $\tau_a$ with $a\in S$ have the same number leaves $n\geq 2$ and define $\mathcal F_\tau$ as above.
The forest category $\mathcal F_\tau$ is left-cancellative, satisfies Ore's property, and the spine of $\mathcal F_\tau$ is equal to $\mathcal F_\tau(2,1)\cup\{C_a(\tau_a)\}$ where $a\in S$ is any colour.
Let $G_\tau:=\Frac(\mathcal F_\tau,1)$ be its fraction group.
When $S$ is finite of cardinal $n$, then $G_\tau$ admits a finite presentation with $4n-2$ generators and $8n^2-4$ relations.
Moreover, $G_\tau$ is of type $F_\infty$ (and so are its $T,V$, and $BV$-versions).
\end{theorem}
\begin{proof}
Fix $\tau:S\to\mathcal{UF}\langle*\rangle$ as above and write $\mathcal F$ for $\mathcal F_\tau$.
By abuse of notation we write $C_a$ for the composition
$$\chi\circ C_a:\mathcal{UF}\langle*\rangle\to\mathcal{UF}\langle S\rangle\to\mathcal F$$
which consists in taking one monochromatic forest $f$, colouring all its vertices by $a$, and considering its image in the quotient forest category $\mathcal F$.
Note that $C_a(\tau_a)\in\mathcal F$ does not depend on $a\in S$ and is equal to a certain tree that we denote by $t$.
{\it Ore's property.}
We start by proving that $\mathcal F$ satisfies Ore's property.
Consider a tree $s\in\mathcal F$. Let us show that $s$ is dominated by a tree in $\mathcal F(t)$ (this latter being the quasi-forest subcategory generated by the tree $t$).
We proceed by induction on the number of carets of $s$.
If $s$ is trivial, then this is obvious.
If not, decompose $s$ as $s_0\circ f_0$ where $f_0$ is an elementary forest having a single caret: say a caret of colour $a$ at the $k$th root.
Consider the forest $f'_0$ with its $k$th tree equal to $t$ and all other trees trivial.
We have that $f_0\leq f'_0$ and thus $s\leq s_0\circ f'_0$.
Consider now the tree $s_0$. If $s_0$ is trivial, then we are done.
If not, $s_0 = s_1\circ f_1$ where $f_1$ is an elementary forest with a caret of a certain colour $b\in S$ at the $r$th root of $f_1.$
Now, the composition $f_1\circ f_0'$ can be interpreted as a forest with only $b$-carets.
Indeed, $f_0'\in\mathcal F(t)$ and thus is a disjoint union of copies of $t$.
Since $t=C_b(\tau_b)$ it can be interpreted as a tree with only $b$-carets and thus so does $f_1\circ f_0'$.
Now, the observation of above implies that $f_1\circ f_0'$ is dominated by an element $f_1'$ of $\mathcal F(t)$ since $f_1\circ f_0'$ is monochromatic.
We deduce that $s=s_1\circ f_1\circ f_0\leq s_1\circ f_1'$ with $f_1'\in\mathcal F(t)$ and $s_1$ is a proper rooted subtree of $s_0$.
By continuing inductively this process until $s_k$ is trivial we deduce that there exists $s'\in\mathcal F(t)$ satisfying $s\leq s'$.
It is now easy to deduce that $\mathcal F$ satisfies Ore's property.
Indeed, define the sequence $(t_n)_{n\geq 1}$ as in the observation. It is a cofinal sequence of trees in $\mathcal F(t)$ and thus in $\mathcal F$ by the proof given just above. We deduce that $\mathcal F$ satisfies Ore's property.
{\it Left-cancellative.}
Let us prove that $\mathcal F$ is left-cancellative.
Observe that the forest presentation of $\mathcal F$ is complemented (see Definition \ref{def:complemented}) since for each pair of distinct colours $(a,b)$ there is a unique relation of the form $a_1\cdots=b_1\cdots$ which is $(C_a(\tau_a),C_b(\tau_b))$ and no relation of the form $a_1\cdots=a_1\cdots.$
By Proposition \ref{prop:LC-complemented}, we are reduced to check that
\begin{equation*}E(a,b,c):=[(a_1\backslash b_1)\backslash (a_1\backslash c_1)]\backslash [(b_1\backslash a_1)\backslash (b_1\backslash c_1)]\end{equation*}
is either undefined or equal to $e$ for all triple of distinct colours $(a,b,c)\in S^3.$
For each colour $a\in S$ we decompose the tree $C_a(\tau_a)$ as $a_1\circ f^a$ where $f^a$ is a forest with two roots that we identify with the element $f^a\otimes I^{\otimes \infty}$ of $\mathcal F_\infty.$
By definition we have that $(a_1\backslash b_1) = f^a$ and note that this does not depend on the colour $b$.
We deduce that
$$E(a,b,c)=[f^a\backslash f^a]\backslash [f^b \backslash f^b] = e\backslash e =e.$$
We conclude that $\mathcal F$ is left-cancellative and thus $\mathcal F$ is a Ore category admitting a fraction group $G:=\Frac(\mathcal F,1).$
{\it Spine and finiteness property.}
We now compute the spine of $\mathcal F$.
By definition $\Sp_0(\mathcal F)=\{Y_a:\ a\in S\}.$
Now, $\mcm(Y_a,Y_b)=\{C_a(\tau_a)\} = \{ t\}$ for all $a,b\in S, a\neq b.$
We deduce that $\Sp_1(\mathcal F)=\{t\}$ is a singleton implying that $\Sp_n(\mathcal F)$ is empty for all $n\geq 2$.
Therefore, the spine of $\mathcal F$ is equal to $$\Sp(\mathcal F)=\{Y_a:\ a\in S\}\cup \{t\}.$$
In particular, the cardinal of $\Sp(\mathcal F)$ is smaller or equal to the cardinal of $S$ and a point.
By Theorem \ref{theo:Finfty}, we deduce that if $S$ is finite, then the fraction group $\Frac(\mathcal F,1)$ is of type $F_\infty$ and so are its $T,V,$ and $BV$-versions.
{\it Presentation.}
Fix one colour $a\in S$.
A forest presentation of $\mathcal F$ is given by
$$(S \ | \ (C_a(\tau_a),C_b(\tau_b)),\ b\in S\setminus\{a\})$$
where the set of relations is in bijection with $S\setminus\{a\}$ (rather than all pairs $(b,c)$ with $b\neq c$).
Consider now the reduced group presentation of $G$ associated to the same colour $a$ in Theorem \ref{theo:groupG-presentation}.
Remove the generators $\widehat a_1$ and $\widehat a_2$ and the third kind of relations.
Assume now that $S$ is finite of cardinal $n.$
We have $4n-2$ generators.
Moreover, the Thompson-like relations provide $2(4n^2 - 2n)=8n^2 - 4n$ group relations for $G$.
The last kind of group relation of $G$ in Theorem \ref{theo:groupG-presentation} provide $4n-4$ relations since we have $n-1$ forest relations.
In conclusion $G$ admits a group presentation with $4n-2$ generators and $8n^2-4$ relations.
\end{proof}
\begin{remark}\label{rem:tau}
Note that if $\tau_a$ has two leaves for all $a\in S$, then $\mathcal F_\tau$ is nothing else than $\mathcal{UF}\langle*\rangle$ the free forest category over one colour and thus $\Frac(\mathcal F_\tau,1)=F.$
When the trees $\tau_a,a\in S$ have at least three leaves, then $S$ is in bijection with $\mathcal F(2,1)$.
We have proven that the spine of $\mathcal F_\tau$ is equal to its trees with two leaves and $t:=C_a(\tau_a)$ for a given $a\in S$.
This is the smallest spine we could expect from a forest category that is not free.
Hence, $\mathcal F_\tau$ has a very low complexity with respect to Thumann's algorithm for proving finiteness properties.
Given $\tau:S\to \mathcal F$ and $\sigma$ a permutation of $S$, note that the two groups $G_\tau$ and $G_{\tau\circ \sigma}$ are isomorphic. Hence, $G_\tau$ only remembers the range of $\tau$ and the cardinal of each pre-image $\tau^{-1}(\{t\})$.
The fact that the forest relations of $\mathcal F_\tau$ are of the form $(t,s)$ with $t,s$ {\it monochromatic} makes possible to prove Ore's property at this level of generality. When the trees involved in a forest presentation are not monochromatic then it makes the analysis much harder. Indeed, there are many forest presentations with two colours and one relation that fails to provide Ore's property.
Consider for instance $$\mathcal F=\FC\langle a,b| a_1b_1a_1= b_1b_1a_1\rangle.$$
Note that this class of examples is very rich thanks to the two-dimensional structures of our diagrams.
If we adapt this construction to classical monoids where equations are written on a line we would obtain much reduced and less interesting class of examples: the monoids
$$\Mon\langle S| a^n=b^n, \text{ for all } a,b\in S\rangle$$
where $S$ is a set and $n$ a natural number.
\end{remark}
{\bf System of groups.}
Fix a map $\tau:S\to\mathcal{UF}\langle *\rangle$ as above.
Now, for any nonempty subset $S_0\subset S$ we have the forest presentation $(S_0,R_{S_0})$ where $R_{S_0}$ is the set of all pairs $(C_a(\tau_a),C_b(\tau_b))$ with $a,b\in S_0, a\neq b$.
This produces a forest category $\mathcal F_{S_0}$ and forest group $G_{S_0}$.
Note that the canonical embedding $S_0\hookrightarrow S$ provides an injective morphism $\mathcal F_{S_0}\hookrightarrow \mathcal F_S$ and group embedding $\iota_{S,S_0}:G_{S_0}\hookrightarrow G_S$.
Moreover, we have $\iota_{S,S_0}\circ \iota_{S_0,S_1}=\iota_{S,S_1}$ for a chain $S_1\subset S_0\subset S$.
We obtain a directed set of groups $(G_{S_0}:\ S_0\subset S, T\neq\emptyset).$
In particular, even if $S$ is infinite we can describe the group $G_S$ as an inductive limit of $G_{S_0}$ with $S_0$ finite.
\subsection{Two colours}\label{sec:two-colours}
Consider the case where we have two colours, i.e.~$S=\{a,b\}$.
Given any pair $(t,s)$ of nontrivial monochromatic trees with the same number of leaves we have a forest group $G_{(t,s)}$ and its $X$-versions $G_{(t,s)}^X$ for $X=T,V,BV$ that are all of type $F_\infty$ by the last section.
This is a very large class of groups.
As observed in Remark \ref{rem:tau} we have that $G_{(t,s)}\simeq G_{(s,t)}$.
Moreover, if $t'$ is mirror image of $t$ we have $G_{(t,s)}\simeq G_{(t',s')}.$
However, it seems there are no easy way to systematically decide if two such groups are isomorphic or not.
Although, they are all of type $F_\infty$ we suspect they may satisfy rather different properties.
Note that a pair $(t,s)$ defines an element $g$ of Thompson's group $F$ but note that if both $(t,s)$ and $(t',s')$ defines $g$ then we don't have in general $G_{(t,s)}\simeq G_{(t',s')}.$
Indeed, one can prove for instance that the family $$(G_{(t,t)}:\ t \text{ tree } )$$
contains infinitely many isomorphism classes of groups.
Although, each pair $(t,t)$ corresponds to the trivial element of $F$.
Here is a short proof of this fact.
\begin{proof}
Consider the monochromatic tree $t_n = x_1^n$ with $n+1$ equal to a left-vine made of $n$ carets.
This produces a Ore forest category $\mathcal F_n:=\FC\langle a,b| a_1^n=b_1^n\rangle$ with forest groups $G_n:=\Frac(\mathcal F_n,1)$ and $H_n:=\Frac((\mathcal F_n)_\infty)$.
Observe that $\mathcal F_n$ satisfies the CGP and thus $G_n\simeq H_n$.
Now, $H_n$ admits the infinite group presentation:
$$H_n=\Gr\langle a_j,b_j, j\geq 1 | x_q y_j = y_j x_{q+1}, a_j^n=b_j^n, 1\leq j<q\rangle$$
with abelianisation isomorphic to $\mathbf{Z}^2 \oplus (\mathbf{Z}/n\mathbf{Z})^2$
implying that the forest groups $(G_n:\ n\geq 2)$ are pairwise non-isomorphic.
\end{proof}
This leads to a question for which we have no answer.
\begin{question}
Is there a clear description of the following equivalence relation $\top$ defined as:
$$(t,s)\top (t',s') \text{ if and only if } G_{(t,s)}\simeq G_{(t',s')}?$$
\end{question}
\newcommand{\etalchar}[1]{$^{#1}$}
|
1,108,101,564,537 | arxiv | \chapter[Lecture \##1]{}}
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\section{Introduction}
In the theory of error-correcting codes,
two of the most-studied channel models are probabilistic channels and worst-case channels.
In probabilistic channels, errors are introduced through stochastic processes,
and the most well-known one is the binary symmetric channel (BSC).
In worst-case (or adversarial) channels, errors are introduced adversarially
by considering the choice of codes and transmitted codewords under the restriction of the error rate.
In his seminal work~\cite{Sha48}, Shannon showed that reliable communication can be achieved over BSC
if the coding rate is less than $1-H_2(p)$, where $H_2(\cdot)$ is the binary entropy function
and $p$ is the crossover probability of BSC.
In contrast, it is known that reliable communication cannot be achieved over worst-case channels
when the error rate is at least $1/4$ unless the coding rate tends to zero~\cite{Plo60}.
If we view the introduction of errors as \emph{computation} of the channel,
probabilistic channels perform low-cost computation with little knowledge about the code and the input,
while worst-case channels perform high-cost computation with the full-knowledge.
As intermediate channels between probabilistic channels and worst-case channels,
Lipton~\cite{Lip94} introduced \emph{computationally-bounded channels},
where errors are introduced by polynomial-time computation.
He showed that reliable communication can be achieved
at the coding rate less than $1-H_2(p)$ in the shared randomness setting, where $p < 1$ is the error rate,
which is the fraction of errors introduced by the channel.
Micali et al.~\cite{MPSW10} presented reliable coding schemes in the public-key infrastructure setting.
Guruswami and Smith~\cite{GS16} showed reliable coding schemes without assuming the shared randomness or
the public-key infrastructure.
Note that these work~\cite{Lip94,MPSW10,GS16} consider the settings in which channels are computationally-bounded
and the error rate is bounded.
In this work, we also focus on computationally-bounded channels.
In particular, we consider \emph{samplable additive channels},
in which errors are sampled by efficient computation and added to the codeword in an oblivious way.
More precisely, errors are sampled by a probabilistic polynomial-time algorithm, but the algorithm does not depend on the choice of the code or the transmitted codeword.
This is stronger than the standard notion of obliviousness, where an oblivious channel can depend on the code, but not the codeword (cf.~\cite{Lan08}).
Furthermore,
in this work, we consider samplable additive channels with \emph{unbounded} error rate.
Namely, the error rate $p$ is not a priori bounded.
Although most of the work in the literature focuses on bounded error-rate settings,
this restriction might not be necessary for modeling errors generated by nature as a result of polynomial-time computation.
We believe it is worth studying unbounded error-rate settings
since exploring the correctability in unbounded error-rate settings can reveal what \emph{error structures} can help to achieve error correction.
In particular, the study on samplable additive errors can reveal what \emph{computational} structures of errors are necessary to be corrected.
Samplable additive channels are relatively simple channel models since the error distributions are identical
for every coding scheme and transmitted codeword.
The binary symmetric channel is an example of samplable additive channels.
Thus, we consider the setting in which coding schemes can be designed with the knowledge of the error distribution
that is generated by an efficient algorithm.
This setting is incomparable to previous notions of error correction against computationally-bounded channels.
Our model is stronger because we do not restrict the error rate,
but is weaker because the channel cannot see the code or the transmitted codeword.
\subsection{Our Results}
We would like to characterize samplable additive channels
regarding the existence of efficient reliable coding schemes.
We use the entropy of the error distributions as a criterion.
The reason is that, if the entropy is zero, it is easy to achieve reliable communication
since the error is a fixed string and this information can be used for designing a reliable coding scheme.
On the other hand, if the error distribution has the full entropy,
we could not achieve reliable communication
since the truly random error will be added to the transmitted codeword.
Thus, there seem to be bounds on the existence of efficient reliable coding schemes
depending on the entropy of the underlying error distribution.
When reliable coding schemes exist,
an important quantity of the scheme is
the \emph{information rate} (or simply \emph{rate)},
which is the ratio of the message length to the codeword length.
We investigate the bounds on the rate when reliable communication is achievable.
Let $Z$ be an error distribution over $\{0, 1\}^n$ associated with a samplable additive channel,
and $H(Z)$ the Shannon entropy of $Z$.
Note that for a \emph{flat} distribution, which is a uniform distribution over its support,
$H(Z)$ is equal to the \emph{min-entropy} of $Z$.
\paragraph{Basic observations.}
First, we observe several basic facts regarding the correctability of samplable additive errors.
Let consider flat distributions $Z$.
It follows from a probabilistic argument that for any flat $Z$,
there is a linear code that corrects $Z$ with error $\epsilon$ for rate $R \leq 1 - H(Z)/n - 2\log(1/\epsilon)/n$.
The decoding complexity is $O(n^22^{H(Z)})$.
Thus, if $H(Z) = O(\log n)$, the code can correct $Z$ in polynomial time.
Conversely, by a simple counting argument, it holds that
every flat distribution $Z$ is not correctable with error $\epsilon$
for rate $R > 1 - H(Z)/n + \log(1/(1-\epsilon))/n$.
In addition, we observe that it is difficult to construct a code that corrects the family of flat distributions with the same entropy.
Specifically, we show that for every code of rate $k/n$ and every $m \leq k$,
there is a flat distribution $Z$ with $H(Z) = m$ that is not correctable by the code.
A positive result can be obtained if we consider much more structured errors.
If the error vectors form a \emph{linear subspace},
there is an efficient coding scheme that corrects them by syndrome decoding
with optimal rate $R = 1 - m/n$, where $m$ is the dimension of the linear subspace.
Regarding efficient coding schemes,
we observe that if errors are pseudorandom (in the cryptographic sense),
then efficient coding scheme cannot correct them.
This implies that assuming the existence of one-way functions,
there exist $Z$ with $H(Z) = n^\epsilon$ for $0 < \epsilon <1$ that are not efficiently correctable.
\ignore{
\paragraph{Errors from flat distributions.}
We investigate the correctability of general flat distributions $Z$.
Namely, $Z$ is a uniform distribution over its support.
By a probabilistic argument, we show that for any flat $Z$,
there is a linear code that corrects $Z$ with error $\epsilon$ for rate $R \leq 1 - H(Z)/n - 2\log(1/\epsilon)/n$.
The decoding complexity is $O(n^32^{H(Z)})$.
Thus, if $H(Z) = O(\log n)$, the code can correct $Z$ in polynomial time.
Conversely, we show that every flat distribution $Z$ is not correctable with error $\epsilon$
for rate $R > 1 - H(Z)/n + \log(1/(1-\epsilon))/n$.
The result comes from a simple counting argument.
We also observe that it is difficult to construct a code that corrects the family of flat distributions with the same entropy.
Specifically, we show that for every code of rate $k/n$ and every $m \leq k$,
there is a flat distribution $Z$ with $H(Z) = m$ that is not correctable by the code.
\paragraph{Errors from linear subspaces.}
A positive result can be shown if we consider much more structured errors.
We show that, if the error vectors form a \emph{linear subspace},
then there is an efficient coding scheme that correct them by syndrome decoding
with optimal rate $R = 1 - m/n$, where $m$ is the dimension of the linear subspace.
\paragraph{Pseudorandom errors.}
Regarding efficient coding schemes,
we observe that if errors are pseudorandom (in the cryptographic sense),
then efficient coding scheme cannot correct them.
This implies that assuming the existence of one-way functions,
there exist $Z$ with $H(Z) = n^\epsilon$ for $0 < \epsilon <1$ that are not efficiently correctable.
}
\paragraph{Errors with membership test.}
To avoid the impossibility of correcting pseudorandom errors,
we consider samplable distributions for which membership test can be done efficiently.
Such distributions are not pseudorandom since the membership test can be used to distinguish them from the uniform distribution.
\ignore{
We show the existence of an uncorrectable distribution with low entropy.
Specifically, we show that there is an oracle relative to which
there exists a samplable distribution with membership test of entropy $\omega(\log n)$
that is not correctable by linear codes that employ efficient \emph{syndrome decoding}
with rate $R > \omega((\log n)/n)$.
For this result, we use the relation between linear codes correcting additive errors and
linear \emph{data compression}.
The result follows by combining this relation and
the result of Wee~\cite{Wee04}, who showed that there is an oracle relative to which
there exists a samplable distribution with membership test of entropy $\omega(\log n)$
that cannot be efficiently compressible to length less than $n - \omega(\log n)$.
}
\ignore{
\paragraph{Uncorrectable errors with low entropy.}
Next, we investigate whether there is an uncorrectable error distribution with much lower entropy.
We show that there is an oracle relative to which
there exists $Z$ with $H(Z) = \omega(\log n)$
that is not correctable by linear coding schemes that employ efficient \emph{syndrome decoding}
with rate $R > \omega((\log n)/n)$.
For this result, we use the relation between linear codes correcting additive errors and
linear \emph{data compression}.
The result follows by combining this relation and
the result of Wee~\cite{Wee04}, who showed that there is an oracle relative to which
there exists a samplable distribution of entropy $\omega(\log n)$
that cannot be efficiently compressible to length less than $n - \omega(\log n)$.
}
\ignore{
The above result implies the impossibility of correcting every samplable errors with membership test by efficient syndrome decoding in a black-box way.
However, this implication is quite restrictive
since the syndrome decoding can be employed only for linear codes,
and there are many efficient decoding algorithms other than syndrome decoding in the coding theory literature.
Furthermore, the general syndrome-decoding problem is known to be NP-hard~\cite{BMT78}.
We tried to derive impossibility results for general coding schemes,
and give an oracle separation result for \emph{low-rate} codes.
Specifically, we show that there is an oracle relative to which there exists a samplable distribution $Z$ with $H(Z) = \omega(\log n)$
that is not correctable by efficient coding schemes of rate $R < 1 - H(Z)/n - \omega({\log n}/{n})$.
Namely, we could remove the restriction of syndrome decoding.
This negative result also implies the impossibility of correcting every samplable errors by low-rate coding schemes in a black-box way.
For codes with a higher rate, as will be stated later, we give a stronger impossibility result.
Specifically, we show that every flat distribution $Z$ is not correctable by codes with rate $R > 1 - H(Z)/n$.
}
We show the existence of an uncorrectable distribution with membership test for \emph{low-rate} codes.
Specifically, we show that there is an oracle relative to which there exists a samplable distribution $Z$ of entropy $\omega(\log n)$
that is not correctable by efficient coding schemes of rate $R < 1 - H(Z)/n - \omega({\log n}/{n})$.
The result complements the impossibility of correcting flat distributions for rate $R > 1 - H(Z)/n + O(1/n)$.
Also, the entropy of $\omega(\log n)$ is optimal since, as in the above observations,
there is a probabilistic construction of a code that corrects $Z$ in polynomial time if $H(Z) = O(\log n)$.
To derive this result, we use the technique of Wee~\cite{Wee04},
which is based on the \emph{reconstruction paradigm} of Gennaro and Trevisan~\cite{GT00}.
We use his technique for the problem of error correction.
We show that if a samplable distribution with a sampler $S$ is efficiently correctable,
then the function of $S$ has a short description, and thus, by a counting argument,
efficient coding schemes cannot correct every samplable distribution with membership test.
This negative result seems counterintuitive.
In general, constructing low-rate codes seems to be easier than high-rate codes.
However, the result implies the impossibility of constructing low-rate codes.
The reason for such a result is that
the reconstruction paradigm crucially uses the fact that some function can be described shortly.
In our case, we use the fact that functions for samplable errors can be described shortly
if the errors are correctable by coding schemes with short descriptions.
Since low-rate codes have short descriptions, the result can be applied to low-rate codes.
\begin{table*}[t]
\begin{center}
\caption{Correctability of Samplable Additive Error $Z$}\label{tb:result}
\smallskip
\begin{tabular}{clll}
\hline
\textbf{$H(Z)$} & \textbf{Correctabilities} & \textbf{Assumptions} & \textbf{References} \\ \hline\hline
& Efficiently correctable with error $\epsilon$ for & \\
\raisebox{1.2ex}[0pt]{$O(\log n)$} & $R \leq 1 - \frac{H(Z)}{n} - \frac{2\log(1/\epsilon)}{n}$ & \raisebox{1.2ex}[0pt]{No}
& \raisebox{1.2ex}[0pt]{Proposition~\ref{prop:random}} \\ \hline
& $\exists$ $Z$ with membership test, & & \\
$\omega(\log n)$ & not efficiently correctable for & Oracle access & Corollary~\ref{cor:lowentropy} \\
& $R < 1 - \frac{H(Z)}{n} - \omega\left(\frac{\log n}{n}\right)$ & & \\ \hline
$n^\epsilon$ & & & \\
\raisebox{0.1ex}[0pt]{$(0 < \epsilon < 1)$} & \raisebox{1.2ex}[0pt]{$\exists \, Z$ not efficiently correctable for any $R$}
& \raisebox{1.2ex}[0pt]{OWF} & \raisebox{1.2ex}[0pt]{Proposition~\ref{prop:prg}}
\\\hline
& $\forall$ code with $R = k/n$, & & \\
\raisebox{1.2ex}[0pt]{$1 \leq m \leq k$} & $\exists \, Z$ not correctable by the code & \raisebox{1.2ex}[0pt]{No} & \raisebox{1.2ex}[0pt]{Proposition~\ref{prop:linear}} \\\hline
& $\forall$ linear subspace $Z$ of dimension $m$, & & \\
\raisebox{1.2ex}[0pt]{---} & \raisebox{0.1ex}[0pt]{$\exists$ code correcting $Z$ with $R \leq 1-\frac{m}{n}$} & \raisebox{1.2ex}[0pt]{No} & \raisebox{1.2ex}[0pt]{Proposition~\ref{prop:subspace}}
\\\hline
& $\forall$ flat $Z$, & & \\
& (1) $\exists$ (non-explicit) code correcting $Z$ & & \\
\raisebox{0ex}[0pt]{---} & \raisebox{0.1ex}[0pt]{\qquad with error $\epsilon$ for $R \leq 1 - \frac{H(Z)}{n} - \frac{2\log(1/\epsilon)}{n}$}
& \raisebox{1.2ex}[0pt]{No} & \raisebox{1.2ex}[0pt]{Proposition~\ref{prop:random}} \\
& (2) not correctable with error $\epsilon$ for & & \\
& \raisebox{0.1ex}[0pt]{\qquad $R > 1 - \frac{H(Z)}{n} + \frac{\log(1/(1-\epsilon))}{n}$} & \raisebox{1.2ex}[0pt]{No} & \raisebox{1.2ex}[0pt]{Proposition~\ref{prop:flatbound}} \\
\hline
& $\forall$ flat $Z$ is efficiently correctable for & & \\
\raisebox{1.2ex}[0pt]{---} & \raisebox{0.1ex}[0pt]{$R \leq 1 - \frac{H(Z)}{n} - O(\frac{\log n}{n})$} & \raisebox{1.2ex}[0pt]{No OWF} & \raisebox{1.2ex}[0pt]{Theorem~\ref{th:noowf}} \\ \hline
\end{tabular}
\end{center}
\end{table*}
\ignore{
\paragraph{Errors from small-biased distributions.}
A distribution is called \emph{small-biased} if it is indistinguishable from a uniform distribution by linear functions.
We show that $\delta$-biased distributions are not correctable for rate $R > 1 - (2\log(1/\delta)+1)/n$.
This result follows from the fact that small-biased distributions can be used
for keys of the one-time pad for high-entropy messages, which was established by Dodis and Smith~\cite{DS05}.
}
\paragraph{Necessity of one-way functions.}
Finally, we show that it is difficult to
prove \emph{unconditional} impossibility results
for coding schemes of rate $R \leq 1 - H(Z)/n$.
Specifically, we show that if one-way functions do not exist, then any samplable flat distribution $Z$
is correctable by an efficient coding scheme of rate $1-H(Z)/n-O(\log n /n)$.
Thus, it is necessary to assume the existence of one-way functions or oracle access to derive
impossibility results for rate $R \leq 1 - H(Z)/n$.
The results are summarized in Table~\ref{tb:result},
where $R$ denotes the rate of coding schemes.
\subsection{Related Work}
The notion of computationally-bounded channel was introduced by Lipton~\cite{Lip94}.
He showed that if the sender and the receiver can share secret randomness,
then the Shannon capacity can be achieved for this channel.
Micali et al.\,\cite{MPSW10} considered a similar channel model in a public-key setting,
and gave a coding scheme based on list-decodable codes and digital signature.
Guruswami and Smith~\cite{GS16} gave constructions of
capacity achieving codes for worst-case additive-error channel
and time/space-bounded channels.
In their setting of additive-error channel,
the error rate is bounded, and the errors are only independent of the encoder's random coins.
They also gave strong impossibility results for bit-fixing channels,
but their results can be applied to channels that use the information on the code
and the transmitted codewords. In this work, we give impossibility results even for channels that
do no use such information.
Samplable distributions were also studied in the context of
data compression~\cite{GS91,TVZ05,Wee04},
randomness extractor~\cite{TV00,Vio11,DW12},
and randomness condenser~\cite{DRV12}.
Samplable distributions with membership test appeared in the study of efficient compressibility of samplable sources~\cite{GS91,TVZ05,Wee04}.
\section{Preliminaries}
For $n \in \mathbb{N}$, we write $[n]$ as the set $\{1, 2, \dots, n\}$.
For a distribution $X$, we write $x \sim X$ to indicate that $x$ is chosen according to $X$.
We may use $X$ also as a random variable distributed according to $X$.
The \emph{support} of $X$ is $\mathrm{Supp}(X) = \{ x : \Pr_X(x) \neq 0\}$,
where $\Pr_X(x)$ is the probability that $X$ assigns to $x$.
The \emph{Shannon entropy} of $X$ is $H(X) = E_{x \sim X}[-\log \Pr_X(x)]$.
The \emph{min-entropy} of $X$ is given by $\min_{x \in \mathrm{Supp}(X)}\{ - \log \Pr_X(x)\}$.
It is known that the min-entropy of $X$ is a lower bound on $H(X)$.
A \emph{flat distribution} is a distribution that is uniform over its support.
For flat distributions, the Shannon entropy is equal to the min-entropy.
Thus, we simply say that a flat distribution $Z$ has entropy $m$ if its Shannon entropy is $m$.
For $n \in \mathbb{N}$, we write $U_n$ as the uniform distribution over $\{0, 1\}^n$.
We define the notion of additive-error correcting codes.
\begin{definition}[Additive-error correcting codes]
For two functions $\Enc : \mathbb{F}^{k} \to \mathbb{F}^n$ and $\Dec: \mathbb{F}^n \to \mathbb{F}^{k}$,
and a distribution $Z$ over $\mathbb{F}^n$, where $\mathbb{F}$ is a finite field,
we say $(\Enc, \Dec)$ \emph{corrects (additive error) $Z$ with error $\epsilon$}
if for any $x \in \mathbb{F}^{k}$, we have that $\Pr_{z \sim Z}[\Dec(\Enc(x) + z) \neq x] \leq \epsilon$.
The \emph{rate} of $(\Enc, \Dec)$ is $k/n$.
\end{definition}
\begin{definition}
A distribution $Z$ is said to be \emph{correctable with rate $R$ and error $\epsilon$}
if there is a pair of functions $(\Enc, \Dec)$ of rate $R$ that corrects $Z$ with error $\epsilon$.
\end{definition}
We call a pair $(\Enc, \Dec)$ a \emph{coding scheme} or simply \emph{code}.
The coding scheme is called \emph{efficient} if $\Enc$ and $\Dec$ can be computed in polynomial-time in $n$.
The code is called \emph{linear}
if $\Enc$ is a linear mapping,
that is, for any $x, y \in \mathbb{F}^n$ and $a, b \in \mathbb{F}$, $\Enc(ax+by) = a\,\Enc(x)+b\,\Enc(y)$.
If $|\mathbb{F}| = 2$, we may use $\{0, 1\}$ instead of $\mathbb{F}$.
Next, we define syndrome decoding for linear codes.
\begin{definition}
For a linear code $(\Enc, \Dec)$,
$\Dec$ is said to be a \emph{syndrome decoder}
if there is a generator matrix $G \in \mathbb{F}^{Rn \times n}$ and a function $\textsf{Rec} : \mathbb{F}^{(n-Rn)} \to \mathbb{F}^n$ such that
$\Dec(y) = (y - \textsf{Rec}(y \cdot H^T)) \cdot G^{-1}$,
where $\Enc(x) = x \cdot G$ for all $x \in \mathbb{F}^{Rn}$,
$G^{-1} \in \mathbb{F}^{n \times Rn}$ is a right inverse matrix of $G$ (i.e., $G G^{-1}=I$),
and $H \in \mathbb{F}^{(n - Rn) \times n}$ is a dual matrix of $G$ (i.e., $GH^T = 0$).
\end{definition}
We consider a computationally-bounded analogue of additive-error channels.
We introduce the notion of samplable distributions.
\begin{definition}
A distribution family $Z = \{Z_n\}_{n \in \mathbb{N}}$ is said to be \emph{samplable} if
there is a probabilistic polynomial-time algorithm $S$ such that
$S(1^n)$ is distributed according to $Z_n$ for every $n \in \mathbb{N}$.
\end{definition}
We consider the setting in which coding schemes can depend on the sampling algorithm of $Z$,
but not on its random coins,
and $Z$ does not use any information on the coding scheme or transmitted codewords.
In this setting, the randomization of coding schemes does not help much.
\begin{proposition}\label{pro:deterministic}
Let $(\Enc, \Dec)$ be a randomized coding scheme
that corrects a distribution $Z$ with error $\epsilon$.
Then, there is a deterministic coding scheme that corrects $Z$ with error $\epsilon$.
\end{proposition}
\begin{proof}
Assume that $\Enc$ uses at most $\ell$-bit randomness.
Since $(\Enc, \Dec)$ corrects $Z$ with error $\epsilon$,
we have that for every $x \in \mathbb{F}^k$, $\Pr_{z \sim Z,r \sim U_\ell}[\Dec(\Enc(x;r)+z) \neq x] \leq \epsilon.$
By the averaging argument, for every $x \in \mathbb{F}^k$, there exists $r_x \in \{0, 1\}^\ell$ such that
$\Pr_{z \sim Z}[\Dec(\Enc(x;r_x) + z)\neq x] \leq \epsilon.$
Thus, by defining $\Enc'(x) = \Enc(x;r_x)$,
the deterministic coding scheme $(\Enc', \Dec)$ corrects $Z$ with error $\epsilon$.
\end{proof}
The fact that the randomization does not help much is contrast to the setting of Guruswami and Smith~\cite{GS16},
where the channels can use the information on the coding scheme and transmitted codewords, but not the random coins for encoding.
They present a randomized coding scheme with optimal rate $1-H_2(p)$ for worst-case additive-error channels,
for which deterministic coding schemes are only known to achieve rate $1-H_2(2p)$,
where $p$ is the error rate of the channels.
\ignore{
Next, we define the notion of data compression.
\begin{definition}
For two functions $\Com : \mathbb{F}^* \to \mathbb{F}^*$ and $\Decom: \mathbb{F}^* \to \mathbb{F}^*$,
and a distribution $Z$,
we say $(\Com, \Decom)$ \emph{compresses} $Z$ to length $m$ if
\begin{enumerate}
\item For any $z \in \mathrm{Supp}(Z)$, $\Decom( \Com(z) ) = z$, and
\item $E[ | \Com(Z) |] \leq m$.
\end{enumerate}
\end{definition}
\begin{definition}
We say a distribution $Z$ is \emph{compressible} to length $m$,
if there are two functions $\Com$ and $\Decom$ such that
$(\Com, \Decom)$ compresses $Z$ to length $m$.
\end{definition}
If $\Com$ is a linear mapping, $(\Com, \Decom)$ is called a \emph{linear} compression.
Finally, we define the notion of lossless condensers.
\begin{definition}
A function $f : \mathbb{F}^n \times \{0, 1\}^d \to \mathbb{F}^r$ is said to be an $(m, \epsilon)$-\emph{lossless condenser}
if for any distribution $X$ of min-entropy $m$,
the distribution $(f(X,Y),Y)$ is $\epsilon$-close to a distribution $(Z, U_d)$ with min-entropy at least $m+d$,
where $Y$ is the uniform distribution over $\{0, 1\}^d$.
A condenser $f$ is \emph{linear} if for any fixed $z \in \{0, 1\}^d$, any $x, y \in \mathbb{F}^n$ and $a, b \in \mathbb{F}$,
$f(ax+by,z) = af(x,z)+bf(y,z)$.
\end{definition}
}
\section{Basic Properties of Samplable-Additive Errors}
We present several basic facts regarding the correctability of samplable additive errors.
Although the claims in this section are elementary or folklore,
we include the proofs for completeness.
\subsection{Errors from Flat Distributions}\label{sec:flat}
For any flat distribution $Z$, a random linear code can correct $Z$ with high probability.
Consider a random linear code of rate $R$ such that the parity check matrix $H$ is chosen uniformly at random from $\mathcal{H}_R = \{0, 1\}^{(n-Rn)\times n}$.
The decoding is done in a brute-force way, namely, for a received word $y$, find $x \in \{0, 1\}^{Rn}$ and $z \in \mathrm{Supp}(Z)$
such that $y = x \cdot G + z$, and output $x$, where $G$ is a generator matrix for $H$.
\begin{proposition}\label{prop:rand}
For any flat distribution $Z$ over $\{0, 1\}^n$,
a random linear code from $\mathcal{H}_R$ corrects $Z$ with error $2^{-(n-Rn-H(Z))}$.
\end{proposition}
\begin{proof}
It is sufficient to show that, for a random $H$ from $\mathcal{H}_R$, every $z \in \mathrm{Supp}(Z)$ has a unique syndrome $z \cdot H^T$ with high probability.
For each $z \in \mathrm{Supp}(Z)$,
\begin{align}
& \Pr_{H \in \mathcal{H}_R}\left[ \exists z' \in \mathrm{Supp}(Z) \setminus \{z\} : z \cdot H^T = z' \cdot H^T \right] \nonumber\\
& = \Pr_{H \in \mathcal{H}_R} \left[ \exists z' \in \mathrm{Supp}(Z) \setminus \{z\} : \forall i \in [n-Rn], h_i\cdot(z - z') = 0 \right] \label{eq:2}\\
& \leq \sum_{z' \in \mathrm{Supp}(Z) \setminus \{z\}} \prod_{i \in [n-Rn]} \Pr_{h_i \in \{0, 1\}^n}\left[ h_i\cdot(z - z') = 0 \right] \nonumber\\
& \leq 2^{-(n-Rn-H(Z))},\nonumber
\end{align}
where $H^T = (h_1^T, \dots, h_{n-Rn}^T)$ in (\ref{eq:2}),
the last inequality follows from the fact that $z-z' \neq 0$ and $|\mathrm{Supp}(Z)| = 2^{H(Z)}$ for flat $Z$.
\end{proof}
By Proposition~\ref{prop:rand}, for a $(1 - 2^{-(n-Rn-H(Z))/2})$-fraction of $H$ in $\mathcal{H}_R$,
the corresponding code corrects $Z$ with error at most $2^{-(n-Rn-H(Z))/2}$.
Thus, we have the following proposition.
\begin{proposition}\label{prop:random}
Let $Z$ be any flat distribution over $\{0, 1\}^n$ of entropy $m$.
There is a linear code of rate $R$ that corrects $Z$ with error $\epsilon$ for $R \leq 1 - m/n - 2\log(1/\epsilon)/n$.
The decoding complexity is at most $O(n^22^m)$.
\end{proposition}
\begin{proof}
The existence of such a code immediately follows from the above argument.
Given a received word $y$, the brute-force decoder checks if $(y - z) \cdot H^T = 0$ for all $z \in \mathrm{Supp}(Z)$,
where $H$ is the parity check matrix. If so, output $x$ satisfying $x \cdot G = y - z$. Thus, the decoding is done in time $O(n^2) \cdot |\mathrm{Supp}(Z)|$.
\end{proof}
Proposition~\ref{prop:random} implies that for any flat $Z$ of entropy $O(\log n)$,
there is a code that corrects $Z$ in polynomial time.
Although the construction is not fully explicit, we can obtain such a code with high probability.
\ignore{
Cheraghchi~\cite{Che09} showed a relation between lossless condensers and linear codes correcting additive errors.
He gave the equivalence between a \emph{linear} lossless condenser for a \emph{flat} distribution $Z$
and a linear code ensemble in which most of them correct additive errors from $Z$.
Based on his result, we can show that, for any flat distribution, there is a linear code that corrects errors from the distribution.
\begin{theorem}\label{th:flatcorrect}
For any $\epsilon > 0$ and flat distribution $Z$ over $\{0, 1\}^n$ of entropy $m$,
there is a linear code of rate $1 - m/n - 4\log(1/\epsilon)/n$ that corrects $Z$ with error ${\epsilon}$ by syndrome decoding.
\end{theorem}
\begin{proof}
Let $f : \{0, 1\}^n \times \{0, 1\}^d\to \{0, 1\}^r$ be a linear $(m,\epsilon)$-lossless condenser.
Define a code ensemble $\{C_u\}_{u \in \{0, 1\}^d}$ such that $C_u$ is a linear code
for which a parity check matrix $H_u$ satisfies that for each $x \in \{0, 1\}^n$, $x \cdot H_u^T = f(x,u)$.
Cheraghchi~\cite{Che09} proved the following lemma.
\begin{lemma}[Lemma~15 of \cite{Che09}]\label{lem:flatcorrect}
For any flat distribution $Z$ of entropy $m$,
at least a $(1-2\sqrt{\epsilon})$ fraction of the choices of $u \in \{0, 1\}^d$,
the code $C_u$ corrects $Z$ with error $\sqrt{\epsilon}$.
\end{lemma}
We use a linear lossless condenser that can be constructed
from a universal hash family consisting of linear functions.
This is a generalization of the Leftover Hash Lemma and the proof is given in~\cite{Che09}.
\begin{lemma}[Lemma~7 of \cite{Che09}]\label{lem:llc}
For every integer $r \leq n, m,$ and $\epsilon > 0$ with $r \geq m + 2\log(1/\epsilon)$,
there is an explicit $(m,\epsilon)$ linear lossless condenser $f: \{0, 1\}^n \times \{0, 1\}^n \to \{0, 1\}^r$.
\end{lemma}
The statement of the theorem immediately follows by combining Lemmas~\ref{lem:flatcorrect} and~\ref{lem:llc}.
\end{proof}
}
Conversely, we can show that the rate achieved in Proposition~\ref{prop:random} is almost optimal.
\begin{proposition}\label{prop:flatbound}
Let $Z$ be any flat distribution over $\{0, 1\}^n$ of entropy $m$.
If a code of rate $R$ corrects $Z$ with error $\epsilon$,
then $R \leq 1 - m/n + \log(1/(1-\epsilon))/n$.
\end{proposition}
\begin{proof}
Let $(\Enc, \Dec)$ be a code that corrects $Z$ with error $\epsilon$.
For $x \in \{0, 1\}^{Rn}$, define $D_x = \{ y \in \{0, 1\}^n : \Dec(y) = x\}$.
That is, $D_x$ is the set of inputs that are decoded to $x$ by $\Dec$.
Since the code corrects the flat distribution $Z$ with error $\epsilon$,
$|D_x| \geq (1-\epsilon)2^m$ for every $x \in \{0, 1\}^{Rn}$.
Since each $D_x$ is disjoint, $\sum_{x \in \{0, 1\}^{Rn}}|D_x| \leq 2^n$.
Therefore, we have that $(1-\epsilon)2^m\cdot 2^{Rn} \leq 2^n$,
which implies the statement.
\end{proof}
By Proposition~\ref{prop:random},
one may hope to construct a \emph{single} code that corrects
errors from any flat distribution with the same entropy,
as constructed in~\cite{Che09} for the case of binary symmetric channels
by using Justesen's construction~\cite{Jus72}.
However, it is impossible to construct such codes.
We show that for every deterministic coding scheme of rate $k/n$,
there is a flat distribution $Z$ of entropy $m$ that is not correctable by the scheme for any $1 \leq m \leq k$.
By combining this result with Proposition~\ref{pro:deterministic},
we can conclude that there is no coding scheme of rate $k/n$ that corrects every flat distribution of entropy $m$ with $1 \leq m \leq k$.
\begin{proposition}\label{prop:linear}
For any deterministic code of rate $k/n$ and any $m$ with $1 \leq m \leq k$,
there is a flat distribution of entropy $m$ that is not correctable by the code with error $\epsilon < 1/2$.
\end{proposition}
\begin{proof}
Define a flat distribution to be a uniform distribution over
$2^m$ distinct codewords $c_1, \dots, c_{2^m}$.
If the input to the decoder is $c_i + c_j$ for $i, j \in [2^m]$, the decoder cannot distinguish the two cases
where the transmitted codewords are $c_i$ and $c_j$.
Thus, the decoder outputs the wrong answer with probability at least $1/2$ for at least one of the two cases.
\end{proof}
\subsection{Errors from Linear Subspaces}
Let $Z = \{z_1, z_2, \dots, z_m\} \subseteq \mathbb{F}^n$ be a set of linearly independent vectors.
We can construct a linear code that corrects additive errors from the linear span of $Z$.
\begin{proposition}\label{prop:subspace}
There is a linear code of rate $1-m/n$ that corrects the linear span of $Z$ by syndrome decoding.
\end{proposition}
\begin{proof}
\ignore{
For a permutation $\pi : \{1, \dots, n\} \to \{1, \dots, n\}$ and a vector $v = (v_1, v_2, \dots, v_n) \in \{0, 1\}^n$,
define $\pi(v) = (v_{\pi(1)}, v_{\pi(2)}, \dots, v_{\pi(n)})$.
It is not difficult to see that there is a matrix $M \in \{0, 1\}^{m \times n}$ such that
(1) $M = [I_m\, M']$, where $I_m \in \{0, 1\}^{m \times m}$ is the identity matrix
and $M' \in \{0, 1\}^{m \times (n-m)}$,
and (2) for some permutation $\pi$, the linear span of $\pi(m_1), \pi(m_2), \dots, \pi(m_m)$
is equal to that of $Z$, where $m_1, \dots, m_m$ are the rows of $M$.
For $i \in \{1, \dots, n\}$,
let $h_i \in \{0, 1\}^n$ be the vector such that the $i$-the element is $1$ and the other elements are $0$.
Note that for $i, j \in \{1, \dots, m\}$, the inner product $m_i \cdot h_j$ is $1$ if $i = j$, and $0$ otherwise.
Let $H \in \{0, 1\}^{m \times n}$ be the matrix consisting of $\pi(h_1), \pi(h_2), \dots, \pi(h_m)$ as rows.
Then, the linear code having $H$ as a parity-check matrix can correct additive errors from the linear span of $Z$.
In syndrome decoding, we define the function $\textsf{Rec}$ such that, for $s = (s_1, \dots, s_m) \in \{0, 1\}^{m}$,
$\textsf{Rec}(s) = \sum_{i=1}^{m} s_i \pi(m_i)$.
Let $z \in \mathbb{F}^n$ be any error from the linear span of $Z$.
It follows from the property~(2) of the matrix $M$ that $z$ can be represented as $\sum_{i=1}^{n}b_i \pi(m_i)$,
where $b_i \in \{0, 1\}$.
Then,
\begin{align*}
z \cdot H^T & = z \cdot \begin{bmatrix} \pi(h_1)\\ \vdots \\ \pi(h_m) \end{bmatrix}^T \\
& = (z \cdot \pi(h_1), \dots, z \cdot \pi(h_m)).
\end{align*}
By the linearity of the inner product,
\begin{align*}
z \cdot \pi(h_j) & = \left(\sum_{i=1}^{n}b_i \pi(m_i) \right) \cdot \pi(h_j) \\
& = \sum_{i=1}^n b_i (\pi(m_i) \cdot \pi(h_j)) \\
& = b_j,
\end{align*}
where the last equality follows from the fact that $m_i \cdot h_j$ is $1$ if $i = j$, and $0$ otherwise.
Hence, $z \cdot H^T = (b_1, \dots, b_m)$,
and thus $\textsf{Rec}(z\cdot H^T)$ can recover the error vector $z$.
Therefore, the syndrome decoding corrects any error from the linear span of $Z$.
Since $H \in \{0, 1\}^{m \times n}$ is a parity check matrix, the rate of the code is $(n- m)/n$.
}
Consider $n-m$ vectors $w_{m+1}, \dots, w_n \in \mathbb{F}^n$ such that the set $\{z_1, z_2, \dots, z_m, w_{m+1}, \dots, w_n\}$ forms a basis of $\mathbb{F}^n$.
Then, there is a linear transformation $T : \mathbb{F}^n \to \mathbb{F}^m$ such that $T(z_i) = e_i$ and $T(w_i) = 0$,
where $e_i$ is the vector with $1$ in the $i$-th position and $0$ elsewhere.
Let $H$ be the matrix in $\mathbb{F}^{m \times n}$ such that $x H^T = T(x)$,
and consider a code with parity check matrix $H$.
Let $z = \sum_{i=1}^m a_i z_i$ be a vector in the linear span of $Z$, where $a_i \in \mathbb{F}$.
Since $z \cdot H^T = (\sum_{i=1}^m a_i z_i) \cdot H^T = \sum_{i=1}^m a_i e_i = (a_1, \ldots, a_m)$,
the code can correct the error $z$ by syndrome decoding.
Since $H \in \mathbb{F}^{m \times n}$ is the parity check matrix, the rate of the code is $(n- m)/n$.
\end{proof}
\subsection{Pseudorandom Errors}
We show that
no efficient coding scheme can correct pseudorandom errors.
\begin{proposition}\label{prop:prg}
Assume that a one-way function exists.
Then, for any $0 < \epsilon < 1$,
there is a samplable distribution $Z$ over $\{0, 1\}^n$ such that
$H(Z) \leq n^\epsilon$ and no polynomial-time algorithms $(\Enc, \Dec)$ can correct $Z$.
\end{proposition}
\begin{proof}
If a one-way function exists,
there is a pseudorandom generator $G : \{0, 1\}^{n^\epsilon} \to \{0, 1\}^n$ secure for any polynomial-time algorithm~\cite{HILL99}.
Then, a distribution $Z = G(U_{n^\epsilon})$ is not correctable by polynomial-time algorithms $(\Enc, \Dec)$.
If so, we can construct a polynomial-time distinguisher for pseudorandom generator by employing $(\Enc, \Dec)$,
and thus a contradiction follows.
\end{proof}
\ignore{
\subsection{Uncorrectable Errors with Low Entropy}
We consider errors from low-entropy distributions that
are not correctable by efficient coding schemes.
We use the relation between error correction by linear code
and data compression by linear compression.
The relation was explicitly presented by Caire et~al.~\cite{CSV04}.
\begin{theorem}[\cite{CSV04}]\label{th:correct_comp}
For any distribution $Z$ over $\mathbb{F}^n$,
$Z$ is correctable with rate $R$ by syndrome decoding if and only if
$Z$ is compressible by linear compression to length $n(1-R)$.
\end{theorem}
Wee~\cite{Wee04} showed that
there is an oracle relative to which
there is a samplable distribution over $\{0, 1\}^n$ of entropy $\omega(\log n)$
that cannot be compressed to length less than $n - \omega(\log n)$ by any efficient compression.
\begin{lemma}[\cite{Wee04}]\label{lem:imp_comp}
For any $m$ satisfying $6 \log s + O(1) < m < n$,
there are a function $f : \{0, 1\}^m \to \{0, 1\}^n$ and an oracle $\mathcal{O}_f$
such that given oracle access to $\mathcal{O}_f$,
\begin{enumerate}
\item $f(U_m)$ is samplable, and has entropy $m$.
\item $f(U_m)$ cannot be compressed to length less than $n - 2 \log s - \log n - O(1)$
by oracle circuits of size $s$.
\end{enumerate}
\end{lemma}
By combining Lemma~\ref{lem:imp_comp} and Theorem~\ref{th:correct_comp},
we obtain the following theorem.
\begin{theorem}\label{th:oracle}
For any $m$ satisfying $6 \log s + O(1) < m < n$,
there are a function $f : \{0, 1\}^m \to \{0, 1\}^n$ and an oracle $\mathcal{O}_f$
such that given oracle access to $\mathcal{O}_f$,
\begin{enumerate}
\item $f(U_m)$ is samplable, and has entropy $m$.
\item $f(U_m)$ is not correctable with rate $R > (2 \log s + 7 \log n + O(1))/n$
by any linear code $(\Enc, \Dec)$ implemented by an oracle circuit of size $s$,
where $\Dec$ is a syndrome decoder.
\end{enumerate}
\end{theorem}
\begin{proof}
Item~1 is the same as Lemma~\ref{lem:imp_comp}.
We prove Item~2 in the rest.
For contradiction, assume that
there is an oracle circuit of size $s$ such that
the circuit implements a linear code $(\Enc, \Dec)$ in which $\Dec$ is a syndrome decoder,
and $(\Enc, \Dec)$ corrects $f(U_m)$ with rate $R > (2 \log s + 7 \log n + O(1) )/n$.
By Theorem~\ref{th:correct_comp},
we can construct $(\Com, \Decom)$ that can compress $f(U_m)$ to length $n(1-R) < n - 2 \log s - 7 \log n - O(1)$,
and is implemented by an oracle circuit of size $s + n^3$,
where the addition term of $n^3$ is due to the computation of $\Com$,
which is defined as $\Com(z) = z \cdot H^T$.
This contradicts Lemma~\ref{lem:imp_comp}.
\end{proof}
The following corollary immediately follows.
\begin{corollary}\label{cor:oracle}
For any $m$ satisfying $\omega(\log n) < m < n$,
there is an oracle relative to which there is a samplable distribution with membership test of entropy $m$
that is not correctable by linear codes of rate $R > \omega((\log n)/n)$ with efficient syndrome decoding.
\end{corollary}
}
\section{Errors with Membership Test}
Since pseudorandom distributions are not correctable by efficient schemes,
we investigate the correctability of distributions that are not pseudorandom.
For such distributions, we consider distributions for which the membership test can be done efficiently.
A distribution $Z$ is called a \emph{distribution with membership test}
if there is a polynomial-time algorithm $D$ such that
$D(z) = 1 \Leftrightarrow z \in \mathrm{Supp}(Z)$.
Since the algorithm $D$ can distinguish $Z$ from the uniform distribution,
$Z$ is not pseudorandom.
We show that there is an oracle relative to which there exists a samplable distribution with membership test
that is not correctable by efficient coding schemes with low rate.
Let $N = 2^n, K = 2^k, M = 2^m$.
Let $\mathcal{F}$ be the set of injective functions $f : \{0, 1\}^m \to \{0, 1\}^n$.
For each $f \in \mathcal{F}$, define an oracle $\mathcal{O}_f$ such that
\begin{equation*}
\mathcal{O}_f(b,y) =
\begin{cases}
\mathcal{O}_f^S(y) & \text{if } b = 0, y \in \{0, 1\}^m\\
\mathcal{O}_f^M(y) & \text{if } b = 1, y \in \{0, 1\}^n\\
\bot & \text{otherwise}
\end{cases},\\
\ \ \mathcal{O}^M_f(y) =
\begin{cases}
1 & \text{if } y \in f(\{0, 1\}^m)\\
0 & \text{if } y \notin f(\{0, 1\}^m)\\
\end{cases},\\
\ \ \mathcal{O}^S_f(y) = f(y).
\end{equation*}
Let $\mathsf{correctf}$ be the set of functions $f \in \mathcal{F}$ for which
there exist oracle circuits $(\Enc, \Dec)$ that make $q$ queries to oracle $\mathcal{O}_f$
and correct $f(U_m)$ with rate $k/n$.
For each $f \in \mathcal{F}$ and the corresponding $(\Enc, \Dec)$, we define
\begin{align*}
\mathsf{invert}_f & = \{ y \in \{0, 1\}^m : \text{for any $x \in \{0, 1\}^k$, on input $\Enc(x) + f(y)$,} \\
& \ \ \ \ \ \
\text{$\Dec$ queries $\mathcal{O}_f^S$ on $y$} \}, \\
\mathsf{forge}_f & = \{ y \in \{0, 1\}^m : \text{for some $x \in \{0, 1\}^k$, on input $\Enc(x) + f(y)$,} \\
& \ \ \ \ \ \
\text{$\Dec$ does not query $\mathcal{O}_f^S$ on $y$} \}.
\end{align*}
Note that $\mathsf{invert}_f$ and $\mathsf{forge}_f$ is a partition of $\{0, 1\}^m$.
We also define
\begin{align*}
\mathsf{invertible} & = \{ f \in \mathsf{correctf} : |\mathsf{invert}_f| > \epsilon \cdot 2^m \},\\
\mathsf{forgeable} & = \{ f \in \mathsf{correctf} : |\mathsf{forge}_f| \geq \delta \cdot 2^m \},
\end{align*}
where $\epsilon$ and $\delta$ are any positive constants satisfying $\epsilon + \delta = 1$.
Note that $\mathsf{correctf} = \mathsf{invertible} \cup \mathsf{forgeable}$.
Intuitively, if $f$ is in $\mathsf{invertible}$, then there is a small circuit that inverts $f$.
This is done by computing $\Enc(x) + f(y)$ and monitoring oracle queries that
$\Dec(\Enc(x)+f(y))$ makes to $\mathcal{O}_f^S$.
Since a random function is one-way with high probability,
we can show that the size of invertible functions, i.e., $\mathsf{invertible}$, is small.
Similarly, if $f$ is in $\mathsf{forgeable}$, then $\Dec$ corrects $f(y)$ without querying $\mathcal{O}_f^S$ on $y$.
This means that
$f(y)$ can be described using $\Dec$ and $\Enc(x)+f(y)$, and thus if $\Enc(x)+f(y)$ has a short description,
the size of $\mathsf{forgeable}$ is small.
To argue the above intuition formally, we use the reconstruction paradigm of~\cite{GT00}.
Then, we show that both $\mathsf{invertible}$ and $\mathsf{forgeable}$ are small.
First, we show that $f \in \mathsf{invertible}$ has a short description.
\begin{lemma}\label{lem:invertible}
Take any $f \in \mathsf{invertible}$ and the corresponding pair of oracle circuits $(\Enc, \Dec)$ that makes at most $q$ queries to $\mathcal{O}_f$ in total
and corrects $f(U_m)$ with rate $k/n$.
Then $f$ can be described using at most
\[ \log{N \choose c} + \log{M \choose c} + \log\left( {N-c \choose M-c}(M-c)!\right) \]
bits, given $(\Enc, \Dec)$, where $c = \epsilon M/q$.
\end{lemma}
\begin{proof}
First, consider an oracle circuit $A$ such that, on input $z$,
$A$ picks any $x \in \{0, 1\}^k$ and simulates $\Dec$ on input $\Enc(x) + z$.
Then, for any $y \in \mathsf{invert}_f$, on input $f(y)$, $A$ outputs $y$ by making at most $q$ queries to $\mathcal{O}_f$.
Next, we show that for any $f \in \mathsf{invertible}$, $f$ has a short description given $A$.
Without loss of generality, we assume that $A$ makes distinct queries to $\mathcal{O}_f^S$.
We also assume that on input $f(y)$, $A$ always queries $\mathcal{O}_f^S$ on $y$ before it outputs $y$.
We will show that there is a subset $T \subseteq f(\mathsf{invert}_f)$ such that $f$ can be described given
$T$, $B(T)$, $f|_{\{0, 1\}^m \setminus B(T)}$, where $B(T) = \{ y \in \{0, 1\}^m : y \leftarrow A(z), z \in T\}$.
We describe how to construct $T$ below.
\medskip
\noindent \textsc{Construct-$T$}:
\begin{enumerate}
\itemsep=1pt\parskip=1pt\topskip=0pt
\item Initially, $T$ is empty, and all elements in $T^* = f(\mathsf{invert}_f)$ are candidates for inclusion in $T$.
\item \label{step:2**} Choose the lexicographically smallest $z$ from $T^*$, put $z$ in $T$, and remove $z$ from $T^*$.
\item \label{step:3*} Simulate $A$ on input $z$, and halt the simulation immediately after $A$ queries $\mathcal{O}_f^S$ on $y$.
Let $y_1', \dots, y_p'$ be the queries that $A$ makes to $\mathcal{O}_f^S$, where $y_p' = y$ and $p \leq q$.
\begin{itemize}
\itemsep=1pt\parskip=1pt\topskip=0pt
\item Remove $f(y_1'), \dots, f(y_{p-1}')$ from $T^*$.
(This means that these elements will never belong to $T$, and in simulating $A(z)$ in the recovering phase,
the answers to these queries are made by using the look-up table for $f$.)
\item Continue to remove the lexicographically smallest $z$ from $T^*$
until we have removed exactly $q-1$ elements in Step~\ref{step:3*}.
\end{itemize}
\item Return to Step~\ref{step:2**}.
\end{enumerate}
Next, we describe how to reconstruct $f$ from $T$, $B(T)$, and $f|_{\{0, 1\}^m \setminus B(T)}$.
We show how to recover the look-up table for $f$ on values in $B(T)$.
\medskip
\noindent \textsc{Recover-$f$}:
\begin{enumerate}
\itemsep=1pt\parskip=1pt\topskip=0pt
\item \label{step:1*} Choose the lexicographically smallest element $z \in T$, and remove it from $T$.
\item Simulate $A$ on input $z$, and halt the simulation immediately after $A$ queries $\mathcal{O}_f^S$ on $y$ for which
the answer does not exist in the look-up table for $f$.
Since the query $y$ satisfies that $y = f^{-1}(z)$, add the entry $(y, z)$ to the look-up table.
In what follows, we explain why we can correctly simulate $A(z)$.
\begin{itemize}
\itemsep=1pt\parskip=1pt\topskip=0pt
\item Since $B(T)$ and $f|_{\{0, 1\}^m \setminus B(T)}$ are given, we can answer all queries to $O_f^M$.
\item For any query $y'$ to $O_f^S$, it must be either (1) $y' \notin B(T)$, or (2) $y'$ is the output of $A$ on input $z'$ such that
$z' \in W$ and $z'$ is lexicographically smaller than $z$.
In either case, the look-up table has the corresponding entry, and thus we can answer the query.
\end{itemize}
\item Return to Step~\ref{step:1*}.
\end{enumerate}
In each iteration in \text{Construct-$T$}, we add one element to $T$ and remove exactly $q$ element from $T^*$.
Since initially the size of $T^* = f(\mathsf{invert}_f)$ is $\epsilon M$, the size of $T$ in the end is $c = \epsilon M/q $.
The sets $T$ and $B(T)$, and the look-up table for $f|_{\{0, 1\}^m \setminus B(T)}$ can be described using
$\log{N \choose c}$, $\log{M \choose c}$, and $\log( {N-c \choose M-c}(M-c)!)$, respectively.
Therefore, the statement follows.
\end{proof}
We show that the fraction of $f \in \mathcal{F}$ for which $f \in \mathsf{invertible}$ and $f(U_m)$ is correctable is small.
\begin{lemma}\label{lem:invertiblecount}
If $m > 3 \log s + \log n + O(1)$,
then the fraction of functions $f \in \mathcal{F}$ such that $f \in \mathsf{invertible}$ and
$f(U_m)$ can be corrected by a pair of oracle circuits $(\Enc, \Dec)$ of total size $s$
is less than $2^{-(sn\log s+1)}$
for all sufficiently large $n$.
\end{lemma}
\begin{proof}
It follows from Lemma~\ref{lem:invertible} that, given $(\Enc, \Dec)$, the fraction is
\begin{equation*}
\frac{|\mathsf{invertible}|}{{N \choose M}M!} \leq \frac{{N \choose c}{M \choose c}{N-c \choose M-c}(M-c)!}{{N \choose M}M!}
= \frac{{M \choose c}}{c!},
\end{equation*}
where $c = \epsilon M/(qK)$.
By using the fact that $q \leq s$ and the inequalities ${n \choose k} < \left( \frac{en}{k} \right)^k$ and $n! > \left( \frac{n}{e} \right)^n$,
the expression is upper bounded by
\begin{equation*}
\left(\frac{eM}{c} \right)^c \left( \frac{e}{c} \right)^c\\
= \left( \frac{e^2 q^2}{\epsilon^2M} \right)^{{\epsilon M}/{q}}\\
< \left( \frac{1}{2} \right)^{ns \log s + 1}
\end{equation*}
for all sufficiently large $n$.
The last inequality follows from the fact that
\begin{equation*}
\frac{e^2 q^2}{\epsilon^2 M} < \frac{e^2 q^2}{\epsilon^2 \, \Omega(s^3 n)} < \frac{1}{2} \text{\quad and \quad}
\frac{\epsilon M}{q} > \frac{\epsilon \, \Omega(s^3 n)}{q} > n s \log s + 1.
\end{equation*}
\end{proof}
Next, we show that $\mathsf{forgeable}$ has a short description.
\begin{lemma}\label{lem:forgeable}
Take any $f \in \mathsf{forgeable}$ and the corresponding pair of oracle circuits $(\Enc, \Dec)$ that make at most $q$ queries to $\mathcal{O}_f$ in total
and corrects $f(U_m)$ with rate $k/n$.
Then $f$ can be described using at most
\[ \log{M \choose d} + \log\left( {N-d \choose M-d}(M-d)!\right) + d(k + m + \log q)\]
bits, given $(\Enc, \Dec)$, where $d = \delta M/q$.
\end{lemma}
\begin{proof}
First, consider an oracle circuit $A$ such that, on input $w$,
$A$ obtains $x$ by simulating $\Dec$ on input $w$,
queries $\mathcal{O}_f^M$ on $w - \Enc(x)$,
and outputs $\bot$ if $\mathcal{O}_f^M(w - \Enc(x)) = 0$, and $x$ otherwise.
Then, $A$ satisfies that, on input $w$, $A$ outputs $\bot$ if $w \notin \Enc(\{0, 1\}^k)+f(\{0, 1\}^m)$,
and $\Dec(w)$ otherwise.
Next, we show that for any $f \in \mathsf{forgeable}$, $f$ has a short description given $A$.
Without loss of generality, we assume that $A$ makes distinct queries to $\mathcal{O}_f^S$ and $\mathcal{O}_f^M$.
We also assume that
for $x \in \{0, 1\}^k$ and $y \in \{0, 1\}^m$, $A(\Enc(x)+f(y))$ always queries $\mathcal{O}_f^M$ on $f(y)$ before it outputs $x$.
Note that for $y \in \mathsf{forge}_f$, there is some $x \in \{0, 1\}^k$ such that, on input $\Enc(x)+f(y)$, $A$ does not query $\mathcal{O}_f^S$ on $y$.
We will show that there is a subset $Y \subseteq \mathsf{forge}_f$
such that $f$ can be described given
$Y$, $f|_{\{0, 1\}^m \setminus Y}$, and $\{ (x_y, a_y, b_y) \in \{0, 1\}^k \times [M] \times [q] : y \in Y\}$ of a set of advice strings.
For $x \in \{0, 1\}^k$, we define $D(x) = \{ \Enc(x) + f(y) : y \in \{0, 1\}^m \}$.
Note that $|D(x)| = M$ for any $x \in \{0, 1\}^k$.
We describe how to construct $Y$ below.
\medskip
\noindent \textsc{Construct-$Y$}:
\begin{enumerate}
\itemsep=1pt\parskip=1pt\topskip=0pt
\item Initially, $Y$ is empty. All elements in $Y^* = \mathsf{forge}_f$ are candidates for inclusion in $Y$.
For every $x \in \{0, 1\}^k$, set $D_x = \{ \Enc(x) + f(y) : y \in \mathsf{forge}_f \}$.
We write $\mathcal{D}_k = \bigcup_{x \in \{0, 1\}^k}D_x$.
\item \label{step:2}
Choose the lexicographically smallest $y$ from $Y^*$, put $y$ in $Y$, and remove $y$ from $Y^*$.
\item
Choose the lexicographically smallest $w$ from
the set of $\Enc(x) + f(y) \in D_x$ such that $A$ does not query $\mathcal{O}_f^S$ on $y$.
If $w = \Enc(x) + f(y)$, set $x_y = x$.
Then, for every $x' \in \{0, 1\}^k$, remove $\Enc(x')+f(y)$ from $D_{x'}$.
(This removal means that hereafter there are no elements in $\mathcal{D}_k$ for which $A$ outputs some $x$ such that $f(y)$ is the error vector.)
When $w$ is the lexicographically $t$-th smallest element in $D(x)$, set $a_y = t$
(so that we can recognize that the $a_y$-th element in $D(x)$ is $w$ in the recovering phase).
\item \label{step:3} Simulate $A$ on input $w$, and halt the simulation immediately after $A$ queries $\mathcal{O}_f^M$ on $f(y)$.
Let $y_1', \dots, y'_p$ be the queries that $A$ makes to $\mathcal{O}^S_f$, and
$z_1', \dots, z_r'=f(y)$ be the queries that $A$ makes to $\mathcal{O}^M_f$.
Set $b_y = r$
(so that we can recognize that the $b_y$-th query that $\Dec$ makes to $\mathcal{O}_f^M$ is $f(y)$ in the recovering phase).
\begin{enumerate}
\itemsep=1pt\parskip=1pt\topskip=0pt
\item For every $x' \in \{0, 1\}^k$,
remove $\Enc(x')+f(y_1'), \dots, \Enc(x')+f(y_p')$ from $D_{x'}$.
\item For every $i \in [p]$, if $z_i' \in f(\mathsf{forge}_f)$, then for every $x' \in \{0, 1\}^k$, remove $\Enc(x')+z_i'$ from $D_{x'}$, and otherwise, do nothing.
\item
Continue to remove the elements $\Enc(x')+f(y)$ from $D_{x'}$ for every $x' \in \{0, 1\}^k$
for the lexicographically smallest $w = \Enc(x)+f(y) \in \mathcal{D}_k$
until we have removed exactly $(q-1)K$ elements from $\mathcal{D}_k$ in Step~\ref{step:3}.
\end{enumerate}
\item Return to Step~\ref{step:2}.
\end{enumerate}
Next, we describe how to construct $f$ from $Y$, $f|_{\{0, 1\}^m \setminus Y}$, and $\{ (x_y, a_y, b_y) \in \{0, 1\}^k \times [M] \times [q] : y \in Y\}$.
We show how to recover the look-up table for $f$ on values in $Y$.
\medskip
\noindent \textsc{Recover-$f$:}
\begin{enumerate}
\itemsep=1pt\parskip=1pt\topskip=0pt
\item \label{step:1}
Choose the lexicographically smallest $y \in Y$, and remove it from $Y$.
Then, choose the lexicographically $a_y$-th smallest element $w$ from $D(x_y)$.
\item Simulate $A$ on input $w$, and halt the simulation immediately after $A$ makes the $b_y$-th query to $\mathcal{O}_f^M$.
Since the $b_y$-th query is $f(y)$, add the entry $(y, f(y))$ to the look-up table.
In what follows, we explain why we can correctly simulate $A(w)$.
\begin{itemize}
\itemsep=1pt\parskip=1pt\topskip=0pt
\item For any query $y'$ to $\mathcal{O}_f^S$, it must be either
(1) $y' \notin Y$ or (2) $y'$ is lexicographically smaller than $y$.
In case (1), we can answer the query by using $f|_{\{0, 1\}^m \setminus Y}$.
In case (2), since $y$ was chosen as the lexicographically smallest element such that $A$ does not query $\mathcal{O}_f^S$ on $y$,
the look-up table has the answer to the query.
\item Consider any of the first $b_y-1$ queries $z'$ to $\mathcal{O}_f^M$.
If $z' \in f(\{0, 1\}^m)$, namely $z' = f(y')$ for some $y'$,
then it must be either (1) $y' \notin Y$ or (2) $y'$ is lexicographically smaller than $y$.
In either case, the look-up table has the entry $(y', z')$.
If $z' \notin f(\{0, 1\}^m)$, there is no entry for $z'$ in the look-up table.
Thus, we can answer the query by saying ``yes'' if $z'$ is in the look-up table, and ``no'' otherwise.
\end{itemize}
\item Return to Step~\ref{step:1}.
\end{enumerate}
In each iteration in~\textsc{Construct-$Y$},
we add one element to $Y$ and remove exactly $qK$ elements from $\mathcal{D}_k$.
Since initially the size of $\mathcal{D}_k$ is at least $\delta KM$,
the size of $Y$ in the end is at least $d = \delta M/q$.
The set $Y$, the look-up table for $f|_{\{0, 1\}^m \setminus Y}$,
the sets $\{ (x_y, a_y, b_y) \in \{0, 1\}^k \times [M] \times [q] : y \in Y\}$
can be described using $M \choose d$,
$\log( {N-d \choose M-d}(M-d)!)$, and $d (k + m +\log q)$ bits respectively.
Therefore, the statement follows.
\end{proof}
We show that the fraction of $f \in \mathcal{F}$ for which $f \in \mathsf{forgeable}$ and $f(U_m)$ is correctable is small.
\begin{lemma}\label{lem:forgeablecount}
If $m > 3 \log s + \log n + O(1)$ and $m < n-k-2\log s-O(1)$,
then the fraction of functions $f \in \mathcal{F}$ such that $f \in \mathsf{forgeable}$ and
$f(U_m)$ can be corrected by a pair of oracle circuits $(\Enc, \Dec)$ of total size $s$
is less than $2^{-(sn\log s+1)}$
for all sufficiently large $n$.
\end{lemma}
\begin{proof}
It follows from Lemma~\ref{lem:forgeable} that, given $(\Enc, \Dec)$, the fraction is
\begin{equation*}
\frac{|\mathsf{forgeable}|}{{N \choose M}M!} \leq \frac{{M \choose d}{N-d \choose M-d}(M-d)!}{{N \choose M}M!}2^{d(k+m+\log q)}
= \frac{{M \choose d}}{{N \choose d}d!}(qKM)^d,
\end{equation*}
where $d = \delta M/q$.
By using the fact that $q \leq s$ and the inequalities ${n \choose k} < \left( \frac{en}{k} \right)^k$, ${n \choose k} > \left( \frac{n}{k} \right)^k$, and $n! > \left( \frac{n}{e} \right)^n$,
the expression is upper bounded by
\begin{equation*}
\left(\frac{eM}{d} \right)^d \left(\frac{d}{N} \right)^d \left( \frac{e}{d} \right)^d (qKM)^d\\
= \left( \frac{e^2 q^2 KM}{\delta N} \right)^{\delta M/q}
< \left( \frac{1}{2} \right)^{ns \log s +1}
\end{equation*}
for all sufficiently large $n$.
The last inequality follows from the fact that
\begin{equation*}
\frac{e^2 q^2 KM}{\delta N} < \frac{e^2 q^2}{\delta \, \Omega(s^2n)} < \frac{1}{2}
\text{\quad and \quad}
\frac{\delta M}{q} > \frac{\delta \, \Omega(s^3 n)}{q} > n s \log s + 1.
\end{equation*}
\end{proof}
We obtain the main result of this section.
\begin{theorem}\label{th:lowentropy}
For any $m$ and $k$ satisfying $3 \log s + \log n + O(1) < m < n - k - 2\log s - O(1)$,
there exist injective functions $f : \{0, 1\}^m \to \{0, 1\}^n$ such that,
given oracle access to $\mathcal{O}_f$,
(1) $f(U_m)$ is a samplable distribution with membership test of entropy $m$,
and (2) $f(U_m)$ cannot be corrected with rate $k/n$ by oracle circuits of size $s$.
\end{theorem}
\begin{proof}
Since $\mathsf{correctf} = \mathsf{invertible} \cup \mathsf{forgeable},$
it follows from Lemmas~\ref{lem:invertiblecount} and~\ref{lem:forgeablecount}
that for a fixed $(\Enc, \Dec)$ of size $s$,
the fraction of functions $f \in \mathcal{F}$ such that $(\Enc, \Dec)$ corrects $f(U_m)$ with rate $k/n$
is less than $2^{-(sn\log s)}$.
Since there are at most $2^{sn \log s}$ circuits of size $s$,
there are functions $f \in \mathcal{F}$ such that $f(U_m)$ cannot be corrected with rate $k/n$ by oracle circuits of size $s$.
Given oracle access to $\mathcal{O}_f$, $f(U_m)$ is samplable. Since $f$ is injective, $f(U_m)$ has entropy $m$.
\end{proof}
The following corollary immediately follows.
\begin{corollary}\label{cor:lowentropy}
For any $m$ and $k$ satisfying $\omega(\log n) < m < n - k - \omega(\log n)$,
there exists an oracle relative to which there exists a samplable distribution with membership test of entropy $m$
that cannot be corrected with rate $k/n$ by polynomial size circuits.
\end{corollary}
\ignore{
\subsection{Errors from Small-Biased Distributions}
A sample space $S \subseteq \{0, 1\}^n$ is said to be \emph{$\delta$-biased}
if for any non-zero $\alpha \in \{0, 1\}^n$, $|\mathbb{E}_{s \sim U_S}[(-1)^{\alpha \cdot s}]| \leq \delta$,
where $U_S$ is the uniform distribution over $S$.
Dodis and Smith~\cite{DS05} proved that
small-biased distributions can be used as sources of keys
of the one-time pad for high-entropy messages.
This result implies that high-rate codes cannot correct errors from small-biased distributions.
\begin{theorem}\label{th:biased}
Let $S$ be a $\delta$-biased sample space over $\{0, 1\}^n$.
If a code of rate $R$ corrects $U_S$ with error $\epsilon < 1/2$,
then $R \leq 1 - (2\log(1/\delta)+1)/n$.
\end{theorem}
\begin{proof}
Assume for contradiction that $(\Enc, \Dec)$ corrects $Z = U_S$ with rate $R$ and error $\epsilon$.
Dodis and Smith~\cite{DS05} give the one-time pad lemma for high-entropy messages.
\begin{lemma}\label{lem:biasedotp}
For a $\delta$-biased sample space $S$,
there is a distribution $G$ such that
for every distribution $M$ over $\{0, 1\}^n$ with min-entropy at least $t$,
$\mathsf{SD}(M \oplus U_S, G) \leq \gamma$ for $\gamma = \delta 2^{(n-t-2)/2}$,
where $\oplus$ is the bit-wise exclusive-or.
\end{lemma}
For $b \in \{0, 1\}$, let $M_b \subseteq \{0, 1\}^{Rn}$ be the uniform distribution over
the set of strings in which the first bit is $b$. Note that the min-entropy of $M_b$ is $Rn-1$.
Define $D_b = \Dec(\Enc(M_b) \oplus U_S)$.
By Lemma~\ref{lem:biasedotp},
\begin{align*}
\mathsf{SD}(D_0, D_1)
& \leq \mathsf{SD}(\Enc(M_0) \oplus U_S, \Enc(M_1) \oplus U_S)\\
& \leq \mathsf{SD}(\Enc(M_0) \oplus U_S,G) + \mathsf{SD}(G,\Enc(M_1) \oplus U_S)\\
& \leq 2\gamma
\end{align*}
for $\gamma = \delta 2^{(n-Rn-1)/2}$.
Since the code corrects $U_S$ with error $\epsilon$, we have that $\mathsf{SD}(D_0,D_1) \geq 1 - 2\epsilon$.
Thus, we have that $2\gamma > 1-2\epsilon > 0$, and hence $R \leq 1 - (2\log(1/\delta)+1)/n$.
\end{proof}
Alon et al.~\cite{AGHP92} give a construction of $\delta$-biased sample spaces of size $O(n^2/\delta^2)$.
This leads to the following corollary.
\begin{corollary}\label{cor:biased}
There is a small-biased distribution of min-entropy at most $m$ that is not corrected by
codes with rate $R > 1 - m/n + (2\log n+O(1))/n$ and error $\epsilon < 1/2$.
\end{corollary}
}
\section{Necessity of One-Way Functions}
We show that if one-way functions do not exist, then any samplable flat distribution of entropy $m$
is correctable by an efficient coding scheme of rate $1-m/n-O(\log n/n)$.
For this, we use a technique used in the proof of~\cite[Theorem~6.3]{Wee04} that shows the necessity of one-way functions
for separating pseudoentropy and compressibility.
We observe that in its proof,
a family of linear hash functions is used for giving an efficient compression function.
Since a linear compression function is a dual object of a linear code that corrects additive errors,
we can use a family of linear hash functions for constructing an efficient decoder.
\begin{definition}[\cite{IL89}]
We say a function $f$ is \emph{distributionally one-way} if it is computable in polynomial time
and there exists a constant $c > 0$ such that
for every probabilistic polynomial-time algorithm $A$, the statistical distance between
$(x, f(x))$ and $(A(f(x)), f(x))$ is at least $1/n^c$, where $x \sim U_n$.
\end{definition}
\begin{theorem}[\cite{IL89}]
If there is a distributionally one-way function, then there is a one-way function.
\end{theorem}
\begin{theorem}\label{th:noowf}
If one-way functions do not exist,
then any samplable flat distribution $Z$ over $\{0, 1\}^n$ of entropy $m$ can be
corrected with rate $1 - m/n - (c\log n)/n$ and error $O(n^{-c})$ for any constant $c > 0$
by polynomial-time coding schemes.
\end{theorem}
\begin{proof}
Let $Z = f(U_m)$ for an efficiently computable function $f$.
Consider a family of linear universal hash functions $\mathcal{H} = \{h : \{0, 1\}^n \to \{0, 1\}^{n+2c\log n}\}$,
where
the universality means that for any distinct $x, x' \in \{0, 1\}^n$, $\Pr_{h \in \mathcal{H}}[ h(x) = h(x')] \leq 2^{-(m+2c\log n)}$,
and the linearity means that for any $x, x' \in \{0, 1\}^n$ and $a, b \in \{0, 1\}$, $h(ax+bx') = ah(x) + bh(x')$.
For each $h \in \mathcal{H}$, we define $C_h = \{ x \in \mathrm{Supp}(Z) : \exists x' \in \mathrm{Supp}(Z) \text{ s.t. } x' \neq x \wedge h(x) = h(x') \}$.
Namely, $C_h$ is the set of inputs with collisions under $h$.
By a union bound, it holds that for any $x \in \mathrm{Supp}(Z)$,
\begin{equation*}
\Pr_{h \in \mathcal{H}}[ \exists x' \in \mathrm{Supp}(Z) : x' \neq x \wedge h(x') = h(x)] \leq \frac{2^m}{2^{m+2c\log n}} = \frac{1}{n^{2c}}.
\end{equation*}
Thus, $E[ |C_h| ] \leq 2^m/n^{2c}$.
We say $h \in \mathcal{H}$ is good if $|C_h| \leq 2^m/n^c$.
By Markov's inequality, we have that $\Pr_{h \in \mathcal{H}}[ |C_h| > 2^m/n^c] < 1/n^c$.
Consider the function $g : \{0, 1\}^m \times \mathcal{H} \to \mathcal{H} \times \{0, 1\}^{m+2\log n}$ given by $g(y,h) = (h, h(f(y)))$.
Note that $g$ is polynomial-time computable.
By the assumption that one-way functions do not exist, and thus distributionally one-way functions do not exist,
there is a polynomial-time algorithm $A$ such that the statistical distance between $(y, h, g(y, h))$ and $(A(g(y,h)), g(y,h))$ is
at most $n^{-c}$,
where $y \sim U_m$ and $h \in \mathcal{H}$.
Then, it holds that
\begin{equation*}
\Pr_{A,y,h}[g(A(g(y,h))) = g(y,h)] \geq 1 - \frac{1}{n^c},
\end{equation*}
where the probability is taken over the random coins of $A$, $y \sim U_m$, and $h \in \mathcal{H}$.
Thus, we have that
\begin{equation*}
\Pr_{A,y,h}[ g(A(g(y,h))) = g(y,h) \wedge \text{$h$ is good}] \geq 1 - \frac{2}{n^c}.
\end{equation*}
By fixing the coins of $A$ and $h \in \mathcal{H}$,
it holds that there are deterministic algorithm $A'$ and $h_0 \in \mathcal{H}$ such that $h_0$ is good and
\begin{equation*}
\Pr_{y}[ g(A'(g(y,h_0))) = g(y,h_0) ] \geq 1 - \frac{2}{n^c}.
\end{equation*}
For $y \in \{0, 1\}^m$ satisfying $g(A'(g(y,h_0))) = g(y, h_0)$,
we write $A'(g(y,h_0)) = (y', h')$, where $A_1'(g(y,h_0)) = y'$ and $A_2'(g(y,h_0)) = h'$.
Then, it holds that $h' = h_0$ and $h_0(f(y)) = h_0(f(y'))$.
Furthermore, since $h_0$ is good, $\Pr_y[ f(y) \notin C_{h_0}] \geq 1 - 1/n^c$.
Let $H_0 \in \{0, 1\}^{(m + c\log n) \times n}$ be a matrix such that $x H_0^T = h_0(x)$ for $x \in \{0, 1\}^n$.
(Such matrices exist since $\mathcal{H}$ is a set of linear hash functions.)
Consider a linear coding scheme in which $H_0$ is employed as the parity check matrix, and $A_1'$ is employed for recovering errors from syndromes.
That is, $\Enc(x) = x G$ for a matrix $G \in \{0, 1\}^{(n-m-c\log n) \times n}$ satisfying $GH_0^T = 0$,
and $\Dec(y) = (y - f(A_1'(h_0, y H_0^T)) ) G^{-1}$,
where $G^{-1} \in \{0, 1\}^{n \times Rn}$ is a right inverse matrix of $G$.
Then, for any $x \in \{0, 1\}^m$,
\begin{align*}
&\Pr_{y \sim U_r}[\Dec( \Enc(x) + f(y)) = x] \\
& = \Pr_{y \sim U_r}[\Enc(x)+f(y) - f(A_1'(h_0, (\Enc(x)+f(y))H_0^T)) = xG]\\
& = \Pr_{y \sim U_r}[f(A_1'(g(y,h_0))) = f(y)],
\end{align*}
where we use the property that $GG^{-1}=I$, $\Enc(x) = xG$, $GH_0^T = 0$, and $x H_0^T = h_0(x)$.
Since the probability that $g(A_0(g(y,h_0))) = g(y,h_0)$ is at least $1 - 2/n^c$,
and for any $y \in \{0, 1\}^m$ satisfying $g(A_0(g(y,h_0))) = g(y,h_0)$, $\Pr_y[ f(y) \notin C_{h_0}] \geq 1 - 1/n^c$,
we have that
\begin{equation*}
\Pr_{y \sim U_m}[f(A_1'(g(y,h_0))) = f(y)] \geq 1 - \frac{3}{n^c}.
\end{equation*}
Hence the statement follows.
\end{proof}
\section{Conclusions}
In this work, we study the correctability of samplable additive errors with unbounded error-rate.
We have considered a relatively simple setting in which the error distribution is identical for every coding scheme and codeword.
The results imply that even when a distribution is not pseudorandom by membership test,
it is difficult to correct every such samplable distribution by efficient coding schemes.
Nevertheless, a positive result can be obtained if we consider much more structured errors such as errors from linear subspaces.
We present some possible future work of this study.
\setcounter{paragraph}{0}
\paragraph{Further study on the correctability.}
In this work, we have mostly discussed impossibility results.
Thus, showing non-trivial possibility results is interesting.
A possible direction is to consider more structured errors than samplable errors.
One can consider \emph{computationally} structured errors
such as errors computed by log-space machines, constant-depth circuits, or monotone circuits.
Also, one can consider other types of structures, e.g., errors are introduced in a \emph{split-state} manner.
Namely, an error vector is split into several parts, and each part is independently computed.
This model has been well-studied in the context of leakage-resilient cryptography~\cite{DP08,LL12} and non-malleable codes~\cite{DPW10,CG17,ADL14}.
BSC can be seen as an extreme of this type of channels in which each error bit is computed by the same biased-sampler.
\ignore{
\paragraph{Generalizing the results of Cheraghchi~\cite{Che09}.}
In Section~\ref{sec:flat}, we have used the results of Cheraghchi~\cite{Che09}
who showed that a linear lossless condenser for a flat distribution $Z$ is equivalent to a linear code ensemble
in which most of them correct additive errors from $Z$.
One possible future work is to generalize this result to more general distributions than flat distributions.
Although the decoding complexity is not considered in the above equivalence,
as presented by Cheraghchi~\cite{Che09} for binary-symmetric channels,
it is possible to construct an efficient coding scheme using
inefficient decoders based on Justesen's concatenated construction~\cite{Jus72}.
It may be interesting to explore other distributions (or characterize distributions) that can be efficiently correctable by Justesen's construction.
}
\paragraph{Characterizing correctability.}
We have investigated the correctability of samplable additive errors
using the Shannon entropy as a criterion.
There may be another better criterion for characterizing the correctability of these errors,
which might be related to efficient computability, to which samplability is directly related.
Since all the results in this paper deal with flat distributions,
the results can be stated using other entropies such as the min-entropy.
Since we have considered general distributions as error distributions,
the information-spectrum approach~\cite{VH94,Han03} may be more plausible.
\section*{Acknowledgment}
This research was supported in part by JSPS/MEXT Grant-in-Aid for Scientific Research Numbers 25106509, 15H00851, and 16H01705.
We thank anonymous reviewers for their helpful comments.
\bibliographystyle{abbrv}
|
1,108,101,564,538 | arxiv | \section*{Introduction}
The large deviation principle (LDP) for Brownian motion $\beta$ on $[0,1]$
- contained in Schilder's theorem (\cite{Schilder}) - describes the exponential decay of the
probabilities with which $\sqrt{\varepsilon} \beta$ takes values in closed
or open subsets of the path space of continuous functions in which the
trajectories of $\beta$ live. The path space is equipped with the topology
generated by the uniform norm. The decay is dominated by a rate function
capturing the 'energy' $\frac{1}{2} \int_0^1 (\dot{f}(t))^2 dt$ of
functions $f$ on the Cameron-Martin space for which a square integrable
derivative exists. Schilder's theorem is of central importance to the
theory of large deviations for randomly perturbed dynamical systems or
diffusions taking their values in spaces of continuous functions (see
\cite{Freidlin}, \cite{Dembo}, and references therein, \cite{Vares}). A
version of Schilder's theorem for a $Q$-Wiener processes $W$ taking values
in a separable Hilbert space $H$ is well known (see \cite{DaPrato},
Theorem 12.7 gives an LDP for Gaussian laws on Banach spaces). Here
$Q$ is a self adjoint positive trace class operator on $H$. If
$(\lambda_i)_{i\ge 0}$ are its summable eigenvalues with respect to an
eigenbasis $(e_k)_{k\ge 0}$ in $H$, $W$ may be represented with respect to
a sequence of one dimensional Wiener processes $(\beta_k)_{k\ge 0}$ by $W =
\sum_{k=0}^\infty \lambda_k \beta_k\, e_k$. The LDP in this framework can
be derived by means of techniques of reproducing kernel Hilbert spaces (see
\cite{DaPrato}, Chapter 12.1). The rate
function is then given by an analogous energy functional for which $\dot{f}^2$
is replaced by $\lVert Q^{-\frac{1}{2}}\dot{F}\rVert^2$ for continuous functions
$F$ possessing square integrable derivatives $\dot{F}$ on $[0,1]$.
Schilder's theorem for $\beta$ may for instance be derived via
approximation of $\beta$ by random walks from LDP principles for discrete
processes (see \cite{Dembo}). \cite{Baldi} give a very elegant alternative
proof of Schilder's theorem, the starting point of which is a Fourier
decomposition of $\beta$ by a complete orthonormal system (CONS) in
$L^2([0,1])$. The rate function for $\beta$ is then simply calculated by
the rate functions of one-dimensional Gaussian unit variables. In this
approach, the LDP is first proved for balls of the topology, and then
generalized by means of exponential tightness to open and closed sets of
the topology. As a special feature of the approach, Schilder's theorem is
obtained in a stricter sense on all spaces of H\"older continuous functions
of order $\alpha<\frac{1}{2}$. This enhancement results quite naturally
from a characterization of the H\"older topologies on function spaces by
appropriate infinite sequence spaces (see \cite{Ciesielski}). Representing
the one-dimensional Brownian motions $\beta_k$ for instance by the CONS of
Haar functions on $[0,1]$, we obtain a description of the Hilbert space
valued Wiener process $W$ in which a double sequence of independent
standard normal variables describes randomness. Starting with this
observation, in this paper we extend the direct proof of Schilder's theorem
by \cite{Baldi} to $Q$-Wiener processes $W$ with values on
$H$. On the way, we also retrieve the enhancement of the LDP to spaces of
H\"older continuous functions on $[0,1]$ of order $\alpha<\frac{1}{2}$. The
idea of approaching problems related to stochastic processes with values in
function spaces by sequence space methods via Ciesielski's isomorphism is
not new: it has been employed in \cite{BenarousGradinaru} to give an
alternative treatment of the support theorem for Brownian motion, in
\cite{BenarousLedoux} to enhance the Freidlin-Wentzell theory from the
uniform to H\"older norms, and in \cite{Eddahbi} and \cite{EddahbiOuknine}
further to Besov-Orlicz spaces.
In Section \ref{preliminaries} we first give a generalization of
Ciesielski's isomorphism of spaces of H\"older continuous functions and
sequence spaces to functions with values on Hilbert spaces. We briefly
recall the basic notions of Gaussian measures and Wiener processes on
Hilbert spaces. Using Ciesielski's isomorphism we give a Schauder
representation of Wiener processes with values in $H$. Additionally we give
a short overview of concepts and results from the theory of LDP needed in
the derivation of Schilder's theorem for $W$. In the main Section \ref{LDP}
the alternative proof of the LDP for $W$ is given. We first introduce a new
norm on the space of H\"older continuous functions $C_\alpha([0,1],H)$ with
values in $H$ which is motivated by the sequence space representation in
Ciesielski's isomorphism, and generates a coarser topology. We adapt the
description of the rate function to the Schauder series setting, and then
prove the LDP for a basis of the coarser topology using Ciesielski's
isomorphism. We finally establish the last ingredient, the crucial property
of exponential tightness, by construction of appropriate compact sets in
sequence space.
\section{Preliminaries}\label{preliminaries}
In this section we collect some ingredients needed for the proof of
a large deviations principle for Hilbert space valued Wiener processes.
We first prove Ciesielski's theorem for Hilbert
space valued functions which translates properties of functions into properties of
the sequences of their Fourier coefficients with respect to complete orthonormal systems
in $L^2([0,1])$. We summarize some basic properties of Wiener
processes $W$ with values in a separable Hilbert space $H$. We then discuss
Fourier decompositions of $W$, prove that its trajectories
lie almost surely in $C_\alpha^0([0,1],H)$ and describe its
image under the Ciesielski isomorphism. We will always denote by $H$ a
separable Hilbert space equipped with a symmetric inner product
$\langle\cdot, \cdot\rangle$ that induces the norm $\lVert \cdot\rVert_H$ and a
countable complete orthonormal system (CONS) $(e_k)$ $k\in \mathbb{N}$.
\subsection{Ciesielski's isomorphism}
The \textbf{Haar functions} $(\chi_n, n \ge 0)$ are defined as $\chi_0 \equiv 1$,
\begin{align} \label{eq:haar functions def}
\chi_{2^k+l}(t) := \begin{cases}
\sqrt{2^k}, & \frac{2l}{2^{k+1}} \le t < \frac{2l+1}{2^{k+1}},\\
-\sqrt{2^k}, & \frac{2l+1}{2^{k+1}} \le t \le \frac{2l+2}{2^{k+1}}, \\
0, & \text{otherwise.}
\end{cases}
\end{align}
The Haar functions form a CONS of $L^2([0,1],dx)$. Note that because of their wavelet structure, the integral $\int_{[0,1]} \chi_n
df$ is well-defined for all functions $f$. For $n=2^k+l$ where $k\in \mathbb{N}$
and $0\le l\le 2^k-1$ we have $\int_{[0,1]} \chi_n
dF=\sqrt{2^k}[2F(\frac{2l+1}{2^{k+1}})-F(\frac{2l+2}{2^{k+1}})-
F(\frac{2l}{2^{k+1}})]$, and it does not matter whether $F$ is a real or
Hilbert space valued function.
The primitives of the Haar functions are called \textbf{Schauder functions}, and they are given by
\begin{align*}
\phi_n (t) = \int_0^t \chi_n(s) ds\text{, }t\in[0,1],\text{ }n\ge 0.
\end{align*}
Slightly abusing notation, we denote the $\alpha$-H\"older seminorms on
$C_\alpha([0,1];H)$ and on $C_\alpha([0,1];\mathbb{R})$ by the same symbols
\begin{align*}
\lVert F\rVert_\alpha &:= \sup_{0 \le s < t \le 1} \frac{ \lVert F(t) - F(s)\rVert_H}{ |t-s|^\alpha}, \quad F \in C_\alpha([0,1];H), \\
\lVert f\rVert_\alpha &:= \sup_{0 \le s < t \le 1} \frac{ |f(t) - f(s)|}{ |t-s|^\alpha}, \quad f \in C_\alpha([0,1];\mathbb{R})
\end{align*}
$C_\alpha([0,1];H)$ is of course the space of all functions $F: [0,1] \rightarrow H$ such that $\lVert F\rVert_\alpha < \infty$, and similarly for $C_\alpha([0,1];\mathbb{R})$. We also denote the supremum norm on $C([0,1]; H)$ and $C([0,1]; \mathbb{R})$ by the same symbol $\lVert \cdot \rVert_\infty$.
Denote in the sequel for an $H$-valued
function $F$ its orthogonal component with respect to $e_k$ by $F_k =
\langle F, e_k\rangle, k\ge 0.$ Further denote by $P_k$ (resp. $R_k$) the
orthogonal projectors on $\mbox{span}(e_1,\cdots, e_k)$ (resp. its
orthogonal complement), $k\ge 0.$ For every $F \in C_\alpha([0,1];H)$, every
$k\ge 0, s,t\in[0,1]$ we have
$$
| \langle F(t), e_k \rangle - \langle F(s), e_k \rangle |
\le \lVert F(t) - F(s)\rVert_H
$$
More generally, for any $k\ge 0, s,t\in[0,1]$ we have
$$ \lVert P_k F(t) - P_k F(s)\rVert_H
\le
\lVert F(t) - F(s)\rVert_H, \quad \lVert R_k F(t) - R_k F(s)\rVert_H
\le
\lVert F(t) - F(s)\rVert_H.
$$
Our approach starts with the observation that we may decompose functions
$F\in C_\alpha ([0,1];H)$ by double series with respect to the system
$(\phi_n\,e_k: n,k\ge 0)$.
\begin{lem}\label{lem:hoelderschauder}
Let $\alpha\in (0,1)$ and $F\in C_\alpha([0,1];H)$. Then we have
\begin{align*}
F=\sum_n \int_{[0,1]} \chi_n dF \phi_n = \sum_{n=0}^\infty \sum_{k=0}^\infty \int_{[0,1]} \chi_n dF_k e_k\phi_n
\end{align*}
with convergence in the uniform norm on $C([0,1];H)$.
\end{lem}
\begin{proof}
For the real valued functions $F_k, k\ge 0,$ the representation
$$F_k = \sum_{n=0}^\infty \int_{[0,1]} \chi_n d F_k\, \phi_n$$
is well known from \citet{Ciesielski}. Therefore we may write for $F\in
C_\alpha([0,1];H)$
\begin{align*}
F&=\sum_{k=0}^\infty F_ke_k\\
&=\sum_{k=0}^\infty e_k\sum_{n=0}^\infty \int_{[0,1]} \chi_n dF_k \phi_n \\
&=\sum_{n=0}^\infty \sum_{k=0}^\infty \int_{[0,1]} \chi_n dF_k e_k\phi_n \\
&=\sum_{n=0}^\infty \int_{[0,1]} \chi_n dF \phi_n.
\end{align*}
To justify the exchange in the order of summation and the convergence in
the uniform norm, we have to show
$$\lim_{N,m\to \infty} \left\rVert\sum_{n\ge N} \int_{[0,1]} \chi_n d R_m F
\phi_n\right\rVert_\infty = 0.$$ For this purpose, note first that by definition of
the Haar system for any $n,m\ge 0, n=2^k + l$, where $0\le l\le 2^k-1$
\begin{align*}
\left\lVert \int_{[0,1]} \chi_n dR_m F\right\rVert_H&=\sqrt{2^k}\left\lVert 2R_m F\left(\frac{2l+1}{2^{k+1}}\right)-R_m F\left(\frac{2l+2}{2^{k+1}}\right)-
R_m F\left(\frac{2l}{2^{k+1}}\right)\right\rVert_H\\
&\leq 2\lVert R_m F\rVert_\alpha 2^{-\alpha(k+1)}2^{\frac{1}{2}k}\\
&=\lVert R_m F\rVert_\alpha 2^{-\alpha(k+1)+\frac{1}{2}k+1}.
\end{align*}
Therefore and for $K\ge 0$ such that $2^K\le N\le 2^{K+1}$, using the fact
that $\phi_{2^k+l}, 0\le l\le 2^k-1$ have disjoint support and that
$\lVert\phi_{2^k+l}\rVert_\infty \le 2^{-\frac{k}{2}-1}$, we obtain
\begin{align*}
\left\lVert \sum_{n\geq N} \int_{[0,1]}\chi_nd R_m F\phi_n\right\rVert_\infty&\leq \sum_{k\geq K}\left\lVert \sum_{0\leq l\leq 2^k-1} \int_{[0,1]}\chi_{2^k+l}dF\phi_{2^k+l}\right\rVert_\infty\\
&\leq \sum_{k\geq K} \sup_{0\leq l\leq 2^k-1} \left\lVert \int_{[0,1]} \chi_{2^k+l}d R_m F \right\rVert_\infty2^{-\frac{k}{2}-1}\\
&\leq \sum_{k\geq K} \lVert R_m F \rVert_\alpha 2^{-\alpha (k+1)}\\
&\le\lVert R_m F \rVert_\alpha \sum_{k\geq K}
(2^\alpha)^{-k}\xrightarrow[K,m\rightarrow \infty]{} 0.
\end{align*}
Here we use $\lVert R_m F\rVert_\alpha \le \lVert F\rVert_\alpha<\infty$ for all $m\ge 0$,
the fact that $\lim_{m\to\infty} R_m F(t)=0$ for any $t\in[0,1]$, and
dominated convergence to obtain $\lim_{m\to\infty} \lVert R_m F\rVert_\alpha = 0$.
\end{proof}
A closer inspection of the coefficients in the decomposition of Lemma
\ref{lem:hoelderschauder} leads us to the following isomorphism, described
by \citet{Ciesielski} in the 1-dimensional case. To formulate it, denote by
$\mathcal{C}_0^H$ the space of $H$-valued sequences $(\eta_n)_{n \in \mathbb{N}}$ such that $\lim_{n\rightarrow \infty}\lVert\eta_n\rVert_H=0$. If we equip $\mathcal{C}_0^H$ with the supremum norm (using again the symbol $\lVert\cdot \rVert_\infty$), it becomes a Banach space.
\begin{thm}[Ciesielski's isomorphism for Hilbert spaces]\label{CiesilskiH}
Let $0 < \alpha < 1$. Let $(\chi_n)$ denote the Haar functions, and $(\phi_n)$ denote the Schauder functions. Let for $0\le n=2^k+l\ge0$, where $0\le l\le 2^k-1$
\begin{align*}
c_0(\alpha) := 1, \quad c_n(\alpha):=2^{k(\alpha-1/2) + \alpha - 1}.
\end{align*}
Define
\begin{align*}
T^H_\alpha: C_\alpha^0([0,1];H) \rightarrow \mathcal{C}_0^H \qquad F \mapsto \left(c_n(\alpha) \int_{[0,1]} \chi_n dF\right)_{n \in \mathbb{N}}
\end{align*}
Then $T^H_\alpha$ is continuous and bijective,
its operator norm is 1, and its inverse is given by
\begin{align*}
(T^H_\alpha)^{-1}: \mathcal{C}_0^H \rightarrow C_\alpha^0([0,1];H), \quad (\eta_n) \mapsto \sum_{n=0}^\infty \frac{\eta_n}{c_n(\alpha)} \phi_n,
\end{align*}
The norm of $(T^H_\alpha)^{-1}$ is bounded by
\begin{align*}
\left\lVert(T^H_\alpha)^{-1}\right\rVert \le \frac{2}{(2^\alpha-1)(2^{1-\alpha}-1)}.
\end{align*}
\end{thm}
\begin{proof}
Observe that for $n\in \mathbb{N}$ with $n=2^{k}+l$, $0\le l\le 2^{k}-1$
\begin{align*}
&\left\lVert \int_{[0,1]} \chi_n dF\right\rVert_H\\
&=\sqrt{2^{k}}\left\lVert 2F\left(\frac{2l+1}{2^{k+1}}\right)-F\left(\frac{2l+2}{2^{k+1}}\right)-F\left(\frac{2l}{2^{k+1}}\right)\right\rVert_H\\
&\leq \frac{1}{2c_\alpha(n)}\left(\frac{\left\lVert F(\frac{2l+2}{2^{k+1}})-F(\frac{2l+1}{2^{k+1}})
\right\rVert_H}{2^{-\alpha(k+1)}}+\frac{\left\lVert F(\frac{2l+1}{2^{k+1}})-F(\frac{2l}{2^{k+1}})
\right\rVert_H}{2^{-\alpha(k+1)}}\right)\\
&\le \frac{1}{c_\alpha(n)}
\sup_{t,s\in[0,1],\,\,|t-s|\le2^{-k-1}}\frac{\lVert
F(t)-F(s)\rVert_H}{|t-s|^\alpha}\\ &\leq \frac{1}{c_\alpha(n)}\lVert
F\rVert_\alpha.
\end{align*}
This gives the desired bound on the norm. Moreover, since $F\in
C_\alpha^0([0,1],H)$ we have
\begin{align*}
\lim_{n\rightarrow \infty} c_\alpha(n)\left\lVert \int_{[0,1]} \chi_n dF\right\rVert_H\le\lim_{n\rightarrow \infty}
\sup_{t,s\in[0,1],\,\,|t-s|\le2^{-k-1}}\frac{\lVert F(t)-F(s)\rVert_H}{|t-s|^\alpha}=0.
\end{align*}
Thus the range of $T^H_\alpha$ is indeed contained in $\mathcal{C}_0^H$.
Taking $F:[0,1]\rightarrow H$ with $F(s)=se_1$ for $s\in[0,1]$ we find that
$T^H_\alpha(F)=(e_1,0,0,...)$, thus $ \lVert F\rVert_\alpha= \lVert
T^H_\alpha (F)\rVert_\infty$. Therefore $\lVert T^H_\alpha \rVert=1$.
Clearly $T^H_\alpha$ is injective.
To see that $T^H_\alpha$ is bijective and that the inverse is bounded as claimed,
define
\begin{align*}
A: \mathcal{C}_0^H \rightarrow C_\alpha^0([0,1];H), \quad (\eta_n) \mapsto \sum_{n=0}^\infty
\frac{\eta_n}{c_n(\alpha)} \phi_n.
\end{align*}
Now a straightforward calculation using the orthogonality of the
$(\chi_n)_{n\ge 0}$ gives for any $(\eta_n)_{n\ge 0}\subset \mathcal{C}^H_0$
\begin{align*}
T^H_\alpha\circ A((\eta_n)_{n\ge 0})&=T^H_\alpha\left(\sum_{n=0}^\infty \frac{\eta_n}{c_n(\alpha)}\phi_n\right)\\
&=\left(\sum_{n,m=0}^\infty\eta_n\int_{[0,1]}\chi_md\phi_n\right)_{m\in\mathbb{N}}\\
&=\left(\sum_{n,m=0}^\infty \eta_n\int\chi_n(t)\chi_m(t)dt\right)_{m\in\mathbb{N}}\\
&=(\eta_m)_{m\ge 0}.
\end{align*}
Consequently we can infer that $A=(T^H_\alpha)^{-1}$.\\
We still have to show that $(T^H_\alpha)^{-1}$ satisfies the claimed norm
inequality and maps every sequence $(\eta_n)_{n\ge 0}\in \mathcal{C}_0^H$
to an element of $C^0_\alpha([0,1],H)$. For this purpose let
$(\eta_n)_{n\ge 0}\in \mathcal{C}_0^H$, set
$F=(T^H_{\alpha})^{-1}((\eta_n))$ and let $s,t\in[0,1]$ be given. Then we have
\begin{align*}
\lVert F(t)-F(s)\rVert_H\leq \lVert (\eta_n)_{n\ge 0}\rVert_{\infty}\left(\lvert t-s\rvert +\sum_{k=0}^{\infty}\sum_{l=0}^{2^k-1}\frac{\lvert \phi_{2^k+l}(t)-\phi_{2^k+l}(s)\rvert}{c_{2^k}(\alpha)}\right).\\
\end{align*}
The term in brackets on the right hand side is exactly the one appearing in
the real valued case (\citet{Ciesielski}). Consequently we have the same
bound, given by
\begin{align*}
\lVert (T^H_{\alpha})^{-1}\rVert\leq \frac{1}{(2^\alpha-1)(2^{\alpha-1}-1)}.
\end{align*}
A more careful estimation yields
\begin{align*}
\lVert F(t)-F(s)\rVert_H\leq \lVert \eta_0\rVert \lvert t-s \rvert + \sum_{k=0}^{\infty}\sum_{l=0}^{2^k-1}\frac{1}{c_{2^k}(\alpha)}\lVert \eta_{2^k+l}\rVert \lvert \phi_{2^k+l}(t)-\phi_{2^k+l}(s)\rvert.\\
\end{align*}
This is the same expression as in the real valued case. Its well known
treatment implies
\begin{align*}
\lim_{\lvert t-s \rvert\rightarrow 0}\frac{\lVert F(t)-F(s)\rVert_H}{\lvert t-s\rvert^\alpha }=0.
\end{align*}
This finishes the proof.
\end{proof}
\subsection{Wiener processes on Hilbert spaces}
We recall some basic concepts of Gaussian random variables and Wiener processes with values in
a separable Hilbert space $H$. Especially we will derive a Fourier sequence decomposition of Wiener processes.
Our presentation follows \citet{DaPrato}.
\begin{defn}Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space, $m\in H$ and $Q:H\rightarrow H$
a positive self adjoint operator. An $H$-valued random variable $X$ such that for every $h\in H$
\begin{align*}
E[\exp(i\langle h, X\rangle)]=\exp\left(i\langle h, m\rangle-\frac{1}{2}\langle Qh,h\rangle\right).
\end{align*}
is called Gaussian with covariance operator $Q$ and mean $m\in H$.
We denote the law of $X$ by $\mathcal{N}(m, Q)$.
\end{defn}
By Proposition 2.15 of \citet{DaPrato}, $Q$ has to be a positive, self-adjoint trace class
operator, i.e. a bounded operator from $H$ to $H$ that satisfies
\begin{enumerate}
\item $\langle Qx,x \rangle \ge 0 \text{ for every } x \in H$,
\item $\langle Qx,x \rangle = \langle x, Qx\rangle \text{ for every } x \in H$,
\item $\sum_{k=0}^\infty \langle Q e_k, e_k \rangle < \infty$ for every CONS $(e_k)_{k\ge 0}$.
\end{enumerate}
If $Q$ is a positive, self-adjoint trace class operator on $H$, then there exists a CONS $(e_k)_{k\ge 0}$ such that $Q e_k = \lambda_k
e_k$, where $\lambda_k \ge 0$ for all $k$ and $\sum_{k=0}^\infty \lambda_k <
\infty$. Note that for such a $Q$, an operator $Q^{1/2}$ can be defined by
setting $Q^{1/2} e_k := \sqrt{\lambda_k} e_k, k \in \mathbb{N}_0$. Then $Q^{1/2}
Q^{1/2} = Q$.
\begin{defn}Let $Q$ be a positive, self-adjoint trace class operator on $H$. A $Q$-Wiener process $(W(t): t \in [0,1])$ is a stochastic process with values in $H$ such that
\begin{enumerate}
\item $W(0) = 0$,
\item $W$ has continuous trajectories,
\item $W$ has independent increments,
\item $\L(W(t) - W(s)) = \mathcal{N}(0, (t-s) Q)$
\end{enumerate}
\end{defn}
In this case $(W(t_1), \dots , W(t_n))$ is $H^n$-valued Gaussian for all
$t_1, \dots, t_n \in [0,1]$. By Proposition 4.2 of \citet{DaPrato} we know
that such a process exists for every positive, self-adjoint trace class
operator $Q$ on H. To get the Fourier decomposition of a $Q$-Wiener process
along the Schauder basis we use a different standard characterization.
\begin{lem}\label{lem:characterisation of Wienerprocess}
A stochastic process $Z$ on $(H, \mathcal{B}(H))$ is a $Q$-Wiener process if and only if
\begin{itemize}
\item $Z_0=0$ $\mathbb{P}$-a.s.,
\item $Z$ has continuous trajectories,
\item $cov(\langle v, Z_t\rangle \langle w, Z_s\rangle)=(t\wedge s) \langle v,Qw \rangle$ $\forall v,w\in H$, $\forall 0\leq s\leq t<\infty$,
\item $\forall (v_1,...,v_n)\in H^n$ $(\langle v_1,Z \rangle,...,\langle v_n,Z\rangle)$
is a $\mathbb{R}^n$-valued Gaussian process.
\end{itemize}
\end{lem}
Independent Gaussian random variables with values in a Hilbert space
asymptotically allow the following bounds.
\begin{lem}\label{lem:estimate for Gaussian RV}
Let $Z_n\sim \mathcal{N}(0,Q)$, $n\in \mathbb{N}$, be independent. Then there exists an a.s. finite real valued random
variable C such that
\begin{align*}
\lVert Z_n\rVert_H\leq C\sqrt{\log n}\text{ }\mathbb{P}\text{ }a.s..
\end{align*}
\end{lem}
\begin{proof}
By using the exponential integrability of $\lambda \lVert Z_n\rVert^2_H$ for small
enough $\lambda$ and Markov's inequality, we obtain that there exist
$\lambda, c\in \mathbb{R}_+$ such that for any $a>0$
\begin{align*}
\P(\lVert Z\rVert_H>a)\le ce^{-\lambda a^2}.
\end{align*}
Thus for $\alpha > 1$ and $n$ big enough
\begin{align*}
\mathbb{P}\left(\lVert Z_n\rVert_H\geq \sqrt{\lambda^{-1}\alpha \log{n}}\right)\leq c n^{-\alpha}.
\end{align*}
We set $A_n=\left\{ \lVert Z_n\rVert_H\geq \sqrt{\lambda^{-1}\alpha \log{n}}\right\}$ and have
\begin{align*}
\sum_{n=0}^{\infty}\mathbb{P}(A_n)< \infty.
\end{align*}
Hence the lemma of Borel-Cantelli gives, that $\mathbb{P}(\limsup_n
A_n)=0$, i.e. $\mathbb{P}- a.s.$ for almost all $n\in\mathbb{N}$ we have
$\lVert Z_n\rVert_H\leq \sqrt{\lambda^{-1}\alpha \log{n}}$. In other words
\begin{align*}
C:=\sup_{n\ge 0} \frac{\lVert Z_n \rVert_H}{\sqrt{\log{n}}}< \infty\text{ }\mathbb{P}-a.s.
\end{align*}
\end{proof}
Using Lemma \ref{lem:estimate for Gaussian RV} and the characterization of
$Q$-Wiener processes of Lemma \ref{lem:characterisation of Wienerprocess},
we now obtain its Schauder decomposition which can be seen as a Gaussian
version of Lemma \ref{lem:hoelderschauder}.
\begin{prop}\label{lem:representation of Wienerprocess}
Let $\alpha \in (0,1/2)$, let $(\phi_n)_{n\ge 0}$ be the Schauder functions and $(Z_n)_{n\ge 0}$ a
sequence of independent, $\mathcal{N}(0,Q)$-distributed Gaussian variables,
where $Q$ is a positive self adjoint trace class operator on $H$. The
series defined process
\begin{align*}
W_t=\sum_{n=0}^{\infty}\phi_n(t)Z_n,\quad t\in[0,1],
\end{align*}
converges $\P$-a.s. with respect to the $\lVert\cdot \rVert_\alpha$-norm on $[0,1]$ and is an $H$-valued $Q$-Wiener
process.
\end{prop}
\begin{proof}
We have to show that the process defined by the series satisfies the
conditions given in Lemma \ref{lem:characterisation of Wienerprocess}. The
first and the two last conditions concerning the covariance structure and
Gaussianity of scalar products have standard verifications. Let us just
argue for absolute and $\lVert\cdot \rVert_\alpha$-convergence of the series, thus proving
H\"older-continuity of the trajectories.
Since $T^H_\alpha$ is an isomorphism and since any single term of the series is even Lipschitz-continuous, it suffices to show that
\begin{align*}
\left(T^H_\alpha \left( \sum_{n=0}^m \phi_n Z_n\right): m \in \mathbb{N}\right)
\end{align*}
is a Cauchy sequence in $\mathcal{C}_0^H$. Let us first calculate the image of term $N$ under $T^H_\alpha$. We have
\begin{align*}
(T^H_\alpha \phi_n Z_n)_N = 1_{\{n = N\}} c_N(\alpha) Z_N.
\end{align*}
Therefore for $m_1, m_2\ge 0, m_1\le m_2$
\begin{align*}
\sum_{n=m_1}^{m_2} (T^H_\alpha \phi_n Z_n)_N = 1_{\{m_1 \le N \le m_2\}}c_N(\alpha) Z_N = \left(T^H_\alpha\left( \sum_{n=m_1}^{m_2} \phi_n Z_n\right)\right)_N.
\end{align*}
So if we can prove that $c_N(\alpha) Z_N$ a.s. converges to $0$ in $H$ as $N \rightarrow \infty$, the proof is complete. But this follows immediately from Lemma \ref{lem:estimate for Gaussian RV}: $c_N(\alpha)$ decays exponentially fast, and $\lVert Z_N \rVert_H \le C \sqrt{\log N}$.
\end{proof}
In particular we showed that for $\alpha < 1/2$ $W$ a.s. takes its trajectories in
\begin{align*}
C_\alpha^0([0,1]; H) := \left\{ F:[0,1] \rightarrow H, F(0) = 0,
\lim_{\delta \rightarrow 0} \sup_{\substack{t \neq s, \\ |t-s|
< \delta}} \frac{\lVert F(t) - F(s) \rVert_H}{|t-s|^\alpha} = 0 \right\}
\end{align*}
By Lipschitz continuity of the scalar product, we also have $\langle F, e_k \rangle \in C_\alpha^0 ([0,1];\mathbb{R})$:
Since $P_k$
and $R_k$ are orthogonal projectors and therefore Lipschitz continuous, we
obtain that for $F \in C_\alpha^0([0,1];H)$
$$
\sup_{k \ge 0} \lVert\langle F, e_k \rangle \rVert_\alpha \le \lVert F \rVert_\alpha.
$$
We also saw that $T^H_\alpha(W)$ is well defined almost surely. As a
special case this is also true for the real valued Brownian motion. We have
by Proposition \ref{lem:representation of Wienerprocess}
\begin{align*}
T^H_\alpha(W)=(c_n(\alpha) Z_n)
\end{align*}
where $(Z_n)_{n\ge 0}$ is a sequence of i.i.d.
$\mathcal{N}(0,Q)-$variables.
Plainly, the representation of the preceding Lemma can be used to prove the
representation formula for $Q$-Wiener processes by scalar Brownian motions
according to \cite{DaPrato}, Theorem 4.3.
\begin{prop}\label{prop: series representation of Wienerprocess}
Let $W$ be a $Q$-Wiener process. Then
\begin{align*}
W(t) = \sum_{k=0}^\infty \sqrt{\lambda_k} \beta_k (t) e_k\text{, }t\in [0,1],
\end{align*}
where the series on the right hand side $\mathbb{P}$-a.s. converges uniformly on $[0,1]$,
and $(\beta_k)_{k\ge 0}$ is a sequence of independent real valued Brownian motions.
\end{prop}
\begin{proof}
Using arguments as in the proof of Theorem \ref{CiesilskiH} and Lemma
\ref{lem:estimate for Gaussian RV} to justify changes in the order of
summation we get
\begin{align*}
W=\sum_{n=0}^{\infty}\phi_n Z_n=\sum_{k\ge 0}\sum_{n\ge 0}\phi_n \langle Z_n,e_k\rangle e_k=\sum_{k\ge 0}
\sqrt{\lambda_k}\sum_{n\ge 0}\phi_n N_{n,k}e_k=\sum_{k\ge 0} \sqrt{\lambda_k}\beta_k e_k,
\end{align*}
where the equivalences are $\P$-a.s. and $(N_{n,k})_{n,k\ge 0},
(\beta_k)_{k\ge 0}$ are real valued iid $\mathcal{N}(0,1)$ random
variables resp. Brownian motions. For the last step we applied Proposition
\ref{lem:representation of Wienerprocess} for the one-dimensional case.
\end{proof}
\subsection{Large deviations}
Let us recall some basic notions of the theory of large deviations that
will suffice to prove the large deviation principle for Hilbert space
valued Wiener processes. We follow \citet{Dembo}. Let $X$ be a topological
Hausdorff space. Denote its Borel $\sigma$-algebra by $\mathcal{B}$.
\begin{defn}[Rate function]
A function $I : X \rightarrow [0, \infty]$ is called a rate function if it is lower semi-continuous, i.e. if for every $C \ge 0$ the set
\begin{align*}
\Psi_I (C) := \{ x \in X: I(x) \le C \}
\end{align*}
is closed. It is called a good rate function, if $\Psi_I(C)$ is compact. For $A \in \mathcal{B}$ we define
$I(A):= \inf_{x \in A} I(x)$.
\end{defn}
\begin{defn}[Large deviation principle]
Let $I$ be a rate function. A family of probability measures $(\mu_\varepsilon)_{\varepsilon > 0}$ on $(X, \mathcal{B})$
is said to satisfy the large deviation principle (LDP) with rate function $I$ if for any closed
set $F \subset X$ and any open set $G \subset X$ we have
\begin{align*}
\limsup_{\varepsilon \rightarrow 0} \varepsilon \log \mu_\varepsilon (F)& \le - I(F) \mbox{ and} \\
\liminf_{\varepsilon \rightarrow 0} \varepsilon \log \mu_\varepsilon (G) & \ge -
I(G).
\end{align*}
\end{defn}
\begin{defn}[Exponential tightness]
A family of probability measures $(\mu_\varepsilon)_{\varepsilon > 0}$ is said to be exponentially tight
if for every $a>0$ there exists a compact set $K_a \subset X$ such that
\begin{align*}
\limsup_{\varepsilon \rightarrow 0} \varepsilon \log \mu_\varepsilon (K_a^c) <
-a.
\end{align*}
\end{defn}
In our approach to Schilder's Theorem for Hilbert space valued Wiener
processes we shall mainly use the following proposition which basically
states that the rate function has to be known for elements of a sub-basis
of the topology.
\begin{prop}\label{prop:exponential und ldp fuer offene
baelle, dann ldp}
Let $\mathcal{G}_0$ be a collection of open sets in the topology of $X$ such that for every
open set $ G \subset X$ and for every $x \in G$ there exists $G_0 \in \mathcal{G}_0$ such that
$x \in G_0 \subset G$. Let $I$ be a rate function and let $(\mu_\varepsilon)_{\varepsilon > 0}$ be an
exponentially tight family of probability measures. Assume that for every $G \in \mathcal{G}_0$ we have
\begin{align*}
- \inf_{x\in G} I(x) = \lim_{\varepsilon \rightarrow 0}\varepsilon \log \mu_\varepsilon (G).
\end{align*}
Then $I$ is a good rate function, and $(\mu_\varepsilon)_\varepsilon$ satisfies an LDP
with rate function $I$.
\end{prop}
\begin{proof}
Let us first establish the lower bound. In fact, let $G$ be an open set. Choose $x\in G$, and a basis set $G_0$ such that $x\in G_0\subset G.$
Then evidently
$$\liminf_{\varepsilon\to 0} \varepsilon\ln \mu_\varepsilon(G) \ge \liminf_{\varepsilon\to 0} \varepsilon\ln \mu_\varepsilon(G_0) = - \inf_{y\in G_0} I(y) \ge -I(x).$$
Now the lower bound follows readily by taking the $\sup$ of $-I(x), x\in G,$
on the right hand side, the left hand side not depending on $x$.
For the upper bound, fix a compact subset $K$ of $X.$ For $\delta > 0$ denote
$$I^\delta(x) = (I(x)-\delta)\wedge \frac{1}{\delta},\quad x\in X.$$
For any $x\in K$, use the lower semicontinuity of $I$, more
precisely that $\{y\in X: I(y)> I^\delta(x)\}$ is open to choose a
set $G_x\in \mathcal{G}_0$ such that
$$- I^\delta(x) \ge \limsup_{\varepsilon\to 0} \varepsilon \ln \mu_\varepsilon(G_x).$$ Use compactness of $K$ to extract from the open cover
$K \subset \cup_{x\in K} G_x$ a finite subcover $K\subset \cup_{i=1}^n G_{x_i}.$
Then with a standard argument we obtain
$$\limsup_{\varepsilon\to 0} \varepsilon \ln \mu_\varepsilon(K) \le \max_{1\le i\le n} \limsup_{\varepsilon\to 0} \varepsilon \ln \mu_\varepsilon(G_{x_i})\le - \min_{1\le i\le n} I^\delta(x_i)
\le - \inf_{x\in K} I^\delta(x).$$
Now let $\delta\to 0.$ Finally use exponential tightness to show that $I$ is a good rate function (see \cite{Dembo}, Section 4.1).
\end{proof}
The following propositions show how large deviation principles are
transferred between different topologies on a space, or via continuous maps
to other topological spaces.
\begin{prop}[Contraction principle]\label{prop:contraction principle}
Let $X$ and $Y$ be topological Hausdorff spaces, and let $I: X \rightarrow [0, \infty]$ be a good rate function. Let $f: X \rightarrow Y$ be a continuous mapping. Then
\begin{align*}
I' : Y \rightarrow [0, \infty], I'(y) = \inf\{ I(x): f(x) = y \}
\end{align*}
is a good rate function, and if $(\mu_\varepsilon)_{\varepsilon > 0}$
satisfies an LDP with rate function $I$ on $X$, then $(\mu_\varepsilon \circ f^{-1})_{\varepsilon > 0}$
satisfies an LDP with rate function $I'$ on $Y$.
\end{prop}
\begin{prop}\label{prop:ldp von grober auf feine topo}
Let $(\mu_\varepsilon)_{\varepsilon > 0}$ be an exponentially tight family of probability
measures on $(X, \mathcal{B}_{\tau_2})$ where $\mathcal{B}_{\tau_2}$ are the Borel sets of $\tau_2$.
Assume $(\mu_\varepsilon)$ satisfies an LDP with rate function $I$ with respect to some
Hausdorff topology $\tau_1$ on $X$ which is coarser than $\tau_2$, i.e. $\tau_2 \subset \tau_1$.
Then $(\mu_\varepsilon)_{\varepsilon > 0}$ satisfies the LDP with respect to $\tau_2$, with good rate
function $I$.
\end{prop}
The main idea of our sequence space approach to Schilder's Theorem for
Hilbert space valued Wiener processes will just extend the following large
deviation principle for a standard normal variable with values in $\mathbb{R}$ to
sequences of i.i.d. variables of this kind.
\begin{prop} \label{prop:ldp standard normal}
Let $Z$ be a standard normal variable with values in $\mathbb{R}$,
\begin{align*}
I: \mathbb{R} \rightarrow [0,\infty), \, x \mapsto \frac{x^2}{2},
\end{align*}
and for Borel sets $B$ in $\mathbb{R}$ let $\mu_\varepsilon(B):= \P(\sqrt{\varepsilon} Z \in B)$.
Then $(\mu_\varepsilon)_{\varepsilon > 0}$ satisfies a LDP with good rate function $I$.
\end{prop}
\section{Large Deviations for Hilbert Space Valued Wiener Processes}\label{LDP}
Ciesielski's isomorphism and the Schauder representation of Brownian motion
yield a very elegant and simple method of proving large deviation
principles for the Brownian motion. This was first noticed by \citet{Baldi}
who gave an alternative proof of Schilder's theorem based on this
isomorphism. We follow their approach and extend it to Wiener processes
with values on Hilbert spaces. In this entire section we always assume $0 <
\alpha < 1/2$. By further decomposing the orthogonal 1-dimensional Brownian
motions in the representation of an $H$-valued Wiener process by its
Fourier coefficients with respect to the Schauder functions, we describe it
by double sequences of real-valued normal variables.
\subsection{Appropriate norms}
We work with new norms on the spaces of $\alpha$-H\"older continuous
functions given by
\begin{align*}
&\lVert F\rVert^{'}_\alpha := \lVert T^H_\alpha F\rVert_\infty = \sup_{k,n} \left| c_n(\alpha) \int_{[0,1]} \chi_n(s)
d\langle F, e_k \rangle(s) \right|, F \in C_\alpha^0([0,1];H), \\
& \lVert f\rVert^{'}_\alpha := \lVert T_\alpha f\rVert_\infty = \sup_{n} \left| c_n(\alpha) \int_{[0,1]} \chi_n(s)df(s)
\right|, f \in C_\alpha^0([0,1];\mathbb{R}).
\end{align*}
Since $T^{H}_\alpha$ is one-to-one, $\lVert.\rVert^{'}_\alpha$ is indeed a norm.
Also, we have $\lVert.\rVert_\alpha^{'} \le \lVert.\rVert_\alpha$. Hence the topology
generated by $\lVert.\rVert_\alpha^{'}$
is coarser than the usual topology on $C_\alpha^0([0,1],H)$.\\
Balls with respect to the new norms $U_\alpha^\delta(F) := \{ G \in
C_\alpha^0([0,1];H): \lVert G - F\rVert_\alpha^{'} < \delta\}$ for $F\in C_\alpha^0([0,1];H), \delta>0$, have a
simpler form for our reasoning, since the condition that for $\delta > 0$ a
function $G\in C^0_\alpha([0,1],H)$ lies in $U_\alpha^\delta(F)$ translates into
the countable set of one-dimensional conditions $|\langle
T^H_\alpha(F)_n-T^H_\alpha(G)_n, e_k\rangle|<\delta$ for all $n,k\ge 0.$
This will facilitate the proof of the LDP for the basis of open balls of
the topology generated by $\lVert.\rVert_\alpha^{'}$. We will first prove the LDP
in the topologies generated by these norms and then transfer the result to
the finer sequence space topologies using Proposition \ref{prop:ldp von
grober auf feine topo}, and finally to the original function space using
Ciesielski's isomorphism and Proposition \ref{prop:contraction principle}.
\subsection{The rate function}
Recall that $Q$ is supposed to be a positive self-adjoint trace-class
operator on H. Let $H_0 :=( Q^{1/2} H, \lVert \cdot \rVert_0)$, equipped
with the inner product
\begin{align*}
\langle x,y \rangle_{H_0} := \langle Q^{-1/2} x, Q^{-1/2} y \rangle_H,
\end{align*}
that induces the norm $\lVert \cdot\rVert_0$ on $H_0$. We define the
Cameron-Martin space of the $Q$-Wiener process $W$ by
\begin{align*}
\H:= \left\{ F \in C([0,1];H): F(\cdot) = \int_0^\cdot U(s) ds \mbox{ for some } U \in L^2([0,1]; H_0)
\right\}.
\end{align*}
Here $L^2([0,1];H_0)$ is the space of measurable functions $U$ from $[0,1]$
to $H_0$ such that $\int_0^1 \lVert U\rVert^2_{H_0} dx < \infty$. Define the
function $I$ via
\begin{align*}
&I: C([0,1];H) \rightarrow [0, \infty] \\
&F\mapsto \inf \left\{ \frac{1}{2}\int_0^1 \lVert U(s)\rVert^2_{H_0} ds: U \in L^2([0,1];H_0), F(\cdot) =
\int_0^\cdot U(s) ds \right\}
\end{align*}
where by convention $\inf \emptyset = \infty$. In the following we will
denote any restriction of $I$ to a subspace of $C([0,1];H)$ (e.g. to
$(C_\alpha([0,1];H)$) by $I$ as well. We will use the structure of $H$ to
simplify our problem. It allows us to compute the rate function $I$ from
the rate function of the one dimensional Brownian by the following Lemma.
\begin{lem}\label{lem:gleichheit rate functions}
Let $\tilde{I}: C([0,1];\mathbb{R})$ be the rate function of the Brownian motion, i.e.
\begin{align*}
\tilde{I}(f) := \left\{\begin{array}{ll} \int_0^1 |\dot{f}(s)|^2 ds, & f (\cdot) = \int_0^{\cdot} \dot{f}(s) ds\,\,\mbox{for a square integrable function}\,\, \dot{f},\\ \infty, & \mbox{otherwise}.\end{array}\right.
\end{align*}
Let $(\lambda_k)_{k\ge 0}$ be the sequence of eigenvalues of $Q$. Then for all $F \in C([0,1];H)$ we have
\begin{align*}
I(F) = \sum_{k=0}^\infty \frac{1}{\lambda_k} \tilde{I}(\langle F, e_k \rangle).
\end{align*}
where we convene that $c/0 = \infty$ for $c > 0$ and $0/0 = 0$.
\end{lem}
\begin{proof}
Let $F\in C([0,1];H)$.
\begin{enumerate}
\item First assume $I(F) < \infty$. Then there exists $U \in L^2([0,1]; H_0)$ such
that $F = \int_0^\cdot U(s) ds$ and thus $\langle F,e_k\rangle=\int_0^\cdot \langle U(s), e_k
\rangle ds$ for $k\ge 0$.
Consequently we have by monotone convergence
\begin{align*}
\frac{1}{2}\int_0^1 \lVert U(s)\rVert^2_{H_0} ds &= \frac{1}{2} \int_0^1 \left\lVert\sum_{k=0}^\infty \langle U(s), e_k \rangle e_k\right\rVert_{H_0} ds\\
&=\frac{1}{2} \int_0^1 \sum_{k=0}^\infty \langle U(s), e_k \rangle^2 \langle Q^{-\frac{1}{2}} e_k, Q^{-\frac{1}{2}}e_k\rangle ds\\
&=\frac{1}{2} \int_0^1 \sum_{k=0}^\infty \frac{1}{\lambda_k} \langle U(s), e_k \rangle^2 ds\\
& = \sum_{k=0}^\infty \frac{1}{\lambda_k}\tilde{I}(\langle F, e_k
\rangle).
\end{align*}
The last expression does not depend on the choice of $U$. Hence we get that
$I(F) < \infty$ implies $I(F)=\sum_{k=0}^\infty \frac{1}{\lambda_k}
\tilde{I}(\langle F, e_k \rangle)$.
\item Conversely assume $\sum_{k=0}^\infty \frac{1}{\lambda_k} \tilde{I}(\langle F, e_k \rangle) < \infty$.
Since $\tilde{I}(\langle F, e_k \rangle) < \infty$ for all $k\ge 0$, we
know that there exists a sequence $(U_k)_{k\ge 0}$ of square-integrable
real-valued functions such that $\langle F, e_k \rangle = \int_0^\cdot
U_k(s) ds$. Further, those functions $U_k$ satisfy by monotone convergence
\begin{align*}
\int_0^1\sum_{k=0}^\infty \frac{1}{\lambda_k} |U_k(s)|^2 ds = \sum_{k=0}^\infty \frac{1}{\lambda_k}
\int_0^1 |U_k(s)|^2 ds = \sum_{k=0}^\infty \frac{2}{\lambda_k} \tilde{I}(\langle F, e_k \rangle)<
\infty.
\end{align*}
So if we define $U(s):= \sum_{k=0}^\infty U_k(s) e_k, s\in[0,1]$, then $U \in L^2([0,1];H_0)$. This follows from
\begin{align*}
U\in L^2([0,1];H_0) &\text{ iff }
\int_0^1 \lVert U(s)\rVert^2_{H_0} ds = \int_0^1\sum_{k=0}^\infty \frac{1}{\lambda_k} |U_k(s)|^2 ds< \infty.
\end{align*}
Finally we obtain by dominated convergence ($\lVert F(t)\rVert_H<\infty$)
\begin{align*}
F(t) = \sum_{k=0}^\infty \langle F(t), e_k \rangle e_k = \sum_{k=0}^\infty e_k \int_0^t U_k(s) ds =
\int_0^t U(s) ds,
\end{align*}
such that
\begin{align*}
I(F) \le \frac{1}{2}\int_0^1 \lVert U(s)\rVert^2_{H_0} ds = \frac{1}{2}\int_0^1 \sum_{k=0}^\infty
\frac{1}{\lambda_k} |U_k(s)|^2 ds < \infty.
\end{align*}
\end{enumerate}
Combining the two steps we obtain $I(F)<\infty$ iff $\sum_{k=0}^\infty
\frac{1}{\lambda_k} \tilde{I}(\langle F, e_k \rangle) < \infty$ and in
this case
\begin{align*}
I(F) = \sum_{k=0}^\infty \frac{1}{\lambda_k} \tilde{I}(\langle F, e_k \rangle).
\end{align*}
This completes the proof.
\end{proof}
Lemma \ref{lem:gleichheit rate functions} allows us to show that $I$ is a rate function.
\begin{lem}\label{lem: rate function}
$I$ is a rate function on $(C_\alpha^0([0,1];H), \lVert.\lVert_\alpha^{'})$.
\end{lem}
\begin{proof}
For a constant $C\ge 0$ we have to prove that if $(F_n)_{n\ge 0} \subset \Psi_I(C) \cap C_\alpha^0([0,1];H)$ converges
in $C_\alpha^0([0,1];H)$ to $F$, then $F$ is also in $\Psi_I(C)$.
It was observed in \citet{Baldi} that $\tilde{I}$ is a rate function for the $\lVert . \rVert_\alpha^{'}$-topology
on $C_0^\alpha([0,1|;\mathbb{R})$. By our assumption we know that for every $k\in\mathbb{N}$, $(\langle F_n, e_k
\rangle)_{n\ge 0}$ converges in $(C_0^\alpha([0,1|;\mathbb{R}),\lVert .\rVert_\alpha^{'})$ to
$\langle F, e_k \rangle$. Therefore
\begin{align*}
\tilde{I} (\langle F, e_k \rangle) \le \liminf_{n \rightarrow \infty} \tilde{I}(\langle F_n,
e_k \rangle),
\end{align*}
so by Lemma \ref{lem:gleichheit rate functions} and by Fatou's lemma
\begin{align*}
C &\ge \liminf_{n \rightarrow \infty} I(F_n) = \liminf_{n \rightarrow \infty} \sum_{k=0}^\infty
\frac{1}{\lambda_k} \tilde{I}(\langle F_n, e_k \rangle) \ge \sum_{k=0}^\infty \frac{1}{\lambda_k}
\liminf_{n \rightarrow \infty} \tilde{I}(\langle F_n, e_k \rangle) \\
& \ge \sum_{k=0}^\infty \frac{1}{\lambda_k} \tilde{I} (\langle F, e_k\rangle) = I(F).
\end{align*}
Hence $F \in \Psi_I (C)$.
\end{proof}
\subsection{LDP for a sub-basis of the coarse topology}
To show that the $Q$-Wiener process $(W(t): t \in [0,1])$ satisfies a LDP
on $(C_\alpha([0,1];H), \lVert .\rVert_\alpha)$ with good rate function $I$ as
defined in the last section we now show that the LDP holds for open balls
in our coarse topology induced
by $\lVert.\rVert_\alpha^{'}$. The proof is an extension of the version of \citet{Baldi} for the real valued Wiener process.
For $\varepsilon>0$ denote by $\mu_\varepsilon$ the law of $\sqrt{\varepsilon} W$, i.e.
$\mu_\varepsilon(A) = \P( \sqrt{\varepsilon}W \in A)$,
$A\in\mathcal{B}(H)$.
\begin{lem}\label{lem: ldp for basis}
For every $\delta > 0$ and every $F \in C_\alpha^0([0,1]; H)$ we have
\begin{align*}
\lim_{\varepsilon \rightarrow 0} \varepsilon \log \mu_\varepsilon (U^\delta_\alpha(F))
= - \inf_{G \in U^\delta_\alpha(F)} I(G).
\end{align*}
\end{lem}
\begin{proof}
1. Write $T^H_\alpha F = (\sum_{k=0}^\infty F_{n,k}e_k)_{n\in\mathbb{N}}$. Then $\sqrt{\varepsilon}W$ is
in $U^\delta_\alpha(F)$ if and only if
\begin{align*}
\sup_{k,n \ge 0} \left| \sqrt{\varepsilon} c_n(\alpha) \int_0^1 \chi_n d \langle W,
e_k\rangle -F_{k,n} \right| < \delta.
\end{align*}
Now for $k\ge 0$ we recall $\langle W, e_k \rangle = \sqrt{\lambda_k} \beta_k$, where $(\beta_k)_{k\ge 0}$ is a sequence of independent standard
Brownian motions. Therefore for $n,k\ge 0$
\begin{align*}
\left| \int_0^1 \chi_n d \langle W, e_k\rangle \right| = \left| \sqrt{\lambda_k} Z_{k,n} \right| ,
\end{align*}
where $(Z_{k,n})_{k,n\ge 0}$ is a double sequence of independent standard normal variables. Therefore by independence
\begin{align*}
\mu_\varepsilon (U^\delta_\alpha(F)) & = \P\left(\bigcap_{k,n \in \mathbb{N}_0}
\left| c_n(\alpha) \sqrt{\varepsilon\lambda_k} Z_{k,n} -F_{k,n} \right| <\delta \right) \\
& = \prod_{k=0}^\infty \prod_{n=0}^\infty \P\left( c_n(\alpha)
\sqrt{\varepsilon\lambda_k} Z_{k,n} \in (F_{k,n} - \delta , F_{k,n} + \delta ) \right).
\end{align*}
To abbreviate, we introduce the notation
\begin{align*}
\P_{k,n} (\varepsilon) = \P\left( c_n(\alpha) \sqrt{\varepsilon\lambda_k} Z_{k,n} \in
(F_{k,n} - \delta , F_{k,n} + \delta ) \right)\text{, }\varepsilon >0, n,k\in\mathbb{N}_0.
\end{align*}
For every $k\ge 0$ we split $\mathbb{N}_0$ into subsets $\Lambda^k_i,$ $i=1,2,3,4$, for each of which we will
calculate $\prod_{k =0}^\infty\prod_{n \in \Lambda^k_i} \P_{n,k} (\varepsilon)$ separately. Let
\begin{align*}
& \Lambda^k_1 = \{ n \ge 0: 0 \notin [ F_{k,n} - \delta , F_{k,n} + \delta ]\} \\
& \Lambda^k_2 = \{ n \ge 0: F_{k,n} = \pm \delta \} \\
& \Lambda^k_3 = \{ n \ge 0: [- \delta/2, \delta/2] \subset [ F_{k,n} - \delta ,
F_{k,n} + \delta ]\} \\
& \Lambda^k_4 = ( \Lambda^k_1 \cup \Lambda^k_2\cup\Lambda^k_3)^c.
\end{align*}
By applying Ciesielski's isomorphism to the real-valued functions $\langle F, e_k\rangle$, we see that for every fixed $k$, $\Lambda_3^k$
contains nearly all $n$. Since $(T_\alpha^H F)_n$ converges to zero in $H$, in particular $\sup_{k \ge 0} |F_{k,n}|$ converges to zero as $n \rightarrow \infty$. But for every fixed $n$, $(F_{k,n})_k$ is in $l^2$ and therefore converges to zero. This shows that for large enough $k$ we must have $\Lambda_3^k = \mathbb{N}_0$, and therefore $\cup_{k}(\Lambda^k_3)^c$ is finite.
2. First we examine $\prod_{k=0}^\infty \prod_{n \in \Lambda^k_3} \P_{k,n} (\varepsilon)$.
Note that for $n \in \Lambda_3^k$ we have
\begin{align*}
[- \delta/2, \delta/2] \subset [ F_{k,n} - \delta , F_{k,n} + \delta ],
\end{align*}
and therefore
\begin{align*}
\prod_{k=0}^\infty \prod_{n \in \Lambda^k_3} \P_{k,n}(\varepsilon) & \ge \prod_{k=0}^\infty
\prod_{n \in \Lambda^k_3} \P\left(Z_{k,n} \in \left(-\frac{\delta}{ 2c_n(\alpha)
\sqrt{\varepsilon\lambda_k} }, \frac{ \delta} {2c_n(\alpha) \sqrt{\varepsilon\lambda_k} }
\right) \right) \\
& = \prod_{k=0}^\infty \prod_{n \in \Lambda^k_3} \left( 1 - \sqrt{\frac{2}{\pi}}
\int_{\delta/(2c_n(\alpha)\sqrt{\varepsilon\lambda_k})}^\infty e^{-u^2/2} du \right).
\end{align*}
For $a > 1$ we have $\int_a^\infty e^{-x^2/2} dx \le e^{-a^2/2}$. Thus for small enough
$\varepsilon$:
\begin{align*}
\prod_{k=0}^\infty \prod_{n \in \Lambda^k_3} \P_{k,n}(\varepsilon) \ge
\prod_{k=0}^\infty \prod_{n \in \Lambda^k_3} \left( 1 - \sqrt{\frac{2}{\pi}}
\exp\left( - \frac{\delta^2}{8 c_n^2(\alpha) \varepsilon \lambda_k} \right)\right).
\end{align*}
This amount will tend to $1$ if and only if its logarithm tends to 0 as $\varepsilon \rightarrow 0$. Since $\log(1-x) \leq - x$ for $x\in (0,1)$, it suffices to prove that
\begin{align} \label{eq:lambda3}
\lim_{\varepsilon \rightarrow 0} \sum_{k=0}^\infty \sum_{n \ge 0}
\exp\left( - \frac{\delta ^2}{8 c_n^2(\alpha) \varepsilon \lambda_k} \right) = 0.
\end{align}
This is true by dominated convergence, because $c_n(\alpha) = 2^{n(\alpha-1/2) + \alpha - 1}$,
and since $(\lambda_k) \in l_1$.
We will make this more precise.
First observe that for $a>0$
\begin{align*}
e^{-a}\leq \frac{1}{a}e^{-1}\\
\mbox{if }\log(a)-a\leq -1.
\end{align*}
For $k,n\ge 0$ we write $\eta_{n,k}=\frac{\delta^2}{8 c_n^2(\alpha) \varepsilon
\lambda_k}$. Clearly there exists a finite set $T\subset \mathbb{N}_0^2$
such that $\log(\eta_{n,k})-\eta_{n,k}\leq -1$ for all $(n,k)\in T^c$. We
set $C=\sum_{(n,k)\in T}e^{-\eta_{n,k}}$ and get
\begin{align*}
\sum_{k=0}^\infty \sum_{n= 0}^\infty \exp\left( - \frac{\delta^2}{8 c_n^2(\alpha)
\varepsilon \lambda_k} \right)&=C+\sum_{(n,k)\in T^c}^\infty e^{-\eta_{n,k}}\\
&\leq C+\sum_{(n,k)\in T^c}^\infty\frac{1}{\eta_{n,k}}e^{-1}\\
&\leq C+\frac{8\varepsilon e^{-1}}{\delta^2}\sum_{k\ge 0}
\lambda_k\sum_{n\ge 0}c_n(\alpha)^2 <\infty.
\end{align*}
3. Since $\cup_{k\ge 0}\Lambda_4^k$ is finite, and since for every $n$ in $\Lambda_4^k$
the interval $(\mathcal{F}_{k,n} - \delta, \mathcal{F}_{k,n} + \delta )$ contains a small
neighborhood of $0$, we have
\begin{align} \label{eq:lambda4}
\lim_{\varepsilon \rightarrow 0}\prod_{k=0}^\infty
\prod_{n \in \Lambda^k_4} \P_{k,n}(\varepsilon) = 1.
\end{align}
4. Again because $\cup_{k\ge 0}\Lambda_2^k$ is finite, we obtain from its definition that
\begin{align}\label{eq:lambda2}
\lim_{\varepsilon \rightarrow 0}\prod_{k=0}^\infty \prod_{n \in
\Lambda^k_2} \P_{k,n}(\varepsilon) = 2^{-|\cup_k \Lambda_2^k|}.
\end{align}
5. Finally we calculate $\lim_{\varepsilon \rightarrow 0} \prod_{k=0}^\infty
\prod_{n \in \Lambda^k_1} \P_{k,n}(\varepsilon)$. For given $k,n$ define
\begin{align*}
\bar{F}_{k,n} = \left\{\begin{array}{ll} F_{k,n} - \delta, &F_{k,n} > \delta, \\ F_{k,n} + \delta, &
F_{k,n} < - \delta. \end{array} \right.
\end{align*}
We know that $Z_{k,n}$ is standard normal, so that by Proposition
\ref{prop:ldp standard normal} for $n \in \Lambda_1^k$
\begin{align*}
\lim_{\varepsilon \rightarrow 0} \varepsilon \log \P_{k,n}^0(\varepsilon)
= - \frac{\bar{F}_{k,n}^2}{2c_n^2(\alpha) \lambda_k},
\end{align*}
and therefore again by the finiteness of $\cup_k \Lambda_1^k$
\begin{align} \label{eq:lambda1}
\lim_{\varepsilon \rightarrow 0} \varepsilon \log \prod_{k=0}^\infty \prod_{n \in \Lambda_1^k}
\P_{k,n}^0(\varepsilon) = - \sum_{k=0}^\infty \sum_{n \in \Lambda_1^k}
\frac{\bar{F}_{k,n}^2}{2c_n^2(\alpha) \lambda_k}.
\end{align}
6. Combining (\ref{eq:lambda3}) - (\ref{eq:lambda1}) we obtain
\begin{align*}
\lim_{\varepsilon \rightarrow 0} \varepsilon \log \mu_\varepsilon (U^\delta_\alpha(F)) = -
\sum_{k=0}^\infty \frac{1}{\lambda_k} \sum_{n \in \Lambda_1^k} \frac{\bar{F}_{k,n}^2}{2c_n^2(\alpha)}.
\end{align*}
So if we manage to show
\begin{align*}
- \sum_{k=0}^\infty \frac{1}{\lambda_k} \sum_{n \in \Lambda_1^k}
\frac{\bar{F}_{k,n}^2}{2c_n^2(\alpha)} = -\inf_{G \in U^\delta_\alpha(F)} I(G),
\end{align*}
the proof is complete. By Ciesielski's isomorphism, every $G \in C_\alpha^0([0,1];H)$ has the
representation
\begin{align*}
G = \sum_{k=0}^\infty e_k \sum_{n=0}^\infty \frac{G_{k,n}}{c_n(\alpha)} \phi_n.
\end{align*}
Its derivative fulfills (if it exists) for any $k\ge 0$
\begin{align*}
\langle \dot G, e_k \rangle = \sum_{n=0}^\infty \frac{G_{k,n}}{c_n(\alpha)} \chi_n.
\end{align*}
Since the Haar functions $(\chi_n)_{n\ge 0}$ are a CONS for $L^2([0,1])$, we see that $\tilde{I}(\langle G,
e_k \rangle) < \infty$ if and only if $(G_{k,n}/ c_n(\alpha)) \in l_2$, and in this case
\begin{align*}
\tilde{I} (\langle G, e_k \rangle) = \frac{1}{2} \int_0^1 \langle \dot G(s), e_k \rangle^2 ds =
\sum_{n=0}^\infty \frac{G_{k,n}^2}{2c_n^2(\alpha)}.
\end{align*}
So we finally obtain with Lemma \ref{lem:gleichheit rate functions} the desired equality
\begin{align*}
\inf_{G \in U^\delta_\alpha(F)} I(G) &= \inf_{G \in U^\delta_\alpha(F)}
\sum_{k=0}^\infty \frac{1}{\lambda_k} \tilde{I}(\langle G, e_k \rangle) =
\inf_{G \in U^\delta_\alpha(F)} \sum_{k=0}^\infty \frac{1}{\lambda_k} \sum_{n=0}^\infty
\frac{G_{k,n}^2}{2c_n^2(\alpha)} \\
& = \sum_{k=0}^\infty \frac{1}{\lambda_k} \sum_{n \in \Lambda_1^k}
\frac{\bar{F}_{k,n}^2}{2c_n^2(\alpha)}.
\end{align*}
\end{proof}
\subsection{Exponential tightness}
The final ingredient needed in the proof of the LDP for Hilbert space valued Wiener processes is exponential tightness.
It will be established in two steps. The first step claims exponential tightness for the family of laws of $\sqrt{\varepsilon}Z, \varepsilon>0,$ where $Z$ is an $H$-valued $\mathcal{N}(0,Q)$-variable.
\begin{lem}\label{lem: exponential tight elementary}
Let $\varepsilon>0$ and $\nu_\varepsilon=\P\circ (\sqrt{\varepsilon}Z)^{-1}$ for a centered Gaussian random variable $Z$ with values in the separable Hilbert space $H$ and covariance operator $Q$. Then $(\nu_\varepsilon)_{\varepsilon\in (0,1]}$ is exponentially tight. More precisely for every $a>0$ there exists a compact subset $K_a$ of $H$, such that for every $\varepsilon \in (0,1]$
\begin{align*}
\nu_\varepsilon(K_a^c) \le e^{-a/\varepsilon}
\end{align*}
\end{lem}
\begin{proof}
We know that for a sequence $(b_k)_{k\ge 0}$ converging to 0, the operator $T_{(b_k)}:=\sum_{k=0}^\infty b_k \langle \cdot, e_k\rangle e_k$ is compact. That is, for bounded sets $A\subset H$ the set $T_{(b_k)}(A)$ is precompact in $H$. Since $H$ is complete, this means that $cl(T_{(b_k)}(A))$ is compact. Let $a' > 0$ to be specified later. Denote by $B(0,\sqrt{a'})\subset H$ the ball of radius $\sqrt{a'}$ in $H$. We will show that there exists a zero sequence $(b_k)_{k\ge 0}$, such that the compact set $K_{a'}=cl(T_{(b_k)}(B(0,\sqrt{a'})))$ satisfies for all $\varepsilon \in (0,1]$
\begin{align}\label{eq: estimate for compact(a')}
\P(\sqrt{\varepsilon}Z\in (K_{a'})^c)\leq c e^{-a'/\varepsilon}.
\end{align}
with a constant $c>0$ that does not depend on $a'$. Thus for given $a$, we can choose $a'>a$ such that for every $\varepsilon \in (0,1]$
\begin{align*}
c \le e^{(a'-a)/\varepsilon}
\end{align*}
and therefore the proof is complete once we proved \eqref{eq: estimate for compact(a')}.
Since $Z$ is Gaussian, $e^{\lambda \lVert Z\rVert_H}$ is integrable for small $\lambda$, and we can apply Markov's inequality to obtain constants $\lambda(Q),c(Q) > 0$ such that $\P(\lVert Z\rVert_H\ge \sqrt{a'})\le c(Q)e^{-\lambda(Q) a'}$.
Note that if $(\lambda_k)_{k\ge 0}\in l^1$, we can always find a sequence $(c_k)_{k\ge 0}$ such that $\lim_{k\to\infty}c_k = \infty$ and $\sum_{k\ge 0} c_k\lambda_k<\infty$. For $\beta>0$ that will be specified later, we set $b_k=\sqrt{\frac{\beta}{c_k}}$ for all $k\ge 0$. We can define $(T_{(b_k)})^{-1}=\sum_{k=0}^\infty \frac{1}{b_k}\langle \cdot, e_k\rangle e_k$. This gives
\begin{align*}
\P(\sqrt{\varepsilon}Z\in (K_{a'})^c)&\le\P(\sqrt{\varepsilon}(T_{(b_k)})^{-1}(Z)\notin B(0,\sqrt{a'}))\\
&=\P(\lVert (T_{(b_k)})^{-1}(Z)\rVert_H^2 \ge \frac{a'}{\varepsilon})\\
&=\P\left(\sum_{k=0}^\infty c_k |\langle Z,e_k\rangle|^2\ge \frac{\beta a'}{\varepsilon}\right)\\
&=\P\left(\lVert \tilde{Z}\rVert_H\ge \sqrt{\frac{\beta a'}{\varepsilon}}\right),
\end{align*}
where $\tilde{Z}$ is a centered Gaussian random variable with trace class covariance operator
\begin{align*}
\tilde{Q}=\sum_{k=0}^\infty c_k\lambda_k\langle \cdot, e_k\rangle e_k.
\end{align*}
Consequently we obtain
\begin{align*}
\P(\sqrt{\varepsilon}Z\in (K_{a'})^c)\le c(\tilde{Q})e^{-\frac{\lambda(\tilde{Q})\beta a'}{\varepsilon}}
\end{align*}
Choosing $\beta=\frac{1}{\lambda(\tilde{Q})}$ proves the claim \eqref{eq: estimate for compact(a')}.
\end{proof}
With the help of Lemma \ref{lem: exponential tight elementary} we are now in a position to prove exponential tightness for the family $(\mu_\varepsilon)_{\varepsilon \in (0,1]}$.
\begin{lem}\label{lem:exponential tightness}
$(\mu_\varepsilon)_{\varepsilon \in (0,1]}$ is an exponentially tight family of probability measures on $(C_\alpha^0([0,1];H)$, $\lVert.\rVert_\alpha)$.
\end{lem}
\begin{proof}
Let $a > 0$. We will construct a suitable set of the form
\begin{align*}
\tilde{K}^a = \prod_{n=0}^\infty K^a_n
\end{align*}
such that
\begin{align*}
\limsup_{\varepsilon \rightarrow 0}\varepsilon \log\mu_{\varepsilon}\left[\left(\left(T^H_\alpha\right)^{-1}\tilde{K}^a\right)^c\right]\leq -a.
\end{align*}
Here each $K^a_n$ is a compact subset of $H$, such that the diameter of $K^a_n$ tends to 0 as $n$ tends to $\infty$. Then $\tilde{K}^a$ will be sequentially compact in $\mathcal{C}^H_0$ by a diagonal sequence argument. Since $\mathcal{C}^H_0$ is a metric space, $\tilde{K}^a$ will be compact. As we saw in Theorem \ref{CiesilskiH}, $(T^H_{\alpha})^{-1}$ is continuous, so that then $K^a:=(T^H_{\alpha})^{-1}(\tilde{K}^a)$ is compact in $(C^0_{\alpha}([0,1],H), \lVert \cdot \rVert_{\alpha})$.
Let $\nu_\varepsilon=\P\circ (\sqrt{\varepsilon}Z)^{-1}$ for a random variable $Z$ on $H$ with $Z\sim \mathcal{N}(0,Q)$. By Lemma \ref{lem: exponential tight elementary}, we can find a sequence of compact sets $(K^a_n)_{n\in\mathbb{N}}\subset H$ such that for all $\varepsilon \in (0,1]$:
\begin{align*}
\nu_\varepsilon((K^a_n)^c)\leq \exp\left(\frac{-(n+1)a}{\varepsilon}\right).
\end{align*}
To guarantee that the diameter of the $K^a_n$ converges to zero, denoting by $\overline{B}(0,d)$ the closed ball of radius $d$ around $0$, we set
\begin{align*}
\tilde{K^a}:=\prod_{n=0}^{\infty}c_n(\alpha)\left(\overline{B}\left(0,\sqrt{\frac{a(n+1)}{\lambda}}\right)\cap K^a_n\right).
\end{align*}
Since $c_n(\alpha)\sqrt{a(n+1)/\lambda}\rightarrow0$ as $n\rightarrow \infty$, this is a compact set in $\mathcal{C}^H_0$. Thus $K^a:=(T^H_{\alpha})^{-1}(\tilde{K^a})$ is compact in $(C^0_{\alpha}([0,1],H), \lVert \cdot \rVert_{\alpha})$.
Remember that by Lemma \ref{lem:representation of Wienerprocess} we have $W=\sum_{n=0}^{\infty}\phi_nZ_n$, where $(Z_n)_{n\ge 0}$ is an i.i.d. sequence of $\mathcal{N}(0,Q)-$ variables. This implies $T_\alpha^H(W)=(c_n(\alpha)Z_n)_{n\ge 0}$ and thus for any $\varepsilon \in (0,1]$
\begin{align*}
\mu_{\varepsilon}((K^a)^c)&=\P\left[\cup_{n\in\mathbb{N}_0}\left\{c_n(\alpha)\sqrt{\varepsilon}Z_n\in \left(c_n(\alpha)\left(B\left(0,\sqrt{\frac{a(n+1)}{\lambda}}\right)\cap K^a_n\right)\right)^c\right\}\right]\\
&\leq \sum_{n=0}^{\infty}\left(\nu_\varepsilon((K^a_n)^c)+\P\left(\lVert Z_n\rVert\geq \sqrt{\frac{a(n+1)}{\varepsilon\lambda}}\right)\right)\\
&\leq \sum_{n=0}^{\infty}\left(e^{\frac{-(n+1)a}{\varepsilon}}+ce^{\frac{-a(n+1)}{\varepsilon}}\right)\\
&=(1+c)\frac{e^{\frac{-a}{\varepsilon}}}{1-e^{\frac{-a}{\varepsilon}}}.
\end{align*}
So we have
\begin{align*}
\limsup_{\varepsilon \rightarrow 0}\varepsilon \log\mu_{\varepsilon}((K^a)^c)\leq -a,
\end{align*}
\end{proof}
We now combine the arguments given so far to obtain an LDP in the H\"older spaces.
\begin{lem}
$(\mu_\varepsilon)_{\varepsilon \in (0,1]}$ satisfies an LDP on $(C_\alpha^0([0,1];H), \lVert.\rVert_\alpha)$ with good rate function $I$.
\end{lem}
\begin{proof}
We know $\lVert.\rVert_\alpha^{'} \le \lVert.\rVert_\alpha$. Therefore the $\lVert.\rVert_\alpha^{'}$-topology is coarser, which in turn implies that every compact set in the $\lVert.\rVert_\alpha$-topology is also a compact set in the $\lVert.\rVert_\alpha^{'}$-topology. From Lemma \ref{lem:exponential tightness} we thus obtain that $(\mu_\varepsilon)_{\varepsilon \in (0,1]}$ is also exponentially tight on $(C_\alpha^0([0,1];H), \lVert.\rVert_\alpha^{0})$.
Proposition \ref{prop:exponential und ldp fuer offene baelle, dann ldp} implies that $(\mu_\varepsilon)_{\varepsilon \in (0,1]}$ satisfies an LDP with good rate function $I$ on $(C_\alpha^0([0,1];H)$, $\lVert.\rVert_\alpha^{0})$.
Finally we obtain from Proposition \ref{prop:ldp von grober auf feine topo} and from Lemma \ref{lem:exponential tightness} that $(\mu_\varepsilon)_{\varepsilon \in (0,1]}$ satisfies an LDP with good rate function $I$ on $(C_\alpha^0([0,1];H), \lVert.\rVert_\alpha)$.
\end{proof}
We may now extend the LDP from $(C_\alpha^0([0,1];H), \lVert.\rVert_\alpha)$ to $(C_\alpha([0,1];H), \lVert.\rVert_\alpha)$. This is an immediate consequence of the contraction principle (Proposition \ref{prop:contraction principle}), since the inclusion map from $C_\alpha^0([0,1];H)$ to $C_\alpha([0,1];H)$ is continuous.
Similarly we can transfer the LDP from $C^0_\alpha([0,1];H)$ to $C([0,1]; H)$, the space of continuous functions on $[0,1]$ with values in $H$, equipped with the uniform norm.
\begin{thm}
Let $(W(t): t \in [0,1])$ be a $Q$-Wiener process and let for $\varepsilon \in (0,1]$ $\mu_\varepsilon$ be the law of $\sqrt{\varepsilon} W$. Then $(\mu_\varepsilon)_{\varepsilon \in (0,1]}$ satisfies an LDP on $(C([0,1];H),\lVert.\rVert_\infty)$ with rate function $I$.
\end{thm}
\begin{proof}
First we can transfer the LDP from $(C^0_\alpha([0,1];H), \lVert.\rVert_\alpha)$ to $(C^0_\alpha([0,1];H)$, $\lVert.\rVert_\infty)$. This is because on $C^0_\alpha([0,1];H)$, $\lVert.\lVert_\infty \le \lVert.\rVert_\alpha$, whence the $\lVert.\rVert_\infty$-topology is coarser. Therefore $I$ is a good rate function for the $\lVert.\rVert_\infty$-topology as well, and $(\mu_\varepsilon)_{\varepsilon \in (0,1]}$ satisfies an LDP on $(C^0_\alpha([0,1];H)$, $\lVert.\rVert_\infty)$ with good rate function $I$.
The inclusion map from $(C^0_\alpha([0,1];H)$, $\lVert.\rVert_\infty)$ to $(C([0,1];H)$, $\lVert.\rVert_\infty)$ is continuous, so that an application of the contraction principle (Proposition \ref{prop:contraction principle}) finishes the proof.
\end{proof}
\vspace{20pt}
\noindent \textbf{Acknowledgement:} Nicolas Perkowski is supported by a Ph.D. scholarship of the Berlin Mathematical School.
\begin{footnotesize}
|
1,108,101,564,539 | arxiv | \section{Introduction}
\label{sec:intro}
Anomaly detection is the task of identifying anomalous observations~\cite{ivanovska2020,wang2018,prado2016,feng2010,reif2008}. In this paper, we focus on anomaly detection applied to machine health monitoring via audio signal. Detecting anomalies is useful for identifying incipient machine faults, condition-based maintenance, and quality assurance, which are integral components towards Industry 4.0. In comparison to direct measurements, audio is a cost-effective, non-intrusive, and scalable sensor modality.
Specifically, we focus on the problem set-up (Figure \ref{fig:framework}a) where our system adapts to new conditions using only a handful of samples (specifically 3 in the experiment). It is clearly not desirable for an observation to be tagged as anomalous due to changes in operating condition or environmental noise. Thus, we aim to develop an anomalous sound detection (ASD) system that can adapt to new conditions quickly with few-shot samples. Additionally, some meta-information may be available, e.g., machine model and operating load.
Due to the practical difficulty of enumerating potential anomalous conditions and generating such samples, it is typically assumed that only normal samples are available for training \cite{ruff2018deep}. This poses the challenge that anomaly detectors need to learn to identify anomalies in the absence of any anomalous samples. Furthermore, deep anomaly detectors perform in unexpected and not well-understood ways under out-of-distribution scenarios \cite{ruff2021unifying}. Last but not least, the unique characteristics of audio add to the challenge.
\begin{figure}
{
\includegraphics[width= 1.\linewidth]{figures/framework_font.pdf}%
}
\caption{Framework. (a) Our anomalous sound detection system adapts to new conditions with 3-shot samples. We adopt a classification-based approach and define the auxiliary classification tasks on available meta-data. (b) In the outer loop, we alternative between the auxiliary classification tasks, such that the model can quickly adapt to new conditions.
(c) For a given auxiliary classification task, the anomaly score is calculated based on distance between each sample ($\cdot$) to the prototypes ($\star$) on the embedding space. This is equivalent to mixture density estimation on normal samples. }
\label{fig:framework}
\vspace{-0.5cm}
\end{figure}
Given its superior performance, especially on similar problems \cite{koizumi2020description, giri2020self, morita2021anomalous}, we adopt a classification-based approach for anomaly detection \cite{bergman2020classification}, where auxiliary classification tasks are defined on the available meta-data. However, existing classification-based methods \cite{bergman2020classification, kawaguchi2021description} are not designed for anomaly detection under domain shift, and thus are not sufficient for competitive performance on their own, which we show in Section \ref{sec:results}. Thus, we propose a novel approach, with bi-level optimization, to tackle the challenging problem of anomaly detection under domain shifts. In the outer loop (Figure \ref{fig:framework}b), we alternate between all available auxiliary classification tasks leveraging a gradient-based meta-learning (GBML) algorithm \cite{nichol2018first}, such that the resulting model can quickly adapt to the target domain. In the inner loop (Figure \ref{fig:framework}c), we train the classification-based anomaly detector with an episodic procedure to match the few-shots setting \cite{snell2017prototypical}.
While classification-based methods show strong empirical results in the literature \cite{golan2018deep, hendrycks2018deep, bergman2020classification}, there is not a convincing explanation for that performance. Thus, we propose a supplementary explanation that classification-based methods are equivalent to mixture density estimation on normal samples.
Finally, we evaluate our proposed approach on a recently-released dataset of audio measurements from a variety of machine types \cite{kawaguchi2021description} and show strong empirical results.
\begin{comment}
\end{comment}
\section{Related Work}
\label{sec:review}
\paragraph{Anomaly Detection}
Recently, there is a surge of interest in using deep learning approaches for anomaly detection to handle complex, high-dimensional data. Anomaly detection methods can be categorized as density estimation-based, classification-based, and reconstruction-based \cite{bergman2020classification,ruff2021unifying}. Density estimation-based methods fit a probabilistic model on normal data, and data from low density region are considered anomalies. For the purpose of anomaly detection, we are only interested in level sets of the distribution, and thus classification-based methods learn decision boundaries that delineate high-density and low-density region. Reconstruction-based methods train a model to reconstruct normal samples, and use reconstruction error as a proxy for anomaly score.
Due to the superior performance of classification-based methods on our application of interest \cite{giri2020self, morita2021anomalous, koizumi2020description}, we focus our review on this class of methods and refer interested readers to comprehensive reviews \cite{chalapathy2019deep, ruff2021unifying} for more information. A representative method is Deep Support Vector Data Description (SVDD) \cite{ruff2018deep}, a one-class classification method, where the neural network learns a representation that enforces the majority of normal samples to fall within a hypersphere. The core challenge of this line of work is learning the decision boundary for a binary classification problem in the presence of only normal samples. Alternatively, outlier exposure \cite{hendrycks2018deep} takes advantage of an auxiliary dataset of outliers, i.e. samples disjoint from both normal and anomalous ones, to improve performance for anomaly detection and generalization at unseen scenarios. Finally, other methods train models on auxiliary classification tasks and use the negative log-likelihood of a sample belonging to the correct class as the anomaly score. Some examples of such auxiliary classification tasks include identifying the geometric transformation applied to the original sample \cite{golan2018deep, bergman2020classification} and classifying the metadata associated with the sample \cite{giri2020self, morita2021anomalous}. This is reminiscent of self-supervised learning (SSL), where learning to distinguish between self and others is conducive to learning salient representation for other downstream tasks.
\paragraph{Meta-Learning}
The objective of meta-learning is to find a model that can generalize to a new, unseen task with a handful of samples \cite{finn2017model}.
Metric-based meta-learning algorithms learn feature representation such that query samples can be classified based on their similarity to the support samples with known labels. For instance, Matching Networks \cite{vinyals2016matching} use attention mechanism to evaluate the similarity. Prototypical Networks \cite{snell2017prototypical} establish class prototypes based on support samples, and assign each query sample to its nearest prototype.
GBML algorithms, popularized by model-agnostic meta-learning (MAML) \cite{finn2017model}, train models that can quickly adapt to new tasks with gradient-based updates, typically using a bi-level optimization procedure. Variants of MAML have been proposed in works such as \cite{nichol2018first, raghu2019rapid, wang2021bridging}.
\begin{comment}
\begin{itemize}
\item Relation Network \cite{sung2018learning}: Use a neural module to learn a metric, instead of using fixed metric, e.g. Euclidean distance and cosine similarity, on the embedding space.
\item Related application: machine health monitoring \cite{ren2020novel, ding2021meta} and speech recognition \cite{winata2020learning}
\end{itemize}
\end{comment}
\paragraph{Anomalous Sound Detection}
In comparison to general anomaly detection, audio has its unique characteristics, e.g., temporal structure \cite{rushe2019anomaly}. On the same task, \cite{lopez2021ensemble} observed that there was limited performance gain by trying a number of common domain adaptation techniques, highlighting the challenge and the need for research on audio-specific methods.
While reconstruction-based \cite{rushe2019anomaly, suefusa2020anomalous} and density estimation-based \cite{yamaguchi2019adaflow} anomaly detection methods have been used for anomalous sound detection, classification-based methods enjoy superior performance on similar / the same tasks \cite{koizumi2020description, giri2020self, morita2021anomalous}. Albeit differences in specific implementation, the core idea is to classify metadata, such as machine identification, and compute the anomaly score from the negative log-likelihood of class assignment. Also relevant to the task are Sniper \cite{koizumi2019sniper} and SpiderNet \cite{koizumi2020spidernet}, which memorize few-shot anomalous samples to prevent overlooking other anomalous samples. In particular, SpiderNet uses attention mechanism to evaluate the similarity with known anomalous samples, similar to Matching Networks.
\section{Approach}
\label{sec:approach}
\subsection{Preliminaries}
By defining the distribution of normal data as $\mathbb{P}^\star$ over the data space $\mathcal{X}\subseteq\mathbb{R}^D$, anomalies may be characterized as the set where normal samples is unlikely to be, i.e. $\mathcal{A} = \{x\in \mathcal{X}\;|\;\mathbb{P}^\star(x)\leq \alpha\}$, where $\alpha$ is a threshold \cite{ruff2021unifying} that controls Type I error.
A fundamental assumption in anomaly detection is \textit{the concentration assumption}, i.e. the region where normal data reside can be bounded \cite{ruff2021unifying}. More precisely, there exists $\alpha \geq 0$, such that
$\mathcal{X} \setminus \mathcal{A} = \{x\in \mathcal{X}\;|\;\mathbb{P}^\star(x) > \alpha \}$ is nonempty and small.
Note that this assumption does not require the support of $\mathbb{P}^\star$ be bounded; only that the support of the high-density region of $\mathbb{P}^\star$ be bounded. In contrast, anomalies need not be concentrated and a commonly-used assumption is that anomalies follow a uniform distribution over $\mathcal{X}$ \cite{ruff2021unifying}.
\subsection{Problem Formulation}
We study the problem of anomaly detection under domain shift, having access to only few-shot samples in the target domain.
Specifically, we have access to a dataset from the source domain $\mathcal{D} = \{(x_i, y_i)\}$, where each $x_i\in \mathbb{R}^D$ is a normal sample and $y_i\in \mathbb{R}^L$ are the labels for auxiliary classification tasks. We also have access to a small number of samples in the target domain $\mathcal{D}' = \{(x'_i, y_i)\}$, where $x'_i\in \mathbb{R}^D$ is a normal sample from a domain-shifted condition. Note that we only access to normal samples in both source and target domain. We denote each auxiliary task as $\mathcal{T}_l$ and the set of $L$ auxiliary tasks as $\mathcal{T}= \{\mathcal{T}_1, \dots, \mathcal{T}_L\}$.
We use a neural network $f_\theta:\mathbb{R}^{D}\rightarrow \mathbb{R}^d$ to map samples from the data space to an embedding space, where $D\gg d$. We denote the embedding that corresponds to each sample as $z_i$, where $z_i =f_\theta(x_i) \in \mathbb{R}^d$, and the distribution of normal data in the embedding space as $\mathbb{P}^\star_z$.
\subsection{Anomaly Detection via Metadata Classification}
To simplify notation, we consider a single auxiliary task $\mathcal{T}_l$ with class labels $1, \dots, K$ in this subsection. We denote $\mathcal{D}_k=\{(x_i, y_i)\in \mathcal{D}\;|\;y_i=k)\}$.
As discussed in Section \ref{sec:review}, anomaly detection methods based on auxiliary classification train models to differentiate between classes, either defined via metadata inherent to the dataset or synthesized by applying various transformations to the sample, and calculate the anomaly score as the negative log-likelihood of a sample belonging to the correct class (Eqn. \ref{eq:anomaly_score}) \cite{hendrycks2016baseline}.
\begin{equation}\label{eq:anomaly_score}
\Omega_\theta(x_i, y_i) =-\log \mathbb{P}_\theta(y=y_i|x_i)
\end{equation}
Following \cite{bergman2020classification}, we define the likelihood by distance on the embedding space (Eqn. \ref{eq:prob}), rather than a classifier head, to handle new, unseen classes in the target domain, where $d$ is a distance metric and $c_k= \frac{1}{|\mathcal{D}_k|}\sum_{(x_i, y_i)\sim \mathcal{D}_k} f_\theta(x_i)$ is the class centroid.
\begin{equation}\label{eq:prob}
\mathbb{P}_\theta(y=k|x_i) = \frac{\exp(-d(f_\theta(x_i), c_k))}{\sum_{k'}\exp(-d(f_\theta(x), c_{k'}))}
\end{equation}
The learning objective is to maximize the log-likelihood of assigning each sample to its correct class, or equivalently minimizing the negative log-likelihood as in Eqn. \ref{eq:ce_loss}, leading to the familiar cross-entropy loss.
\begin{equation}\label{eq:ce_loss}
\mathcal{L}_{\tau_l}(x_i, y_i; \theta)
= -\log\frac{\exp(-d(f_\theta(x_i), c_{y_i}))}{\sum_{k}\exp(-d(f_\theta(x), c_{k}))}
\end{equation}
An explanation for the strong empirical performance of anomaly detection based on auxiliary classification is that the auxiliary classification tasks are conducive to learning salient feature useful for anomaly detection \cite{golan2018deep, giri2020self}. However, this explanation provides limited insight in how to define the auxiliary classification tasks and what kind of performance may be expected.
Here, we provide an alternative explanation that the classification objective (Eqn. \ref{eq:ce_loss}) is equivalent to mixture density estimation on normal samples. In Deep SVDD \cite{ruff2018deep}, it is assume that the neural network can find a latent representation such that the majority of normal points fall within a single hypersphere or equivalently $\mathbb{P}_z^\star$ follows an isotropic Gaussian distribution. Based on the same intuition, we
assume $\mathbb{P}_z^\star$ can be characterized by a mixture model (Equation \ref{eq:mixture_model}), where $\pi$ is the prior distribution for class membership. As an example, a machine may operate normally under different operating loads. Instead of assuming that normal data can be modeled by a single cluster, the distribution of normal data can be better characterized with a number of clusters each corresponding to an operating load. We hypothesize that this more flexible representation enables the model to learn fine-granular features conducive to anomaly detection, and extrapolate better to unseen scenarios.
\begin{equation}\label{eq:mixture_model}
\hat{\mathbb{P}}^\star_z (z) = \sum_{k=1}^K \pi_k \mathbb{P}(z|y=k)
\end{equation}
We also assume that $\mathbb{P}(z|y=k)$ can be characterized by a distribution in the exponential family, $\mathbb{P}(z|y=k) \propto \exp(-d(z, c_k)) $
where $d$ is a Bregman divergence \cite{banerjee2005clustering}. The choice of $d$ dictates the modeling assumption on the conditional distribution, $\mathbb{P}(z|y)$. For instance, by choosing squared Euclidean distance, i.e. $d(z, c_k) = ||z-c_k||^2_2$, one models each cluster as an isotropic Gaussian. Given these modeling assumptions, it is easy to see that
\begin{equation}\label{eq:cluster_assignment}
\mathbb{P}_\theta(y=k|x_i) = \frac{\pi_k\exp(-d(f_\theta(x_i), c_k))}{\sum_{k'}\pi_{k'}\exp(-d(f_\theta(x), c_{k'}))}
\end{equation}
Eqn. \ref{eq:cluster_assignment} is the same as Eqn. \ref{eq:prob}, by assuming a flat prior on class membership, which can be satisfied by sampling mini-batches with balanced classes. This shows that learning the auxiliary classification task is equivalent to performing mixture density estimation with exponential family, where each cluster corresponds to a class.
\begin{algorithm}[tb]
\caption{Learning to Adapt with Few-shot Samples}
\label{alg:algorithm}
\begin{algorithmic}[1]
\State \textbf{input}: Data $\mathcal{D}$, $\mathcal{D}'$; Test set $\mathcal{D}_{\text{test}}$; Model $f_\theta$;
\State \textbf{parameter}: Learning rate $\alpha$; Outer step-size $\epsilon$; Number of inner and fine-tuning iterations $T$ and $T_{\text{test}}$
\Function{ComputeLoss}{$\mathcal{S}$, $\mathcal{Q}$, $l$}
\State// \emph{input: support set, query set, task index}
\State // \emph{Compute prototypes from the support set}
\State $c_k= \frac{1}{|\mathcal{S}_k|}\sum_{(x_i, y_i)\sim \mathcal{S}_k}f_\theta (x_i), \; \forall k$
\State // \emph{Calculate task loss from the query set}
\State \Return $\frac{1}{|\mathcal{Q}|}\sum_{(x_i, y_i)\sim \mathcal{Q}} \mathcal{L}_{\tau_l}(x_i, y_i; \theta)$
\EndFunction
\State
\Procedure{Train}{$f_\theta$, $\mathcal{D}$}
\State// \emph{input: model, training data}
\State \textbf{initialize} $\theta$
\While{not done}
\For{$\mathcal{T}_l\sim \mathcal{T}$}
\State \textbf{set} $\theta^{(0)} = \theta$
\For{$t = 0, \dots, T-1$}
\State // \emph{Sample support and query set}
\State $\mathcal{S}, \mathcal{Q}\sim \mathcal{D}$
\State $\mathcal{L}_{\mathcal{T}_l}$ = \Call{ComputeLoss}{$\mathcal{S}$, $\mathcal{Q}$, $l$}
\State \textbf{update} $\theta^{(t+1)} = \theta^{(t)} - \alpha \nabla_\theta \mathcal{L}_{\mathcal{T}_l}(\theta^{(t)})$
\EndFor
\State \textbf{meta update} $\theta \leftarrow\theta + \epsilon\left[\theta^{(T)}-\theta\right]$
\EndFor
\EndWhile
\EndProcedure
\State
\Procedure{Inference}{$f_{\theta^\star}$, $\mathcal{D}'$, $\mathcal{D}_{\text{test}}$}
\State // \emph{input: trained model, few-shot examples, test set}
\State \textbf{set} $\theta^{(0)} = \theta^\star$
\State // \emph{Fine-tuning on few-shot examples}
\For{$t = 0, \dots, T_{\text{test}}-1$}
\State $\mathcal{L}_{\mathcal{T}_0}$ = \Call{ComputeLoss}{$\mathcal{D}'$, $\mathcal{D}'$, 0}
\State \textbf{update} $\theta^{(t+1)} = \theta^{(t)} - \alpha \nabla_\theta \mathcal{L}_{\mathcal{T}_0}(\theta^{(t)})$
\EndFor
\State // \emph{Compute anomaly score for test samples}
\State \textbf{compute} $\Omega_{\theta^{(T_{\text{test}})}}(x_i, y_i), \;\; \forall (x_i, y_i)\sim \mathcal{D}_{\text{test}}$
\EndProcedure
\end{algorithmic}
\end{algorithm}
\paragraph{Adaptation with Few-shot Samples} \label{sec:Protonet} A distinction from the typical anomaly detection problem set-up is that we need to adapt to new conditions and only have access to few-shot samples from the target domain. Thus, we draw from the few-shot classification literature, and modify the classification-based anomaly detector for the few-shot setting. Specifically, Prototypical networks (ProtoNet) \cite{snell2017prototypical} is a few-shot learning method that also classifies samples based on distance on the embedding space. During inference, ProtoNet
uses the few-shot examples to establish the new class centroid (i.e. prototype) under domain-shifted conditions, $c'_k$, i.e.
$c'_k = \frac{1}{|\mathcal{D}'_k|}\sum_{(x'_i, y'_i)\sim \mathcal{D}'_k}f_\theta (x'_i)$. Each test sample is assigned to the nearest class prototype, with the same probability as defined by Eqn. \ref{eq:prob}.
ProtoNet adopts an episodic training procedure (see \textsc{ComputeLoss} in Algorithm \ref{alg:algorithm}), such that the training condition matches the test condition. During training, ProtoNet splits each mini-batch into support set, $\mathcal{S}$, and query set, $\mathcal{Q}$, where the support set simulates the few-shot examples, and the query set simulates the test samples during inference. In other words, the prototypes are computed on the support set and the loss is evaluated on the query set.
\paragraph{Outlier Exposure} We also use the outlier exposure (OE) technique \cite{hendrycks2018deep} to boost performance. It is commonly assumed that anomalies follow a uniform distribution over the data space \cite{ruff2021unifying}. Thus, the auxiliary loss for outlier exposure $\mathcal{L}_{\text{OE}}$ is defined as cross-entropy to the uniform distribution, and added to the learning objective with weight $\lambda$.
$$\mathcal{L} = \mathcal{L}_{\mathcal{T}_l} + \lambda\mathcal{L}_{\text{OE}}$$
\begin{comment}
\cite{bergman2020classification} suggested that triplet-center loss \cite{he2018triplet} empirically performs better than cross entropy loss.
$$\mathcal{L} = \frac{1}{N}\sum_{i=1}^{N} \max{\left(d(f_\theta(x_i), c_{y_i})-\min_{k\neq y_i} d(f_\theta(x_i, c_{k})+m, 0\right)}$$
\textcolor{teal}{Ablation: Cross-entropy loss as in \cite{snell2017prototypical} vs. triplet-center loss as suggested in \cite{bergman2020classification}}
\end{comment}
\subsection{Multi-objective Meta-learning}
So far, we have focused the discussion on the case of there being a single auxiliary classification task. But, more meta-information regarding operating conditions may be available. Since the samples may be subject to different changes due to operating condition, machine load, or environment noise, we hypothesize that training on a variety of tasks is conducive to generalizing well to different domain shifts. Also, it is not known a priori which auxiliary classification task would be most effective for anomaly detection. Thus, it is sensible to train on all available auxiliary classification tasks. Empirically, we show in Section \ref{sec:results} that training on all auxiliary classification tasks does outperform training on any single one.
Recall that meta-learning trains the model on a distribution of tasks such that it can quickly learn new, unseen tasks with few-shot samples. Thus, these auxiliary classification tasks can be naturally incorporated into meta-learning algorithms. Wang et al. in \cite{wang2021bridging} draw the close connections between multi-task learning and meta-learning.
A distinction, however, is that meta-learning typically trains on different tasks of the same nature, e.g. 5-way image classification, while multi-task learning may train on functionally related tasks of different nature, e.g. image reconstruction and classification. Our approach falls under the latter case.
We use Reptile \cite{nichol2018first}, a first-order variant of MAML, which learns a parameter initialization that can be fine-tuned quickly on a new task. Reptile repeatedly samples a task $\mathcal{T}_l\sim \mathcal{T}$, trains on it, and moves the model parameter towards the trained weights on that task (see \textsc{Train} in Algorithm \ref{alg:algorithm}) following
$$\theta \leftarrow\theta + \epsilon\left[\theta^{(T)}-\theta\right]$$
where $\theta^{(T)}$ are the model parameters after training on $\mathcal{T}_l$ for $T$ gradient steps and $\epsilon$ is the step-size for meta-update. Based on Taylor series analysis, \cite{nichol2018first} shows that Reptile simultaneously minimizes expected loss over all tasks, and maximizes within-task generalization, i.e. taking a gradient step for a specific task on one mini-batch, also improves performance on other tasks.
During inference, the trained model is first fine-tuned on few-shot examples from the target domain, and the anomaly score (Eqn. \ref{eq:anomaly_score}) can be computed on the test set (see \textsc{Inference} in Algorithm \ref{alg:algorithm}).
\begin{table*}[ht]
\centering
\begin{tabular}{c C{5.0cm} C{5.5cm} C{3.8cm}}
\hline \noalign{\smallskip}
\textbf{Machine Type} & \textbf{Anomalous Conditions} & \textbf{Variations across Domains} & \textbf{Auxiliary Classification Tasks} \\\hline \noalign{\smallskip}
\textbf{Toy Car}& Bent shaft; Deformed / melted gears; Damaged wheels & Car model; Speed; Microphone type and position & Section ID; Model No.; Speed \\ \noalign{\smallskip
\textbf{Toy Train} & Flat tire; Broken shaft; Disjoint railway track; Obstruction on track & Train model; Speed; Microphone type and position & Section ID; Model No.; Speed \\ \noalign{\smallskip}
\textbf{Fan} & Damaged / unbalanced wing; Clogging; Over-voltage& Wind strength; Size of the fan; Background noise & Section ID \\\noalign{\smallskip}
\textbf{Gearbox} & Damaged gear; Overload; Over-voltage & Voltage; Arm-length; Weight& Section ID; Voltage; Arm length; Weight; \\ \noalign{\smallskip
\textbf{Pump} & Contamination; Clogging; Leakage; \newline Dry run & Fluid viscosity; Sound-to-Noise Ratio (SNR); Number of pumps & Section ID \\ \noalign{\smallskip
\textbf{Slider} &Damaged rail; Loose belt; No grease & Velocity; Operation pattern; Belt material & Section ID; Slider type; Velocity; Displacement; \\ \noalign{\smallskip}
\textbf{Valve} &Contamination&Operation pattern; Existence of pump in background; Number of valves & Section ID; Pattern; Existence of pump; Multiple valves\\ \hline
\end{tabular}
\caption{Summary of anomalous conditions, domain shifts, and auxiliary classification tasks}
\label{tab:dataset}
\vspace{-0.5cm}
\end{table*}
\section{Experiment}
\label{sec:experiments}
In this section, we describe the experimental set-up, including the dataset, evaluation metrics, our implementation details, baselines, and ablation study.
\paragraph{Dataset}
We use the dataset from Detection and Classification of Acoustic Scenes and Events (DCASE) Challenge 2021 Task 2 \cite{kawaguchi2021description}, which is composed of subsets of ToyADMOS2 \cite{harada2021toyadmos2} and MIMII DUE \cite{tanabe2021mimii}. The dataset consists of normal and anomalous samples from seven distinct machine types, i.e. toy car, toy train, fan, gearbox, pump, slider, and valve. Anecdotally, we found it extremely challenging to distinguish normal and anomalous with untrained human ears. The anomalous samples are generated by intentionally damaging the machines in different ways (see Table \ref{tab:dataset}). Note that the anomalous samples are used for evaluation only.
Each sample is a 10s audio clip at 16kHz sampling rate, including both machine sound and environment noise.
For each machine type, the data is grouped into 6 sections, where a \texttt{section} is a unit for performance evaluation and roughly corresponds to a distinct machine. In each section, the samples are collected from two different conditions, which we refer to as the source and target domain. The domain shifts are different across sections. A notable challenge is that there are only 3 normal samples for the target domain in each section, while there are 1000 for the source domain. Table \ref{tab:dataset} summarizes the domain shifts in the dataset.
For evaluation, each section has 100 normal samples and 100 anomalous samples for both source and target domain. Section 0, 1, 2 are designated as the validation set, and Section 3, 4, 5 are designated as the the test set.
\paragraph{Evaluation Metrics} We follow the same evaluation procedure as the DCASE Challenge, and report the Area Under the Receiver Operating Characteristic curve (AUROC) and the partial AUROC (pAUROC), which controls the false positive rate at 0.1. The metrics are aggregated over sections and machines types with harmonic mean. We focus on the results on the target domain.
\paragraph{Implementation Details}
Our preprocessing and model architecture follows the 2\textsuperscript{nd} baseline from the challenge organizer \cite{kawaguchi2021description}. The raw audio is preprocessed into log-mel-spectrogram with a frame size of 64ms, a hop size of 50\%, and 128 mel filters. After preprocessing, 64 consecutive frames in a context window are treated as a sample, and samples are generated from a audio clip by shifting the context window by 8 frames. Each 10s audio clip results in 32 samples, with each sample, $x_i\in \mathbb{R}^{64\times 128}$. We use MobileNetV2 \cite{sandler2018mobilenetv2} as backbone, and set the bottleneck size to 128. Thus, $z_i = f_\theta(x_i) \in \mathbb{R}^{128}$.
We follow the optimization procedure in \cite{nichol2018first}. In the inner loop, we use ADAM as the optimizer with a learning rate $\alpha = 0.001$, $\beta_1 = 0$, and $\beta_2 = 0.999$. Each mini-batch is sampled to be have balanced class, with 3 audio clips as support and 5 as query for each class, simulating the 3-shot setting during inference. For the outer loop, we use SGD with step-size, $\epsilon$, linearly annealing from 1 to 0. The number of iterations for inner loop and fine-tuning are $T = 8$, and $T_{\text{test}} = 50$ respectively. We have a model for each machine type and train it for 10K steps. For outlier exposure, we generate outliers by 1) taking samples from machines other than the one being trained, and 2) synthesizing samples via frequency warping following \cite{giri2020self}. We use $\lambda=0.2$.
The implementation is in \texttt{PyTorch}\cite{pytorch}
and trained on a machine with Intel\textsuperscript{\textregistered} Core\texttrademark~i9-10900KF [email protected] and NVIDIA GeForce RTX\texttrademark~3090 GPU.
\paragraph{Baselines and Ablation Study} We compare our model to the two baselines provided by the challenge organizer \cite{kawaguchi2021description}. The 1\textsuperscript{st} baseline adopts a reconstruction-based approach with an autoencoder. The 2\textsuperscript{nd} baseline uses a classification-based approach with MobileNetV2 as the backbone. The auxiliary classification task is defined as section ID. Note that our preprocessing and the neural architecture is the same as that of the 2\textsuperscript{nd} baseline. We also compare our model to the best-performing system \cite{lopez2021ensemble} among the 77 submissions to the challenge, which used an ensemble of two classification-based models and one density estimation-based model. We conduct ablation study to analyze the contribution of individual components.
\section{Results}\label{sec:results}
\begin{figure}
\centering
\includegraphics[width = \linewidth]{figures/by_task.pdf}
\caption{Performance comparison of using individual vs. all auxiliary classification tasks (evaluated on validation set). For individual tasks, the bar length indicates the averaged score over tasks and error bar indicates the minimum and the maximum over tasks.}
\label{fig:by_task}
\vspace{-0.5cm}
\end{figure}
\paragraph{Choice of Auxiliary Classification Tasks}
While in prior work \cite{giri2020self, morita2021anomalous, lopez2021ensemble}, it is popular to use machine / section ID as the auxiliary classification task, we hypothesize that training on multiple auxiliary classification tasks performs better than training on any individual one. We define auxiliary classification tasks by parsing the meta-information associated with each audio clip, also summarized in Table \ref{tab:dataset}. Take \texttt{Toy Car} as an example, we have access to information on car model and speed. Intuitively, being able to differentiate between car models/speeds, is conducive for the model to adapt to new model/speed and potentially other new conditions.
\begin{figure*}[ht]
\centering
\includegraphics[width=\linewidth]{figures/results.pdf}
\caption{Performance comparison with baselines and ablation (evaluated over the test set). The proposed method (Ours) is compared against Baseline 1 and 2, DCASE 2021 Task2 winner, and ProtoNet, and our proposed method without outlier exposure (ProtoNet + Reptile).}
\label{fig:results}
\vspace{-0.5cm}
\end{figure*}
To confirm the hypothesis, we train the anomaly detector on individual classification tasks with ProtoNet, and compare its performance with our proposed approach of alternating between all auxiliary classification tasks using Reptile. We report the score on the validation set in Figure \ref{fig:by_task}. As mentioned in Section \ref{sec:experiments}, we follow the evaluation procedure in the DCASE challenge and report the performance as harmonic mean over AUROC and [email protected] of relevant data sections.
\texttt{Fan} and \texttt{Pump} are not compared here, as only one auxiliary classification task is available for these two machine types. As expected, training on all tasks performs consistently better across different machine types, confirming the hypothesis.
\paragraph{Overall Results}
The overall results evaluated on the test set are summarized in Figure \ref{fig:results}. In summary, our model (\texttt{Ours}) improved on average upon Baseline 1 and 2 by 11.7\% and 9.6\% respectively, and is on par with the best-performing system \cite{lopez2021ensemble}, despite being a third of its model complexity (measured by the number of model parameters). While Baseline 1 and Baseline 2 perform similarly based on the harmonically-averaged scores, there are significant variations across machines. It appears reconstruction-based Baseline 1 and classification-based Baseline 2 excel on different machines. Our proposed model outperforms both baselines for \texttt{ToyCar}, \texttt{Pump}, \texttt{Slider}, and \texttt{Valve}.
In comparison to vanilla classification-based approach in Baseline 2, we trained the classifier episodically to match few-shot setting (\texttt{ProtoNet}), and iterated among different auxiliary classification tasks (\texttt{ProtoNet + Reptile}) to improve generalization. Finally, we augmented the dataset via OE, adding up to our proposed approach (\texttt{Ours}). On average, the episodic training procedure improved performance by 1.9\%, GBML applied to auxiliary classification improved performance by 4.1\% and data augmentation by OE improved performance by 3.6\%. The most significant improvement comes from training on all auxiliary classification tasks via Reptile. There is no improvement from \texttt{Fan} or \texttt{Pump} as these two machine types have only one auxiliary classification task.
\paragraph{Low-Dimensional Visualization} Qualitatively, we show the embedding of the test sample from the trained model for \texttt{Toy Car}, before and after fine-tuning. The stars indicate the prototypes established by the 3-shot samples.
\begin{figure}
\centering
\subcaptionbox{Before Fine-tuning \label{fig:Before}}{\includegraphics[width = \linewidth]{figures/ToyCar-Before_edited.png}}
\subcaptionbox{After Fine-tuning \label{fig:After}}{\includegraphics[width=\linewidth]{figures/ToyCar-After_edited.png}}
\caption{Trained Embedding of Toy Car (via t-SNE) on the test set before \textit{(top)} and after \textit{(bottom)} fine-tuning on the target domain $\mathcal{D'}$. Fine-tuning improves discriminability of normal vs. anomalous samples \textit{(bottom-right)} on the target domain.}
\vspace{-0.5cm}
\label{fig:my_label}
\end{figure}
The trained model has not seen any samples from the target domain prior to fine-tuning. Regardless, the model has already learned meaningful embedding (Figure \ref{fig:Before}), where the samples are naturally separated by section. But, the normal and anomalous samples are not yet well separated. During fine-tuning, the prototypes attract normal samples, and as a result the normal and anomalous samples become better separated.
\begin{comment}
Feature Normalization
\begin{equation}
\frac{1}{|\mathcal{F}|} \sum_i \sum_t \log|S_{i, (t\times f)}|
\end{equation}
PSD Normalization
\begin{equation}
\log [ \frac{1}{|\mathcal{F}|} \sum_t |S_{i, (t\times f)}|^2]
\end{equation}
\end{comment}
\section{Conclusions}
In this work, we tackle the challenging task of adapting unsupervised anomaly detector to new conditions using few-shot samples. To achieve this objective, we leverage approaches from meta-learning literature. We train a classification-based anomaly detector in an episodic procedure to match the few-shot setting during inference. We use a GBML algorithm to find parameter initialization that can quickly adapt to new conditions with gradient-based updates. Finally, we boost our model with outlier exposure.
Grounded in the application of machine health monitoring, we evaluate our proposed method on a recently-released dataset of audio measurements from different machine types, used for DCASE challenge 2021. We model is on par with best-performing system among the 77 submissions to the challenge. We conduct ablation to analyze the contribution of each component, and the GBML procedure leads to the most significant improvement.
\bibliographystyle{IEEEtran}
\section{Response to Reviewers \#5 (\textcolor{Orange!80!Black}{\textbf{R5}})}
\paragraph{Novelty of the Proposed Approach} Existing classification-based methods are not designed for anomaly detection under domain shift, and deep anomaly detectors perform in unexpected and not well-understood ways under out-of-distribution scenarios. To substantiate this point, we show in Section V that a classification-based method (Baseline 2) is unable to do well on its own in out-of-distribution scenarios.
To tackle the challenging task of adapting an anomaly detector to new conditions with few-shot samples, we draw inspiration from few-shot and meta-learning literature. Starting with a classification-based baseline, we 1) incorporate the episodic training procedure proposed in ProtoNet to match the few-shot condition during testing, 2) adopt a gradient-based meta-learning algorithm, Reptile, to improve generalization to different domain shifts. Thus, the novelty of our approach lies in adapting techniques from few-shot and meta-learning literature to the problem of anomaly detection under domain shifts. Empirically, we show through the ablation study (Section V) that using ProtoNet or Reptile on their own is not sufficient for competitive performance in this problem set-up.
We have edited the last paragraph in Section I to make these points more clear.
\paragraph{Support-Query Split} As we stated in the second paragraph of Section IV(c),
\begin{quote}
\textit{Each mini-batch is sampled to be have balanced class, with 3 audio clips as support and 5 as query for each class, simulating the 3-shot setting during inference.}
\end{quote}
\paragraph{Comparison with \cite{ivanovska2020}}
As per reviewer's comment:
\begin{quote}
\textit{The considered Loss is very similar to \cite{ivanovska2020} and a comparison with that approach is needed.}
\end{quote}
To the best of our understanding, \cite{ivanovska2020} introduces an evaluation metric for real-world, GAN-based image anomaly detection in unbalanced datasets. We certainly recognize this to be of paramount importance for a realistic evaluation of the performance of an anomaly detection system in the wild.
The dataset we use to evaluate our proposed method is, however, a balanced dataset, as stated in Section IV. As per \cite{ivanovska2020} - Table II, the correlation between AUC and \%TN@90\%TP is high for balanced dataset, independently on the anomaly detection method being used. This leads us to the conclusion that adding \%TN@90\%TP as a further metric to the manuscript would not provide additional information to the reader.
GAN-based sound anomaly detection systems were proposed on the task under analysis (DCASE challenge 2021, Task 2) by \cite{LiuCQUPT2021} and \cite{DiniTAU2021}, ranking respectively 64th and 68th in the leaderboard\footnote{https://dcase.community/challenge2021/task-unsupervised-detection-of-anomalous-sounds-results}. As we are already comparing our proposed method against the best-performing system \cite{lopez2021ensemble}, we don't believe adding a comparison to GAN-based methods would provide the reader with additional information of value.
\section{Response to Reviewers \#7 (\textcolor{violet}{\textbf{R7}})}
We appreciate your positive feedback. We have increased the font size in Figure 1 to improve readability.
\newpage
\bibliographystyle{IEEEtran}
|
1,108,101,564,540 | arxiv | \section{Introduction}
\label{Section: Introduction}
\IEEEPARstart{T}{he} evolution of multiple chronic conditions (MCC) follows a complex stochastic process. This path of evolution is often influenced by several factors, including inter-relationship of existing conditions, patient-level modifiable and non-modifiable risk factors \cite{rappaport_genetic_2016}.
MCCs are associated with 66\% of the total healthcare costs in the United States, and approximately one in four Americans and 75\% of Americans aged 65 years are burdened with MCC \cite{goodman_defining_2013, campbell_reducing_2017}.
Furthermore, people with MCCs have an increased risk of mortality \cite{prados-torres_multimorbidity_2014}. Thus having MCC is one of the biggest challenges of the 21st century in healthcare \cite{baker_crossing_2001}.
Several aspects of MCC have been studied in literature over the years. Lippa et al. \cite{lippa_deployment-related_2015} conducted a structured clinical interview of a sample of 255 previously deployed Post-9/11 service members and veterans. They found over 90\% of them suffer from psychiatric conditions. Approximately half of them had three or more conditions, and 76.9\% of them suffer from four clinically relevant psychiatric and behavioral factors, including deployment trauma, somatic, anxiety, and substance abuse. Alaeddini et al. \cite{alaeddini_mining_2017} identified major transitions of four MCC that include hypertension (HTN), depression, PTSD, and back pain in a cohort of 601,805 Iraq and Afghanistan war Veterans (IAVs). They also developed a Latent Regression Markov Mixture Clustering (LRMCL) algorithm that can predict the exact status of comorbidities about 48\% of the time. In a separate study, Cai et al. \cite{cai_analysis_2015} developed algorithms to identify the relationships between factors influencing hepatocellular carcinoma after hepatectomy. Lappenschaar et al.\cite{lappenschaar_multilevel_2013} and Faruqui et al.\cite{faruqui_mining_2018} separately used a large dataset to develop a multilevel temporal Bayesian network (MTBN) to model the progression of MCCs. Several studies have also covered the prevalence of MCC and their rate of increase \cite{wolff_prevalence_2002, vogeli_multiple_2007, anderson_growing_2004, schneider_prevalence_2009, freid_multiple_2012, lehnert_review_2011, ward_prevalence_2013, lochner_county-level_2015, ward_multiple_2014, cabassa_race_2013}; health consequences of MCC and their complications \cite{bayliss_predicting_2004, tinetti_contribution_2011, grembowski_conceptual_2014, hempstead_fragmentation_2014, gijsen_causes_2001}; cost and quality of life \cite{friedman_hospital_2006, chen_health-related_2010, boyd_clinical_2005, fried_health_2011, zhong_effect_2015, min_multimorbidity_2007, domino_heterogeneity_2014}; patient support, intervention and complications \cite{wyatt_out_2014, beadles_medical_2015}; and assessment, prediction, and decision making \cite{pugh_complex_2014, miotto_deep_2016, alaeddini_mining_2017, faruqui_mining_2018}. However, most of the existing literature is cross-sectional, considers single chronic conditions, or studies a short period of time. Moreover, while these methods describe general comorbidity phenotypes, they do not provide insight into
the impact of existing comorbid conditions, and lifestyle behaviors of individual patients on dynamics of MCC emergence and progression \cite{faruqui_mining_2018}. Therefore, the dynamic interactions between MCCs and lifestyle behaviors of an individual patient on the complex evolution pathway of
MCC are not precisely known and need further investigation.
In this study, we first represent the complex stochastic relationship between MCC as a functional continuous time Bayesian network (FCTBN) \cite{faruqui_functional_2020} to take into account the impact of the patients' risk factors on the MCC emergence and progression. We then develop a dynamic FCTBN (D-FCTBN) to capture the dynamic impact of modifiable risk factors and their interaction with existing conditions on the emergence of new MCC. This is done by formulating the conditional dependencies of FCTBN using a non-linear state-space model based on Extended Kalman Filter (EKF). Next, We develop a tensor control chart to monitor the changes in the estimated parameters of the proposed D-FCTBN model, which may have a potentially significant impact on the risk of developing a new MCC. Finally, we validate the proposed approach using a combination of simulation and real data. The overall schema for the proposed method is shown in Figure \ref{Figure:Overall_Planning}.
\Figure[t!](topskip=0pt, botskip=0pt, midskip=0pt)[width=.90\textwidth]{Figure_1_.png}
{Overall schema of the proposed approach for dynamic prediction and monitoring of the emergence and progression of MCC.}\label{Figure:Overall_Planning}
The proposed methodology has the following contributions:
\begin{enumerate}
\item We propose to formulate the conditional dependencies of FCTBN as a non-linear state-space model based on EKF to create a dynamic network (D-FCTBN) that captures the dynamics of modifiable risk factors on the structure and parameters of the network.
\item We propose a tensor control chart to monitor the evolution of the D-FCTBN network parameters over time proactively and signal when there is a significant change in the estimated parameters, which can result in an increased risk of developing new conditions.
\item We validate the proposed methodology for dynamic prediction and monitoring of the emergence of multiple chronic conditions based on a combination of a simulated and real dataset from the Cameron County Hispanic Cohort Cohort (CCHC).
\end{enumerate}
The remainder of the paper is structured as follows.
Section \ref{Section: Relevant_Background} presents the preliminaries and background for the CTBN and FCTBN. Specifically, Section \ref{Subsection:Continuous_Time_Bayesian_Network} describes the details of the CTBN, and Section \ref{Subsubsection:Functional_CTBN} explains the functional CTBN and the regularized regression model for learning its structure and parameters.
Section \ref{Section:Proposed_Approach} details the proposed approach for developing the Dynamic FCTBN (D-FCTBN) and the tensor control chart for monitoring the evolution of D-FCTBN. In particular, Section \ref{Subsection:Extended_Kalman_filter_Main} describes the details of the proposed EKF model for modeling the dynamics of edges of the D-FCTBN based on the changes in the modifiable risk factors and their interaction with existing conditions.
Also, Section \ref{Subsection:Monitoring_control_events} describes the building blocks of the proposed tensor control chart for monitoring the estimated parameters of the proposed D-FCTBN.
Section \ref{Section:Results_and_Discussion} presents the study population, the resulting model structure and parameters, and the tensor control chart to detect network changes. Finally, Section \ref{Section:Conclusion} provides the concluding remarks.
\section{Relevant Background}
\label{Section: Relevant_Background}
In this section, we review some of the major components of the proposed approach, including the CTBN for modeling MCC evolution as a finite-state continuous-time conditional Markov process over a factored state
\cite{nodelman_continuous_2012, nodelman_expectation_2012, nodelman_learning_2012}, and functional CTBN (FCTBN) \cite{faruqui_functional_2020} for extending CTBN edges based on Poisson regression of some exogenous risk factors.
\subsection{Continuous Time Bayesian Network (CTBN)}
\label{Subsection:Continuous_Time_Bayesian_Network}
\subsubsection{CTBN Components}
\label{Subsubection:CTBN1}
Continuous time Bayesian networks (CTBNs) are Bayesian networks that models time explicitly by defining a graphical structure over continuous time Markov processes \cite{nodelman_continuous_2012}.
Let $X = \{x_1,x_2,....,x_n\}$ denotes the state space of a set of random variables with
discrete states $x_i=\{1,...,l\}$, such as MCC like Diabetes, Obesity, Hypertension, Heyperlipidemia, and Cognitive Impairment.
A CTBN consists of a set of conditional intensity matrices (CIM) under a given graph structure \cite{nodelman_continuous_2012, norris_markov_1998}. The components of a CTBN are -
\begin{enumerate}
\item An initial distribution $(P_x^0)$, which formulates the structure of the (conditional) relationship among the random variables and is specified as a Bayesian network, where each edge $x_i \to x_j$ on the network implies the impact of the parent condition $x_i$ on the child condition $x_j$.
\item A state transition model $(Q_{X_i|\textbf{u}})$, which describes the transient behavior of each variable $x_i\in X$ given the state of parent variables $\textbf{u}$, and is specified based on CIMs -
\end{enumerate}
\[
\textbf{Q}_{X|\textbf{u}} = \begin{bmatrix}
-q_{x_1|\textbf{u}} & q_{x_1x_2|\textbf{u}} & \dots & q_{x_1x_n|\textbf{u}} \\
q_{x_2x_1|\textbf{u}} & -q_{x_2|\textbf{u}} & \dots & q_{x_2x_n|\textbf{u}} \\
\vdots & \vdots & \ddots & \vdots\\
q_{x_nx_1|\textbf{u}} & q_{x_nx_2|\textbf{u}} & \dots & -q_{x_n|\textbf{u}}
\end{bmatrix}
\]
\noindent where $q_{x_ix_j|\textbf{u}}$ represents the intensity of the transition from state $x_i$ to state $x_j$ given a parent set of node $\textbf{u}$, and $q_{x_i} = \sum_{j \neq i} q_{x_ix_j}$.
Conditioning the transitions on parent conditions sparsifies the intensity matrix considerably, which is especially helpful for modeling large state spaces. When no parent variable is present, the CIM will be the same as the classic intensity matrix.
The probability density function ($f$) and the probability distribution function ($F$) for staying at the same state (say, $x_i$), which is exponentially distributed with parameter $q_{x_i}$, are calculated as-
\begin{align}
\label{Equation:Probability density}
f(q_x,t) &= q_{x_i} exp(-q_{x_i}t),&{t \geq 0}\\
F(q_x,t) &= 1 - exp(-q_{x_i}t), &{t \geq 0}
\end{align}
\subsubsection{CTBN Parameter Estimation}
\label{Subsubsection:Parameter_Estimation1}
Given a dataset $\mathcal{D} = \{\tau_{h=1}, \tau_{h=2},....,\tau_{h=H}\}$ of $H$ observed transitions, where $\tau_h$ represents the time at which the $h^{th}$ transition has occurred, and $\mathcal{G}$ is a Bayesian network defining the structure of the (conditional) relationship among variables, we can use maximum likelihood estimation (MLE) (equation \eqref{Equation:Likelihood_Function}) to estimate parameters of the as defined in Nodelman et al \cite{nodelman_continuous_2012, nodelman_learning_2012}-
\begin{align}
\label{Equation:Likelihood_Function}
\begin{split}
L_x(q_{x|\textbf{u}}:\mathcal{D}) &= \prod_{\textbf{u}} \prod_x q_{x|\textbf{u}}^{M{[x|\textbf{u}]}} exp(-q_{x|\textbf{u}}T{[x|\textbf{u}]})
\end{split}
\end{align}
\noindent where, $T[x|\textbf{u}]$ is the total time $X$ spends in the same state $x$, and $M{[x|\textbf{u}]}$ the total number of time $X$ transits out of state $x$ given, $x = x'$ .The log-likelihood function can be then written as-
\begin{align}
\label{Equation:Log-Likelihood_Function}
\begin{split}
l_x(q_{x|\textbf{u}}:\mathcal{D}) &= \sum_{\textbf{u}} \sum_x M[x|\textbf{u}]\hspace{1pt} ln(q_{x|\textbf{u}}) - q_{x|\textbf{u}}\hspace{1pt} T[x|\textbf{u}]
\end{split}
\end{align}
Maximizing equation \eqref{Equation:Log-Likelihood_Function}, provides the maximum likelihood estimate of the paramters of the FCTBN.
\subsection{Functional CTBN (FCTBN)}
\label{Subsubsection:Functional_CTBN}
\subsubsection{FCTBN with Conditional Intensities as Poisson Regression}
\label{Subsubection:FCTN_PoissonRegression}
In reality, the progression of state variables, such as chronic conditions, not only depends on the state of their parents, such as preexisting chronic conditions but some exogenous variables, such as patient level risk factors like age, gender, etc.
Using Poisson regression to represent the impact of exogenous variables on the conditional dependencies, the rate of transition between any pair of MCC states can be derived as \cite{faruqui_functional_2020}-
\begin{subequations}
\begin{align}
\label{Equation:CTBN_Poisson_1}
\log {q}_{x_i,x_j|\textbf{u}} &= \beta_{0_{x_i,x_j|\textbf{u}}} + \beta_{1_{x_i,x_j|\textbf{u}}} + ... ... + z_m \beta_{m_{x_i,x_j|\textbf{u}}} \\
&= \boldsymbol{z} \boldsymbol{\beta}_{\textbf{x}_i,\textbf{x}_j|\textbf{u}}
\end{align}
\end{subequations}
\noindent where, $\textbf{z}=\{z_1,z_2,...,z_m\}$ is the set of exogenous variables (e.g. patient-level risk factors such as age, gender, race, education, marital status, etc.), and $\beta_{k_{x_i|\textbf{u}}}=\sum_{j\neq i}\beta_{k_{x_ix_j|\textbf{u}}}, k=0,\ldots,m$ is the set of coefficients (parameters) associated with the exogenous variables.
Also, the rate of staying in the same state is modeled as-
\begin{subequations}
\begin{align}
\label{Equation:CTBN_Poisson_2}
\log q_{x_i|\textbf{u}} &= \beta_{0_{x_i|\textbf{u}}} + z_1 \beta_{1_{x_i|\textbf{u}}} + ... ... + z_m \beta_{m,{x_i|\textbf{u}}} \\
&= \boldsymbol{z} \boldsymbol{\beta}_{\textbf{x}_i|\textbf{u}}
\end{align}
\end{subequations}
When the state space of the random variables is binary, as in our case study on MCC transitions, where MCC states include having/not having each of the conditions, the conditional intensities in $\textbf{Q}_{x_i |\textbf{u}_i}$, can be estimated just using Equation \ref{Equation:CTBN_Poisson_2} because for Markov processes with binary states $ q_{x_i |\textbf{u}} = - \sum_{j\neq i} q_{(x_i x_j |\textbf{u})}$.
This feature considerably simplifies the estimation of the functional CTBN conditional intensity matrix based on Poisson regression.
\subsubsection{Parameter Estimation}
\label{Subsubsection:Parameter_Estimation}
Having the dataset $D=\left\{\tau_{\left(p=1,h=1\right)},\ldots,\tau_{\left(P,H\right)}\right\}$ of MCC trajectories, where $\tau_{\left(p,h\right)}$ represents the time at which the $h^{th}$ (MCC) transition of the $p^{th}$ patient has occurred, we use maximum likelihood estimation to estimate parameters of the proposed FCTBN.
The likelihood of $D$ can be decomposed as the product of the likelihood for individual transitions. Let $d=\langle \textbf{z},\textbf{u},x_i|\textbf{u},t_d,x_j|\textbf{u} \rangle$ be the transition of patient $p$ with risk factors $\textit{z}$ and existing conditions $\textbf{u}$, who made the transition to state $x_{j|\textbf{u}}$ after spending the amount of time $t_d=\tau_{\left(p,h\right)}-\tau_{\left(p,h-1\right)}$ in state $x_{i|\textbf{u}}$.
By multiplying the likelihoods of all conditional transitions during the entire trajectory for all patients $p=1,\ldots,P$, and taking the log, we obtain the overall log-likelihood function as-
\begin{align}
\label{Equation:FCTBNLikelihood_Function}
\begin{split}
l_N\left(\textbf{q}:\mathcal{D}\right)\
&=\sum_{p}\sum_{h}\sum_{\textbf{u}}\sum_{x_j}\sum_{x_i} \\
& \left\{{t_d}_p\left[x_ix_j|\textbf{u}\right]\exp{\left(\boldsymbol{z}_{p,h}\boldsymbol{\beta}_{x_ix_j|\textbf{u}}\right)}\right\}
\end{split}
\end{align}
which is a convex function and can be maximized efficiently using a convex optimization algorithm such as Newton Raphson to estimate parameters $\textit{\beta}_{x_i|\textbf{u}}$.
\subsubsection{Group regularization for structure learning of the FCTBN}
\label{Subsubsection:Group regularization}
The parameter estimation approach presented above requires the parent set of each condition to be known, which is equivalent to knowing the Bayesian network structure. Given that FCTBN has a special structure based on a conditional intensity matrix that allows for cycles, group regularization can be used to penalize groups of parameters pertaining to each specific conditional transition (each edge) \cite{faruqui_functional_2020} as-
\begin{equation}
\label{Equation:CTBN_Minimization_1}
\centering
\min -l_N(\textbf{q}:\mathcal{D}) + k \sum_{x_{i}|\textbf{u}}\lambda_j\|\boldsymbol{\beta}_{x_{i}|\textbf{u}}\|
\end{equation}
\noindent where, $\|\boldsymbol{\beta}_{x_{i}|\textbf{u}}\| = \sqrt{\sum_{\textbf{u}} \sum_{x_{i}} (\boldsymbol{\beta}_{x_{i|\textbf{u}}}.\boldsymbol{\beta}^T_{x_{i|\textbf{u}} )}}$ is the $L_1$-norm of the group of parameters associated with each conditional transition. $k$ is the groups size which is based on the number of coefficients in the Poisson regression for each conditional intensity. $\lambda_j = \lambda \|\Tilde{\boldsymbol{\beta}_j}\|^{-1}$ is the tuning parameters (of the adaptive group regularization) that control the amount of shrinkage, where $\lambda$ is inversely weighted based on the unpenalized estimated value of the regression coefficients $\Tilde{\boldsymbol{\beta}_j}$ \cite{wang_note_2008}.\\
\section{Proposed Approach}
\label{Section:Proposed_Approach}
In this section, we first propose an extended Kalman filter to capture the effects of the dynamics of modifiable risk factors on the parameters, edges, and structure of the FCTBN (D-FCTBN). Next, we develop a tensor control chart to monitor the evolution of the dynamic FCTBN (D-FCTBN).
\subsection{An extended Kalman filter for dynamic prediction of FCTBN parameters}
\label{Subsection:Extended_Kalman_filter_Main}
The conditional dependencies (edges) of FCTBN provide the rate of transitioning from one state to another given the parents' state and exogenous variables, i.e., the rate of a new chronic condition such as obesity emergence during the next $t$ years given the preexisting conditions such as diabetes and patient's level risk factors such as gender, age, etc.
However, in reality, conditional dependencies dynamically change based on a person's modifiable behavioral factors, i.e., diet, exercise, and interaction with non-modifiable risk factors and existing conditions. To capture the dynamics of the changes in the conditional intensities (risk) of MCC, we propose to transform the parameters (coefficients) of the regression functions, which represent the edges of the MCC (FCTBN) network, into an extended Kalman filter (EKF) \cite{gahrooei_change_2018}.
EKF consists of an observation equation and a state transition equation. The observation equation describes the most recent observation of state variables using system dynamics, namely Poisson regression coefficients associated with the emergence and progression of MCC. The transition equation predicts how the state variables evolve to the next period, namely how the coefficients associated with MCC will progress/emerge in the next period (See Fig. \ref{Figure:EKF_Illustration}).
\noindent \textit{\textbf{Observation equation:}} Each edge/connecting in the MCC FCTBN network represents, the rate of occurrence of a chronic condition such as diabetes, based on a Poisson regression function with parameters $\left[\textit{\beta}_{x_ix_j|\textit{u}}\right]_t$. We consider $\left[\textit{\beta}_{x_ix_j|\textit{u}}\right]_t$ as the state variables of the dynamical system which describe the (noisy) sequences of MCC observations. This results in the observation equation given by-
\begin{align}
\label{Equation:Observation_0}
\left[q_{x_i|\textbf{u}}\right]_t=exp\left(\boldsymbol{z}_\textit{t}\left[\boldsymbol{\beta}_{x_i|\textbf{u}}\right]_t\right)\
\end{align}
The observation equation \eqref{Equation:Observation_0} is non-linear and thus we will employ the extended Kalman filter (EKF) \cite{fahrmeir_kalman_1991} instead of general Kalman filters (KF) \cite{kalman_new_1960}. EKFs similar to KF follows a recursive procedure where it performs predictions based on a given observation and updates the estimates iteratively \cite{fahrmeir_kalman_1991}.
\noindent \textit{\textbf{State transition equation:}} As a patient changes her lifestyle behaviors, the state variables of the proposed dynamical model evolve in time to best predict the MCC emergence and progression. This results in a state transition equation given by-
\begin{align}
\label{Equation:Statetransition_0}
\left[\boldsymbol{\beta}_{x_i|\textbf{u}}\right]_t=\boldsymbol{F}\left[\boldsymbol{\beta}_{x_i|\textbf{u}}\right]_{t-1}+\textit{\varepsilon}_t
\end{align}
\noindent where $\boldsymbol{F}$ is the state transition matrix, and $\varepsilon_t$ is the white noise assumed to follow a Gaussian distribution with mean zero and covariance $\sigma^2\mathbf{I}$. The transition matrix $\boldsymbol{F}$ can be approximated from stream of data $\boldsymbol{X}_{i|t}$ utilizing the FCTBN model evaluated at different point in time or by utilizing some system identification techniques \cite{ljung_system_2017}.
\Figure[t!](topskip=0pt, botskip=0pt, midskip=0pt)[scale = 0.2]{Figure_2_.png}
{Illustration of the impact of behavioral risk factors dynamics on the conditional intensities/dependencies and risk trajectory of developing new MCC conditions, i.e. Diabetes, at three time points, including baseline, 5-year follow up, and 10-year follow up, using extended Kalman filter; The nodes with thick outlines represent the existing or developed conditions over time. (The nodes, OB: Obesity, HP: High Blood Pressure, DI: Diabetes, HL: Hyperlipidemia, and CI: Cognitive Impairment).\label{Figure:EKF_Illustration}}
\noindent\textit{\textbf{Dynamic Prediction via EKF:}} EKF takes the most recent estimate of the state variables with information of changes in the (modifiable) behavioral risk factors up to time $t$ and uses the system dynamics to predict the future state of the variables and prediction of the MCC as \cite{fahrmeir_kalman_1991}-
\begin{align}
\label{Equation:Prediction_1}
[\boldsymbol{\beta}_{x_i|\textbf{u}}]_{t|t-1} &= \boldsymbol{F}[\boldsymbol{\beta}_{x_i|\textbf{u}}]_{t-1|t-1} + \sigma^2 I\\
\label{Equation:Prediction_2}
P_{t|t-1} &= \boldsymbol{F}[\boldsymbol{\beta}_{x_i|\textbf{u}}]_{t-1|t-1}\boldsymbol{F}^T + Q_t
\end{align}
where $[\boldsymbol{\beta}_{x_i|\textbf{u}}]_{t|t-1}$ and $P_{t|t-1}$ are the extended Kalman prediction of the matrix of estimated coefficients and their covariance
respectively given a set of observations $\boldsymbol{X}_{i,t}$. The observation equation is linearized using the Taylor Expansion to achieve a sub-optimal estimate of the state value.
\\
\\
\noindent\textit{\textbf{Dynamic estimation via EKF:}} When new observations of MCC are obtained, the error between the observation and the EKF predictions is used to update the posterior mean of the state variable as-
\begin{align}
\label{Equation:Update_1}
[\boldsymbol{\beta}_{x_i|\textbf{u}}]_{t|t-1} &= [\boldsymbol{\beta}_{x_i|\textbf{u}}]_{t|t-1} + K_t (\boldsymbol{X_{x_i|\textbf{u},t}} - \boldsymbol{X}_{x_i|\textbf{u},t|t-1})
\end{align}
\noindent where $K_t$ is the Kalman gain and calculated using the following equations-
\begin{align}
\label{Equation:Update_2}
P_{t|t-1} &= (I - K_t G_t) P_{t|t-1}\\
\label{Equation:Update_3}
K_t &= P_{t|t-1} G_t (G_t^T P_{t|t-1} G_t + R_t ) ^{-1}
\end{align}
\noindent where, $G_t$ denotes the Jacobian of $g$ evaluated at $[\boldsymbol{\beta}_{x_i|\textbf{u}}]_{t|t-1}$ i.e. $G_t = \frac{\partial{g}}{\partial{[\boldsymbol{\beta}_{x_i|\textbf{u}}]_t}}|_{[{\boldsymbol{\beta
}_{x_i|\textbf{u}}]}_{t|t-1}} = \boldsymbol{Z}_t^T diag(exp(\boldsymbol{Z}_t[\boldsymbol{\beta}_{x_i|\textbf{u}}]_{t|t-1})) $, $R_t$ represents the variance of observations and is estimated based on the underlying network distribution and the observation prediction. The estimated parameter $[\boldsymbol{\beta}_{x_i|\textbf{u}}]_{t|t}$ provides a sub-optimal estimate of the network parameters at time $t$ \cite{fahrmeir_kalman_1991}.
\subsection{Monitoring of events}
\label{Subsection:Monitoring_control_events}
In this section, we propose a monitoring scheme to determine meaningful changes in the exogenous variables (modifiable risk factors) that can have an impact on the risk of developing new chronic conditions. For this purpose, we propose a statistical control chart that automatically signals when there is a meaningful change in the predicted value of the coefficients associated with the patient level (modifiable) risk factors, namely $[\boldsymbol{\beta}_{x_i|\textbf{u}}]_{t|t-1}$, which is dynamically updated by the D-FCTBN. The idea behind monitoring the $[\boldsymbol{\beta}_{x_i|\textbf{u}}]_{t|t-1}$ is that the predicted value of the risk factors coefficients are directly related to the network edges (conditional intensities) and the risk of developing new MCC conditions.
Given the dynamic prediction of D-FCTBN (MCC network) parameters for a time point ${t|t-1}$, the coefficients form a 3-dimensional tensor including the parents, children, and risk factors dimensions (modes). To effectively monitor the tensor of predicted coefficients for any potential changes, we propose to use multilinear principal component analysis (MPCA) to extract the most salient features of the data for building the control chart. It is worth mentioning that, there are other alternative methods that can also be used for extracting major features of the coefficients tensor, including Unfolded PCA \cite{liu_extraction_2007, bharati_image_2004}, Multilinear Principal Component Analysis (MPCA) \cite{lu_mpca_2008}, Uncorrelated MPCA \cite{lu_uncorrelated_2009}, Robust MPCA \cite{dehaan_adaptive_2007, inoue_robust_2009} , and Non-negative MPCA \cite{zass_nonnegative_2006}. Interested readers may also refer to Kruger et al \cite{kruger2008developments}, Lu et al \cite{lu_mpca_2008} and Paynabar et al \cite{paynabar2013monitoring} for comprehensive reviews of tensor feature extraction methods.
\subsubsection{Multilinear Principal Component Analysis}
\label{subsubsection:Multilinear_Principal_Component_Analysis}
Lu et al. \cite{lu_mpca_2008} introduce the MPCA framework for tensor feature extraction. They decompose the original tensor into a series of multiple projection sub-problems and solves them iteratively.
\Figure[h!](topskip=0pt, botskip=0pt, midskip=0pt)[width=.45\textwidth]{Figure_8_v2.png}
{Visual illustration of multilinear projection; projection in the 1-mode vector space.\label{Figure:MPCA_Illustrated2}}
For an in-control training set of $N$-th order tensor denoted as $\mathcal{A} \in \mathbb{R}^{I_1 \times I_2, \times I_3, ... ..., \times I_N}$, the MPCA projection can be denoted by $y \in \mathbb{R}^{I_1 \times I_2 \times I_3}$, where $n = 1, 2, ... ..., N$, $I_1, I_2$ and $I_3$ are the dimensions of the coefficient tensor and $N$ is the number of updated coefficients attained from EKF.
Lu et al. \cite{lu_mpca_2008} find the set of orthogonal transformation matrices,
$\textbf{U}^n \in \mathbb{R}^{I_n \times P_n}, n = 1, 2, 3, ...., N$,
where the dimensionality $P_n$ ($P_n \leq I_n$) for each mode is predetermined or known for the application of interest.
They have also developed methods for adaptive determination of $P_n$ in case it's not pre-determined. The transformation is performed such that it captures the most variations of the original tensor. To keep the original estimated values of the coefficients, in this work we utilized the non-centered version of the MPCA. Therefore, the low dimensional features after applying MPCA will be,
\begin{equation}
\textbf{U}^n = \arg\max_{{{\bf U}}^{(1)},{{\bf U}}^{(2)},\ldots,{{\bf U}}^{(N)}} \sum_{i=1}^N || \mathcal{Z}_i ||^2_F
\end{equation}
\noindent where, $n = 1, 2, 3$, $\mathcal{Z}_i = \mathcal{A} \times_1\textbf{U}^{(1)} \times_2\textbf{U}^{(2)} \times_3\textbf{U}^{(3)}$ and $\mathcal{A}$ is non-centered tensor data. In case of centered data, $\mathcal{A}$ can be replaced with $\widetilde{\mathcal{A}}$, where $\widetilde{\mathcal{A}} = {\mathcal{A}} - \overline{\mathcal{A}}$. For a new feature tensor, $\mathcal{A}_{new} \in \mathbb{R}^{I_1 \times I_2, \times I_3}$, the features are calculated as,
\begin{equation}
\centering
\label{Equation:MPCA_New_Feature}
\mathcal{Z}_{new} = \mathcal{A}_{new} \times_1 \textbf{U}^{(1)} \times_2 \textbf{U}^{(2)} \times_3 \textbf{U}^{(3)}
\end{equation}
and residuals of reconstruction can be calculated as
\begin{equation}
\centering
\label{Equation:MPCA_New_Error_Feature}
\mathcal{E}_{new} = \mathcal{A}_{new} - \mathcal{Z}_{new} \times_1 \textbf{U}^{(1)T} \times_2 \textbf{U}^{(2)T} \times_3 \textbf{U}^{(3)T}
\end{equation}
The errors at every time step can also be vectorized by calculating the \textit{norm} of all the data, i.e. $\mathcal{E}_{new\_vector} = ||\mathcal{E}_{new}||_2$.
\subsubsection{Monitoring Scheme}
\label{subsubsection: Monitoring_Scheme}
Here, we propose a tensor control chart to monitor the changes in the D-FCTBN network edges caused by changes in the patient modifiable risk factors, namely lifestyle behavioral changes. Given the estimate of FCTBN parameters $\sigma^2 I$ and $[\boldsymbol{\beta}_{x_i|\textbf{u}}]_{t|t-1}$ based on Section \ref{Subsubsection:Functional_CTBN}, for any new observation of patients (modifiable and non modifiable) risk factors and MCC conditions, the tensor of new network parameters (risk factors' coefficients) are predicted using the EKF detailed in Section \ref{Subsection:Extended_Kalman_filter_Main}, and the relevant features are extracted using MPCA discussed in Section \ref{subsubsection:Multilinear_Principal_Component_Analysis}.
When there is no significant change in the patients' lifestyle behaviors, the reconstruction error in Equation \ref{Equation:MPCA_New_Error_Feature} will be small, as patients' historical/past behavior can accurately estimate the F-FCTBN parameters. However, when there is a significant change in the patients' lifestyle behaviors, the distribution of reconstruction error will change, and the observed value will supposedly increase. Therefore, for new predictions of the D-FCTBN parameters,
$[\boldsymbol{\beta}_{x_i|\textbf{u}}]_{t+1|t} \approx \mathcal{A}_{new} \in \mathbb{R}^{I_1 \times I_2, \times I_3}$, the reconstruction error can be used to identify potential high-impact changed in patients modifiable risk factors. The proposed monitoring scheme is based on a Multivariate Exponential Weighted Moving Average (MEWMA) \cite{lowry_multivariate_1992} control chart of the vectorized reconstruction error:
\begin{equation}
\centering
\label{Equattion: MEWMA_Smoothing}
\boldsymbol{Z}_i = \lambda \textbf{x}_i + (1 - \lambda) \boldsymbol{Z}_{i-1}
\end{equation}
\noindent where $0 \leq \lambda \leq 1$ and $\boldsymbol{Z}_0 = 0$. In case of MEWMA the quantity plotted on the control chart is -
\begin{equation}
\centering
\label{Equation:T2}
T_i^2 = \boldsymbol{Z}_i^T S_I^{-1} \boldsymbol{Z}_i
\end{equation}
\noindent where the covariance matrix is,
\begin{equation}
\centering
\label{Equation:Covariance}
S_I^{-1} = \frac{\lambda}{2 - \lambda} [1 - (1 - \lambda)^{2i}] S
\end{equation}
\noindent which is equivalent to the variance of the univariate EWMA, and $S$ is the sample covariance matrix calculated of the features estimated by $N$ in-control samples \cite{montgomery_introduction_2020}. The control limits of control chart can be calculated as follows (for the univariate case)-
\begin{align}
\begin{split}
\label{Equation: Limits_of_EWMA}
UCL &= \mu_0 + L \sigma \sqrt{\frac{\lambda}{(2 - \lambda)} [1 - (1 - \lambda)^{2i}]} \\
CL &= \mu_0 \\
LCL &= \mu_0 - L \sigma \sqrt{\frac{\lambda}{(2 - \lambda)} [1 - (1 - \lambda)^{2i}]}
\end{split}
\end{align}
\noindent Where, $L$ is the width of the control limits, $\sigma^2$ is the variance of the data, and $i$ represents the observation number of the EWMA statistics.
The selection of the features to be used is determined based on the percentage of the total variance explained by the extracted features using MPCA.
\section{Results and Discussion}
\label{Section:Results_and_Discussion}
Long-lasting diseases, otherwise known as chronic conditions, can be considered a degradation process that progresses over time and contributes to the development of other new chronic conditions. The presence of two or more chronic medical conditions in an individual is commonly defined as multimorbidity, or multiple chronic conditions (MCC) \cite{faruqui_mining_2018, pugh_complex_2014}. Here, we use the proposed dynamic FCTBN (D-FCTBN) to find the impact of patient level risk factors, specifically lifestyle behaviors, on the conditional dependencies of MCC over time. In addition, we use the proposed tensor control chart to monitor the risk of new MCC emergence based on the dynamics of patients' lifestyle behaviors.
\subsection{Study Population and Demographics}
\label{Subsection:Study_Population}
\Figure[h!](topskip=0pt, botskip=0pt, midskip=0pt)[width=.45\textwidth]{Figure_9_DatasetSelection.png}
{Flow diagram of sample selection and the final number of patients included in the analysis.\label{Figure:Study_Population_Selection}}
Our case study is based on the Cameron County Hispanic Cohort (CCHC) dataset for comorbidity analyses. The CCHC is a cohort study comprised of mainly Mexican Americans (98\% of cohort) randomly recruited from a population with severe health disparities on the Texas-Mexico border and started in 2004.
The CCHC is employing a rolling recruitment strategy and currently numbers 4,546 adults. Inclusion criteria: (1) participating in the study between 2004 and 2020, (2) having at least three 5-year follow-up visits during that period. 385 patients met these criteria, which include the dataset of our study (see Figure \ref{Figure:Study_Population_Selection}). The survey includes participants’ socio-demographic factors (age, gender, marital status, education, etc.) and lifestyle behavioral factors (diet, exercise, tobacco use, alcohol use, etc.).
\subsection{Diagnosed Health Condition and Patient Associated Risk Factors}
\label{Subsection:Study_Population_2}
For this study, we considered some of the most common MCCs present in the Hispanic community, including diabetes, obesity, hypertension, hyperlipidemia, and mild cognitive impairment. The positive criteria (considering the condition to be active) for the conditions selected as below-
\begin{itemize}
\item \textbf{Diabetes}: Fasting Glucose >= 126 mg/dL, HbA1c >= 6.5\%, or take diabetes medication \cite{association_diagnosis_2010}.
\item \textbf{Obesity}: Body mass index (BMI, kg/m2)>= 30 \cite{fruhbeck_abcd_2019}.
\item \textbf{Hypertension}: Systolic blood pressure (BP) >= 130 mmHg, Diastolic BP >= 80 mmHg, or take anti-hypertensive medication \cite{pk_2018}.
\item \textbf{Hyperlipidemia}: Total cholesterol > 200 mg/dL, triglycerides >= 150 mg/dLl, HDLC < 40 mg/dL (for male)/ HDLC < 50 mg/dL (for female), LDLC >= 130 mg/dL, or take medication for hyperlipidemia \cite{grundy_scott_m_2018_2019}.
\item \textbf{Mild Cognitive Impairment}: Mini-Mental State Score < 23 (out of 30) \cite{wu_association_2018}.
\end{itemize}
\Figure[h!](topskip=0pt, botskip=0pt, midskip=0pt)[width=1\textwidth]{Figure_5.png}
{The estimated parameters of FCTBN based on the optimal value of tuning parameters. The matrix contains all the possible combinations of parent and child interaction. For example, the first row set (first 32 rows) of the matrix represents the parameters learned child node 1 while considering the parents' node are 2, 3, 4, and 5. The right side of the Figure shows all the possible condition possible (1 for the presence of a condition and 0 for no presence of no condition).\label{Figure:Transition_Explain}}
For the risk factors, the dataset includes the participant's non-modifiable risk factors based on socio-demographic information (age, gender, and education history) and modifiable risk factors based on lifestyle behavioral factors (diet, exercise, tobacco use, alcohol use). Diet and exercise are categorized according to the U.S. Healthy Eating Guideline, and U.S Physical Activity Guideline \cite{wu_fruit_2019}.
\subsection{FCTBN Structure and Parameter Learning}
\label{Subsection:Learned_Bayesian_Structure}
To identify the optimal value of the tuning parameter ($\lambda$) of the adaptive group regularization method for FCTBN structure and parameter learning, we use cross-validation error based on several $\lambda$ values. We attain the structure of FCTBN and the parameters using the optimal value of $\lambda = 10^2$. Figure \ref{Figure:Transition_Explain} provides the heatmap of the estimated parameters for the based FCTBN model. These learned parameters will be used as the initial parameters of EKF for estimating the dynamic FTCBN and monitoring possible changes in the risk of acquiring a new MCC condition.
\subsection{D-FCTBN Dynamic Estimation and Prediction Using EKF}
\label{Subsection:EKF_Update}
The estimated parameters of FCTBN provide the baseline/initial values of the D-FCTBN. As new (dynamic) observations of patient's (modifiable) risk factors and MCC status are made available, we use EKF to capture the dynamics of a patient's modifiable risk factors and MCC update, as detailed in Section \ref{Subsection:Extended_Kalman_filter_Main}. Figure \ref{Figure:Change_Coefficients}, visualizes the estimated parameter of the D-FCTNB for time, $t+1$ given the parameter information at time, $t$ and base parameter, $\beta_{t+1|t}$ using the proposed EKF module for 5 patients over 11 consecutive year. The proposed model provides a near-optimal approach to estimate and update the D-FCTBN parameters. The illustrated changes in D-FCTBN network parameters demonstrate the dynamics of patient's lifestyle behaviors and their impact on the MCC.
\Figure[t!](topskip=0pt, botskip=0pt, midskip=0pt)[width=.9\textwidth]{Figure_6_.png}
{The EKF estimation of the parameters of the propsoed D-FCTBN for 5 patients patients over 11 consecutive periods. The illustration shows the change in parameters with respect to base year ($t=0$).\label{Figure:Change_Coefficients}}
\subsubsection{Stability Analysis of EKF for Estimating Parameters of FCTBN}
\label{Subsubsection:Stability_EKF}
In this section, we will discuss the stability of the EKF model derived in Section \ref{Subsection:Extended_Kalman_filter_Main}. For the measurement error, $\textbf{e} = \textbf{X}_t - \textbf{X}_{t|t-1}$ Konrad et al. \cite{reif1999stochastic} showed that the estimation error remains bounded if the following conditions hold-
\begin{enumerate}
\item $||\textbf{F}(\textbf{X}_t) || \leq \alpha $, \\
$||G_t(\textbf{X}_t) || \leq \beta $, \\
where $\alpha, \beta \geq 0$ positive real number for each, $t$.
\item \textbf{F} is non-singular for every $t$.
\item The estimation error, $\textbf{e}$ is exponentially bounded in mean square error. This also bounds the probability to one. This is only true when the estimates satisfy the condition $||\textbf{e}|| \leq \epsilon$ and the covariance matrices of the noise terms are bounded via, $\sigma^2 I \leq \delta I$ and $QI \leq \delta I$, where $\delta , \epsilon > 0$.
\end{enumerate}
\noindent EKF model generally needs additional steps to correct the estimation of the future state. These additional steps are necessary to make sure estimated parameters do not diverge over time.
We consider mean squared error (MSE) for stability analysis of EKF.
Figure \ref{Figure:MSE_Error_Stability} shows the stability check for the proposed EKF model of D-FCTBN parameters. As shown in the Figure, the mean square error rapidly decreases with respect to the time and iterations, showing an acceptable level of stability for the analysis.
\Figure[ht!](topskip=0pt, botskip=0pt, midskip=0pt)[width=.9\textwidth]{Figure_MSE_Stability_.png}
{
The stability check of the the D-FCTBN model for estimating the model parameters in the presence of new data.
The Figure shows the MSE of predictions for two patients. The time axis shows the parameters estimated at each time step, and the iteration axis shows the steps to minimize the error at each time step.).\label{Figure:MSE_Error_Stability}
}
\subsection{Monitoring Events in a D-FCTBN}
\label{Subsection:Tensor_MPCA}
To demonstrate the effectiveness of the proposed tensor control chart for event detection for a change in D-FCTBN model parameter discussed in Section \ref{subsubsection: Monitoring_Scheme}, we setup two experiments, (1) a simulated experiment, where the patient's data and the behavioral data are synthetically generated to represent similar characteristics of the actual data, and (2) a real experiment, where we use the previously introduced data.
In the model setup, we have three non-modifiable demographic conditions (age, gender, and years of education) and four modifiable behavioral factors (healthy diet, exercise, smoking habits, and drinking habits). We conducted the experiments in two stages. We utilize the data generated (the estimated $\beta_t$ coefficients) to build phase I of the control chart. During this period, the behavioral factors are controlled (for the simulated data). In phase II, out-of-control samples are randomly generated for the simulated data (by altering the modifiable factors). For each sample, the monitoring features and the residuals are calculated. The error features are then plotted on the corresponding control charts.
To demonstrate the generative capability of the proposed approach, for simulated experiments, we consider/generate monthly data/observations. However, for the real experiments, we consider yearly time intervals, given the availability of data.
For the simulated experiments, two scenarios are considered. In the first scenario, we change only one of the behavioral factors, and in the second scenario, we change two behavioral factors simultaneously. Meanwhile, for real experiments, we consider the case where two behavioral factors change.
\Figure[t!](topskip=0pt, botskip=0pt, midskip=0pt)[width=0.9\textwidth]{Figure_ControlChart_.png}
{EWMA Control chart of the reconstruction error obtained from the proposed model. (a) In-control control chart (simulation case) (b-c) shows case 1 where only one of the behavioral factors are modified (simulation case), (d) shows case 2 where we randomly change more than one behavioral factor (simulation case), and (e) shows (uncontrolled) changes in more than one behavioral factor (real case)).\label{Figure:Control_Chart_Example}}
\subsubsection{Simulated Case Study: Changing One of the Behavioral Factors}
\label{Subsubsection:Case_1}
We consider the following setup for estimation of the parameters of the control chart (Phase I). The patient is considered to have diabetes as a prior chronic condition. The patient's fall is in the age range of 31-35 and male. The lifestyle behaviors during the in-control phase are [Healthy Diet, Exercise, Smoke, Drink] = [Yes, Yes, No, No]. We assume no extreme behavioral change during the phase I period (12 months). For evaluating the performance of the control chart (phase II) (after the first 12 months), we modify one or more of the behavioral factor/s of the patients at some (random) points in time. The control chart parameters are set to $\lambda$ = .15 and $L$ = 1.5 based on simulation analysis.
\noindent \textbf{Case (a): In-control behavior:} Figure \ref{Figure:Control_Chart_Example}\textcolor{blue}{(a)} shows the control chart for an in-control case where there is no behavioral change in either phase I or phase II (in 48 months). As a result, the control chart doesn't produce any out-of-control signal, which verifies its low type I error.
\noindent \textbf{Case (b): Change in eating habits:} Figure \ref{Figure:Control_Chart_Example}\textcolor{blue}{(b)} represents the case, where the patient changes his diet from healthy eating to unhealthy eating, i.e. [Healthy Diet, Exercise, Smoke, Drink] = [No, Yes, No, No]. We introduce this change in the eating habit in the 17th month. The out-of-control events can be noticed in the control chart after three observations (in month 20). The quick diagnosis of the change in the behavioral change by the control chart can be attributed to the significant effect of eating habits on the parameters of the D-FCTBN.
\noindent \textbf{Case (c): Change in drinking habits:} For the next out-of-control scenario, we assume the patient picks up drinking alcoholic beverages i.e. [Healthy Diet, Exercise, Smoke, Drink] = [Yes, Yes, No, Yes]. This change was made on 17th month.
The control chart picks up this change after four observations (around the 21st month).
Similar to eating habits, the quick detection of the change in the drinking habits by the control chart can be related to the significant effect of eating habits on the parameters of the D-FCTBN.
\subsubsection{Simulated Case Study: Changing More than One Behavioral Factors}
\label{Subsubsection:Case_2}
\noindent \textbf{Case (d): Change in drinking and exercise habits:}
Here, we consider the same behavioral factors and set up as the phase I analysis.
Meanwhile, for phase II analysis, we modify two behavioral factors (instead of one) simultaneously at on 19th month.
The factors considered for change are [Healthy Diet, Exercise, Smoke, Drink] = [No, Yes, No, Yes].
As shown in Figure \ref{Figure:Control_Chart_Example}\textcolor{blue}{(c)}, the first out-of-control signal is produced by the control chart in month 39 (after 20 months). This prolonged time for diagnosis can be because of the the complex interaction between the modifiable risk factors, which have changed in the opposite directions (stop exercise and stop drinking simultaneously).
\subsubsection{Real Case Study: Change in Two Behavioral Factors}
\label{Subsubsection:Case_3}
For the real case study, we consider the patients' data presented in Section \ref{Subsection:EKF_Update}. Due to limited number of consecutive visits data available, we conducted this experiment in a hybrid setting.
The patient considered has Hyperlipidemia and Obesity as a prior chronic condition/s. The lifestyle behaviors during the in-control phase are [Healthy Diet, Exercise, Smoke, Drink] = [No, Yes, No, Not Provided]. We estimate the data for 12 years using the initial learned model and build phase I control chart with this prior setup. Next, we utilize the patient actual behavioral factors in their follow-up meeting for phase II analysis.
\noindent \textbf{Case (e): Real behavioral change:} Figure \ref{Figure:Control_Chart_Example}\textcolor{blue}{(e)} shows the control chart with the real patient data in phase II. The figure shows the patient's behavior change in year 13 to [Healthy Diet, Exercise, Smoke, Drink] = [No, No, Yes, Not Provided]. Consequently, the chart produces an out-of-control signal after one observation (at year 14), which shows the sensitivity of the proposed control scheme when a significant behavioral change or multiple changes in the same (negative/positive) direction occurs.
\section{Conclusion}
\label{Section:Conclusion}
In this paper, we proposed extending functional continuous time Bayesian network based on an extended Kalman filter to build a dynamic functional continuous time Bayesian network to capture the dynamic (vs static) effect of changes of modifiable variables on the structure and parameters of the FCTBNs. We also utilized a low-rank tensor decomposition method based on multilinear principal component analysis to extract meaningful features of the tensor of D-FCTBN parameters. We developed a monitoring scheme based on the concept of a multivariate exponentially weighted moving average (MEWMA) control chart to identify changes in the modifiable variables that can significantly affect the structure and/or parameters of the D-FCTBN. We validated the proposed approach using both real data from Cameron County Hispanic Cohort (CCHC) as well as simulations. The results demonstrate the effectiveness of the proposed D-FCTBN and tensor control chart for dynamic prediction and monitoring the impact of patient's modifiable lifestyle behaviors on the emergence of multiple chronic conditions.
\bibliographystyle{IEEEtran}
|
1,108,101,564,541 | arxiv | \section{Abstract}
The reconstruction of transmission trees for epidemics from genetic data has been the subject of some recent interest. It has been demonstrated that the transmission tree structure can be investigated by augmenting internal nodes of a phylogenetic tree constructed using pathogen sequences from the epidemic with information about the host that held the corresponding lineage. In this paper, we note that this augmentation is equivalent to a correspondence between transmission trees and partitions of the phylogenetic tree into connected subtrees each containing one tip, and provide a framework for Markov Chain Monte Carlo inference of phylogenies that are partitioned in this way, giving a new method to co-estimate both trees. The procedure is integrated in the existing phylogenetic inference package BEAST.
\section{Introduction}
The increasing availability of faster and cheaper sequencing technologies is making it possible to acquire genetic data on the pathogens involved in outbreaks and epidemics at a very fine resolution. It is likely that in future outbreaks where most or all infected hosts can be identified, one or more pathogen nucleotide sequences will be available from each one as a matter of course. Identification of a high proportion of hosts is plausible in several scenarios, such as agricultural outbreaks, where the infected unit will usually be taken to be the farm and considerable government resources will be employed to identify every one, HIV, where almost all infected individuals will eventually seek treatment, and epidemics involving a population that can be closely monitored, such as those occurring in hospitals or prisons. As a result, much recent work has been performed to develop computational methods to analyse data of this kind, combining it with more traditional epidemiological data \cite{cottam_integrating_2008, aldrin_modelling_2011, ypma_unravelling_2011, jombart_reconstructing_2011, morelli_bayesian_2012, ypma_relating_2013, jombart_bayesian_2014, didelot_bayesian_2014, mollentze_bayesian_2014}. A Bayesian Markov Chain Monte Carlo (MCMC) approach is almost always employed, as the probability spaces involved are of very high dimension and mathematically complicated; the only exception is the study by Aldrin et al. \cite{aldrin_modelling_2011}, which used a maximum-likelihood method.
The most frequent approach to this problem has been to attach a mutation model to a model of transmission, making simplifications that link the process of nucleotide substitution to host-to-host transmission events. Commonly, transmission events are assumed to coincide with times of most recent common ancestor of isolates, ignoring any within-host diversity; the assumption being, in effect, that the phylogeny of the pathogen samples and the transmission tree of the epidemic coincide. No coexistence of separate lineages within the same host is permitted which, over the short timespan of an epidemic, might not be realistic. The alternative is to treat the phylogenetic and transmission trees as separate, although related, entities, and explicitly model a phylogeny occurring within each host. The initial exploration of this was performed by Ypma et al. \cite{ypma_relating_2013}, who linked up individual within-host phylogenies according to a transmission tree structure to build a single tree describing the history of the pathogen lineages for an entire epidemic. They applied the principle to simulated measles outbreaks and data from the 2001 UK foot and mouth disease outbreak, using rather different mathematical formulations for each. Our objective here is to build a general framework for an analysis of this sort, that is publicly available and easily modifiable for different models of host-to-host transmission, within-host pathogen population dynamics and nucleotide substitution.
The MCMC procedure used by Ypma et al. \cite{ypma_relating_2013} treated every individual within-host phylogeny as a distinct entity and modified them individually. Two previous papers have noted that, instead, a transmission history can be reconstructed by augmenting the internal nodes of a single phylogenetic tree for the entire epidemic with information about the host in which the corresponding lineage was located. Cottam et al. \cite{cottam_integrating_2008} were the first to identify this, and it was recently revisited and refined by Didelot et al. \cite{didelot_bayesian_2014}. These studies, however, have been constrained by the lack of a method to co-estimate the complete phylogeny simultaneously with its node labels; they have instead used a fixed tree pre-generated by a standard phylogenetic method. (Another recent paper, by Vrancken et al. \cite{vrancken_genealogical_2014}, encountered the opposite difficulty, and estimated a phylogeny consistent with a fixed transmission history.) This leads to two problems. Firstly, the use of a single tree will ignore any uncertainty in estimates of the phylogeny. If a Bayesian phylogeny reconstruction method is used, this can be mitigated to some extent by using the same method on each one of a sample of trees drawn from the posterior distribution, but at the cost of greater computation time. Secondly, and more seriously, a time-resolved tree constructed using such a method will usually have been built using assumptions about the pathogen population structure that are incompatible with what we know about an epidemic. Commonly, all viral lineages are assumed to be part of a single, freely mixing population, the probability of a tree calculated based on the assumption that it was generated by a coalescent process in this population. The result is that phylogenies may display features that are not epidemiologically plausible. For example, while mutation rates for, particularly, RNA viruses are fast, it remains true that many sequences collected over the short timescale of an epidemic will be identical \cite{bataille_evolutionary_2011}. If this is the case for two isolates, they are likely to form a ``cherry'' in the reconstructed phylogeny whose time of most recent common ancestor (TMRCA) can take values very close to the sampling time of the earlier isolate, because in a panmictic population, there is no reason to rule this out. In an epidemic situation where each sample is taken from a different host, we know that this is impossible, as there must have been at least one infection event since that TMRCA, and in the time from infection to sampling, a host will have gone through an incubation period and probably also a period from manifestation of symptoms to sampling. If a single tree with these short terminal branch lengths is then used to estimate epidemiological parameters, estimates of times from infection to sampling are unlikely to be reliable.
Our contribution here is threefold. Firstly, we formally establish that the procedure for augmenting internal nodes in a phylogeny identified by Didelot et al. \cite{didelot_bayesian_2014} does indeed allow simultaneous exploration of the complete space of both phylogenies and transmission trees. Secondly, we provide a full Bayesian MCMC framework for estimation of phylogenies using a model of the pathogen population that is consistent with host-to-host transmission during an epidemic, integrating relevant epidemiological data. Thirdly, as our method is fully integrated into the existing phylogenetics application BEAST \cite{drummond_bayesian_2012}, it provides a freely-available implementation of a method of this type for use by the research community, as well as platform for future development that has access to all the models and methods that are already implemented in that package.
\section{Method}
\subsection{Transmission trees as phylogenetic tree partitions}
We take as our dataset $D$ a set of $N$ sequences, each taken from a different infected unit (be it an infected organism or infected premise - from now on we use the word ``host'', but it need not be a single organism) in an infectious disease outbreak or epidemic, such that the total number of infections was also $N$. Let our set of hosts be $\mathbf{A}=\{a_1,\ldots,a_N\}$. Let $\mathcal{T}$ be a genealogy describing the ancestral relationship between those $N$ isolates, with branch lengths in units of time. It consists of two components:
\begin{itemize}
\item{A rooted, binary tree $T$ with a set $\mathbf{E}_T$ of $N$ labelled tips (labelled with the elements of $\mathbf{A}$) and a set $\mathbf{I}_T$ of $N-1$ internal nodes. Let $\mathbf{N}_T=\mathbf{E}_T\cup\mathbf{I}_T$ be the complete set of nodes. Let $\Gamma_\mathbf{A}$ be the set of all such trees.}
\item{A length function $l:\mathbf{N}_T\rightarrow(0,\infty)$ that takes each non-root node of $T$ to the difference in calendar time (in whatever units we choose) between the event represented by that node and the event represented by its parent. The event represented by an element of $\mathbf{E}_T$ is the sampling of the isolate from the host corresponding to $u$'s label; the event represented by an element of $\mathbf{I}_T$ is the existence of the most common ancestor of the isolates that correspond to $v$'s descendants. In contrast to the convention in most phylogenetic methods, we do indeed define a nonzero $l(r)$ for the root node $r$ of $T$. Its value is largely arbitrary, but it must be greater than any plausible value for the time between the existence of the event (generally an ancestor) represented by $r$ and the infection event that seeded the entire outbreak.}
\end{itemize} The length function $l$ allows us to also define a height function $h:\mathbf{N}_T\rightarrow[0,\infty)$ that takes each node to the difference in time between the event represented by that node and the time at which the last isolate was sampled.
For our purposes, we define a transmission tree on $\mathbf{A}$ to be a rooted tree with $N$ nodes labelled with the elements of $\mathbf{A}$. The root node of such a tree is labelled with the first case in the outbreak, and the children of a node are labelled with the hosts that were directly infected by that node's label. In this framework, transmission trees do not contain timing information and consist solely of a description of which host infected which others. They are not binary and a node can have any number of children. In fact, if $\mathcal{N}$ is such a tree, it can be thought of as a map $\mathcal{N}:\mathbf{A}\rightarrow\mathbf{A}\cup\emptyset$ taking each host $a_i$ to its infector $\mathcal{N}(a_i)$, or to $\emptyset$ if $a_i$ is the first host, and we will use this notation henceforth.
Let $\Pi_\mathbf{A}$ be the set of all transmission trees on $\mathbf{A}$. ($\Pi_\mathbf{A}$ has cardinality $N^{N-1}$ by Cayley's formula, as there are $N^{N-2}$ such trees and $N$ choices of root for each.) Take $T$ be a phylogenetic tree as above, describing the ancestry of $\mathbf{A}$, and assume no reinfection of hosts. We are interested in the set of transmission trees in $\Pi_\mathbf{A}$ that are consistent with the ancestry represented by $T$. Let $\Omega^T$ be the set of partitions of the set of nodes of $T$ such that:
\begin{itemize}
\item{If $\mathcal{P}\in\Omega^T$ and $p\in \mathcal{P}$, then the removal from $T$ of all nodes in $\mathbf{N}_T$ that are not in $p$, and all edges adjacent to at least one of them, leaves a connected graph.}
\item{All elements of $\Omega^T$ contain one and only one tip of $T$.}
\end{itemize}
For $\mathcal{P}\in\Omega^T$, define a map $\delta_\mathcal{P}:\mathbf{N}_T\to\mathbf{A}$ that takes each node of $T$ to the label of the tip that is in the same element of $\mathcal{P}$ as itself. For each $a_i\in\mathbf{A}$, let $S_{\mathcal{P},i}$ be the subtree of $T$ constructed by removing all nodes, and edges adjacent to them, that do not map to $a_i$ under $\delta_\mathcal{P}$. Because $S_{\mathcal{P},i}$ is connected, it has a single root node. Define a second map $\epsilon_\mathcal{P}:\mathbf{A}\to\mathbf{N}_T$ taking each $a_i$ to this root node. For brevity write $s_i=\epsilon_\mathcal{P}(a_i)$. All $s_i$ have a parent $s_iP$ in $T$, except for the root $r$ of $T$ (which must be the root of one such subtree). We also define a map $\gamma:\mathbf{A}\rightarrow \mathbf{E}_T$ taking a host to the tip of $T$ which is labelled with it.
If $T$ does indeed describe the ancestral relationships between the isolates collected from the elements of $\mathbf{A}$, and we know that we have sampled every host and that there is no reinfection, it is quite intuitively clear (see figure~\ref{sameoldsameold}) that an element $\mathcal{P}$ of $\Omega^T$ corresponds to a transmission history for the epidemic. The preimage of $a_i\in\mathbf{A}$ under $\delta_\mathcal{P}$ is the set of nodes that make up $S_{\mathcal{P},i}$. Infection events occur along branches of $T$ whose start and end nodes are in different elements of $\mathcal{P}$. The assumption of no reinfection mandates the connectedness requirement (or there would be multiple introductions to the same host) and the assumption that all hosts in the outbreak were sampled mandates that each element of $\mathcal{P}$ contains a tip (because one that did not would correspond to an unsampled host).
To formalise the correspondence, we construct a map $z:\Omega^T\rightarrow\Pi_\mathbf{A}$ such that if $\mathcal{P}\in\Omega^T$ and $a_i\in\mathbf{A}$,
\begin{eqnarray*}
z(\mathcal{P})(a_i) = \begin{cases}
\delta_\mathcal{P}(s_iP) & s_i\neq r\\
\emptyset & s_i= r
\end{cases}
\end{eqnarray*}
\begin{proposition}
For $\mathcal{P}\in\Omega_T$, the directed graph given by drawing an edge from $z(\mathcal{P})(a_i)$ to $a_i$ for all $a_i\in\mathbf{A}$ is a tree, and if $r$ is the root of $T$, the directionality coincides with that given by taking $\delta_\mathcal{P}(r)$ to be its root.
\end{proposition}
\begin{proof}
For the first part, we must show that the graph is simple, connected, and has no cycles. For simplicity, the construction will never give a node with indegree greater than 1, so if two edges join the same two nodes then their directionality is different. Suppose $a_i,a_j\in\mathbf{A}$ are such that $a_i=\delta_\mathcal{P}(s_jP)$ and $a_j=\delta_\mathcal{P}(s_iP)$. Now $s_i$ and $s_jP$ (which may not be distinct) are nodes of $S_{\mathcal{P},i}$, and $s_j$ as a descendant of $s_jP$ is also a descendant of $s_i$ in $T$. Similarly, $s_i$ is a descendant of $s_j$. This contradicts the fact that $T$, as a tree, has no cycles, or, if $s_j=s_iP$ and $s_i=s_jP$, that it is simple.
For connectedness, again suppose $a_i\in\mathbf{A}$ and let $a_j=\delta_\mathcal{T}(r)$; the root $a_j$ of $S_{\mathcal{P},j}$ is the root of $T$. It may be that $a_i=a_j$. If not, the path in $T$ from $a_i$ to $a_j$ passes through $n\geq2$ elements of $\mathcal{P}$ whose elements map under $\delta_\mathcal{P}$ to the hosts $a_{o(1)},\ldots,a_{o(n)}\in\mathbf{A}$, where $o$ is some permutation of $\{1,\ldots,N\}$ with $o(1)=i$ and $o(n)=j$. In particular it must pass through the root nodes of all these subtrees, $s_{o(1)},\ldots,s_{o(n)}$, implying that $z(\mathcal{P})(a_{o(k)})=a_{o(k+1)}$ for all $1\leq k \leq n-1$. It follows that $(z(\mathcal{P}))^{n-1}(a_i)=a_j$; thus all hosts in $\mathbf{A}$ are connected to $a_j$ and each other.
Suppose $z(\mathcal{P})$ has a cycle. It must be a directed cycle or else $z(\mathcal{P})$ has a node with indegree greater than 1. With $o$ denoting a permutation of $\{1,\ldots,N\}$ as before, suppose the cycle has $n\geq 3$ (if $n=2$ then the graph is not simple) elements $a_{o(1)},\ldots,a_{o(n)}$ such that $z(\mathcal{P})(a_{o(k)})=a_{o(k+1)}$ for all $1\leq k \leq n-1$ and $z(\mathcal{P})(a_{o(n)})=a_{o(1)}$. If $i\geq2$, the $S_{\mathcal{P},o(i)}$ is a subtree of $T$ containing a root node $s_{o(i)}$ and the parent $s_{o(i-1)}P$ of the root node of the subtree $S_{\mathcal{P},o(i-1)}$; similarly $S_{\mathcal{P},o(1)}$ contains $s_{o(n)}P$. Since $S_{\mathcal{P},o(i)}$ for each $i$ contains a sequence of nodes, following the directedness of $T$ induced by its root, running from $s_{o(i)}$ to $s_{o(i-1)}P$ to and there is a directed link from each $s_{o(i)}P$ to $s_{o(i)}$ in $T$, the concatenation of all of these forms a cycle in $T$, contradicting the fact it is a tree.
For the second part, there is no node $z(\mathcal{P})(\delta_\mathcal{P}(r))$ by construction, and we have already shown that our construction produces a directed path from each $a\in\mathbf{A}$ to $\delta_\mathcal{P}(r)$. As we have shown $z(\mathcal{P})$ is a tree, this is the only such path, hence the directedness of all edges is towards $\delta_\mathcal{P}(r)$.
\end{proof}
\begin{proposition}\label{inj}
$z$ is injective.
\end{proposition}
\begin{proof}
We suppose the we have two partitions $\mathcal{P},\mathcal{P}'$ that have the same image under $z$, i.e. for all $a_i\in\mathbf{A}$, $z(\mathcal{P})(a_i)=z(\mathcal{P}')(a_i)$. If $\mathcal{P}\neq\mathcal{P}'$ then there exists some node $u$ of $T$ that has $a_i=\delta_\mathcal{P}(u)\neq a_j=\delta_{\mathcal{P}'}(u)$. We can assume that either $u$ is the root of $T$ or $\delta_\mathcal{P}(uP)=\delta_\mathcal{P'}(uP)$ for the parent $uP$ of $u$ (or else we move down $T$ to find a new $u$ for which this is true).
If $u$ is the root of $T$, then it is the root of the subtrees $S_{\mathcal{P},i}$ and $S_{\mathcal{P}',j}$. This implies $z(\mathcal{P})(a_i)=\emptyset$ but $z(\mathcal{P}')(a_i)\neq\emptyset$ because $z(\mathcal{P}')(a_j)=\emptyset$; only one element of $\mathbf{A}$ can be sent to $\emptyset$ by $z(\mathcal{P}')$ since the root of $T$ is unique. So $uP$ exists.
Let $a_k=\delta_\mathcal{P}(uP)=\delta_\mathcal{P'}(uP)$. First suppose $k\neq i$ and $k\neq j$. Then $z(\mathcal{P})(a_i)=a_k$. We show that $z(\mathcal{P'})(a_i)=a_k$ is not possible. Let $v=\gamma(h)$. Now $v$ is a descendant of $u$ because $u$ is the root node of the subtree $S_{\mathcal{P},i}$, and $S_{\mathcal{P},i}$ includes $v$. $\mathcal{P'}$ gives rise to another subtree of $T$, $S_{\mathcal{P}',i}$, all of whose nodes map to $a_i$ under $\delta_{\mathcal{P}'}$. This $S_{\mathcal{P}',i}$ has a root node $s'_i$ which is \emph{not} $u$ because $\delta_{\mathcal{P}'}(u)=a_j$. It must, in fact, also be a descendant of $u$; if it were not, $S_{\mathcal{P}',i}$ would be disconnected by $u$. The parent $s'_iP$ cannot have $\delta_{\mathcal{P}'}(s'_i)=a_k$ because either a) $s'_iP=u$ and $\delta_{\mathcal{P}'}(u)=a_j$ by construction or b) $s'_iP\neq u$ and if $\delta_{\mathcal{P}'}(s'_iP)=a_k$ were true, the subtree of nodes that map to $a_k$ under $\delta_{\mathcal{P}'}$ would be disconnected by $u$. Hence $z(\mathcal{P'})(a_i)\neq a_k$.
So without loss of generality suppose $k\neq i$ but $k= j$. Again $z(\mathcal{P})(a_i)=a_k$. Let $v$ be the unique tip of $T$ that has $\delta_\mathcal{P}(v)=\delta_\mathcal{P'}(v)=a_k$. Now, $v$ is not a descendant of $u$. If it were, then $S_{\mathcal{P},k}$, the subtree of $T$ whose nodes are mapped to $a_k$ by $\delta_{\mathcal{P}}$, would be disconnected by $u$, which maps to $a_i$. This implies that there is a descendant $w$ of $u$ in $T$, possibly $u$ itself, which maps to $a_k$ under $\delta_{\mathcal{P}'}$ but neither of whose children $wC_1$ and $wC_2$ do. (If this were not true, a second tip would map to $a_k$ under $\delta_{\mathcal{P}'}$). Whether it is $u$ or not, $w$ cannot map to $a_k$ under $\delta_\mathcal{P}$; if it is $u$ then it does not by construction, and if is not, it would have an ancestor, $u$, which did not, and an earlier ancestor, $uP$, which did, breaking connectedness. This implies that $z(\mathcal{P}')(wC_1)=z(\mathcal{P}')(wC_2)=a_k$ but $z(\mathcal{P})(wC_1)=z(\mathcal{P})(wC_2)\neq a_k$.
\end{proof}
For the next proposition, we need the following:
\begin{lemma}\label{ancestors}
If $a_i,a_j\in\mathbf{A}$ and $\mathcal{N}\in\Pi_\mathbf{A}$ is a transmission tree in which $a_i$ is an ancestor of $a_j$, then if $\mathcal{P}\in\Omega^T$ with $z(\mathcal{P})=\mathcal{N}$ and $u$ is a node of $T$ with $\delta_\mathcal{P}(i)=a_j$, $u$ has an ancestor $v$ in $T$ with $\delta_\mathcal{P}(j)=a_i$.
\end{lemma}
\begin{proof}
Strong induction on the number $n$ of intervening hosts between $a_i$ and $a_j$ in $\mathcal{N}$. If $n=0$, this is true by definition of $u$, as the node $r_{h_2}$ is an ancestor of $i$ and its parent maps to $h_1$. If the lemma is true for all $n\leq m$ and the set of intervening hosts has size $m+1$, let $a_k$ be an arbitrary member of that set. The number of intervening hosts between $a_k$ and $a_j$ in $\mathcal{N}$ is less than $m+1$, so $i$ has an ancestor $v$ in $T$ with $\mathcal{P}(k)=a_k$. The number of intervening hosts between $a_i$ and $a_k$ in $\mathcal{N}$ is also less than $m+1$, so $v$ has an ancestor $w$ in $T$ with $\mathcal{P}(k)=a_i$. It follows that $w$ is the ancestor of $u$ that we need.
\end{proof}
\begin{proposition}\label{notsurj}
$z$ is not surjective for $N>2$.
\end{proposition}
\begin{proof}
For $N=2$, $|\Pi_\mathbf{A}|=2$ and $|\Omega^T|=2$ since the latter is simply the number of assignments for the single internal node of $T$ to a subgraph containing one tip or the other. The map's injectiveness ensures its surjectiveness. If $N>2$, then let $a_i,a_j,a_k\in\mathbf{A}$ be any three hosts. In $T$, $\gamma(a_i)$, $\gamma(a_j)$ and $\gamma(a_k)$ have a most recent common ancestral node $u$ and two of them, without loss of generality $\gamma(a_j)$ and $\gamma(a_k)$, have a most recent common ancestral node $v$ which is a descendant of $u$. We show that there is no element of $\Omega^T$ which will map to any member of $\Pi_\mathbf{A}$ in which any of the following are true:
\begin{itemize}
\item{$a_j$ is an ancestor of $a_i$, which is an ancestor of $a_k$.}
\item{$a_j$ is an ancestor of $a_k$, which is an ancestor of $a_i$.}
\item{$a_k$ is an ancestor of $a_i$, which is an ancestor of $a_j$.}
\item{$a_k$ is an ancestor of $a_j$, which is an ancestor of $a_i$.}
\end{itemize}
Let $\mathcal{P}$ be a partition such that $z(\mathcal{P})$ is a transmission tree in which $a_j$ is an ancestor of both $a_i$ and $a_k$. Now $\delta_{\mathcal{P}}(u)=a_j$. To see this, note that since $u$ is an ancestor of $\gamma(a_j)$, if it does not map to $a_j$ under $\delta_{\mathcal{P}}$ then neither do any of its ancestors, by connectedness. Nor do any descendants of the child of $u$ which is not an ancestor of $\gamma(a_j)$ and $\gamma(a_k)$, a set which includes $\gamma(a_i)$. All ancestors of $\gamma(a_i)$ apart from $u$ belong to one of those categories. But this contradicts lemma~\ref{ancestors} because $\gamma(a_i)$ has no ancestor which maps to $a_j$ under $\delta_\mathcal{P}$ despite the fact that $a_j$ is an ancestor of $a_i$.
Now $\gamma(a_i)$ has no ancestor in $T$ that maps to $a_k$ under $\delta_{\mathcal{P}}$, because the node $u$ breaks connectedness between $\gamma(a_k)$ and any position that such a node could be. The contrapositive of lemma~\ref{ancestors} then says that $a_k$ is not an ancestor of $a_i$. Similarly $a_i$ is not an ancestor of $a_k$. Likewise, if $z(\mathcal{P})$ is such that $a_k$ is an ancestor of both $a_i$ and $a_k$, $a_i$ is not an ancestor of $a_j$ nor vice versa.
\end{proof}
Let the image of $\Omega^T$ under $z$ be $\Lambda^T_\mathbf{A}\subseteq\Pi_\mathbf{A}$. The actual cardinality of $\Lambda^T_\mathbf{A}$ varies with the topology of $T$, which can be clearly seen in the case $N=4$ (figure~\ref{4partitions}).
Proposition~\ref{inj} states that no two partitions of the internal nodes of $T$ correspond to the same transmission history; the set of partitions and the set of compatible transmission trees are equivalent. Proposition~\ref{notsurj} shows, however, that not every possible transmission tree on $\mathbf{A}$ actually corresponds to a partition of the nodes of a fixed $T$. If we are interested in exploring the complete space of transmission trees using this construction, we need to vary the phylogeny as well.
Let the set $\mathbf{\Omega}=\{\Omega^T:T\in\Gamma_\mathbf{A}\}$ consist of all partitions of all phylogenies with tips labelled with $\mathbf{A}$. The map $z$ can be extended to a map $Z:\mathbf{\Omega}\to\Pi_\mathbf{A}$ in the obvious way.
\begin{proposition}\label{zsurj}
$Z$ is surjective. In other words, any transmission tree on $\mathbf{A}$ arises as a partition of \emph{some} phylogenetic tree $T\in\Gamma_\mathbf{A}$.
\end{proposition}
\begin{proof}
Let $\mathcal{N}\in\Pi_\mathbf{A}$. Use the following procedure to construct an element of $\mathbf{\Omega}$. If each $a_i\in\mathbf{A}$ has $n_i$ children in $\mathcal{N}$, take $n_i+1$ nodes $v_{i,1},\ldots,v_{i,n_i},v_{i,n_i+1}$. Pick an arbitrary ordering of the children of each $a_i$ and make a graph $T$ by drawing two edges from each $v_{i,k}$ to $v_{i,k+1}$ and from $v_{i,k}$ to $v_{j,1}$ where $j$ is such that $a_j$ is the $k$th child of $a_i$ in the ordering. (Notice that $v_{i,n+1}$ gets no children either way.) If $r\in\{1,\ldots,N\}$ is such that $a_r$ is the root of $\mathcal{N}$, let the root of $T$ be $v_{r,1}$.
It is clear that $T$ is a rooted binary tree, its tips are the $v_{i,n_i+1}$ and if each of these is labelled with the corresponding $a_i$ then they are in one-to-one correspondence with $\mathbf{A}$. The set of nodes $v_{i,1},\ldots,v_{i,n_i},v_{i,n_i+1}$ for each $h_i$ are by construction connected in $T$ and contain the single tip $v_{i,n_i+1}$; hence this partitioning of the nodes of $T$ is an element $\mathcal{P}$ of $\Omega^T$. It is easily checked that $z(\mathcal{P})=\mathcal{N}$.
\end{proof}
As an aside, $Z$ is not injective, as is clear from the arbitrary choice of ordering for the children of each $a_i$. (In fact, some elements of $\mathbf{\Omega}$ cannot be produced by this construction at all, for example, the bottom right example in figure~\ref{sameoldsameold}.) The upshot of proposition~\ref{zsurj} is that a MCMC procedure that fully explores the space of these partitioned phylogenies is also fully exploring the space of transmission trees amongst the elements of $\mathbf{A}$. We outline such a procedure in the next section.
So far, we have only dealt with the phylogenetic tree topology $T$. If this construction is to be useful for epidemic reconstruction, we must now consider branch lengths. Let $\mathcal{P}$ be a partition of $T$, and suppose $T$ is the topology of a genealogy $\mathcal{T}$ with length function $l$ and height function $h$. Suppose $a_i\in\mathbf{A}$ and that $z(\mathcal{P})(a_i)\neq\emptyset$. Let $u=\epsilon_\mathcal{P}(a_i)$, and let $uP$ be the parent of $u$. An infection event occurs on the branch between $uP$ and $u$, which means, assuming that internal nodes of $T$ and transmissions do not occur at exactly the same time, that it occurs at a height in the interval $(h(u),h(uP))$. In what follows it will be convenient to use a forwards timescale, so let $C:\mathbb{R}\to\mathbb{R}$ be a function converting between tree height and such a timescale (in the same units, so branch lengths are maintained). Let $t^{\textrm{inf}}_i$ be this time of infection in \emph{forwards} time. Let $q_i\in(0,1)$ be such that $t^{\textrm{inf}}_i = C(h(uP)) + q_i(C(h(u))-C(h(uP))) = C(h(uP)) + q_il(u)$. If $z(\mathcal{P})(a_i)=\emptyset$, i.e. $a_i$ is the first host in the epidemic, then $t^{\textrm{inf}}_i$ is between $C(h(r)+l(r))$ ($r$ being the root node of $T$) and $C(h(r))$ (remembering that we gave $r$ a finite branch length) we can similarly define $q_i$ such that $t^{\textrm{inf}}_i = C(h(r)+l(r)) + q_il(r)$.
The combination of a genealogy $\mathcal{T}$, partition $\mathcal{P}$ and a set of $q_i$s for all elements $a_i\in\mathbf{A}$ then entirely determines the transmission history of the epidemic, describing which host infected which others and when. No assumptions are made at this, conceptual, stage about when hosts cease to be infectious; a host can continue to infect others at any time following the time at which is sample was acquired. If, as will often be the case, this is an unreasonable assumption, the likelihood of such partitions can be evaluated to zero in the calculation of the posterior probability.
\section{MCMC procedure}
The most common methods for estimation of time-resolved phylogenies involve the use of Bayesian MCMC to sample from the probability distribution of phylogenetic trees given the available sequence data. The previous section demonstrates that, if the sequence data is such that one sample is taken from each host, such procedures can be extended to simultaneously sample from the probability distribution of reconstructed epidemics each sampled tree is augmented a partition of its nodes as well as the values of each $q_i$. We have implemented this procedure in the package BEAST \cite{drummond_bayesian_2012}. Because of the special requirements of this type of augmentation, the standard moves on the phylogenetic tree topology cannot be used. Nor are the structured tree operators developed by Vaughan et al. \cite{vaughan_efficient_2014} suitable, as those are designed for the exploration of the space of trees where every point on every branch can be freely assigned a ``type'' from a finite set. This condition is much less restrictive than than connectedness requirements that we have outlined above and the result of such a move on a tree of our type would not necessarily meet our requirements for partitions. Instead, specialised moves have been devised to alter the partitioned phylogeny in such a way that the transmission tree structure is maintained. In addition, we give an operator to alter the transmission tree while keeping the phylogenetic tree fixed, by changing node labels.
Note that these moves do not simultaneously change the value of any of the $q_i$s, as moves on these are proposed and evaluated separately. Nevertheless, changes to either tree may involve resampling the times of infection of some hosts. If $a_i\in\mathbf{A}$, changing partition from $\mathcal{P}$ to $\mathcal{P}'$ may mean that $\epsilon_\mathcal{P}(a_i)$ and $\epsilon_{\mathcal{P}'}(a_i)$ are different nodes with different heights, and so while $q_i$ will not change, $t^{\mathrm{inf}}_i$ will. Even a move has no effect on the partition or phylogenetic tree topology, such as a change to branch lengths, may also alter the height of $\epsilon_\mathcal{P}(a_i)$ and/or its parent, which will also modify $t^{\mathrm{inf}}_i$ while $q_i$ remains fixed.
\begin{definition}
For a partition $\mathcal{P}$ of a phylogeny $\mathcal{T}$, if $u$ is a phylogenetic tree node with $\delta_\mathcal{P}(u)=a_i\in\mathbf{A}$ we say $u$ is \emph{ancestral under $\mathcal{P}$} if it is an ancestor of the only member of the subtree $S_{\mathcal{P},i}$ which is a tip of $\mathcal{T}$.
\end{definition}
\begin{definition}
For a partition $\mathcal{P}$ of a phylogeny $\mathcal{T}$, the \emph{infection branch} for $a_i\in\mathbf{A}$ is the branch of $\mathcal{T}$ ending in $\epsilon_\mathcal{P}(a_i)$.
\end{definition}
\subsection{Infection branch operator}
We randomly select a host $a_i$ that is not the first host in the outbreak (i.e. $\epsilon_\mathcal{P}(a_i)$ is not the root of $\mathcal{T}$). Consider $\epsilon_\mathcal{P}(a_i)$. The operator performs both ``downward'' and ``upward'' moves, but if $\epsilon_\mathcal{P}(a_i)$ is a tip then the move must be downwards. If it is internal, then we select upwards or downwards each with probability 0.5. Let $u=\epsilon_\mathcal{P}(a_i)$ and $uP$ be the parent of $u$ (which must exist as we avoided the root). It must be that $u$ and $uP$ are in different elements of $\mathcal{P}$, and this implies that $u$ is ancestral under $\mathcal{P}$ because the path from any node $v$ that is not a descendant of $u$ to $u$ must pass through $uP$ and if $\delta_\mathcal{P}(v)=a_i$ this would violate the connectedness requirement. Suppose $\delta_{\mathcal{P}}(uP)=a_j$.
\paragraph{Upward move} We create a new partition $\mathcal{P}'$ that has $\delta_{\mathcal{P}'}(u)=a_j$, moving the infection branch of $a_i$ up the tree. Consider the two children $uC_1$ and $uC_2$ of $u$ (as this is the upward move, $u$ is not a tip). At least one of these is mapped to the same element of $\mathbf{A}$ as $u$ by $\delta_\mathcal{P}$ because $u$ must be in the same element of $\mathcal{P}$ as the tip $\gamma\circ\delta_\mathcal{P}(u)$ and the path from $u$ to this tip in the subtree will intersect one of its children. If this is true of only one child then without loss of generality say it is $uC_1$. In this case we can simply make $\mathcal{P}'$ by setting $\delta_{\mathcal{P}'}(i)=a_j$ and leaving the rest of the partition unchanged; this is clearly still a valid partition because all subtrees remain connected. So suppose also $\delta_{\mathcal{P}}(uC_2)=\delta_{\mathcal{P}}(u)$. At most one of $uC_1$ and $uC_2$ is ancestral under $\mathcal{P}$ (as siblings, they cannot both be ancestors of the same tip) so, again without loss of generality, say it is $uC_1$. If we again set $\delta_{\mathcal{P}'}(u)=a_j$, the removal of $u$ from the subtree $S_{\mathcal{P},i}$ splits the nodes of the latter into two sets, $V_1$ containing $uC_1$ and $\gamma\circ\delta_\mathcal{P}(u)$, and $V_2$ containing $uC_2$. The nodes of both sets and the edges between them form connected subtrees of $T$, but their union is not connected. We complete the construction of $\mathcal{P}'$ by setting $\delta_{\mathcal{P}'}(v)=a_j$ for all $v\in V_2$. $S_{\mathcal{P}',i}$ and $S_{\mathcal{P}',j}$ are then connected.
The effect on the transmission tree is that all $a_k\in\mathbf{A}$ that have $z(\mathcal{P})(a_k)=a_i$ and $\gamma(a_k)$ a descendant of $uC_2$ have $z(\mathcal{P}')(a_k)=a_j$ instead.
\paragraph{Downward move} We create a new partition $\mathcal{P}'$ that has $\delta_{\mathcal{P}'}(uP)=a_i$, moving the infection branch of $a_i$ down the tree. We need to consider the grandparent $uG$ of $u$ if it exists, and the child $uS$ of $uP$ that is not $u$. At least one of $uG$ and $uS$ must be in the same element of $\mathcal{P}$ as $uP$ (or else $uP$ is not in a partition element containing a tip). If $uG$ does not exist then this must be $uS$.
If $\delta_\mathcal{P}(uS)=a_j$ and either $\delta_\mathcal{P}(uG)\neq a_j$ or $uG$ does not exist, then setting $\delta_{\mathcal{P}'}(uP)=a_i$ is all that is required to make $\mathcal{P}'$ a valid partition. The two or three nodes joined to $uP$ by edges were all in different elements of $\mathcal{P}$ and remain so; $uP$ was in the element of $\mathcal{P}$ containing one of its children and is moved to the one containing the other child in $\mathcal{P}'$. Similarly, if $\delta_\mathcal{P}(uG)=a_j$ and $\delta_\mathcal{P}(uS)\neq\delta_\mathcal{P}(uP)$, then $\mathcal{P}'$ then all we need do is set $\delta_{\mathcal{P}'}(uP)=a_i$; the situation is the same except that the $uP$ has moved from the element of $\mathcal{P}$ that contains of one of its children to the one containing its parent.
If $uG$ exists and $\delta_\mathcal{P}(uS)=\delta_\mathcal{P}(uG)=a_j$, then the removal of $uP$ from the subtree $S_{\mathcal{T},j}$ splits into two subtrees whose union is again not a connected subtree of $T$. Let the node sets of these two subtrees be $V_1$ and $V_2$, with $V_1$ containing $uG$ and $V_2$ containing $uS$. If $uP$ is ancestral under $\mathcal{P}$ then $V_2$ also contains the tip $\gamma(a_j)$, and if it is not then $V_1$ does. We complete $\mathcal{P}'$ by setting $\delta_{\mathcal{P}'}(uS)=a_i$ for all $v$ in the set that does not contain $\gamma(a_j)$. $S_{\mathcal{P}',i}$ and $S_{\mathcal{P}',j}$ are now connected. Note that $V_1$ may contain the root node and if it does not contain $\gamma(a_j)$ then the root's image under $\delta_\mathcal{P}$ is different from that under $\delta_{\mathcal{P}'}$, which is how this move may change the first host in the outbreak even though the root host is never chosen by the move. This can be seen in example 7) of figure~\ref{ttoperator}.
If $uP$ is not ancestral under $\mathcal{P}$, then the effect on the transmission tree is that all $a_k\in\mathbf{A}$ that have $z(\mathcal{P})(a_k)=a_j$ and $\gamma(a_k)$ a descendant of $uS$ have $z(\mathcal{P}')(a_k)=a_i$ instead. If $uP$ is ancestral under $\mathcal{P}$ then, in $z(\mathcal{P}')$, $a_i$ is the infector of $a_j$ instead of vice versa, and all $a_k\in\mathbf{A}$ that have $z(\mathcal{P})(a_k)=a_j$ and $\gamma(a_k)$ \emph{not} a descendant of $uS$ have $z(\mathcal{P}')(a_k)=a_i$ instead.
\paragraph{Hastings ratio} We observe that:
\begin{itemize}
\item{The upward move on $u$ is reversed by the downward move on the child $uC_1$ of $u$ that is ancestral under $\mathcal{P}$. Thus the Hastings ratio is 1 if $uC_1$ is not a tip and 2 if it is.}
\item{If $uP$ is not ancestral under $\mathcal{P}$, then the downward move on $u$ is reversed by the upward move on $uP$. The Hastings ratio is 1 if $u$ is not a tip and $1/2$ if it is.}
\item{If $uP$ is ancestral under $\mathcal{P}$, then the downward move on $u$ is reversed by the downward move on its sibling $uS$. The Hastings ratio is 1 if $u$ and $uS$ are both tips or both not tips, 1/2 if $u$ is but $uS$ is not, and 2 if $uS$ is but $u$ is not. }
\end{itemize}
The various variations of this move are depicted in figure~\ref{ttoperator}. If the initial partition is that depicted as 1), the downward moves depicted as 2), 4), 7), 9), 10) and 12) involve a parent that is ancestral under $\mathcal{P}$ and 5) and 6) involve one that is not.
\subsection{Phylogenetic tree operators}
We have adapted the three standard tree moves used in BEAST (exchange, subtree slide, and Wilson-Balding \cite{wilson_genealogical_1998, drummond_estimating_2002, hohna_clock-constrained_2008}) such that they respect the transmission tree structure induced by partitioning the internal nodes. We give two versions of each:
\begin{itemize}
\item{A ``type A'' operator which does not alter the transmission tree at all; all parental relationships remain the same.}
\item{A ``type B'' operator which performs phylogenetic tree modifications which simultaneously rearrange the transmission tree by assigning new parents to one or two hosts.}
\end{itemize}
\subsubsection{Type A operators}
\paragraph{Type A exchange}
Select a random node $u$ that is not the root $r$ of the phylogenetic tree $\mathcal{T}$, and then randomly selects a second node $v$, also not $r$ and not the sibling $uS$ of $u$, such that the parents $uP$ and $vP$ of $u$ and $v$ are in the same element of $\mathcal{P}$, $h(uP)>h(v)$, and $h(vP)>h(u)$. If there is no such $v$ then the operator fails. Otherwise, $u$ and $v$ exchange parents to obtain a new phylogenetic tree $\mathcal{T}'$ with the same partition of nodes $\mathcal{P}$. $\mathcal{P}$ is still valid in terms of connectedness, because if $\delta_\mathcal{P}(u)\neq\delta_\mathcal{P}(uP)$ then all nodes in the element of $\mathcal{P}$ containing $u$ are descendants of $u$ and the move has not affected them, whereas if $\delta_\mathcal{P}(u)=\delta_\mathcal{P}(uP)$ then changing $u$'s parent to $vP$ means that after the move it is still adjacent to a node with the same image under $\delta_\mathcal{P}$ as itself; the same goes for $v$. The transmission tree structure is unchanged: if $\delta_\mathcal{P}(u)\neq\delta_\mathcal{P}(uP)$ then $\delta_\mathcal{P}(u)$ is infected by $\delta_\mathcal{P}(uP)$ before the move and is by $\delta_\mathcal{P}(vP)=\delta_\mathcal{P}(uP)$ afterwards, whereas if $\delta_\mathcal{P}(u)=\delta_\mathcal{P}(uP)$ then $\delta_\mathcal{P}(u)$'s infection branch was not affected at all. Again, the same goes for $v$.
For the Hastings ratio, note that the partitioned tree obtained by selecting $u$ and then $v$ is exactly the same as that obtained by selecting $v$ and then $u$. If a node $w$ is selected first, let $c_\mathcal{P}(w)$ be the number of eligible nodes to be selected as the second (this is explicitly calculated every time the operator acts). The denominator of the Hastings ratio is then $\frac{1}{2N-2}(\frac{1}{c_\mathcal{P}(u)}+\frac{1}{c_\mathcal{P}(v)})$. The move is reversed by selecting the same two nodes again (in either order) hence we calculate $c_{\mathcal{P}'}(u)$ and $c_{\mathcal{P}'}(v)$ and the ratio's numerator is $\frac{1}{2N-2}(\frac{1}{c_{\mathcal{P}'}(u)}+\frac{1}{c_{\mathcal{P}'}(v)})$. Cancellation gives $\frac{\frac{1}{c_{\mathcal{P}'}(u)}+\frac{1}{c_{\mathcal{P}'}(v)}}{\frac{1}{c_\mathcal{P}(u)}+\frac{1}{c_\mathcal{P}(v)}}$.
\paragraph{Type A subtree slide} Select a random node $u$ under the conditions that $u\neq r$ and either $u$'s grandparent $uG$ or sibling $uS$ (or both) is in the same element of $\mathcal{P}$ as its parent $uP$. Draw a distance $\Delta\in\mathbb{R}$ from some probability distribution that is symmetric about 0. We aim to change the height of $uP$ to $h(uP)+\Delta$. If $\Delta>0$, examine $uP$'s ancestors to find a node $v$ such that either $v=r$ or $h(v)<h(uP)+\Delta$ but $h(vP)>h(uP)+\Delta$; if no such ancestor exists then let $v=uS$ and this is true. If $\delta_\mathcal{P}(v)\neq\delta_\mathcal{P}(uP)$ then the move fails. If $v=uS$ then simply change the height of $uP$ to $h(uP)+\Delta$ and the topology is unchanged. Otherwise, modify the tree such that $uP$ has height $h(uP)+\Delta$, parent $vP$ (or no parent if $v=r$ in which case $uP$ is now the root node) and child $v$, and $uS$ has parent $uG$. Again, do not change $\mathcal{P}$. Connectedness rules are still obeyed because, in the new tree $\mathcal{T}'$, $uP$ is adjacent to $v$, which is in the same element of $\mathcal{P}$ as itself. The transmission tree structure is unchanged as:
\begin{itemize}
\item{The move does not change the partition, so any infection branches have not changed if the particular phylogenetic tree branch was not modified by the move. This applies to the branch between $u$ and $uP$ as well as all branches adjacent to nodes other than $u$, $uP$, $uG$, $uS$, $v$, and $vP$.}
\item{If $uS$ and $uP$ are in different elements of $\mathcal{P}$ then $uP$ and $uG$ are in the same one, so the infector of $\delta_\mathcal{P}(uS)$ remains the same.}
\item{If $uG$ and $uP$ are in different elements of $\mathcal{P}$ then the move fails if $h(uP)+\Delta>h(uG)$ so the phylogenetic tree topology is unchanged.}
\item{If $v$ and $vP$ are in different elements of $\mathcal{P}$ then $uP$, instead of $v$, is now the end of $\delta_\mathcal{P}(uP)$'s infection branch, but $\delta_\mathcal{P}(uP)=\delta_\mathcal{P}(v)$ and its infector is still $\delta_\mathcal{P}(vP)$.}
\end{itemize}
If $\Delta<0$, then if $h(uP)+\Delta<h(u)$ the move fails. Otherwise, we select a node $v$ at random from the set $W$ which consists of nodes $w$ that:
\begin{enumerate}
\item{Are descendants of $uP$ but not descendants of $u$.}
\item{Have $h(k)<h(uP)+\Delta$ but $h(kP)>h(uP)+\Delta$.}
\item{Have $\delta_{\mathcal{P}}(wP)=\delta_{\mathcal{P}}(uP)$.}
\end{enumerate}
If $W$ is empty the move fails. In the case that $W$ consists only of $uS$ then simply set $h(uP)=h(uP)+\Delta$ and the topology is unchanged. Otherwise, modify the tree such that $uP$ has height $h(uP)+\Delta$, parent $vP$ and child $v$, and $uS$ has parent $uG$. connectedness rules are still obeyed because there is an edge from $uP$ to a node ($vP$) in the same element of the partition. The transmission tree structure is unchanged as:
\begin{itemize}
\item{Again, the move does not change the partition, so any infection branches have not changed if the particular phylogenetic tree branch was not modified by the move.}
\item{If $uS$ and $uP$ are in different elements of $\mathcal{P}$ then the move fails if $h(uP)+\Delta<h(uS)$ so the topology is unchanged.}
\item{If $uG$ and $uP$ are in different elements of $\mathcal{P}$ then $uP$ and $uS$ were in the same one, so the infector of $\delta_\mathcal{P}(uP)$ remains the same; $uS$ is now the end of its infection branch.}
\item{If $v$ and $vP$ are in different elements of $\mathcal{P}$ then the infector of $\delta_\mathcal{P}(v)$ is still $\delta_\mathcal{P}(vP)=\delta_\mathcal{P}(uP)$.}
\end{itemize}
Suppose there are $d_{\mathcal{T}}$ nodes eligible for this move before it occurs and $d_{\mathcal{T}'}$ afterwards. If the topology did not change then the Hastings ratio is $\frac{d_{\mathcal{T}'}}{d_{\mathcal{T}}}$. Otherwise, it is $\frac{|W|d_{\mathcal{T}'}}{d_{\mathcal{T}}}$ if $\Delta<0$ and $\frac{d_{\mathcal{T}'}}{|W'|d_{\mathcal{T}}}$ if $\Delta>0$, where the $W'$ is the set of nodes $w$ that:
\begin{enumerate}
\item{Are descendants of $vP$ (in the original tree) but not descendants of $u$.}
\item{Have $h(w)<h(uP)$ but $h(wP)>h(uP)$.}
\item{Have $\delta_{\mathcal{P}}(wP)=\delta_{\mathcal{P}}(v)$.}
\end{enumerate}
\paragraph{Type A Wilson-Balding move} Pick a node $u$ under the same conditions as for the type A subtree slide. Pick a second node $v$ at random from amongst all nodes that are in the same element of $\mathcal{P}$ as $uP$, or whose parents are, and such that $h(vP)>h(u)$. The move fails if $uP=vP$, or $v=uP$. The node $uP$ is pruned and reattached as a child of $vP$ and the parent of $v$ as with the standard Wilson-Balding move \cite{wilson_genealogical_1998, drummond_estimating_2002}. As before, do not change $\mathcal{P}$. Connectedness rules are obeyed because there is an edge from $uP$ to a node (either $v$ or $vP$) in the same element of $\mathcal{P}$ as itself. The transmission tree structure is unchanged because if there was an infection event between $uG$ and $uC$ (and there was at most one by construction) then there still is and it involves the same hosts, and likewise if there was one between $vP$ and $v$ then there still is and it involves the same hosts. If there was no infection event in either case then the removal or insertion of $uP$ does not add one.
Notice that if $u$ is subsequently selected for this move again, then the set of candidates for the second node is the same except that it excludes the original $v$ and includes the original $uG$; in particular it has the same cardinality, as it did for the standard Wilson-Balding move. So only the choice of first node affects the Hastings ratio. It follows that this is the ratio from the standard Wilson-Balding move multiplied by $\frac{e_{\mathcal{T}}}{e_{\mathcal{T}'}}$, where $e_{\mathcal{T}}$ is the number of nodes eligible for this move before it occurs and $e_{\mathcal{T}'}$ is the number afterwards.
\subsubsection{Type B operators}
\paragraph{Type B exchange}
Select a random node $u$, not $r$, whose parent $uP$ is in a different element of $\mathcal{P}$ to itself. Pick a second node $v$, also not $r$ and not $uS$, whose parent $uP$ is also in a different element of $\mathcal{P}$ to itself (but this time the elements containing $uP$ and $vP$ do not have to be the same), such that $h(uP)>h(v)$, and $h(vP)>h(u)$. If there is no such $v$ then the operator fails. Otherwise, $u$ and $v$ exchange parents as with the type A operator. That it preserves connectedness of subtrees is clear. The effect on the transmission tree is that $\delta_\mathcal{P}(u)$ and $\delta_\mathcal{P}(v)$ exchange parents (if their parents are different).
The Hastings ratio is calculated in effectively the same way as for the type A version, noting that the number of choices for $u$ is just $N-1$. If $f_\mathcal{P}(w)$ is the number of eligible choices for a second node if $w$ is chosen first, then the ratio is $\frac{\frac{1}{f_{\mathcal{P}'}(u)}+\frac{1}{f_{\mathcal{P}'}(v)}}{\frac{1}{f_\mathcal{P}(u)}+\frac{1}{f_\mathcal{P}(v)}}$.
\paragraph{Type B subtree slide} This time, $u$ is a random node whose parent exists and is in a different element of $\mathcal{P}$ to itself. This implies that $uP$ is in the same element as either $uS$ or $uG$ (if the latter exists) because otherwise $uP$ would not be in a partition element containing a tip. The operator performs the standard subtree slide move \cite{hohna_clock-constrained_2008} on $u$, inserting $uP$ as the parent of another node $v$ and (if $v$ was not the root node), the child of $vP$. $\mathcal{P}$ is changed to a new partition $\mathcal{P}'$ as follows: if $vP$ does not exist or $v$ and $vP$ are in the same element of $\mathcal{P}$, $uP$ is moved to the element containing $v$. Otherwise, it is moved to either the element containing $v$ or that containing $vP$ with equal probability. This reallocation is enough to ensure that $\mathcal{P}'$ obeys connectedness rules. The effect on the transmission tree is that $\delta_\mathcal{P}(u)$ is moved to become a child of either $\delta_\mathcal{P}(v)$ or $\delta_\mathcal{P}(vP)$. If $\delta_\mathcal{P}(uS)\neq\delta_\mathcal{P}(uG)$ then $\delta_\mathcal{P}(uS)$ was the child of $\delta_\mathcal{P}(uG)$ before the move and remains so.
Noting that there are always $N-1$ choices for $u$, the Hastings ratio is the same as the standard subtree slide move, except that the denominator is multiplied by $\frac{1}{2}$ if $vP$ exists and $v$ and $vP$ are not in the same element of $\mathcal{P}$, and the numerator is multiplied by $\frac{1}{2}$ if $uG$ exists and $uG$ and $uS$ are not in the same element of $\mathcal{P}$.
\paragraph{Type B Wilson-Balding move}
In a similar way, $u$ is randomly picked from the set of nodes whose parents exist and are in different subtrees to themselves, and the standard Wilson-Balding move is performed on it, inserting $uP$ as a parent of another node $v$ and a child of its parent if that exists. The reassignment of $uP$ to a new subtree is performed in the same was as for type B subtree slide, and the adjustment to the Hastings ratio is identical. The effect on the transmission tree is also the same.
\subsection{Irreducibility of the chain}
Suppose $\mathcal{P}$ is a partition of a phylogeny $\mathcal{T}$ with root node $r$. First, notice the following about the infection branch operator described above:
\begin{itemize}
\item{For any $a_i\in\mathbf{A}$, if $\delta_\mathcal{P}(r)\neq a_i$, a series of downward moves, starting with one on $\epsilon_\mathcal{P}(a_i)$, eventually results in a new partition $\mathcal{P}'$ which has $\delta_{\mathcal{P}'}(r)=a_i$.}
\item{If $\delta_\mathcal{P}(r)=a_i$, a series of upward moves on $\epsilon_\mathcal{P}(a_j)$ for all $a_j\neq a_i$ will eventually give a partition $\mathcal{P}'$ in which $\delta_{\mathcal{P}'}(u)=a_i$ for all internal nodes $u$ of $\mathcal{T}$. As all such moves are reversible, we can get from $\mathcal{P}'$ to any partition $\mathcal{P}''$ that has $\delta_{\mathcal{P}''}(r)=a_i$.}
\end{itemize}
The above demonstrates that a MCMC chain made up of these moves on the space of partitions of a single phylogeny is irreducible. If $\mathcal{P}$ and $\mathcal{P}'$ are two partitions such that $\delta_\mathcal{P}(r)=\delta_{\mathcal{P}'}(r)$ then there is a series of moves taking $\mathcal{P}$ to $\mathcal{P}'$, and if $\delta_\mathcal{P}(r)\neq\delta_{\mathcal{P}'}(r)$ then there is a series of moves taking $\mathcal{P}$ to a partition $\mathcal{P}''$ that has $\delta_{\mathcal{P}''}(r)=\delta_{\mathcal{P}'}(r)$ and then a series of moves taking $\mathcal{P}''$ to $\mathcal{P}'$.
To extend this to a variable phylogenetic tree, we use the fact that in the space of standard, unpartitioned phylogenies, the Wilson-Balding move on its own is sufficient for irreducibility \cite{drummond_estimating_2002}. Suppose $\mathcal{P}$ is a partition of $\mathcal{T}$ such that $\delta_{\mathcal{P}'}(u)=\delta_\mathcal{P}(r)$ for all internal nodes $u$ of $\mathcal{T}$. Now every node of $\mathcal{T}$ is eligible to be the first node chosen by the type A Wilson-Balding move, as is true with the standard Wilson-Balding move on an unpartitioned tree, and subsequently, the set of nodes that is eligible to the the second node chosen is the same for both moves too. In addition, after this move, the new tree is still partitioned such that all internal nodes are in the same element of the partition. As a result, every move on an unpartitioned phylogeny that can be made by the standard move is also possible on the space of partitioned phylogenies with all internal nodes in the same partition element as a unique tip. Hence the chain is irreducible under the type A move when restricted to phylogenies with partitions of this type, and we have already shown that the infection branch operator is sufficient to move from a partition of this type to any other partition of the same tree. This is sufficient to establish irreducibility on the entire space of partitioned phylogenies using just these two moves.
\section{Bayesian decomposition}
Having established the correspondence between partitioned phylogenetic trees and transmission trees, we now show how the likelihood of such a partitioned phylogeny can be calculated given models of between-host transmission dynamics, of the duration of the infection within each host, of the population dynamics of the ``agents'' (which can be taken to be pathogens or infected individuals) within each host, and of sequence evolution.
In contrast to the previous work of Didelot et al. \cite{didelot_bayesian_2014}, whose underlying model of transmission was a compartmental SIR model, we use an individual-based model similar to those employed in previous work on agricultural outbreaks \cite{cottam_integrating_2008, morelli_bayesian_2012, ypma_unravelling_2011, ypma_relating_2013}. This much more readily allows for the accommodation of host heterogeneity, and makes no assumption of random mixing. Instead, the force of infection of a host $a_i$ on another $a_j$ is given by a basic transmission rate $\beta$ multiplied by a positive real number $d(h_1,h_2)$ from a function $d:\mathbf{A}\times\mathbf{A}\rightarrow[0,\infty)$ describing some relationship between $a_i$ and $a_j$. Possible choices for $d$ are a spatial kernel function, a network metric, or a function modifying $\beta$ based on shared membership in some class of host.
As in previous work \cite{ypma_relating_2013, didelot_bayesian_2014} we take the model of the dynamics of the ``agents'' to be a coalescent process amongst lineages in a freely-mixing population within each host. If the hosts are single organisms, the agents will naturally be individual pathogens. If, on the other hand, the hosts are infected locations, they could instead be considered to be infected organisms. In either case, only a miniscule proportion of the total agent population are represented by lineages in the tree, and the assumption of a low sampling fraction required for use of the coalescent process is satisfied.
We use the following notation:
\begin{itemize}
\item{The sequence data, $D$}
\item{The phylogenetic tree, $\mathcal{T}$}
\item{The transmission tree structure, $\mathcal{N}$}
\item{The set $\mathbf{T}^{\mathrm{inf}}$ of times of infection of each host}
\item{The times of sampling $\mathbf{T}^{\mathrm{exam}}$ of the sequence from each host}
\item{The times of becoming noninfectious $\mathbf{T}^{\mathrm{end}}$ of each host.}
\item{Data $L$ describing the relationship between hosts that is used to define the function $d$ (for example, spatial locations).}
\item{The basic transmission rate $\beta$.}
\item{The parameters $\phi$ of the distance function $d$.}
\item{The parameters $\psi$ of the population dynamics of the agents within each host.}
\item{The parameters $\omega$ of the nucleotide substitution model and molecular clock.}
\end{itemize}
We condition on $\mathbf{T}^{\mathrm{exam}}$, $\mathbf{T}^{\mathrm{end}}$, and $L$. We assume that $\mathbf{T}^{\mathrm{exam}}$ and $\mathbf{T}^{\mathrm{end}}$ are not contradictory; no sample was taken after a host became noninfectious. $\mathbf{T}^{\mathrm{end}}$ can be the same set of times as $\mathbf{T}^{\mathrm{exam}}$, or a separate set of later times. If any or all hosts are known to have remained infectious indefinitely, their values of $\mathbf{T}^{\mathrm{end}}$ can be set to the time at which the last sample was taken.
The posterior probability we are interested in calculating is $p(\mathcal{T}, \mathcal{N}, \mathbf{T}^{\mathrm{inf}}, \beta, \phi, \psi, \omega | D, \mathbf{T}^{\mathrm{exam}}, \mathbf{T}^{\mathrm{end}}, L)$. By Bayes' Theorem this is equal to:
\begin{eqnarray*}
\frac{p(D|\mathcal{T}, \mathcal{N}, \mathbf{T}^{\mathrm{inf}}, \beta, \phi, \psi, \omega, \mathbf{T}^{\mathrm{exam}}, \mathbf{T}^{\mathrm{end}}, L)p(\mathcal{T}, \mathcal{N}, \mathbf{T}^{\mathrm{inf}}, \beta, \phi, \psi, \omega, \mathbf{T}^{\mathrm{exam}}, \mathbf{T}^{\mathrm{end}}, L)}{p(D|\mathbf{T}^{\mathrm{exam}}, \mathbf{T}^{\mathrm{end}}, L)}
\end{eqnarray*}
As usual, we need not calculate the denominator if we are uninterested in model comparison as it does not vary. We assume that mutations occur neutrally over the the phylogenetic tree in a process that ignores the host structure, so $D$ depends only on $\mathcal{T}$ and $\omega$ and the likelihood reduces to $p(D|\mathcal{T},\omega)$, which can be calculated using the Felsenstein pruning algorithm and a molecular clock model in the normal way \cite{felsenstein_evolutionary_1981, drummond_estimating_2002, drummond_relaxed_2006}. It remains to calculate the prior probability $p(\mathcal{T}, \mathcal{N}, \mathbf{T}^{\mathrm{inf}}, \phi, \psi, \omega, \mathbf{T}^{\mathrm{exam}}, \mathbf{T}^{\mathrm{end}}, L)$. The full decomposition is as follows:
\begin{eqnarray*}
p(\mathcal{T}, \mathcal{N}, \mathbf{T}^{\mathrm{inf}}, \phi, \psi, \omega, \mathbf{T}^{\mathrm{exam}}, \mathbf{T}^{\mathrm{end}}, L) &=& p(\beta|\mathcal{T}, \mathcal{N}, \mathbf{T}^{\mathrm{inf}}, \phi, \psi, \omega, \mathbf{T}^{\mathrm{exam}}, \mathbf{T}^{\mathrm{end}}, L)\\
&& \times p(\mathcal{T} | \mathcal{N}, \mathbf{T}^{\mathrm{inf}}, \phi, \psi, \omega, \mathbf{T}^{\mathrm{exam}}, \mathbf{T}^{\mathrm{end}}, L)\\
&& \times p(\mathcal{N} | \mathbf{T}^{\mathrm{inf}}, \phi, \psi, \omega, \mathbf{T}^{\mathrm{exam}}, \mathbf{T}^{\mathrm{end}}, L)\\
&& \times p(\mathbf{T}^{\mathrm{inf}} | \phi, \psi, \omega, \mathbf{T}^{\mathrm{exam}}, \mathbf{T}^{\mathrm{end}}, L)\\
&& \times p(\phi, \psi, \omega | \mathbf{T}^{\mathrm{exam}}, \mathbf{T}^{\mathrm{end}}, L)
\end{eqnarray*}
The following assumptions of independence are then made:
\begin{itemize}
\item{All parameters in the decomposition are independent of $\omega$.}
\item{$\beta$, the base transmission rate, is (at least) conditionally independent of $\mathcal{T}$ and $\psi$ given $\phi$, $\mathcal{N}$, $\mathbf{T}^{\mathrm{exam}}$, $\mathbf{T}^{\mathrm{inf}}$, $\mathbf{T}^{\mathrm{end}}$, and $L$. This is intuitive given that the latter set of parameters completely describe the epidemic and the distance-based modification of $\beta$.}
\item{$\mathcal{T}$, the phylogenetic tree, is (at least) conditionally independent of $\phi$, $\mathbf{T}^{\mathrm{end}}$, and $L$ given $\psi$, $\mathcal{N}$, $\mathbf{T}^{\mathrm{inf}}$, and $\mathbf{T}^{\mathrm{exam}}$.}
\item{$\mathcal{N}$, the transmission tree structure, is (at least) conditionally independent of $\mathbf{T}^{\mathrm{exam}}$ and $\psi$ given $\phi$, $\mathbf{T}^{\mathrm{inf}}$, $\mathbf{T}^{\mathrm{end}}$ and $L$.}
\item{$\mathbf{T}^{\mathrm{inf}}$, the times of infection, is (at least) conditionally independent of $\phi$, $\mathbf{T}^{\mathrm{exam}}$, $\psi$ and $L$ given $\mathbf{T}^{\mathrm{end}}$.}
\item{$\phi$, $\psi$ and $\omega$ are independent of $\mathbf{T}^{\mathrm{inf}}$, $\mathbf{T}^{\mathrm{end}}$, $L$, and each other.}
\end{itemize}
The decomposition then reduces to:
\begin{eqnarray*}
p(\mathcal{T}, \mathcal{N}, \mathbf{T}^{\mathrm{inf}}, \phi, \psi, \omega, \mathbf{T}^{\mathrm{exam}}, \mathbf{T}^{\mathrm{end}}, L) &=& p(\beta|\mathcal{N}, \mathbf{T}^{\mathrm{inf}}, \phi, \mathbf{T}^{\mathrm{exam}}, \mathbf{T}^{\mathrm{end}}, L)\\
&& \times p(\mathcal{T} | \mathcal{N}, \mathbf{T}^{\mathrm{inf}}, \psi, \mathbf{T}^{\mathrm{exam}})\\
&& \times p(\mathcal{N} | \mathbf{T}^{\mathrm{inf}}, \phi, \mathbf{T}^{\mathrm{end}}, L)\\
&& \times p(\mathbf{T}^{\mathrm{inf}} | \mathbf{T}^{\mathrm{end}})\\
&& \times p(\phi)p(\psi)p(\omega)
\end{eqnarray*}
For calculation of $p(\beta|\mathcal{N}, \mathbf{T}^{\mathrm{inf}}, \phi, \mathbf{T}^{\mathrm{exam}}, \mathbf{T}^{\mathrm{end}}, L)$, we use Bayes' Theorem again:
\begin{eqnarray*}
p(\beta|\mathcal{N}, \mathbf{T}^{\mathrm{inf}}, \phi, \mathbf{T}^{\mathrm{exam}}, \mathbf{T}^{\mathrm{end}}, L)&=&\frac{p(\mathcal{N}, \mathbf{T}^{\mathrm{inf}}|\beta, \phi, \mathbf{T}^{\mathrm{exam}}, \mathbf{T}^{\mathrm{end}}, L)p(\beta |\phi, \mathbf{T}^{\mathrm{exam}}, \mathbf{T}^{\mathrm{end}}, L)}{p(\mathcal{N},\mathbf{T}^{\mathrm{inf}}|\phi,\mathbf{T}^{\mathrm{exam}}, \mathbf{T}^{\mathrm{end}}, L)}
\end{eqnarray*}
The denominator $p(\mathcal{N},\mathbf{T}^{\mathrm{inf}}|\phi,\mathbf{T}^{\mathrm{exam}}, \mathbf{T}^{\mathrm{end}}, L)$ can be evaluated as
\begin{eqnarray*}
\int_\beta p(\mathcal{N}, \mathbf{T}^{\mathrm{inf}}|\beta, \phi, \mathbf{T}^{\mathrm{exam}}, \mathbf{T}^{\mathrm{end}}, L)p(\beta |\phi, \mathbf{T}^{\mathrm{exam}}, \mathbf{T}^{\mathrm{end}}, L)d\beta
\end{eqnarray*}
by the law of total probability. This will not in general have a closed form solution and we use numerical integration to estimate it. The term $p(\beta |\phi,\mathbf{T}^{\mathrm{exam}}, \mathbf{T}^{\mathrm{end}}, L)$ is our prior belief in the value of $\beta$ given $\phi$ and the background information; in the absence of other information we take $\beta$ to be independent of these and simply give it any prior distribution $p(\beta)$ that we please.
It remains to calculate $p(\mathcal{N}, \mathbf{T}^{\mathrm{inf}}|\beta, \phi, \mathbf{T}^{\mathrm{exam}}, \mathbf{T}^{\mathrm{end}}, L)$. The calculation here is along the same lines of that introduced by Gibson and Austin \cite{gibson_fitting_1996}, but heavily modified. Given a particular set of $\mathbf{T}^{\mathrm{inf}}$, we reorder the indexes of $\mathbf{A}$ to be in increasing order of infection. As before, the infection time of $a_i$ is $t^{\mathrm{inf}}_i$. The probability that $a_1$ was infected at time $t^{\mathrm{inf}}_1$ given that it was first in the epidemic is effectively unknowable and we set it to 1. For notational simplicity now treating $\mathcal{N}$ as a map from the index of a case to the index of its infector, we need the probability that $a_{\mathcal{N}(i)}$ infected $a_i$ at $t^{\mathrm{inf}}_i$, which is made up of:
\begin{itemize}
\item{The probability that $a_{\mathcal{N}(i)}$ infected $a_i$ at $t^{\mathrm{inf}}_i$, but not before:
\begin{eqnarray*}
\beta d(a_i,a_{\mathcal{N}(i)})\times\mathrm{exp}\left(-\beta d(a_i,a_{\mathcal{N}(i)})(t^{\mathrm{inf}}_i-t^{\mathrm{inf}}_{\mathcal{N}(i)})\right)
\end{eqnarray*}
}
\item{The probability that no other host infected $a_i$ before $t^{\mathrm{inf}}_i$. As we assume no reinfection, infection events that would occur after this time are ignored. Noting that the last possible time that an $a_j$ could have infected $a_i$ for this to be true is the smaller of $t^{\mathrm{inf}}_i$ and the end of of $a_j$'s infectiousness, $t^{\mathrm{end}}_j$, this is given by:
\begin{eqnarray*}
\prod_{j\in\{1,\ldots,i-1\}\setminus \mathcal{N}(i)}\mathrm{exp}\left({-\beta d(a_i,a_j)(\textrm{min}\{t^{\mathrm{inf}}_i,t^{\mathrm{end}}_j\}-t^{\mathrm{inf}}_j)}\right)
\end{eqnarray*}
}
\item{As we are conditioning on $\mathbf{T}^{\mathrm{exam}}$ and $\mathbf{T}^{\mathrm{end}}$, we implicitly assume that each $a_i$ was, in fact, infected and was infected before its time of sampling $t^{\mathrm{exam}}_i$. As a result, we need to normalise by the probability that an infection did happen before this date, which is one minus the probability that none did. The last possible time that an $a_j$ could have infected $a_i$ at all is the smaller of $t^{\mathrm{exam}}_i$ and $t^{\mathrm{end}}_j$, so this expression is:
\begin{eqnarray*}
1-\prod_{\substack{j\in\{1,\ldots,N\}\\t^{\mathrm{inf}}_j<t^{\mathrm{exam}}_i}} \mathrm{exp}\left({-\beta d(a_i,a_j)(\textrm{min}\{t^{\mathrm{exam}}_i,t^{\mathrm{end}}_j\}-t^{\mathrm{inf}}_j)}\right)
\end{eqnarray*}
}
\end{itemize}
Thus the full expression for $p(\mathcal{N}, \mathbf{T}^{\mathrm{inf}}|\beta, \phi, \mathbf{T}^{\mathrm{exam}}, \mathbf{T}^{\mathrm{end}}, L)$ is:
\begin{eqnarray*}
\mathlarger{\prod}_{i\in\{2,\ldots,N\}}\left(\frac{\beta d(a_i,a_{\mathcal{N}(i)})\prod_{j\in\{1,\ldots,i-1\}}\mathrm{exp}\left(-\beta d(a_i,a_j)(\textrm{min}\{t^{\mathrm{inf}}_i,t^{\mathrm{end}}_j\}-t^{\mathrm{inf}}_j)\right)}{1-\prod_{\substack{j\in\{1,\ldots,N\}\\t^{\mathrm{inf}}_j<t^{\mathrm{exam}}_i}} \textrm{exp}\left({-\beta d(a_i,a_j)(\textrm{min}\{t^{\mathrm{exam}}_i,t^{\mathrm{end}}_j\}-t^{\mathrm{inf}}_j)}\right)}\right)
\end{eqnarray*}
If one of the $d(a_i,a_j)$ terms in this expression is zero, then the whole thing is zero, representing an impossible transmission history. Otherwise this is undefined for $\beta=0$ but exists and is positive for all other $\beta\in(0,\infty)$, as the denominator is always greater than zero. Let this expression, as a function of $\beta$ alone with all other variables constant, be $I(\beta)$. The integral $\int_0^\infty I(\beta)p(\beta)d\beta$, where $p(\beta)$ is the prior probability of $\beta$, can estimated by numerical methods if we show that it is in fact finite.
\begin{proposition}
Let $a,b\in(0,\infty)$. The improper integral $\int_a^b I(\beta)d\beta$ converges as $a\to0$ and $b\to\infty$.
\end{proposition}
\begin{proof}
For the lower limit, we use:
\begin{lemma}\label{lop}
Suppose $A$ and $B$ are positive real numbers. Then:
\begin{eqnarray*}
\lim_{x\to0} \frac{xe^{-Ax}}{1-e^{-Bx}}=\frac{1}{B}
\end{eqnarray*}
\end{lemma}
\begin{proof}
Let $f(x)=xe^{-Ax}$ and $g(x)=1-e^{-Bx}$. Then $f'(x)=(1-Ax)e^{-Ax}$ and $g'(x)=Be^{-Bx}$. Hence:
\begin{eqnarray*}
\frac{f'(x)}{g'(x)}&=&\frac{(1-Ax)e^{-Ax}}{Be^{-Bx}}\\
&=& \frac{1-Ax}{B}e^{(B-A)x}\\
\end{eqnarray*}
This shows that $\lim_{x\to0}f'(x)/g'(x)=1/B$ and the result follows by l'H\^{o}pital's rule.
\end{proof}
Lemma~\ref{lop} shows that each individual term in the product that makes up $I(\beta)$ does not have 0 as an asymptote, hence they are all bounded on $(0,a]$ as they clearly have no others. Hence, on this interval, $I(\beta)$, as the product of bounded functions, is bounded and the integral converges.
For the upper limit, we can write:
\begin{eqnarray*}
I(\beta) = \frac{A\beta^{n-1}\textrm{exp }(-B\beta)}{\prod_{i=2}^N(1-\textrm{exp }(-C_i\beta))}\\
\end{eqnarray*}
where $A$, $B$ and each $C_i$ is a positive real number. If we let $J(\beta)=A\beta^{n-1}\textrm{exp }(-B\beta)$ then $J(\beta)/I(\beta)=\prod_{i=2}^N(1-\textrm{exp }(-C_i\beta))$ whose limit as $\beta\to\infty$ is 1. The limit comparison test then says that $\int_a^bI(\beta)d\beta$ converges as $b\to\infty$ if and only if $\int_a^bJ(\beta)d\beta$ does. Recursive integration by parts gives:
\begin{eqnarray*}
\int_a^bJ(\beta)d\beta = \left[ \frac{A}{B}\left(\sum_{k=0}^{n-1}\left(\frac{-1}{B}\right)^{n-1-k}\beta^{k}\right)\textrm{exp }(-B\beta)\right]_a^b\\
\end{eqnarray*}
$\int_a^\infty J(\beta)d\beta = \lim_{b\to\infty}\int_a^b J(\beta)d\beta$, and $\int_a^\infty J(\beta)d\beta$ can thus be expressed as a constant expression involving $a$, plus the sum of $n$ limits of the form $\lim_{\beta\to\infty}D\beta^k\textrm{exp }(-B\beta)$ where $D$ is a constant and $k\in\mathbb{N}$. It is a standard result that each of these is 0. Hence $J(\beta)$ converges and so does $I(\beta)$.
\end{proof}
\begin{corollary}
Let $a,b\in(0,\infty)$. If $p(\beta)$ is a proper prior distribution whose support is a subset of $(0,\infty)$, the improper integral $\int_a^b I(\beta)p(\beta)d\beta$ converges as $a\to0$ and $b\to\infty$.
\end{corollary}
\begin{proof}
If $p(\beta)$ has finite support, then $I(\beta)p(\beta)$ is bounded on a finite interval and zero elsewhere, and the intergral of such a function must converge. If not, then use of the limit comparison test with numerator $I(\beta)p(\beta)$ and denominator $I(\beta)$ gives that, because $\lim_{\beta\to\infty}p(\beta)=0$, $\int_a^b I(\beta)p(\beta)d\beta$ converges if $\int_a^b I(\beta)d\beta$ does, and we know this to be true.
\end{proof}
\begin{remark}
Notice that if $p(\beta)$ is, for example a uniform infinite improper prior or a gamma distribution, $I(\beta)p(\beta)$ takes the form $D\beta^k\textrm{exp }(-E\beta)f(\beta)$ for a function $f$ where $D$ and $E$ are positive constants and $k>1$. Generalised Gauss-Laguerre quadrature is therefore a natural choice for the estimation of $\int_0^\infty I(\beta)p(\beta)d\beta$ for such a $p(\beta)$.
\end{remark}
Next, we need to calculate $p(\mathcal{T} | \mathcal{N}, \mathbf{T}^{\mathrm{inf}}, \psi, \mathbf{T}^{\mathrm{exam}})$. We extend the procedure outlined by Didelot et al\cite{didelot_bayesian_2014} to allow for the use of any of the standard models of deterministic population growth, and the possibility of host heterogeneity. The latter is accomplished by dividing the set of hosts into categories and assigning a separate demographic model to all the hosts in each one. Categories can be assigned from known epidemiological data about the hosts; for example, in a livestock disease outbreak, they may reflect the size of farm. Formally, let $\mathbf{C}^{\textrm{coal}}$, a finite set of size $p$, be the set of categories, and $cc:\{1,\ldots,N\}\to\mathbf{C}^{\textrm{coal}}$ the map assigning them to the index of each host in $A$. If it is not desired to accommodate heterogeneity in this way, $p$ can be 1. Every element $\mathbf{c}\in\mathbf{C}^{\textrm{coal}}$ corresponds to a separate demographic function $N_\mathbf{c}:\mathbb{R}\rightarrow[0,\infty)$ with parameters $\psi_\mathbf{c}$ where $N_\mathbf{c}(t)$ is the product of the effective population size and the generation time at time $t$.
Given a host $a_i\in\mathbf{A}$ which is infected at time $t^{\mathrm{inf}}_i$, sampled at time $t^{\mathrm{exam}}_i$ and ceases to be infectious at time $t^{\mathrm{end}}_i$, and has $n$ children $a_{o(1)},\ldots,a_{o(n)}$ (for some permutation $o$ of $\{1,\ldots,N\}$) infected at times $t^{\mathrm{inf}}_{o(1)},\ldots,t^{\mathrm{inf}}_{o(n)}$, suppose $\mathcal{S}_i$ is a phylogenetic tree that describes the part of the outbreak that took place within $a_i$. It has has $n+1$ tips, one for each infection event and one for its own sampling event. If $m=\textrm{max }\{t^{\mathrm{inf}}_{o(1)},\ldots,t^{\mathrm{inf}}_{o(n)},t^{\mathrm{exam}}_{i}\}$, the height (in the tree $\mathcal{S}_i$) $h_i(r)$ of its root node $r$ is less than $m-t^{\mathrm{inf}}_i$ and we can give it a root branch of length $m-h_i(r)-t^{\mathrm{inf}}_i$. If we have a $\mathcal{S}_i$ for each $h$, and we know $\mathcal{N}$, we can build a phylogenetic tree for the entire epidemic by attaching the root node of each $\mathcal{S}_i$ to the tip of $\mathcal{S}_{\mathcal{N}(i)}$ that corresponds to the infection of $a_i$, by a branch with length equal to the root branch length of $\mathcal{S}_i$. If $\mathcal{T}$ cannot be built up from $\mathcal{S}_i$s in this way, $p(\mathcal{T} | \mathcal{N}, \mathbf{T}^{\mathrm{inf}}, \psi, \mathbf{T}^{\mathrm{exam}})=0$. Otherwise, we calculate it as:
\begin{eqnarray*}
p(\mathcal{T} | \mathcal{N}, \mathbf{T}^{\mathrm{inf}}, \psi, \mathbf{T}^{\mathrm{exam}})= \prod_{i\in\{1,\ldots,N\}}p(\mathcal{S}_{i}|\psi_{cc(i)})
\end{eqnarray*}
In the standard coalescent model \cite{slatkin_pairwise_1991}, the probability density function for the for the time $t$ of the first coalescence of $K\geq2$ lineages after $t_0$ where the demographic function is $N_\mathbf{c}$ is given by:
\begin{eqnarray*}
p(t)&=&\frac{K(K-1)}{2N_\mathbf{c}(t)}\textrm{ exp}\left(-\int_{t_0}^t\frac{K(K-1)}{2N_\mathbf{c}(s)}ds\right)\\
\end{eqnarray*}
and as usual, if we know the two specific lineages that converged, the $K(K-1)/2$ cancels.
The cumulative density function of this is:
\begin{eqnarray*}
P(t)&=&\int_{t_0}^t\frac{K(K-1)}{2N_\mathbf{c}(r)}\textrm{ exp}\left(-\int_{t_0}^r\frac{K(K-1)}{2N_\mathbf{c}(s)}ds\right)dr\\
&=&1-\textrm{exp}\left(-\int_{t_0}^t\frac{K(K-1)}{2N_\mathbf{c}(s)}ds\right)
\end{eqnarray*}
and the probability that there were no coalescences between $t_0$ and $t$ is 1 minus this.
As Didelot et al. \cite{didelot_bayesian_2014} note, this is not quite sufficient for our purposes because we have a maximum height for the last coalescence. If this is $t_{\mathrm{max}}$, the normalised probability distribution for the time of first coalescence is:
\begin{eqnarray*}
p(t|T) = \begin{cases} \frac{\frac{K(K-1)}{2N_\mathbf{c}(t)}\textrm{exp}\left(-\int_{t_0}^t\frac{K(K-1)}{2N_\mathbf{c}(s)}ds\right)}{1-\textrm{exp}\left(-\int_{t_0}^{t_{\mathrm{max}}}\frac{K(K-1)}{2N_\mathbf{c}(s)}ds\right)} & t_0\leq t<t_{\mathrm{max}}\\
0 & \mbox{otherwise}
\end{cases}
\end{eqnarray*}
This is the probability of an interval in $\mathcal{S}_h$ ending in a coalescent event. The probability of an interval ending in a transmission or sampling event is the probability that no events occur in the interval, which is one minus the cumulative distribution function of the above, $P(t|T)$:
\begin{eqnarray*}
1-P(t|T)&=&1-\frac{1-\textrm{exp}\left(-\int_{t_0}^t\frac{K(K-1)}{2N_\mathbf{c}(s)}ds\right)}{1-\textrm{exp}\left(-\int_{t_0}^{t_{\mathrm{max}}}\frac{K(K-1)}{2N_\mathbf{c}(s)}ds\right)}\\
&=&\frac{\textrm{exp}\left(-\int_{t_0}^t\frac{K(K-1)}{2N_\mathbf{c}(s)}ds\right)-\textrm{exp}\left(-\int_{t_0}^{t_{\mathrm{max}}}\frac{K(K-1)}{2N_\mathbf{c}(s)}ds\right)}{1-\textrm{exp}\left(-\int_{t_0}^{t_{\mathrm{max}}}\frac{K(K-1)}{2N_\mathbf{c}(s)}ds\right)}\\
\end{eqnarray*}
Note that while in the case of no maximum root height, the formula happens to work for $K=1$, here it does not as the denominator is 0, and we instead set the probability of any coalescent interval with one lineage to 1. In particular, if $a_i$ has no children then $p(\mathcal{S}_{i}|\psi_{cc(i)})=1$.
If $t_{\mathrm{max}}=m-t^{\mathrm{inf}}_i$, these formulae can be used to calculate $p(\mathcal{S}_i|\psi_{cc(i)})$ for every $\mathcal{S}_{i}$ in the established way for a tree with temporally offset tips \cite{drummond_estimating_2002}, and the product of these is the full probability of the complete phylogeny. It is most intuitive to standardise the timescale within each $\mathcal{S}_{i}$ such that the effective population size at the point of the infection (the maximum root height) is the same across all hosts. As a result, we depart from the normal convention of making height 0 the time of the last tip (which will occur at a different point in the course of infection in different hosts), and instead put it at the point of infection, with all later events occurring at negative heights.
The choice of each demographic function $N_\mathbf{c}$ is wide. For an epidemic situation, exponential or logistic growth \cite{slatkin_pairwise_1991, pybus_epidemic_2001} would be most appropriate. Different categories $\mathbf{c}\in\mathbf{C}$ may be assigned the same family of demographic model but a different set of parameters $\psi_\mathbf{c}$. As our method is integrated within BEAST, any of the functions already implemented in that package can be used without additional programming work.
The next term in the decomposition is $p(\mathcal{N} | \mathbf{T}^{\mathrm{inf}}, \phi, \mathbf{T}^{\mathrm{end}}, L)$. The instantaneous probability that host $a_i$ was infected by host $a_{\mathcal{N}(i)}$ at time $t^{\mathrm{inf}}_i$ is $\beta d(a_i,a_{\mathcal{N}(i)})$, and if we condition on the fact that $a_i$ was indeed infected by \emph{some} host at $t^{\mathrm{inf}}_i$ then we normalise by the sum $\sum_{a_j\in\mathbf{A}_i}\beta d(a_i,a_j)$ where $\mathbf{A}_i$ is the subset of $\mathbf{A}$ whose elements have infection times before $t^{\mathrm{inf}}_i$ and noninfectiousness times after it. In this normalisation the $\beta$s cancel, leaving an expression solely in terms of the distance function. The probability of the infection of the first host $a_1$ is once again set to 1. The expression is:
\begin{eqnarray*}
p(\mathcal{N} | \mathbf{T}^{\mathrm{inf}}, \phi, \mathbf{T}^{\mathrm{end}}, L) = \prod_{a_i\in\mathbf{A}\setminus a_1}\frac{d(a_i,a_{\mathcal{N}(i)})}{\sum_{a_j\in\mathbf{A}_i}d(a_i,a_j)}
\end{eqnarray*}
There are many possible choices for the function $d$. If we assume no spatial structure or heterogeneity then we can just take $d(a_i,a_j)=1$ for all $a_i,a_j\in\mathbf{A}$. Otherwise, it can be based on Euclidean distance, or on a network metric. It can also be used to state prior information about the transmission tree structure; if it is known \emph{a priori} that $a_i$ did not infect $a_j$, then $d(a_i,a_j)$ can be set to zero. While we have assumed it up to this point, there is also no requirement that $d$ be symmetric.
The calculation of $p(\mathbf{T}^{\mathrm{inf}} | \mathbf{T}^{\mathrm{end}})$, the probability of the times of infection, can be handled in a number of ways. It is effectively the calculation of the probability of the time from infection to noninfectiousness, $t^{\mathrm{end}}_i - t^{\mathrm{inf}}_i$, of each host $a_i$. Previous work on foot-and-mouth disease virus \cite{cottam_integrating_2008, morelli_bayesian_2012} has used clinical data to estimate times of infection, and if this kind of information is available, it can be used to determine a separate prior distribution for each $t^{\mathrm{end}}_i - t^{\mathrm{inf}}_i$. If we cannot use information of this type, we take a similar approach to that in the coalescent calculations above and assign each host $a_i$ to a category $ic(a_i)$ from a finite set $\mathbf{C}^{\mathrm{inf}}$ of size $q$. This again allows us to accommodate known host heterogeneity; for example in an agricultural outbreak it is likely that times from infection to noninfectiousness decrease as time goes by and control measures are brought to bear. Once again, if we do not want to incorporate such heterogeneity we can set $q=1$. If the infectious period of the disease is well understood, we can assign a single prior distribution for $t^{\mathrm{end}}_i - t^{\mathrm{inf}}_i$ for all hosts in each category.
It may be, however, that we want to estimate the distribution of infectious periods from the genetic data. In this case we take each $t^{\mathrm{end}}_i - t^{\mathrm{inf}}_i$ within a category as a draw from a probability distribution with unknown parameters, and then put hyperpriors on those parameters. A noninformative option is to regard each as a draw from an unknown normal distribution whose mean we are uninterested in (as we can just as well calculate the mean of the sampled values of each infectious period post-hoc) and use the Jeffreys prior on its standard deviation, such that $p(\mathbf{T}^{\mathrm{inf}} | \mathbf{T}^{\mathrm{end}})$ is proportional to the reciprocal of the standard deviation of all the infectious periods in the category. This can obviously be done on the logarithm of the infectious periods instead, if we prefer the assumption that they are lognormally distributed. Alternatively, we can use an informative prior. To avoid having to use MCMC to estimate both the parameters $\chi$ of a probability distribution $D$ and a series of draws from that distribution, we integrate out the actual values of $\chi$ by using $D$'s conjugate prior for them and then calculating the marginal likelihood of the infectious periods given the hyperpriors. Any continuous probability distribution with a prior whose marginal likelihood is analytically tractable can be considered. A normal distribution is not absolutely ideal as infectious periods are non-negative parameters, but it does have the useful property that its mean and variance are independent, unlike most other candidates for $D$ (such as lognormal, exponential or gamma). We suggest it still be considered as an option if infectious periods are expected to be sufficiently long, and their variance sufficiently small, that the probability density contained in the area less than 0 would negligible if a normal distribution were used.
Finally, all that remains is to place prior distributions on the parameters making up $\phi$, $\psi$, and $\omega$.
\subsection*{Latent periods}
The above formulation has taken the course of infection to follow a SIR structure; hosts are assumed to be infectious as soon as they are infected. It is straightforward to replace this with a SEIR structure instead. We add an extra set of parameters $\mathbf{T}^{\textrm{trans}}$ consisting of the time of infectiousness $t^{\textrm{trans}}_i$ of each host $a_i\in\mathbf{A}$. In the MCMC procedure these are calculated by adding a $r_{i}\in[0,1]$ such that, if $t^{\textrm{maxtrans}}_i=\textrm{min }(\{t^{\textrm{end}}_i\}\cup\{t^{\textrm{inf}}_j : \mathcal{N}(j)=i\})$ ($t^{\textrm{maxtrans}}_i$ being the upper bound on $t^{\textrm{trans}}_i$ determined by $\mathbf{T}^{\textrm{inf}}$ and $\mathbf{T}^{\textrm{end}}$), then $t^{\textrm{trans}}_i = t^{\textrm{inf}}_i + r_i(t^{\textrm{maxtrans}}_{i}-t^{\textrm{inf}})$. (We assume that hosts are infectious at the time they cease to be infected, but not that they necessarily are at the time of sampling.) Simple MCMC moves on numerical parameters are then employed to sample values of each $r_{i}$. The phylogeny $\mathcal{T}$ is assumed to be conditionally independent of $\mathbf{T}^{\textrm{trans}}$ given $\mathbf{T}^{\textrm{inf}}$.
The decomposition becomes:
\begin{eqnarray*}
p(\mathcal{T}, \mathcal{N}, \mathbf{T}^{\mathrm{inf}}, \mathbf{T}^{\textrm{trans}}, \phi, \psi, \omega, \mathbf{T}^{\mathrm{exam}}, \mathbf{T}^{\mathrm{end}}, L) &=& p(\beta|\mathcal{N}, \mathbf{T}^{\mathrm{inf}}, \mathbf{T}^{\textrm{trans}}, \phi, \mathbf{T}^{\mathrm{exam}}, \mathbf{T}^{\mathrm{end}}, L)\\
&& \times p(\mathcal{T} | \mathcal{N}, \mathbf{T}^{\mathrm{inf}}, \psi, \mathbf{T}^{\mathrm{exam}})\\
&& \times p(\mathcal{N} | \mathbf{T}^{\mathrm{inf}}, \mathbf{T}^{\textrm{trans}}, \phi, \mathbf{T}^{\mathrm{end}}, L)\\
&& \times p(\mathbf{T}^{\mathrm{inf}}, \mathbf{T}^{\textrm{trans}} | \mathbf{T}^{\mathrm{end}})\\
&& \times p(\phi)p(\psi)p(\omega)
\end{eqnarray*}
The only modifications to the existing procedure that are needed to calculate the first and third elements in this product involve accounting for the fact that infectious pressure is now only applied after the end of a host's latent period. The term $p(\mathbf{T}^{\mathrm{inf}}, \mathbf{T}^{\textrm{trans}} | \mathbf{T}^{\mathrm{end}})$ is calculated by retaining the existing procedure for infectious periods and also picking a suitable distribution, or possibly set of distributions according to a set $\mathbf{C}^{\mathrm{lat}}$ of categories, for latent periods. The marginal likelihood of the set of infectious periods $\{t^{\mathrm{end}}_i - t^{\mathrm{trans}}_i|i\in\{1,\ldots,N\}\}$ and latent periods $\{t^{\mathrm{trans}}_i - t^{\mathrm{inf}}_i|i\in\{1,\ldots,N\}\}$ is then calculated as before.
\section{Conclusions and future work}
The most obvious limitation to the method outlined here is the requirement that the tree contain a tip from every host involved in the outbreak. Note that this is not, in fact, a requirement that a sequence be taken from every host, as, if we have no sample from some but are aware of their existence, we can use epidemiological data for them along with a noninformative sequence (consisting entirely of the nucleotide code `N') and integrate over their unknown true sequences. Their existence would then still contribute to estimation of epidemiological parameters, and their estimated placement in the transmission tree would be informed by their geographical locations if those were included in the model. The performance of this procedure for varying numbers of unknown sequences warrants investigation in simulations. Nevertheless, this is a solution only where all unsampled hosts are actually known to investigators, which will not always be the case. Non-phylogenetic methods for transmission tree reconstruction using genetic data have started to consider the case of unsampled hosts \cite{jombart_bayesian_2014,mollentze_bayesian_2014} and work is needed to introduce this element to our framework. This could be done by introducing a variable number of partitions containing no tips, possibly in a reversable-jump MCMC framework, or by simply assigning tree nodes to ``nonsampled'' subtrees with no specific enumeration of how many extra hosts these represent.
Another enhancement, potentially useful in HIV studies where multiple samples are taken from the same patient over time, would be to relax the partition rules to allow each subtree to contain more than one tip. The adjustments to the method needed to accomplish this are likely considerably less onerous than allowing for unsampled hosts.
In conclusion, in this document we have outlined the framework for co-estimation of transmission trees as part of an analysis performed in one of the most widely-used software packages for phylogeny reconstruction. Results from analyses of both simulated and real data will follow. It is, as of the time of writing, implemented in current development builds of BEAST.
\section{Acknowledgements}
We would like to thank Trevor Bedford, Samantha Lycett and Melissa Ward for contributions to the development of this model. MH was supported by a PhD studentship from the Scottish Government-funded EPIC programme, and the research leading to these results has received funding from the European Union Seventh Framework Programme for research, technological development and demonstration under Grant Agreement no. 278433-PREDEMICS
\begin{figure*}
\centering
\includegraphics[width=16.0cm]{extended_painting_drawing.pdf}
\caption{The five compatible transmission tree structures of a phylogenetic tree with three tips, depicted as partitions of the phylogeny (above) and as directed graphs amongst the hosts A B and C (below)}\label{sameoldsameold}
\end{figure*}
\begin{figure*}
\centering
\includegraphics[width=16.0cm]{4_partitions.pdf}
\caption{Above: the twelve valid partitions of the phylogeny ((A,B),(C,D)). Below: the thirteen valid partitions of the phylogeny (A,(B,(C,D)))}\label{4partitions}
\end{figure*}
\begin{figure*}
\centering
\includegraphics[width=18.0cm]{ttoperator.pdf}
\caption{Illustration of the effects of the infection branch operator on the partition $\mathcal{P}$ of a phylogeny of samples from the set of hosts A-I, and corresponding effects on the transmission tree. 1. Original partition. 2. Downward move on B. 3. Upward move on B. 4. Downward move on C. 5. Downward move on D. 6. Downward move on E. 7. Downward move on F. 8. Upward move on F. 9. Downward move on G. 10. Downward move on H. 11. Upward move on H. 12. Downward move on I.}\label{ttoperator}
\end{figure*}
\newpage
|
1,108,101,564,542 | arxiv | \section{Conclusion}
\label{sec:con}
RecSys-DAN is a novel framework for cross-domain collaborative filtering,
particularly, the real-world cold-start recommendation problem. It learns to
adapt the user, item and user-item interaction representations from a source
domain to a target domain in an unsupervised and adversarial fashion.
Multiple generators and discriminators are designed to adversarially learn
target generators for generating domain-invariant representations. Four
RecSys-DAN instances, namely, UI-DAN, U-DAN, I-DAN, and H-DAN, are explored
by considering different scenarios characterized by the overlap of users and
items in both unimodal and multimodal settings. Experimental results
demonstrates that RecSys-DAN has a competitive performance compared to
state-of-the-art supervised methods for the rating prediction task, even with
absent preference information.
\section{Introduction}
\label{sec:intro}
\IEEEPARstart{R}{ecommender} systems (RS) generate predictions based on the
customers' preferences and purchasing histories. Collaborative filtering
(CF) and content-based filtering
(CBF) are popular techniques used in
such systems~\cite{koren2009matrix}. CF-based methods generate recommendations by computing latent
representations of users and products with matrix factorization (MF) methods~\cite{LiCYM17}. Although CF-based
approaches perform well in several application domains, they
are based solely on the \emph{sparse} user-item rating matrix and, therefore,
suffer from the so-called \textit{cold-start} problem \cite{schein2002methods}. For new
users without a rating history and newly added products with few or no ratings
(i.e., sparse historical data), the systems fail to generate high-quality
personalized recommendations.
Alternatively, CBF approaches
leverage auxiliary information such as product
descriptions~\cite{WangNL18},
locations~\cite{Lu17HBGG} and social network~\cite{LiWTM15} to generate
recommendations. These methods are in principle more robust to cold-start
problem as they can utilize different modalities. However, a pure CBF
approach will face difficulties in learning sharable and transferable
information of users and items across different product domains (e.g., ``book"
or ``movie")~\cite{pan2010survey}.
A typical example of this scenario is cross-domain
recommendation. Large online retailers such as
Amazon and eBay often obtain user-item preferences from multiple domains so
that the quality of recommendation could be improved by transferring knowledge
acquired in a source domain to a target domain. The source-target data domain
pairs in cross-domain recommendation are typically \emph{imbalanced} in two
aspects: \textit{cross-domain imbalance} and \textit{within-domain imbalance}. The former means
that the numbers of users, items or labels in two domains are imbalanced (as shown in Tab.~\ref{tab:dataset}),
The latter refers to the problem that the distribution of categorical
labels (i.e., rating scores) within one domain is imbalanced.
Fig.~\ref{fig:rating_dis} presents the imbalanced scenarios in 5-score based
cross-domain recommendation. In this example, both cross-domain imbalance and
within-domain imbalance exist.
\begin{figure}[!t]
\begin{center}
\includegraphics[width=0.65\columnwidth]{rating_distribution1.pdf}
\end{center}
\vspace{-2mm}
\caption{\label{fig:rating_dis} The illustration of cross-domain imbalance and within domain imbalance problems in cross-domain recommendation problem. The \textit{source domain} represents product domain ``digital music" and \textit{target domain} stands for product domain ``music instrument'' in Amazon dataset (see Section~\ref{sec:data} for the detailed explanation of the dataset).}
\vspace{-5mm}
\end{figure}
Alleviating the aforementioned data sparsity and data imbalance problems is a non-trivial issue for the cross-domain recommendation. However, existing CF-based and CBF-based approaches may fail to handle the problems when data becomes more and more sparse. One possible solution is to shift the learning schema from supervised to semi-supervised with limited labeled data. When it comes to a target domain in which the labeled data are completely unavailable, the only way to make a recommendation is transfer learning, particularly domain adaptation, by leveraging the knowledge from other domains.
To address the limitation of existing methods, in this paper, we propose a method called \emph{Discriminative Adversarial Networks for Cross-Domain Recommendation (RecSys-DAN)}
to learn the transferable latent representations of users, items and user-item pairs across different product domains. RecSys-DAN is rooted in the recent success of imbalanced learning~\cite{he2008adasyn, he2009learning, he2013imbalanced, ming2015unsupervised, tsai2016domain}, transfer learning~\cite{li2009transfer} and adversarial learning~\cite{goodfellow2014generative},
It adopts unsupervised adversarial loss function in combination with a discriminative
objective.
A related research field to RecSys-DAN is domain adaptation~\cite{mansour2009domain}. Although domain adaptation has shown the capability to mitigate the
rating sparsity problem, we argue that adversarial domain adaptation~\cite{ganin2015unsupervised} for recommender systems has two distinct advantages.
First, with unsupervised adversarial domain adaptation, we can learn a recommendation model when labels in the target domain
are entirely not available, the typical domain adaption usually week or even not work in this case~\cite{man2017cross,liu2018transferable,yu2018svms}. Second, we can observe the performance improvements that brought from adversarial domain adaptation as compared to traditional domain adaption, and we reported the evidence in Tab.~\ref{tab:modality}.
Moreover, RecSys-DAN incorporates not only rating information but
also additional user and item features such as product images and review texts.
Fig.~\ref{fig:adaptation} demonstrates how RecSys-DAN aligns objects with
different types and their existing preference relationships in order to
predict new preference relationships in the target domain.
\begin{figure}
\begin{center}
\includegraphics[width=0.375\textwidth]{adaptation.pdf}
\end{center}
\caption{\label{fig:adaptation}Unsupervised adversarial adaptation for cross-domain recommendation. Each square presents a user and circle presents an item, the links between users and items present the preference information (rating) that users express on items. The rating scores are not available in the target domain. The dash links are generated by our proposed method with adversarial adaptation.}
\vspace{-4mm}
\end{figure}
RecSys-DAN targets at the cold-start scenarios where no or only very few user-item preferences are available in the target domain. Existing supervised
methods~\cite{lee2001algorithms, koren2008factorization,
mcauley2013hidden, iwata2015cross, li2014matching,
liu2015non, he2017neural, zheng2017joint} fail in this setting. We
evaluate RecSys-DAN on real-world datasets and explore various scenarios where
the information in the source and target domains are in the form of uni-modality or multi-modality. The experimental results show that RecSys-DAN
achieves competitive performance compared to a variety of state-of-the-art
supervised methods which have access to ratings in the target domain.
In summary, RecSys-DAN makes the following contributions:
\begin{itemize}
\item RecSys-DAN is the first neural framework adopting an adversarial loss for the cold-start problem that caused by data sparsity and imbalance in cross-domain recommender systems. It learns domain indistinguishable representations of different types of objects (users and items) and their interactions.
\item RecSys-DAN is a highly flexible framework, which incorporates data in various modalities such as numerical, image and text.
\item RecSys-DAN addresses the cross-domain data imbalance issue as well as imbalanced preferences in recommender systems by using representation learning and adversarial learning.
\item RecSys-DAN achieves very competitive performance to the state-of-the-art supervised methods on real-world datasets where the target labels are completely not available.
\end{itemize}
The rest of this paper is organized as follows: Section~\ref{sec:related}
provides background and discusses related work. We present the motivation and
problem statement in Section~\ref{sec:motivation}. The details of our proposed
approach, RecSys-DAN, are illustrated in Section~\ref{sec:method}. Experiments
on real datasets that demonstrate the practicality and effectiveness of
RecSys-DAN are presented in Section~\ref{sec:exp}. Section~\ref{sec:con}
concludes our work.
\section{Related Work}
\label{sec:related}
This work is related to
four lines of work: cross-domain recommendation, imbalanced learning,
adversarial learning and domain adaptation.
\subsection{Cross-domain recommendation}
Cross-domain recommendation (CDR) offers
recommendations in a target domain by exploiting knowledge from source
domains. To some extent, CDR can overcome the
limitations of traditional recommendation approaches. It has been viewed as a
potential solution to mitigate the cold-start and sparsity problem in
recommender systems. Some methods have been proposed~\cite{tang2012cross,
iwata2015cross} along this line. EMCDR~\cite{man2017cross} is proposed to
learn a mapping function across domains. TCB~\cite{liu2018transferable} learns
transferable contextual bandit policy for CDR.
Sheng et al.~\cite{gao2013cross} propose
ONMTF, which is a non-negative matrix tri-factorization based method. Xu et
al.~\cite{yu2018svms} recently propose a two-side cross-domain model
(CTSIF\_SVMs) which assumes that there are some objects (users and/or items)
which can be shared in the user-side domain and item-side domain. Different
to these methods, RecSys-DAN considers that target domain has completely
unlabeled data (i.e., no ratings). Existing methods will encounter
difficulties in learning effective models for such a scenario.
\subsection{Imbalanced Learning}
Recently, Imbalanced learning~\cite{he2008adasyn, he2009learning,
he2013imbalanced, xue2015does} has been adapted to cross-domain
data~\cite{ming2015unsupervised, tsai2016domain}. Xue et.
al~\cite{xue2015does} explore the theoretical explanations for re-balancing
imbalanced data. Hsu et al.~\cite{ming2015unsupervised} propose a Closest
Common Space Learning (CCSL) algorithm by exploiting both label and structural
information for data within and across domains. This is achieved by learning
data correlations~\cite{ma2016decorrelation} and related latent source-target
domain pairs. RecSys-DAN is similar to CCSL, but it distinguishes itself by
integrating representation learning and adversarial learning in recommender
system domain. While the typical cross-domain recommendation is in line with data
imbalance problem, RecSys-DAN aims to transfer knowledge from a domain with
abundant data to a domain with scarce data instead of directly re-balancing
data.
\subsection{Generative Adversarial Network (GANs)}
Generative Adversarial Network (GANs)~\cite{goodfellow2014generative} is the
most successful method in adversarial learning. Recently, many GAN-based
extensions are proposed in different areas: image generation (e.g.,
DCGAN~\cite{radford2015unsupervised} and Wasserstein
GAN~\cite{arjovsky2017wasserstein}), NLP (e.g., SeqGAN~\cite{yu2017seqgan}
) and domain transfer
problem~\cite{tzeng2017adversarial}. In recommender systems
community, IRGAN \cite{wang2017irgan} is the first work to integrate GANs into
item-based recommendation. Differently, RecSys-DAN can be viewed as the first
work which explores the power of GAN in the context of cross-domain
recommender systems.
\subsection{Domain Adaptation}
Transfer learning~\cite{li2009transfer,pan2013transfer} has been
recently proposed to address the data sparsity problem in recommender
systems~\cite{pan2016survey,zhao2013active}. Domain adaptation, as a special
form of transfer learning, arises with the hypothesis that large amounts of
labeled data from a source domain are somehow similar to that in the unlabeled
target domain. It has been applied to learn domain transferable representation
in a variety of computer vision tasks~\cite{ganin2015unsupervised, sener2016learning, taigman2016unsupervised,
tzeng2017adversarial, xu2018cross}. Domain-Adversarial Neural
Network (DANN)~\cite{ganin2015unsupervised} learns domain-invariant features
with adversarial training. Domain Transfer Network
(DTN)~\cite{taigman2016unsupervised} translates images across domains. E.
Tzeng et al. propose a unified framework, Adversarial Discriminative Domain
Adaptation (ADDA)~\cite{tzeng2017adversarial}, for object classification task.
RecSys-DAN is partly inspired by ADDA, though there are many differences
between ADDA and RecSys-DAN. RecSys-DAN is different to existing adaptation
methods mainly in two aspects: RecSys-DAN adopts multi-level generators and
discriminator for user/item features and their interactions, and it can
captures features from multimodal data~\cite{WangMM}.
\newcommand*{\Scale}[2][4]{\scalebox{#1}{$#2$}}%
\section{Motivation and Problem Statement}
\label{sec:motivation}
\begin{figure*}
\begin{subfigure}[t]{0.245\textwidth}
\includegraphics[width=4.5cm, height=2.25cm]{UI-GAN.pdf}
\caption{UI-DAN}
\label{fig:framework:a}
\end{subfigure}
\begin{subfigure}[t]{0.24\textwidth}
\includegraphics[width=4.3cm, height=2.25cm]{U-GAN.pdf}
\caption{U-DAN}
\label{fig:framework:b}
\end{subfigure}
\begin{subfigure}[t]{0.25\textwidth}
\includegraphics[width=4.6cm, height=2.25cm]{I-GAN.pdf}
\caption{I-DAN}
\label{fig:framework:c}
\end{subfigure}
\begin{subfigure}[t]{0.245\textwidth}
\includegraphics[width=4.2cm, height=2.25cm]{H-GAN.pdf}
\caption{H-DAN}
\label{fig:framework:d}
\end{subfigure}
\caption{RecSys-DAN instantiations. $U^k$, $V^k$, $k\in\{s,t\}$ are user and item sets in domain $k$. The overlaps show that the shared user set of $U^s$ and $U^t$, or shared item set of $V^s$ and $V^t$ . $G_u$, $G_v$, $G_f$ ($D_u$, $D_v$, $D_f$) are corresponding to user, item and interaction feature generators (discriminators). The goal is: learning to align the latent representations between a source domain and a target domain that discriminators cannot distinguish. $G_y^s$ is the scoring function in the source domain.}
\label{fig:framework}
\end{figure*}
\vspace{5pt}
\subsubsection{Motivation} Motivated by the success of GANs and domain adaptation, RecSys-DAN aims to
address the data sparsity and data imbalance problem in a target domain by
adapting the object (user or item) and their interactions from a source
domain, i.e., learning to align user, item and user-item preference
representations across domains via discriminative adversarial domain
adaptation.
\vspace{5pt}
\subsubsection{General Problem} We first formalize the typical setting of a recommender system. Let $\mathcal{D}$ be a dataset consisting of $N$ users $U=\{\mathcal{U}_1,...,\mathcal{U}_N\}$ and $M$ items $V=\{\mathcal{V}_1,...,\mathcal{V}_M\}$. The user-item preferences can be represented as a rating matrix $\mathcal{Y}\in \mathbb{R}^{N\times M }$, where $\mathcal{Y}_{uv}$ is user $\mathcal{U}$'s preference rating on item $\mathcal{V}$.
We denote by $\mathcal{U}=V(\mathcal{U})=\{\mathcal{V} \in V| \mathcal{Y}_{uv}\neq 0\}$, the set of items on which user $\mathcal{U}$ has non-zero preference values. Similarly, we use $\mathcal{V}=U(\mathcal{V})=\{\mathcal{U} \in U| \mathcal{Y}_{uv}\neq 0\}$ to indicate the set of users who have non-zero ratings on item $\mathcal{V}$. The task of recommender systems is to learn a function $h$ to predict the preference rating $\hat{\mathcal{Y}}_{uv}$ of user $\mathcal{U}$ for item $\mathcal{V}$ so that $\hat{\mathcal{Y}}_{uv}$ approximates ground-truth preference score $\mathcal{Y}_{uv}$. The function $h$ often has the following form:
\begin{equation}
\label{eq:scoring}
\hat{\mathcal{Y}}_{uv}=h(\mathcal{U},\mathcal{V}; \Theta_h),
\end{equation}
where $\Theta_h$ are the learnable parameters of $h$. The users and items are associated with existing features such as product metadata when available. The denser the user-item preference matrix $\mathbf{P}$ is, the less challenging the learning and prediction problems are.
However, $\mathbf{P}$ can be very sparse in practice.
\vspace{5pt}
\subsubsection{Adversarial Cross-Domain Alignment} To address this type of data
sparsity problem, we propose to perform domain adaptation going from a
\textit{source} domain with several user-item preference values to a
\textit{target} domain with \emph{no} user-item preferences. Specifically,
the proposed approach learns a function $G$ that maps the following objects to latent vector representations:
the set of items that represented as $\mathcal{U}$; the set of users that represented as $\mathcal{V}$; the set of user-item pairs ($\mathcal{U}, \mathcal{V}$). The $G$ is learned in a way that a discriminator $D$
cannot distinguish the latent representations generated for the target domain
from the latent representations generated for the source domain. We achieve
this by introducing an adversarial learning loss involving $G$ and $D$. For
the sake of readability, we refer to $G$ as a generator and write $G^{k}_{j}$
to denote different types of generators with $k \in \{s, t\}$ (source or
target) and $j \in \{u, v, f\}$ (user, item or item-user pairs).
Contrary to existing work, we formulate the adversarial loss for different
types of objects (users and items) and their interactions. The adversarial
loss, therefore, aligns distributions of latent items and user representations
\emph{as well as} their relationships given by the user-item preferences. The
latent representations computed by the generators, therefore, fall into three
categories: (1) user representations; (2) item representation;
and (3) interaction representations of user-item pairs.
\vspace{5pt}
\subsubsection{Shared Cross-Domain Objects} Learning across domains
requires the existence of some relations in the participating domains.
Usually, this relation is formed when objects (users, items) are found to be
common in both domains \cite{khan2017cross}.
To cover the different scenarios, RecSys-DAN includes four different adversarial cross-
domain adaptation scenarios as below. They are classified according to whether a subset of user set
$U$ and item set $V$ exists in both source and target domains:
\begin{itemize}
\item Interaction adaptation: ~~~~~~$U^s \cap U^t=\varnothing$ and $V^s\cap V^t=\varnothing$.
\item User adaptation: ~~~~~~~~~~~~~~~$U^s \cap U^t = \varnothing$ and $V^s\cap V^t\neq\varnothing$.
\item Item adaptation: ~~~~~~~~~~~~~~~$U^s \cap U^t \neq \varnothing$ and $V^s\cap V^t=\varnothing$.
\item Hybrid adaptation: ~~~~~~~~~~~$U^s \cap U^t \neq \varnothing$ and $V^s\cap V^t\neq\varnothing$.
\end{itemize}
Correspondingly, we proposed UI-DAN, U-DAN, I-DAN and H-DAN as shown in Fig.~\ref{fig:framework}. The additional discriminators (in green) are
introduced for shared objects. For instance, in the user adaptation scenario
(U-DAN) where the set of users in the source and target domain are disjoint,
we introduce a discriminator $D_u$ attempting to distinguish between latent
user representations from the source and target domain in order to align those
representations in latent space.
\section{Discriminative Adversarial Networks for Cross-Domain Recommendation}
\label{sec:method}
We firstly describe the learning of representations of
objects (i.e., user and item) and their interactions. Then we elaborate
on the objectives of learning to align the representations across domains.
Finally, we introduce RecSys-DAN as a generalized adversarial adaptation
framework.
\subsection{Learning Domain Representations}
Given a set of users, items and ratings in the source domain, we can learn a
latent representation space $\mathcal{X}^s\in \mathbb{R}^d$ by computing a
supervised loss on the given input $X^s$ and ratings $Y^s$. Since
the target domain has no or only few ratings, we do not directly learn the
representations for the target domain. Instead, we learn mappings from the
source representation space $\mathcal{X}^s$ to the target representation
space $\mathcal{X}^t\in \mathbb{R}^d$ so as to minimize the distance between
them. This can be achieved by first parameterizing source and target mapping
functions, $M^s: X^s\rightarrow \mathcal{X}^s$ and $M^t: X^t\rightarrow
\mathcal{X}^t$, and then minimizing the distance between the empirical source
and target mapping distributions: $M^s(X^s)$ and
$M^t(X^t)$~\cite{tzeng2017adversarial}. In this work, $M^k=\{G_u^k,
G_v^k,G_f^k\}, k \in {\{s, t\}}$ is a set consisting of user mapping function $G_u^k$ and item mapping function
$G_v^k$, and user-item pair mapping function $G_f^k$.
For learning textual representations, the $G_u^k$ is a recurrent neural network (RNN), specifically, RecSys-DAN adopts Long Short-Term Memory (LSTM)~\cite{HochreiterS97}:
\begin{align}
\begin{split}
\label{equ:input}
& i_t=\sigma (W_{xi}x_t+W_{hi}h_{t-1}+b_i)\\
& f_t=\sigma (W_{xf}x_t+W_{hf}h_{t-1}+b_f)\\
& o_t=\sigma (W_{xo}x_t+W_{ho}h_{t-1}+b_o)\\
& g_t=\tanh (W_{xc}x_t+W_{hc}h_{t-1}+b_c)\\
& c_t=f_t \odot c_{t-1}+i_t\odot g_t\\
& h_t=o_t\odot\tanh(c_t)
\end{split}
\end{align}
where $i_t$, $f_t$ and $o_t$ are input, forget and output gate respectively, $c_t$ is memory cell.
$G_v^k$ can be either RNN-based (when review texts are used to represent an item) or convolutional neural network (CNN)-based for visual representations (when product image is used to represent an item). As shown in Fig.~\ref{fig:framework}, the mapping function $G_u^k (\mathcal{U}^k; \Theta_u^k):
\mathcal{U}^k \rightarrow \mathcal{X}^k_u\in\mathbb{R}^d$ that maps a user
sample to a $d$ dimensional vector $\mathcal{X}^k_u$ and is parameterized by
$\Theta_u^k$. Similarly, we have the item mapping function $G_v^k
(\mathcal{V}^k; \Theta_v^k): \mathcal{V}^k \rightarrow
\mathcal{X}^k_v\in\mathbb{R}^d$. Given user and item representations
($\mathcal{X}_u^k, \mathcal{X}_v^k$), mapping function
$G_f^k(\mathcal{X}^k_u, \mathcal{X}^k_v; \Theta_f^k):(\mathcal{X}^k_u,
\mathcal{X}^k_v) \rightarrow \mathcal{X}^k_f\in\mathbb{R}^d$ learns user-item
interaction representation $\mathcal{X}^k_f$. The prediction
$\hat{\mathcal{Y}}=h^k\Big(G_f^k(\mathcal{X}^k_u, \mathcal{X}^k_v;
\Theta_f^k);\Theta_h^k\Big)$, where $h^k$ is the scoring function in
Eq.~\ref{eq:scoring}. Since there in only one scoring function can be learned in supervised way, i.e. $G_y^s$, and $h^s=h^t=G_y^s$, we use $G_y^s$ to represent the scoring function.
In a source domain, the parameters $\Theta^s=\{\Theta_u^s, \Theta_v^s,
\Theta_f^s, \Theta_h^s\}$ are learned by optimizing the objective:
\begin{equation}
\min_\mathbf{\Theta^s}\left [\frac{1}{\left |\mathcal{D} \right |}\sum_{i=1}^{|\mathcal{D}|}\mathcal{L}^s(\mathcal{U}_i^s, \mathcal{V}_i^s, \mathcal{Y}_i^s )+\lambda \left \| \Theta^s \right \|\right ]
\label{equ:s_loss}
\end{equation}
where $\langle$$\mathcal{U}_i^s$, $\mathcal{V}_i^s$,
$\mathcal{Y}_i^s$$\rangle$ presents raw $\langle$user, item, truth
score$\rangle$ triple, and $\mathcal{L}^s(\mathcal{U}_i^s,
\mathcal{V}_i^s,\mathcal{Y}_i^s)=\parallel \hat{\mathcal{Y}}^s_i-\mathcal{Y}_i^s\parallel ^2$.
$\left |\mathcal{D} \right |$ is the size of training set. $\lambda$ is the
regularization parameter. By minimizing the objective function
(\ref{equ:s_loss}), the mapping functions $G_u^s$, $G_v^s$ and $G_f^s$ can be
learned and used for extracting user, item and user-item features
respectively in source domain by fixing corresponding parameter. For the
unlabeled target domain, the corresponding target mapping functions $G_u^t$,
$G_v^t$ and $G_f^t$ can be learned adversarially as we will explain in the
next section.
\subsection{Adversarial Representation Adaptation}
One of the algorithmic principles of domain adaptation is to learn a space in
which source and target domains are close to each other while keeping good
performances on the source domain task~\cite{pan2010survey}. Following the
settings of standard GAN~\cite{goodfellow2014generative}, domain
discriminators $D_u$, $D_v$ and $D_f$ in RecSys-DAN are designed to perform
min-max games and adversarially learn target generators (i.e., mapping
functions) $G_u^t (\mathcal{U}^t; \Theta_u^t)$, $G_v^t (\mathcal{V}^t;
\Theta_v^t)$ and $G_f^t(\mathcal{X}_u^t, \mathcal{X}_v^t;
\Theta_f^t)$ with unlabeled samples. The loss functions of each
instantiation of RecSys-DAN are as follows:
\begin{itemize}
\setlength\itemsep{3pt}
\item UI-DAN: $\Scale[1]{ \min\limits_{G_f^t}\max\limits_{D_f}\mathcal{L}(D_f, G_f^t)}$\\ $\Scale[1]{\ \ \ \ \ \ \ \ \ \ \ \ s.t. ~U^s\cap U^t=\varnothing$ and $V^s\cap V^t=\varnothing}$.
\item U-DAN: $\Scale[1]{ \min\limits_{G_f^t, G_u^t }\max\limits_{D_f, D_u}\mathcal{L}(D_f, D_u, G_f^t, G_u^t)}$\\ $\Scale[1]{\ \ \ \ \ \ \ \ \ \ \ \ s.t. ~ U^s\cap U^t = \varnothing$ and $V^s\cap V^t\neq\varnothing}$
\item I-DAN: $\Scale[1]{\min\limits_{G_f^t, G_v^t }\max\limits_{D_f, D_v}\mathcal{L}(D_f, D_v, G_f^t, G_v^t,)}$\\ $\Scale[1]{\ \ \ \ \ \ \ \ \ \ \ \ s.t. ~ U^s\cap U^t \neq \varnothing$ and $V^s\cap V^t=\varnothing}$
\item H-DAN: $\Scale[1]{\min\limits_{G_f^t, G_u^t , G_v^t }\max\limits_{D_f, D_u, D_v}\mathcal{L}(D_f, D_u, D_v, G_f^t, G_u^t , G_v^t)}$ \\ $\Scale[1]{\ \ \ \ \ \ \ \ \ \ \ \ s.t. ~ U^s\cap U^t \neq \varnothing$ and $V^s\cap V^t\neq\varnothing}$
\end{itemize}
The objectives are learning generators in the target domain to generate features
$\mathcal{X}^t\in\{\mathcal{X}_u^t, \mathcal{X}_v^t, \mathcal{X}_f^t\}$ which
are intended to be close to the source latent representations
$\mathcal{X}^s\in \{ \mathcal{X}_u^s, \mathcal{X}_v^s, \mathcal{X}_f^s\}$.
More specifically, $G_f$ generates interaction-level domain indistinguishable
features, while $G_u$/$G_v$ generates indistinguishable user/item features for overlapping users/items.
Formally, the source generators $M^s=\{G_f^s, G_u^s, G_v^s\}$ and predictor
$G_y^s$ is learned in a supervised way:
\begin{align}
\begin{split}
\min_{G_y^s, M^s}\mathcal{L}_s(U^s, V^s,Y^s) \\
&\hspace{-25mm}= \mathbb{E}_{{(\mathcal{U}^s, \mathcal{V}^s,\mathcal{Y}^s)} \sim (U^s, V^s, Y^s)}[(G_y^s(M^s, \mathcal{U}^s, \mathcal{V}^s,\mathcal{Y}^s)] \\
&\hspace{-25mm}=\frac{1}{\left |\mathcal{D}^s \right |}\sum_{i=1}^{|\mathcal{D}^s|}\mathcal{L}^s(\mathcal{U}_i^s, \mathcal{V}_i^s,\mathcal{Y}_i^s)+\lambda \left \| \Theta^s \right \| \\
&\hspace{-25mm}=\frac{1}{\left |\mathcal{D}^s \right |}\sum_{i=1}^{|\mathcal{D}^s|}(\hat{\mathcal{Y}}^s_i-\mathcal{Y}_i^s )^2+\lambda \left \| \Theta^s \right \|
\end{split}
\end{align}
The optimization of source weights $\Theta^s$ is formulated as a regression task which
minimizes the mean squared error (MSE) over samples. In learning target
generators $M^t=\{G_f^t, G_u^t, G_v^t\}$, $M^s$ is used as a domain
regularizer with fixed parameters. This is similar to the original
GAN~\cite{goodfellow2014generative} where a generated space is updated with a
fixed real space. To simplify, we take UI-DAN as an exemplary illustration,
the learning objective is:
\begin{align}
\begin{split}
\max_{D_f}\mathcal{L}_f(U^s, V^s, U^t, V^t, M^s, M^t)\\
&\hspace{-40mm}=\mathbb{E}_{{(\mathcal{U}^s, \mathcal{V}^s)} \sim (U^s_u, V^s_v)}[\log D_f(M^s(\mathcal{U}^s, \mathcal{V}^s))]\\
&\hspace{-40mm} + \mathbb{E}_{{(\mathcal{U}^t, \mathcal{V}^t)} \sim (U^t_u, V^t_v)}[\log(1-D_f(M^t(\mathcal{U}^t, \mathcal{V}^t)))]
\end{split}
\end{align}
\begin{align}
\begin{split}
\min_{M^t}\mathcal{L}_{m}(U^t, V^t, D_f)\\
&\hspace{-20mm}=\mathbb{E}_{{(\mathcal{U}^t, \mathcal{V}^t)} \sim (U^t, V^t)}[\log(1-D_f(M^t(\mathcal{U}^t, \mathcal{V}^t)))]
\end{split}
\end{align}
where $M^t$ is initialized with $M^s$.
With learned $M^t$, user, item, interaction representations $\mathcal{X}_u^t$, $\mathcal{X}_v^t$, $\mathcal{X}_f^t$ can be extracted as inputs for scoring function $G_y^s$, which makes
preference predictions. Note that one of the essential differences between RecSys-DAN
and prior recommendation methods is that ResSys-DAN takes the cross-domain
overlap users (items) into account to learn indistinguishable user (item)
representation as shown in Fig.~\ref{fig:framework:b}, Fig.~\ref{fig:framework:c}, and Fig.~\ref{fig:framework:d}. With shared users and
items across domain, additionally, $D_u$ and $D_v$ are designed and lead to
scenarios that have interaction-level $D_f$, $G_f$ and feature-level $D_u$,
$D_v$, $G_u$, $G_v$:
\begin{align}
\begin{split}
\max_{D_z}\mathcal{L}_z(U^s, V^s, U^t, V^t, G_u^s, G_v^s, G_u^t, G_v^t) \\
\min_{G^t_z}\mathcal{L}_{m}(U^t, V^t, D_z)\\
s.t. ~~D_z=D_u~~\text{if}~ U^s \cap U^t = \varnothing~\text{and} ~V^s\cap V^t\neq \varnothing \\
~~ D_z=D_v~~\text{if}~U^s \cap U^t \neq \varnothing~\text{and} ~V^s\cap V^t = \varnothing \\
~~D_z=D_u,D_v~~\text{if}~ U^s \cap U^t \neq \varnothing~\text{and} ~V^s\cap V^t\neq \varnothing
\end{split}
\end{align}
The optimization of the additional discriminators and generators is achieved by fine-tuning $G_u^t$ ($G_v^t$) on cross-domain shared user/item subset.
\begin{algorithm}
\small
\caption{Learning algorithm for UI-DAN}
\label{alg}
\KwIn{source set $\mathcal{D}^s=\{X_u^s,X_v^s, Y^s\}$,
target set $\mathcal{D}^t=\{X_u^t, X_v^t\}$, dummy domain label $Y^d\in\{0,1\}$, batch size $\mathcal{B}$.}
\textbf{Initilize: $M^s, M^t, G_y^s, D_f$} \\
$\mathcal{N}^s=|\mathcal{D}^s|$, $\mathcal{N}^t=|\mathcal{D}^t|$ \\
\textit{pre-train on source domain:} \\
\Repeat{\text{stopping criterion is met}}{
\For {$b\leq \frac{\mathcal{N}^s}{\mathcal{B}}$}
{ mini batch $(\mathcal{U}_b^s, \mathcal{V}_b^s, \mathcal{Y}_b^s) \in (X_u^s, X^s_v, Y^s)$ \\
$M^s, G_y^s \Leftarrow \min \mathcal{L}_s(\mathcal{U}_b^s, \mathcal{V}_b^s,\mathcal{Y}_b^s)$
}
}
\textit{train generators on target domain}: \\
set $M^t\Leftarrow M^s$, and fix $M^s$ \\
\Repeat{\text{stopping criterion is met}}{
\For {$b\leq \frac{\mathcal{N}^s}{\mathcal{B}}$}
{
mini batch $(\mathcal{U}_b^s, \mathcal{V}_b^s) \in (X_u^s, X^s_v)$ \\
\For {$k\leq \frac{\mathcal{N}^t}{\mathcal{B}}$}
{
mini batch $(\mathcal{U}_k^t, \mathcal{V}_k^t) \in (X_u^t, X_v^t)$ \\
$D_f\Leftarrow \max \mathcal{L}_f(\mathcal{U}_b^s, \mathcal{V}_b^s,\mathcal{U}_k^t, \mathcal{V}_k^t,\mathcal{Y}^d)$ \\
$M_t\Leftarrow \min \mathcal{L}_m(\mathcal{U}_k^t, \mathcal{V}_k^t)$ \\
}
}
}
\KwOut{$M^t$}
\textit{inference on target domain}: \\
$\hat{y}_t \Leftarrow G_y^s(M^t(x_u^t, x_v^t))$
\end{algorithm}
\vspace{-2mm}
\subsection{Generalized Framework}
RecSys-DAN is a generalized framework. The choice of RecSys-DAN
instantiations is based on considering the following questions: (1) Which
type of modalities (e.g. numerical rating, review or image) are used to represent
$\mathcal{U}$ and $\mathcal{V}$? (2) Are there shared users and/or items
across domains? (3) Which adversarial objective is used?
The training procedure of each instantiation is different to each other, but
they also share some similarities. Algorithm 1 summaries the learning
procedure of UI-DAN in which two training stages are involved. First, the pre-training in
the source domain for obtaining source generators $M^s$ and scoring
function $G_y^s$. The update of parameters $\Theta_u^s, \Theta_v^s, \Theta_f^s$ are achieved by:
\begin{align}
\small
\begin{split}
\Theta_j^s:=\Theta_j^s -\eta \nabla_{\Theta_j^s}\frac{1}{\mathcal{B}}\sum_{i=1}^\mathcal{B}\mathcal{L}_s(\mathcal{U}_i^s, \mathcal{V}_i^s,\mathcal{Y}_i^s),~~j\in\{u,v,f\}
\end{split}
\end{align}
where $\mathcal{B}$ is a min-batch of training samples, $\eta$ is learning rate.
Similarly, the optimal weights for scoring function $G_y^s(G_f^s;
\Theta_y^s)$ can be learned. Second, cross-domain adversarial learning, the goal is to learn the
target generators $M^t$ in an adversarial way. By using dummy domain labels,
$y^d=1$ presents the data from source domain and $y^d=0$ for target domain.
The domain discriminator $D_f(G_f^s, G_f^t; \Theta_d)$ is obtained by
ascending stochastic gradients~\cite{goodfellow2014generative} at each batch
using the following update rule:
\begin{align}
\Theta_d:= \Theta_d+\eta \nabla_{\Theta_d}\frac{1}{\mathcal{B}}\sum_{i=1}^\mathcal{B}\mathcal{L}_f(\mathcal{U}_i^s, \mathcal{V}_i^s,\mathcal{U}_i^t, \mathcal{V}_i^t,\mathcal{Y}_i^d)
\end{align}
Note that target generators $M^t$ is initialized with and updated in similar
way as $M^s$. By doing this, $M^t$ tries to push the user-item interaction
representations in the target domain as close as possible to the source
domain. Additionally, the ratings (i.e., labels) in the target domain are
never accessed in learning procedures of RecSys-DAN. As a comparison,
existing recommendation methods fail to handle this scenario. With learned
$M^t$, the rating regression can be performed with source score function
$G_y^s$ for a given user-item pair in the target domain:
\begin{align}
\hat{\mathcal{Y}}_{uv} \Leftarrow G_y^s\big(M^t(\mathcal{U}^t, \mathcal{V}^t)\big).
\end{align}
The learning procedures of U-DAN, I-DAN and
H-DAN have additional fine-tuning stage with training samples of shared
users/items.
\vspace{-2mm}
\begin{algorithm}
\small
\caption{Learning for U-DAN and I-DAN}
\label{alg1}
\KwIn{$\mathcal{D}^s=\{X_u^s,X_v^s, Y^s\}$, $\mathcal{D}^t=\{X_u^t, X_v^t\}$, \\
shared item set $\mathcal{D}^o_u=\{X_v^o, X_u^s,X_u^t\}$,\\
shared user set $\mathcal{D}^o_v=\{X_u^o, X_v^s,X_v^t\}$, $Y^d\in\{0,1\}$.}
\textbf{Initialize: $M^s, M^t, G_y^s, D_u,D_v,D_f$} \\
\textit{call Algorithm \ref{alg} to obtain $M^t$, learning rate $\eta \times 0.001$} \\
\textit{learning U-DAN}: \\
\Repeat{\text{stopping criterion is met}}{
\For {each batch $b$, $(\mathcal{V}_b^o, \mathcal{U}_b^s,\mathcal{U}_b^t) \in (X_v^o,X_u^s, X^t_u)$}
{
$D_u^t\Leftarrow \max \mathcal{L}_f(\mathcal{V}_b^o, \mathcal{U}_b^s,\mathcal{U}_b^t,\mathcal{Y}^d_b)$ \\
$G_u^t\Leftarrow \min \mathcal{L}_m(\mathcal{V}_b^o,\mathcal{U}_b^t)$ \\
}
}
\textit{learning I-DAN}: \\
\Repeat{\text{stopping criterion is met}}{
\For {each batch $b$, $(\mathcal{U}_b^o, \mathcal{V}_b^s,\mathcal{V}_b^t) \in (X_u^o, X_v^s, X^t_v)$}
{
$D_v^t\Leftarrow \max \mathcal{L}_f(\mathcal{U}_b^o, \mathcal{V}_b^s,\mathcal{V}_b^t,\mathcal{Y}^d_b)$ \\
$G_v^t\Leftarrow \min \mathcal{L}_m(\mathcal{U}_b^o,\mathcal{V}_b^t)$ \\
}
}
\KwOut{$G_u^t, G_v^t$}
\end{algorithm}
\vspace{-2mm}
Algorithm \ref{alg1} presents the learning for U-DAN and I-DAN while H-DAN is a combination of them.
\section{Experiments}
\label{sec:exp}
This section evaluates the performance of RecSys-DAN on both unimodal and multimodal scenarios.
\begin{table}[!t]
\footnotesize
\centering
\caption{Overview of the datasets ($\dagger$ presents training samples for shared users and items respectively)}
\label{tab:dataset}
\begin{tabular}{|l|p{0.75cm}lp{1.8cm}|p{0.85cm}|}
\hline
$D^s \rightarrow D^t$ & User & Item & Sample & $\mathcal{|VOC|}$ \\ \hline\hline
DM & 5540 & 3558 & 64544 & \multicolumn{1}{l|}{\multirow{3}{*}{4696}} \\
MI & 1429 & 891 & 10156 & \multicolumn{1}{l|}{} \\
DM $\cap$ MI & 23 & 0 & 23 & \multicolumn{1}{l|}{} \\ \hline
HK & 14285 & 3227 & 41810 & \multirow{3}{*}{3651} \\
OP & 4773 & 1312 & 28044 & \\
HK $\cap$ OP & 1709 & 0 & 1709 & \\ \hline
CDs & 41437 & 9650 & 84432 & \multirow{3}{*}{10355} \\
DM & 5540 & 3558 & 6615 & \\
CDs $\cap$ DM & 4394 & 829 & 19529/6216$^\dagger$ & \\ \hline
\end{tabular}
\end{table}
\begin{table*}[!t]
\footnotesize
\caption{\small \label{tab:comparison_S_H} The results for UI-DAN and I-DAN in the unimodal and multimodal settings (s: source-only, a: adaptation, u: unimodal, m: multimodal). The best (supervised) baselines are in {\color{blue!95}\textbf{blue}}, and the best unimodal (multimodal) results of RecSys-DAN are in {\color{OliveGreen!95}\textbf{green}} ({\color{red!95}\textbf{red}}) . $\Delta = (2\left | S_{-}^{*}-S_{+}^{*} \right |)/(S_{-}^{*}+S_{+}^{*})$ presents the percentage differences between the best result of ours ($S_{-}^*$, in green) and that of baselines ($S_{+}^*$, in blue). It demonstrates how close the performance of (unsupervised) RecSys-DAN to the performance of (supervised) baselines.}
\label{tab:modality}
\centering
\begin{center}
\begin{tabular}{|l|cccc|ccc|}
\hline
$\mathcal{D}^s\rightarrow\mathcal{D}^t$ & \multicolumn{2}{c}{DM$\rightarrow$MI } & \multicolumn{2}{c|}{ HK $\rightarrow$ OP}& \multicolumn{3}{c|}{ Target Domain Training Data} \\
Models & RMSE & MAE & RMSE & MAE & Rating& Review& Image \\ \hline \hline
Normal & 1.165 $\pm$ 0.022 & 0.843 $\pm$ 0.025 & 1.194 $\pm$ 0.024 & 0.894 $\pm$ 0.023 & Yes & No& No \\
KNN & 1.040 $\pm$ 0.000 & 0.709 $\pm$ 0.000 & 0.957 $\pm$ 0.000 & 0.710 $\pm$ 0.000 & Yes & No& No \\
NMF & 0.922 $\pm$ 0.009 & 0.644 $\pm$ 0.007 & 0.866 $\pm$ 0.003 & 0.637 $\pm$ 0.005 & Yes & No& No \\
SVD++ & 0.891 $\pm$ 0.008 & 0.648 $\pm$ 0.006 & \color{blue!95}\textbf{0.844 $\pm$ 0.002} & 0.642 $\pm$ 0.002 & Yes & No& No \\
HFT & 0.914 $\pm$ 0.000 & 0.704 $\pm$ 0.000 & 0.917 $\pm$ 0.000 & 0.735 $\pm$ 0.000 & Yes & Yes& No \\
DeepCoNN & \color{blue!95}\textbf{0.868 $\pm$ 0.002} & \color{blue!95}\textbf{0.599 $\pm$ 0.003} & 0.875 $\pm$ 0.001 & \color{blue!95} \textbf{0.634 $\pm$ 0.001} & Yes & Yes& No \\
\hline \hline
UI-DAN (s, u) & 1.087$\pm$ 0.180 & 0.918$\pm$ 0.002 & 0.959$\pm$ 0.028 & 0.684$\pm$ 0.003 & No & Yes& No \\
I-DAN (s, u) & 1.052$\pm$ 0.220 & 0.884$\pm$ 0.264 & 0.957$\pm$ 0.033 & 0.684$\pm$ 0.002 &No & Yes& No \\
UI-DAN (s, m) & 1.043$\pm$ 0.056 & 0.879$\pm$ 0.089 & 1.037$\pm$ 0.008 & 0.875$\pm$ 0.011 & No & Yes& Yes \\
I-DAN (s, m) & 1.450$\pm$ 0.291 & 1.296$\pm$ 0.308 & 1.953$\pm$ 0.290 & 1.759$\pm$ 0.286 & No & Yes& Yes \\ \hline \hline
UI-DAN (a, u) & 0.920$\pm$ 0.223 & \color{OliveGreen}\textbf{0.674$\pm$ 0.021} & 0.917$\pm$ 0.005 & 0.674$\pm$ 0.002& No & Yes& No \\
I-DAN (a, u) & \color{OliveGreen} \textbf{0.914$\pm$ 0.002} & 0.675$\pm$ 0.021 & \color{OliveGreen}\textbf{0.911$\pm$ 0.002} & \color{OliveGreen}\textbf{0.670$\pm$ 0.002} & No & Yes& No \\
UI-DAN (a, m) & \color{red!95}\textbf{0.991$\pm$ 0.077} & \color{red!95}\textbf{0.765$\pm$ 0.143} & \color{red!95}\textbf{0.934$\pm$ 0.004} & \color{red!95}\textbf{0.745$\pm$ 0.006} & No & Yes& Yes \\
I-DAN (a, m) & 1.078$\pm$ 0.033 & 0.795$\pm$ 0.027 & 1.144$\pm$ 0.078 & 0.868$\pm$ 0.039 & No & Yes& Yes \\ \hline
$\Delta$ & 5.16\%$\pm$ 0.22\% & 11.78\%$\pm$ 1.88\% & 7.64\%$\pm$0.23\% & 5.52\% $\pm$0.23\% & - & -& -\\ \hline
\end{tabular}
\end{center}
\end{table*}
\begin{figure*}[!]
\vspace{-5mm}
\centering
\includegraphics[width=0.8\linewidth]{gan_loss_new5.pdf}
\caption{\small Learning unimodal UI-DAN and I-DAN. It plots the changes of loss and accuracy of interaction-level (a-b) discriminator $D_f$/generator $G_f$ and item-level discriminator $D_v$/generator $G_v$ (c-d) on two dataset pairs against training epochs. The dash vertical lines in (c-d) denote the starting point for fine-tuning I-DAN. The X-axis presents the number of training epochs.}
\label{fig:unimodal_training}
\end{figure*}
\subsection{Dataset and Evaluation Metric}
\label{sec:data}
We evaluated RecSys-DAN on multiple sets on the Amazon
dataset~\cite{mcauley2015image}\footnote{\url{http://jmcauley.ucsd.edu/data/amazon/}},
which is widely used for evaluating recommender
systems~\cite{mcauley2013hidden,zheng2017joint}. It contains
different item and user modalities such as review text, product images and
ratings. We selected 5 categories to form three (source
$\rightarrow$ target) domain pairs: Digital Music$\rightarrow$Music
Instruments (DM$\rightarrow$MI), Home \& Kitchen$\rightarrow$Office Products
(HK$\rightarrow$OP) and CDs \& Vinyl$\rightarrow$Digital Music
(CDs$\rightarrow$DM). Some statistics of the datasets are listed in
Tab.~\ref{tab:dataset}. $\mathcal{|VOC|}$ is the size of the vocabulary of
words used in reviews in the source and target training sets. Words which
occurred less than 5 times were removed. We randomly split each dataset into
80\%/10\%/10\% for training/validation/test. The training reviews
associated with a user/item were concatenated to present the user/item
following previous work~\cite{zheng2017joint}. We aligned users (items) that
occured in both the source and target domains to ensure an equal number of
training reviews for both domains.
We evaluated all the models on the rating prediction task using both the root
mean squared error (RMSE)
and the mean average error (MAE):
\begin{equation}
\scriptsize
\text{RMSE}=\sqrt{\frac{1}{\mathcal{|D|}}\sum_{(\mathcal{U}, \mathcal{V})\in \mathcal{D}} ({\hat{\mathcal{Y}}_{uv}}-\mathcal{Y}_{uv})^2},\,\,\,\text{MAE}=\frac{1}{\mathcal{|D|}}\sum_{(\mathcal{U}, \mathcal{V})\in \mathcal{D}} |\hat{\mathcal{Y}}_{uv}-\mathcal{Y}_{uv}|
\end{equation}
where $\hat{\mathcal{Y}}_{uv}$ and $\mathcal{Y}_{uv}$ are predicted and truth rating, respectively.
\subsection{Baseline Methods}
We compare RecSys-DAN against a variety of methods.
Naive: \textbf{Normal} is a random rating predictor which gives predictions based on the (norm) distribution of the training set.
Matrix factorization: \textbf{NMF}~\cite{lee2001algorithms}, Non-negative Matrix Factorization that only uses ratings. And \textbf{SVD++}~\cite{koren2008factorization}, extended SVD for latent factor modeling.
Nearest neighbors: \textbf{KNN}~\cite{koren2010factor}.
Topic modeling: \textbf{HFT}~\cite{mcauley2013hidden}.
Deep learning methods: \textbf{DeepCoNN}~\cite{zheng2017joint}, which is the current state-of-the-art approach. Additionally, we compared RecSys-DAN with typical cross-domain recommendation methods.
Following previous work~\cite{sener2016learning,tzeng2017adversarial},
\textbf{source-only} results for applying a source domain models to the
target domain are also reported. Note that rating information in the target
domain is accessible to the baseline methods (except source-only), while
RecSys-DAN has no access to ratings in the target domain.
\subsection{Implementations}
We implemented RecSys-DAN with Theano\footnote{\url{http://www.deeplearning.net/software/theano/}}. The discriminators $D_f$, $D_u$, $D_v$ are formed
with following layers: Dense(512)$\rightarrow$Relu($\cdot$)$\rightarrow$Dense(2)$\rightarrow$Softmax($\cdot$). The architecture of generators varies
according to different scenarios. For unimodal scenario (textual user and
item representations), $G_u^s$, $G_v^s$, $G_u^t$, $G_v^t$ are formed by:
Embedding($\mathcal{|VOC|}$)$\rightarrow$LSTM (256)$\rightarrow$Average
Pooling, and $G_f^s$, $G_f^t$ are constructed using:
Dense(512)$\rightarrow$Dropout (0.5). For multimodal scenario
(textual user representation and visual item representation),
the main architecture of $G_v^s$, $G_v^t$ is: CNN$\rightarrow$Dense
(4096)$\rightarrow$ Dense(256), and other configurations remain unchanged as
in unimodal scenario. The weights of LSTM are orthogonally
initialized~\cite{saxe2013exact}. We used a batch size of 512. The models
were optimized with ADADELTA \cite{zeiler2012adadelta} and the initial
learning rate $\eta$ is 0.0001 (decreased by $\times$0.001 for
U-DAN, I-DAN and H-DAN). We implemented KNN, NMF and SVD++ using
SurPrise package\footnote{\url{http://surpriselib.com/}} and used authors'
implementations for
HFT\footnote{\url{http://cseweb.ucsd.edu/~jmcauley/code/code_RecSys13.tar.gz}}
and DeepCoNN\footnote{\url{https://github.com/chenchongthu/DeepCoNN}}. To make a fair comparison, implemented baselines are trained with grid search (for NMF and SVD++, regularization [0.0001, 0.0005, 0.001], learning rate [0.0005, 0.001, 0.005, 0.01]. For HFT, regularization [0.0001, 0.001, 0.01, 0.1, 1], lambda [0.1, 0.25, 0.5, 1]). For DeepCoNN, we use the suggested default parameters. The best scores are reported.
\subsection{Results and Discussions}
\label{sec:uni_modal}
We first evaluated two RecSys-DAN instances: UI-DAN (applied to the scenario
where source and target domains have neither overlapping users nor items) and
I-DAN (applied to the scenario where the source and target domains only shared some users) in the unimodal and multimodal scenarios. The results are
summarized in Tab.~\ref{tab:modality}.
\subsubsection{Unimodal RecSys-DAN}
The results listed in Tab.~\ref{tab:modality} show that both UI-DAN and I-DAN improve the source-only
baselines. For instance, UI-DAN reduces the source-only error by $\sim$15\%
(RMSE) and $\sim$27\% (MAE) on DM$\rightarrow$MI. On HK$\rightarrow$OP, it
improves the source-only baselines by $\sim$4\% (RMSE) and $\sim$1.5\% (MAE),
respectively. In the scenario where source and target domains share users,
I-DAN can improve UI-DAN on both dataset pairs ($\sim$0.4\% on average across
metrics). Compared to its source-only baselines, I-DAN achieves improvements
similar to those of UI-DAN.
Fig.~\ref{fig:unimodal_training}a and Fig.~\ref{fig:unimodal_training}b show
the changes of the loss/accuracy of the interaction discriminator $D_f$ and
the loss of $G_f$ against the number of epochs with the UI-DAN. On both
dataset pairs, the equilibrium points are reached at $\sim$100 epochs where
binary classification accuracy of discriminator is 50\%. It suggests that the
user-item interaction representation from generator is indistinguishable to
discriminator. When training I-DAN with shared user samples, we first trained
interaction-level $D_f$ and $G_f$ and then fine-tuned item-level $D_v$ and
$G_v$ by decreasing learning rate to 0.001$\times \eta$.
We adopted small learning rate $\eta$ to ensure that $G_v$ could generate
indistinguishable item representation for shared users while maintaining
interaction-level representations. Figures~\ref{fig:unimodal_training}c
and~\ref{fig:unimodal_training}d present the training procedure of I-DAN. On
DM$\rightarrow$MI, $D_v$ and $G_v$ had difficulty to converge due
to limited shared user samples. On the contrary, with more shared samples, I-DAN was able to converge
on both interaction-level and item-level on HK$\rightarrow$OP. From
experimental results, we can observe that item-level representations are not
as important as interaction-level representation on rating prediction task.
Similar findings are reported in Tab.~\ref{tab:CDs}.
\subsubsection{Multimodal RecSys-DAN}
The task becomes more challenging when both ratings and reviews are not
available. In this scenario, we replaced the review text of an item with its
image, if available, which leads to a multimodal unsupervised adaptation
problem. The correlations between textual user embeddings and visual item
embeddings need to be adapted across the given domains. The results of UI-DAN
(a, m) and I-DAN (a, m) in the multimodal settings can be found in
Tab.~\ref{tab:modality}. We find that it is more difficult to learn
user-item correlations across modalities, compared to the unimodal setting.
Fig.~\ref{fig:multi_train} presents the learning of multimodal adversarial
adaptation paradigm. Although the performance of multimodal UI-DAN and I-DAN
is not as good as the unimodal ones, it is still robust when addressing the
item-based cold-start recommendation problem. UI-DAN (a, m) and I-DAN (a, m),
however, significantly improve UI-DAN (s, m) and I-DAN (s, m). For instance,
I-DAN (a, m) outperforms I-DAN (s, m) by $\sim$26\% (RMSE)/$\sim$39\% (MAE)
for DM$\rightarrow$MI and $\sim$41\% (RMSE)/$\sim$51\% (MAE) for
HK$\rightarrow$OP, respectively.
\begin{table}
\footnotesize
\center
\caption{RecSys-DAN Results on CDs $\rightarrow$DM }
\label{tab:CDs}
\begin{tabular}{|l|cc|}
\hline
$\mathcal{D}^s\rightarrow\mathcal{D}^t$ & \multicolumn{2}{c|}{CDs$\rightarrow$DM } \\
Models & RMSE & MAE \\ \hline \hline
Normal & 1.452 $\pm$ 0.021 & 1.100 $\pm$ 0.022 \\
KNN & 1.110 $\pm$ 0.000 & 0.870 $\pm$ 0.000 \\
NMF & 1.062 $\pm$ 0.001 & 0.861 $\pm$ 0.001 \\
SVD++ & 1.061 $\pm$ 0.000 & 0.841 $\pm$ 0.001 \\
HFT & 1.099 $\pm$ 0.000 & 0.869 $\pm$ 0.000 \\
DeepCoNN & \color{blue!95}\textbf{1.038 $\pm$ 0.004} & \color{blue!95}\textbf{0.805 $\pm$ 0.003} \\
\hline \hline
Source Only & 1.131$\pm$ 0.028 & 0.857$\pm$ 0.080 \\
UI-DAN & 1.076$\pm$ 0.002 & 0.791$\pm$ 0.019 \\
U-DAN & 1.071$\pm$ 0.005 & 0.784$\pm$ 0.002 \\
I-DAN & 1.068$\pm$ 0.006 & 0.781$\pm$ 0.002 \\
H-DAN & \color{OliveGreen!95}\textbf{1.068$\pm$ 0.002} & \color{OliveGreen!95}\textbf{0.779$\pm$ 0.002} \\ \hline
$\Delta$ & 2.85\%$\pm$ 0.28\% & \textbf{3.28\%$\pm$ 0.32\%} \\ \hline
\end{tabular}
\end{table}
\begin{figure}[!]
\vspace{-4mm}
\center
\begin{subfigure}{0.75\linewidth}
\centering
\includegraphics[width=\linewidth]{multimodal_training.pdf}
\end{subfigure}
\caption{Learning multimodal UI-DAN. The labels and legends are the same as Fig.~\ref{fig:unimodal_training}.}
\label{fig:multi_train}
\vspace{-10pt}
\end{figure}
\subsubsection{Compare Different Instances of RecSys-DAN}
An experiment was conducted on CDs$\rightarrow$DM (unimodal) where both
shared users and items existed to further explore the different instances of
RecSys-DAN. The results in Tab.~\ref{tab:CDs} illustrate that unsupervised
domain adaptation models improve source-only baseline by $\sim$4.8\% (RMSE)
and $\sim$7.7\% (MAE). We find that U-DAN, I-DAN and H-DAN did not bring
significant improvements over UI-DAN. This is similar to the results of I-DAN
and UI-DAN in Tab.~\ref{tab:modality}. We conjecture the main reason is that
the rating prediction task is primarily based on the user-item interactions
(e.g., users express preferences on items). The interaction representations
are therefore of crucial importance as compared to user-level and item-level
representations, though the shared users/items could be beneficial when connecting domains.
\subsubsection{Compare to Cross-domain Recommendation Models}
We now compare our proposed architectures with the state-of-the-art
supervised models.
As the first attempt to utilize unsupervised adversarial domain adaptation
for the (cold-start) cross-domain recommendation, it is difficult to directly
compare RecSys-DAN with previous methods. Existing cross-domain (e.g.,
EMCDR~\cite{man2017cross}, CrossMF~\cite{iwata2015cross}, HST~\cite{liu2015non}) or hybrid collaborative filtering (e.g.,
DeepCoNN~\cite{zheng2017joint}, cmLDA~\cite{tan2014cross}) methods
are \textit{NOT} able to learn models in the scenarios where ratings and/or
review texts are completely not available for training. The Tab.~\ref{tab:dis} suggests previous methods' limitations, which are addressed by
our proposed adversarial domain adaptation method. Therefore, we compare
RecSys-DAN with supervised baselines indirectly.
\subsubsection{Compare to Supervised Models}
We trained the baselines directly on the target domain with labeled samples (Normal, KNN, NMF and SVD++ were trained
with user-item ratings, while HFT and DeepCoNN were trained with both ratings
and reviews). The goal is to examine how close the performance of
unsupervised RecSys-DAN without labeled target data to those supervised
methods which can access labeled target data. The results are reported in
Tab.~\ref{tab:modality} and Tab.~\ref{tab:CDs}. By purely transferring the
representations learned in the source domain to the target domain, our
methods achieve competitive performance compared to strong baselines.
Specifically, RecSys-DAN is able to achieve similar performance as NMF and
SVD++ with unsupervised adversarial adaptation and it outperforms baselines
on MAE in Tab.~\ref{tab:CDs}. From the aforementioned analysis, we can conclude
that ResSys-DAN has much better generalization ability and it is more
suitable to address practical problems such as cold-start recommendation.
\begin{table}[!t]
\footnotesize
\center
\caption{The comparison with RecSys-DAN and existing cross-domain recommendation methods. Existing methods have difficulties in learning a recommendation model when ratings on the target domain are completely missing.}
\label{tab:dis}
\begin{tabular}{|l|lc|} \hline
Methods & Required Target Inputs &Target Learning \\ \hline \hline
EMCDR~\cite{man2017cross} & rating & supervised \\
DeepCoNN~\cite{zheng2017joint} & rating, review &supervised \\
DLSCF~\cite{jiang2017deep} & rating, binary rating &supervised \\
CrossMF~\cite{iwata2015cross} & rating & supervised \\
CTSIF\_SVMs~\cite{yu2018svms} & rating & supervised \\
HST~\cite{liu2015non} & ratings & supervised \\
cmLDA~\cite{tan2014cross} & rating, review, description & supervised \\
RecSys-DAN & {review or image} & adversarial \\ \hline
\end{tabular}
\vspace{-5mm}
\end{table}
\begin{table*}[!t]
\centering
\caption{\small Exemplary predictions of RecSys-DAN (UI-DAN) on the target test set of ``office product'' with HK$\rightarrow$OP cross-domain recommendation. The first two examples are unimodal and the last two examples are multimodal based prediction. The predictions are purely based on transferring the representations of user-item interaction in the source domain (``home \& kitchen'') via an unsupervised and adversarial way. ``$<$UNK$>$'' means the word is not included in built vocabulary dictionary $\mathcal{VOC}$. We removed punctuations in reviews.}
\label{tab:examplel}
\scalebox{0.95}{
\begin{tabular}{|p{6.cm}|p{9cm}|p{1cm}p{1cm}|}
\hlinewd{1pt}
Reviews written by user & Reviews and/or Images associated to item & Prediction& Truth \\ \hline
has four internal pockets which is a nice addition round rings but with the better $<$UNK$>$ closure handy but could use slight improvement (...) & just what we needed good item great organizer less useful than i thought although may be just right for some colorful organizing okay (...) & \textbf{4.58} & \textbf{5} \\ \hline
worked well very cool great product great product works great awesome product well very easy to use & need a computer excellent for keeping organized in class durable easy to use super nice for presentations great quality and price great idea to (...) & \textbf{5.08} & \textbf{5} \\ \hline
good tape $<$UNK$>$ not very good flow good boxes but they come $<$UNK$>$ &
\begin{wrapfigure}{L}{0.07\textwidth}
\vspace{-5mm}
\begin{minipage}{0.85\textwidth}
\includegraphics[width=0.1\linewidth, height=0.05\textheight]{B0006HVPTW.jpg}
\end{minipage}
\end{wrapfigure}
\vspace{-0.5mm}
efficient tool best value for price while it lasted no frills sturdy sharpener for frequent pencil $<$UNK$>$ sharp works as it should noisy but good excellent maybe not perfect for your use (...)
& \textbf{4.15} & \textbf{4} \\ \hline
make sure you are on 24 $<$UNK$>$ wifi nice little printer must have unit cost too high nice $<$UNK$>$ &
\begin{wrapfigure}{L}{0.06\textwidth}
\vspace{-5.0mm}
\begin{minipage}{0.6\textwidth}
\includegraphics[width=0.1\linewidth, height=0.0375\textheight]{B002GHBUTU.jpg}\\\\
\end{minipage}
\end{wrapfigure}
\vspace{-1mm}
great a really nice little remote what a treat for powerpoint presentations only on some $<$UNK$>$ simple perfection (...) & \textbf{4.67} & \textbf{5} \\
\hlinewd{1pt}
\end{tabular}
}
\end{table*}
\subsubsection{Representation Alignment}
To examine the extent to which the adversarial objective aligns the source
and target latent representations, we randomly selected 2,000 test samples (1,000 from the source
and 1,000 from the target domain) for extracting latent
representations with $G_f$ at different epochs.
Fig.~\ref{fig:vis_embeddings} visualizes the source and target domain
representations. The source domain models' parameters are not updated during
the adversarial training of the target generators. Comparing the
representations at the 0$\textsuperscript{th}$ epoch (no adaptation) and
50$\textsuperscript{th}$, 100$\textsuperscript{th}$,
200$\textsuperscript{th}$ epochs, we can find that the distance between
the latent representations of the source and target domains is decreasing during adversarial learning, making target representations more indistinguishable to source representations. Fig.~\ref{fig:vis_weights} shows the visualization of weights for source and target domains after training. We can observe that the weights of the target mapping function $G_f^t$ approximate those of source mapping function $G_f^s$, which again demonstrates that RecSys-DAN succeeds in aligning the representations of the source and target domains through adversarial learning.
\begin{figure}[t]
\vspace{-5mm}
\begin{subfigure}{\linewidth}
\centering
\includegraphics[width=7.5cm,height=1.5cm]{digital_music_vis2.pdf}
\end{subfigure}
\begin{subfigure}{\linewidth}
\centering
\includegraphics[width=7.5cm,height=1.5cm]{home_office_vis1.pdf}
\end{subfigure}
\caption{t-SNE~\cite{maaten2008visualizing} visualizations of source (blue) and target (red) domain representations from UI-DAN (DM$\rightarrow$MI (top), HK$\rightarrow$OP (bottom)) at the 0$\textsuperscript{th}$ (no adaptation), the interaction representation adaptation at the 50$\textsuperscript{th}$, 100$\textsuperscript{th}$ and 200$\textsuperscript{th}$ epochs.}
\label{fig:vis_embeddings}
\end{figure}
\begin{figure}[t]
\center
\includegraphics[width=0.7\columnwidth]{weights1.pdf}\\
\caption{\small The visualization of weights $\Theta_f^s, \Theta_f^t \in \mathbb{R}^{512\times512}$ for source (top) and target (bottom) domains. The model is trained with DM$\rightarrow$MI domain pair, we only show the top-left 16$\times$64 of weight matrix for readability. The red circles highlight the patterns that shared by the weights of source and target domains.}
\label{fig:vis_weights}
\vspace{-5pt}
\end{figure}
\subsubsection{Cold-Start Recommendation}
Tab.~\ref{tab:examplel} presents some random rating prediction examples with
pre-trained RecSys-DAN models in unimodal and multimodal scenarios. We can
observe that representing users and items with reviews can effectively
alleviates the cold-start recommendation problem when ratings are completely
not available, since the proposed adversarial adaptation transfers the user,
item and their interaction representations from a labeled source domain to an
unlabeled target domain. It demonstrates the superiority of RecSys-DAN in
making preference prediction without the access to label information (i.e.,
ratings in this example). The existing recommendation
methods~\cite{lee2001algorithms, koren2008factorization, koren2010factor,
mcauley2013hidden, zheng2017joint} fail in this scenario.
\subsubsection{Running Time}
The pre-training of RecSys-DAN in the source domain took $\sim$10 epochs (avg.
69s/epoch). The adversarial training in both source and target domains took
$\sim$100 epochs to reach an equilibrium point. For inference, our model
performs as fast as baseline models, since RecSys-DAN directly adapts the
source scoring function.
\section{Conclusion}
\label{sec:con}
RecSys-DAN is a novel framework for cross-domain collaborative filtering,
particularly, the real-world cold-start recommendation problem. It learns to
adapt the user, item and user-item interaction representations from a source
domain to a target domain in an unsupervised and adversarial fashion.
Multiple generators and discriminators are designed to adversarially learn
target generators for generating domain-invariant representations. Four
RecSys-DAN instances, namely, UI-DAN, U-DAN, I-DAN, and H-DAN, are explored
by considering different scenarios characterized by the overlap of users and
items in both unimodal and multimodal settings. Experimental results
demonstrates that RecSys-DAN has a competitive performance compared to
state-of-the-art supervised methods for the rating prediction task, even with
absent preference information.
\ifCLASSOPTIONcaptionsoff
\newpage
\fi
\begin{small}
\bibliographystyle{unsrt}
|
1,108,101,564,543 | arxiv | \section{Introduction}
The observed thermal states of isolated neutron stars have become the primary
source to glean useful and interesting information about the internal structure
of neutron stars (NSs). Reconciliation of theoretical cooling curves with
observations of nearby isolated cooling NSs is a challenging task
\citep[see for e.g.][for a comprehensive review]{YakovlevPethick2004,Pageetal2006}. On
the theoretical front, the problem arises from incomplete knowledge of the composition
and equation of state (EOS) of matter in the NS core at supernuclear densities
($\rho > 10^{14}\rmn{ g cm}^{-3}$). Many possibilities can be realized: Depending
on the composition the EOS can be either \textit{soft} or \textit{stiff}, where
the stiffness characterizes the compressibility of matter, and strongly depends
on the internal degrees of freedom of the system \cite[e.g.][]{Schaabetal1996}. For
example, for a polytropic EOS, $P=K\rho^\Gamma$, a larger adiabatic index $\Gamma$
yields stiffer EOS. Such an EOS generally produces larger maximum masses and
radii of NSs than its softer counterpart. Softer EOSs can be obtained by the
introduction of phase transitions in the theory where the core of the NS may
be composed of boson condensates, quark or hyperonic matter. However, the presence
of these at nuclear densities has been ruled out from the observation of a
$1.97\pm0.04~\rmn{M}_\odot$ NS \citep{Demorestetal2010}. The cooling behavior
of NSs depends very sensitively on the composition of the inner core, such that
it affects the choice of the neutrino cooling process that is dominant in the first
$10^4 - 10^5~\rmn{yrs}$ of its evolution \citep[see for e.g.][for a detailed
description of all the neutrino emission processes in NSs]{Yakovlevetal2001}.
X-ray observations of isolated cooling NSs have been crucial in
their discovery \citep[see for e.g.][]{Mereghetti2011}. However, these observations
present its own set of challenges in determining their
cooling rate. Here, one is interested in detecting radiation emanating from the surface of the
NS which, in the case of pulsars, is complicated by the non-thermal emission from the
magnetosphere. Also, the non-uniform
heating of the crust due to energetic particles accelerated in the magnetosphere render accurate
determination of effective temperatures hard \citep[see for e.g. the review by][on
surface emission from NSs]{Ozel2013}. The unknown composition of NS atmospheres leads to
the overestimation of their temperatures when fitting their spectra with a blackbody
\citep[e.g.][]{Lloydetal2003}. Apart from the
uncertainties involved in the spectral modelling of NS surface emission, uncertainties
in their distances can also contribute to poor effective temperature and luminosity
estimates. Furthermore, to place observed isolated sources on cooling curves, accurate ages
are needed that may be hard to obtain as we discuss below.
In this study, we show that, for a large sample, ages of isolated thermal emitters can
be derived from statistical arguments. As the sample size grows, the statistical error
diminishes. We look at a small group of thermally emitting NSs discovered in the
ROSAT all-sky survey in Sec. \ref{sec:M7} and derive their ages statistically in Sec.
\ref{sec:ages}. Lastly, we discuss the possibility of improving the meager sample of
isolated thermal emitters by detecting more such objects in the upcoming eROSITA
all-sky survey.
\section{The Magnificent Seven}\label{sec:M7}
Nearby isolated cooling neutron stars are particularly important for confronting
cooling models with observations. They were discovered by Einstein, ROSAT,
and ASCA space telescopes \citep{BeckerPavlov2002}, and were further observed
with the extremely sensitive and high resolution high-energy space telescopes Chandra
and XMM-Newton. Among all the discovered objects, the most interesting are the seven
radio-quiet thermally emitting isolated NSs (a.k.a. the magnificent seven, M7)
discovered in the ROSAT all-sky survey (RASS) \citep[see for e.g.][for a review]{Haberl2007}.
These radio-quiet objects radiate predominantly in X-rays with
high X-ray to optical flux ratios, $f_X/f_{opt} > 10^{4 - 5}$. Their soft X-ray
spectra are reasonably well fit by an absorbed blackbody-like spectrum with
$kT \la 100$ eV and a hydrogen column density $n_H \sim 10^{20}\mbox{ cm}^{-2}$,
indicating small distances $d \sim \mbox{ few }\times 100$ pc. Astrometric measurements
of some of the member objects independently confirm the distances inferred from
column densities (see references in Table \ref{tbl:m7table}). That the thermal
emission is coming from majority of the stellar surface is confirmed by the small
pulse fractions $\la 20 \%$ of the X-ray light curves. Spin periods ranging
from 3 - 12 s have been measured for all but one (RX J1605.3+3249)
of the M7 objects \citep[see for e.g.][Table \ref{tbl:m7table}]{Mereghetti2011}.
This in conjunction with the measured spin-down rates,
$\dot{P} \sim 10^{-14} - 10^{-13}\mbox{ s s}^{-1}$, yields an estimate of the polar
magnetic field strengths $B_p\sim 10^{13}$ G and the characteristic spin-down ages
$\tau_c\sim 10^6$ years \citep{KaplanKerkwijk2005b,KaplanKerkwijk2005a,KaplanKerkwijk2009b,
KaplanKerkwijk2009a,KaplanKerkwijk2011,KerkwijkKaplan2008}.
Measurements of the high proper motions of three of the M7
objects and their association thus established to the Sco OB2 complex comprising the
Gould Belt yield kinematic and/or dynamical ages that are smaller than ages inferred from
spin down \citep{Kaplanetal2002,Kaplanetal2007,WalterLattimer2002,Motchetal2005,
Motchetal2009,Tetzlaffetal2010,Tetzlaffetal2011,Mignanietal2013}.
The discrepancy between characteristic and kinematic ages strongly suggests
that the spin-down ages are overestimates and the M7 objects in reality are much younger
\citep[see for e.g.][]{KaplanKerkwijk2005b,KaplanKerkwijk2009b}.
Even when considering simple cooling models \citep{HeylHernquist1998, Ponsetal2009},
one finds the spin-down ages to be $3-4$ times in excess of the cooling ages of $\sim 0.5~\rmn{Myr}$
\citep{Kaplanetal2002b}.
\subsection{Spin-Down Ages: Poor Age Estimators}
According to the standard magnetic dipole model of pulsars \citep{ShapiroTeukolsky1983},
a rotating NS with a polar magnetic field spins down over time by emitting magnetic dipole
radiation. From the rate of change of the angular frequency $\dot{\Omega}$, the spin-down age of the NS can
be readily determined, with the assumption that the initial angular frequency is much larger than the
present value ($\Omega_0 \gg \Omega(t)$),
\begin{equation}
\tau = \frac{\Omega}{2\|\dot{\Omega}\|}
\label{eq:sdage}
\end{equation}
This assumption is invalid in the case of CCOs as these objects are believed to have their initial periods
very close to the current values \citep[e.g.][]{HalpernGotthelf2010}.
The spin-down law implicitly assumes that none
of the other physical characteristics of the pulsar vary over time. This may not be the
case and, in general, the spin-down law \citep{lynesmith2006} written as the following
can be allowed to include variation of $B_p$, the moment of inertia $I$, and the angle
between the rotation axis and the magnetic dipole axis $\alpha$, so that
\begin{equation}
\frac{d\Omega}{dt} = -\kappa(t)\Omega(t)^n
\end{equation}
where $\kappa(t)$ is usually assumed to be a constant and $n = 3$ is the \textit{braking index}
for magnetic dipole braking. Any change in $\kappa$ with
time naturally yields ages of pulsars that are in conflict with their spin-down ages;
Generally, the spin-down age should only be taken as a rough estimate to aid in calculations.
An independent age estimate is provided by the age of the associated supernova remnant
(SNR) or massive star cluster for younger objects. Establishing such an association for
older NSs may prove to be difficult since SNRs fade away in $\sim 60$ kyr, and in the
same time, due to natal kicks ($\sim 500\mbox{ km s}^{-1}$), NSs may move significantly
far away from their birth sites \citep{Frailetal1994}. We plot the spin-down
and the estimated SNR ages for young pulsars ($\tau < 10^5$ yrs), central compact
objects (CCOs), and magnetars (SGRs and AXPs) along with their timing properties
\citep[see for e.g.][for a review]{Becker2009} in Fig. \ref{fig:snrVtc_fdecay}, and it is clear
that for many NSs, that are not the typical spin-down powered radio pulsars,
the characteristic age is a poor age estimator. The objects that have SNR
ages smaller than their spin-down ages can be explained by having a braking index less
than the \textit{canonical} value, $n < 3$. An excellent example supporting this notion
is the Vela pulsar which has a very small breaking index $n = 1.4\pm0.2$ estimated from
an impressive 25-year long observation \citep{Lyneetal1996}, albeit under the assumption
that $\kappa$ is still a constant. This yields a spin-down age of 25.6 kyr, making it
appear more than twice as old as its age inferred from the standard magnetic braking
scenario. This result is well supported by the estimated age of the Vela SNR
($t_{\rmn{SNR}}\sim18 - 31$ kyr) \citep{Aschenbachetal1995}.
On the other hand, for objects that have spin-down ages larger than that of their true ages,
that may be inferred from their associated SNR ages, it
can be argued that the magnetic moments decrease in strength over time
\citep{HeylKulkarni1998}. There are
three main mechanisms by which magnetic fields can decay in isolated NS, namely
Ohmic dissipation, ambipolar diffusion, and Hall drift \citep{GoldreichReisenegger1992}.
The timescale over which the field decays substantially due to these processes
\citep[see for e.g.][]{HeylKulkarni1998}
\begin{figure*}
\centering
\includegraphics[width=0.8\textwidth]{snrVtc.eps}
\caption{Change in the spin-down or characteristic age of an isolated NS due to various
magnetic field decay mechanisms, namely the ambipolar (I)rrotational
($a = 0.01$, $\alpha = 5/4$, dotted) and
(S)olenoidal ($a = 0.15$, $\alpha = 5/4$, dashed) modes,
and the Hall cascade ($a = 10$, $\alpha = 1$, dot-dashed)
\citep[See Eq. 2 - 4 of][]{Colpietal2000}.
The solid black line denotes $t = \tau$, meaning no field decay. This assumes
an initial field strength $B_0 = 10^{16}$ G and period $P_0 = 1$ ms. The Crab pulsar
is shown here with a black diamond.}
\label{fig:snrVtc_fdecay}
\end{figure*}
determines the dominating process at different stages in the evolution of an isolated NS.
An important consequence of field decay is that it leads to an overestimation of the
real age of the NS. Following the analysis of \citep[][see Eq. 2, 3, and 4]{Colpietal2000}, we
plot in Fig. \ref{fig:snrVtc_fdecay} the change in the spin-down age of the object over time due to the decaying strength of
the magnetic field by the aforementioned processes.
\begin{equation}
\tau(t) = \frac{P(t)^2}{2bB(t)^2}
\end{equation}
where $b \approx 10^{-39}$ in cgs units. The effect of field decay at late times
is apparent from the divergence of the characteristic age from the real age.
\begin{table*}
\caption{Properties of isolated NSs with SNR or massive star cluster associations}
\begin{minipage}{\textwidth}
\centering
\begin{threeparttable}
\begin{tabular}{ccccccc}
\hline
\hline
Object & $P$ & $\dot{P}$ & $t_{\rmn{sd}}$ & $t_{\rmn{snr}}$ & SNR/Cluster & References\\
& (s) & $(10^{-15}~\rmn{s~s}^{-1})$ & $(\rmn{kyr})$ & $(\rmn{kyr})$ & & \\
\hline
\multicolumn{6}{c}{Young Pulsars$^{\dagger}$} \\
\hline
0531+21 & 0.033 & 421 & 1.3 & 0.90 & Crab Nebula & \citetalias{Hester2008}, \citetalias{Nugent1998} \\
1509-58 & 0.150 & 1536 & 1.691 & $6-20$ & MSH 15-52 & \citetalias{SewardHarnden1982},
\citetalias{Manchesteretal1982}, \citetalias{Weisskopfetal1983}, \citetalias{Kaspietal1994}, \citetalias{Sewardetal1983} \\
0833-45 & 0.089 & 125 & 11.3 & $18 - 31$ & Vela XYZ & \citetalias{Largeetal1968}, \citetalias{Dodsonetal2002}, \citetalias{Aschenbachetal1995} \\
1853+01 & 0.267 & 208 & 20.3 & $15-20$ & W44 & \citetalias{Wolszczanetal1991}, \citetalias{DermerPowale2013}\\
0540-69 & 0.050 & 479 & 1.66 & $0.76-1.66$ & SNR 0540-693 & \citetalias{Sewardetal1984}, \citetalias{Parketal2010} \\
1610-50 & 0.232 & 495 & 7.4 & $\sim 3.0$ & Kes 32 & \citetalias{Johnstonetal1992}, \citetalias{Vink2004} \\
1338-62 & 0.193 & 253 & 12 & $\sim 32.5$ & G308.8-0.1 & \citetalias{Manchesteretal1985}, \citetalias{Caswelletal1992} \\
1757-24 & 0.125 & 128 & 15.5 & $\ga 70$ & G5.4-1.2 & \citetalias{Manchesteretal1985}, \citetalias{Hobbsetal2004}, \citetalias{Blazeketal2006} \\
1800-21 & 0.134 & 134 & 15.8 & $15 - 28$ & W30 & \citetalias{CliftonLyne1986}, \citetalias{Yuanetal2010}, \citetalias{FinleyOegelman1994} \\
1706-44 & 0.102 & 93 & 17.5 & $8-17$ & G343.1-2.3 & \citetalias{Johnstonetal1992}, \citetalias{Dodsonetal2002} \\
1930+22 & 0.144 & 57.6 & 40 & $\sim 486$ & G57.1+1.7 & \citetalias{HulseTaylor1975}, \citetalias{Hobbsetal2004}, \citetalias{Kovalenko1989} \\
2334+61 & 0.495 & 193 & 41 & $7.7$ & G114.3+0.3 & \citetalias{Deweyetal1985}, \citetalias{Yuanetal2010}, \citetalias{Yaretal2004} \\
1758-23 & 0.416 & 113 & 59 & $33-150$ & W28 & \citetalias{Manchesteretal1985}, \citetalias{Yuanetal2010}, \citetalias{SawadaKoyama2012} \\
\hline
\multicolumn{6}{c}{CCOs} \\
\hline
RX J0822.0-4300 & 0.122 & $0.00928\pm0.00036$ & $2.54\times10^3$ & $3.7-4.45$ & Pupppis A & \citetalias{Gotthelfetal2013}, \citetalias{WinklerKirshner1985},
\citetalias{Beckeretal2012} \\
1E 1207.4-5209 & 0.424 & $0.02224\pm0.00016$ & $3.02\times10^3$ & $\sim 10$ & G296.5+10.0 & \citetalias{Gotthelfetal2013}, \citetalias{Vasishtetal1997} \\
CXOU J185238.6 & 0.105 & $0.00868\pm0.00009$ & 7 & $3-7.8$ & Kes 79 & \citetalias{HalpernGotthelf2010}, \citetalias{Sunetal2004} \\
\hline
\multicolumn{6}{c}{Magnetars$^{\dagger\dagger}$} \\
\hline
CXOU J164710.2 & 10.6 & $<400$ & $>420$ & $(4\pm1)\times10^3$ & Westerlund 1${}^\star$ & \citetalias{Anetal2013}, \citetalias{Munoetal2006} \\
CXOU J171405.7 & 3.83 & $6.4\times10^4$ & 0.95 & $\sim1.5$ & CTB 37B & \citetalias{Satoetal2010}, \citetalias{HalpernGotthelf2010a} \\
1E 1841-045 & 11.78 & $3.93\times10^4$ & 4.8 & $\sim 2$ & Kes 73 & \citetalias{Dibetal2008}, \citetalias{VasishtGotthelf1997} \\
1E 2259+586 & 6.98 & 484 & 230 & 14 & CTB 109 & \citetalias{GavriilKaspi2002}, \citetalias{Saakietal2013} \\
1E 1048.1-5937 & 6.46 & $2.25\times10^4$ & 4.5 & $(1.1\pm0.5)\times10^3$ & GSH 288.3-0.5-2.8 & \citetalias{Dibetal2009}, \citetalias{Gaensleretal2005} \\
SGR 0526-66 & 8.05 & $3.8\times10^4$ & 3.4 & $\sim 4.8$ & N49 & \citetalias{Tiengoetal2009}, \citetalias{Parketal2012} \\
SGR 1627-41 & 2.59 & $1.9\times10^4$ & 2.2 & $\sim 5$ & G337.0-0.1 & \citetalias{Espositoetal2009}, \citetalias{Espositoetal2009a}, \citetalias{Corbeletal1999} \\
SGR 1900+14 & 5.2 & $9.2\times10^4$ & 0.90 & $> 6\pm1.8$ & Massive star cluster & \citetalias{Mereghettietal2006}, \citetalias{Tendulkaretal2012} \\
SGR 1806-20 & 7.6 & $7.5\times10^5$ & 0.16 & $> 0.65\pm0.3$ & Massive star cluster & \citetalias{Nakagawaetal2009}, \citetalias{Tendulkaretal2012} \\
\hline
\end{tabular}
\begin{tablenotes}
\setcounter{footnote}{0}
\item $^\dagger$ http://www.atnf.csiro.au/research/pulsar/psrcat (\citet{Manchesteretal2005});
$^{\dagger\dagger}$ http://www.physics.mcgill.ca/~pulsar/magnetar/main.html;
$^\star$ Westerlund 1 is a massive star cluster;
\citepalias{Hester2008} \citet{Hester2008} and references therein;
\citepalias{Nugent1998} \citet{Nugent1998};
\citepalias{SewardHarnden1982} \citet{SewardHarnden1982};
\citepalias{Manchesteretal1982} \citet{Manchesteretal1982};
\citepalias{Weisskopfetal1983} \citet{Weisskopfetal1983};
\citepalias{Kaspietal1994} \citet{Kaspietal1994};
\citepalias{Sewardetal1983} \citet{Sewardetal1983};
\citepalias{Largeetal1968} \citet{Largeetal1968};
\citepalias{Dodsonetal2002} \citet{Dodsonetal2002};
\citepalias{Aschenbachetal1995} \citet{Aschenbachetal1995};
\citepalias{Wolszczanetal1991} \citet{Wolszczanetal1991};
\citepalias{DermerPowale2013} \citet{DermerPowale2013};
\citepalias{Sewardetal1984} \citet{Sewardetal1984};
\citepalias{Parketal2010} \citet{Parketal2010};
\citepalias{Johnstonetal1992} \citet{Johnstonetal1992};
\citepalias{Vink2004} \citet{Vink2004};
\citepalias{Manchesteretal1985} \citet{Manchesteretal1985};
\citepalias{Caswelletal1992} \citet{Caswelletal1992};
\citepalias{Hobbsetal2004} \citet{Hobbsetal2004};
\citepalias{Blazeketal2006} \citet{Blazeketal2006};
\citepalias{CliftonLyne1986} \citet{CliftonLyne1986};
\citepalias{Yuanetal2010} \citet{Yuanetal2010};
\citepalias{FinleyOegelman1994} \citet{FinleyOegelman1994};
\citepalias{Dodsonetal2002} \citet{Dodsonetal2002};
\citepalias{HulseTaylor1975} \citet{HulseTaylor1975};
\citepalias{Kovalenko1989} \citet{Kovalenko1989};
\citepalias{Deweyetal1985} \citet{Deweyetal1985};
\citepalias{Yaretal2004} \citet{Yaretal2004};
\citepalias{SawadaKoyama2012} \citet{SawadaKoyama2012};
\citepalias{Gotthelfetal2013} \citet{Gotthelfetal2013};
\citepalias{WinklerKirshner1985} \citet{WinklerKirshner1985};
\citepalias{Beckeretal2012} \citet{Beckeretal2012};
\citepalias{Vasishtetal1997} \citet{Vasishtetal1997};
\citepalias{HalpernGotthelf2010} \citet{HalpernGotthelf2010};
\citepalias{Sunetal2004} \citet{Sunetal2004};
\citepalias{Anetal2013} \citet{Anetal2013};
\citepalias{Munoetal2006} \citet{Munoetal2006};
\citepalias{Satoetal2010} \citet{Satoetal2010};
\citepalias{HalpernGotthelf2010a} \citet{HalpernGotthelf2010a};
\citepalias{Dibetal2008} \citet{Dibetal2008};
\citepalias{VasishtGotthelf1997} \citet{VasishtGotthelf1997};
\citepalias{GavriilKaspi2002} \citet{GavriilKaspi2002};
\citepalias{Saakietal2013} \citet{Saakietal2013};
\citepalias{Dibetal2009} \citet{Dibetal2009};
\citepalias{Gaensleretal2005} \citet{Gaensleretal2005};
\citepalias{Tiengoetal2009} \citet{Tiengoetal2009};
\citepalias{Parketal2012} \citet{Parketal2012};
\citepalias{Espositoetal2009} \citet{Espositoetal2009};
\citepalias{Espositoetal2009a} \citet{Espositoetal2009a};
\citepalias{Corbeletal1999} \citet{Corbeletal1999};
\citepalias{Mereghettietal2006} \citet{Mereghettietal2006};
\citepalias{Tendulkaretal2012} \citet{Tendulkaretal2012};
\citepalias{Nakagawaetal2009} \citet{Nakagawaetal2009};
\end{tablenotes}
\end{threeparttable}
\end{minipage}
\label{tab:youngpulsars}
\end{table*}
\section{True Age Estimates of Isolated Neutron Stars}\label{sec:ages}
The M7 objects don't have any SNR or massive star cluster associations. Therefore,
ages for these objects have been derived from their $P$ and $\dot{P}$ measurements.
In addition, since they are nearby objects ($d \la 500$ pc), and due to their large
proper motions, kinematic ages became a possibility and have been estimated for only
four of the group members (see Table \ref{tbl:m7table}). In this
case, one finds that the spin-down ages are larger by a factor of $3 -
10$ than the kinematic ages. Accurate age estimates are extremely important in determining
the cooling behavior of isolated NSs. Overestimated ages used to fit model
cooling curves can obscure the determination of the true thermal state of these objects.
\subsection{Age Estimates from Population Synthesis}
In the following, we estimate the true ages of the M7 members by a method that is
motivated by another method devised by \citet{Schmidt1968} and then applied by
\citet{HuchraSargent1973} to calculate the luminosity function of field galaxies.
The original idea is implemented as follows. An apparent magnitude limited sample
is first obtained and it is assumed that the objects in the sample, field galaxies
for instance, are distributed uniformly in Euclidean space, such that the
luminosity function is independent of the distance. Then,
to each object an \textit{accessible volume} $V_{\rmn{max}}(M)$ is assigned \citep{AvniBahcall1980}.
This volume depends on the absolute magnitude of the object and gives a measure of the volume
surveyed for a given object with an absolute magnitude $M$. Basically, $V_{\rmn{max}}(M)$ is
the maximum volume in which the object would be detected had its apparent magnitude
been equal to the limiting magnitude of the survey. The whole sample is then divided
into bins of size $dM$ with objects having absolute magnitude in the range $[M-dM/2,M+dM/2]$.
The luminosity function for this bin is estimated by adding the inverse of the
accessible volumes for each object in that bin
\begin{equation}
\Phi(M) (\mbox{ Mpc}^{-3}\mbox{ mag}^{-1}) = \sum_i \frac{1}{V_{i,\rmn{max}}(M)}.
\end{equation}
This method provides a non-parametric way of estimating the luminosity function
and it exactly reproduces the true luminosity function within statistical
errors \citep{HartwickSchade1990}. Furthermore, this is a very general method
which relies on only one underlying assumption that the objects are distributed
according to their intrinsic brightnesses.
In the case of neutron stars (unlike galaxies) it is reasonable to
assume that the birthrate has been constant over the past few million
years, and furthermore, the neutron stars progress from bright to
faint luminosities as they age in the same (albeit unknown) way. Under
these assumptions we can deduce the age of a neutron star of a given
absolute magnitude
\begin{equation}
t(M) = \frac{1}{\beta}\int_{-\infty}^{M} \Phi(M') dM'
\end{equation}
where $\beta$ is the neutron-star birthrate per volume.
\subsection{RASS and Population Synthesis}
In the following, we develop a slight variant of the
\citet{Schmidt1968} estimator to calculate the true ages of the
members of the M7 family. The method we develop cannot be completely
model independent as the distribution of NSs in space, unlike that of galaxies
over large scales, is not uniform. Since the progenitors of NSs mainly
reside in the arms of a spiral galaxy, and for a natal kick velocity
of, say $\sim 500 \mbox{ km s}^{-1}$, the NSs only travel a distance
of $\sim 50$~pc from their birth sites within $\sim 10^5$ years. This
is small compared to the scale height of the thin disk $\sim 300$~pc
\citep{BinneyMerrifield1998}. As a result, the \citet{Schmidt1968}
estimator cannot be used here. Instead, we look at the NS progenitor
population and calculate the weight, that is used to determine the
statistical age (see Eq. \ref{eq:age}), for each M7 member by
counting the number of massive OB stars that are found in the
accessible volume $V_{\mathrm{max}}$ for that object. The population
synthesis method is given in our earlier study \citep{GillHeyl2007},
and essentially requires the assumption of the luminosity function and
spatial distribution of massive OB stars in the galaxy
\citep{BahcallSoneira1980}, and the distribution of HI, which we model
as a smooth exponential disk both radially and vertically
\citep{FosterRoutledge2003}. As pointed out in \citet{Posseltetal2007},
local clumpiness of the ISM will affect the level of absorption. Thus,
distances derived from assuming a homogeneous model will also be affected. This presents
a very small degree of uncertainty in $V_\rmn{max}$, that is insignificant
compared to other uncertainties in the model, e.g. the uncertainty in
the SN rate used to derive statistical ages (see below).
All M7 objects were discovered by ROSAT which scanned the whole sky
with a limiting count rate of $0.015 \mbox{ cts s}^{-1}$ in the energy
range $\sim 0.12 - 2.4$ keV \citep[see][for more details]{Hunschetal1999}. The complete survey covers $92\%$ of the sky for a count
rate of $0.1 \mbox{ cts s}^{-1}$ \citep{Vogesetal1999} and has yielded
the most complete and sensitive survey of the soft X-ray sky. Therefore, it
provides a perfect flux-limited sample for our study. Next, we
calculate the weights for each object in the sample by simulating the
RASS and finding the total number of massive OB stars in the volume
$V_{\rmn{max}}$, such that the statistically estimated age is given in terms of the
typical age of their progenitors $t_{\mathrm{OB}}$,
\begin{equation}
t_{i,\rmn{stat}} \sim t_{\mathrm{OB}}\sum_{j=1}^{j=i}\frac{N_j}{N_{j\mathrm{,OB}}}
\rightarrow t_{\mathrm{OB}}\sum_{j=1}^{j=i}\frac{1}{N_{j\mathrm{,OB}}}
\label{eq:age}
\end{equation}
where $N_j$ is the number of M7 objects in a small absolute magnitude bin of
size $dM$ centered at $M_j$. Since the sample is of marginal size,
$N_j = 1$ in this case. Then, $1/N_{j\rmn{,OB}}$ gives the number of
massive OB stars per $j^{\rmn{th}}$ object in the sample. We first
calculate the number of OB stars that lie in the volume $V_{\rmn{max}}$
for each M7 member. Then, we rank the M7 objects with respect to their
effective temperatures with the hottest member ranked first $(i=1)$ and the
coolest ranked last $(i=7)$. According to Eq. \ref{eq:age}, the weight for
the first object is $N^{-1}_{1,\rmn{OB}}$, the second is
$(N^{-1}_{1,\rmn{OB}}+N^{-1}_{2,\rmn{OB}})$, and so on.
The ages of NS progenitors are highly uncertain and are usually
obtained by estimating the main sequence turn-off ages of the massive
star cluster to which the NS may be associated \citep[see for
e.g.][for a review]{Smartt2009}. Typical ages of $\sim 3 - 15$
Myr have been estimated for the progenitors of NSs and
magnetars. \citet{Figeretal2005} report the age of the cluster of
massive stars, containing three Wolf-Rayet stars and a post
main-sequence OB supergiant, associated to the magnetar SGR $1806 -
20$ to be roughly $3.0 - 4.5$ Myr. Also, \citet{Munoetal2006} report
an age of $4\pm1$ Myr for the cluster Westerlund 1 which seems to be
the birth site of another magnetar CXOU J$164710.2 - 455216$. In yet
another study, \citet{Daviesetal2009} find the age of the cluster
associated to the magnetar SGR $1900 + 14$ to be $14\pm1$ Myr. Since
the spin-down ages of magnetars are much smaller ($\sim 10^{3 - 4}$ yr)
than that of the clusters, the notion that the cluster age reflects
the age of the progenitor, under the assumption of coevality of its
members, is a valid one. In the case of SGR $1806 - 20$ and CXOU
J$164710.2 - 455216$, both groups find that the progenitor must be a
massive star with $M > 40 M_\odot$, except in the last study where the
progenitor of SGR $1900 + 14$ is claimed to be a lower mass star with
initial MS mass of $17\pm2 M_\odot$. Notwithstanding this last result,
it has been claimed that magnetars may be the progeny of only
sufficiently massive stars ($M \ga 25 M_\odot$)
\citep{Gaensleretal2005} that would, otherwise, have resulted in the
formation of a black hole. Although the members of the M7 family are
endowed with fields an order of magnitude higher than the normal radio
PSRs, they are not magnetars and can be argued to be the descendants
of progenitors not much more massive than that of the normal radio
PSRs. In that case, it is expected that the progenitor age
$t_{\rmn{OB}}$ will be considerably longer in comparison to that of
magnetar progenitors. An upper limit on the ages of M7 progenitors
can be placed from the age of the Gould Belt,
$t_{\rmn{OB}}\leq t_{\rmn{GB}}\sim 30 - 60~\rmn{Myr}$ \citep{Torraetal2000}.
From Eq. \ref{eq:age} the ages of the sample objects are proportional
to $t_{\rmn{OB}}$, thus significant uncertainty in the progenitor age
will yield erroneous ages. We estimate $t_{\rmn{OB}}$ by considering
the total number of OB stars in the Galaxy and the supernova rate
corresponding to type Ib/c and type II supernovae,
\begin{equation}
t_{\rmn{OB}} \approx \frac{N_{\rmn{OB,Gal}}}{\beta_{\rmn{SN}}}
\label{eq:tob}
\end{equation}
From our modeling of OB stars in the Galaxy \citep{GillHeyl2007}, we
estimate $N_{\rmn{OB,Gal}} \sim 5.2\times10^5$, and using the supernova (SN)
rate reported by \citet{Diehletal2006} $\beta_{\rmn{SN}} = 1.9\pm1.1$ per
century, we find $t_{\rmn{OB}} \approx 27\pm16~\rmn{Myr}$.
\begin{table*}
\setcounter{footnote}{0}
\caption{Properties of nearby thermally emitting isolated NSs}
\begin{threeparttable}
\begin{tabular}{llllllllll}
\hline
\hline
Object & $P$ & $\dot{P}$ & $D$ & $T_{bb}$ & $N_H$ & $F_x$ & $N_{OB}$ & $t_{\rmn{stat}}\pm\Delta t_\rmn{sys}$ & $t_{\rmn{kin/dyn}}$ \\
& (s) & $(10^{-14}\mbox{ s s}^{-1})$ & (pc) & (eV) & $(10^{20}\mbox{ cm}^{-2})$ &
& & (Myr) & (Myr) \\
\hline
RBS 1223\tnote{1} & 10.31\tnote{2} & 11.20\tnote{2} & $\geq 525$\tnote{3} & $118\pm13$ &
$0.5 - 2.1$ & 4.5 & $368^{+32}_{-28}$ & $0.073\pm0.005$ & $0.5-1$\tnote{25} \\
2XMM J$104608.7^\star$\tnote{4} & - & - & 2000 & $117\pm14$ & $35\pm11$ &
0.097 & $1702^{+668}_{-477}$ & $0.089\pm0.008$ \\
RX J$1605.3 + 3249$\tnote{5} & - & - & $325 - 390$\tnote{6} & $86 - 98$ &
$0.6 - 1.5$ & 1.15 & $181^{+414}_{-23}$ & $0.23\pm0.18$ & $0.45-3.5$\tnote{26} \\
RBS 1774\tnote{7} & 9.437\tnote{8} & $4.1\pm1.8$\tnote{9} & $390 - 430$\tnote{10} & $92^{+19}_{-15}$
& $4.6\pm0.2$ & 8.7 & $588^{+138}_{-75}$ & $0.28\pm0.18$ \\
RX J$0806.4 - 4123$\tnote{11} & 11.37\tnote{12} & $5.5\pm3.0$\tnote{12} & $240\pm25$\tnote{13} &
$78\pm7$ & $2.5\pm0.9$ & 2.9 & $211^{+26}_{-27}$ & $0.41\pm0.18$ \\
RX J$0720.4 - 3125$\tnote{14} & 8.39\tnote{15} & $6.98\pm0.02$\tnote{15} & $330^{+170}_{-80}$\tnote{16} &
$79\pm4$ & $1.3\pm0.3$ & 11.5 & $278^{+42}_{-35}$ & $0.51\pm0.18$ & $0.5-1$\tnote{17} \\
RX J$0420.0 - 5022$\tnote{18} & 3.45\tnote{19} & $2.8\pm0.3$\tnote{19} & 350\tnote{20} & $57^{+25}_{-47}$
& 1.7 & 0.69 & $713^{+99}_{-89}$ & $0.55\pm0.18$ \\
RX J$1856.5 - 3754$\tnote{21} & 7.06\tnote{22} & $2.97\pm0.07$\tnote{22} & $161^{+18}_{-14}$\tnote{23}
& $57\pm1$ & $1.4\pm0.1$ & 14.6 & $604^{+256}_{-115}$ & $0.59\pm0.18$ & $\sim 0.4$\tnote{24} \\
\hline
\end{tabular}
\begin{tablenotes}[flushleft]
\setcounter{footnote}{0}
\item $F_x (10^{-12}\mbox{ erg cm}^{-2}\mbox{ s}^{-1})$ - The absorbed X-ray flux in the ROSAT energy band ($0.12 - 2.4$ keV).
\item ${}^\star$ 2XMM J$104608.7 - 594306$
\item [1] \citet{Schwopeetal1999}
\item [2] \citet{KaplanKerkwijk2005a}
\item [3] \citet{Posseltetal2007}
\item [4] \citet{Piresetal2009}
\item [5] \citet{Motchetal1999}
\item [6] \citet{Posseltetal2007}
\item [7] \citet{Zampierietal2001}
\item [8] \citet{Zaneetal2005}
\item [9] \cite{KaplanKerkwijk2009a}
\item [10] \citet{Posseltetal2007}
\item [11] \citet{Haberletal1998}
\item [12] \citet{KaplanKerkwijk2009b}
\item [13] \citet{Motchetal2008}
\item [14] \citet{Haberletal1997}
\item [15] \citet{KaplanKerkwijk2005b}
\item [16] \citet{Kaplanetal2007}
\item [17] \citet{Kaplanetal2007,Tetzlaffetal2011}
\item [18] \citet{Haberletal1999}
\item [19] \citet{KaplanKerkwijk2011}
\item [20] \citet{Posseltetal2007}
\item [21] \citet{Walteretal1996}
\item [22] \citet{KerkwijkKaplan2008}
\item [23] \citet{Kaplanetal2007}
\item [24] \citet{Mignanietal2013,Tetzlaffetal2011,Kaplanetal2002,WalterLattimer2002}
\item [25] \citet{Tetzlaffetal2010}
\item [26] \citet{Tetzlaffetal2012}
\end{tablenotes}
\end{threeparttable}
\label{tbl:m7table}
\end{table*}
Over the last few years, two new candidates have been added to the M7
group. The first object, 1RXS J$141256.0+792204$ dubbed
\textit{Calvera} \citep{Rutledgeetal2008}, was actually cataloged in
the RASS Bright Source Catalog \citep{Vogesetal1999} for having a high
X-ray to optical flux ratio $F_X/F_V > 8700$. However, its large height above the Galactic plane
($z\approx5.1$ kpc), requiring a space velocity $v_z\ga5100\rmn{ km
s}^{-1}$, presents a challenge for its interpretation as an isolated
cooling NS like the M7 members \citep[see][for a detailed discussion]{Rutledgeetal2008}.
Also, recent X-ray observations of Calvera done
with the XMM-Newton space telescope found unambiguous evidence for
pulsations with period $P = 59.2~\mathrm{ms}$
\citep{Zaneetal2011}. The authors of this study argued that Calvera is
most probably a CCO or a slightly recycled pulsar. The uncertainty in
its nature \citep[see][]{halpern2011} doesn't warrant inclusion into
our sample of radio-quiet isolated NSs. The second object 2XMM
J$104608.7 - 594306$ \citep{Piresetal2009}, discovered serendipitously
in an XMM-Newton pointed observation of the Carina Nebula hosting the
binary system Eta Carinae, appears to be a promising candidate (see
Table \ref{tbl:m7table} for properties). This object was not detected
in the RASS due to its larger distance ($\approx 2.3$ kpc, based on
its association to the Carina nebula) and higher neutral hydrogen
absorption column density ($N_H = 3.5\pm1.1\times10^{21}\mbox{
cm}^{-2}$). Therefore, the accessible volume $V_{\rmn{max}}$ is the
ROSAT surveyed volume plus the additional volume probed by the
XMM-Newton's pointed observation.
In Table \ref{tbl:m7table}, we provide all the relevant data on the
sample objects including the spectral fit parameters that were used to
simulate the RASS to obtain $V_{\mathrm{max}}$. We take the calculated
ages and plot them against the effective temperatures observed at
infinity in Fig. \ref{fig:m7cool}. The errorbars on the ages
correspond to the maximum of the difference in ages obtained due to
uncertainties in $T_{bb}$, $N_H$ (these two parameters are
covariant), and the distance. The blackbody temperature $T_{bb}$ is
obtained by fitting a blackbody spectrum to that observed from the source.
The temperature, thus, corresponds to the color temperature of the object
and is an overestimation of the effective temperature $T_{eff}$ due to strong
energy dependence of the free-free and bound-free opacities of the photosphere
\citep[see for e.g.][]{Lloydetal2003}. The effective temperature is obtained from
$T_{bb}$ using a color correction factor $f_c = T_{bb}/T_{eff}$ where
$1\la f_c\la 1.8$ \citep[e.g.][]{Ozel2013}. For comparison, we also plot some
cooling curves from \citet{YakovlevPethick2004}, where the
non-superfluid (No SF) model for a $1.3 M_\odot$ cannot explain the
data. Other model curves show NS cooling behavior if proton
superfluidity in the core is taken into account
\citep[see][for more details on the 1P and 2P models]{YakovlevPethick2004}.
\subsection{\label{sec:errors}Statistical Ages Vs The True Ages}
The ages of isolated NSs have been estimated using different methods, namely from the
spin-down law, cooling models, and kinematics. The method we propose in this study to
estimate the true ages of these objects has only been applied, in its original form,
to estimate the ages of white dwarfs from their cumulative luminosity
function in globular cluster \citep[see for e.g.][]{Goldsburyetal2012}. The method
itself is purely a statistical one, for which the underlying assumption is that the
objects in the sample follow a Poisson distribution \citep{Felton1976} with a constant production rate.
The important question to ask here is
how good of an estimate of the true age is the statistical age. What is
the inherent statistical and other errors associated to this method of predicting ages?
\begin{figure}
\centering
\includegraphics[width=0.5\textwidth]{M7cool.eps}
\caption{Cooling curve of nearby thermally emitting isolated NSs obtained from
model temperatures and statistical ages. The errors in the ages reflect only the
systematic error (see Sec. \ref{sec:errors} for details). We plot
sample theoretical cooling curves from \citet{YakovlevPethick2004} for comparison.}
\label{fig:m7cool}
\end{figure}
\subsubsection{Statistical Error}
Consider
the youngest object, RBS 1223, which is also the hottest among the eight isolated NSs.
The kinematic age of RBS 1223 has been
found based on its association to possible OB associations and young star clusters to
be $\sim0.5-1$ Myr \citep{Tetzlaffetal2010},
with large uncertainties. The statistical age of RBS 1223 that we find
in our study, given that the sample only contains eight such objects,
is much smaller $\sim 0.073\pm0.005$ Myr, with uncertainties
corresponding to systematic errors. The statistical error is of course
much larger than the systematic one. Since the discovery of an
object in a given volume follows Poisson statistics, the relative
error scales as $1/\sqrt{N}$ where $N$ is the sample size with ages
$t_i \leq t_N$. Therefore, the error in the age of the first object
is $\pm t_1$ where $t_1$ is its statistical age. Likewise, the error
in the age of the eighth object is $\pm t_8/\sqrt{8}$. Evidently,
one needs a much larger sample to reduce the statistical errors
to that comparable to the systematic errors.
\subsubsection{Other Errors}
The grand assumption made in this work is that all M7 members are essentially
the same object but with different ages. Thus, they all follow one cooling
curve. That may not be the complete story. These objects may have different
masses, surface composition, magnetic fields, and they also may be located
with a line of sight with an inhomogeneous HI column density. All of these
parameters affect the inferred temperatures and observability of these objects, and
hence their statistical ages \citep[see for e.g. the review by][]{YakovlevPethick2004},
as the objects were ordered from hottest to coldest in Eq. \ref{eq:age}. Even in a larger sample
of isolated NSs, the dominant errors come from systematics and the uncertainty in the progenitor age $t_\rmn{OB}$.
In the case of RX J1856.5-3754,
the statistical age with its systematic error is $t_\rmn{stat}=0.59\pm0.18~\rmn{Myr}$.
Including the statistical error $\Delta t_\rmn{stat} = 0.59/\sqrt{8}~\rmn{Myr}$, since it's the
$8^\rmn{th}$ object in the temperature ordered list, and the error due to the uncertainty
$\Delta t_\rmn{OB}=16~\rmn{Myr}$, the total error is $\Delta t_\rmn{tot} = 0.44~\rmn{Myr}$.
Systematic errors can be reduced by having better estimates of distances, senstive surveys that
yield accurate surface temperatures, and an improved model of the HI column density. Accurate
determination of these parameters is crucial to obtain accurate estimates of $V_\rmn{max}$.
For RX J0720.4-3125, we find that the total error is $\Delta t_\rmn{tot} = 0.41~\rmn{Myr}$. Although
we find this object to be younger than RX J1856.5-3754, the large uncertainties in the statistical
ages of both objects make it hard to distinguish which is younger. The kinematic ages of these
objects tell a different story, where RX J0720.4-3125 is older than RX J1856.5-3754. This
discrepancy results from the ordering of objects based on their model temperatures. If
both objects are indeed very similar in their mass and surface composition, and have
cooled not much differently, then the statistical ages would agree with RX J0720.4-3125
being younger.
\section{Discussion}
The ages of isolated NSs are primarily important for constraining their
inner structure. In addition, they can be useful for accurately
determining the birthplace of the object, if its proper motion
is known. Similarly, if the SNR-NS association has been made,
then the age of the object can reveal its space velocity and the
kinematics of the SNR. The knowledge of ages of such objects is also
useful for population synthesis models which rely on the spatial and
velocity distributions, and the birthrates of NSs. In this study,
we propose a statistical method to estimate the true ages of the
ROSAT discovered sample comprising the M7. We then use the age estimates
along with the derived spectral temperatures to compare the data
with some cooling models. The strength of this technique, as
discussed earlier, lies in obtaining a larger sample of \textit{coolers}.
With only eight objects, the statistical error is indubitably much larger.
We have used the statistical ages and model temperatures of the nearby thermally
emitting isolated NSs to derive a cooling curve, under the assumption that all
of the objects in the sample are similar in their mass and surface composition, and
have also cooled in a similar fashion. Due to the marginal size of the sample, with
large statistical errors, it is not yet meaningful to differentiate between the
different cooling models. However, this whole exercise shows that ages of such
objects can be inferred statistically and provides another way of determining
accurate ages in addition to ages obtained from kinematics and/or spin-down.
The statistics can be improved by locating more of these objects in the disk of the Galaxy.
In a recent population synthesis study, \citet{Posseltetal2010} find that young isolated NSs,
that are both hot and bright, with ROSAT count rates below $0.1\mbox{ cts s}^{-1}$ (the ROSAT bright source catalog had a limiting count rate of $0.05\mbox{ cts s}^{-1}$) should be located in OB associations beyond the Gould belt. They
also remark on the possibility of finding new isolated cooling NSs by
conducting yet another careful search of such objects in the RASS and
the XMM-Newton Slew Survey \citep{Esquejetal2006}. However, they note that ROSAT
observations are incapable at locating the isolated sources with sufficient
spatial accuracy, such that many optical counterparts can be found in its large
positional error circle. On the other hand, although XMM-Newton is much more sensitive,
albeit with strong inhomogeneities, and can probe deeper into the Galactic plane,
the slew survey only covers $15\%$ of the sky currently. Searching the RASS for new
isolated NSs may appear to be a promising avenue \citep[e.g.][]{Turneretal2010}.
What is needed at the moment is another all-sky survey
that is able to surpass ROSAT in both sensitivity and positional
accuracy. In that regard, the upcoming eROSITA mission \citep{Cappellutietal2011} shows
a lot of promise, and its planned launch in 2014 makes it very timely. The X-ray
instrument eROSITA will be part of the Russian Spectrum-Roentgen-Gamma (SRG) satellite,
equipped with seven Wolter-I telescope modules with an advanced version of the XMM-Newton
pnCCD camera at its prime focus. The telescope will operate with
an energy range of $0.5 - 10$ keV, a field of view (FOV) of $1.03^\circ$, an angular
resolution of $28''$ averaged over the FOV, and a limiting flux of
$\sim10^{-14}\mbox{ erg cm}^{-3}$ in the $0.5 - 2$ keV
energy range and $\sim3\times10^{-13}\mbox{erg cm}^{-3}$ in the $2 - 10$
keV energy range. The all-sky survey will reach sensitivities that are $\sim 30$ times
that of the RASS where the entire sky will scanned over a period of four years.
\section{acknowledgements}
We would like to thank the referee for significantly
improving the quality of this work. The Natural
Sciences and Engineering Research Council of Canada,
Canadian Foundation for Innovation and the British
Columbia Knowledge Development Fund supported this
work. Correspondence and requests for materials should
be addressed to J.S.H. ([email protected]). This research
has made use of NASAs Astrophysics Data System
Bibliographic Services
\bibliographystyle{mn2e}
|
1,108,101,564,544 | arxiv | \section{Introduction}
This work aims to solve the following linearly constrained separable convex problem with $n$ blocks of variables
\begin{equation}
\begin{split}
\min\limits_{\mathbf{x}_1,\cdots,\mathbf{x}_n} & \ f(\mathbf{x})=\sum\limits_{i=1}^nf_i(\mathbf{x}_i)=\sum_{i=1}^n\left( g_i(\mathbf{x}_i)+h_i(\mathbf{x}_i)\right), \\
\text{s.t.} & \quad
\mathcal{A}(\mathbf{x})=\sum\limits_{i=1}^n \mathcal{A}_i(\mathbf{x}_i)=\b,\label{eq:model_problem_multivar}
\end{split}
\end{equation}
where $\mathbf{x}_i$'s and $\b$ can be vectors or matrices and both $g_i$ and $h_i$ are convex and lower semi-continuous. For $g_i$, we assume that $\nabla g_i$ is Lipschitz continuous with the Lipschitz constant $L_i>0$, i.e, $\left\| \nabla g_i(\mathbf{x}_i)- \nabla g_i(\mathbf{y}_i)\right\| \leq L_i \left\| \mathbf{x}_i-\mathbf{y}_i\right\|, \forall \mathbf{x}_i,\mathbf{y}_i$. For $h_i$, we assume that it may be nonsmooth and it is simple, in the sense that the proximal operator problem $\min_{\mathbf{x}} h_i(\mathbf{x}) + \frac{\alpha}{2}||\mathbf{x} - \a||^2$ ($\alpha>0$) can be cheaply solved. The bounded mappings $\mathcal{A}_i$'s are linear (e.g., linear transformation or the sub-sampling operator in matrix completion \cite{candes2009exact}). For the simplicity of discussion, we denote $\mathbf{x} = [\mathbf{x}_1 ; \mathbf{x}_2; \cdots ; \mathbf{x}_n ]$, $\mathcal{A} = [\mathcal{A}_1, \mathcal{A}_2, \cdots, \mathcal{A}_n]$ and
$\sum_{i=1}^{n} \mathcal{A}_i(\mathbf{x}_i) = \mathcal{A}(\mathbf{x})$, $f_i=g_i+h_i$. For any compact set $X$, let $D_{{X}} =
\sup_{\mathbf{x}_1,\mathbf{x}_2 \in {X}} ||\mathbf{x}_1 - \mathbf{x}_2||$ be
the diameter of ${X}$. We also denote $\mathbf{D}_{\mathbf{x}^{*}} =
||\mathbf{x}^{0} - \mathbf{x}^{*}||$. We assume there exists a saddle point $(\mathbf{x}^{*}, \bm{\lambda}^{*}) \in X \times {\Lambda}$ to (\ref{eq:model_problem_multivar}), i.e., $\mathcal{A}(\mathbf{x}^{*}) = \b$ and $- \mathcal{A}_i^T(\bm{\lambda}^{*}) \in \partial f_i(\mathbf{x}_i^{*}), \ i = 1,\cdots,n$, where $\mathcal{A}^T$ is the adjoint
operator of $\mathcal{A}$, $X$ and $\Lambda$ are the feasible sets of the primal variables and dual variables, respectively.
By using different $g_i$'s and $h_i$'s, a variety of machine learning problems can be cast into (\ref{eq:model_problem_multivar}), including Lasso \cite{tibshirani1996regression} and its variants \cite{lu2013correlation,jacob2009group}, low rank matrix decomposition \cite{RPCA}, completion \cite{candes2009exact} and representation model \cite{lu2012robust,liu2011latent} and latent variable graphical model selection~\cite{chandrasekaran2010latent}. Specifically, examples of $g_i$ are: (i) the square loss $\frac{1}{2}||\mathbf{D} \mathbf{x} - \mathbf{y}||^2$ , where $\mathbf{D}$ and $\mathbf{y}$ are of compatible dimensions. A more special case is the known Laplacian regularizer $\text{Tr}(\mathbf{X} \mathbf{L} \mathbf{X}^T)$, where $\mathbf{L}$ is the Laplacian matrix which is positive semi-definite; (ii) Logistic loss $\sum^{m}_{i=1} \log (1 + \exp(-y_i \mathbf{d}_i^T \mathbf{x}))$, where $\mathbf{d}_i$'s and $y_i$'s are the data points and the corresponding labels, respectively; (iii) smooth-zero-one loss $\sum^{m}_{i=1} \frac{1}{1+ \exp(c y_i \mathbf{d}_i^T \mathbf{x})}$, $c > 0$.
The possibly nonsmooth $h_i$ can be many norms, e.g., $\ell_1$-norm $||\cdot||_{1}$ (the sum of absolute values of all entries), $\ell_2$-norm $|| \cdot ||$ or Frobenius norm $||\cdot||_F$ and nuclear norm $|| \cdot ||_{*}$ (the sum of the singular values of a matrix).
This paper focuses on the popular approaches which study problem (\ref{eq:model_problem_multivar}) from the aspect of the augmented Lagrangian function $L(\mathbf{x},\bm{\lambda}) = f(\mathbf{x}) + \langle \bm{\lambda}, \mathcal{A}(\mathbf{x}) - \b \rangle + \frac{\beta}{2} ||\mathcal{A}(\mathbf{x}) - \b ||^2$, where $\bm{\lambda}$ is the Lagrangian multiplier or dual variable and $\beta > 0$. A basic idea to solve problem (\ref{eq:model_problem_multivar}) based on $L(\mathbf{x},\bm{\lambda})$ is the Augmented Lagrangian Method (ALM) \cite{hestenes1969multiplier}, which is a special case of the Douglas-Rachford splitting \cite{douglas1956numerical}.
\begin{table*}[t]
\begin{tabular}{c|c|c|c}
\hline
{ PALM } & { Fast PALM} & { PL-ADMM-PS} & { Fast PL-ADMM-PS} \\
\hline
\begin{minipage}{2.5cm} \centering \vspace{0.2cm}
$O\left(\frac{ D^2_{\mathbf{x}^{*}} + D^2_{\bm{\lambda}^{*}} }{K} \right)$
\vspace{0.1cm}\end{minipage} &
\begin{minipage}{2.5cm} \centering \vspace{0.2cm} $O\left(\frac{ D^2_{\mathbf{x}^{*}} + D^2_{\bm{\lambda}^{*}} }{K^2} \right)$
\vspace{0.1cm}\end{minipage} & \begin{minipage}{3.5cm} \centering \vspace{0.2cm} $O\left( \frac{D^2_{{\mathbf{x}}^*}}{K} + \frac{D^2_{\mathbf{x}^*}}{K} + \frac{D^2_{\bm{\lambda}^*}}{ K} \right)$
\vspace{0.1cm}\end{minipage} &
\begin{minipage}{3.5cm} \centering \vspace{0.2cm} $O \left( \frac{ D^2_{{\mathbf{x}}^*}}{K^2} + \frac{D^2_{{X}}}{K} + \frac{D^2_{\Lambda}}{ K} \right)$ \vspace{0.1cm}\end{minipage} \\
\hline
\end{tabular}
\centering \caption{Comparison of the convergence rates of previous methods and our fast versions} \label{all_convergence_rates}
\vspace{-0.6em}
\end{table*}
An influential variant of ALM is the Alternating Direction Mehtod of Multiplier (ADMM) \cite{boyd2011distributed}, which solves
problem (\ref{eq:model_problem_multivar})
with $n = 2$ blocks of variables.
However, the cost for solving the subproblems in ALM and ADMM in each iteration is usually high when $f_i$ is not simple and $\mathcal{A}_i$ is non-unitary ($\mathcal{A}_i^T\mathcal{A}_i$ is not the identity mapping). To alleviate this issue, the Linearized ALM (LALM) \cite{yang2013linearized} and Linearized ADMM (LADMM) \cite{LADMAP} were proposed by linearizing the augmented term $\frac{\beta}{2}||\mathcal{A} (\mathbf{x}) - \b||^2 $ and thus the subproblems are easier to solve. For (\ref{eq:model_problem_multivar}) with $n>2$ blocks of variables, the Proximal Jacobian ADMM \cite{Mintao} and Linearized ADMM with Parallel Splitting (L-ADMM-PS) \cite{LADMPS}
guaranteed to solve (\ref{eq:model_problem_multivar}) when $g_i = 0$ with convergence guarantee. To further exploit the Lipschitz continuous gradient property of $g_i$'s in (\ref{eq:model_problem_multivar}), the
work \cite{LADMPS} proposed a Proximal Linearized ADMM with Parallel Splitting (PL-ADMM-PS) by further linearizing the smooth part $g_i$. PL-ADMM-PS requires lower per-iteration cost than L-ADMM-PS for solving the general problem (\ref{eq:model_problem_multivar}).
Beyond the per-iteration cost, another important way to measure the speed of the algorithms is the convergence rate. Several previous work proved the convergence rates of the augmented Lagrangian function based methods \cite{he20121,Mintao,LADMPS}. Though the convergence functions used to measure the convergence rate are different, the
convergence rates of all the above discussed methods for (\ref{eq:model_problem_multivar}) are all $O({1}/{K})$, where $K\textsc{}$ is the number of iterations.
However, the rate $O(1/K)$ may be suboptimal in some cases. Motivated by the seminal work \cite{nesterov1983method}, several fast first-order
methods with the optimal rate $O(1/K^2)$ have been developed for unconstrained
problems \cite{beck2009fast,tseng}. More recently, by applying a similar accelerating technique, several fast ADMMs have been proposed to solve a special case of problem (\ref{eq:model_problem_multivar}) with $n=2$ blocks of variables
\begin{equation}\label{pro2222}
\min_{\mathbf{x}_1,\mathbf{x}_2} g_1(\mathbf{x}_1) + h_2(\mathbf{x}_2),\ \
\text{s.t.}\ \ \mathcal{A}_1 (\mathbf{x}_1) + \mathcal{A}_2 (\mathbf{x}_2) = \b.
\end{equation}
A fast ADMM proposed in \cite{azadi2014towards}\footnote{The method in \cite{azadi2014towards} is a fast stochastic ADMM. It is easy to give the corresponding deterministic version by computing the gradient in each iteration exactly to solve (\ref{pro2222}).} is able to solve (\ref{pro2222}) with the convergence rate $O\left(\frac{D_{X}^2}{K^2} + \frac{D_\Lambda^2}{K}\right)$. But their
result is a bit weak since their used function to characterize the
convergence can be negative. The work \cite{ouyang2014accelerated} proposed another
fast ADMM with the rate $O\left(\frac{ D_{\mathbf{x}^{*}}^2}{K^2} + \frac{ D_{\mathbf{x}^{*}}^2}{K}\right)$
for primal residual and $O\left(\frac{ D_{\mathbf{x}^{*}}^2}{K^{3/2}} + \frac{D_{\mathbf{x}^{*}} + D_{\bm{\lambda}^{*}}}{K}\right)$ for feasibility
residual. However, their result requires that the number of iterations $K$ should be predefined, which is not reasonable in practice. It is usually difficult in practice to determine the optimal $K$ since we usually stop the algorithms when both the primal and feasibility residuals are sufficiently small \cite{LADMPS}. The fast ALM proposed in \cite{he2010acceleration} owns the convergence rate $O(1/K^2)$, but it requires the objective $f$ to be differentiable. This limits its applications for nonsmooth optimization in most compressed sensing problems. Another work \cite{goldstein2012fast} proved a better convergence rate than $O(1/K)$ for ADMM. But their method requires much stronger assumptions, e.g., strongly convexity of $f_i$'s, which are usually violated in practice. In this work, we only consider (\ref{eq:model_problem_multivar}) whose objective is not necessarily strongly convex.
In this work, we aim to propose fast ALM type methods to solve the general problem (\ref{eq:model_problem_multivar}) with optimal convergence rates. The contributions are summarized as follows:
\begin{itemize}
\item First, we consider (\ref{eq:model_problem_multivar}) with $n=1$ (or one may regard all $n$ blocks as a superblock) and propose the Fast Proximal Augmented Lagrangian Method (Fast PALM). We prove that Fast PALM converges with the rate
$O\left(\frac{ D^2_{\mathbf{x}^{*}} + D^2_{\bm{\lambda}^{*}}}{K^2}\right)$, which is a significant improvement of ALM/PALM\footnote{PALM is a variant of ALM proposed in this work.} with rate $O\left(\frac{ D^2_{\mathbf{x}^{*}} + D^2_{\bm{\lambda}^{*}}}{K}\right)$. To the best of our knowledge, Fast PALM is the first improved ALM/PALM which achieves the rate $O(1/K^2)$ for the nonsmooth problem (\ref{eq:model_problem_multivar}).
\item Second, we consider (\ref{eq:model_problem_multivar}) with $n>2$ and propose the Fast Proximal Linearized
ADMM with Parallel Splitting (Fast PL-ADMM-PS), which converges with rate $O \left( \frac{ D^2_{{\mathbf{x}}^*}}{K^2} + \frac{D^2_{{X}}}{K} + \frac{D^2_{\Lambda}}{ K} \right)$. As discussed in Section 1.3 of \cite{ouyang2014accelerated}, such a rate is optimal and thus is better than PL-ADMM-PS with rate $O\left( \frac{D^2_{{\mathbf{x}}^*}}{K} + \frac{D^2_{\mathbf{x}^*}}{K} + \frac{D^2_{\bm{\lambda}^*}}{ K} \right)$ \cite{LADMPS}. To the best of our knowledge, Fast PL-ADMM-PS is the first fast Jacobian type (update the variables in parallel) method to solve (\ref{eq:model_problem_multivar}) when $n>2$ with convergence guarantee.
\end{itemize}
Table \ref{all_convergence_rates} shows the comparison of the convergence rates of previous methods and our fast versions. Note that Fast PALM and Fast PL-ADMM-PS have the same pter-iteration cost as PALM and PL-ADMM-PS, respectively. But the per-iteration cost of PL-ADMM-PS and Fast PL-ADMM-PS may be much cheaper than PALM and Fast PALM.
\section{Fast Proximal Augmented Lagrangian Method}
In this section, we consider (\ref{eq:model_problem_multivar}) with $n=1$ block of variable,
\begin{equation}\label{pro_one}
\min_{\mathbf{x}} f(\mathbf{x}) = g(\mathbf{x}) + h(\mathbf{x}),\quad
\text{s.t.}\quad \mathcal{A} (\mathbf{x}) = \b,
\end{equation}
where $g$ and $h$ are convex and $\nabla g$ is Lipschitz continuous with the Lipschitz constant $L$. The above problem can be solved by the traditional ALM which updates $\mathbf{x}$ and $\bm{\lambda}$ by
\begin{equation}
\left\{
\begin{aligned}
\mathbf{x}^{k+1}=&\arg\min_{\mathbf{x}} g(\mathbf{x})+h(\mathbf{x})+\langle \bm{\lambda}^k, \mathcal{A}(\mathbf{x}) - \b \rangle \\
& +\frac{{\beta}^{(k)}}{2} ||\mathcal{A}(\mathbf{x}) - \b ||^2 ,\label{updatexalm}\\
\bm{\lambda}^{k+1} =& \bm{\lambda}^k + {\beta}^{(k)} (\mathcal{A}(\mathbf{x}^{k+1}) - \b),
\end{aligned}
\right.
\end{equation}
where ${\beta}^{(k)}>0$. Note that $\nabla g$ is Lipschitz continuous. We have \cite{nesterov2004introductory}
\begin{equation}\label{propg}
g(\mathbf{x})\leq g(\mathbf{x}^k)+\langle \nabla g(\mathbf{x}^k), \mathbf{x} - \mathbf{x}^k \rangle+\frac{L}{2} ||\mathbf{x} - \mathbf{x}^k||^2.
\end{equation}
This motivates us to use the right hand side of (\ref{propg}) as a surrogate of $g$ in (\ref{updatexalm}). Thus we can update $\mathbf{x}$ by solving the following problem which is simpler than (\ref{propg}),
\begin{equation*}
\begin{split}
&\mathbf{x}^{k+1}= \arg\min_{\mathbf{x}} g(\mathbf{x}^k)+\langle \nabla g(\mathbf{x}^k), \mathbf{x} - \mathbf{x}^k \rangle + h(\mathbf{x}) \\
&+ \langle \bm{\lambda}^k, \mathcal{A}(\mathbf{x}) - \b \rangle +
\frac{{\beta}^{(k)}}{2} ||\mathcal{A}(\mathbf{x}) - \b ||^2+ \frac{L}{2} ||\mathbf{x} - \mathbf{x}^k||^2.
\end{split}
\end{equation*}
We call the method by using the above updating rule as Proximal Augmented Lagrangian Method (PALM). PALM can be regarded as a special case of Proximal Linearized Alternating Direction Method of Multiplier with Parallel Splitting in \cite{LADMPS} and it owns the convergence rate $O\left({1}/{K}\right)$, which is the same as the traditional ALM and ADMM. However, such a rate is suboptimal. Motivated by
the technique from the accelerated proximal gradient method \cite{tseng}, we propose the Fast PALM as shown in Algorithm~\ref{alg1}. It uses the interpolatory sequences $\mathbf{y}^k$ and $\mathbf{z}^k$ as well as the stepsize
${\theta}^{(k)}$. Note that if we set ${\theta}^{(k)} = 1$ in each iteration, Algorithm \ref{alg1} reduces to PALM. With careful choices of ${\theta}^{(k)}$ and
${\beta}^{(k)}$ in Algorithm \ref{alg1}, we can accelerate the convergence rate of PALM from $O\left({1}/{K}\right)$ to $O({1}/{K^2})$.
\begin{algorithm}[t]\label{alg1}
\caption{Fast PALM Algorithm}
\hrule
\hrule
\vspace{0.1cm}
\textbf{Initialize}: $\mathbf{x}^0$, $\mathbf{z}^0$, $\bm{\lambda}^0$, $\beta^{(0)}=\theta^{(0)}=1$.\\
\For{$k = 0, 1, 2,\cdots$}{
\begin{align}
\mathbf{y}^{k+1}&=(1-{\theta}^{(k)})\mathbf{x}^k+{\theta}^{(k)}\mathbf{z}^{k};\\
\mathbf{z}^{k+1}&=\argmin\limits_{\mathbf{x}} \left \langle\nabla g(\mathbf{y}^{k+1}),\mathbf{x}\right \rangle+h(\mathbf{x})\notag\\
&+ \left \langle\bm\lambda^k,\mathcal{A}(\mathbf{x}) \right \rangle+\frac{{\beta}^{(k)}}{2}\|\mathcal{A}(\mathbf{x})-\b\|^2 \notag\\
&+\frac{L{\theta}^{(k)}}{2}\|\mathbf{x}-\mathbf{z}^k\|^2;\label{updatexkfpalm} \\
\mathbf{x}^{k+1}&=(1-{\theta}^{(k)})\mathbf{x}^k+{\theta}^{(k)}\mathbf{z}^{k+1}; \label{ddefinex}\\
\bm{\lambda}^{k+1}&=\bm{\lambda}^k+{\beta}^{(k)}(\mathcal{A}(\mathbf{z}^{k+1})-\b);\label{update_lambda_one} \\
{\theta}^{(k+1)}&=\frac{-({\theta}^{(k)})^2+\sqrt{({\theta}^{(k)})^4+4({\theta}^{(k)})^2}}{2}; \\
{\beta}^{(k+1)}&= \frac{1}{{\theta}^{(k+1)}}.
\end{align}
}
\hrule
\hrule\hrule
\vspace{0.1cm}
\end{algorithm}
\begin{proposition}\label{prop_one}
In Algorithm \ref{alg1}, for any $\mathbf{x}$, we have
\begin{align}
&\frac{1-{\theta}^{(k+1)}}{({\theta}^{(k+1)})^2} \left(f(\mathbf{x}^{k+1})-f(\mathbf{x})\right)-\frac{1}{{\theta}^{(k)}}\left \langle\mathcal{A}^T(\bm{\lambda}^{k+1}),\mathbf{x}-\mathbf{z}^{k+1} \right \rangle\qquad \notag\\
&\leq\frac{1-{\theta}^{(k)}}{({\theta}^{(k)})^2}\left(f(\mathbf{x}^k)-f(\mathbf{x})\right) \label{eqpro1}\\
& \ \ \ \ + \frac{L}{2}\left(\|\mathbf{z}^{k}-\mathbf{x}\|^2-\|\mathbf{z}^{k+1}-\mathbf{x}\|^2\right).\notag
\end{align}
\end{proposition}
\begin{theorem}\label{conv_rate_one}
In Algorithm \ref{alg1}, for any $K>0$, we have
\begin{align}\label{eqconrateone}
& f(\mathbf{x}^{K+1})-f(\mathbf{x}^*)+\left \langle\bm\lambda^*,\mathcal{A}(\mathbf{x}^{K+1})-\b \right \rangle+\frac{1}{2}\|\mathcal{A} (\mathbf{x}^{K+1})-\b\|^2 \notag\\
&\leq\frac{2}{(K+2)^2}\left(LD_{\mathbf{x}^*}^2+D_{\bm{\lambda}^*}^2\right).
\end{align}
\end{theorem}
We use the convergence function, i.e., the left hand side of (\ref{eqconrateone}), in \cite{LADMPS} to measure the convergence rate of the algorithms in this work. Theorem \ref{conv_rate_one} shows that our Fast PALM achieves the rate $O\left(\frac{LD_{\mathbf{x}^*}^2+D^2_{\bm{\lambda}^*}}{K^2}\right)$, which is much better than $O\left(\frac{LD^2_{\mathbf{x}^*}+\frac{1}{\beta}D^2_{\bm{\lambda}^*}}{K}\right)$ by PALM\footnote{It is easy to achieve this since PALM is a special case of Fast PALM by taking ${\theta}^{(k)}=1$. }.
The improvement of Fast PALM over PALM is similar to the one of Fast ISTA over ISTA \cite{beck2009fast,tseng}. The difference is that Fast ISTA targets for unconstrained problem which is easier than our problem (\ref{eq:model_problem_multivar}). Actually, if the constraint in (\ref{eq:model_problem_multivar}) is dropped (i.e., $\mathcal{A}=\mathbf{0}$, $\b=\mathbf{0}$), our Fast PALM is similar as the Fast ISTA.
We would like to emphasize some key differences between our Fast PALM and previous fast ALM type methods \cite{azadi2014towards,ouyang2014accelerated,he2010acceleration}. First, it is easy to apply the two blocks fast ADMM methods in \cite{azadi2014towards,ouyang2014accelerated} to solve problem (\ref{pro_one}). Following their choices of parameters and proofs, the convergence rates are still $O({1}/{K})$. The key improvement of our method comes from the different choices of
${\theta}^{(k)}$ and ${\beta}^{(k)}$ as shown in Theorem \ref{conv_rate_one}. The readers can refer to the detailed proofs in the supplementary material. Second, the fast ADMM in \cite{ouyang2014accelerated} requires predefining
the total number of iterations, which is usually difficult
in practice. However, our Fast PALM has no such a limitation. Third, the fast ALM in \cite{he2010acceleration} also owns the rate $O(1/K^2)$. But it is restricted to differentiable objective minimization and thus is not applicable to our problem (\ref{eq:model_problem_multivar}). Our method has no such a limitation.
A main limitation of PALM and Fast PALM is that their per-iteration cost may be high when $h_i$ is nonsmooth and $\mathcal{A}_i$ is non-unitary. In this case, solving the subproblem (\ref{updatexkfpalm}) requires calling other iterative solver, e.g., Fast ISTA \cite{beck2009fast}, and thus the high per-iteration cost may limit the application of Fast PALM. In next section, we present a fast ADMM which has lower per-iteration cost.
\section{Fast Proximal Linearized ADMM with Parallel Splitting}
In this section, we consider problem (\ref{eq:model_problem_multivar}) with $n>2$ blocks of variables. The state-of-the-art solver for (\ref{eq:model_problem_multivar}) is the Proximal Linearized ADMM with Parallel Splitting (PL-ADMM-PS) \cite{LADMPS} which updates each $\mathbf{x}_i$ in parallel by
\begin{align}
\mathbf{x}_i^{k+1}&=\argmin\limits_{\mathbf{x}_i} g_i(\mathbf{x}_i^{k})+ \left \langle\nabla g_i(\mathbf{x}_i^{k}),\mathbf{x}_i-\mathbf{x}_i^{k} \right \rangle+h_i(\mathbf{x}_i) \notag \\
&+ \left \langle\bm{\lambda}^k,\mathcal{A}_i(\mathbf{x}_i) \right \rangle+\left\langle {\beta}^{(k)}\mathcal{A}_i^T\left( \mathcal{A}(\mathbf{x}^k)-\b\right),\mathbf{x}_i-\mathbf{x}_i^k\right\rangle \notag\\
&+\frac{L_i+{\beta}^{(k)}\eta_i}{2}\|\mathbf{x}_i-\mathbf{x}_i^k\|^2,\label{updatexladmmpsss2}
\end{align}
where $\eta_i>n||\mathcal{A}_i||^2$ and ${\beta}^{(k)}>0$. Note that the subproblem (\ref{updatexladmmpsss2}) is easy to solve when $h_i$ is nonsmooth but simple. Thus PL-ADMM-PS has much lower per-iteration cost than PALM and Fast PALM. On the other hand, PL-ADMM-PS converges with the rate $O(1/K)$ \cite{LADMPS}. However, such a rate is also suboptimal. Now we show that it can be further accelerated by a similar technique as that in Fast PALM.
See Algorithm \ref{alg2} for our Fast PL-ADMM-PS.
\begin{proposition}\label{prop3}
In Algorithm \ref{alg2}, for any $\mathbf{x}_i$, we have
\begin{align}
&\frac{1-{\theta}^{(k+1)}}{({\theta}^{(k+1)})^2}\left(f_i(\mathbf{x}_i^{k+1})-f_i(\mathbf{x}_i)\right) \notag\\
& -\frac{1}{{\theta}^{(k)}} \left \langle\mathcal{A}_i^T({\hat{\bm{\lambda}}^{k+1}}),\mathbf{x}_i-\mathbf{z}_i^{k+1} \right \rangle\notag\\
\leq&\frac{1-{\theta}^{(k)}}{({\theta}^{(k)})^2}\left(f_i(\mathbf{x}_i^k)-f_i(\mathbf{x}_i)\right) \label{pro331} \\
& +\frac{L_i}{2}\left(\|\mathbf{z}_i^{k}-\mathbf{x}_i\|^2-\|\mathbf{z}_i^{k+1}-\mathbf{x}_i\|^2\right)\notag\\
&+\frac{{\beta}^{(k)}\eta_i}{2{\theta}^{(k)}}\left(\|\mathbf{z}_i^{k}-\mathbf{x}_i\|^2-\|\mathbf{z}_i^{k+1}-\mathbf{x}_i\|^2-\|\mathbf{z}_i^{k+1}-\mathbf{z}_i^k\|^2\right),\notag
\end{align}
where
$\hat{\bm\mathbf{\lambda}}^{k+1}=\bm\lambda^k+{\beta}^{(k)}\left(\mathcal{A}(\mathbf{z}^{k})-\b\right)$.
\end{proposition}
\begin{theorem}\label{con_fastPS}
In Algorithm \ref{alg2}, for any $K>0$, we have
\begin{align}
& f(\mathbf{x}^{K+1})-f(\mathbf{x}^*)+\left \langle\bm{\lambda}^*,\mathcal{A}(\mathbf{x}^{K+1})-\b \right \rangle\notag\\
& +\frac{\beta\alpha}{2}\left\|\mathcal{A}(\mathbf{x}^{K+1})-\b\right\|^2 \label{con_FastPS}\\
\leq & \frac{2L_{\max}D^2_{\mathbf{x}^*}}{(K+2)^2}+\frac{2\beta\eta_{\max} D^2_{X}}{K+2}+\frac{2D^2_{\Lambda}}{\beta(K+2)},\notag
\end{align}
where $\alpha=\min\left\{\frac{1}{n+1},\left\{\frac{\eta_i-n\|\mathcal{A}_i\|^2}{2(n+1)\|\mathcal{A}_i\|^2},i=1,\cdots,n\right\}\right\}$, $L_{\max}=\max\{L_i,i=1,\cdots,n\}$ and $\eta_{\max}=\max\{\eta_i,i=1,\cdots,n\}$.
\end{theorem}
\newcommand{\mathbin{\!/\mkern-5mu/\!}}{\mathbin{\!/\mkern-5mu/\!}}
\begin{algorithm}[!t]\label{alg2}
\DontPrintSemicolon
\caption{Fast PL-ADMM-PS Algorithm}
\hrule
\hrule
\vspace{0.1cm}
\textbf{Initialize}: $\mathbf{x}^0$, $\mathbf{z}^0$, $\bm{\lambda}^0$, $\theta^{(0)}=1$, fix ${\beta}^{(k)}=\beta$ for $k\geq0$, $\eta_i>n\|\mathcal{A}_i\|^2$, $i=1,\cdots,n$,\\
\For{$k = 0, 1, 2, \cdots$}{
\hspace*{0.1cm}$\mathbin{\!/\mkern-5mu/\!}$ Update $\mathbf{y}_i$, $\mathbf{z}_i$, $\mathbf{x}_i$, $i=1,\cdots,n$, in parallel by
\begin{align}
\mathbf{y}_i^{k+1}&=(1-{\theta}^{(k)})\mathbf{x}_i^k+{\theta}^{(k)}\mathbf{z}_i^k;\\
\mathbf{z}_i^{k+1}&=\argmin\limits_{\mathbf{x}_i} \left \langle\nabla g_i(\mathbf{y}_i^{k+1}),\mathbf{x}_i \right \rangle +h_i(\mathbf{x}_i)\notag \\
&+ \left \langle\bm{\lambda}^k,\mathcal{A}_i(\mathbf{x}_i) \right \rangle+\left\langle {\beta}^{(k)}\mathcal{A}_i^T\left( \mathcal{A}(\mathbf{z}^k)-\b\right),\mathbf{x}_i\right\rangle \notag \\
&+\frac{L(g_i){\theta}^{(k)}+{\beta}^{(k)}\eta_i}{2}\|\mathbf{x}_i-\mathbf{z}_i^k\|^2; \label{update_z}\\
\mathbf{x}_i^{k+1}&=(1-{\theta}^{(k)})\mathbf{x}_i^k+{\theta}^{(k)}\mathbf{z}_i^{k+1}; \\
\mathbf{\bm{\lambda}}^{k+1}&=\mathbf{\bm{\lambda}}^k+\beta^k\left(\mathcal{A}(\mathbf{z}^{k+1})-\b\right);\label{update_lambda}\\
{\theta}^{(k+1)}&=\frac{-({\theta}^{(k)})^2+\sqrt{({\theta}^{(k)})^4+4({\theta}^{(k)})^2}}{2}.
\end{align}
}
\hrule
\hrule
\hrule
\vspace{0.1cm}
\end{algorithm}
From Theorem \ref{con_fastPS}, it can be seen that our Fast PL-ADMM-PS \emph{partially} accelerates the convergence rate of PL-ADMM-PS from $O\left(\frac{L_{\max}D_{\mathbf{x}^{*}}^2}{K} + \frac{\beta \eta_{\max}D^2_{\mathbf{x}^{*}}}{K} + \frac{D^2_{\bm{\lambda^{*}}}}{\beta K} \right)$ to $O\left(\frac{L_{\max} D_{\mathbf{x}^{*}}^2}{K^2} + \frac{\beta\eta_{\max} D^2_{{X}}}{K} + \frac{D^2_{\Lambda}}{\beta K} \right)$.
Although the improved rate is also $O(1/K)$, what makes it more attractive is that it allows very large Lipschitz constants $L_i$'s. In particular, $L_i$ can be as large as $O(K)$, without affecting the rate of convergence (up to a constant factor).
The above improvement is the same as fast ADMMs \cite{ouyang2014accelerated} for problem (\ref{pro2222}) with only $n=2$ blocks. But it is inferior to the Fast PALM over PALM. The key difference is that Fast PL-ADMM-PS further linearizes the augmented term $\frac{1}{2}||\mathcal{A}(\mathbf{x})-\b||^2$.
This improves the efficiency for solving the subproblem, but slows down the convergence. Actually, when linearizing the augmented term, we have a new term with the factor ${\beta}^{(k)}\eta_i/{\theta}^{(k)}$ in (\ref{pro331}) (compared with (\ref{eqpro1}) in Fast PALM). Thus (\ref{con_FastPS}) has a new term by comparing with that in (\ref{eqconrateone}). This makes the choice of ${\beta}^{(k)}$ in Fast PL-ADMM-PS different from the one in Fast PALM. Intuitively, it can be seen that a larger value of ${\beta}^{(k)}$ will increase the second terms of (\ref{con_FastPS}) and decrease the third term of (\ref{con_FastPS}). Thus ${\beta}^{(k)}$ should be fixed in order to guarantee the convergence. This is different from the choice of ${\beta}^{(k)}$ in Fast PALM which is adaptive to the choice of ${\theta}^{(k)}$.
Compared with PL-ADMM-PS, our Fast PL-ADMM-PS achieves a better rate, but with the price on the boundedness of the feasible primal set
${X}$ and the feasible dual set $\Lambda$. Note that many previous work, e.g., \cite{he20121,azadi2014towards}, also require such a boundedness assumption when proving the convergence of ADMMs. In the following, we give some conditions which guarantee such a boundedness assumption.
\begin{figure*}[t]
\setcounter{subfigure}{0}
\subfigure[$m=100$, $n=300$]{
\includegraphics[width=0.24\textwidth]{figs/fig_ALM_obj1.pdf}}
\subfigure[$m=300$, $n=500$]{
\includegraphics[width=0.24\textwidth]{figs/fig_ALM_obj2.pdf}}
\subfigure[$m=500$, $n=800$]{
\includegraphics[width=0.24\textwidth]{figs/fig_ALM_obj3.pdf}}
\subfigure[$m=800$, $n=1000$]{
\includegraphics[width=0.24\textwidth]{figs/fig_ALM_obj4.pdf}}\\
\vspace{-0.3cm}
\caption{Plots of the convergence function values of (\ref{eqconrateone}) in each iterations by using PALM and Fast PALM for (\ref{probl1}) with different sizes of $\mathbf{A}\in\mathbb{R}^{m\times n}$.}
\label{fig_res_fastALM}
\vspace{-0.4cm}
\end{figure*}
\begin{theorem}\label{bound_lambda}
Assume the mapping $\mathcal{A}(\mathbf{x}_1,\cdots,\mathbf{x}_n)=\sum_{i=1}^n\mathcal{A}_i(\mathbf{x}_i)$ is onto\footnote{This assumption is equivalent to that the matrix $A\equiv(A_1,\cdots,A_n)$ is of full row rank, where $A_i$ is the matrix representation of $\mathcal{A}_i$.}, the sequence $\{\mathbf{z}^k\}$ is bounded, $\partial h(\mathbf{x})$ and $\nabla g(\mathbf{x})$ are bounded if $\mathbf{x}$ is bounded, then $\{\mathbf{x}^k\}$, $\{\mathbf{y}^k\}$ and $\{\bm{\lambda}^k\}$ are bounded.
\end{theorem}
Many convex functions, e.g., the $\ell_1$-norm, in compressed sensing own the bounded subgradient.
\section{Experiments}
In this section, we report some numerical results to demonstrate the effectiveness of our fast PALM and PL-ADMM-PS. We first compare our Fast PALM which owns the optimal convergence rate $O(1/K^2)$ with the basic PALM on a problem with only one block of variable. Then we conduct two experiments to compare our Fast PL-ADMM-PS with PL-ADMM-PS on two multi-blocks problems. The first one is tested on the synthesized data, while the second one is for subspace clustering tested on the real-world data. We examine the convergence behaviors of the compared methods based on the convergence functions shown in (\ref{eqconrateone}) and (\ref{con_FastPS}). All the numerical experiments are run on a PC with 8 GB of RAM and Intel Core 2 Quad CPU Q9550.
\subsection{Comparison of PALM and Fast PALM}
We consider the following problem
\begin{equation}\label{probl1}
\min_{\mathbf{x}} ||\mathbf{x}||_1+\frac{\alpha}{2}||\mathbf{A}\mathbf{x}-\b||_2^2, \ \ \text{s.t.} \ \ \mathbf{1}^T\mathbf{x}=1,
\end{equation}
where $\alpha>0$, $\mathbf{A}\in\mathbb{R}^{m\times n}$, $\b\in\mathbb{R}^m$, and $\mathbf{1}\in\mathbb{R}^n$ is the all one vector. There may have many fast solvers for problem (\ref{probl1}). In this experiment, we focus on the performance comparison of PALM and Fast PALM for (\ref{probl1}). Note that the per-iteration cost of these two methods are the same. Both of them requires solving an $\ell_1$-minimization problem in each iteration. In this work, we use the SPAMS package \cite{mairal2010online} to solve it which is very fast.
The data matrix $\mathbf{A}\in\mathbb{R}^{m\times n}$, and $\mathbf{b}\in\mathbb{R}^m$ are generated by the Matlab command \mcode{randn}. We conduct four experiments on different sizes of $\mathbf{A}$ and $\mathbf{b}$. We use the left hand side of (\ref{eqconrateone}) as the convergence function to evaluate the convergence behaviors of PALM and Fast PALM. For the saddle point $(\mathbf{x}^*,\bm{\lambda}^*)$ in (\ref{eqconrateone}), we run the Fast PALM with 10,000 iterations and use the obtained solution as the saddle point. Figure \ref{fig_res_fastALM} plots the convergence functions value within 1,000 iterations. It can be seen that our Fast PALM converges much faster than PALM. Such a result verifies our theoretical improvement of Fast PALM with optimal rate $O(1/K^2)$ over PALM with the rate $O(1/K)$.
\begin{figure*}[t]
\setcounter{subfigure}{0}
\subfigure[$m=100$]{
\includegraphics[width=0.32\textwidth]{figs/fig_m=100_objs.pdf}}
\subfigure[$m=300$]{
\includegraphics[width=0.31\textwidth]{figs/fig_m=300_objs.pdf}}
\subfigure[$m=500$]{
\includegraphics[width=0.32\textwidth]{figs/fig_m=500_objs.pdf}}
\vspace{-0.2cm}
\caption{Plots of the convergence function values of (\ref{con_FastPS}) in each iterations by using PL-ADMM-PS and Fast PL-ADMM-PS for (\ref{eqmulti}) with different sizes of $\mathbf{X}\in\mathbb{R}^{m\times m}$.}
\label{fig_res_fALMps}
\end{figure*}
\begin{figure*}[t]
\setcounter{subfigure}{0}
\subfigure[5 subjects]{
\includegraphics[width=0.32\textwidth]{figs/fig_yaleb5_objs.pdf}}
\subfigure[8 subjects]{
\includegraphics[width=0.32\textwidth]{figs/fig_yaleb8_objs.pdf}}
\subfigure[10 subjects]{
\includegraphics[width=0.32\textwidth]{figs/fig_yaleb10_objs.pdf}}
\caption{Plots of the convergence function values of (\ref{con_FastPS}) in each iterations by using PL-ADMM-PS and Fast PL-ADMM-PS for (\ref{eqsub}) with different sizes of data $\mathbf{X}$ for subspace clustering.}
\label{fig_res_ALMpsclu}
\vspace{-0.5cm}
\end{figure*}
\subsection{Comparison of PL-ADMM-PS and Fast PL-ADMM-PS}
In this subsection, we conduct a problem with three blocks of variables as follows
\begin{equation}\label{eqmulti}
\begin{split}
\min_{\mathbf{X}_1,\mathbf{X}_2,\mathbf{X}_3} & \ \ \sum_{i=1}^{3} \left(||\mathbf{X}_i||_{\ell_i}+\frac{\alpha_i }{2}||\mathbf{C}_i\mathbf{X}_i-\mathbf{D}_i||_F^2\right), \\
\text{s.t.} & \ \ \sum_{i=1}^{3}\mathbf{A}_i\mathbf{X}_i=\mathbf{B},
\end{split}
\end{equation}
where $||\cdot||_{\ell_1}=||\cdot||_1$ is the $\ell_1$-norm, $||\cdot||_{\ell_2}=||\cdot||_*$ is the nuclear norm, and $||\cdot||_{\ell_3}=||\cdot||_{2,1}$ is the $\ell_{2,1}$-norm defined as the sum of the $\ell_2$-norm of each column of a matrix. We simply consider all the matrices with the same size ${\mathbf{A}}_i, {\mathbf{C}}_i, {\mathbf{D}}_i, \mathbf{B}, {\mathbf{X}}_i\in\mathbb{R}^{m\times m}$. The matrices ${\mathbf{A}}_i, {\mathbf{C}}_i, {\mathbf{D}}_i, i=1,2,3$, and $\mathbf{B}$ are generated by the Matlab command \mcode{randn}. We set the parameters $\alpha_1=\alpha_2=\alpha_3=0.1$. Problem (\ref{eqmulti}) can be solved by PL-ADMM-PS and Fast PL-ADMM-PS, which have the same and cheap per-iteration cost. The experiments are conducted on three different values of $m=$100, 300 and 500. Figure \ref{fig_res_fALMps} plots the convergence function values of PL-ADMM-PS and Fast PL-ADMM-PS in (\ref{con_FastPS}). It can be seen that Fast PL-ADMM-PS converges much faster than PL-ADMM-PS. Though Fast PL-ADMM-PS only accelerates PL-ADMM-PS for the smooth parts $g_i$'s, the improvement of Fast PL-ADMM-PS over PL-ADMM-PS is similar to that in Fast PALM over PALM. The reason behind this is that the Lipschitz constants $L_i$'s are not very small (around 400, 1200, and 2000 for the cases $m=100$, $m=300$, and $m=500$, respectively). And thus reducing the first term of (\ref{con_FastPS}) faster by our method is important.
\subsection{Application to Subspace Clustering}
In this subsection, we consider the following low rank and sparse representation problem for subspace clustering
\begin{equation}\label{eqsub}
\begin{split}
\min_{\mathbf{Z}} & \ \ \alpha_1||\mathbf{Z}||_*+\alpha_2||\mathbf{Z}||_1+\frac{1}{2}||\mathbf{X}\mathbf{Z}-\mathbf{X}||^2,\\
\text{s.t.}&\ \ \mathbf{1}^T\mathbf{Z}=\mathbf{1}^T,
\end{split}
\end{equation}
where $\mathbf{X}$ is the given data matrix. The above model is motivated by \cite{zhuang2012non}. However, we further consider the affine constraint $\mathbf{1}^T\mathbf{Z}=\mathbf{1}^T$ for affine subspace clustering \cite{6482137}. Problem (\ref{eqsub}) can be reformulated as a special case of problem (\ref{eq:model_problem_multivar}) by introducing auxiliary variables. Then it can be solved by PL-ADMM-PS and Fast PL-ADMM-PS.
Given a data matrix $\mathbf{X}$ with each column as a sample, we solve (\ref{eqsub}) to get the optimal solution $\mathbf{Z}^*$. Then the affinity matrix $\mathbf{W}$ is defined as $\mathbf{W}=(|\mathbf{Z}|+|\mathbf{Z}^T|)/2$. The normalized cut algorithm \cite{shi2000normalized} is then performed on $\mathbf{W}$ to get the clustering results of the data matrix $\mathbf{X}$. The whole clustering algorithm is the same as \cite{6482137}, but using our defined affinity matrix $\mathbf{W}$ above.
\renewcommand{\arraystretch}{1.3}
\begin{table}[t]
\small
\vspace{-0.2cm}
\caption{Comparision of subspace clustering accuracies ($\%$) on the Extended Yale B database.}\vspace{0.1cm}
\centering
\begin{tabular}{c|c|c|c}
\hline
Methods & 5 subjects & 8 subjects & 10 subjects \\ \hline
PL-ADMM-PS & 94.06 & 85.94 & 75.31 \\ \hline
Fast PL-ADMM-PS & 96.88 & 90.82 & 75.47 \\ \hline
\end{tabular}\label{tabacc}
\vspace{-0.5cm}
\end{table}
We conduct experiments on the Extended Yale B database \cite{georghiades2001few}, which is challenging for clustering. It consists of 2,414 frontal face images of 38 subjects under various lighting, poses and illumination conditions. Each subject has 64 faces. We construct
three matrices $\mathbf{X}$ based on the first 5, 8 and
10 subjects. The data matrices $\mathbf{X}$ are first
projected into a $5\times 6$, $8\times 6$, and $10\times 6$-dimensional subspace
by PCA, respectively. Then we run PL-ADMM-PS and Fast PL-ADMM-PS for 1000 iterations, and use the solutions $\mathbf{Z}$ to define the affinity matrix $\mathbf{W}=(|\mathbf{Z}|+|\mathbf{Z}^T|)/2$. Finally, we can obtain the clustering results by normalized cuts. The accuracy, calculated by the best matching rate of the predicted label and the ground
truth of data, is reported to measure the performance. Table \ref{tabacc} shows the clustering accuracies based on the solutions to problem (\ref{eqsub}) obtained by PL-ADMM-PS and Fast PL-ADMM-PS. It can be seen that Fast PL-ADMM-PS usually outperfoms PL-ADMM-PS since it achieves a better solution than PL-ADMM-PS within 1000 iterations. This can be verified in Figure \ref{fig_res_ALMpsclu} which shows the convergence function values in (\ref{con_FastPS}) of PL-ADMM-PS and Fast PL-ADMM-PS in each iteration. It can be seen that our Fast PL-ADMM-PS converges much faster than PL-ADMM-PS.
\section{Conclusions}
This paper presented two fast solvers for the linearly constrained convex problem (\ref{eq:model_problem_multivar}). In particular, we proposed the Fast Proximal Augmented Lagragian Method (Fast PALM) which achieves the convergence rate $O(1/K^2)$. Note that such a rate is theoretically optimal by comparing with the rate $O(1/K)$ by traditional ALM/PALM. Our fast version does not require additional assumptions (e.g. boundedness of $X$ and $\Lambda$, or a predefined number of iterations) as in the previous works~\cite{azadi2014towards,ouyang2014accelerated}. In order to further reduce the per-iteration complexity and handle the multi-blocks problems ($n>2$), we proposed the Fast Proximal Linearized ADMM with Parallel Splitting (Fast PL-ADMM-PS). It also achieves the optimal $O(1/K^2)$ rate for the smooth part of the objective. Compared with PL-ADMM-PS, though Fast PL-ADMM-PS requires additional assumptions on the boundedness of $X$ and $\Lambda$ in theory, our experimental results show that significant improvements are obtained especially when the Lipschitz constant of the smooth part is relatively large.
\section*{Acknowledgements}
This research is supported by the Singapore National Research Foundation under its International Research Centre
@Singapore Funding Initiative and administered by the IDM Programme Office. Z. Lin is supported by NSF China
(grant nos. 61272341 and 61231002), 973 Program of China
(grant no. 2015CB3525) and MSRA Collaborative Research
Program.
{
\footnotesize
\bibliographystyle{aaai}
|
1,108,101,564,545 | arxiv | \section{Introduction}
\noindent
In recent times the study of the spiral structure of the outer
Galactic disk has witnessed a renewed interest
(see e.g. Levine et al. 2006, V\'azquez et al. 2008, Benjamin 2008),
in part due to the claimed discovery of structures in the form of star
over-densities which would be produced by accretion/merging events.
Examples are those in Monoceros (Yanny et al. 2003), Canis Major (Martin
et al. 2004) and Argus (Rocha Pinto et al. 2006). To assess the reality
and properties of these over-densities, a detailed investigation of the
outer Galactic disk is a basic requirement. Such an investigation would
also improve our knowledge of the extreme periphery of the Milky Way (MW).\\
\noindent
Spiral features can be detected by using a variety of tracers, namely
HI (see e.g. Levine et al. 2006), HII (see e.g. Russeil 2003), CO (see e.g.
Luna et al. 2006), and optical objects (see e.g. V\'azquez et al 2008).
These diverse techniques have provided a coherent picture of the spiral
structure in the third Galactic quadrant (Moitinho et al. 2008). Our group
has contributed to this effort providing optical information for a large
sample of young open clusters (Moitinho et al. 2006), in the field of which
we have recognized distant and reddened sequences which allowed us to define
the shape and extent of spiral arms up to 20 kpc from the Galactic center.\\
\begin{figure*}
\centering
\includegraphics[width=18.5cm]{AA10800fig1.eps}
\caption{Left panel: $V$-band 100 sec exposure image of the field observed
in the area of the star cluster Shorlin~1. Field shown is $20\arcmin$ on a side.
North is up and East to the left. Right panel: Zoom of the cluster region,
in which the two WR stars (WR38 and WR38a) are indicated. Field shown is
$\sim3\arcmin$ on a side.}%
\end{figure*}
\noindent
This paper is the first result of a study of selected low-absorption regions in
the fourth Galactic quadrant, aimed at finding distant optical spiral features.
We present an investigation of the field towards the Wolf-Rayet (WR) stars WR38 and
WR38a ($RA = 11^{h}06^{m}$, $DEC = -61^{o}14^{\prime}$, Shorlin et al. 2004, hereafter Sh04; Wallace et al. 2005, hereafter Wa05),
which attracted our attention for several reasons.\\
Sh04 and Wa05 discuss the discovery of a distant, young, compact
star cluster, at a distance of $\sim$12 kpc, associated with these two WR stars.
These two studies, however, reach quite different conclusions about the relationship
of this cluster with known spiral arms. While Sh04 propose that the cluster
is associated with an extension of the Perseus arm beyond the Carina-Sagittarius arm,
Wa05, more conservatively, suggest that it lies in the more distant part of the Carina
arm. In fact, the line-of-sight in the direction to WR38 and WR38a crosses twice the
Carina-Sagittarius arm. In a quite different interpretation, Frinchaboy et al. (2004)
associate this cluster to the Galactic Anticenter Stellar Structure (GASS), a
population of old star clusters, possibly members of the Monoceros Ring.\\
With the purpose of clarifying these issues, we undertook an observational campaign
which addresses the following questions: Is the group of stars close to WR38 and WR38a
(i.e. Shorlin 1) a real star cluster? Does this group belong to the most distant
part of the Carina arm or to the Perseus arm? Are we looking in this direction at
signatures of the Monoceros ring, or the Argus system?\\
\noindent
The paper is organized as follows. In Sect~2 we describe the observations and the
reduction procedure and compare our data-set with previous investigations,
and in Sect.~3 we discuss the reddening law in the Galactic direction under consideration.
In Sect.~4 and 5 we thoroughly discuss the properties of the three different stellar
populations detected in the field observed. Finally, Sec.~6 summarizes the results of
our study.
\section{Observations and Data Reduction}
\subsection{Observations}
The region of interest (see Fig.~1) was observed with the Y4KCAM camera
attached to the 1.0m telescope, which is operated by the SMARTS consortium\footnote{{\tt http://http://www.astro.yale.edu/smarts}}
and located at Cerro Tololo Inter-American Observatory (CTIO). This camera
is equipped with an STA 4064$\times$4064 CCD with 15-$\mu$ pixels, yielding a scale of
0.289$^{\prime\prime}$/pixel and a field-of-view (FOV) of $20^{\prime} \times 20^{\prime}$ at the
Cassegrain focus of the CTIO 1.0m telescope. The CCD was operated without binning, at a nominal
gain of 1.44 e$^-$/ADU, implying a readout noise of 7~e$^-$ per quadrant (this detector is read
by means of four different amplifiers). QE and other detector characteristics can be found at:
http://www.astronomy.ohio-state.edu/Y4KCam/detector.html.\\
\noindent
The observational material was obtained in three observing runs, resumed in Table~1. In our
first run we took deep $UBVI$ images of Shorlin~1, under good seeing, but non-photometric
conditions. In our second run we took medium and short exposures of Shorlin~1, and
observed Landolt's SA~98 $UBVRI$ standard stars area (Landolt 1992), to tie our $UBVI$
instrumental system to the standard system. In a final third run we secured an additional
set of $U$-band images of Shorlin~1. Average seeing was 1.2$\arcsec$.\\
\begin{table}
\tabcolsep 0.1truecm
\caption{Log of $UBVI$ photometric observations.}
\begin{tabular}{lcccc}
\hline
\noalign{\smallskip}
Target& Date & Filter & Exposure (sec) & airmass\\
\noalign{\smallskip}
\hline
\noalign{\smallskip}
Shorlin~1 & 24 May 2006 & U & 60, 1500 &1.28$-$1.33\\
& & B & 30, 100, 1200 &1.20$-$1.35\\
& & V & 30, 100, 900 &1.22$-$1.34\\
& & I & 100, 700 &1.25$-$1.34\\
Shorlin~1 & 29 January 2008 & U & 5, 15 &1.19$-$1.22\\
& & B & 3, 5, 10 &1.19$-$1.23\\
& & V & 3, 5, 10 &1.18$-$1.22\\
& & I & 3, 5, 10 &1.18$-$1.23\\
SA~98 & 29 January 2008 & U & 2x10, 200, 300, 400 &1.17$-$1.89\\
& & B & 2x10, 100, 2x200 &1.17$-$1.99\\
& & V & 2x10, 25, 50, 2x100 &1.20$-$2.27\\
& & I & 2x10, 50, 100, 2x150 &1.18$-$2.05\\
Shorlin~1 & 05 June 2008 & U & 30x3, 1800 &1.19$-$1.20\\
\noalign{\smallskip}
\hline
\end{tabular}
\end{table}
Our $UBVI$ instrumental photometric system was defined by the use of a standard broad-band
Kitt Peak $BVR_{kc}I_{kc}$ set in combination with a U+CuSO4 $U$-band filter. Transmission
curves for these filters can be found at: http://www.astronomy.ohio-state.edu/Y4KCam/filters.html.
To determine the transformation from our instrumental system to the standard Johnson-Kron-Cousins
system, we observed 46 stars in area SA~98 (Landolt 1992) multiple times, and with different
airmasses ranging from $\sim$1.2 to $\sim$2.3. Field SA~98 is very advantageous, as it
includes a large number of well observed standard stars, and it is completely covered by
the CCD's FOV. Furthermore, the standard's color coverage is very good, being:
$-0.5 \leq (U-B) \leq 2.2$; $-0.2 \leq (B-V) \leq 2.2$ and $-0.1 \leq (V-I) \leq 6.0$.\\
\subsection{Reductions}
Basic calibration of the CCD frames was done using the IRAF\footnote{IRAF is distributed
by the National Optical Astronomy Observatory, which is operated by the Association
of Universities for Research in Astronomy, Inc., under cooperative agreement with
the National Science Foundation.} package CCDRED. For this purpose, zero-exposure
frames and twilight sky flats were taken every night. Photometry was then performed
using the IRAF DAOPHOT and PHOTCAL packages. Instrumental magnitudes were extracted
following the point spread function (PSF) method (Stetson 1987). A quadratic, spatially
variable, Master PSF (PENNY function) was adopted. The PSF photometry was finally
aperture-corrected, filter by filter. Aperture corrections were determined performing
aperture photometry of a suitable number (typically 10 to 20) of bright stars in the field.
These corrections were found to vary from 0.120 to 0.215 mag, depending on the filter.\\
\subsection{The photometry}
Our final photometric catalog consists of 7425 entries having $UBVI$ measures down to
$V \sim $ 20, and 12250 entries having $BVI$ measures down to $V \sim 22$.\\
After removing saturated stars, and stars having only a few measurements in Landolt's (1992)
catalog, our photometric solution for a grand-total of 183 measurements in $U$ and $B$, and of 206
measurements in $V$ and $I$, turned out to be:\\
\noindent
$ U = u + (3.290\pm0.010) + (0.47\pm0.01) \times X - (0.010\pm0.016) \times (U-B)$ \\
$ B = b + (2.220\pm0.012) + (0.29\pm0.01) \times X - (0.131\pm0.012) \times (B-V)$ \\
$ V = v + (1.990\pm0.007) + (0.16\pm0.01) \times X + (0.021\pm0.007) \times (B-V)$ \\
$ I = i + (2.783\pm0.011) + (0.08\pm0.01) \times X + (0.043\pm0.008) \times (V-I)$ \\
The final {\it r.m.s} of the fitting was 0.073, 0.069, 0.035 and 0.030 in $U$, $B$, $V$ and $I$,
respectively.\\
\noindent
Global photometric errors were estimated using the scheme developed by Patat \& Carraro
(2001, Appendix A1), which takes into account the errors resulting from the PSF fitting
procedure (e.i. from ALLSTAR), and the calibration errors (corresponding to the zero point,
color terms and extinction errors). In Fig.~2 we present global photometric error trends
plotted as a function of $V$ magnitude. Quick inspection shows that stars brighter than
$V \approx 20$ mag have errors lower than 0.10~mag in magnitude and lower than 0.20~mag in
all colors. \\
\begin{figure}s
\centering
\includegraphics[width=\columnwidth]{AA10800fig2.eps}
\caption{Photometric errors in V, (B-V), (U-B) and (V-I) as a function of V mag.}%
\end{figure}
The only previous $UBV$ (Johnson-Morgan) photometric study of this region is that of Sh04,
who present photoelectric and CCD photometry for roughly 140 stars brighter than $V \sim$ 16.5
in the field of WR38 and WR38a. In Fig.~3, we compare our photometry with that of Sh04 for
$V$, $B-V$ and $U-B$, in the sense ours minus theirs, as a function of our $V$ magnitude.
This comparison was possible only for 103 of their stars because, from the material presented
in Sh04, it was not possible to identify all objects measured by them (this issue was addressed
with D. Turner, private communication). From the stars in common we obtain:
$\Delta V = 0.017 \pm 0.065$, $\Delta (B-V) = 0.014 \pm 0.066$ and $\Delta (U-B) = 0.034 \pm 0.199$.\\
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{AA10800fig3.eps}
\caption{Comparison of our photometry with that of Sh04 for $V$, $B-V$ and $U-B$ (in the sense
this work minus Sh04), as a function of our $V$ magnitude.}%
\end{figure}
In spite of the significant scatter in the comparison for $U-B$, it can be concluded that the two
studies are in nice agreement. The poor match in this color may be ascribed to problems in their
$U-$band photometry, as extensively discussed in Sh04. Granted this, as a by-product we confirm that
the discrepancies between the photometry Sh04 and that of Wa05 is possibly due to the transformation
of the F336 ($U$), F439W ($B$), F555W ($V$) HST photometry to the Johnson-Morgan system.
\subsection{Astrometry}
For two-hundred stars in our photometric catalog there are J2000.0 equatorial coordinates
from the Guide Star Catalogue \footnote{Space telescope Science Institute, 2001, The Guide
Star Catalogue Version 2.2.02.}, version 2 (GSC-2.2, 2001). Using the SkyCat tool at ESO, and
the IRAF tasks ccxymatch and ccmap, we first established the transformation between our $(X,Y)$
pixel coordinates (from ALLSTAR) and the International Celestial Reference Frame (Arias et al.
1995). These transformations turned out to have an ${\it r.m.s.}$ value of 0.15$^{\prime\prime}$.
Finally, using the IRAF task cctran, we computed J2000.0 coordinates for all objects in our
catalog.
\section{The reddening law toward l=290.63 and b=-0.903}
A basic requirement before analyzing our photometric material is to investigate the reddening law
in the direction of WR38 and WR38a. Sh04 emphasize that according to their analysis the extinction
in this direction follows the normal law, namely that the ratio of total over selective absorption
$\frac{A_V}{E(B-V)}$ is 3.1. This value has already been found to be of general validity towards
Carina (Carraro 2002, Tapia et al. 2003). The $(V-I)$ vs. $(B-V)$ color-color diagram show in Fig.~4.,
constructed using our $UBVI$ photometry, confirms the above statements. Most stars lie close to the
reddening-free Schmidt-Kaler (1982) Zero Age Main Sequence (ZAMS) line (dashed line), which runs
almost parallel to the reddening vector plotted in the upper-right corner of this figure. This
vector has been drawn following the normal extinction ratio (Dean et al. 1978), which suggests that
in this specific Galactic direction absorption follows the normal law. In the subsequent discussion
we will therefore adopt $\frac{A_V}{E(B-V)} = 3.1$ and $\frac{E(U-B)}{E(B-V)} = 0.72$.
\begin{figure}
\centering
\includegraphics[width=9.5truecm]{AA10800fig4.eps}
\caption{$(V-I)$ vs. $(B-V)$ color-color diagram for stars in our field with $UBVI$ photometry.
The reddening vector for the normal extinction law is plotted in the upper-right corner. The
the dashed and dotted lines represent the reddening-free Schmidt-Kaler (1982) Zero Age Main
Sequence relations for dwarf and giant stars, respectively.}%
\end{figure}
\section{Stellar populations in the field}
In Fig.~5. we present a $(B-V)$ vs. $(U-B)$ color-color diagram for those stars in our photometric
catalog which have photometric errors lower than 0.05 magnitudes in both $(B-V)$ and $(U-B)$.
This figure is similar to Fig.~3 of Sh04, except for the much larger number of stars resulting
from our larger area coverage and deeper photometry. The solid line is an empirical reddening-free
ZAMS for dwarf stars, from Schmidt-Kaler (1982). In agreement with what was discussed in the
previous section, we have adopted a normal reddening law for this region. The corresponding
reddening vector has been plotted in the bottom of the figure.\\
\begin{figure*}
\centering
\includegraphics[width=15truecm]{AA10800fig5.eps}
\caption{$(B-V)$ vs. $(U-B)$ color-color diagram for stars in our photometric catalog
with photometric errors lower than 0.05 magnitudes in both $(B-V)$ and $(U-B)$. The
reddening vector for the normal extinction law has been plotted in the bottom of the
figure. The solid line is an empirical reddening-free ZAMS for dwarf stars, from
Schmidt-Kaler (1982). The same ZAMS (dashed lines) has been displaced along the reddening
line for three different amounts of reddening, 0.25, 0.52 and 1.45. The approximate
location of stars with spectral type O6, B5 and A0 is also indicated. The two crosses are
here used to show the position of WR38 and WR38a. See text for more details.}%
\end{figure*}
\noindent
Shifting the ZAMS by different amounts in the direction of the reddening vector, we can fit the
most obvious stellar sequences, which has led us to identify three distinct stellar populations.
This displacement is illustrated by dotted lines (in black)
for three spectral type along the reddening direction. This permits to define
the spectral type of reddened stars.\\
In what follows we discuss how we have chosen these amounts and why we claim we are distinguishing
three different populations.\\
\noindent
The yellow dot-dashed ZAMS has been shifted by $E(B-V)$ = 0.25, and fits a conspicuous sequence
of early type stars. As indicated in the figure, this group is composed by stars of spectral
type from as early as O8 up to A0. Beyond A0, the definition of spectral type becomes ambiguous
due to the crossing of different reddening ZAMS close to the A0V knee. This low-reddening
group of stars clearly suffers from variable extinction, as indicated by the significant
spread around the $E(B-V)$ = 0.25 ZAMS.
We estimate in fact that the mean reddening
for this group is 0.25$\pm$0.10. Sh04 identified this group as stars belonging to the Carina
branch of the Carina-Sagittarius spiral arm. We will refer to this group of stars as group
{\bf A}.\\
\noindent
The blue short-dashed ZAMS has been shifted by $E(B-V)$ = 0.52. It passes through a group of
young stars, with spectral types ranging from B2 to A0, which exhibit almost no reddening
variation. In fact, we estimate that the mean reddening for this group is 0.52$\pm$0.05, namely
the scatter around the mean value of 0.52 is small, and compatible with the typical photometric
error in the (U-B) and (B-V) colors.
In Fig.~3 of
Sh04 this group is barely visible, and the authors make no comments about it. Because of the
wider area covered by our study, it could be readily detected in our $(B-V)$ vs. $(U-B)$
color-color diagram. We will refer to this group of stars as group {\bf B}.\\
Similar features (namely secondary sequences in two-color diagrams) have been found in a
variety of two-color diagrams of stellar clusters and field stars in the third Galactic
Quadrant (see e.g. Carraro et al. 2005). They are though to be produced by early type stars stars belonging
to distant spiral features in the outer Galactic disk (see also Moitinho et al. 2008).\\
\noindent
The two red long-dash ZAMSs have been shifted by $E(B-V)$ = 1.10 and $E(B-V)$=1.50,
and identify a group of
early-type (from O6 to B5) stars suffering strong reddening with a significant dispersion.
Since the region in the color-color diagram between the two red ZAMS is continuously occupied by stars,
we propose they basically belong to the same distant population.
We estimate that the mean reddening for this group is 1.30$\pm$0.20.
Within this group of stars,
Sh04 identify a cluster of young stars
surrounding the two WR stars (the two crosses in Fig.~5).
We will refer to this group of stars as group {\bf C} or
Shorlin~1 (also known as C1104-610a).\\
\noindent
As anticipated, the two crosses in Fig.~5 identify the two WR stars; WR38 (WC4) and WR38a (WN5). As exhaustively
discussed by Sh04, $U$-band photometry of this type of star is difficult, but they do provide
corrections to transform their colors to the Johnson-Morgan standard system. Given that our
uncorrected colors (see Table~2) are in good agreement with theirs, we adopted their corrections
(see the Table~3 in Sh04).
\begin{table}
\tabcolsep 0.1truecm
\caption{Photometry of the stars WR38 and WR38a}
\begin{tabular}{lccc}
\hline
\noalign{\smallskip}
& V & (B-V) & (U-B) \\
\noalign{\smallskip}
\hline
\noalign{\smallskip}
WR38 & 14.684 & 1.216 & 0.679\\
WR38a & 15.113 & 1.191 & 0.077\\
\noalign{\smallskip}
\hline
\end{tabular}
\end{table}
\section{Properties of groups A, B and C}
To be more quantitative, we first measured the individual reddening of the stars in groups {\bf A},
{\bf B} and {\bf C} using the well known reddening free $Q-$ parameter (see for instance Johnson 1966),
which allows to establish the membership of a certain
star to a group on the basis of common reddening (see e.g. V\'azquez et al. 1996, V\'azquez et al. 2005, and Carraro 2002).
Having provided evidences that the reddening law is normal, the Q-parameter is therefore defined as:
\[
Q = (U-B) - 0.72 \times (B-V)
\]
\noindent
The value of Q is a function of spectral type and absolute magnitude (Schmidt-Kaler 1982).
\\
This technique permits to identify {\it bona fide} members of similar spectral type as late as
A0V. We have identified 146 stars in our sample with a mean $E(B-V)$ of 0.25$\pm$0.10 (group {\bf A}),
96 stars with a mean $E(B-V)$ of 0.52$\pm$0.05 (group {\bf B}), and 212 stars with a mean $E(B-V)$
of 1.30$\pm$0.20 (group {\bf C}).\\
\noindent
Then, to obtain an estimate of the mean distance of the three groups, we made use of the
variable extinction diagram shown in Fig.~6, where to derive ($V-M_V$) we
adopt M$_V$ as a function of sprectral type from Schmidt-Kaler 1982.
Yellow triangles, blue squares and red circles
depict members of groups {\bf A}, {\bf B}, and {\bf C}, respectively. The four lines plotted
have been drawn adopting the normal extinction law ($R_V$ = 3.1), and correspond to four different
absolute distance moduli (V$_{0}$ - M$_{V}$): 10.0 (solid), 12.0 (dotted), 14.0 (short-dashed) and
16.0 (long-dashed). The histograms in the right panel of this figure give the apparent distance
moduli distribution, from which we infer the mean distance to the three groups.\\
One can in fact extrapolate the absolute
distance modulus (m-M)$_{V,o}$ reading the value along the $Y$-axis, where the reddening is zero.\\
\begin{figure}
\centering
\includegraphics[width=9.0truecm]{AA10800fig6.eps}
\caption{Variable extinction diagram for all early type star for which individual reddening
was measured. The four lines plotted have been drawn adopting the normal extinction law
($R_V$ = 3.1), and correspond to four different absolute distance moduli (V$_{0}$ - M$_{V}$):
10.0 (solid), 12.0 (dotted), 14.0 (short-dash) and 16.0 (long-dash). The histograms in the
right panel of this figure give the apparent distance moduli distribution, from which we
infer the mean distance of the three groups.}%
\end{figure}
\noindent
In the left panels of figures 7, 8 and 9 we present $(B-V)$ vs. $(U-B)$ color-color diagrams
enhancing the stars of groups {\bf A} (yellow triangles), {\bf B} (red squares) and {\bf C}
(blue circles), respectively.
The right panels of these three figures show the distribution on the plane of the sky for the objects
in each group. In these latter panels stars have been plotted as circles scaled according to their
$V$ magnitude. The fields shown in these figures are centered at RA = $11^{h}05^{m}48.5^{sec}$,
DEC = $-61^{o}11^{\prime}5.9^{\prime \prime}$.
North is up, East to the left as in Fig.~1.\\
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{AA10800fig7.eps}
\caption{Stars belonging to group {\bf A} (yellow triangles). Left panel (as well as left
panel of Fig.~6) illustrates how this group was selected on a reddening basis. See text for
details.}%
\end{figure}
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{AA10800fig8.eps}
\caption{Same as Fig.~6, for stars belonging to group {\bf B} (red squares). See text for
details.}%
\end{figure}
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{AA10800fig9.eps}
\caption{Same as Fig.~6, for stars belonging to group {\bf C} (blue circles). See text for
details.}%
\end{figure}
\subsection{Group A}
Inspection of the right panel of Fig.~7 shows that the objects in group {\bf A} are evenly distributed
on the sky, with no hints for any particular concentration.\\
The sequence defined by this group in variable reddening diagram (Fig.~6) has a mean intrinsic
distance modulus of 12.0 magnitudes, with a large spread. Still, most of the stars lie at a mean
distance of 2.5$\pm$1.5 kpc. This mean distance is compatible with this group lying in the part
of the Carina branch of the Carina-Sagittarius arm closest to the Sun. The observed distance spread
is compatible with the typical size of inner Galactic spiral arms (1.5-2.0 kpc, Bronfman et al. 2000).\\
We therefore conclude that group {\bf A} is composed of young field stars belonging to the Carina-Sagittarius
arm, confirming the suggestion by Sh04. We dispute however the possibility that these stars form a
star cluster as suggested by Shorlin (1998), and claim that this conclusion was possibly the result of
the small FOV covered by the photometry of Shorlin (1998). Our wider coverage clearly shows that we
are looking at a population of field stars having the same reddening and age, but located at different
distances in the Carina arm.
\subsection {Group B}
As was the case of group {\bf A}, the objects in group {\bf B} are also evenly distributed on the
plane of the sky, but they are, on the average, fainter (see right panel of Fig.~8).\\
In the variable reddening diagram (Fig.~6) they trace a sequence of stars which lie at larger distance than
those of group {\bf A}. They have a mean absolute distance modulus of 14.0 magnitudes, which implies
a bulk distance of 6.0$\pm$2.0 kpc. The stars in this group are also young, and, according to modern
descriptions of the MW spiral structure (Vall\'ee 2005, Russeil 2003) they are most probably
located in the part of the Carina branch more distant from the Sun. In fact, the tangent to the
Carina-Sagittarius arm points to $l\sim280^o$, in a way that the line of sight to these groups is
expected to cross twice the Carina portion of this arm.\\
This group had not been noticed before, which may be the origin of the different conclusions
suggested by Sh04 and Wa05 on the location, and distance, of the putative star cluster associated to
WR38 and WR38a (see discussion below).
\subsection {Group C}
This is the most interesting group of faint young stars with common reddening. Our mean $E(B-V)$
value (1.30$\pm$0.20) is smaller than the value found by Sh04, but still marginally compatible
within the errors declared. The 212 stars extracted within this reddening range have a broad distance
distribution (see Fig.~6), are faint, and fairly evenly distributed across the field of view
(see Fig.~9). Careful inspection of Fig.~9 indicates the existence of a concentration of stars close
to the center of the field. This concentration corresponds to Shorlin~1 (C1104-610a, Sh04).\\
\noindent
The point now is whether this is really a cluster or just a chance concentration of bright stars-
and where it lies exactly. As Sh04 argue, the reality of the cluster can be assessed only with much
deeper photometry. To address these two issues we will make use of the variable reddening diagram
(Fig.~6) and of the CMDs presented in Fig~10 and Fig.~11, constructed with the objects belonging to
group {\bf C} having the largest reddening (E(B-V) $>$ 1.45), as Shorlin~1 (see Sh04).
Filled symbols in Figs.~10 and 11 indicate stars within a $0.6\arcmin$ wide circle
centered on Shorlin~1 (nominal center RA = $11^{h}05^{m}46.52^{sec}$; DEC=$-61^{o}13^{\prime}49.1^{\prime\prime}$),
which is the zone where the cluster seems to be located.\\
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{AA10800fig10.eps}
\caption{Color-Magnitude Diagrams for stars belonging to group {\bf C}, for different color
combinations. Filled symbols indicate stars inside the Shorlin~1 area}%
\end{figure}
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{AA10800fig11.eps}
\caption{Reddening free Color-Magnitude Diagram for stars belonging to group {\bf C}, for
different color combinations. Filled symbols indicate stars inside Shorlin~1 area. In this figure only
stars having E(B-V) larger than 1.45 are plotted. See text for more details.}%
\end{figure}
\noindent
Given that these common reddening stars of group {\bf C} distribute evenly in the field (see Fig.~9),
we can compare the position (in a CMD) of objects in the general field with that of objects in
the Shorlin~1 region. As shown by the CMDs presented in Fig.~10, general field stars (open symbols)
occupy the same region as Shorlin~1 stars (filled symbols). This suggests that they share the same
distance range and age, and therefore conform a common properties, evenly distributed, population.\\
The same conclusion can be drawn from the reddening-free CMDs presented in Fig.~11. In these
plots the stars within the putative cluster area and those in the general field define the same
sequence. The $V_o$ vs. $(B-V)_o$ CMD presented in the right panel of Fig.~11 has been plotted
in the same scale as Fig.~5 of Sho04. Note that, in spite of our photometry being six magnitudes
deeper in $V$, we detect only one fainter object in the cluster area. We can be confident therefore
that there are no fainter stars associated to the cluster.\\
The ZAMS superposed in the both panels of Fig.~11 is for (m-M)$_{V,o}$ = 15.5, which suggests that
the distance of group {\bf C} is about 12.7$\pm$3.0 kpc. Both Shorlin~1 members and field stars exhibit
however a significant distance spread (see Fig.~6)\\
For the reasons summarized below, we conclude that there is no clear evidence of a star
cluster at the location of WR38/WR38a.
\begin{description}
\item $\bullet$ a cluster hosting two WR stars would have to be a relatively massive one, with
numerous stars of spectral type later than B; on the contrary to what we see,
\item $\bullet$ we do not see any special concentration of stars at the putative cluster location
(see Fig.~1),
\item $\bullet$ the sequence defined by Shorlin~1 in the CMDs is also occupied by field stars which
have the same reddening, and are evenly distributed in the field (see Figs. 9, 10 and 11),
\item $\bullet$ the stars defining the cluster (the 8 stars discovered by Sh04, plus one fainter
object from this work) exhibit a significant distance spread (see Fig.~6), incompatible with a
physical, although loose, star cluster.
\end{description}
\noindent
To summarize our findings, we report in Table~3 the main properties of the three
groups. As for the age we only provide an upper limit, since the populations under
investigations are surely young, and a precise age estimate - by the way quite difficutl-
is beyond the scope of this work.
\section{Conclusions}
The large angular coverage, and depth, of our $UBVI$ photometry of the region around the
WR38/WR38a has allowed us to clarify the stellar populations towards Galactic longitude
$\approx$ 290$^o$, in the fourth Galactic quadrant.\\
\noindent
Our results can be visualized with the aid of Fig.~12, where the spiral structure of the
Galaxy (Vall\'ee 2005) is presented in the $X-Y$ plane, where $X$ points in the direction of
Galactic rotation and $Y$ points towards the anti-center. The Carina and Perseus arms are
indicated with the symbols $I$ and $II$, respectively. The Galactic center and the position
of the Sun are also indicated, at (0.0,0.0) and (0.0,8.5), respectively. The position, pitch
angle, and extension of the arms are clearly model dependent; to a lesser extent for the Carina arm
(which is well known) and significantly more for the less-known Perseus arm (both HI observations
(Levine et al. 2006) and HII observations (Russeil 2003) coincide however on the approximate
location and extent of the Perseus arm). Beyond $\sim$9 kpc from the Sun, it is not expected
to find spiral features related to the Carina arm (Georgelin et al. 2000); more distant structures
can be associated with the Perseus arm. According to Levine et al. (2006, their Fig.~4) at l=290$^o$
the Perseus arm is located in the range $ -18.0\leq X \leq -13.0 $ and $0.0 \leq Y \leq 2.0$ Kpc,
respectively, in nice agreement with Vall\'ee model.\\
\noindent
With a dashed arrow we indicate the line-of-sight in the direction of our field, and with thick
segments the distance range we derived for the three groups ({\bf A}, {\bf B} and {\bf C}) we
identified in this Galactic direction (see also Table~3). Fig.~12 shows that the groups detected remarkably fit
the position of the Carina and Perseus arm. This confirms Sh04's suggestion that the most distant
group ({\bf C}) is most probably associated with the extension of the Perseus arm in the fourth
Galactic quadrant, and constitutes the first optical detection of this arm. However, on the
contrary to Sh04, here we propose that this extreme group is not a star cluster, but simply a group
of young stars uniformly distributed within the Perseus arm.\\
The direction we are investigating is about $\sim1^o$ below the Galactic plane. At distances of
2.5, 6.0 and 12.7 kpc this implies heights below the Galactic plane of about 50, 100, and 210 pc,
respectively. These are sizeable values, and reflect the trend of the Galactic warp in this
zone of the disk (Momany et al. 2006).\\
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{AA10800fig12.eps}
\caption{Schematic view of the spiral structure of the Galaxy (Vall\'ee 2005). $X$ points in
the direction of Galactic rotation, and $Y$ points towards the anti-center. The position of the Sun
(0.0,8.5) and of the Galactic center (0.0,0.0) are indicated. The symbols I and II indicate the
Carina and Perseus spiral arms, respectively. Thick segments illustrate the location of the three
groups ({\bf A}, {\bf B} and {\bf C}) discussed in the text. See text for details}%
\end{figure}
\begin{table}
\tabcolsep 0.1truecm
\caption{Properties of the three distinct groups we detected in the field toward WR~38 and WR~38a.}
\begin{tabular}{lcccc}
\hline
\noalign{\smallskip}
Group& E(B-V) & d$_{\odot}$ & age & Note\\
\noalign{\smallskip}
\hline
\noalign{\smallskip}
A & 0.25$\pm$0.10 & 2.5$\pm$1.5 & $<$ 100 Myr& Carina\\
B & 0.52$\pm$0.05 & 6.0$\pm$2.0 & $<$ 100 Myr& Carina\\
C & 1.30$\pm$0.20 & 12.7$\pm$3.0 & $<$ 100 Myr& Perseus\\
\noalign{\smallskip}
\hline
\end{tabular}
\end{table}
\noindent
We finally wish to discuss our photometric database in the framework of a possible detection
of the Argus over-density discussed in Rocha-Pinto et al. (2006), or the Monoceros Ring,
which Shorlin~1 was suggested to be associated with (Frinchaboy et al. 2004). With this aim, in
Fig.~13 we present $V$ vs. $(B-V)$ CMD for all the stars having BV photometry.\\
\begin{figure}
\centering
\includegraphics[width=9.5truecm]{AA10800fig13.eps}
\caption{CMD of all stars with BV photometry in the field under study}%
\end{figure}
The distribution of stars in this diagram is relatively easy to describe. Apart from the
prominent blue sequence of young stars brighter than V$\approx$ 14.0, which are the objects of
{\bf A}, we can distinguish two other remarkable features. The first is a thick Main Sequence (MS)
downward of V$\approx$ 14.0, widened by photometric errors and binaries, but mostly reflecting the
fact we are sampling objects at very different distances, and with different reddening, in the
Galactic disk. Along this thick MS we do not find any evidence of a blue Turn Off Point, typical
of the metal poor, intermediate age, population of the Monoceros Ring (Conn et al 2007).\\
Therefore we conclude that no indication of the presence of the Monoceros
Ring or the Argus system is detected in our field.\\
\noindent
The second is a Red Giant Branch, significantly widened and somewhat bent by variable reddening,
composed of giant stars at different distances. The relatively high number of giants can be easily accounted for,
because we are sampling a field located 1$^o$ below the Galactic plane.\\
\noindent
Our results confirm the effectiveness of multicolor optical photometry in the study of the
structure of the MW disk. More fields need to be observed to better constrain the
spiral structure in the fourth Galactic quadrant, in particular the shape and extent of the
Perseus arm, and, possibly, detect the more distant Norma-Cygnus arm.\\
At odds with radio observations, which has to rely on the poorly known Galactic rotation
curve, optical observation, especially in low absorption directions,
can better constraint the distance to spiral
features.
\begin{acknowledgements}
Tomer Tal and Jeff Kenney are deeply thanked for securing part of the observations
used in this paper. GC acknowledges R.A.V\'azquez for very
fruitful discussions. EC acknowledges the Chilean Centro de Astrof\'isica FONDAP
(No. 15010003).
\end{acknowledgements}
|
1,108,101,564,546 | arxiv | \section{Completely Sidon sets}
Let $U_n$ be the unitary generators in $C^*(\bb F_\infty)$.\\
Let $A$ be a $C^*$-algebra.
Let $(\psi_n)$ be a bounded sequence in $A$.
\begin{dfn} We say that $(\psi_n)$ is completely Sidon if
there is $C$ such that
for any matricial coefficients $(a_n)$
$$\|\sum a_n \otimes U_n\|_{\min}\le C \|\sum a_n \otimes \psi_n\|_{\min}.$$
Equivalently, the operator space spanned by $(\psi_n)$ in $A$
is completely isomorphic to $\ell_1$ equipped with the maximal operator space structure.\\
Fix an integer $k\ge 1$.
We say that $(\psi_n)$ is completely $\otimes_{\max}^k$-Sidon
in $A$
if the sequence $(\psi_n\otimes \cdots\otimes \psi_n)$ ($k$-times)
is completely Sidon in $A \otimes_{\max} \cdots\otimes_{\max} A$ ($k$-times).
\end{dfn}
\begin{rem} It is important to note that for $k>1$
the notion of completely $\otimes_{\max}^k$-Sidon
is relative to the ambient $C^*$-algebra $A$.
If $A$ is a $C^*$-subalgebra of a $C^*$-algebra $B$,
and $(\psi_n)$ is completely $\otimes_{\max}^k$-Sidon
in $A$, it does not follow in general that
$(\psi_n)$ is completely $\otimes_{\max}^k$-Sidon
in $B$. This does hold nevertheless if there is
a c.p. or decomposable projection from $B$ to $A$.
It obviously holds without restriction if $k=1$,
but for $k>1$ the precision $\otimes_{\max}^k$-Sidon
``in $A$" is important.
However, when there is no risk of confusion
we will omit ``in $A$".
\end{rem}
\begin{pro}\label{R3} The following are equivalent:
\item{(i)} The sequence $(\psi_n)$ is completely Sidon.
\item{(ii)} There is $C$ such that for any ${K},m,k$,
any $(a_n)$ in $M_m$, and
any $(u_n)$ in $U(M_k)$ we have
$$\|\sum\nolimits_1^{K} a_n\otimes u_n\| \le C \|\sum\nolimits_1^{K} a_n\otimes \psi_n\|.$$
\item{(ii)'} Same as (ii) but for $(u_n)$ in the unit ball of $M_k$.
\item{(ii)''} There is $C$ such that for any
$C^*$-algebras $B$ and $D$,
any $(a_n)$ in $B$, and
any $(u_n)$ in the unit ball of $D$ we have
$$\|\sum\nolimits_1^{K} a_n\otimes u_n\|_{B\otimes_{\min} D} \le C \|\sum\nolimits_1^{K} a_n\otimes \psi_n\|_{B\otimes_{\min} A}.$$
\item{(iii)} Same as (ii) but with ${K}$ even, say ${K}=2m$ and the $u_n$'s restricted
to be such that $u_{m+j}=u_j^{-1}$ for $1\le j\le m$.
\item{(iv)} Same as (ii) but with the $u_n$'s restricted
to be selfadjoint unitaries.
\end{pro}
\begin{proof}[Sketch]
The equivalence (i) $\Leftrightarrow$ (ii) is just an explicit reformulation of the preceding definition.
To justify (iii) $\Rightarrow$ (ii) we can use $(z u_j, \bar z u_{m+j}^{-1})_{1\le j\le m}$. Then after integrating in $z\in \bb T$, we can separate the two
parts of the sum appearing in (ii).
This gives us for the sup over all the $u_n$'s as in (iii)
$$ {\sup\nolimits_{(iii)}\|\sum\nolimits_1^{2m} a_n\otimes u_n\|} \ge
\max\{ \|\sum\nolimits_1^{m} a_n\otimes u_n\|, \|\sum\nolimits_{m+1}^{2m} a_n\otimes u_n\| \} .$$
and hence (triangle inequality)
\begin{equation}\label{e89} {\sup\nolimits_{(iii)}\|\sum\nolimits_1^{2m} a_n\otimes u_n\|} \ge
(1/2) \sup\nolimits_{(ii)}\|\sum\nolimits_1^{2m} a_n\otimes u_n\| ,\end{equation}
where the last sup runs over all $(u_n)$ as in (ii).
We then deduce
(ii) from (iii) possibly with a different constant.\\
To justify (iv) $\Rightarrow$ (ii) we can use
a $2\times 2$-matrix trick: if $(u_n)$ is an arbitrary
sequence in $U(k)$, $(\begin{matrix}0 &u_n\\ u_n^* &0 \end{matrix})$
are selfadjoint in $U(2k)$. We then deduce
(ii) from (iv) with the same constant.\\
Lastly the equivalence (ii) $\Leftrightarrow$ (ii)'
is obvious by an extreme point argument,
and (ii)' $\Leftrightarrow$ (ii)''
(which reduces to $B=B(H)$ and hence to the matricial case)
follows by Russo-Dye and standard operator space arguments
(see \cite[p. 155-156]{P4} for more background).
\end{proof}
\begin{rem} For simplicity we state our results
for sequences indexed by $\bb N$, but actually they hold with obvious adaptation of the proofs
for families indexed by an arbitrary set, finite or not, with bounds independent
of the number of elements.
\end{rem}
\noindent{\bf Examples :} \\
(i) The fundamental example of a completely Sidon set (with $C=1$) is of course
any free subset in a group.
More precisely, if we add the unit to a free set, the resulting augmented set
is still completely Sidon with $C=1$.
Moreover, any left or right translate of a
completely Sidon set is completely Sidon (with the same $C$).
Therefore any left translate of a free set augmented by the unit
is completely Sidon with $C=1$.
The converse also holds and is easy to prove, see \cite{Pifz}.\\
(ii) It is proved in \cite[Th. 8.2 p.150]{P4} that for any $G$
the diagonal mapping $t\mapsto \lambda_G(t) \otimes \lambda_G(t) $
defines an isometric embedding of $C^*(G)$ into
$C_\lambda^*(G)\otimes_{\max} C_\lambda^*(G)$.
It follows that a subset $\Lambda \subset G$
is completely Sidon iff the set
$\{\lambda_G(t) \mid t\in \Lambda\}$ is completely $\otimes_{\max}^2$-Sidon in $C_\lambda^*(G)$.
Let $M_G$ be the von Neumann algebra of $G$ (i.e. the one generated by
$ \lambda_G$).
Similar arguments show that the same diagonal embedding
embeds $C ^*(G)$ also into $M_G\otimes_{\max} M_G$.
In particular the set of free generators
is a completely $\otimes_{\max}^2$-Sidon set in $C_\lambda^*(\bb F_\infty)$
(and also in $M_{\bb F_\infty}$).
\medskip
We will be interested in another property, namely the following one:
Let $X_1,X_2$ be preduals of $C^*$-algebras (so-called non-commutative $L_1$-spaces).
We say that a bounded linear map $v: X_2 \to X_1$ is
completely positive (in short c.p.) if $v^*: X_1^* \to X_2^*$ is c.p..\\
Let $A,B$ be $C^*$-algebras.
Let $CP(A,B)$ be the set of c.p. maps from $A$ to $B$.
We say that a bounded linear map $u:A\to B$
is decomposable if
there are $u_j \in CP(A,B)\quad(j=1,2,3,4)$ such that
$$u=u_1 -u_2 +i(u_3 -u_4 ).$$
We use the dec-norm as defined by Haagerup \cite{Haa}.
We denote
\begin{equation}\label{d11}\| u\|_{dec}=\inf\{\max\{\| S_1\|,\| S_2\|\}\}
\end{equation}
where the infimum runs over all maps $S_1 ,S_2\in CP(A,B)$ such that the map
\begin{equation}\label{d12} V: x\to \left(
\begin{matrix}
S_1 (x) & u(x)\\
u(x^* )^* & S_2 (x)
\end{matrix}
\right) \end{equation}
is in $CP(A,M_2 (B))$.
A mapping $v: X_2 \to X_1$ is said to be decomposable
if its adjoint $v^*: X_1^* \to X_2^*$ is decomposable in the preceding sense
(linear combination of c.p. maps),
and we set by convention
$$\|v\|_{dec} =\|v^*\|_{dec}.$$
We use the term $c$-decomposable
for maps that are decomposable with dec-norm $\le c$.
The crucial property of a decomposable map
$v: A\to B$ between
$C^*$-algebras is that for any other $C^*$-algebra
$C$ the mapping $id_C\otimes v$ extends to a bounded (actually decomposable) map from $C\otimes_{\max} A$
to
$C\otimes_{\max} B$. Moreover we have
$$ \| id_C\otimes v: C\otimes_{\max} A\to C\otimes_{\max} B \|
\le \|v\|_{dec}.
$$
Consequently, for any pair
$v_j: A_j\to B_j$ ($j=1,2$)
of decomposable maps
between
$C^*$-algebras, we have
\begin{equation}\label{e8}
\| v_1\otimes v_2: A_1\otimes_{\max} A_2\to B_1\otimes_{\max} B_2\|_{cb} \le
\| v_1\otimes v_2: A_1\otimes_{\max} A_2\to B_1\otimes_{\max} B_2\|_{dec} \end{equation}
\begin{equation}\label{e8'}
\le \|v_1\|_{dec}\|v_2\|_{dec}.
\end{equation}
\begin{dfn}\label{dom} Let $I,J$ be any sets.
(i) Let $(x_n^1)_{n\in I } $ (resp. $(x_n^2)_{n\in I } $ )
be a family in $X_1$ (resp. $X_2$).
Let us say that $(x_n^1)_{n\in I } $ is $c$-dominated
(or ``decomposably $c$-dominated")
by $(x_n^2)_{n\in I} $ if there is a decomposable mapping
$v: X_2 \to X_1$
with $\|v\|_{dec}\le c$ such that $v( x_n^2 ) = x_{n}^1 $
for any $n\in I$.\\
We simply say ``dominated" for $c$-dominated for some $c$.\\
(ii) We say that $(x_n^1)_{n\in I } $ and $(x_n^2)_{n\in J } $
are `` decomposably equivalent "
if there is a bijection $f: I \to J$ such that
each of the families
$(x_n^1)_{n\in I } $ and $(x_{f(n)}^2)_{n\in I} $ is dominated by the other.
\end{dfn}
Let $Y$ be the predual of a $C^*$-algebra.
The positive cone in $M_k(Y)$ is the polar of the
positive cone $M_k(Y^*)_+$ in the $C^*$-algebra $M_k(Y^*)$.
More precisely
$y\in M_k(Y)_+$
iff
$$\forall a\in M_k(Y^*)_+\quad \sum\nolimits_{ij} a_{ij} (y_{ij} )\ge 0. $$
Clearly $v: X_2 \to X_1$ is c.p.
iff for any $k$ the mapping $id_{M_k} \otimes v: M_k(X_2)
\to M_k(X_1)$ is positivity preserving.
More generally, since we have positive cones on
both $M_k(X^*)$ and $M_k(Y)$,
we can extend the definition of complete positivity
to maps from a $C^*$-algebra to $Y$
or from $Y$ to a $C^*$-algebra. In particular,
a map $T: X^*\to Y$
is called c.p. if
$id_{M_k} \otimes T: M_k(X^*)\to M_k(Y)$ is positivity preserving
for any $k$.
\begin{rem}\label{rop}[Opposite von Neumann algebra]
The opposite von Neumann ${M^{op}}$ is the same linear space
as $M$ but with the reverse product.
Let $\Phi: {M^{op}} \to M$ be the identity map,
viewed as acting from ${M^{op}}$ to $ M$,
so that $\Phi^*: M^* \to {M^{op}}^*$ also acts as the identity.\\
When $M$ is a von Neumann algebra
equipped with a normal faithful tracial state $\tau$,
there is a minor problem that needs clarification.
We have a natural inclusion $J: M\to M^*$ denoted by $y\mapsto J y$ and
defined by $J{y}(x)=\tau(yx)$.
In general this is not c.p, but
it is c.p. when viewed as a mapping
either from $M^{op}\to M^*$ or from $M\to {M^{op}}^*$.
Indeed, for all $x,y\in M_k(M)_+$
we have $[{\rm tr}\otimes\tau] (xy)=\sum_{ij} \tau ( x_{ij} y_{ji}) \ge 0$
but in general it is \emph{not true} that for $\sum_{ij} \tau ( x_{ij} y_{ij})=
\sum_{ij} J{y_{ij}} ( x_{ij} ) $.\\
Then the content of the preceding observation is that
$\Phi^* J: M \to {M^{op}}^*$ is c.p. (but $J$ in general fails this).
\end{rem}
\begin{rem}\label{R6}[About preduals of finite vN algebras]
Let $(M^1,\tau^1)$ be here any noncommutative probability space,
i.e. a von Neumann algebra equipped with
a normal faithful tracial state.
The predual ${M^1}_*$ is the subset of ${M^1}^*$ formed of the
weak* continuous functionals on $M^1$. It can be
isometrically identified with the space
$L_1(\tau_1)$ defined as the completion of $M_1$ for the
norm $\|x\|_1=\tau_1(|x|)$. Thus
we have a natural inclusion
with dense range $M^1\subset L_1(\tau_1)$.
We need to observe
the following fact. Let $({M}^2, \tau^2)$
be another noncommutative probability space.
Let $V: L_1(\tau_1)\to L_1(\tau_2)$
be a linear map that is a $*$-homomorphism
from $M^1$ to $M^2$
when restricted to $M^1$.
Then $V$ is
completely positive and hence 1-decomposable.
\end{rem}
\section{Analysis of the free group case}\label{sf0}
We denote by $M$ the von Neumann algebra of the free group
$\bb F_{\infty}$ equipped
with its usual trace $\tau$.
We denote by $(\varphi_n)_{ n\ge 1}$ the elements of $M=\lambda_{\bb F_{\infty}} (\bb F_{\infty})''$
corresponding
to the free generators $(g_n)$ in $\bb F_{\infty}$,
i.e. $\varphi_n=\lambda_{\bb F_{\infty}}(g_n)$.
For convenience we set
$$\forall n\ge 1\quad \varphi_{-n}=\varphi_n^{-1}.$$
Although this is a bit pedantic, it is wise
to distinguish the elements of $M$ from the linear functionals
on $M$ that they determine. Thus we let
$(y_n)_{n\in \bb Z_*}$ be the sequence in $M_*\subset M^*$
that is biorthogonal to the sequence $({\varphi}_n)_{n\in \bb Z_*}$, and defined
for all $n\in \bb Z_*=\bb Z\setminus \{0\}$ by
\begin{equation}\label{e11}\forall a\in M\quad {y_n}(a)=\tau (\varphi_n^* a).\end{equation}
We also define $y_n^*\in M_*\subset M^*$ as follows
\begin{equation}\label{e12}\forall a\in M\quad {y^*_n}(a)=\tau (\varphi_n a).\end{equation}
Again let $J: M \to M^*$ be the inclusion
mapping defined by
$Ja(b)=\tau(ab)$. With this notation
$$y_n=J(\varphi_n^*) \text{ and } y_n^*=J(\varphi_n) .$$
For future reference, we record here a simple observation:
\begin{lem}\label{dom1}
Recall $\bb Z_*=\bb Z\setminus \{0\}$. The families
$(y_n)_{ n\ge 1}$ and $(y_n)_{ n\in \bb Z_*}$
are decomposably equivalent in the sense of Definition \ref{dom}.
\end{lem}
\begin{proof} Let $(z_n)_{n\in \bb Z_*}$ be a sequence
such that each $(z_n)_{n>0}$ and $(z_n)_{n<0}$
are mutually free, each one being a free Haar unitary sequence.
Then $(z_n)_{n\in \bb Z_*}$ and $(\varphi_n)_{ n\ge 1}$ are trivially
decomposably equivalent.
Let $L$ be a copy of the von Neumann algebra of $\bb Z$.
Let $N=M\ast L$. Let $U$ denote the generator of $L$
viewed as a subalgebra of $N$.
We also view $M\subset N$.
Then the family $(Uy_n)_{ n\in \bb Z_*}$
viewed as sitting in $N_*$
is a family of free Haar unitaries.
Therefore $(Uy_n)_{ n\in \bb Z_*}$ and
$(y_n)_{ n\ge 1}$ are
decomposably equivalent. But $(Uy_n)_{ n\in \bb Z_*}$
and $(y_n)_{ n\in \bb Z_*}$
are also decomposably equivalent in $N_*$, because the multiplication
by $U$ or $U^{-1}$ is decomposable
(roughly because, since $x\mapsto a xa^*$ is c.p.,
$x\mapsto a xb^*$ is decomposable by the polarization formula).
Lastly using conditional expectations
it is easy to see that the families
$(y_n)_{ n\in \bb Z_*}\subset N_* $
and $(y_n)_{ n\in \bb Z_*}\subset M_* $ (identical families
viewed as sitting in $N_* $ or $M_* $)
are decomposably equivalent.
\end{proof}
Let $\cl A$ be the algebra generated by $(\varphi_n)_{n\in \bb Z_*}$.
Note for further reference
that the orthogonal projection $P_1$
onto the closed span in $L_2(\tau)$ of $(\varphi_n)_{n\in \bb Z_*}$
is defined by
$$\forall a\in \cl A\quad P_1(a)=\sum \tau(\varphi_n^*a) \varphi_n. $$
We use ingredients analogous to those of \cite{Pi3}
but in \cite{Pi3} the free group is replaced by the free Abelian group,
and an ordinary gaussian sequence is used
(we could probably use analogously a free semicircular sequence here).
Let $U_n\in C^*(\bb F_\infty)$ be the unitaries coming from the
free generators. We set again by convention
$U_{-n}=U_n^{-1}$ ($ n\ge 1$).\\
Let $\cl E={\rm span}[U_n\mid n\in \bb Z_*] \subset C^*(\bb F_\infty)$.
Consider the
natural
linear map
$\pi: \cl E\to M$
such that
$$\forall n\in \bb Z_*\quad\pi(U_n)={\varphi}_n.$$
Its key property is that
for some Hilbert space $H$ there is a factorization
of the form
$$\cl E {\buildrel \pi_1\over
\longrightarrow } B(H) {\buildrel \pi_2\over
\longrightarrow } M$$
such that
$$\forall n\in {\bb Z_*}\quad \pi_2\pi_1(U_n)=\pi(U_n)={\varphi}_n$$
where $\|\pi_1\|_{cb}\le 1$, and $\pi_2$ is a
\emph{decomposable} map
with
$ \|\pi_2\|_{dec}=1$.
To check this note that $M$ embeds in a trace preserving way into
an ultraproduct $\cl M$ of matrix algebras, and there is a c.p. conditional
expectation from $\cl M$ onto $M$.
Therefore
there is a completely positive surjection $\pi_2$
from $B=\prod\nolimits_{k} M_k$ to $M$ and a $*$-homomorphism
$\pi_1: C^*(\bb F_\infty) \to B$ such that ${\pi_2 \pi_1}_{| {\cl E }} =\pi$.
To complete the argument we need to replace $B$ by $B(H)$. Since
$B$ embeds in $B(H)$ for some $H$
and there is a conditional expectation from
$B(H)$ to $B$, this is immediate.
We refer the reader to \cite[\S 9.10]{P4} for more details.
The following statement
on the free group factor $M$
is the key for our results.
\begin{thm}\label{key4}
The sequence $(\varphi_n)_{ n\in {\bb Z_*}}$ in $ M$
satisfies the following property:\\
any bounded sequence $(z_n)_{ n\in {\bb Z_*}}$ in $M$ that is
biorthogonal to $(\varphi_n)_{ n\in {\bb Z_*}}$
in $L_2(\tau)$ meaning that
$$\tau(z_n \varphi_m^*)=0 \text{ if } n\not = m \text{ and } \tau(z_n \varphi_n^*)=1,$$
is completely $\otimes^2_{\max}$-Sidon.
More generally, if $(z^1_n)_{ n\in {\bb Z_*}}$
and $(z^2_n)_{ n\in {\bb Z_*}}$ are bounded in $M$ and each biorthogonal to $(y_n)_{ n\in {\bb Z_*}}$,
then $(z^1_n\otimes z^2_n)_{ n\in {\bb Z_*}}$ is completely Sidon in $M \otimes_{\max} M$.
\end{thm}
Let $(z^1_n)$
and $(z^2_n)$ be as in Theorem \ref{key4}.
Assume $\|z^j_n\|\le C'_j$ for all $n\in {\bb Z_*}$ ($j=1,2$).
Fix integers $k,k'\ge 1$. Let $(a_n)$ be a family in $M_k$ with only finitely many $n$'s
for which $a_n\not= 0$.
Let $(u_n)_{n\in {\bb Z_*}}$ be unitaries in $M_k$ such that
$u_{-n}=u_n^{-1}$ for all $n$.
Our goal is to show that there is a constant $\alpha$ depending only
on $C'_1,C'_2$ such that
$$ \|\sum u_n \otimes a_n \|_{M_{k'}\otimes_{\min} M_k}\le \alpha\| \sum a_n \otimes z^1_n \otimes {z^2_n}^* \|_{M_k( M \otimes_{\max} M^{op})}.$$
This will prove the key Theorem \ref{key4}
with $M \otimes_{\max} M^{op}$ instead of
$M \otimes_{\max} M$. Then a simple elementary argument will allow us
to replace $M^{op}$ by $M$.
\begin{rem}\label{r1}
Let $T: M \to M^*$ be a c.p. map such that $T(1)(1)=1$.
We associate to it a state $f$ on $M\otimes_{\max} M$
by setting
$$f(x\otimes y)= T(x)(y).$$
A matrix $x\in M_k (M^*)$
is defined as $\ge 0$ if $\sum_{ij} x_{ij}(y_{ij}) \ge 0$
for all $y\in M_k(M)_+$.\\
More generally, any decomposable
operator $T$ on $M$ (in particular any finite rank one)
determines an element $\Phi_{T}$ of $(M\otimes_{\max} M^{op})^*$,
defined by for $x,y\in M$ by
$$\langle \Phi_{T}, x\otimes y\rangle=
\tau(T(x)y).
$$
Indeed, the bilinear form
$(x,y)\mapsto \tau(xy)$ is of unit norm in
$(M\otimes_{\max} M^{op})^*$ and
$$\|T\otimes id_{M^{op}}: M\otimes_{\max} M^{op}\to M\otimes_{\max} M^{op} \|\le \|T\|_{dec}.$$
Furthermore, for any pair of $C^*$-algebras $A,B$, we have a 1-1-correspondence between
the set of decomposable maps $T: A\to B^*$ and
$(A\otimes_{\max} B)^*$.
\end{rem}
\begin{rem}\label{r2}
We will need the free analogue of Riesz products.\\
Recall we set $M=M_{\bb F_\infty}$.
Let $0\le \varepsilon\le 1$.
Let $P_\ell$ the orthogonal projection on $L_2(\tau)$ onto the span of the words
of length $\ell$ in $\bb F_\infty$. Let ${\theta}_\varepsilon =\sum_{\ell\ge 0} \varepsilon^\ell P_\ell$.
By Haagerup's well known result \cite{Hainv},
${\theta}_\varepsilon$ is a c.p. map on $M$. Composing it with the inclusion
$M\subset M_*$, we find a unital c.p. map
from $M $ to $ {M^{op}}^*$, and hence ${\theta}_\varepsilon$
determines a state $f_\varepsilon$ on $M\otimes_{\max} M^{op}$.
We view $\theta_\varepsilon$ as acting
from $M$ to $L_2(\tau)$.
We can also consider it as a map taking $\cl A$ to itself.
We will crucially use the decomposition
$ ({\theta}_\varepsilon-{\theta}_0)/\varepsilon= P_1+ \sum_{{\ell}\ge 2} \varepsilon^{{\ell}-1} P_{\ell}.$
We set $$T_\varepsilon=({\theta}_\varepsilon-{\theta}_0)/\varepsilon\text{ and }
R_\varepsilon= - \sum_{{\ell}\ge 2} \varepsilon^{{\ell}-1} P_{\ell},$$
so that
\begin{equation}\label{e0} P_1= T_\varepsilon+R_\varepsilon.\end{equation}
We have
\begin{equation}\label{e1}\|T_\varepsilon\|_{dec}\le 2/\varepsilon\end{equation}
and
\begin{equation}\label{e2}\|R_\varepsilon: M \to L_2(\tau)\|\le \|R_\varepsilon: L_2(\tau) \to L_2(\tau)\|\le \varepsilon.\end{equation}
\end{rem}
\begin{lem}\label{l1}
With the preceding notation, we have
\begin{equation}\label{e3} \| \sum a_n \otimes T_\varepsilon(z^1_n) \otimes {z^2_n}^* \|_{M_k( M \otimes_{\max} M^{op})}
\le (2/\varepsilon) \| \sum a_n \otimes z^1_n \otimes {z^2_n}^* \|_{M_k( M \otimes_{\max} M^{op})}
.\end{equation}
\end{lem}
\begin{proof}
This follows from \eqref{e8'}. \end{proof}
\begin{proof}[Proof of Theorem \ref{key4}]
Fix $\varepsilon<1$ (to be determined later). We have decompositions
$$T_\varepsilon(z^1_n) = \varphi_n + r^1_n $$
$$ z^2_n = \varphi_n + r^2_n $$
where $r^1_n= -R_\varepsilon(z^1_n )$ and $r^2_n $
are orthogonal to $(\varphi_n)_{n\in {\bb Z_*}}$ and moreover
$$\|r^1_n\|_2=\|R_\varepsilon(z^1_n )\|_2 \le \varepsilon \|z^1_n\|_2 \le \varepsilon C'_1,$$
$$\|r^2_n\|_2\le \| z^2_n\|_2 \le C'_2.$$
We have
$$T_\varepsilon(z^1_n) \otimes {z^2_n}^*
=(\varphi_n + r^1_n) \otimes (\varphi_n + r^2_n)^*.$$
The idea will be to reduce this product to the simplest term $\varphi_n \otimes \varphi_n^*$.\\
Let $V: M\to M_k(M)$ be the isometric $*$-homomorphism
taking $\varphi_n$ to $u_n\otimes \varphi_n$.
Note that $V$ is decomposable with $\|V\|_{dec}=1$.
We observe
$$(V\otimes id_{M^{op}} )(\varphi_n \otimes \varphi_n ^* ) =u_n \otimes \varphi_n \otimes \varphi_n^*.$$
Let $\gamma: M\otimes M^{op} \to \bb C$ be the bilinear
form defined by $\gamma(a\otimes a')=\tau(aa')$.
It is a classical fact that $\gamma$ is a state on $M\otimes_{\max} M^{op} $.
We claim
\begin{equation}\label{e88}\|(id_{M_k} \otimes \gamma)(V\otimes id_{M^{op}} )(r^1_n \otimes {r^2_n}^* ) \|_{M_k}
\le \varepsilon C'_1C'_2.\end{equation}
Indeed, let $\bb F=\bb F_\infty$ for simplicity.
We may develop in $L_2(\tau)$
$$r^1_n= \sum\nolimits_{t\in \bb F} r^1_n(t) \lambda_\bb F(t) \text{ and }
r^2_n= \sum\nolimits_{t\in \bb F} r^2_n(t) \lambda_\bb F(t).$$
Let $\sigma$ be the unitary representation
on $\bb F$ taking $g_n$ to $u_n\in M_{k'}$. For simplicity
we denote $u_t=\sigma(t)$ for any $t\in \bb F$. With this notation
$V(\lambda_\bb F(t))= u_t\otimes \lambda_\bb F(t).$
Then
$$(V\otimes id_{M^{op}} )(r^1_n)= \sum\nolimits_{t\in \bb F} r^1_n(t) u_t\otimes \lambda_\bb F(t),$$
$$(id_{M_{k'}} \otimes \gamma)(V\otimes id_{M^{op}} )(r^1_n \otimes {r^2_n}^* )=
\sum\nolimits_{t\in \bb F} r^1_n(t) \overline{ r^2_n(t) } u_t $$
and hence (triangle inequality and Cauchy-Schwarz)
$$\|(id_{M_{k'}} \otimes \gamma)(V\otimes id_{M^{op}} )(r^1_n \otimes {r^2_n}^* ) \|_{M_{k'}}
\le\|r^1_n\|_2 \|r^2_n\|_2 \le \varepsilon C'_1C'_2 .
$$
This proves our claim.
Let
\begin{equation}\label{e20}\delta_n= (id_{M_{k'}} \otimes \gamma)(V\otimes id_{M^{op}} )(r^1_n \otimes {r^2_n}^* ).\end{equation}
Recalling the orthogonality relations $\varphi_n \perp r^1_n$ and $\varphi_n \perp r^2_n$
we see that
$$(id_{M_{k'}} \otimes \gamma)(V\otimes id_{M^{op}} ) ( T_\varepsilon(z^1_n) \otimes {z^2_n}^* )
)
= (id_{M_{k'}} \otimes \gamma)(V\otimes id_{M^{op}} )(\varphi_n \otimes {\varphi_n}^* )
+(id_{M_{k'}} \otimes \gamma)(V\otimes id_{M^{op}} )(r^1_n \otimes {r^2_n}^* )$$
$$= u_n + \delta_n.$$
We now go back to \eqref{e3}:
we have
$$
(id_{M_k} \otimes id_{M_{k'}} \otimes \gamma)(V\otimes id_{M^{op}} )\sum a_n \otimes T_\varepsilon(z^1_n) \otimes {z^2_n}^* =
\sum a_n \otimes (u_n+ \delta_n). $$
Therefore (the norm $\|\ \|$ is the norm in ${M_{k'}\otimes_{\min} M_k}$)
$$\| \sum a_n \otimes (u_n+ \delta_n)\|\le \| \sum a_n \otimes T_\varepsilon(z^1_n) \otimes {z^2_n}^* \|_{M_k( M \otimes_{\max} M^{op})}
$$
and hence by \eqref{e3}
$$\| \sum a_n \otimes (u_n+ \delta_n)\|\le (2/\varepsilon) \| \sum a_n \otimes z^1_n \otimes {z^2_n}^* \|_{M_k( M \otimes_{\max} M^{op})}.
$$
By the triangle inequality
$$\| \sum a_n \otimes u_n \| -\| \sum a_n \otimes \delta_n \| \le (2/\varepsilon) \| \sum a_n \otimes z^1_n \otimes {z^2_n}^* \|_{M_k( M \otimes_{\max} M^{op})} .
$$
Recalling \eqref{e88} and \eqref{e20}
we find
$$\| \sum a_n \otimes u_n \| -\varepsilon C'_1C'_2 \sup\nolimits_{b_n\in B_{M_{k'}}}\| \sum a_n \otimes b_n \|\le (2/\varepsilon) \| \sum a_n \otimes z^1_n \otimes {z^2_n}^* \|_{M_k( M \otimes_{\max} M^{op})} .
$$
Taking the sup over all $u_n$'s and using \eqref{e89} (recall $B_{M_{k'}}$ is the convex hull
of $U(k')$)
we find
$$(1/2-\varepsilon C'_1C'_2) \sup\nolimits_{b_n\in B_{M_{k'}}}\| \sum a_n \otimes b_n \|\le (2/\varepsilon) \| \sum a_n \otimes z^1_n \otimes {z^2_n}^* \|_{M_k( M \otimes_{\max} M^{op})} .
$$
This completes the proof for $M \otimes_{\max} M^{op}$, since
if we choose, say, $\varepsilon=\varepsilon_0$ with $\varepsilon_0=(4 C'_1C'_2)^{-1}$
we obtain the announced result with $\alpha= 8/\varepsilon_0=32 C'_1C'_2$.
It remains to justify the replacement of $M^{op}$ by $M$.
For this it suffices to exhibit
a (normal) $\bb C$-linear $*$-isomorphism $\chi: M^{op}\to M$
such that $\chi(\varphi^*_n)= \varphi_n$ for all $n\in {\bb Z_*}$.
Indeed, let us view $M\subset B(H)$
with $H=\ell_2(\bb F_\infty)$.
Then
since ${}^t\varphi_n=\varphi_n^*$ for all $n\in {\bb Z_*}$
(these are matrices with real entries), the matrix transposition $ x\mapsto {}^t x$
is the required $*$-isomorphism $\chi: M^{op}\to M$.
\end{proof}
\begin{cor}\label{key2} Let $(z^1_n)_{ n\ge 1}$
and $(z^2_n)_{ n\ge 1}$ be bounded in $M$ and each biorthogonal to $(y_n)_{ n\ge 1}$,
then $(z^1_n\otimes z^2_n)_{ n\ge 1}$ is completely Sidon in $M \otimes_{\max} M$.
\end{cor}
\begin{proof} By Lemma \ref{dom1} we know that
$(y_n)_{ n\ge 1}$ is dominated by $(y_n)_{ n\in {\bb Z_*}}$.
Let $v: M_* \to M_*$ decomposable taking $(y_n)_{ n\in {\bb Z_*}}$
to $(y_n)_{ n\ge 1}$ (modulo a suitable bijection $f: {\bb Z_*}\to \bb N_*$).
Then $(v^*(z^j_{f(n)}))_{ n\in {\bb Z_*}}$ ($j=1,2$)
is biorthogonal to $(y_{f(n)})_{ n\in {\bb Z_*}}$.
By Theorem \ref{key4},
$(v^*(z^1_{f(n)}) \otimes v^*(z^2_{f(n)}))_{ n\in {\bb Z_*}}$
is completely Sidon in $M\otimes_{\max} M$.
By \eqref{e8'}
$( z^1_{f(n)} \otimes z^2_{f(n)} )_{ n\in {\bb Z_*}}$
is completely Sidon in $M\otimes_{\max} M$.
Equivalently since this is obviously invariant under permutation,
we conclude
$( z^1_{n} \otimes z^2_{n} )_{ n\in \bb N_*}$
is completely Sidon in $M\otimes_{\max} M$.
\end{proof}
\section{Main results. Free unitary domination}\label{sf}
We start with a simple but crucial observation that links
completely Sidon sets with
the free analogues of Rademacher functions or independent gaussian
random variables.
\begin{pro}\label{p1}
Let $\Lambda=\{\psi_n\mid n\ge 1\}$
be a completely Sidon set in $A$ with constant $C$.
Then there is a biorthogonal system
$(x_n)_{n\ge 1}$ in $A^*$
that is $C$-dominated
by $(y_n)_{n\ge 1}$.
\end{pro}
\begin{proof}
Let $E\subset A$ be the linear span
of $\{\psi_n\} $.
Let $\alpha: E \to \cl E $
be the linear map such that $\alpha (\psi_n)= U_n$.
By our assumption $\|\alpha\|_{cb}\le C$.
We have $\|\pi_1 \alpha: E\to B(H)\|_{cb}\le C$.
By the injectivity of $B(H)$,
$\pi_1 \alpha$ admits an extension
$\beta: A \to B(H)$ with $\|\beta\|_{cb}\le C$.
Note (see \cite{Haa}) that
$\|\beta\|_{dec}=\|\beta\|_{cb}$.
Let $V=\pi_2\beta: A \to M$.
Then $V$ is a $C$-decomposable map.
Its adjoint $V_*: M_* \to A^*$ is also $C$-decomposable.
Let $ {y_n}\in M_*$ be the functionals
biorthogonal to the sequence $({\varphi}_n)$
defined above in $M$.
We have $ {y_n}({\varphi}_m)=\delta_{nm}$.
Therefore since $V(\psi_m) =\pi_2\beta(\psi_m)
=\pi_2\pi_1 \alpha(\psi_m)={\varphi}_m$
$$ {y_n} (V(\psi_m))=\delta_{nm}.$$
Thus setting $x_n=V_*( {y_n})$
we find $ {x_n} (\psi_m)=\delta_{nm}.$
This shows that $(x_n)$,
which is by definition $C$-dominated
by $(y_n)$, is biorthogonal to $(\psi_n)$.
\end{proof}
\begin{thm}\label{t1}
Let $A_1$, $A_2$ be $C^*$-algebras.
Let $(\psi^1_n)_{n\ge 1}$, $(\psi^2_n)_{n\ge 1}$ be bounded sequences in $A_1$, $A_2$
bounded by $C_1'$ and $C_2'$ respectively. Let
$(x^1_n)_{n\ge 1}$ be a sequence in $A_1^*$ biorthogonal to $(\psi^1_n)_{n\ge 1}$,
and let $(x^2_n)_{n\ge 1}$ be a sequence in $A_2^*$ biorthogonal to $(\psi^2_n)_{n\ge 1}$.
If both are dominated by $(y_n)_{n\ge 1}$, then
$(\psi^1_n\otimes \psi^2_n)_{n\ge 1}$ is completely Sidon in $A_1\otimes_{\max} A_2$.\\
More precisely, if $(x^j_n)_{n\ge 1}$ is $c_j$-dominated by $(y_n)_{n\ge 1}$,
$(\psi^1_n\otimes \psi^2_n)_{n\ge 1}$ is completely Sidon in $A_1\otimes_{\max} A_2$ with a constant $C$ depending only on $C_1',C_2', c_1,c_2$.
\end{thm}
\begin{proof}
The key ingredient is Corollary \ref{key2}.
Assume $(x_n^j)$ dominated by $(y_n)$.
Let $v_j: M_* \to A^*_j$ be decomposable such that
$v_j(y_n)=x_n^j$ ($j=1,2$), with $(x_n^j)$
biorthogonal to $(\psi_n^j)$ and $\|v_j\|_{dec}\le c_j$.
Moreover let $w_j: A_j \to M$ be the restriction of
$v_j^{*}: A_j^{**} \to M$ to $A_j$.
Note that $(v_j^{*} (\psi^j_n))$, or equivalently
$(w_j (\psi^j_n))$, is obviously biorthogonal
to $(y_n)$ for each $j=1,2$.
Let $z_n^j=w_j (\psi^j_n)$.
By Corollary \ref{key2}
the sequence $(z_n^1 \otimes z_n^2)$
is completely Sidon in $M \otimes_{\max} M$.
But since $w_j\in D(A_j,M)$
we see by \eqref{e8'}
that this implies that
$(\psi^1_n\otimes \psi^2_n)$ is completely Sidon in $A_1\otimes_{\max} A_2$.
The assertion on the constants is easy to check by going over the argument.
\end{proof}
\begin{rem}[On ``pseudo-free" sequences]\label{ps}
Let us say that a sequence $({\kappa}_n)$ in the predual $N_*$ of a von Neumann algebra $N$
is pseudo-free if $(y_n)$ and $({\kappa}_n)$ are decomposably equivalent.
Clearly, we may replace $(y_n)$ by any other pseudo-free sequence
in what precedes.
Note that any sequence of free Haar unitaries,
(or free Rademacher) or of free semicircular variables
is pseudo-free.
More generally,
any free sequence $({\kappa}_n)$ with mean 0 in a non-commutative tracial
probability space
such that $\inf\|{\kappa}_n\|_1>0$ and $\sup\|{\kappa}_n\|_\infty<\infty$
is pseudo-free.\\
Indeed, this can be deduced
from the fact that trace preserving unital c.p. maps
extend to trace preserving c.p. maps on reduced free products.
The latter fact reduces the problem to the commutative case
(one first checks the result for a single variable
with unital c.p. maps instead of decomposable ones).
\end{rem}
\section{The union problem}
It is high time to formalize a bit more the central notion
of this paper.
\begin{dfn}\label{d1} Let $(x_n) _{n\ge 1}$
be a sequence in the predual $X$ of a von Neumann algebra.
Let $(y_n)$ be as before in $M_*$.
We will say that $(x_n)_{n\ge 1} $
is free-gaussian dominated in $X$ (or dominated by free-gaussians in $X$)
if it is (decomposably) dominated by the sequence
$(y_n)$ in $M_*$, or equivalently (see Remark \ref{ps})
if it is (decomposably) dominated by a free-gaussian sequence
(or any pseudo-free sequence)
in $M_*$. Here ``(decomposably) dominated" is meant
in the sense of Definition \ref{dom}.\\
For convenience we define the associated constant using the
(unitary) sequence $(y_n)$:
we say that $(x_n)_{n\ge 1} $ is free-gaussian $c$-dominated in $X$
if it is $c$-dominated by $\{y_n\mid n\ge 1\} \subset M_* $, so that we have
$T: M_*\to X$ with $\|T\|_{dec}\le c$
such that $T(y_n)=x_n$.
\end{dfn}
\begin{rem} By classical results (see \cite[p. 126]{Tak})
for any von Neumann algebra
${\cl M}$,
there is a c.p. projection
(with dec-norm equal to 1) from ${\cl M}^*=({\cl M}_*)^{**}$ to ${\cl M}_*$.
Therefore the notions of domination
in ${\cl M}_*$ or in ${\cl M}^*$ are equivalent for sequences sitting
in ${\cl M}_*$.
\end{rem}
Of course we frame the preceding definition
to emphasize the analogy with the sequences dominated by
i.i.d gaussians in \cite{Pi3}. Note that in the latter,
with independence in place of freeness,
dominated by gaussians does not imply
dominated by i.i.d. Haar unitaries,
(indeed gaussians themselves fail this)
but it holds in the free case
because free-gaussians are bounded.
Note in passing that bounded linear maps
between $L_1$-spaces of commutative (and hence injective) von Neumann algebras
are automatically decomposable.
\begin{lem}\label{L9} Let ${\cl M},{\cl N} $
be von Neumann algebras. Assume
that ${\cl N}$ is equipped
with a normal faithful tracial state $ \tau' $.
For $a\in {\cl N}$ we denote by $\widehat a \in {\cl N}_*$
the associated linear form on ${\cl N}$ defined by
$\widehat a(x)=\tau'(a^*x)$.
Let $(v_n)_{n\ge 1}$ be unitaries in ${\cl N}$, so that
$\widehat v_n\in N_* $.
Let $(x_n)_{n\ge 1}\in {\cl M}_*$ be
free-gaussian $c$-dominated.
Then the sequence $(x_n\otimes \widehat v_n )\in ({\cl M}\overline\otimes {\cl N})_*$
is also $c$-dominated by $(y_n)$.
\end{lem}
\begin{proof}
Let $T$ be as in Definition \ref{d1} (here $X= {\cl M}_*$).
Since $(y_n\otimes \widehat v_n)$ and $(y_n )$ have the same
$*$-distribution,
the linear mapping $W$ taking $y_n $ to $y_n\otimes \widehat v_n$
extends to a c.p. (isometric, unital and trace preserving) map
$W$ from ${M}_*$ to $({ M}\overline\otimes {\cl N})_*$ (see Remark \ref{R6}).
Then the composition $T_1=(T \otimes id_{{\cl N}_*}) W$
takes
$y_n$ to $x_n \otimes \widehat v_n$.
Since $W^*$ is c.p.
and $ (T \otimes id_{{\cl N}_*})^*=T^*\otimes id_{{\cl N}}$,
with dec-norm $\le c$,
$T_1$ is $c$-decomposable.
\end{proof}
\begin{rem}\label{dye} By the Russo-Dye theorem
the unit ball of ${\cl N}$ is the closed convex hull of its unitaries.
Actually, for any fixed $0<\delta<1$
there is an integer $K_\delta$
such that any $v\in {\cl N}$ with norm $<\delta$
can be written as an average of $K_\delta$ unitaries,
this is due to Kadison and Pedersen, see \cite{KaPe} for a proof with $\delta=1-2/n$ and $K_\delta=n$.
Using this, we can extend Lemma \ref{L9}
to sequences $(v_n) $ in the unit ball of ${\cl N}$.
Indeed, the set of sequences $(v_n)$ in ${\cl N}$
such that the sequence $(x_n\otimes \widehat v_n )$ in $({\cl M}\overline\otimes {\cl N})_*$
is $c$-dominated by $(y_n)$ is obviously a convex set.
By Lemma \ref{L9} it contains the set of sequences
of unitaries in ${\cl N}$.
By the Kadison-Pedersen result
it contains any family $(v_n)$ with
$\sup\|v_n\|<1$.
Therefore if $\|v_n\|\le 1$ for all $n$
the sequence $(x_n\otimes \widehat v_n )\in ( {\cl M} \overline\otimes {\cl N})_*$
is $c(1+\varepsilon)$-dominated by $(y_n) $ for any $\varepsilon>0$.
\end{rem}
\begin{lem}\label{L8}
Let $\Lambda=\{\psi_n\}$
be a completely Sidon set in ${\cl M}$ with constant $C$.
There is a biorthogonal system
$(x_n)$ in ${\cl M}^*$
such that, for any $(\cl N,\tau')$ as before and any $(v_n)$ with
$\sup\nolimits_n\|v_n\|<c'$,
the sequence
$(x_n\otimes \widehat v_n )\in ( {\cl M}^{**} \overline\otimes {\cl N})_*$
is free-gaussian $Ccc'$-dominated.
\end{lem}
\begin{proof}
This follows from Proposition \ref{p1}
and Lemma \ref{L9} with the variant described in Remark \ref{dye}, applied
to $(v_n/c')$.
\end{proof}
\begin{rem}\label{R9}
At this point it is useful to observe the following:
consider two sequences in $X$ (a von Neumann algebra predual)
each of which is free-gaussian $C$-dominated,
we claim that their union
is free-gaussian $2C$-dominated.
Indeed, if we have $x_n^j=T_j(y_n)$, $x_n^j\in X$,
with $T_j: M_* \to X$, $\|T_j\|_{dec}\le C$
($j=1,2$).
Let $M \ast M$ be the free product,
and let $E_j$ ($j=1,2$) be the conditional expectation
onto each copy of $M_*$ in
$(M \ast M)_*$.
We can form the operator
$T: (M \ast M)_*\to X$ defined by
$T(a)= T_1E_1 (x)+ T_2E_2 (x)$.
Clearly $T$ is decomposable with $\|T\|_{dec}\le 2C$.
Let $(y^1_n)$ and $(y^2_n)$ denote the sequences
corresponding to $(y_n)$ in each copy of $M_*$ in
$(M \ast M)_*$. We have
$T(y^j_n)=x^j_n$ for all $n$ and all $j=1,2$.
But since the sequence $\{y^1_n\} \cup \{y^2_n\}$
is clearly equivalent to our original sequence $ \{y_n\}$,
this proves the claim.
\end{rem}
We now come to a non-commutative
generalization of our result from \cite{Pi3}.
\begin{thm}\label{t2} Let ${\cl M},{\cl N} $
be von Neumann algebras, with $\tau'$ as before.
Suppose $\Lambda_1=\{\psi^1_n\}$ and $\Lambda_2=\{\psi^2_n\}$
are two completely Sidon sets in
a $C^*$-subalgebra $A\subset \cl M$.
Assume
there is a representation $\pi: A \to \cl N$
such that for some $\delta>0$ we have
$$\forall \psi \in \Lambda_1\cup\Lambda_2\quad \|\pi(\psi)\|_2\ge \delta.$$
We assume that $\pi(\Lambda_1)$ and $\pi(\Lambda_2)$ are mutually orthogonal in $L_2(\tau')$.
Then the union $ \Lambda_1\cup\Lambda_2$ is completely
$\otimes_{\max}^4$-Sidon.
\end{thm}
\begin{proof}
We first observe that
since $\pi$ extends to a (normal) representation
from $A^{**}$ to $\cl N$, we may assume without loss
of generality that $\cl M=A^{**}$ and that $\pi$ is extended to $\cl M$.
Note that by our assumption
$\Lambda_1\cup\Lambda_2$ is bounded in $\cl M$.
By a simple homogeneity
argument, we may assume without loss of generality that
$$\forall \psi \in \Lambda_1\cup\Lambda_2\quad \|\pi(\psi)\|_2=1.$$
By Lemma \ref{L8}
there are $x_n^1\in \cl M^*$
biorthogonal to $(\psi^1_n)$
such that $(x_n^1\otimes \widehat{\pi(\psi^1_n)})$
is free-gaussian dominated in $( {\cl M}^{**} \overline\otimes {\cl N})_*$.
Note that the latter is also biorthogonal to $(\psi^1_n\otimes \widehat{\pi(\psi^1_n)})$.
Similarly
there are $x_n^2\in \cl M^*$ such that the same holds for
$(x_n^2\otimes \widehat{\pi(\psi^2_n)})$.
By Remark \ref{R9}, the union
$\{x_n^1\otimes\widehat{\pi(\psi^1_n)}\}\cup \{x_n^2\otimes \widehat{\pi(\psi^2_n)}\}
\subset ( {\cl M}^{**} \overline\otimes {\cl N})_*$
is free-gaussian dominated.
But now the latter system
is biorthogonal to
$\{\psi_n^1\otimes {\pi(\psi^1_n)} \}\cup \{\psi_n^2\otimes {\pi(\psi^2_n)}\}\subset {\cl M}^{**}\overline\otimes {\cl N}$.
Indeed, this holds because,
by our orthogonality assumption, $\widehat{\pi\psi^1_n}(\pi \psi_n^2)=\widehat{\pi\psi^2_n}( \pi\psi_n^1)=0$ for \emph{all}
$m,n$. By Theorem \ref{t1}
we conclude that
the latter system,
which can be described as $([id:{\cl M}\to {\cl M} ^{**}] \otimes \pi)(\{\psi_n^1\otimes \psi_n^1\}\cup \{\psi_n^2\otimes \psi_n^2\})$,
is completely
$\otimes_{\max}^2$-Sidon.
Using \eqref{e8'} to remove $$[id:{\cl M}\to {\cl M} ^{**}] \otimes \pi: \cl M \otimes_{\max} \cl M \to {\cl M }^{**} \otimes_{\min} \cl N \subset {\cl M} ^{**} \overline\otimes {\cl N},$$ we see that this implies that
$ \Lambda_1\cup\Lambda_2$ itself is completely
$\otimes_{\max}^4$-Sidon.
\end{proof}
\begin{rem} As the reader may have noticed the preceding proof actually
shows that $\{\psi\otimes \psi \otimes \psi \otimes \psi \mid \psi \in \Lambda_1\cup\Lambda_2 \}$ is completely Sidon
in $(A\otimes_{\min} A) \otimes_{\max} (A\otimes_{\min} A)$.
\end{rem}
Let $G$ be any discrete group.
We say that subset $\Lambda\subset G$ is completely Sidon
if the
set $\{U_G(t)\mid t\in \Lambda\}$ is completely Sidon in $C^*(G)$.
In this setting we recover our recent generalization \cite{Pifz} of Drury's classical commutative result.
\begin{cor}\label{coru} Let $G$ be any discrete group.
The union of two completely Sidon subsets of $G$
is completely Sidon.
\end{cor}
\begin{proof} We claim that any completely $\otimes^4_{\max}$-Sidon
set in $G$ is completely Sidon.
With this claim, the Corollary follows from
Theorem \ref{t2} applied with $A_1=A_2=C^*(G)$.
To check this claim, we use (ii) in Proposition \ref{R3}.
Let $U_G$ be the universal representation on $G$.
Assuming $\Lambda=\{t_n\}$. Let $\psi_n=U_G(t_n)$.
For any unitary representation
$\pi$ on $G$ with values in a unital $C^*$ algebra $A_\pi$,
with the same notation as in (ii), we have obviously
(since $\pi$ extends completely contractively to $C^*(G)$)
$$\|\sum a_n\otimes \pi(t_n) \|_{M_k(A_\pi)}\le \|\sum a_n\otimes U_G(t_n) \|_{M_k(C^*(G))} .$$
Applying this with $\pi=U_G\otimes U_G\otimes U_G\otimes U_G$,
and $A_\pi=C^*(G)\otimes_{\max} C^*(G) \otimes_{\max}C^*(G)\otimes_{\max} C^*(G)$,
the claim becomes immediate.
\end{proof}
We refer to \cite{Pifz} for several complementary results, in particular for ``completely Sidon" versions
of the interpolation and Fatou-Zygmund properties of Sidon sets, and for a discussion
of the closed span of a completely Sidon
set in the \emph{reduced} $C^*$-algebra of $G$.
\begin{rem}\label{Rf} By analogy with the commutative case,
we propose the following definition:
Let
$(y_n)$ be a free-gaussian (i.e. free semicircular) sequence in $M_*$. We say that $(x_n)$
in $A^*$ is free-subgaussian if there is $C$ such that
for any $k$ the union of the sequences $\{x_n^1\},\cdots,\{x_n^k\}$
in $(A^{\ast k})^*$
is $C$-dominated by
$(y_n)$.
Here $A{\ast} \cdots \ast A$ is the (full) free product of $k$ copies of $A$,
and $x_n^1,\cdots,x_n^k$ are the copies of $x_n$ in each of the
free factors of $A\ast \cdots \ast A$.
Note that with the same notation
the sequence ${y_n^1,\cdots,y_n^k}$
in $(M{\ast} \cdots \ast M )^*$ has the same distribution as
the original sequence $(y_n)$.\\
In the commutative case, when $(x_n)$ lies in $L_1$
over some probability space and freeness is replaced by independence, this is the same as subgaussian
in the usual sense,
see \cite[Prop. 2.10]{Pi3} for details.
See \cite{Pis} for a survey on subgaussian systems.
\end{rem}
\medskip
\medskip
\noindent\textit{Acknowledgement.} Thanks are due to Marek Bo\.zejko,
Simeng Wang
and Mateusz Wasilewski for useful communications.
|
1,108,101,564,547 | arxiv | \section{Introduction}\label{section_introduction}
The valuation and risk management of derivative portfolios form a challenging task in banks, insurance companies, and other financial institutions. The issues are formally explained as follows. Most economic scenario generators can be represented as stochastic models with finitely many time periods $t=0,1,\dots,T$, where randomness is generated by some underlying stochastic driver $X=(X_1,\dots,X_T)$. The components $X_t$ are mutually independent, but not necessarily identically distributed, taking values in ${\mathbb R}^{d}$, for some $d\in{\mathbb N}$.\footnote{We endow ${\mathbb R}^d$ with the Borel $\sigma$-algebra ${\mathcal B}({\mathbb R}^d)$.} We assume that $X$ is realized on the path space ${\mathbb R}^{d\times T}$ such that $X_t(x)=x_t$ for a generic sample point $x=(x_1,\dots,x_T)$. We denote the distribution of $X$ by ${\mathbb Q} = {\mathbb Q}_1\times\dots\times {\mathbb Q}_T$, and we assume that ${\mathbb Q}$ represents the risk-neutral pricing measure with respect to some fixed numeraire, such as the money market account. All financial values and cash flows henceforth are discounted by this numeraire, if not otherwise stated. The stochastic driver $X$ generates the filtration ${\mathcal F}_t={\mathcal B}({\mathbb R}^{d})^{\otimes t}$ which represents the flow of information.\footnote{Henceforth, we let ${\mathcal F}_0$ denote the trivial $\sigma$-algebra. However, we could easily extend the setup to include randomness at $t=0$, by setting $X=(X_0,X_1,\dots,X_T)$. Here $X_0$ could include cashflow specific values that parametrize the cumulative cashflow function $f(X)$, such as the strike price of an embedded option or the initial values of underlying financial instruments. We could then sample $X_0$ from a Bayesian prior ${\mathbb Q}_0$.}
We consider a portfolio whose cumulative cash flow is modeled by some measurable function $f:{\mathbb R}^{d\times T}\to{\mathbb R}$ such that $f\in L^2_{\mathbb Q}$. Its dynamic value process $V$ is then given by the martingale
\begin{equation}\label{eqcondY_EU}
\textstyle V_t={\mathbb E}_{\mathbb Q}[f(X) \mid{\mathcal F}_t] , \quad t=0,\dots,T.
\end{equation}
Computing $V$ is challenging, because the conditional expectations in \eqref{eqcondY_EU} are not available in closed form in general. This is the case for most exotic and path-dependent options, such as barrier reverse convertibles, or the following max-call.
For illustration, let us consider the multivariate Black--Scholes model, where $X_t$ are i.i.d.\ standard normal on ${\mathbb R}^d$. There are $d$ nominal stock prices given by
\begin{equation}\label{bs_model}
\textstyle
S_{i,t}=S_{i,t-1}\exp[ \sigma_i^\top X_t \sqrt{\Delta_t} + (r- \|\sigma_i\|^2 /2)\Delta_t],\quad t=1, \dots, T,
\end{equation}
for some initial values $S_{i,0}$, volatility vectors $\sigma_i\in{\mathbb R}^d$, $i=1,\dots,d$, constant risk-free rate $r$, and time step size in units of a year $(\Delta_1, \dots, \Delta_T)$. Then there exists no closed-form expression for the value process $V$ of the max-call option whose discounted payoff at $T$ is
\begin{equation}\label{intro_example}
\textstyle f(X) ={\rm e}^{-r\sum_{t=1}^T \Delta_t}(\max_{i} S_{i, T} - K)^+,
\end{equation}
for some strike price $K$.
We solve this issue via a novel method to learn the portfolio value process $V$. First, we use ensemble estimators with regression trees to learn the function $f$ from a finite sample $\bm X=(X^{(1)},\dots,X^{(n)})$, drawn from ${\mathbb Q}$, along with the corresponding function values $\bm f = (f(X^{(1)}), \dots, f(X^{(n)}))$.\footnote{More precisely, $\bm X$ consists of i.i.d.\ ${\mathbb R}^{d\times T}$-valued random variables $X^{(i)}\sim {\mathbb Q}$ defined on the product probability space $(\bm E , \bm{\mathcal E}, {\bm Q})$ with $ \bm E = {\mathbb R}^{d\times T} \otimes {\mathbb R}^{d\times T} \otimes\cdots $, $ \bm {\mathcal E} ={\mathcal B}({\mathbb R}^{d})^{\otimes T} \otimes{\mathcal B}({\mathbb R}^d)^{\otimes T}\otimes\cdots $, and $ {\bm Q} = {\mathbb Q} \otimes{\mathbb Q}\otimes\cdots $. \label{footnote_1}} Here we consider the two most popular ensemble estimators, namely Random Forest defined in \cite{bre_2001}, and Gradient Boosting defined in \cite{fri_2001}. We denote these estimators of $f$ by $f_{\bm X}$. In either case, the expression of $f_{\bm X}$ is of the form
\begin{equation}\label{generic_el_estimator}
\textstyle f_{\bm X} = \sum_{i=1}^{N} \beta_i {\mathbbm 1}_{\bm A_i},
\end{equation}
where $\beta_i$ are real coefficients, and $\bm A_i$ are hyperrectangles of ${\mathbb R}^{d\times T}$ that cover ${\mathbb R}^{d\times T}$, but are not necessarily disjoint. See \eqref{box_notation} for the definition of a hyperrectangle. Second, we use $f_{\bm X}$ to define the process $V_{\bm X}$ as follows,
\begin{equation}\label{eqYhat}
\textstyle
V_{\bm X,t}={\mathbb E}_{\mathbb Q}[f_{\bm X}(X) \mid{\mathcal F}_t ],\quad t=0,\dots, T.
\end{equation}
The process $V_{\bm{X}}$ in \eqref{eqYhat} is an estimator of the value process $V$ in \eqref{eqcondY_EU}. This estimator is fast to construct for two reasons. First, the training of $f_{\bm{X}}$ is fast, which is due to the use of readily accessible and highly optimized implementations of Random Forest and Gradient Boosting. Specifically, for Random Forest we use the RandomForestRegressor class in scikit-learn \cite{scikit_learn}, and for Gradient Boosting we use the XGBRegressor class in XGBoost (eXtreme Gradient Boosting) \cite{che_gue_2016}. Second, the conditional expectations in \eqref{eqYhat} are given in closed form, in the sense that they can be efficiently evaluated at very low
computational cost, see Section \ref{section_closed_form_Veu_Vus} for details. Furthermore, there is empirical evidence showing that $f_{\bm X}$ is an accurate estimator of $f$, i.e., $f_{\bm{ X}}$ achieves a small $L^2_{\mathbb Q}$-error $\|f-f_{\bm{ X}}\|_{2, {\mathbb Q}}$, in many problems arising from different fields (scientific fields, and machine learning and data mining challenges). See, e.g., \cite{bia_sco_2016} and references therein for Random Forest, and \cite{che_gue_2016} for Gradient Boosting. This implies that $V_{\bm{X}}$ is an accurate estimator of $V$. Indeed, thanks to Doob's maximal inequality, see, e.g., \cite[Corollary II.1.6]{rev_yor_94}, the path-wise maximum $L^2_{\mathbb Q}$-error is bounded by
\begin{equation}\label{doobineq}
\textstyle
\|\max_{t=0,\dots,T} | V_t-V_{\bm{X},t}|\|_{2,{\mathbb Q}}\le 2\| f -f_{\bm{X}} \|_{2,{\mathbb Q}}.
\end{equation}
There are many risk management tasks building on the dynamic value process $V$. In this paper, we focus on risk measurement as a generic example.\footnote{Another important task in portfolio risk management is hedging, which is sketched in more detail in our previous paper \cite{bou_fil_22}.} For two dates $t_0<t_1$, we denote by $\Delta V_{t_0,t_1} = V_{t_1}-V_{t_0}$ the gain from holding the portfolio over the period $[t_0,t_1]$. Portfolio risk managers and financial market regulators alike quantify the risk of the portfolio $V$ over $[t_0, t_1]$ by means of an ${\mathcal F}_{t_0}$-conditional risk measure, such as value at risk or expected shortfall, evaluated at $\Delta V_{t_0,t_1}$.\footnote{For the definition of value at risk and expected shortfall (also called conditional value at risk or average value at risk), we refer to \cite[Section~4.4]{foe_sch_04}, and Section \ref{secexamples} below.} In practice these risk measures are applied under the equivalent real-world measure ${\mathbb P}\sim{\mathbb Q}$. Using the Cauchy--Schwarz inequality and \eqref{doobineq}, we obtain $\|\max_{t=0,\dots,T} | V_t-V_{\bm X,t}|\|_{1,{\mathbb P}}\le \| \frac{d{\mathbb P}}{d{\mathbb Q}}\|_{2,{\mathbb Q}} \|\max_{t=0,\dots,T} | V_t-V_{\bm X,t}|\|_{2,{\mathbb Q}}\le2\| \frac{d{\mathbb P}}{d{\mathbb Q}}\|_{2,{\mathbb Q}}\|f-f_{\bm X}\|_{2,{\mathbb Q}}$, so that $V_{\bm X}$ and $V$ are close in $L^1_{\mathbb P}$ as soon as $f_{\bm X}$ and $f$ are close in $L^2_{\mathbb Q}$. Hence risk measures that are continuous with respect to the $L^1_{\mathbb P}$-norm, such as value at risk (under mild technical conditions) and expected shortfall, see, e.g., \cite[Section 6]{cam_fil_17}, return similar values when applied to $V_{\bm X}$ instead of $V$.
Related literature on portfolio risk measurement includes \cite{bro_du_moa_15} who introduce a regression-based nested Monte Carlo simulation method for the estimation of the unconditional expectation of a Lipschitz continuous function $f(L)$ of the 1-year loss $L=-\Delta V_{0,1}$. They also provide a comprehensive literature overview of nested simulation problems, including \cite{gor_jun_10} who improve the speed of convergence of the standard nested simulation method using the jackknife method. Our method is different as it learns the entire value process $V$ in one go, as opposed to any method relying on nested Monte Carlo simulation, which estimates $V_t$ for one fixed $t$ at a time. Our method shares some similarities with the kernel-based method in our previous paper \cite{bou_fil_22}. There we applied kernel ridge regression to derive a closed-form estimator of the value process $V$. That kernel-based estimator satisfies asymptotic consistency and finite sample guarantees. However, due to cubic training time complexity, it cannot be applied to high dimensional problems, i.e., problems where the sample size $n$ or the path space dimension $d\times T$ are very large. The ensemble estimators in the present paper scale better. Our method also share similarities with the GPR-EI (Gaussian Process Regression-Exact Integration) method in \cite{gou_et_al_2020}, which gives a closed-form estimator of the entire value process $V$ of an American option under the Black--Scholes and Rough--Bergomi models.
Here and throughout we use the following conventions and notation. For any $p\in [1,\infty)$ and measurable function $f:{\mathbb R}^{d\times T}\to{\mathbb R}$, we denote $ \|f\|_{p,{\mathbb Q}}=
(\int_{{\mathbb R}^{d\times T}} |f(x)|^p{\mathbb Q}(dx))^{1/p}$. We denote by $L^p_{\mathbb Q}$ the space of \emph{${\mathbb Q}$-equivalence classes} of measurable functions $f:{\mathbb R}^{d\times T}\to{\mathbb R}$ with $\|f\|_{p,{\mathbb Q}}<\infty$. If not otherwise stated, we will use the same symbol, e.g., $f$, for a function and its equivalence class. Let $ a=(a_1,\dots, a_T)$ and $ b=(b_1, \dots, b_T)$, where $a_t,b_t\in\overline{{\mathbb R}^d}$, so that $ a, b\in \overline{{\mathbb R}^{d\times T}}$. Assume that $a_t<b_t$, i.e., $a_{j,t}<b_{j,t}$ for every $j=1,\dots, d$, for every $t=1,\dots, T$. A hyperrectangle $\bm A=( a, b]$ of ${\mathbb R}^{d\times T}$ is a subset of ${\mathbb R}^{d\times T}$ of the form
\begin{equation}\label{box_notation}
\textstyle
\bm A= \{(x_1, \dots, x_T)\in {\mathbb R}^{d\times T}\mid a_t < x_t \le b_t,\, t=1,\dots, T\}.
\end{equation}
It is convenient to write $\bm A = \prod_{t=1}
^T ( a_t,b_t]$, where $( a_t, b_t] = \prod_{j=1}^d ( a_{j, t}, b_{j, t}]$ is a subset of ${\mathbb R}^{d}$.\footnote{If $b_{j,t}=\infty$, then $(a_{j,t}, b_{j,t}]$ is defined as $(a_{j,t}, \infty)$.}
The remainder of the paper is as follows. Section~\ref{section_new_estimators} presents the ensemble estimators we use to learn the function $f$. Section~\ref{section_closed_form_Veu_Vus} shows that the value process estimator $V_{\bm X}$ is in closed form for two large classes of financial models. Section~\ref{secexamples} provides numerical examples for the valuation of exotic and path-dependent options in the multivariate Black--Scholes model. Section \ref{sec_outlook} discusses future research directions. Section~\ref{secconc} concludes. Appendix~\ref{sec_regnow_reglat} compares our method to its regress-now variant. And Appendix \ref{sec_optimal_stopping} shows how our method can be applied to Bermudan options.
\section{Ensemble estimators based on regression trees}\label{section_new_estimators}
Following up on Section \ref{section_introduction}, we let $f\in L^2_{\mathbb Q}$. We now present the construction of the estimator $f_{\bm X}$ in \eqref{generic_el_estimator}, which is used to define the value process estimator $V_{\bm X}$ in \eqref{eqYhat}. As mentioned above, $f_{\bm X}$ is either a Random Forest or a Gradient Boosting. Random Forest is defined in \cite{bre_2001} using CART regression trees (CART stands for Classification And Regression Trees). The CART method is defined in \cite{bre_et_al_1984}. Gradient Boosting is defined in \cite{fri_2001} using ``a small regression tree, such as those produced by CART''. In order to make the paper self-contained, we first recap the CART method.\footnote{According to the survey \cite{loh_2014}, the first regression tree method is called AID (Automatic Interaction Detector) and was defined in \cite{mor_son_1963}. However, it is \cite{bre_et_al_1984} that has been the most influential in what we now call the regression tree literature. In this literature there are many different methods that produce regression trees. To the best of our knowledge, today the two most popular regression tree methods are CART defined in \cite{bre_et_al_1984}, and C4.5 defined in \cite{qui_1993}.} We then recap the construction of a Random Forest and Gradient Boosting. Throughout we assume as given a finite i.i.d.\ sample $\bm X = (X^{(1)}, \dots, X^{(n)})$ drawn from ${\mathbb Q}$, along with the function values $\bm f = (f(X^{(1)}), \dots, f(X^{(n)}))$. We denote the corresponding empirical distribution by ${\mathbb Q}_{\bm X}=\frac{1}{n}\sum_{i=1}^n \delta_{X^{(i)}}$.
\subsection{CART regression tree}
The CART method gives a piece-wise constant function that has the following form,
\begin{equation}\label{reg_tree}
\textstyle
f_{\bm X, \Pi} = \sum_{\bm A \in \Pi} {\mathbb E}_{{\mathbb Q}_{\bm X}}[f(X^{(0)}) \mid X^{(0)} \in \bm A] {\mathbbm 1}_{\bm A},
\end{equation}
where $ \Pi$ is a finite hyperrectangle partition of ${\mathbb R}^{d\times T}$, $X^{(0)}$ has distribution ${\mathbb Q}_{\bm X}$, and we have ${\mathbb E}_{{\mathbb Q}_{\bm X}}[f(X^{(0)}) \mid X^{(0)} \in \bm A] = {\mathbb E}_{{\mathbb Q}_{\bm X}}[f(X^{(0)}) {\mathbbm 1}_{\bm A} (X^{(0)})]/{\mathbb Q}_{\bm X}[\bm A]$ if ${\mathbb Q}_{\bm X}[\bm A]>0$, and we use the convention ${\mathbb E}_{{\mathbb Q}_{\bm X}}[f(X^{(0)}) \mid X^{(0)} \in \bm A] = 0$ if no point $X^{(i)}$ lies in $\bm A$. The partition $ \Pi$ is constructed recursively by refining the trivial partition $ \Pi_0 = \{{\mathbb R}^{d\times T}\}$ in the following way. Assume at step $t\ge0$ the partition $ \Pi_{t}$ is of size $K_t\ge 1$. Then pick $\bm A \in \Pi_t$ and perform an axis-aligned split, denoted by $(j, s, z)$, to obtain the two hyperrectangles $\bm A_{L} = \{x \mid x \in \bm A,\, x_{j, s} \le z\}$ and $\bm A_{R} = \{x \mid x \in \bm A,\, x_{j,s} > z\}$. This gives a new partition $\Pi_{t+1} = (\Pi_t \setminus \bm A) \cup \{\bm A_{L}, \bm A_{R}\}$ of size $K_{t+1} = K _t + 1$. To find the optimal split $(j,s, z)$ of $\bm A$, one minimizes the within-group variance
\begin{equation}\label{fct_to_max}
\begin{aligned}
(j,s,z)\mapsto V_{\bm A}(j,s, z) &={\mathbb E}_{{\mathbb Q}_{\bm X}}[{\mathbbm 1}_{\bm A_{L}}(X^{(0)})(f(X^{(0)}) - {\mathbb E}_{{\mathbb Q}_{\bm X}}[f(X^{(0)}) \mid X^{(0)} \in \bm A_L])^2] \\
&\quad + {\mathbb E}_{{\mathbb Q}_{\bm X}}[{\mathbbm 1}_{\bm A_{R}}(X^{(0)}) (f(X^{(0)}) - {\mathbb E}_{{\mathbb Q}_{\bm X}}[f(X^{(0)}) \mid X^{(0)} \in \bm A_R])^2]
\end{aligned}
\end{equation}
over ${\mathcal S} = \{(j,s, z) \mid j=1, \dots, d,\, s=1, \dots, T,\, z\in{\mathbb R}\}$.
When to stop refining the partition $\Pi_t$?\footnote{Other regression tree methods construct the partition $\Pi_t$ relying on other minimization criteria than the within-group variance in \eqref{fct_to_max}. For instance, C4.5 \cite{qui_1993} relies on the gain ratio.} In practice, standard stopping rules include to not split a hyperrectangle $\bm A$ if it contains less than a certain number \textbf{nodesize} of points. Another rule is to stop refining $\Pi_t$ when it reaches a certain size $K$. Then, according to \cite{bre_et_al_1984}, $\Pi_t$ should be pruned by looking for an optimal sub-partition $\Pi_{t'}\subseteq \Pi_{t}$, $t'\le t$, so that $f_{\bm X, \Pi}$ in \eqref{reg_tree} is defined with $\Pi = \Pi_{t'}$. However we omit this step and define $f_{\bm X, \Pi}$ with $\Pi=\Pi_t$. In fact, as discussed in \cite[Section~4]{bre_2001} and \cite[Section~8]{fri_et_al_2002}, a CART regression tree should not be pruned when it is used to define a Random Forest or Gradient Boosting.
The CART regression tree is known for its interpretability, and ability to perform dimensionality reduction and handle outliers. However this estimator is very sensitive to the sample $\bm X$: small perturbations of the sample $\bm X$ can lead to large changes in $f_{\bm X, \Pi}$. In response to this issue, Bagging (from bootstrap and aggregating) has been introduced in \cite{bre_1996}. Bagging is the aggregation, i.e., the average, of $M$ CART regression trees. The $m$-th tree is constructed using a sample $\bm{X}_m =(X^{(1)}_m, \dots, X^{(n)}_m)$, obtained by bootstrapping from $\bm X$, and the corresponding function values $ \bm f_m = (f(X^{(1)}_m), \dots, f(X^{(n)}_m))$. Bagging gives significantly better results than a single CART regression tree. Five years later, Random Forest was introduced in \cite{bre_2001}. Random Forest is an enhancement of Bagging. This will be our first ensemble estimator.
\subsection{Random Forest}\label{sec_rf}
\cite{bre_2001} defines a class of estimators called Random Forest. The same paper gives an example of Random Forest termed Random Forest-RI, where RI stands for Random Inputs. As highlighted in the survey \cite{gen_pog_2016}, nowadays the name Random Forest very often refers to Random Forest-RI. Therefore, we will call Random Forest-RI simply Random Forest. This estimator is the aggregation of $M$ regression trees, which are grown slightly differently than CART. Below we detail its construction.
Fix $\widetilde{n} \le n$ and let $(\bm{X}_1, \dots, \bm{X}_M)$ be $M$ samples, where each sample $\bm{X}_m =(X^{(1)}_m, \dots, X^{(\widetilde{n})}_m)$ is constructed by resampling $\widetilde{n} $ points from $\bm X$. The resampling can be with or without replacement. The resampling is called bootstrapping when it is done with replacement and $\widetilde{n}=n$, otherwise it is called subsampling (with or without replacement). Fix $p \in \{1, \dots, d\times T\}$, and grow $M$ regression trees, where the $m$-th tree $f_{\bm{X}_m, \Pi_m}$ is constructed as follows. Instead of $\bm X$ and $\bm f$, use the sample $\bm X_m$ and the corresponding function values $ \bm f_m = (f(X^{(1)}_m), \dots, f(X^{(\widetilde{n})}_m))$. And for every hyperrectangle $\bm A$ to split, draw uniformly $p$ coordinates $(j_1, s_1), \dots, (j_p, s_p)$ from $\{1, \dots, d\}\times\{1, \dots, T\}$, and minimize $(j,s,z)\mapsto V_{\bm A}(j,s,z)$ in \eqref{fct_to_max} over $\{(j_i, s_i,z)\mid i=1, \dots, p,\, z\in {\mathbb R}\}$ instead of ${\mathcal S}$. Then Random Forest, denoted by $f_{\bm{X}, \bm \Pi}$ with $\bm \Pi = ({\Pi}_1, \dots, {\Pi}_M)$, is the aggregation of the $M$ regression trees $f_{\bm{X}_m, \Pi_m}$,
\begin{equation}\label{rand_forest}
\textstyle
f_{\bm X, \bm \Pi} = \frac{1}{M} \sum_{m=1}^M f_{\bm{X}_m, {\Pi}_m}.
\end{equation}
In the case where $p= d\times T$ and the sampling scheme is bootstrapping, Random Forest is just Bagging.
\subsection{Gradient Boosting}\label{sec_gb}
The idea of Boosting goes back to a theoretical question posed in \cite{kea_1988} and \cite{kea_val_1994}, called the ``Hypothesis Boosting Problem''. In the context of binary classification problems, the authors asked whether there exist a process able to turn a weak learner into a strong one. Such a process would be called Boosting. Here a weak learner is a classifier that performs only slightly better than random guessing. And a strong learner is a classifier that achieves a nearly perfect classification. A positive answer to this question was given in \cite{Sch_1990}. However the algorithm in \cite{Sch_1990} could not be implemented in practice, and it is AdaBoost, the algorithm defined in \cite{fre_sch_1996}, that is usually considered as the first workable Boosting algorithm. The success of AdaBoost with classification trees was such that Breiman called it ``best off-the-shelf classifier in the world", see \cite{fri_et_al_2002}. In order to better understand the performance of AdaBoost, a lot of research has been done. This includes the statistical framework developed in \cite{fri_et_al_2002}, which was further developed in \cite{fri_2001} to cover both classification and regression problems. In the later paper, Gradient Boosting is defined. This estimator is based on CART regression trees, and is constructed recursively, for $t\ge 1$, as follows,
\begin{equation}\label{gradient_descent}
\textstyle
\begin{cases}
f_{\bm X, t}(x) &= f_{\bm X, t-1}(x) - \gamma_t g_t(x), \quad x \in {\mathbb R}^{d\times T},\\
\gamma_t &\in \arg\min_{\gamma\in {\mathbb R}_+}{\mathbb E}_{{\mathbb Q}_{\bm X}}\left[\psi(f(X^{(0)}),f_{\bm X, t-1}(X^{(0)}) - \gamma g_t(X^{(0)}))\right],
\end{cases}
\end{equation}
where $f_{\bm X, 0}=\frac{1}{n}\sum_{i=1}^nf(X^{(i)})$, and $\psi :{\mathbb R}^2 \mapsto {\mathbb R}$, $(x,y)\mapsto \psi(x,y)$ is a given loss function. The function $g_t$ is a CART regression tree that estimates the function $x\mapsto \partial_y\psi(f(x), f_{\bm X, t-1}(x))$ using the sample $\bm X$, along with the function values $(\partial_y\psi(f(X^{(1)}), f_{\bm X, t-1}(X^{(1)})), \dots, \partial_y\psi(f(X^{(n)}), f_{\bm X, t-1}(X^{(n)})))$.\footnote{For the sake of brevity, we write $\partial_y\psi(f(x), f_{\bm X, t-1}(x))$ instead of $\frac{\partial \psi(\cdot, \cdot)}{\partial y}\mid_{(f(x), f_{\bm X, t-1}(x))}$.} And $\gamma_t$ is called the optimal step-size.
Often in regression problems, one picks the squared error loss function $ \psi(x, y)=\frac{1}{2}(x-y)^2$, which is what we do in Section \ref{secexamples}. In this case, $x\mapsto \partial_y\psi\big(f(x), f_{\bm X, t-1}(x)\big) = f_{\bm X, t-1}(x) - f(x)$. Thus at step $t\ge 1$ of Gradient Boosting, a CART regression tree is used to estimate the residual function $f_{\bm X, t-1}-f$. In practice other loss functions $\psi$ could be considered. The only requirement is that $\psi$ be differentiable with respect to its second variable, see \cite{fri_2001}.\footnote{There are several popular open-source software libraries that provide implementations of Gradient Boosting \cite{fri_2001}. The most popular ones are XGBoost (for eXtreme Gradient Boosting) \cite{che_gue_2016}, LightGBM (for Light Gradient Boosting Machine) \cite{guo_et_al_2017}, and CatBoost (for Categorical Boosting) \cite{pro_et_al_2018}. Since these libraries are based on several engineering optimizations, they provide estimators that are not exactly as $f_{\bm X, t}$ in \eqref{gradient_descent}. In these libraries, there is a wide range of loss functions available, and it is also possible to implement one's own loss function.}
When to stop increasing the number of boosing iterations $t$? A standard approach to find the optimal $t$ is to use early stopping techniques, which is what we do in Section \ref{secexamples}. However, in practice Gradient Boosting is known to be resistant to overfitting, see, e.g., \cite{sch_etal_1997}. This means that, in general, the $L^2_{\mathbb Q}$-error $\|f_{\bm X, t}-f\|_{2,{\mathbb Q}}$, does not increase as $t$ becomes very large.
Henceforth and throughout, $f_{\bm X}$ is a placeholder for either the Random Forest $f_{\bm X, \bm \Pi}$ in \eqref{rand_forest}, or the Gradient Boosting $f_{\bm X, t}$ in \eqref{gradient_descent}. The generic expression of $f_{\bm X}$ is given in \eqref{generic_el_estimator}.
\section{Closed-form estimators for $V$}\label{section_closed_form_Veu_Vus}
In the previous section we presented the construction of an ensemble estimator $f_{\bm X}$ of $f$ of the form \eqref{generic_el_estimator}. Now we use $f_{\bm X}$ to define an estimator $V_{\bm X}$ of $V$ of the form \eqref{eqYhat}. From \eqref{generic_el_estimator} and \eqref{eqYhat}, we derive the following expression
\begin{equation}\label{hat_Vt_closed_form}
\textstyle
V_{\bm X, t} = \sum_{i=1}^N \beta_i {\mathbb E}_{\mathbb Q}[ {\mathbbm 1}_{\bm A_i}(X) \mid {\mathcal F}_t], \quad t = 0, \dots, T.
\end{equation}
Now recall that $X$ has distribution ${\mathbb Q}(dx) = {\mathbb Q}_1(dx_1) \times \dots \times{\mathbb Q}_T(dx_T)$. Thus for a hyperrectangle $\bm A=\prod_{t=1}
^T (a_t, b_t]$ in \eqref{box_notation}, we have\footnote{For $t=0$, we set $\prod_{s=1}^0\cdot=1$.}
\begin{equation}\label{prob_decomp}
\textstyle {\mathbb E}_{\mathbb Q}[ {\mathbbm 1}_{\bm A}(X) \mid {\mathcal F}_t] = \prod_{s=1}^t{\mathbbm 1}_{( a_{s}, b_{s}]}(X_s)\prod_{s=t+1}^T{\mathbb Q}_s[( a_{s}, b_{s}]].
\end{equation}
Accordingly, we deduce that the value process estimator $V_{\bm X}$ is in closed form as soon as the probability
\begin{equation}\label{closed_form_Qs}
{\mathbb Q}_s[( a_{s},b_{s}]] \text{ is in closed form for every } a_s < b_s \in \overline{{\mathbb R}^d}, \, s=1, \dots, T.
\end{equation}
We say an expression is in closed form if it can be efficiently evaluated at very low computational cost. Below, we present two common cases where property \eqref{closed_form_Qs} is satisfied.
\subsection{Cross-sectional independence}\label{subsec_Qs_as_prod}
The first case is when ${\mathbb Q}_s$ can be factorized as ${\mathbb Q}_s(dx_s) = {\mathbb Q}_{1, s}(dx_{1,s}) \times \dots \times {\mathbb Q}_{d,s}(dx_{d,s})$. In this case, for $ a_s < b_s \in \overline{{\mathbb R}^d}$, we have ${\mathbb Q}_s[( a_{s}, b_{s}]] = {\mathbb Q}_s[ \prod_{j=1}^d ( a_{j, s}, b_{j, s}]] = \prod_{j=1}^d (F_{j, s}(b_{j, s}) - F_{j, s}(a_{j, s}))$, where $F_{j, s}$ denotes the cumulative distribution function of ${\mathbb Q}_{j, s}$. Thus property \eqref{closed_form_Qs} holds as soon as $F_{j,s}$ is in closed form for every $j=1, \dots, d$. There are many such examples. For its extensive use in financial modelling we mention the standard normal distribution ${\mathbb Q}_s={\mathcal N}(0, I_d)$. Examples include the discrete-time multivariate Black--Scholes model in \eqref{bs_model} and many more time-series models, such as the GARCH models in \cite{boll_1986}.
\subsection{Closed-form copulas}\label{subsec_Qs_with_copula}
The second case, generalizing the above, is when ${\mathbb Q}_{s}$ is defined in terms of a copula. As above, we denote by $F_{j, s}$ the cumulative distribution function of the marginal ${\mathbb Q}_{j, s}$. We now assume as given a copula $C_s$ on $[0,1]^d$ such that
\[ \textstyle {\mathbb Q}_s[(-\infty,x_s]]= C_s(F_{1,s}(x_{1,s}), \dots, F_{d,s}(x_{d,s})).\]
In fact, it is well known that any multivariate distribution on ${\mathbb R}^d$ can be expressed in terms of a copula, see \cite{skl_1959} and \cite[Theorem~1]{emb_2009}. Moreover, the copula is unique if the marginals $F_{j, s}$ are continuous.
Now property \eqref{closed_form_Qs} holds as soon as the copula $C_s$ and the marginals $F_{j,s}$ are in closed form. Indeed, for any hyperrectangle $(a,b]=\prod_{j=1}^d(a_j,b_j]$ of ${\mathbb R}^d$, we have
\[
\textstyle{\mathbb Q}_s[(a,b]] = \sum_{z\in \prod_{j=1}^d \{ a_{j}, b_{j}\}} (-1)^{N(z)} C_s(F_{1,s}(z_1), \dots, F_{d,s}(z_d)),\quad N(z) =\mathrm{card}(\{k\mid z_k = a_{k}\}).
\]
The first case corresponds to the independence copula $C_s(y)=\prod_{j=1}^d y_j$. Copula models are widespread in financial risk management, as they allow to design tailor-made dependence structures between the underlying assets. See, e.g., \cite{mcn_et_al_2015} for a thorough discussion.
\section{Numerical experiments}\label{secexamples}
We follow up on the introductory example with the Black–Scholes model with $d$ nominal stock price processes $S_{i, t}$ given by \eqref{bs_model}. In particular, we assume that $X_t$ are i.i.d.\ standard normal on ${\mathbb R}^d$.
As for the portfolios, we fix a strike price $K$ and consider the following European style exotic options with payoff functions
\begin{itemize}
\item Min-put $f(X)= {\rm e}^{-r\sum_{t=1}^T \Delta_t} (K-\min_{i} S_{i,T})^+$;
\item Max-call $f(X)={\rm e}^{-r\sum_{t=1}^T \Delta_t} (\max_{i} S_{i,T} - K)^+$.
\end{itemize}
We also consider a genuinely path-dependent product with the payoff function
\begin{itemize}
\item Barrier reverse convertible (BRC) $f(X)= {\rm e}^{-r\sum_{t=1}^T \Delta_t}\left( C + F \left( 1 - 1_{\{\min_{i,t} S_{i,t}\le B\}} \left(1 - \min_{i} \frac{ S_{i,T}}{S_{i,0} K} \right)^+\right)\right)$,
\end{itemize}
for some barrier $B<K$, coupon $C$, and face value $F$. At maturity $T$, the holder of this structured product receives the coupon $C$. She also receives the face value $F$ if none of the nominal stock prices falls below the barrier $B$ at any time $t=1,\dots,T$. Otherwise, the face value $F$ is reduced by the payoff of $F/K$ min-puts on the normalized stocks $S_{i,T}/S_{i,0}$ with strike price $K$. These payoff functions are inspired from those given in \cite{bec_et_al_2019}. Note that the payoff functions of the min-put and BRC are bounded, while the payoff of the max-call is unbounded.
For our numerical experiments we choose the following parameter values: risk-free rate $r=0$, initial stock prices $S_{i, 0}=1$, volatilities $\sigma_i = 0.2\bm e_i$, where $\bm e_i$ denote the standard basis vectors in ${\mathbb R}^d$, so that stock prices are independent, strike price $K=1$ (at the money), barrier $B=0.6$, coupon $C=0$, and face value $F=1$. For the min-put and max-call, $(d, T) = (6, 2)$ and $(\Delta_1, \Delta_2) = (1/12, 11/12)$; for the BRC, $(d, T) = (3, 12)$ and $(\Delta_1, \dots, \Delta_{12}) = (1/12, \dots, 1/12)$. Thus the path space ${\mathbb R}^{d\times T}$ is of dimension $12$ for the min-put and max-call, and it is of dimension $36$ for the BRC.
Under the parameter specification above, we generate a training sample $\bm X$ of size $n= 20{,}000$. We use $\bm X$, along with the corresponding function values $\bm f$, to construct the ensemble estimator $f_{\bm X}$ in \eqref{generic_el_estimator}. To find the optimal hyperparameter value for this estimator we use a validation sample $\bm X_{\mathrm{valid}}$ of size $0.4\times n=8{,}000$, along with its corresponding function values $\bm f_{\mathrm{valid}}$. Both the optimal hyperparameter value search and the construction of $f_{\bm X}$ are done using the programming language Python and readily accessible machine learning libraries.
Specifically, when $f_{\bm X}$ is the Random Forest $f_{\bm X, \bm\Pi}$ in \eqref{rand_forest}, we use the RandomForestRegressor class of the library scikit-learn \cite{scikit_learn}. We find the optimal hyperparameter value by validation on the set of hyperparameter values ${\mathcal P}_{\mathrm{RF}} =\{(M, \textbf{nodesize}, p)\mid M \in \{100, 250, 500\},\, \textbf{nodesize}\in\{2, 3, 5\},\, p \in \{\lceil d\times T/3\rceil, d\times T\} \}$ using $\bm X_{\mathrm{valid}}$ and $\bm f_{\mathrm{valid}}$. In ${\mathcal P}_{\mathrm{RF}}$ there are three default hyperparameter values. The RandomForestRegressor (Python) default hyperparameter value $(100, 2, d\times T)$, the randomForest \cite{rf_r_pack} (R programming language) default hyperparameter value in regression $(500, 5, \lceil d\times T/3\rceil)$, and our default hyperparameter value $(100, 5, d\times T)$.\footnote{Our default hyperparameter value is an intermediary choice between the default hyperparameter values in RandomForestRegressor and randomForest.} Table \ref{table_parameters_rf} shows the normalized $L^2_{\mathbb Q}$-error $\|f_{\bm X}-f\|_{2,{\mathbb Q}}/V_0$, computed using the validation sample $\bm X_{\mathrm{valid}}$, and the number of hyperrectangles $N$ in the Random Forest $f_{\bm X}$ in \eqref{generic_el_estimator} for these three default hyperparameter values as well as the optimal hyperparameter value in ${\mathcal P}_{\mathrm{RF}}$. For the min-put and max-call, we observe that our default hyperparameter value gives normalized $L^2_{\mathbb Q}$-error comparable to that given by the optimal hyperparameter value. Besides it has the advantage to give, on average, 8 times less hyperrectangles than the optimal hyperparameter value. This implies that the evaluation of $V_{\bm X}$ is 8 times faster with our default hyperparameter value than with the optimal hyperparameter value for the min-put and max-call examples. Thus for computational reason we use our default hyperparameter value $(100,5, 12)$ for min-put and max-call. However for BRC we use the optimal hyperparameter value $(500, 5, 12)$, because here the number of hyperrectangles is relatively small ($N<500{,}000$). For the three payoff functions we use $\textbf{sampling regime}=$bootstrapping. In the class RandomForestRegressor, the variables $M$, \textbf{nodesize}, $p$, \textbf{sampling regime} correspond to \textbf{n\_estimators}, \textbf{min\_samples\_split}, \textbf{max\_features}, \textbf{bootstrap}, respectively.
\begin{table}
\centering
\begin{tabular}{|l|l|l|l|}
\hline
& Min-put & BRC & Max-call \\
\hline
Optimal hyperparameter value & (500, 2, 12) & (500, 5, 12) &(250, 3, 12) \\
Normalized $L^2_{\mathbb Q}$-error in \% &6.864 & 6.884 & 10.26 \\
Number of hyperrectangles &6{,}279{,}290 & 187{,}710 & 2{,}027{,}347 \\
\hline
Default hyperparameter value in RandomForestRegressor (Python) & (100, 2, 12) & (100, 2, 36) &(100, 2, 12) \\
Normalized $L^2_{\mathbb Q}$-error in \% & 6.894 & 6.973 & 10.36 \\
Number of hyperrectangles &1{,}255{,}344 & 52{,}805 & 1{,}237{,}737 \\
\hline
Default hyperparameter value in randomForest (R) & (500, 5, 4) & (500, 5, 12) &(500, 5, 4) \\
Normalized $L^2_{\mathbb Q}$-error in \% & 8.124 & 6.884 & 12.61 \\
Number of hyperrectangles &2{,}575{,}215 & 187{,}710 & 2{,}556{,}448 \\
\hline
\textbf{Our default hyperparameter value} &(100, 5, 12) & (100, 5, 36) &(100, 5, 12) \\
Normalized $L^2_{\mathbb Q}$-error in \% & 6.917& 6.965 & 10.39 \\
Number of hyperrectangles &494{,}118 & 34{,}807 & 489{,}747 \\
\hline
\end{tabular}
\caption{Random Forest validation step: normalized $L^2_{\mathbb Q}$-error $\|f_{\bm X}-f\|_{2, {\mathbb Q}}/V_0$, computed using the validation sample $\bm X_{\mathrm{valid}}$ and expressed in \%, and number of hyperrectangles $N$ in the Random Forest $f_{\bm X}$ in \eqref{generic_el_estimator}, for the optimal hyperparameter value in ${\mathcal P}_{\mathrm{RF}}$, and three default hyperparameter values, for the payoff functions min-put, BRC, and max-call.}\label{table_parameters_rf}
\end{table}
When $f_{\bm X}$ is the Gradient Boosting $f_{\bm X, t}$ in \eqref{gradient_descent}, we use the XGBRegressor class of XGBoost \cite{che_gue_2016}. Similarly to what we did for Random Forest, we use $\bm X_{\mathrm{valid}}$ and $\bm f_{\mathrm{valid}}$ to perform a validation on the set of hyperparameter values ${\mathcal P}_{\mathrm{XGB}}=\{(t_{\mathrm{optimal\_stopping}}(\textbf{nodesize}, \textbf{max depth}),\textbf{nodesize}, \textbf{max depth})\mid \textbf{nodesize} \in \{5, 15, 25, 35, 45\},\, \textbf{max depth} \in \{40, 50, \dots, 90\} \}$ to find the optimal hyperparameter value. The hyperparameter \textbf{max depth} controls the number of hyperrectangles in the regression tree $g_t$ in \eqref{gradient_descent}. Given the values \textbf{nodesize}, \textbf{max depth}, the number of iterations $t_{\mathrm{optimal\_stopping}}(\textbf{nodesize}, \textbf{max depth})$ is determined by early stopping using the validation sample $\bm X_{\mathrm{valid}}$. Table \ref{table_parameters_xgb} shows the normalized $L^2_{\mathbb Q}$-error $\|f_{\bm X}-f\|_{2,{\mathbb Q}}/V_0$, computed using the validation sample $\bm X_{\mathrm{valid}}$, and the number of hyperrectangles $N$ in the Gradient Boosting $f_{\bm X}$ in \eqref{generic_el_estimator} for the optimal hyperparameter value in ${\mathcal P}_{\mathrm{XGB}}$. Furthermore, for these three payoff functions we also considered other hyperparameters in XGBRegressor for which we took standard values: $\textbf{booster}=\text{gbtree}$, $\textbf{learning\_rate}=0.1$, $\textbf{tree\_method}=\text{hist}$, $\textbf{objective}=\text{reg:squarederror}$, and $\textbf{base\_score}=0.5$. Note that in XGBRegressor, the variables \textbf{nodesize} and \textbf{max depth} correspond to \textbf{min\_child\_weight} and \textbf{max\_depth}, respectively.
\begin{table}
\centering
\begin{tabular}{|l|l|l|l|}
\hline
& Min-put & BRC & Max-call \\
\hline
Optimal hyperparameter value & (120, 40, 15) & (256, 50, 15) &(152, 60, 35) \\
Normalized $L^2_{\mathbb Q}$-error in \% &5.856 & 6.284 & 9.753 \\
Number of hyperrectangles &88{,}127 & 189{,}864 & 66{,}225 \\
\hline
\end{tabular}
\caption{XGBoost validation step: normalized $L^2_{\mathbb Q}$-error $\|f_{\bm X}-f\|_{2, {\mathbb Q}}/V_0$, computed using the validation sample $\bm X_{\mathrm{valid}}$ and expressed in \%, and number of hyperrectangles $N$ in the Gradient Boosting $f_{\bm X}$ in \eqref{generic_el_estimator}, for the optimal hyperparameter value in ${\mathcal P}_{\mathrm{XGB}}$, for the payoff functions min-put, BRC, and max-call.}\label{table_parameters_xgb}
\end{table}
Next we use our ensemble estimator $f_{\bm X}$ to construct $V_{\bm X}$ in \eqref{eqYhat}. As discussed in Section \ref{section_closed_form_Veu_Vus}, $V_{\bm X}$ is given in closed form. We then evaluate $V_{\bm X,t}$ at times $t \in \{0, 1, T\}$ on a test sample $\bm X_{\mathrm{test}}$ of size $n_{\mathrm{test}} = 100{,}000$. We benchmark $V_{\bm X}$ to the ground truth value process $V$, which we obtain by means of Monte Carlo schemes using $\bm X_{\mathrm{test}}$. More specifically, we obtain $V_0$ as simple Monte Carlo estimate of $\{f(X)\mid X \in \bm X_{\mathrm{test}}\}$. For $V_1$, we use a nested Monte Carlo scheme, where we estimate $V_1(X_1)$ using $n_{\mathrm{inner}} = 1{,}000$ inner simulations of $(X_2,\dots,X_T)$, for each $X_1$ in $\bm X_{\mathrm{test}}$. Then we carry out the following three evaluation tasks.
First, we compute the absolute relative error of $V_{\bm X, 0}$, $|V_0 - V_{\bm X, 0}|/V_0$, and the normalized $L^2_{{\mathbb Q}}$-errors of $V_{\bm X, t}$, $\|V_t - V_{\bm X, t}\|_{2, {\mathbb Q}}/V_0$, for $t=1, T$. Table \ref{norm_l2_errors_table} shows that normalized $L^2_{\mathbb Q}$-error of $V_{\bm X, t}$ decreases substantially for increasing time-to-maturity $T-t$. More specifically, for XGBoost, the normalized $L^2_{\mathbb Q}$-error of $V_{\bm X, 1}$ is on average 9-times smaller than that of $V_{\bm X, T}$, and the relative absolute error of $V_{\bm X, 0}$ is on average 14-times smaller than the normalized $L^2_{\mathbb Q}$-error of $V_{\bm X, 1}$. For Random Forest these values are 6 and 6, respectively. These findings are in line with \eqref{doobineq}, which has useful practical implications. Indeed, despite the lack of theoretical bounds on the error $\|V_t-V_{\bm X, t}\|_{2, {\mathbb Q}}$, in concrete applications one can always estimate the normalized $L^2_{{\mathbb Q}}$-error of $V_{\bm X, T}$ by a simple Monte Carlo scheme as we do here. This error then serves as upper bound on the normalized $L^2_{{\mathbb Q}}$-errors of $V_{\bm X, t}$, for any $t<T$. Table \ref{norm_l2_errors_table} also reveals that XGBoost outperforms Random Forest in the estimation of $V_{0}$, $V_{1}$ and $V_{T}$ in most cases (8 cases out of 9). In this table we also report the normalized $L^2_{\mathbb Q}$-errors obtained with the kernel-based method in our previous paper \cite[Table~2]{bou_fil_22}. We see that the kernel-based method outperforms our ensemble learning method only in the BRC example. Figures \ref{error_V1_minput_rho0}, \ref{error_V1_barrier_rho0}, and \ref{error_V1_maxcall_rho0} show the decrease of the normalized $L^2_{\mathbb Q}$-error of $V_{\bm X, 1}$ with respect to the training sample size $n$. Figures \ref{error_VT_minput_rho0}, \ref{error_VT_barrier_rho0}, and \ref{error_VT_maxcall_rho0} illustrate the same phenomenon for $V_{\bm X, T}$. In these figures we also recognize the outperformance of XGBoost over Random Forest.
Second, we compute and compare quantiles of $V_{\bm X, 1}$ and $V_1$, and $V_{\bm X, T}$ and $V_T$ using the test and training sample. Thereto, for $t\in \{1, T\}$, we compute the empirical left quantiles of $V_{\bm X, t}$ and $V_t$ at levels $\{0.001\%, 0.002\%, \dots, 0.009\%\}$, $\{0.01\%, 0.02\%,\dots, 0.99\%\}$, $\{1\%,2\%,\dots,99\%\}$, $\{99.01\%, 99.02\%, \dots, 99.99\%\}$, and $\{99.991\%, 99.992\%,\dots, 100\%\}$.\footnote{Note that for the test sample of size $n_{\mathrm{test}}=10^5$, the left $0.001\%$-quantile ($100\%$-quantile) corresponds to the smallest (largest) sample value. For the training sample of size $n_{\mathrm{train}}=2\times 10^4$, the same holds, while the ten left- and right-most quantiles collapse to two values, respectively.} The detrended quantiles (estimated quantiles minus true quantiles) are then plotted against the true quantiles, the produced plot is called a detrended Q-Q plot. Figures \ref{qqplot_V1_minput_rf}, \ref{qqplot_V1_barrier_rf}, \ref{qqplot_V1_maxcall_rf}, and Figures \ref{qqplot_V1_minput_xgb}, \ref{qqplot_V1_barrier_xgb}, \ref{qqplot_V1_maxcall_xgb} show the detrended Q-Q plots of $V_{\bm X, 1}$ with Random Forest and XGBoost, respectively. These figures show that overall the distribution of $V_{1}$ is better estimated with XGBoost than with Random Forest. Figures \ref{qqplot_VT_minput_rf}, \ref{qqplot_VT_barrier_rf}, \ref{qqplot_VT_maxcall_rf}, and Figures \ref{qqplot_VT_minput_xgb}, \ref{qqplot_VT_barrier_xgb}, \ref{qqplot_VT_maxcall_xgb} show the detrended Q-Q plots of $V_{\bm X, T}$ with Random Forest and XGBoost, respectively. Notably, the detrended Q-Q plots of $V_{\bm X, T}$ in Figures \ref{qqplot_VT_barrier_rf} and \ref{qqplot_VT_barrier_xgb} reveal that for less than $3\%$ of the training sample (that is, less than $600$ points out of $n= 20{,}000$), the embedded min-put options in the BRC are triggered and in the money. For the remaining sample points the payoff is equal to the face value, $F=1$. And yet, as Figures \ref{qqplot_V1_barrier_rf} and \ref{qqplot_V1_barrier_xgb} show, this is enough for our ensemble learning method to learn the payoff function such that $V_{\bm X, 1}$ is remarkably close to the ground truth, with a normalized $L^2_{\mathbb Q}$-error less than $0.60\%$, as reported in Table \ref{norm_l2_errors_table}. In \cite{bou_fil_22} we also compute the same detrended Q-Q plots as here. Overall, the detrended Q-Q plots drawn with the kernel-based method, see \cite[Figures~1-3]{bou_fil_22}, and those drawn with the ensemble learning method are of comparable quality.
Third, as risk management application, we compute the value at risk and expected shortfall of long and short positions of the above portfolios. Thereto, we recall the definitions that can also be found in \cite[Chapter~4]{foe_sch_04}. For a confidence level $\alpha \in (0,1)$ and random loss $\mathrm{L}$, the value at risk of $\mathrm{L}$ is defined as left $\alpha$-quantile $\mathrm{VaR}_\alpha(\mathrm{L})=\inf\{y \mid {\mathbb P}[\mathrm{L}\le y] \ge \alpha\}$, and the expected shortfall of $L$ is given by $\mathrm{ES}_{\alpha}(\mathrm{L}) = \frac{1}{1-\alpha}{\mathbb E}_{\mathbb P}[(\mathrm{L}-\mathrm{VaR}_\alpha(\mathrm{L}))^+] + \mathrm{VaR}_\alpha(\mathrm{L})$. Both value at risk and expected shortfall are standard risk measures in practice. For instance, insurance companies have to compute the value at risk at level $\alpha=99.5\%$ and the expected shortfall at level $\alpha=99\%$, under Solvency II and the Swiss Solvency Test, respectively. For more discussion on these two risk measures we refer to \cite{mcn_et_al_2015}. Henceforth, we assume the real-world measure ${\mathbb P}={\mathbb Q}$, for simplicity. For the three payoff functions above, we compute value at risk and expected shortfall of the 1-period loss $\mathrm{L}=V_{0}-V_{1}$ and its estimator $\mathrm{L}_{\bm X}=V_{\bm X, 0}-V_{\bm X, 1}$ for a long position, namely $\mathrm{VaR}_{99.5\%}(\mathrm{L})$, $\mathrm{ES}_{99\%}(\mathrm{L})$, $\mathrm{VaR}_{99.5\%}(\mathrm{L}_{\bm X})$, and $\mathrm{ES}_{99\%}(\mathrm{L}_{\bm X})$. We compute the same risk measures for a short position, namely $\mathrm{VaR}_{99.5\%}(-\mathrm{L})$, $\mathrm{ES}_{99\%}(-\mathrm{L})$, $\mathrm{VaR}_{99.5\%}(-\mathrm{L}_{\bm X})$, and $\mathrm{ES}_{99\%}(-\mathrm{L}_{\bm X})$. And in Tables \ref{var_table} and \ref{es_table}, we report the relative errors of risk measures, namely $(\text{estimated risk measure minus true risk measure})/\text{true risk measure}$, which are expressed in \%. From these two tables, we notice that all risk measures are more accurately estimated with XGBoost than with Random Forest. This echoes the detrended Q-Q plots of $V_{\bm X, 1}$, where we see in Figures \ref{qqplot_V1_minput_rf}, \ref{qqplot_V1_barrier_rf}, \ref{qqplot_V1_maxcall_rf} and Figures \ref{qqplot_V1_minput_xgb}, \ref{qqplot_V1_barrier_xgb}, \ref{qqplot_V1_maxcall_xgb} that the left and right tails of the distribution of $V_1$ are better estimated with XGBoost than with Random Forest. With XGBoost the estimates of risk measures are satisfactory. In Tables \ref{var_table} and \ref{es_table}, we also compute the relative errors of risk measures with the kernel-based method using the risk measurements in \cite[Tables~3-4]{bou_fil_22}. We observe that risk measures are better estimated with the kernel-based method than with the ensemble learning method. Nevertheless, one should keep in mind that these risk measures are a tough metric for our estimators, because they focus on the tails of the distribution beyond the $1\%$- and $99\%$-quantiles.
\begin{table}
\centering
\begin{tabular}{|l|l|l|l|l|}
\hline
Payoff & Estimator & $V_{\bm X, 0}$ & $V_{\bm X, 1}$ & $V_{\bm X, T}$ \\
\hline
Min-put & XGBoost & \textbf{0.1701} & \textbf{1.525} & \textbf{5.814} \\
& Random Forest & 0.2933 & 2.300 & 6.811 \\
& Kernel-based method & 0.1942 & 1.827 & 10.05 \\
\hline
BRC & XGBoost & 0.05519 & 0.3530 & 6.276 \\
& Random Forest & 0.2008 & 0.5276 & 6.660 \\
& Kernel-based method & \textbf{0.02198}& \textbf{0.2506} & \textbf{5.745} \\
\hline
Max-call & XGBoost & \textbf{0.08016} & \textbf{2.217} & 9.923 \\
& Random Forest & 0.3845 & 3.155 & \textbf{9.868} \\
& Kernel-based method & 0.1031& 2.315& 11.65 \\
\hline
\end{tabular}
\caption{Normalized $L^2_{\mathbb Q}$-error $\|V_t-V_{\bm X, t}\|_{2, {\mathbb Q}}/V_0$, computed using the test sample and expressed in \%, at steps $t = 0,1,T$, using XGBoost and Random Forest, for the payoff functions min-put, BRC, and max-call.}\label{norm_l2_errors_table}
\end{table}
\begin{table}
\centering
\begin{tabular}{|l|l|l|l|}
\hline
Payoff & Estimator & $\mathrm{VaR}(\mathrm{L}_{\bm X})$& $\mathrm{VaR}(-\mathrm{L}_{\bm X})$\\
\hline
Min-put & XGBoost & -9.658 & -6.912 \\
& Random Forest & -25.69 & -20.57 \\
& Kernel-based method & \textbf{0.9695}& \textbf{3.158}
\\
\hline
BRC & XGBoost & 2.533 & -42.85 \\
& Random Forest & -3.510 & -75.87 \\
& Kernel-based method & \textbf{0.1893}& \textbf{-13.91} \\
\hline
Max-call & XGBoost & -7.103 & -4.140 \\
& Random Forest & -23.87 & -20.51 \\
& Kernel-based method & \textbf{0.07143}& \textbf{-3.582} \\
\hline
\end{tabular}
\caption{
Relative errors of value at risk $\mathrm{VaR}_{99.5\%}(\mathrm{L}_{\bm X})$ and $\mathrm{VaR}_{99.5\%}(-\mathrm{L}_{\bm X})$, computed as $(\text{estimated VaR minus true VaR})/\text{true VaR}$ using the test sample and expressed in \%, using XGBoost and Random Forest, for the payoff functions min-put, BRC, and max-call.}\label{var_table}
\end{table}
\begin{table}
\centering
\begin{tabular}{|l|l|l|l|}
\hline
Payoff & Estimator & $\mathrm{ES}(\mathrm{L}_{\bm X})$& $\mathrm{ES}(-\mathrm{L}_{\bm X})$\\
\hline
Min-put & XGBoost & -10.23 & -7.434 \\
& Random Forest & -26.41 & -21.07 \\
& Kernel-based method & \textbf{1.261}& \textbf{4.769} \\
\hline
BRC & XGBoost & 3.940 & -43.79
\\
& Random Forest & 16.93 & -76.33 \\
& Kernel-based method & \textbf{-0.5269}& \textbf{-14.40} \\
\hline
Max-call & XGBoost & -7.808 & -4.507 \\
& Random Forest & -24.67 & -21.31 \\
& Kernel-based method & \textbf{-0.3460}& \textbf{-3.588} \\
\hline
\end{tabular}
\caption{Relative errors of value at risk $\mathrm{ES}_{99\%}(\mathrm{L}_{\bm X})$ and $\mathrm{ES}_{99\%}(-\mathrm{L}_{\bm X})$, computed as $(\text{estimated ES minus true ES})/\text{true ES}$ using the test sample and expressed in \%, using XGBoost and Random Forest, for the payoff functions min-put, BRC, and max-call.}\label{es_table}
\end{table}
\begin{figure}[p]
\centering
\begin{subfigure}{0.45\textwidth}
\includegraphics[width=\linewidth]{normalized_l2_error_V1_later_min_put_ntrain20000_ntest100000_ninner1000_d6_T2_rho00_rf_xgb_learnwithXTrue.png}
\caption{Normalized $L^2_{{\mathbb Q}}$-errors of $V_{\bm X, 1}$ in \% with Random Forest and XGBoost.}
\label{error_V1_minput_rho0}
\end{subfigure}\hfil
\begin{subfigure}{0.45\textwidth}
\includegraphics[width=\linewidth]{normalized_l2_error_VT_later_min_put_ntrain20000_ntest100000_ninner1000_d6_T2_rho00_rf_xgb_learnwithXTrue.png}
\caption{Normalized $L^2_{{\mathbb Q}}$-errors of $V_{\bm X, T}$ in \% with Random Forest and XGBoost.}
\label{error_VT_minput_rho0}
\end{subfigure}
\medskip
\begin{subfigure}{0.45\textwidth}
\includegraphics[width=\linewidth]{qqplot_V1_later_random_forest_approx_min_put_ntrain20000_ntest100000_ninner1000_d6_T2_rho00.png}
\caption{Detrended Q-Q plot of $V_{\bm X, 1}$ with Random Forest.}
\label{qqplot_V1_minput_rf}
\end{subfigure}\hfil
\begin{subfigure}{0.45\textwidth}
\includegraphics[width=\linewidth]{qqplot_VT_laterrandom_forest_approx_min_put_ntrain20000_ntest100000_ninner1000_d6_T2_rho00.png}
\caption{Detrended Q-Q plot of $V_{\bm X, T}$ with Random Forest.}
\label{qqplot_VT_minput_rf}
\end{subfigure}
\medskip
\begin{subfigure}{0.45\textwidth}
\includegraphics[width=\linewidth]{qqplot_V1_later_xgboost_approx_min_put_ntrain20000_ntest100000_ninner1000_d6_T2_rho00.png}
\caption{Detrended Q-Q plot of $V_{\bm X, 1}$ with XGBoost.}
\label{qqplot_V1_minput_xgb}
\end{subfigure}\hfil
\begin{subfigure}{0.45\textwidth}
\includegraphics[width=\linewidth]{qqplot_VT_laterxgboost_approx_min_put_ntrain20000_ntest100000_ninner1000_d6_T2_rho00.png}
\caption{Detrended Q-Q plot of $V_{\bm X, T}$ with XGBoost.}
\label{qqplot_VT_minput_xgb}
\end{subfigure}
\caption{Results for the min-put with Random Forest and XGBoost. The normalized $L^2_{\mathbb Q}$-error of $V_{\bm X, 1}$, $\|V_1-V_{\bm X, 1}\|_{2, {\mathbb Q}}/V_0$, is computed using the test sample and expressed in \%. In the detrended Q-Q plots, the blue, cyan, and lawngreen (red, orange, and pink) dots are built using the test (training) sample. $[0\%, 0.01\%)$ refers to the quantiles of levels $\{0.001\%,0.002\%, \dots, 0.009\%\}$, $[0.01\%, 1\%)$ refers to the quantiles of levels $\{0.01\%,0.02\%, \dots, 0.99\%\}$, $[1\%, 99\%]$ refers to the quantiles of levels $\{1\%,2\%, \dots, 99\%\}$, $(99\%, 99.99\%]$ refers to the quantiles of levels $\{99.01\%,99.02\%, \dots, 99.99\%\}$, and $(99.99\%, 100\%]$ refers to the quantiles of levels $\{99.991\%,99.992\%, \dots, 100\%\}$.}
\label{fig_minput}
\end{figure}
\begin{figure}[p]
\centering
\begin{subfigure}{0.45\textwidth}
\includegraphics[width=\linewidth]{normalized_l2_error_V1_later_barrier_ntrain20000_ntest100000_ninner1000_d3_T12_rho00_rf_xgb_learnwithXTrue.png}
\caption{Normalized $L^2_{{\mathbb Q}}$-errors of $V_{\bm X, 1}$ in \% with Random Forest and XGBoost.}
\label{error_V1_barrier_rho0}
\end{subfigure}\hfil
\begin{subfigure}{0.45\textwidth}
\includegraphics[width=\linewidth]{normalized_l2_error_VT_later_barrier_ntrain20000_ntest100000_ninner1000_d3_T12_rho00_rf_xgb_learnwithXTrue.png}
\caption{Normalized $L^2_{{\mathbb Q}}$-errors of $V_{\bm X, T}$ in \% with Random Forest and XGBoost.}
\label{error_VT_barrier_rho0}
\end{subfigure}
\medskip
\begin{subfigure}{0.45\textwidth}
\includegraphics[width=\linewidth]{qqplot_V1_later_random_forest_approx_barrier_ntrain20000_ntest100000_ninner1000_d3_T12_rho00.png}
\caption{Detrended Q-Q plot of $V_{\bm X, 1}$ with Random Forest.}
\label{qqplot_V1_barrier_rf}
\end{subfigure}\hfil
\begin{subfigure}{0.45\textwidth}
\includegraphics[width=\linewidth]{qqplot_VT_laterrandom_forest_approx_barrier_ntrain20000_ntest100000_ninner1000_d3_T12_rho00.png}
\caption{Detrended Q-Q plot of $V_{\bm X, T}$ with Random Forest.}
\label{qqplot_VT_barrier_rf}
\end{subfigure}
\medskip
\begin{subfigure}{0.45\textwidth}
\includegraphics[width=\linewidth]{qqplot_V1_later_xgboost_approx_barrier_ntrain20000_ntest100000_ninner1000_d3_T12_rho00.png}
\caption{Detrended Q-Q plot of $V_{\bm X, 1}$ with XGBoost.}
\label{qqplot_V1_barrier_xgb}
\end{subfigure}\hfil
\begin{subfigure}{0.45\textwidth}
\includegraphics[width=\linewidth]{qqplot_VT_laterxgboost_approx_barrier_ntrain20000_ntest100000_ninner1000_d3_T12_rho00.png}
\caption{Detrended Q-Q plot of $V_{\bm X, T}$ with XGBoost.}
\label{qqplot_VT_barrier_xgb}
\end{subfigure}
\caption{Results for the BRC with Random Forest and XGBoost. The normalized $L^2_{\mathbb Q}$-error of $V_{\bm X, 1}$, $\|V_1-V_{\bm X, 1}\|_{2, {\mathbb Q}}/V_0$, is computed using the test sample and expressed in \%. In the detrended Q-Q plots, the blue, cyan, and lawngreen (red, orange, and pink) dots are built using the test (training) sample. $[0\%, 0.01\%)$ refers to the quantiles of levels $\{ 0.001\%,0.002\%, \dots, 0.009\%\}$, $[0.01\%, 1\%)$ refers to the quantiles of levels $\{0.01\%,0.02\%, \dots, 0.99\%\}$, $[1\%, 99\%]$ refers to the quantiles of levels $\{1\%,2\%, \dots, 99\%\}$, $(99\%, 99.99\%]$ refers to the quantiles of levels $\{99.01\%,99.02\%, \dots, 99.99\%\}$, and $(99.99\%, 100\%]$ refers to the quantiles of levels $\{99.991\%,99.992\%, \dots, 100\%\}$.}
\label{fig_barrier}
\end{figure}
\begin{figure}[p]
\centering
\begin{subfigure}{0.45\textwidth}
\includegraphics[width=\linewidth]{normalized_l2_error_V1_later_max_call_ntrain20000_ntest100000_ninner1000_d6_T2_rho00_rf_xgb_learnwithXTrue.png}
\caption{Normalized $L^2_{{\mathbb Q}}$-errors of $V_{\bm X, 1}$ in \% with Random Forest and XGBoost.}
\label{error_V1_maxcall_rho0}
\end{subfigure}\hfil
\begin{subfigure}{0.45\textwidth}
\includegraphics[width=\linewidth]{normalized_l2_error_VT_later_max_call_ntrain20000_ntest100000_ninner1000_d6_T2_rho00_rf_xgb_learnwithXTrue.png}
\caption{Normalized $L^2_{{\mathbb Q}}$-errors of $V_{\bm X, T}$ in \% with Random Forest and XGBoost.}
\label{error_VT_maxcall_rho0}
\end{subfigure}
\medskip
\begin{subfigure}{0.45\textwidth}
\includegraphics[width=\linewidth]{qqplot_V1_later_random_forest_approx_max_call_ntrain20000_ntest100000_ninner1000_d6_T2_rho00.png}
\caption{Detrended Q-Q plot of $V_{\bm X, 1}$ with Random Forest.}
\label{qqplot_V1_maxcall_rf}
\end{subfigure}\hfil
\begin{subfigure}{0.45\textwidth}
\includegraphics[width=\linewidth]{qqplot_VT_laterrandom_forest_approx_max_call_ntrain20000_ntest100000_ninner1000_d6_T2_rho00.png}
\caption{Detrended Q-Q plot of $V_{\bm X, T}$ with Random Forest.}
\label{qqplot_VT_maxcall_rf}
\end{subfigure}
\medskip
\begin{subfigure}{0.45\textwidth}
\includegraphics[width=\linewidth]{qqplot_V1_later_xgboost_approx_max_call_ntrain20000_ntest100000_ninner1000_d6_T2_rho00.png}
\caption{Detrended Q-Q plot of $V_{\bm X, 1}$ with XGBoost.}
\label{qqplot_V1_maxcall_xgb}
\end{subfigure}\hfil
\begin{subfigure}{0.45\textwidth}
\includegraphics[width=\linewidth]{qqplot_VT_laterxgboost_approx_max_call_ntrain20000_ntest100000_ninner1000_d6_T2_rho00.png}
\caption{Detrended Q-Q plot of $V_{\bm X, T}$ with XGBoost.}
\label{qqplot_VT_maxcall_xgb}
\end{subfigure}
\caption{Results for the max-call with Random Forest and XGBoost. The normalized $L^2_{\mathbb Q}$-error of $V_{\bm X, 1}$, $\|V_1-V_{\bm X, 1}\|_{2, {\mathbb Q}}/V_0$, is computed using the test sample and expressed in \%. In the detrended Q-Q plots, the blue, cyan, and lawngreen (red, orange, and pink) dots are built using the test (training) sample. $[0\%, 0.01\%)$ refers to the quantiles of levels $\{0.001\%,0.002\%, \dots, 0.009\%\}$, $[0.01\%, 1\%)$ refers to the quantiles of levels $\{0.01\%,0.02\%, \dots, 0.99\%\}$, $[1\%, 99\%]$ refers to the quantiles of levels $\{1\%,2\%, \dots, 99\%\}$, $(99\%, 99.99\%]$ refers to the quantiles of levels $\{99.01\%,99.02\%, \dots, 99.99\%\}$, and $(99.99\%, 100\%]$ refers to the quantiles of levels $\{99.991\%,99.992\%, \dots, 100\%\}$.}
\label{fig_maxcall}
\end{figure}
\section{Outlook}\label{sec_outlook}
In this section we discuss two future research directions. The first one is computational: how to deal with higher dimensional problems? The second one is theoretical: is the value process estimator $V_{\bm X}$ asymptotically consistent?
\subsection{Scalability}\label{sec_scalability}
To apply our ensemble learning method to high dimensional problems, where both the sample size $n$ and the path space dimension $d\times T$ are very large, two conditions must be satisfied. First, fast training of the ensemble estimator $f_{\bm X}$ in \eqref{generic_el_estimator}. Second, fast evaluation of the value process estimator $V_{\bm X}$ in \eqref{eqYhat}.
Fortunately, the first condition is already satisfied. As shown in Table \ref{table_training_time_fx}, the training of $f_{\bm X}$ is extremely fast. This speed comes from two sources: a relatively small training time complexity, and the exploitation of parallel processing. In fact, the training time complexity for building a Random Forest $f_{\bm X, \bm \Pi}$ in \eqref{rand_forest}, under the bootstrapping sampling regime, is $\Theta(Mp \tilde{n}\log(\tilde{n}))$, where $\tilde{n}=0.632 n$, see \cite[Table~5.1]{lou_2014}. And if we consider the XGBoost implementation of Gradient Boosting $f_{\bm X, t}$ in \eqref{gradient_descent}, this complexity becomes $\Theta(M\textbf{max\_depth}dT n\log(n))$, see \cite[Section~Time Complexity Analysis]{che_gue_2016}. The notation $\Theta(g(n))$, for some function $g$, means that there exist constants $0<c<C$, such that the training time complexity is bounded from below by $cg(n)$, and bounded from above by $Cg(n)$, as $n\to \infty$. Furthermore, the Random Forest implementation RandomForestRegressor in scikit-learn \cite{scikit_learn}, and the Gradient Boosting implementation XGBRegressor in XGBoost \cite{che_gue_2016} take advantage of parallel processing.
Now we shall discuss the second condition. The estimator $f_{\bm X}$ provided by a machine learning library, such as scikit-learn or XGBoost, should be rewritten into a suitable format. We recall the expression of $f_{\bm X}$ in \eqref{generic_el_estimator}. Thus $f_{\bm X}$ is determined by the hyperrectangles $\bm A_1, \dots,\bm A_N$, and the real coefficients $\beta_1, \dots, \beta_N$. And from \eqref{box_notation}, we know that every hyperrectangle $\bm A_i$ is characterized by two matrices $a^{(i)}, b^{(i)} \in {\mathbb R}^{d\times T}$ such that $\bm A_i=( a^{(i)}, b^{(i)}]$. In summary, $f_{\bm X}$ is determined by the tuple $(\mathbf{cells}, \mathbf{values})$, where $\mathbf{cells} = (( a^{(1)}, b^{(1)}), \dots, (a^{(N)},b^{(N)}))$, and $\mathbf{values} = (\beta_1, \dots, \beta_N)$. The rewriting of $f_{\bm X}$ as $(\mathbf{cells}, \mathbf{values})$ forms a processing step, and it requires a careful look at how $f_{\bm X}$ is encoded in the corresponding library.\footnote{Python codes corresponding to the processing steps of RandomForestRegressor of scikit-learn, and XGBRegressor of XGBoost are available from the authors upon request.} After this processing step, the pseudo-code in Algorithm \ref{algo_eval_Vxt} shows how to evaluate $V_{\bm X, t}$ at some point $(x_1, \dots, x_t) \in {\mathbb R}^{d\times t}$. Thus fast evaluation of the value process $V_{\bm X}$ can be achieved by writing a fast code of Algorithm \ref{algo_eval_Vxt}.
Recall that in Section \ref{secexamples}, for each payoff function, we evaluate the function $V_{\bm X, 1}$ on a test sample of size $n_{\mathrm{test}}=100{,}000$. To achieve this we parallelized Algorithm \ref{algo_eval_Vxt} using the packages MPI for Python \cite{mpi4py} and joblib \cite{joblib}. Specifically, MPI for Python allowed us to parallelize the $100{,}000$ evaluations using several compute nodes. And joblib allowed us to parallelize the for loop in Algorithm \ref{algo_eval_Vxt}. Table \ref{table_time_vx} shows the time to evaluate $V_{\bm X, 1}$. We observe that evaluations of $V_{\bm X, 1}$ are faster with XGBoost than with Random Forest, which is mainly due to the difference in the number of hyperrectangles $N$ in these two estimators. The time to evaluate $V_{\bm X, 1}$ in Table \ref{table_time_vx} can be shortened further using, e.g., a GPU code of Algorithm \ref{algo_eval_Vxt}.
\begin{algorithm}[H]
\SetAlgoLined
\textbf{Input: $\mathbf{cells}$, $\mathbf{values}$, $t$, $(x_1, \dots, x_t)$}\;
{$\mathrm{value} = 0$}\;
{$\mathrm{ncells} = \mathrm{length}(\mathbf{values})$}\;
\For{$i=1, \dots, \mathrm{ncells}$}{
$(a,b) = \mathbf{cells}[i]$\;
\If{$a_{j,s}< x_{j,s}\le b_{j,s}$, $j=1,\dots, d,$ and $s=1,\dots,t$}{
$\mathrm{value} = \mathrm{value} + \mathbf{values}[i] \times \prod_{s=t+1}^T{\mathbb Q}_s[(a_{s},b_{s}]]$
}}{
\textbf{return} $\mathrm{value}$\;
}
\caption{Evaluation of $V_{\bm X, t}$}\label{algo_eval_Vxt}
\end{algorithm}
\begin{table}
\centering
\begin{tabular}{|l|l|l|}
\hline
Payoff & Estimator & Training time (s) \\
\hline
Min-put & XGBoost & 2 \\
& Random Forest& 1 \\
\hline
BRC &XGBoost & 8 \\
& Random Forest& 12 \\
\hline
Max-call &XGBoost & 2 \\
& Random Forest&1 \\
\hline
\end{tabular}
\caption{Training time, in seconds, of $f_{\bm X}$, using XGBoost and Random Forest, using the training sample $\bm X$ and the function values $\bm f$. Computation is performed on 5 compute nodes, each has 2 Skylake processors running at 2.3 GHz, with 18 cores per processor. And we used 188 GB of RAM. We needed a large amount of RAM in order to evaluate $V_1$ and $V_{\bm X,1}$ on a test sample of size $n_{\mathrm{test}}=100{,}000$, see Section \ref{secexamples}. However, for the training of $f_{\bm X}$ one just needs a sufficient amount of memory to store $\bm X$ and $\bm f$.}\label{table_training_time_fx}
\end{table}
\begin{table}
\centering
\begin{tabular}{|l|l|l|l|}
\hline
Payoff & Estimator & Number of hyperrectangles $N$ & Time to evaluate $V_{\bm X, 1}$ (s) \\
\hline
Min-put & XGBoost& 88{,}127 & 413 \\
& Random Forest&494{,}118 &1{,}629 \\
\hline
BRC &XGBoost&189{,}864 & 10{,}947 \\
& Random Forest&187{,}710 &13{,}472 \\
\hline
Max-call &XGBoost& 66{,}225 & 353 \\
& Random Forest&489{,}747 &1{,}644 \\
\hline
\end{tabular}
\caption{Number of hyperrectangles $N$ in the ensemble estimator $f_{\bm X}$ in \eqref{generic_el_estimator}, using XGBoost and Random Forest, and time in seconds to evaluate $V_{\bm X,1}$ on a test sample of size $n_{\mathrm{test}}=100{,}000$. The number of hyperrectangles are copied from Tables \ref{table_parameters_rf} and \ref{table_parameters_xgb}. Computation is performed on 5 compute nodes, each has 2 Skylake processors running at 2.3 GHz, with 18 cores per processor. And we used 188 GB of RAM. We needed a large amount of RAM in order to evaluate $V_1$ and $V_{\bm X,1}$ on a test sample of size $n_{\mathrm{test}}=100{,}000$, see Section \ref{secexamples}. However, for the training of $f_{\bm X}$ one just needs a sufficient amount of memory to store $\bm X$ and $\bm f$.}\label{table_time_vx}
\end{table}
\subsection{Consistency}\label{sec_theory}
In Section \ref{secexamples}, we saw that our estimator $V_{\bm X}$ in \eqref{eqYhat} gives an accurate estimation of the value process $V$ in \eqref{eqcondY_EU}. In order to theoretically characterize the goodness of the estimator $V_{\bm X}$, from Doob’s maximal inequality in \eqref{doobineq}, we see that it is enough to study the ensemble estimator $f_{\bm X}$. In particular, we would be interested in consistency results of the form ${\mathbb E}_{\bm{Q}}[ \|f-f_{\bm X}\|^2_{2,{\mathbb Q}}]\to 0$, as $n\to \infty$, and finite sample guarantees of the form $\bm Q[\|f-f_{\bm X}\|^2_{2,{\mathbb Q}}<c(\eta, n)]\ge 1-\eta$, for all $n\ge n_0(\eta)$, for $\eta\in (0, 1]$. However, theoretical analysis of Random Forest, when $f_{\bm X}$ is $f_{\bm X, \bm \Pi}$ in \eqref{rand_forest}, and Gradient Boosting, when $f_{\bm X}$ is $f_{\bm X, t}$ in \eqref{gradient_descent}, is difficult in general, especially in our financial framework, where the function $f$ typically is neither bounded nor compactly supported, see example \eqref{intro_example}. Below we mention two recent consistency results available in the machine learning literature that drew our attention and that could be investigated further.
Despite the plethora of empirical works on Random Forest, see, e.g., \cite{gen_et_al_2008}, \cite{arc_kim_2008}, and \cite{gen_et_al_2010}, to name a few, little is known on the theoretical side. The sampling scheme, the split criterion \eqref{fct_to_max}, and the sampling of $p$ coordinates for each hyperrectangle to split make the trees highly and non-trivially $(\bm X, \bm f)$-dependent. This renders the Random Forest difficult to analyse mathematically. To better understand theoretically the good performance of Random Forest in practice, less $(\bm X, \bm f)$-dependent versions of Random Forest have been studied, e.g., $(\bm X, \bm f)$-independent in \cite{bia_2012}, or $\bm f$-independent in \cite{sco_2016}. A recent consistency result for the asymptotic Random Forest, $\lim_{M \to \infty} f_{\bm X, \bm \Pi}$ in \eqref{rand_forest}, has been given in \cite{sco_et_al_2015}. They show that ${\mathbb E}_{\bm Q}[\|\lim_{M\to \infty}f_{\bm X, \bm \Pi} - f\|_{2,{\mathbb Q}}^2] \to 0 $, as $n\to \infty$, under the following assumptions: the path space is the unit cube $[0,1]^{d\times T}$ instead of ${\mathbb R}^{d\times T}$, $X$ is uniformly distributed on $[0,1]^{d\times T}$, and $f$ is continuous and additive. The last assumption reads in our case that $f(x) = \sum_{t=1}^{T} \sum_{j=1}^d f_{j, t}(x_{j, t})$, where each $f_{j, t}$ is continuous. These assumptions are too stringent in applications in finance. We recommend the survey \cite{bia_sco_2016} for an overview on the theoretical work on Random Forest.
As for Gradient Boosting, to the best of our knowledge, there is no consistency result for Gradient Boosting with CART in the context of regression problems in the literature. Nevertheless, we shall mention the recent paper \cite{bia_cad_2017}, where the authors study Gradient Boosting in both classification and regression problems. Their result \cite[Theorem~4.1]{bia_cad_2017} holds in the case where the base estimator is a certain type of regression trees. However, it does not hold in the case where the base estimator is a CART regression tree.
\section{Conclusion}\label{secconc}
We introduce a unified framework for quantitative portfolio risk management based on the dynamic value process of the portfolio. We use ensemble estimators with regression trees to learn the value process from a finite sample of the cumulative cash flow of the portfolio. Our portfolio value process estimator is fast to construct, given in closed form, and accurate. The last means that the normalized $L^2_{\mathbb Q}$-error $\|\max_{t=0,\dots,T} | V_t-V_{\bm X,t}|\|_{2,{\mathbb Q}}/V_0$ is relatively small. In fact, numerical experiments for exotic and path-dependent options in the multivariate Black–Scholes model in moderate dimensions show good results for a moderate training sample size. In contrast to the kernel-based method in \cite{bou_fil_22}, our ensemble learning method can be scaled to deal with high dimensional problems.
\begin{appendix}
\section{Comparison with regress-now}\label{sec_regnow_reglat}
The method we develop in this paper gives an estimation of the entire value process $V$. In practice, one could be interested in the estimation of the portfolio value $V_t$ only at some fixed time $t$, e.g., $t=1$. In \cite{gla_yu_2004} two least squares Monte Carlo methods are described to deal with this problem in the context of American options pricing. Their first method, termed ``regress-later'', consists in estimating the payoff function $f$ by means of a projection on a finite number of basis functions. The basis functions are chosen such that their conditional expectation at time $t=1$ is in closed form. Our method can be seen as a double extension of this, because it covers the case where both the basis functions and their number are not known a priori, and it gives closed-form estimation of the portfolio value $V_t$ at any time $t$. Their second method, termed ``regress-now'', consists in estimating $V_1$ by means of a projection on a finite number of basis functions that depend solely on the variable of interest $x_1 \in {\mathbb R}^d$.
We compare our method, which corresponds to ``regress-later'', and which gives the estimator $V_{\bm X, t}$ in \eqref{hat_Vt_closed_form} for $t=1$, to its regress-now variant, whose estimator we denote by $V_{\bm X, 1}^{\text{now}}$. Thereto we briefly discuss how to construct $V_{\bm X, 1}^{\text{now}}$ in the context of the three payoff functions studied in Section \ref{secexamples}.
The construction of $V_{\bm X, 1}^{\text{now}}$ is simpler than that of $V_{\bm X, 1}$. First, instead of the whole sample $\bm X$, one only needs the $(t=1)$-cross-section $\bm X_1 = (X^{(1)}_1,\dots, X^{(n)}_1)$. Second, with the input $\bm X_1$ and $\bm f$, the estimators Random Forest in Section \ref{sec_rf}, and Gradient Boosting in Section \ref{sec_gb} give directly $V_{\bm X, 1}^{\text{now}}$. As we did in Section \ref{secexamples}, we use the validation sample $\bm X_{\mathrm{valid}}$, along with its corresponding function values $\bm f_{\mathrm{valid}}$, to find the optimal hyperparameter values for Random Forest and Gradient Boosting by validation on the sets of hyperparameter values ${\mathcal P}_{\mathrm{RF}}$ and ${\mathcal P}_{\mathrm{XGB}}$, respectively. Table \ref{table_parameters_rf_now} shows the normalized $L^2_{\mathbb Q}$-error $\|V_{\bm X, 1}^{\mathrm{now}}-f\|_{2,{\mathbb Q}}/V_0$, computed using the validation sample $\bm X_{\mathrm{valid}}$, and the number of hyperrectangles $N$ in the Random Forest and Gradient Boosting $f_{\bm X}$ in \eqref{generic_el_estimator} for the optimal hyperparameter values in ${\mathcal P}_{\mathrm{RF}}$ and ${\mathcal P}_{\mathrm{XGB}}$. Unlike in Section \ref{secexamples}, here we choose the optimal hyperparameter value for Random Forest, although the number of hyperrectangles $N$ induced is very large ($N> 500{,}000$). This is because the evaluation of $V_{\bm X, 1}^{\mathrm{now}}$ is very fast irrespectively of $N$, thanks to the highly optimized implementations of Random Forest and Gradient Boosting, RandomForestRegressor in scikit-learn \cite{scikit_learn} and XGBRegressor in XGBoost \cite{che_gue_2016}, respectively.
\begin{table}
\centering
\begin{tabular}{|l|l|l|l|}
\hline
& Min-put & BRC & Max-call \\
\hline
Optimal hyperparameter value for Random Forest & (500, 5, 2) & (500, 5, 1) &(500, 5, 2) \\
Normalized $L^2_{\mathbb Q}$-error in \% &31.98 & 7.195 & 46.58 \\
Number of hyperrectangles &2{,}619{,}900 & 323{,}896 & 2{,}620{,}048 \\
\hline
Optimal hyperparameter value for Gradient Boosting & (44, 40, 45) & (52, 40, 45) &(41, 50, 45) \\
Normalized $L^2_{\mathbb Q}$-error in \% & 32.43 & 6.886 & 46.62 \\
Number of hyperrectangles &13{,}401 & 9{,}606 & 13{,}165 \\
\hline
\end{tabular}
\caption{
Regress-now Random Forest and regress-now XGBoost validation steps: normalized $L^2_{\mathbb Q}$-error $\|V_{\bm X,1}^{\mathrm{now}}-f\|_{2, {\mathbb Q}}/V_0$, computed using the validation sample $\bm X_{\mathrm{valid}}$ and expressed in \%, and number of hyperrectangles $N$ in the Random Forest and Gradient Boosting $f_{\bm X}$ in \eqref{generic_el_estimator}, for the optimal hyperparameter values in ${\mathcal P}_{\mathrm{RF}}$ and ${\mathcal P}_{\mathrm{XGB}}$, for the payoff functions min-put, BRC, and max-call.}\label{table_parameters_rf_now}
\end{table}
Table \ref{norm_l2_errors_nowVSlater} shows the normalized $L^2_{{\mathbb Q}}$-errors of $V_{\bm X, 1}$ and $V_{\bm X, 1}^{\text{now}}$. The former values are copied from Table \ref{norm_l2_errors_table} for convenience. We observe that our regress-later estimators always perform better than their regress-now variants. This finding is confirmed by the normalized $L^2_{\mathbb Q}$-errors of $V_{\bm X, 1}$ and $V_{\bm X, 1}^{\text{now}}$ as function of the training sample in Figures \ref{error_V1_minput_rho0_nowVSlater_rf}, \ref{error_V1_barrier_rho0_nowVSlater_rf}, \ref{error_V1_maxcall_rho0_nowVSlater_rf}, \ref{error_V1_minput_rho0_nowVSlater_xgb}, \ref{error_V1_barrier_rho0_nowVSlater_xgb}, \ref{error_V1_maxcall_rho0_nowVSlater_xgb}. In \cite{bou_fil_22} we also compare regress-later and regress-now with the kernel-based method. In Table \ref{norm_l2_errors_nowVSlater} we report, from \cite[Table~5]{bou_fil_22}, the $L^2_{\mathbb Q}$-errors corresponding to the kernel-based method. We see that also with the kernel-based method, regress-later outperforms regress-now.
Now we discuss the detrended Q-Q plots in Figures \ref{qqplot_minput_rho0_nowVSlater_rf}, \ref{qqplot_barrier_rho0_nowVSlater_rf}, \ref{qqplot_maxcall_rho0_nowVSlater_rf} for Random Forest, and in Figures \ref{qqplot_minput_rho0_nowVSlater_xgb}, \ref{qqplot_barrier_rho0_nowVSlater_xgb}, \ref{qqplot_maxcall_rho0_nowVSlater_xgb} for XGBoost. Their construction is detailed in the second-to-last paragraph of Section \ref{secexamples}. In Figures \ref{qqplot_minput_rho0_nowVSlater_xgb}, \ref{qqplot_barrier_rho0_nowVSlater_xgb}, \ref{qqplot_maxcall_rho0_nowVSlater_xgb}, we see that for XGBoost, the detrended Q-Q plots with regress-later are of much better quality, i.e., they are more aligned with the horizontal black line, than the detrended Q-Q plots with regress-now. As for Random Forest, the outperformance of regress-later over regress-now in terms of normalized $L^2_{\mathbb Q}$-error, seen in Figures \ref{error_V1_minput_rho0_nowVSlater_rf}, \ref{error_V1_barrier_rho0_nowVSlater_rf}, \ref{error_V1_maxcall_rho0_nowVSlater_rf}, does not clearly appear in the detrended Q-Q plots in Figures \ref{qqplot_minput_rho0_nowVSlater_rf}, \ref{qqplot_barrier_rho0_nowVSlater_rf}, \ref{qqplot_maxcall_rho0_nowVSlater_rf}. By comparing Figures \ref{qqplot_minput_rho0_nowVSlater_rf} and \ref{qqplot_minput_rho0_nowVSlater_xgb}; Figures \ref{qqplot_barrier_rho0_nowVSlater_rf} and \ref{qqplot_barrier_rho0_nowVSlater_xgb}; Figures \ref{qqplot_maxcall_rho0_nowVSlater_rf} and \ref{qqplot_maxcall_rho0_nowVSlater_xgb}, we see that regress-later XGBoost gives the best detrended Q-Q plots, i.e., the detrended Q-Q plots that are the most aligned with the horizontal black line.
We finish this section by discussing the risk measure estimates in Tables \ref{var_table_nowVlater} and \ref{es_table_nowVlater}. Their construction is detailed in the last paragraph of Section \ref{secexamples}. Consistently with the last comment in the above paragraph, it is regress-later XGBoost that gives best estimates of risk measures in most cases, 10 cases out of 12. In the 2 left cases, which correspond to the estimation of risk measures of the short position of the BRC, it is regress-now Random Forest that is the most accurate. The last is consistent with the detrended Q-Q plots in Figures \ref{qqplot_barrier_rho0_nowVSlater_rf} and \ref{qqplot_barrier_rho0_nowVSlater_xgb}, where we observe that regress-now Random Forest gives the best estimation of the right tail distribution of $V_1$ among all estimators.
\begin{table}
\centering
\begin{tabular}{|l|l|l|l|}
\hline
Payoff & Estimator & $V_{\bm X, 1}$ & $V_{\bm X, 1}^{\text{now}}$ \\
\hline
Min-put & XGBoost & \underline{\bf{1.525}}& 9.539 \\
& Random Forest & \bf{2.300} & 6.487 \\
& Kernel-based method & \textbf{1.827}& 1.946 \\
\hline
BRC & XGBoost & \bf{0.3530} & 1.492 \\
& Random Forest & \bf{0.5276} & 1.499 \\
& Kernel-based method & \underline{\textbf{0.2506}}& 0.2806 \\
\hline
Max-call & XGBoost & \underline{\bf{2.217}}& 14.24 \\
& Random Forest &\bf{3.155} & 9.863 \\
& Kernel-based method & \textbf{2.315}& 2.606 \\
\hline
\end{tabular}
\caption{
Normalized $L^2_{\mathbb Q}$-error $\|V_1 - \widehat{V}_{1} \|_{2, {\mathbb Q}}/V_0$, computed using the test sample and expressed in \%, for $\widehat{V}_{1}\in \{V_{\bm X, 1}, V_{\bm X, 1}^{\text{now}}\}$, for the payoff functions min-put, BRC, and max-call.}\label{norm_l2_errors_nowVSlater}
\end{table}
\begin{table}
\centering
\begin{tabular}{|l|l|l|l|l|l|}
\hline
Payoff & Estimator & $\mathrm{VaR}(\mathrm{L}_{\bm X})$& $\mathrm{VaR}(\mathrm{L}_{\bm X}^{\text{now}})$ & $\mathrm{VaR}(-\mathrm{L}_{\bm X})$ & $\mathrm{VaR}(-\mathrm{L}_{\bm X}^{\text{now}})$\\
\hline
Min-put & XGBoost & \textbf{-9.658} & 50.94 & \textbf{-6.912} & 48.21 \\
& Random Forest & -25.69 & \textbf{21.38} & \textbf{-20.57} & 23.30 \\
& Kernel-based method & \underline{\textbf{0.9695}}& 1.697& \underline{\textbf{3.158}}& -9.184 \\
\hline
BRC & XGBoost & \textbf{2.533} & 123.3 & \textbf{-42.85} & 141.8 \\
& Random Forest & \textbf{-3.510} & 197.6 & -75.87 & \textbf{24.91} \\
& Kernel-based method & \underline{\textbf{0.1893}}&
-16.81&
-13.91&
\underline{\textbf{-7.342}} \\
\hline
Max-call & XGBoost & \textbf{-7.103} & 50.97 & \textbf{-4.140} & 57.89 \\
& Random Forest & -23.87 & \textbf{21.88} & \textbf{-20.51} & 40.00 \\
& Kernel-based method & \underline{\textbf{0.07143}}&
5.357&
-3.582&
\underline{\textbf{1.237}} \\
\hline
\end{tabular}
\caption{
Relative errors of value at risk $\mathrm{VaR}_{99.5\%}(\mathrm{L}_{\bm X})$, $\mathrm{VaR}_{99.5\%}(\mathrm{L}_{\bm X}^{\mathrm{now}})$, $\mathrm{VaR}_{99.5\%}(-\mathrm{L}_{\bm X})$, $\mathrm{VaR}_{99.5\%}(-\mathrm{L}_{\bm X}^{\mathrm{now}})$, computed as $(\text{estimated VaR minus true VaR})/\text{true VaR}$ using the test sample and expressed in \%, using XGBoost and Random Forest, for the payoff functions min-put, BRC, and max-call.}\label{var_table_nowVlater}
\end{table}
\begin{table}
\centering
\begin{tabular}{|l|l|l|l|l|l|}
\hline
Payoff & Estimator & $\mathrm{ES}(\mathrm{L}_{\bm X})$& $\mathrm{ES}(\mathrm{L}_{\bm X}^{\text{now}})$ & $\mathrm{ES}(-\mathrm{L}_{\bm X})$ & $\mathrm{ES}(-\mathrm{L}_{\bm X}^{\text{now}})$\\
\hline
Min-put & XGBoost & \textbf{-10.23} & 49.75 & \textbf{-7.434} & 47.22 \\
& Random Forest & -26.41 & \textbf{20.46} & \textbf{-21.07} & 23.68 \\
& Kernel-based method & \underline{\textbf{1.261}}& 1.775& \underline{\textbf{4.769}}& -8.546 \\
\hline
BRC & XGBoost & \textbf{3.940} & 113.5 & \textbf{-43.79} & 151.1 \\
& Random Forest & \textbf{16.93} & 203.1 & -76.33 & \textbf{22.27} \\
& Kernel-based method & \underline{\textbf{-0.5269}}&
-18.58&
-14.40&
\underline{\textbf{-8.271}} \\
\hline
Max-call & XGBoost & \textbf{-7.808} & 50.12 & \textbf{-4.507} & 55.32 \\
& Random Forest & -24.67 & \textbf{20.90} & \textbf{-21.31} & 42.63 \\
& Kernel-based method & \underline{\textbf{-0.3460}}&
5.329&
-3.588&
\underline{\textbf{0.8112}} \\
\hline
\end{tabular}
\caption{Relative errors of value at risk $\mathrm{ES}_{99\%}(\mathrm{L}_{\bm X})$, $\mathrm{ES}_{99\%}(\mathrm{L}_{\bm X}^{\mathrm{now}})$, $\mathrm{ES}_{99\%}(-\mathrm{L}_{\bm X})$, $\mathrm{ES}_{99\%}(-\mathrm{L}_{\bm X}^{\mathrm{now}})$, computed as $(\text{estimated ES minus true ES})/\text{true ES}$ using the test sample and expressed in \%, using XGBoost and Random Forest, for the payoff functions min-put, BRC, and max-call.}\label{es_table_nowVlater}
\end{table}
\begin{figure}[p]
\centering
\begin{subfigure}{0.45\textwidth}
\includegraphics[width=\linewidth]{normalized_l2_error_V1_laterVnow_random_forest_approx_min_put_ntrain20000_ntest100000_ninner1000_d6_T2_rho00_learnwithXTrue.png}
\caption{Min-put with Random Forest: normalized $L^2_{{\mathbb Q}}$-errors of $V_{\bm X, 1}$ and $V_{\bm X, 1}^{\text{now}}$ in \%.}
\label{error_V1_minput_rho0_nowVSlater_rf}
\end{subfigure}\hfil
\begin{subfigure}{0.45\textwidth}
\includegraphics[width=\linewidth]{qqplot_V1_nowVlater_random_forest_approx_min_put_ntrain20000_ntest100000_ninner1000_d6_T2_rho00.png}
\caption{Min-put with Random Forest: detrended Q-Q plots of $V_{\bm X, 1}$ and $V_{\bm X, 1}^{\text{now}}$.}
\label{qqplot_minput_rho0_nowVSlater_rf}
\end{subfigure}
\medskip
\begin{subfigure}{0.45\textwidth}
\includegraphics[width=\linewidth]{normalized_l2_error_V1_laterVnow_random_forest_approx_barrier_ntrain20000_ntest100000_ninner1000_d3_T12_rho00_learnwithXTrue.png}
\caption{BRC with Random Forest: normalized $L^2_{{\mathbb Q}}$-errors of $V_{\bm X, 1}$ and $V_{\bm X, 1}^{\text{now}}$ in \%.}
\label{error_V1_barrier_rho0_nowVSlater_rf}
\end{subfigure}\hfil
\begin{subfigure}{0.45\textwidth}
\includegraphics[width=\linewidth]{qqplot_V1_nowVlater_random_forest_approx_barrier_ntrain20000_ntest100000_ninner1000_d3_T12_rho00.png}
\caption{BRC with Random Forest: detrended Q-Q plots of $V_{\bm X, 1}$ and $V_{\bm X, 1}^{\text{now}}$.}
\label{qqplot_barrier_rho0_nowVSlater_rf}
\end{subfigure}
\medskip
\begin{subfigure}{0.45\textwidth}
\includegraphics[width=\linewidth]{normalized_l2_error_V1_laterVnow_random_forest_approx_max_call_ntrain20000_ntest100000_ninner1000_d6_T2_rho00_learnwithXTrue.png}
\caption{Max-call with Random Forest: normalized $L^2_{{\mathbb Q}}$-errors of $V_{\bm X, 1}$ and $V_{\bm X, 1}^{\text{now}}$ in \%.}
\label{error_V1_maxcall_rho0_nowVSlater_rf}
\end{subfigure}\hfil
\begin{subfigure}{0.45\textwidth}
\includegraphics[width=\linewidth]{qqplot_V1_nowVlater_random_forest_approx_max_call_ntrain20000_ntest100000_ninner1000_d6_T2_rho00.png}
\caption{Max-call with Random Forest: detrended Q-Q plots of $V_{\bm X, 1}$ and $V_{\bm X, 1}^{\text{now}}$.}
\label{qqplot_maxcall_rho0_nowVSlater_rf}
\end{subfigure}
\caption{
Results for the min-put, BRC, and max-call with Random Forest. The normalized $L^2_{\mathbb Q}$-errors of $V_{\bm X, 1}$ and $V_{\bm X, 1}^{\text{now}}$, $\|V_1-V_{\bm X, 1}\|_{2, {\mathbb Q}}/V_0$ and $\|V_1-V_{\bm X, 1}^{\text{now}}\|_{2, {\mathbb Q}}/V_0$, are computed using the test sample and expressed in \%. In the detrended Q-Q plots, the blue, cyan, and lawngreen (red, orange, and pink) dots are built using regress-later Random Forest (regress-now Random Forest) and the test sample. $[0\%, 0.01\%)$ refers to the quantiles of levels $\{0.001\%,0.002\%, \dots, 0.009\%\}$, $[0.01\%, 1\%)$ refers to the quantiles of levels $\{0.01\%,0.02\%, \dots, 0.99\%\}$, $[1\%, 99\%]$ refers to the quantiles of levels $\{1\%,2\%, \dots, 99\%\}$, $(99\%, 99.99\%]$ refers to the quantiles of levels $\{99.01\%,99.02\%, \dots, 99.99\%\}$, and $(99.99\%, 100\%]$ refers to the quantiles of levels $\{99.991\%,99.992\%, \dots, 100\%\}$.}
\label{fig_nowVSlater_rf}
\end{figure}
\begin{figure}[p]
\centering
\begin{subfigure}{0.45\textwidth}
\includegraphics[width=\linewidth]{normalized_l2_error_V1_laterVnow_xgboost_approx_min_put_ntrain20000_ntest100000_ninner1000_d6_T2_rho00_learnwithXTrue.png}
\caption{Min-put with XGBoost: normalized $L^2_{{\mathbb Q}}$-errors of $V_{\bm X, 1}$ and $V_{\bm X, 1}^{\text{now}}$ in \%.}
\label{error_V1_minput_rho0_nowVSlater_xgb}
\end{subfigure}\hfil
\begin{subfigure}{0.45\textwidth}
\includegraphics[width=\linewidth]{qqplot_V1_nowVlater_xgboost_approx_min_put_ntrain20000_ntest100000_ninner1000_d6_T2_rho00.png}
\caption{Min-put with XGBoost: detrended Q-Q plots of $V_{\bm X, 1}$ and $V_{\bm X, 1}^{\text{now}}$.}
\label{qqplot_minput_rho0_nowVSlater_xgb}
\end{subfigure}
\medskip
\begin{subfigure}{0.45\textwidth}
\includegraphics[width=\linewidth]{normalized_l2_error_V1_laterVnow_xgboost_approx_barrier_ntrain20000_ntest100000_ninner1000_d3_T12_rho00_learnwithXTrue.png}
\caption{BRC with XGBoost: normalized $L^2_{{\mathbb Q}}$-errors of $V_{\bm X, 1}$ and $V_{\bm X, 1}^{\text{now}}$ in \%.}
\label{error_V1_barrier_rho0_nowVSlater_xgb}
\end{subfigure}\hfil
\begin{subfigure}{0.45\textwidth}
\includegraphics[width=\linewidth]{qqplot_V1_nowVlater_xgboost_approx_barrier_ntrain20000_ntest100000_ninner1000_d3_T12_rho00.png}
\caption{BRC with XGBoost: detrended Q-Q plots of $V_{\bm X, 1}$ and $V_{\bm X, 1}^{\text{now}}$.}
\label{qqplot_barrier_rho0_nowVSlater_xgb}
\end{subfigure}
\medskip
\begin{subfigure}{0.45\textwidth}
\includegraphics[width=\linewidth]{normalized_l2_error_V1_laterVnow_xgboost_approx_max_call_ntrain20000_ntest100000_ninner1000_d6_T2_rho00_learnwithXTrue.png}
\caption{Max-call with XGBoost: normalized $L^2_{{\mathbb Q}}$-errors of $V_{\bm X, 1}$ and $V_{\bm X, 1}^{\text{now}}$ in \%.}
\label{error_V1_maxcall_rho0_nowVSlater_xgb}
\end{subfigure}\hfil
\begin{subfigure}{0.45\textwidth}
\includegraphics[width=\linewidth]{qqplot_V1_nowVlater_xgboost_approx_max_call_ntrain20000_ntest100000_ninner1000_d6_T2_rho00.png}
\caption{Max-call with XGBoost: detrended Q-Q plots of $V_{\bm X, 1}$ and $V_{\bm X, 1}^{\text{now}}$.}
\label{qqplot_maxcall_rho0_nowVSlater_xgb}
\end{subfigure}
\caption{Results for the min-put, BRC, and max-call with XGBoost. The normalized $L^2_{\mathbb Q}$-errors of $V_{\bm X, 1}$ and $V_{\bm X, 1}^{\text{now}}$, $\|V_1-V_{\bm X, 1}\|_{2, {\mathbb Q}}/V_0$ and $\|V_1-V_{\bm X, 1}^{\text{now}}\|_{2, {\mathbb Q}}/V_0$, are computed using the test sample and expressed in \%. In the detrended Q-Q plots, the blue, cyan, and lawngreen (red, orange, and pink) dots are built using regress-later XGBoost (regress-now XGBoost) and the test sample. $[0\%, 0.01\%)$ refers to the quantiles of levels $\{0.001\%,0.002\%, \dots, 0.009\%\}$, $[0.01\%, 1\%)$ refers to the quantiles of levels $\{0.01\%,0.02\%, \dots, 0.99\%\}$, $[1\%, 99\%]$ refers to the quantiles of levels $\{1\%,2\%, \dots, 99\%\}$, $(99\%, 99.99\%]$ refers to the quantiles of levels $\{99.01\%,99.02\%, \dots, 99.99\%\}$, and $(99.99\%, 100\%]$ refers to the quantiles of levels $\{99.991\%,99.992\%, \dots, 100\%\}$.}
\label{fig_nowVSlater_xgb}
\end{figure}
\section{Bermudan options pricing}\label{sec_optimal_stopping}
So far we showed that ensemble estimators with regression trees can be employed to learn a value process of the form \eqref{eqcondY_EU}. In this section, we sketch how these estimators can also be applied to deal with another important and difficult problem in finance, the pricing of Bermudan options under discrete-time local volatility models.\footnote{For the sake of presentation, we only consider discrete-time local volatility models. However, the method we detail below can be adapted to also deal with discrete-time stochastic volatility models.} A standard solution to this problem is to recursively estimate the continuation value, see, e.g., \cite{tsi_roy_1999}, and \cite{lon_sch_2001}. Recent solutions based on machine learning techniques include \cite{bec_et_al_2019}, who apply deep neural networks to learn the optimal stopping rule, and \cite{gou_et_al_2020}, who apply Gaussian Process Regression to learn the value function at every time step. We add to this literature by showing that ensemble estimators with regression trees give closed-form estimators of the entire value process of Bermudan options.
Similarly to what we discussed in Appendix \ref{sec_regnow_reglat} for European style options pricing, we first apply regress-later and then regress-now.
\subsection{Regress-later}\label{sec_us_option_reglater}
Assume the stochastic driver $X=(X_1, \dots, X_T)$ has standard normal distribution on ${\mathbb R}^{d\times T}$, i.e., ${\mathbb Q}_t = {\mathcal N}(0, I_{d})$ for every $t=1, \dots, T$. Let $m \in {\mathbb N}$, and $\alpha_t : {\mathbb R}^{m} \to {\mathbb R}^{m}$ and $\beta_t : {\mathbb R}^{m} \to {\mathbb R}^{m\times d}$ be measurable functions, for $t=0,\dots, T-1$. Let $(Z_t)_{0\le t \le T}$ be an $m$-dimensional stochastic process that represents the evolution of the log-price of $m$ underlying assets. We assume that it follows the following discrete-time local volatility model,
\begin{equation}\label{lv_model}
\textstyle
\begin{cases}
Z_{t} &= \alpha_{t-1}(Z_{t-1}) + \beta_{t-1}(Z_{t-1}) X_{t},\quad t =1, \dots, T,\\
Z_{0} &= z_0,
\end{cases}
\end{equation}
where $z_0 \in {\mathbb R}^{m}$ is the initial log-price vector. We denote by $\mathbb G=({\mathcal G}_t)_{0\le t\le T}$ the natural filtration of $Z$, ${\mathcal G}_t =\sigma(Z_s\mid 0\le s\le t)$. Let $g_t: {\mathbb R}^{m} \to {\mathbb R}$ be measurable functions such that ${\mathbb E}_{ {\mathbb Q}}[g_t(Z_t)^2]< \infty$, for $t=0, \dots, T$. Let ${\mathcal T}_t$ be the set of $\mathbb G$-stopping times $\tau$ taking values in $\{t,t+1,\dots,T\}$, for $t=0,\dots, T$. We are interested in the following optimal stopping problem,
\begin{equation}\label{eqcondY_US_markov}
\textstyle
V_t = \sup_{\tau\in{\mathcal T}_t} {\mathbb E}_{{\mathbb Q}}[g_{\tau}( Z_{\tau})\mid {\mathcal G}_t], \quad t=0, \dots, T.\footnote{We look at Markovian problems, in the sense that $g_t$ depends only on the state variable $Z_t$. However, our method can be adapted to also deal with non Markovian problems, where $g_t$ could depend on the whole past trajectory $Z_0, \dots, Z_t$.}
\end{equation}
It is well known, see, e.g., \cite[Section~1]{pes_shi_2006}, problem \eqref{eqcondY_US_markov} can be solved by backward induction as follows. For time $t=T$, set $V_T= g_T(Z_T)$. By induction, for any time $t= T-1,\dots,0$, define the continuation value $C_{t} = {\mathbb E}_{ {\mathbb Q}}[V_{t+1}(Z_{t+1}) \mid Z_{t} ]$, so that $V_{t} = \max(g_{t}(Z_{t}), C_{t})$. Then, an optimal stopping time $\tau^\star_t\in{\mathcal T}_t$, for which $V_t = {\mathbb E}_{ {\mathbb Q}}[g_{\tau^\star_t}(Z_{\tau^\star_t})\mid {\mathcal G}_t]$, is given by $\tau^\star_t = \inf\{t\le s \le T \mid V_s= g_s(Z_s)\}$.
Our method to solve \eqref{eqcondY_US_markov} is based on a backward induction, where for each time $t=T-1,\dots, 0$ we estimate the value function $V_t$ by an ensemble estimator $V_{\bm X, t}$. Specifically, assume available a finite i.i.d.\ sample $\bm{ Z} = (Z^{(1)}, \dots, Z^{(n)})$ drawn from \eqref{eqcondY_US_markov}. And let $\bm{Z}_t = (Z^{(1)}_t, \dots, Z^{(n)}_t)$ be the $t$-cross-section sample of $\bm{Z}$, for $t=1, \dots, T$. Then proceed backward as follows:
\begin{enumerate}
\item For time $t=T$: set $V_{\bm X, T} = g_T(Z_T)$.
\item For any time $t= T-1,\dots,0$: let $\widehat{V}_{\bm X,t+1}$ be an ensemble estimator of $V_{\bm X,t+1}$, obtained using the sample $\bm{Z}_{t+1}$, along with the function values $\bm{V}_{\bm X, t+1} = (V_{\bm X, t+1}(Z_{t+1}^{(1)}), \dots, V_{\bm X, t+1}(Z_{t+1}^{(n)}))$. Then, we claim that
\begin{equation}\label{ct_markov}
\textstyle
C_{\bm X, t} = {\mathbb E}_{{\mathbb Q}}[\widehat{V}_{\bm X, t+1}(Z_{t+1})\mid Z_{t}] \text{ is in \textbf{closed form}}.
\end{equation}
Then set $V_{\bm X, t} = \max( g_{t}(Z_{t}), C_{\bm X, t})$, which is therefore also in closed form.
\end{enumerate}
Now let us explain why $C_{\bm X, t}$ in \eqref{ct_markov} is in closed form. The function $\widehat{V}_{\bm X, t+1}$ is an ensemble estimator, whose expression can be brought into the form \eqref{generic_el_estimator}, i.e., $\widehat{V}_{\bm X, t+1} = \sum_{i=1}^{N} \beta_i {\mathbbm 1}_{\bm A_i}$, for some real coefficients $\beta_i$ and some hyperrectangles $\bm A_i$ of ${\mathbb R}^m$. Subsequently, by independence of $Z_{t}$ and $X_{t+1}$, we readily obtain that
\[\textstyle C_{\bm X,t}( z_{t}) = \sum_{i=1}^N \beta_{i} {\mathbb Q}_{t+1}[\alpha_{t}( z_{t}) + \beta_{t}( z_{t}) X_{t+1} \in \bm A_i ], \quad z_{t} \in {\mathbb R}^m.\]
Now observe that $\alpha_{t}( z_{t}) + \beta_{t}( z_{t}) X_{t+1}$ is normally distributed under ${\mathbb Q}_{t+1}$, with mean $\alpha_{t}( z_{t})$ and covariance matrix $\beta_{t}( z_{t}) \beta_{t}( z_{t})^\top$. Thanks to \cite{genz_2000}, ${\mathbb Q}_{t+1}[\alpha_{t}( z_{t}) + \beta_{t}( z_{t}) X_{t+1} \in \bm A_i ]$ is in closed form for any hyperrectangle $\bm A_i$. In fact, functions to integrate a multivariate normal distribution on a hyperrectangle are readily accessible on scientific languages, such as Python (see the mvn function in the sub-package stats of the library SciPy \cite{scipy}), and R (see the function pmvnorm in the mvtnorm package \cite{genz_et_al_mvtnorm}). This became possible thanks to the Fortran code of \cite{genz_web}. Matlab code to integrate a multivariate normal distribution on a hyperrectangle can also be found in \cite{genz_web}.
After the construction of the value process estimator $V_{\bm X}$, we define the optimal stopping time estimator
\begin{equation}\label{opt_stop_rule_est}
\tau^{\star}_{\bm X, t} = \inf\{t\le s \le T \mid V_{\bm X, s} = g_s(Z_s)\}.
\end{equation}
Now as a simple numerical example, let $S_t$ represent the nominal price of an underlying asset, whose dynamics is given in \eqref{bs_model} with $d=1$. We denote by $Z_t = \log(S_t)$ its log-price. Then $Z_t$ follows the dynamics
\[\textstyle\begin{cases}
Z_{t} &= Z_{t-1} + (r-\sigma^2/2)\Delta_t + \sigma \sqrt{\Delta_t} X_{t},\quad t =1, \dots, T,\\
Z_{0} &= z_0,
\end{cases}\]
which is of the form \eqref{lv_model}. We are interested in estimating the value process $V$ in \eqref{eqcondY_US_markov}, where the payoff function $g_t$ is
\begin{itemize}
\item Put $g_t(Z_t) = {\rm e}^{-r\sum_{s=1}^t \Delta_s} (K-{\rm e}^{Z_t})^+$.
\end{itemize}
Here we set the following parameter values, $z_0 = 0$, $r=0$, $\sigma=0.2$, $T=7$, $(\Delta_1, \dots, \Delta_T)=(1/T, \dots, 1/T)$, and $K=1$. Under this parameter specification, we generate a training sample $\bm X$ of size $n= 5{,}000$, and a test sample $\bm X_{\mathrm{test}}$ of size $n_{\mathrm{test}}=100{,}000$. When $V_{\bm X}$ is the Random Forest estimator of $V$, we use the RandomForestRegressor class of the library scikit-learn \cite{scikit_learn}, with the following hyperparameter values: $M$=10, \textbf{nodesize} = 2, $p=1$, \textbf{sampling
regime}=bootstrapping.\footnote{Except for the number of trees $M$, the other hyperparameter values are the default values in RandomForestRegressor. We picked a small value for $M$ for computational reasons.} When $V_{\bm X}$ is the Gradient Boosting estimator of $V$, we use the XGBRegressor class of XGBoost \cite{che_gue_2016}, with the default hyperparameter values: $t=100$, $\textbf{nodesize}=1$, $\textbf{max\_depth}=6$.
With the Bermudan put example and the parameter specification, $r=0$, one can readily show that early stopping is not optimal, so that $V_t={\mathbb E}[(K-{\rm e}^{Z_{T}})^+\mid Z_t]$ equals the European put option price, and $V_t$ is given in closed form thanks to Black's formula. In fact, for $t=0, \dots, T-1$, $V_t = -{\rm e}^{Z_t} \Phi(-d_1) + K \Phi(-d_2)$, where $d_1= \frac{1}{\sigma\sqrt{\Delta_{t+1}+\dots+\Delta_T}}\left(\ln({\rm e}^{Z_t}/K) + (r+\sigma^2/2)(\Delta_{t+1} +\dots+\Delta_T) \right)$ and $d_2 = d_1-\sigma \sqrt{\Delta_{t+1} +\dots+\Delta_T}$. We thus have exact ground truth benchmark for our estimator $V_{\bm X}$.
After the construction of the estimated value process $V_{\bm X}$, we evaluate $V_{\bm X, t}$ and $V_t$ on the test sample $\bm X_{\mathrm{test}}$, for $t=0,\dots, T-1$. Then we carry out the following four evaluation tasks.
First, we compute the normalized $L^2_{\mathbb Q}$-error $\|V_{\bm X, t}-V_t\|_{2, {\mathbb Q}}/V_0$, for $t=0, \dots, T-1$. Figures \ref{error_us_Vt_put_rf} and \ref{error_us_Vt_put_xgb} show the evolution of the normalized $L^2_{\mathbb Q}$-error of $V_{\bm X, t}$ as function of $t=0, \dots, T-1$. First, we notice that all normalized $L^2_{\mathbb Q}$-errors are below $0.2\%$ and $0.6\%$ with Random Forest and XGBoost, respectively. Second, with the exception of $t=0$, the normalized $L^2_{\mathbb Q}$-errors have a tendency to increase with time to maturity $T-t$. There seems to be an accumulation of errors, due to the estimation of $V_t$ at each induction step $t+1\to t$. Whereas at $t=0$ the errors seem to cancel out across the sample, as $V_{\bm X,0}=C_{\bm X,0}$ is given by the unconditional expectation \eqref{ct_markov}.
Second, we compute the detrended Q-Q plots of $V_{\bm X}$. Figures \ref{qqplot_us_Vt_put_rf} and \ref{qqplot_us_Vt_put_xgb} show detrended Q-Q plots of $V_{\bm X, t}$, for $t=1, \dots, T-1$, using Random Forest and XGBoost, respectively. They are constructed as the detrended Q-Q plots in Section \ref{secexamples}. Specifically, here we draw the detrended Q-Q plots of $V_{\bm X, t}$ using the test sample, for every $t=1, \dots, T-1$. Thereto, for $t\in \{1,\dots, T-1\}$, we compute the empirical left quantiles of $V_{\bm X, t}$ and $V_t$ at levels $\{0.001\%, 1\%, 2\%,\dots,100\%\}$. The detrended quantiles (estimated quantiles minus true quantiles) are then plotted against the true quantiles. We notice that, as function of $t$, the decrease of the normalized $L^2_{\mathbb Q}$-errors of $V_{\bm X, t}$ translates into a flattening of the detrended Q-Q plots. In fact, for illustration, the almost zero normalized $L^2_{\mathbb Q}$-errors of $V_{\bm X, 4}$, $V_{\bm X, 5}$, and $V_{\bm X, 6}$ in Figure \ref{error_us_Vt_put_rf} correspond to almost perfect detrended Q-Q plots of $V_{\bm X, 4}$, $V_{\bm X, 5}$, and $V_{\bm X, 6}$ in Figure \ref{qqplot_us_Vt_put_rf}, i.e., they correspond to detrended Q-Q plots that are almost perfectly aligned with the horizontal black line. Overall, both Random Forest and XGBoost give excellent detrended Q-Q plots of $V_{\bm X, t}$, for every $t=1,\dots, T-1$.
Third, we use our value process estimator $V_{\bm X}$ to compute the value at risk at level $\alpha=99.5\%$, and the expected shortfall at level $\alpha = 99\%$ of $V_{\bm X, t}-V_{\bm X, t+1}$ and $V_{\bm X, t+1}-V_{\bm X, t}$, for every $t=0, \dots, T-1$. $V_{\bm X, t}-V_{\bm X, t+1}$ and $V_{\bm X, t+1}-V_{\bm X, t}$ are the 1-period losses at time $t$ of long position and short position, respectively. We perform the same risk measure computations with our benchmark $V$. Then, we compute the relative errors of risk measures, computed as $(\text{estimated risk measure minus true risk measure})/\text{true risk measure}$ and expressed in \%. Figure \ref{normalized_errors_risk_measures_rf} shows the evolution of relative errors of risk measures $\mathrm{ES}_{99\%}$ and $\mathrm{VaR}_{99.5\%}$ for both long and short positions with Random Forest. Figure \ref{normalized_errors_risk_measures_xgb} shows the same computations with XGBoost. Let's focus on Figure \ref{normalized_errors_risk_measures_rf}. Relative errors of risk measures of long and short positions are all in the intervals $[-0.3\%, 0\%]$ and $[-1.2\%,0.1\%]$, respectively. Furthermore, we highlight that the relative errors for $t\in \{3, 4, 5, 6\}$ are equal to 0\%. The last is in line with the detrended Q-Q plots at $t\in\{3,4, 5, 6\}$ in Figure \ref{qqplot_us_Vt_put_rf}, which are almost perfectly aligned with the horizontal black line.
Fourth, we compute the optimal stopping rule estimator $\tau_{\bm X, 0}^{\star}$ in \eqref{opt_stop_rule_est} for the $n_{\mathrm{test}}$ simulations in $\bm X_{\mathrm{test}}$. Table \ref{table_opt_stop} shows the distribution of $\tau_{\bm X, 0}^{\star}$ using the test sample $\bm X_{\mathrm{test}}$. The distribution of the true optimal stopping rule $\tau_0^\star$ is the Dirac distribution $\delta_{7}(dx)$. We observe that estimation of the distribution of $\tau_0^\star$ is accurate with both Random Forest and XGBoost, and it is XGBoost that outperforms Random Forest.
\begin{table}
\centering
\begin{tabular}{|l|l|l|l|l|l|l|l|l|}
\hline
Estimator $|$ time $t$ & 0 &1 &2 &3&4&5&6&7 \\
\hline
XGBoost & 0& 0.00009& 0.00037& 0.00036& 0.00064& 0.00030& 0.00104& \underline{\textbf{0.99720}} \\
\hline
Random Forest& 0& 0.00026& 0.00308& 0.00197& 0.00693& 0.00097& 0.00534& \textbf{0.98145}\\
\hline
\end{tabular}
\caption{Distribution of $\tau_{\bm X, 0}^{\star}$, constructed with $V_{\bm X}$ using the test sample $\bm X_{\mathrm{test}}$, using XGBoost and Random Forest. The true distribution of $\tau^\star_0$ is the Dirac distribution $\delta_7(d x)$.}\label{table_opt_stop}
\end{table}
\begin{figure}[p]
\centering
\begin{subfigure}{0.45\textwidth}
\includegraphics[width=\linewidth]{normalized_l2_errors_Vt_us_put_ntrain5000_ntest100000_d1_T7_random_forest_approx.png}
\caption{With Random Forest: normalized $L^2_{{\mathbb Q}}$-errors of $V_{\bm X, t}$ as function of $t=0, \dots, T-1$. Values are expressed in \%.}
\label{error_us_Vt_put_rf}
\end{subfigure}\hfil
\begin{subfigure}{0.45\textwidth}
\includegraphics[width=\linewidth]{normalized_l2_errors_Vt_us_put_ntrain5000_ntest100000_d1_T7_xgboost_approx.png}
\caption{With XGBoost: normalized $L^2_{{\mathbb Q}}$-errors of $V_{\bm X, t}$ as function of $t=0, \dots, T-1$. Values are expressed in \%.}
\label{error_us_Vt_put_xgb}
\end{subfigure}
\medskip
\begin{subfigure}{0.45\textwidth}
\includegraphics[width=\linewidth]{qqplot_Vt_us_put_ntrain5000_ntest100000_d1_T7_random_forest_approx.png}
\caption{With Random Forest: detrended Q-Q plot of $V_{\bm X, t}$ for $t=1, \dots, T-1$.}
\label{qqplot_us_Vt_put_rf}
\end{subfigure}\hfil
\begin{subfigure}{0.45\textwidth}
\includegraphics[width=\linewidth]{qqplot_Vt_us_put_ntrain5000_ntest100000_d1_T7_xgboost_approx.png}
\caption{With XGBoost: detrended Q-Q plot of $V_{\bm X, t}$ for $t=1, \dots, T-1$.}
\label{qqplot_us_Vt_put_xgb}
\end{subfigure}
\medskip
\begin{subfigure}{0.45\textwidth}
\includegraphics[width=\linewidth]{normalized_errors_risk_measures_us_put_ntrain5000_ntest100000_d1_T7_random_forest_approx.png}
\caption{With Random Forest: relative errors of $\mathrm{ES}_{99\%}(V_t-V_{t+1})$, $\mathrm{VaR}_{99.5\%}(V_t-V_{t+1})$, $\mathrm{ES}_{99\%}(V_{t+1}-V_t)$, and $\mathrm{VaR}_{99.5\%}(V_{t+1}-V_t)$ for the estimator $V_{\bm X}$. Values are expressed in \%.}
\label{normalized_errors_risk_measures_rf}
\end{subfigure}\hfil
\begin{subfigure}{0.45\textwidth}
\includegraphics[width=\linewidth]{normalized_errors_risk_measures_us_put_ntrain5000_ntest100000_d1_T7_xgboost_approx.png}
\caption{With XGBoost: relative errors of $\mathrm{ES}_{99\%}(V_t-V_{t+1})$, $\mathrm{VaR}_{99.5\%}(V_t-V_{t+1})$, $\mathrm{ES}_{99\%}(V_{t+1}-V_t)$, and $\mathrm{VaR}_{99.5\%}(V_{t+1}-V_t)$ for the estimator $V_{\bm X}$. Values are expressed in \%.}
\label{normalized_errors_risk_measures_xgb}
\end{subfigure}
\caption{
Results for the Bermudan put with regress-later Random Forest and regress-later XGBoost. The normalized $L^2_{\mathbb Q}$-errors of $V_{\bm X, t}$, $\|V_t-V_{\bm X, t}\|_{2, {\mathbb Q}}/V_0$, for $t=0, \cdots, T-1$, are computed using the test sample and expressed in \%. The detrended Q-Q plots are built using the test sample. They show the detrended quantiles of levels $\{0.001\%, 1\%,2\%,\dots, 100\%\}$ of $V_{\bm X, t}$, for $t=1\cdots, T-1$. The relative errors of value at risk and expected shortfall, computed as $(\text{estimated risk measure minus true risk measure})/\text{true risk measure}$ using the test sample, are expressed in \%.}
\label{fig_us}
\end{figure}
\subsection{Regress-now}
We now compare the above regress-later method for Bermudan options pricing to its regress-now variant, as discussed in Appendix \ref{sec_regnow_reglat} for European style options.
Let us start by introducing the regress-now estimator of the value process of Bermudan options. We shall denote this estimator by $V_{\bm X}^{\mathrm{now}}$. Its construction is as follows. To solve \eqref{eqcondY_US_markov}, proceed backward, where at each time $t=T-1, \dots, 0$ estimate the continuation value function $C_t$ by an ensemble estimator $C_{\bm X, t}^{\mathrm{now}}$. Specifically, assume available a finite i.i.d.\ training sample $\bm{ Z} = (Z^{(1)}, \dots, Z^{(n)})$ drawn from \eqref{eqcondY_US_markov}. And let ${\bm Z}_t = (Z^{(1)}_t, \dots, Z^{(n)}_t )$ denote the $t$-cross-section samples of ${\bm Z}$. Then proceed backward as follows:
\begin{enumerate}
\item For time $t=T$: set $V_{\bm X, T}^{\mathrm{now}} = g_T(Z_Z)$.
\item For any time $t= T-1,\dots,0$: let $C_{\bm X,t}^{\mathrm{now}}$ be an ensemble estimator of $C_{t}$, obtained using the sample ${\bm Z}_{t}$, along with the function values $\bm{V}_{\bm X, t+1}^{\mathrm{now}} = (V_{\bm X, t+1}^{\mathrm{now}}(Z_{t+1}^{(1)}), \dots, V_{\bm X, t+1}^{\mathrm{now}}(Z_{t+1}^{(n)}))$. Then, we set $V_{\bm X, t}^{\mathrm{now}} = \max( g_{t}(Z_{t}), C_{\bm X, t}^{\mathrm{now}})$.
\end{enumerate}
When $V_{\bm X}^{\mathrm{now}}$ is the Random Forest estimator of $V$, we use the RandomForestRegressor class of the library scikit-learn \cite{scikit_learn}, with the hyperparameter values: $M$=500, \textbf{nodesize} = 20, $p=1$, \textbf{sampling
regime}=bootstrapping. When $V_{\bm X}$ is the Gradient Boosting estimator of $V$, we use the XGBRegressor class of XGBoost \cite{che_gue_2016}, with the hyperparameter values: $t=300$, $\textbf{nodesize}=20$, $\textbf{max\_depth}=50$.\footnote{These hyperparameter values give much better numerical results than the default hyperparameter values we used in Section~\ref{sec_us_option_reglater}.}
Just as in Section \ref{sec_us_option_reglater} above, we use the test sample $\bm X_{\mathrm{test}}$ to compute the normalized $L^2_{\mathbb Q}$-error $\|V_{\bm X, t}^{\mathrm{now}}-V_t\|_{2, {\mathbb Q}}/V_0$, for $t=0, \dots, T-1$. Figures \ref{now_error_us_Vt_put_rf} and \ref{now_error_us_Vt_put_xgb} show the evolution of the normalized $L^2_{\mathbb Q}$-error of $V_{\bm X, t}^{\mathrm{now}}$ as function of $t=0, \dots, T-1$. First, we notice that all normalized $L^2_{\mathbb Q}$-errors are below $10\%$ and $11\%$ with Random Forest and XGBoost, respectively. Second, with the exception of $t=0$, the normalized $L^2_{\mathbb Q}$-errors have a tendency to increase with time to maturity $T-t$. Again, there seems to be an accumulation of errors, due to the estimation of $C_t$ at each induction step $t+1\to t$. Whereas at $t=0$ the errors seem to cancel out across the sample, as $V_{\bm X,0}=C_{\bm X,0}$ is given by the unconditional expectation \eqref{ct_markov}. Third, by comparing Figures \ref{now_error_us_Vt_put_rf} and \ref{now_error_us_Vt_put_xgb} to Figures \ref{error_us_Vt_put_rf} and \ref{error_us_Vt_put_xgb}, we highlight the outperformance of our regress-later method over its regress-now variant in terms of normalized $L^2_{\mathbb Q}$-errors.
Then we compute the detrended Q-Q plots of $V_{\bm X}^{\mathrm{now}}$. Figures \ref{now_qqplot_us_Vt_put_rf} and \ref{now_qqplot_us_Vt_put_xgb} show detrended Q-Q plots of $V_{\bm X, t}^{\mathrm{now}}$, for $t=1, \dots, T-1$. They are the counterpart of the detrended Q-Q plots of $V_{\bm X}$ in Figures \ref{qqplot_us_Vt_put_rf} and \ref{qqplot_us_Vt_put_xgb}. By comparing these four figures, we see that the detrended Q-Q plots are of much better quality with regress-later than with regress-now.
Next, we use our value process estimator $V_{\bm X}^{\mathrm{now}}$ to compute the value at risk at level $\alpha=99.5\%$, and expected shortfall at level $\alpha = 99\%$ of $V_{\bm X, t}^{\mathrm{now}}-V_{\bm X, t+1}^{\mathrm{now}}$ and $V_{\bm X, t+1}^{\mathrm{now}}-V_{\bm X, t}^{\mathrm{now}}$, for $t=0, \dots, T-1$. Figures \ref{now_normalized_errors_risk_measures_rf} and \ref{now_normalized_errors_risk_measures_xgb} show relative errors of risk measures with $V_{\bm X}^{\mathrm{now}}$. They are the counterpart of Figures \ref{normalized_errors_risk_measures_rf} and \ref{normalized_errors_risk_measures_xgb}. By comparing these four figures, we see the outperformance of regress-later over regress-now in terms of relative errors of risk measures.
We finish this section by computing the optimal stopping rule estimator $\tau_{\bm X, 0}^{\star, \mathrm{now}}= \inf\{0\le s \le T \mid V_{\bm X, s}^{\mathrm{now}} = g_t(Z_s)\}$ for the $n_{\mathrm{test}}$ simulations in $\bm X_{\mathrm{test}}$. The Table \ref{now_table_opt_stop} is the counterpart of Table \ref{table_opt_stop}. Here again we see the outperformance of regress-later over regress-now in terms of accuracy in the estimation of the optimal stopping rule distribution.
\begin{table}
\centering
\begin{tabular}{|l|l|l|l|l|l|l|l|l|}
\hline
Estimator $|$ time $t$ & 0 &1 &2 &3&4&5&6&7 \\
\hline
XGBoost & 0& 0.00235& 0.00803& 0.004043& 0.05611& 0.07934& 0.12134& \underline{\textbf{0.69240}} \\
Random Forest& 0& 0.00320& 0.0077& 0.03035& 0.05756& 0.06859& 0.27567& \textbf{0.56386}\\
\hline
\end{tabular}
\caption{Distribution of $\tau_{\bm X, 0}^{\star, \mathrm{now}}$, constructed with $V_{\bm X}^{\mathrm{now}}$ using the test sample $\bm X_{\mathrm{test}}$, using XGBoost and Random Forest. The true distribution of $\tau^\star_0$ is the Dirac distribution $\delta_7(d x)$.
}\label{now_table_opt_stop}
\end{table}
\begin{figure}[p]
\centering
\begin{subfigure}{0.45\textwidth}
\includegraphics[width=\linewidth]{now_normalized_l2_errors_Vt_us_put_ntrain5000_ntest100000_d1_T7_random_forest_approx.png}
\caption{With Random Forest: normalized $L^2_{{\mathbb Q}}$-errors of $V_{\bm X, t}^{\mathrm{now}}$ as function of $t=0, \dots, T-1$. Values are expressed in \%.}
\label{now_error_us_Vt_put_rf}
\end{subfigure}\hfil
\begin{subfigure}{0.45\textwidth}
\includegraphics[width=\linewidth]{now_normalized_l2_errors_Vt_us_put_ntrain5000_ntest100000_d1_T7_xgboost_approx.png}
\caption{With XGBoost: normalized $L^2_{{\mathbb Q}}$-errors of $V_{\bm X, t}^{\mathrm{now}}$ as function of $t=0, \dots, T-1$. Values are expressed in \%.}
\label{now_error_us_Vt_put_xgb}
\end{subfigure}
\medskip
\begin{subfigure}{0.45\textwidth}
\includegraphics[width=\linewidth]{now_qqplot_Vt_us_put_ntrain5000_ntest100000_d1_T7_random_forest_approx.png}
\caption{With Random Forest: detrended Q-Q plot of $V_{\bm X, t}^{\mathrm{now}}$ for $t=1, \dots, T-1$.}
\label{now_qqplot_us_Vt_put_rf}
\end{subfigure}\hfil
\begin{subfigure}{0.45\textwidth}
\includegraphics[width=\linewidth]{now_qqplot_Vt_us_put_ntrain5000_ntest100000_d1_T7_xgboost_approx.png}
\caption{With XGBoost: detrended Q-Q plot of $V_{\bm X, t}^{\mathrm{now}}$ for $t=1, \dots, T-1$.}
\label{now_qqplot_us_Vt_put_xgb}
\end{subfigure}
\medskip
\begin{subfigure}{0.45\textwidth}
\includegraphics[width=\linewidth]{now_normalized_errors_risk_measures_us_put_ntrain5000_ntest100000_d1_T7_random_forest_approx.png}
\caption{With Random Forest: relative errors of $\mathrm{ES}_{99\%}(V_t-V_{t+1})$, $\mathrm{VaR}_{99.5\%}(V_t-V_{t+1})$, $\mathrm{ES}_{99\%}(V_{t+1}-V_t)$, and $\mathrm{VaR}_{99.5\%}(V_{t+1}-V_t)$ for the estimator $V_{\bm X}^{\mathrm{now}}$. Values are expressed in \%.}
\label{now_normalized_errors_risk_measures_rf}
\end{subfigure}\hfil
\begin{subfigure}{0.45\textwidth}
\includegraphics[width=\linewidth]{now_normalized_errors_risk_measures_us_put_ntrain5000_ntest100000_d1_T7_xgboost_approx.png}
\caption{With XGBoost: relative errors of $\mathrm{ES}_{99\%}(V_t-V_{t+1})$, $\mathrm{VaR}_{99.5\%}(V_t-V_{t+1})$, $\mathrm{ES}_{99\%}(V_{t+1}-V_t)$, and $\mathrm{VaR}_{99.5\%}(V_{t+1}-V_t)$ for the estimator $V_{\bm X}^{\mathrm{now}}$. Values are expressed in \%.}
\label{now_normalized_errors_risk_measures_xgb}
\end{subfigure}
\caption{
Results for the Bermudan put with regress-now Random Forest and regress-now XGBoost. The normalized $L^2_{\mathbb Q}$-errors of $V_{\bm X,t}^{\mathrm{now}}$, $\|V_t-V_{\bm X,t}^{\mathrm{now}}\|_{2, {\mathbb Q}}/V_0$, for $t=0, \cdots, T-1$, are computed using the test sample and expressed in \%. The detrended Q-Q plots are built using the test sample. They show the detrended quantiles of levels $\{0.001\%, 1\%,2\%,\dots, 100\%\}$ of $V_{\bm X,t}^{\mathrm{now}}$, for $t=1\cdots, T-1$. The relative errors of value at risk and expected shortfall, computed as $(\text{estimated risk measure minus true risk measure})/\text{true risk measure}$ using the test sample, are expressed in \%.}
\label{now_fig_us}
\end{figure}
\end{appendix}
\clearpage
\bibliographystyle{apalike}
|
1,108,101,564,548 | arxiv | \section{Introduction}
In scheduling environments, uncertainty is a common consideration for optimization problems. Commonly, results are either based on worst case considerations or a random distribution over the input. These approaches are known as robust optimization and stochastic optimization, respectively. However, it is often the case that unknown information can be attained through investing some additional resources, e.g.\ time, computing power or money. In his seminal paper, Kahan \cite{Kahan1991} has first introduced the notion of explorable or queryable uncertainty to model obtaining additional information for a problem at a given cost during the runtime of an algorithm. Since then, these kind of problems have been explored in different optimization contexts, for example in the framework of combinatorial, geometric or function value optimization tasks.
Recently, D\"urr \text{et al.\ } \cite{Duerr2018} have introduced a model for scheduling with testing on a single machine within the framework of explorable uncertainty. In their approach, a number of jobs with unknown processing times are given. Testing takes one unit of time and reveals the processing time. If a job is executed untested, the time it takes to run the job is given by an upper bound. The novelty of their approach lies in having tests executed directly on the machine running the jobs as opposed to considering tests separately.
In view of this model, a natural extension is to consider non-uniform testing times to allow for a wider range of problems. D\"urr \text{et al.\ } state that for certain applications it is appropriate to consider a broader variation on testing times and leave this question up for future research.
Situations where a preliminary action, operation or test can be executed before a job are manifold and include a wide range of real-life applications. In the following, we discuss a small selection of such problems and emphasize cases with heterogeneous testing requirements. Consider first a situation where an online user survey can help predict market demand and production times. The time needed to produce the necessary amount of goods for the given demand is only known after conducting the survey. Depending on its scope and size, the invested costs for the survey may vary significantly.
As a second example, we look at distributed computing in a setting with many distributed local databases and one centralized master server. At the local stations, only estimates of some data values are stored; in order to obtain the true value one must query the master server. It depends on the distance and connection quality from any localized database to the master how much time and resources this requires. Olston and Widom \cite{Olston2000} have considered this setting in detail.
Another possible example is the acquisition of a house through an agent giving us more information about its value, location, condition, etc., but demanding a price for her services. This payment could vary based on the price of the house, the amount of work of the agent or the number of competitors.
In their paper, D\"urr \text{et al.\ } \cite{Duerr2018} mention fault diagnosis in maintenance and medical treatment, file compression for transmissions, and running jobs in an alternative fast mode whose availability can be determined through a test. Generally, any situation involving diverse cost and duration estimates, like e.g.\ in construction work, manufacturing or insurance, falls into our category of possible applications.
In view of all these examples, we investigate non-uniform testing in the scope of explorable uncertainty on a single machine as introduced by \cite{Duerr2018}. We study whether algorithms can be extended to this non-uniform case and if not, how we can find new methods for it.
\subsection{Problem Statement}
We consider $n$ jobs to be scheduled on a single machine. Every job $j$ has an unknown processing time $p_j$ and a known upper bound $u_j$. It holds $0 \le p_j \le u_j$ for all $j$. Each job also has a testing time $t_j \ge 0$. A job can either be executed untested, which takes time $u_j$, or be tested and then executed, which takes a total time of $t_j + p_j$. Note that a tested job does not necessarily have to be executed right after its test, it may be delayed arbitrarily while the algorithm tests or executes other jobs.
Since only the upper bounds are initially known to the algorithm, the task can be viewed as an online problem with an adaptive adversary. The actual processing times $p_j$ are only realized after job $j$ has been tested by the algorithm. In the randomized case, the adversary knows the distribution of the random input parameters of an algorithm, but not their outcome.
We denote the completion time of a job $j$ as $C_j$ and primarily consider the objective of minimizing the total sum of completion times $\sum_j C_j$. As a secondary objective, we also investigate the simpler goal of minimizing the makespan $\max_j C_j$. We use competitive analysis to compare the value produced by an algorithm with an optimal offline solution.
Clearly, in the offline setting where all processing times are known, an optimal schedule can be determined directly: If $t_j + p_j \le u_j$ then job $j$ is tested, otherwise it is run untested. For the sum of completion times, the jobs are therefore scheduled in order of non-decreasing $\min(t_j+p_j,u_j)$. Any algorithm for the online problem not only has to decide whether to test a given job or not, but also in which order to run all tests and executions of both untested and tested jobs. For a solution to the makespan objective, the ordering of the jobs does not matter and an optimal offline algorithm decides the testing by the same principle as above.
\subsection{Related Work}
Our setting is directly based on the problem of scheduling uncertain jobs on a single machine with explorable processing times, introduced by D\"urr \text{et al.\ } \cite{Duerr2018} in 2018. They only consider the special case where $t_j \equiv 1$ for all jobs. For deterministic algorithms, they give a lower bound of 1.8546 and an upper bound of 2. In the randomized case, they give a lower bound of 1.6257 and a 1.7453-competitive algorithm. For several deterministic special case instances, they provide upper bounds closer to the best possible ratio of 1.8546. Additionally, tight algorithms for the objective of minimizing the makespan are given for both the deterministic and randomized cases.
Testing and executing jobs on a single machine can be viewed as part of the research area of \emph{queryable uncertainty} or \emph{explorable uncertainty}. The first seminal paper on dealing with uncertainty by querying parts of the input was published in 1991 by Kahan \cite{Kahan1991}. In his paper, Kahan considers a set of elements with uncertain values that lie in a closed interval. He explores approximation guarantees for the number of queries necessary to obtain the maximum and median value of the uncertain elements.
Since then, there has been a large amount of research concerned with the objective of minimizing the number of queries to obtain a solution. A variety of numerical, geometric and combinatorial problems have been studied in this framework, the following is a selection of some of these publications: Next to Kahan, Feder \text{et al.\ } \cite{Feder2003}, Khanna and Tan \cite{Khanna2001}, and Gupta \text{et al.\ } \cite{Gupta2011} have also considered the objective of determining different function values, in particular the k-smallest value and the median. Bruce \text{et al.\ } \cite{Bruce2005} have analysed geometric tasks, specifically the Maximal Points and Convex Hull problems. They have also introduced the notion of \emph{witness sets} as a general concept for queryable uncertainty, which was then generalized by Erlebach \text{et al.\ } \cite{Erlebach2008}. Olston and Widom \cite{Olston2000} researched caching problems while allowing for some inaccuracy in the objective function. Other studied combinatorial problems include minimum spanning tree \cite{Erlebach2008,Megow2017}, shortest path \cite{Feder2007}, knapsack \cite{Goerigk2015} and boolean trees \cite{Charikar2002}. See also the survey by Erlebach and Hoffmann \cite{Erlebach2015} for an overview over research in this area.
A related type of problems within optimization under uncertainty are settings where the cost of the queries is a direct part of the objective function. Most notably, the paper by D\"urr \text{et al.\ } \cite{Duerr2018} falls into this category. There, the tests necessary to obtain additional information about the runtime of the jobs are executed on the same machine as the jobs themselves. Other examples include Weitzman's original Pandora's Box problem \cite{Weitzman1979}, where $n$ independent random variables are probed to maximize the highest revealed value. Every probing incurs a price directly subtracted from the objective function. Recently, Singla \cite{Singla2018} introduced the 'price of information' model to describe receiving information in exchange for a probing price. He gives approximation ratios for various well-known combinatorial problems with stochastic uncertainty.
\subsection{Contribution}
In this paper, we provide the first algorithms for the more general scheduling with testing problem where testing times can be non-uniform. Consult Table \ref{tab:results_expl_uncer} for an overview of results for both the non-uniform and uniform versions of the problem. All ratios provided without citation are introduced in this paper. The remaining results are presented in \cite{Duerr2018}.
\begin{table}[htb]
\centering
\caption{Overview of results}
\label{tab:results_expl_uncer}
\begin{tabulary}{\textwidth}{C | C | C | C}
\textbf{Objective Type} & \textbf{General tests} & \textbf{Uniform tests} & \textbf{Lower bound} \\ \midrule
$\sum C_j$ - deterministic & $4$ & $2$ \cite{Duerr2018} & $1.8546$ \cite{Duerr2018} \\
$\sum C_j$ - randomized & $3.3794$ & $1.7453$ \cite{Duerr2018} & $1.6257$ \cite{Duerr2018} \\
$\sum C_j$ - determ. preemptive & $2 \varphi \approx 3.2361$& - & $1.8546$ \\
$\max C_j$ - deterministic & $\varphi \approx 1.6180$ & $\varphi$ \cite{Duerr2018} & $\varphi$ \cite{Duerr2018} \\
$\max C_j$ - randomized & $\frac{4}{3}$ & $\frac{4}{3}$ \cite{Duerr2018} & $\frac{4}{3}$ \cite{Duerr2018} \\
\end{tabulary}
\end{table}
For the problem of scheduling uncertain jobs with non-uniform testing times on a single machine, our results are the following: A deterministic 4-competitive algorithm for the objective of minimizing the sum of completion times and a randomized 3.3794-competitive algorithm for the same objective. If we allow preemption - that is, to cancel the execution of a job at any time and start working on a different job - then we can improve the deterministic case to be $2\varphi$-competitive. Here, $\varphi \approx 1.6180$ is the golden ratio.
For the objective of minimizing the makespan, we adopt and extend the ideas of D\"urr \text{et al.\ } \cite{Duerr2018} to provide a tight $\varphi$-competitive algorithm in the deterministic case and a tight $\frac{4}{3}$-competitive algorithm in the randomized case.
Our approaches handle non-uniform testing times in a novel fashion distinct from the methods of \cite{Duerr2018}. As we show in Appendix \ref{append:small_upper_limits}, the idea of scheduling untested jobs with small upper bounds in the beginning of the schedule, which works well in the uniform case, fails to generalize to non-uniform tests. Additionally, describing parameterized worst-case instances becomes intangible in the presence of an arbitrary number of different testing times.
In place of these methods, we compute job completion times by cross-e\-xa\-min\-ing contributions of other jobs in the schedule. We determine tests based on the ratio between the upper bound and the given test time and pay specific attention to sorting the involved executions and tests in an suitable way.
The paper is structured as follows: Sections \ref{sec:deterministic} and \ref{sec:randomized} examine the deterministic and randomized cases respectively. Various algorithms are presented and their competitive ratios proven. We extend the optimal results for the objective of minimizing the makespan from the uniform case to general testing times in Section \ref{sec:makespan}. Finally, we conclude with some open problems.
\section{Deterministic Setting}
\label{sec:deterministic}
In this section, we introduce our basic algorithm and prove deterministic upper bounds for the non-preemptive as well as the preemptive case. The basic structure introduced in Section \ref{subsec:basic_alg_and_proof} works as a framework for other algorithms presented later. We give a detailed analysis of the deterministic algorithm and prove that it is $4$-competitive if parameters are chosen accordingly. In Section \ref{subsec:det_preemption} we prove that an algorithm for the preemptive case is 3.2361-competitive and that no preemptive algorithm can have a ratio better than $1.8546$.
\subsection{Basic Algorithm and Proof of 4-Competitiveness}
\label{subsec:basic_alg_and_proof}
We now present the elemental framework of our algorithm, which we call \emph{$(\alpha,\beta)$-SORT}. As input, the algorithm has two real parameters, $\alpha \ge 1$ and $\beta \ge 1$.
\begin{algorithm}[ht]
$T \leftarrow \emptyset$, $N \leftarrow \emptyset$, $\sigma_j \equiv 0$\;
\ForEach{$j\in[m]$}{
\eIf{$u_j \ge \alpha t_j$}
{add $j$ to $T$\; set $\sigma_j \leftarrow \beta t_j$\;}
{add $j$ to $N$\; set $\sigma_j \leftarrow u_j$\;}
}
\While{$N \cup T \neq \emptyset$}{
choose $j_{\min} \in \argmin_{j \in N \cup T} \sigma_j$\;
\uIf{$j_{\min} \in N$}{execute $j_{\min}$ untested\; remove $j_{\min}$ from $N$\;}
\uElseIf{$j_{\min} \in T$}{
\eIf{$j_{\min}$ not tested}
{test $j_{\min}$\; set $\sigma_{j_{\min}} \leftarrow p_{j_{\min}}$\;}
{execute $j_{\min}$\; remove $j_{\min}$ from $T$\;}
}
}
\caption{$(\alpha,\beta)$-SORT}
\label{alg:alpha_beta_sort}
\end{algorithm}
The algorithm is divided into two phases. First, we decide for each job whether we test this job or not based on the ratio $\frac{u_j}{t_j}$. This gives us a partition of $[m]$ into the disjoint sets ${T = \{j\ \in [m]: \text{ALG tests } j\}}$ and $N = \{j\ \in [m]: \text{ALG runs } j \text{ untested}\}$. In the second phase, we always attend to the job $j_{\min}$ with the current smallest \emph{scaling time} $\sigma_j$. The scaling time is the time needed for the next step of executing $j$:
\begin{itemize}
\item If $j$ is in $N$, then $\sigma_j = u_j$.
\item If $j$ is in $T$ and has not been tested, then $\sigma_j = \beta t_j$.
\item If $j$ is in $T$ and has already been tested, then $\sigma_j = p_j$.
\end{itemize}
Note that in the second case above, we 'stretch' the scaling time by multiplying with ${\beta \ge 1}$. The intention behind this stretching is that testing a job, unlike executing it, does not immediately lead to a job being completed. Therefore the parameter $\beta$ artificially lowers the relevance of testing in the ordering of our algorithm. Note that the actual time needed for testing remains $t_j$.
In the following, we show that the above algorithm achieves a provably good competitive ratio. The parameters are kept general in the proof and are then optimized in a final step. We present the computations with general parameters for a clearer picture of the proof structure, which we will reuse in later sections. In the final optimization step it will turn out that setting $\alpha = \beta = 1$ yields a best-possible competitive ratio of $4$.
\begin{theorem}
\label{thm:1_1_SORT}
The $(1,1)$-SORT algorithm is $4$-competitive for the objective of minimizing the sum of completion times.
\end{theorem}
\begin{proof}
For the purpose of estimating the algorithmic result against the optimum, let $\rho_j := {\min(u_j,t_j + p_j)}$ be the optimal running time of job $j$. Without loss of generality, we order the jobs \text{s.t.\ } ${\rho_1 \ge \hdots \ge \rho_n}$. Hence the objective value of the optimum is
\begin{equation}
\label{eq:opt_value}
\OPT = \sum_{j=1}^n j \cdot \rho_j
\end{equation}
Additionally, let
\begin{equation}
\label{eq:alg_proc_time}
p_j^A := \begin{cases}
t_j + p_j & \text{if } j \in T,\\
u_j & \text{if } j \in N,
\end{cases}
\end{equation}
be the \emph{algorithmic running time} of $j$, i.e.\ the time the algorithm spends on running job $j$.
We start our analysis by comparing $p_j^A$ to the optimal runtime $\rho_j$ for a single job, summarized in the following Proposition:
\begin{proposition}
\label{prop:claims_case_dist}
\begin{enumerate}[label=\emph{(\alph*)}]
\item $\forall j \in T$: $t_j \le \rho_j$, $p_j \le \rho_j$
\item $\forall j \in T$: $p_j^A \le \left(1+\frac{1}{\alpha}\right) \rho_j$
\item $\forall j \in N$: $p_j^A \le \alpha \rho_j$
\end{enumerate}
\end{proposition}
Part (a) directly estimates testing and running times of tested jobs against the values of the optimum. We will use this extensively when computing the completion time of the jobs.
The proof of parts (b) and (c) is very similar to the proof of Theorem 14 in \cite{Duerr2018} for uniform testing times. We refer to the appendix for a complete write-down of the proof. Note that instead of considering a single bound, we split the upper bound of the algorithmic running time $p_j^A$ into different results for tested (b) and untested jobs (c). This allows us to differentiate between different cases in the proof of Lemma \ref{lem:contribution_lemma} in more detail. We will often make use of this Proposition to upper bound the algorithmic running time in later sections.
To obtain an estimate of the completion time $C_j$, we consider the \emph{contribution} $c(k,j)$ of all jobs $k \in [n]$ to $C_j$. We define $c(k,j)$ to be the amount of time the algorithm spends scheduling job $k$ before the completion of $j$. Obviously it holds that $c(k,j) \le p_k^A$. The following central lemma computes an improved upper bound on the contribution $c(k,j)$, using a rigorous case distinction over all possible configurations of $k$ and $j$:
\begin{lemma}[Contribution Lemma]
\label{lem:contribution_lemma}
Let $j \in [n]$ be a given job. The completion time of $j$ can be written as
\begin{equation*}
C_j = \sum_{k \in [n]} c(k,j).
\end{equation*}
Additionally, for the contribution of $k$ to $j$ it holds that
\begin{equation*}
c(k,j) \le \max\left(\left(1+\frac{1}{\beta}\right)\alpha,1+\frac{1}{\alpha},1+\beta\right) \rho_j.
\end{equation*}
\end{lemma}
Refer to Appendix \ref{append:contribution_lemma} for the proof. Depending on whether $j$ and $k$ are tested or not, the lemma computes various upper bounds on the contribution using estimates from Proposition \ref{prop:claims_case_dist}. Finally, the given bound on $c(k,j)$ is achieved by taking the maximum over the different cases.
Recall that the jobs are ordered by non-increasing optimal execution times $\rho_j$, which by Proposition \ref{prop:claims_case_dist} are directly tied to the algorithmic running times. Hence, the jobs $k$ with small indices are the 'bad' jobs with possibly large running times. For jobs with $k \le j$ we therefore use the independent upper bound from the Contribution Lemma. Jobs with large indices $k>j$ are handled separately and we directly estimate them using their running time $p_k^A$.
By Lemma \ref{lem:contribution_lemma} and Proposition \ref{prop:claims_case_dist}(b),(c) we have
\begin{equation*}
\begin{aligned}
C_j &= \sum_{k>j} c(k,j) + \sum_{k \le j} c(k,j)\\
&\le \sum_{k>j} p_k^A + \sum_{k \le j} \max\left(\left(1+\frac{1}{\beta}\right)\alpha,1+\frac{1}{\alpha},1+\beta\right) \rho_j\\
&= \sum_{k>j} \max\left(\alpha,1+\frac{1}{\alpha}\right) \rho_k + \max\left(\left(1+\frac{1}{\beta}\right)\alpha,1+\frac{1}{\alpha},1+\beta\right) j \cdot \rho_j.\\
\end{aligned}
\end{equation*}
Finally, we sum over all jobs $j$:
\begin{equation*}
\begin{aligned}
\sum_{j=1}^n C_j &= \sum_{j=1}^n \sum_{k=j+1}^n \max\left(\alpha,1+\frac{1}{\alpha}\right) \rho_k\\
&\qquad + \sum_{j=1}^n \max\left(\left(1+\frac{1}{\beta}\right)\alpha,1+\frac{1}{\alpha},1+\beta\right) j \cdot \rho_j\\
&= \max\left(\alpha,1+\frac{1}{\alpha}\right) \sum_{j=1}^n (j-1) \rho_j\\
&\qquad + \max\left(\left(1+\frac{1}{\beta}\right)\alpha,1+\frac{1}{\alpha},1+\beta\right) \sum_{j=1}^n j \cdot \rho_j\\
&\le \underbrace{ \left( \max\left(\alpha,1+\frac{1}{\alpha}\right) + \max\left(\left(1+\frac{1}{\beta}\right)\alpha,1+\frac{1}{\alpha},1+\beta\right) \right) }_{=: f(\alpha,\beta)} \sum_{j=1}^n j \cdot \rho_j\\
&= f(\alpha,\beta) \cdot \OPT\\
\end{aligned}
\end{equation*}
Minimizing $f(\alpha,\beta)$ on the domain $\alpha,\beta \ge 1$ yields optimal parameters $\alpha = \beta = 1$ and a value of $f(1,1) = 4$. We conclude that $(1,1)$-SORT is $4$-competitive.
\end{proof}
The parameter selection $\alpha = 1$, $\beta = 1$ is optimal for the closed upper bound formula we obtained in our proof. It is possible and somewhat likely that a different parameter choice leads to better overall results for the algorithm. In the optimal makespan algorithm (see Section \ref{sec:makespan}) the value of $\alpha$ is higher, suggesting that $\alpha = 1$, which leads to testing all non-trivial jobs, might not be the best choice. The problem structure and the approach by D\"urr \text{et al.\ } \cite{Duerr2018} also motivate setting $\beta$ to some higher value than $1$. For our proof, setting parameters like we did is optimal.
In the appendix, we take advantage of this somewhat unexpected parameter outcome to prove that $(1,1)$-SORT cannot be better than $3$-competitive. Additionally, we show that for \emph{any} choice of parameters, $(\alpha,\beta)$-SORT is not better than 2-competitive.
\subsection{A Deterministic Algorithm with Preemption}
\label{subsec:det_preemption}
The goal of this section is to show that if we allow jobs to be preempted there exists a $3.2361$-competitive algorithm. In his book on Scheduling, Pinedo \cite{Pinedo2016} defines preemption as follows: "The scheduler is allowed to interrupt the processing of a job (preempt) at any point in time and put a different job on the machine instead."
The idea for our algorithm in the preemptive setting is based on the so-called \emph{Round Robin} rule, which is used frequently in preemptive machine scheduling \cite[Chapters 3.7, 5.6, 12.4]{Pinedo2016}. The scheduling time frame is divided into very small equal-sized units. The Round Robin algorithm then cycles through all jobs, tending to each job for exactly one unit of time before switching to the next. It ensures that at any time the amount every job has been processed only differs by at most one time unit \cite{Pinedo2016}.
The Round Robin algorithm is typically applied when job processing times are completely unknown. In our setting, we are actually given some upper bounds for our processing times and may invest testing time to find out the actual values. Despite having more information, it turns out that treating all job processing times as unknown in a Round Robin setting gives a provably good result. The only way we employ upper bounds and testing times is again to decide which jobs will be tested and which will not. We again do this at the beginning of our schedule for all given jobs. The rule to decide testing is exactly the same as in the first phase of Algorithm \ref{alg:alpha_beta_sort}: If $u_j/t_j \ge \alpha$, then test $j$, otherwise run $j$ untested. Again, $\alpha$ is a parameter that is to be determined. It will turn out that setting $\alpha = \varphi$ gives the best result.
The pseudo-code for the \emph{Golden Round Robin} algorithm is given in Algorithm~\ref{alg:golden-round-robin}.
\begin{algorithm}[ht]
$T \leftarrow \emptyset$, $N \leftarrow \emptyset$, $\sigma_j \equiv 0$\;
\ForEach{$j\in[m]$}{
\eIf{$u_j \ge \varphi t_j$}
{add $j$ to $T$\; set $\sigma_j \leftarrow t_j$\;}
{add $j$ to $N$\; set $\sigma_j \leftarrow u_j$\;}
}
\While{$\exists j\in[m]$ not completely scheduled}{
run Round Robin on all jobs using $\sigma_j$ as their processing time\;
let $j_{\min}$ be the first job to finish during the current execution\;
\If{$j_{\min} \in T$ and $j_{\min}$ tested but not executed}{set $\sigma_{j_{\min}} \leftarrow p_{j_{\min}}$ and keep $j_{\min}$ in the Round Robin rotation\;}
}
\caption{Golden Round Robin}
\label{alg:golden-round-robin}
\end{algorithm}
Essentially, the algorithm first decides for all jobs whether to test them and then runs a regular Round Robin scheme on the algorithmic testing time $p_j^A$, which is defined as in \eqref{eq:alg_proc_time}.
\begin{theorem}
\label{thm:det_preempt}
The Golden Round Robin algorithm is $3.2361$-competitive in the preemptive setting for the objective of minimizing the sum of completion times. This analysis is tight.
\end{theorem}
We only provide a sketch of the proof here, the full proof can be found in Appendix \ref{append:det_preemption}.
\begin{proof}[Proof sketch]
We set $\alpha = \varphi$ and use Proposition \ref{prop:claims_case_dist}(b),(c) to bound the algorithmic running time $p_j^A$ of a job $j$ by its optimal running time $\rho_j$.
\begin{equation*}
p_j^A \le \varphi \rho_j.
\end{equation*}
We then compute the contribution of a job $k$ to a fixed job $j$ by grouping jobs based on their finishing order in the schedule. This allows us to estimate the completion time of job $j$:
\begin{equation*}
C_j \le \sum_{k > j} p_k^A + j \cdot p_j^A
\end{equation*}
Finally, we sum over all jobs to receive $\ALG \le 2\varphi \cdot \OPT$.
To show that the analysis is tight, we provide an example where the algorithmic solution has a value of $2\varphi \cdot \OPT$ if we let the number of jobs approach infinity.
\end{proof}
The following theorem additionally provides a lower bound for the deterministic preemptive setting, giving us a first simple lower bound for this case. The proof is based on the lower bound provided in \cite{Duerr2018} for the deterministic non-preemptive case. We refer to Appendix \ref{append:lower_bound_preempt} for this proof.
\begin{theorem}
\label{thm:lower_bound_preempt}
No algorithm in the preemptive deterministic setting can be better than $1.8546$-competitive.
\end{theorem}
\section{Randomized Setting}
\label{sec:randomized}
In this section we introduce randomness to further improve the competitive ratio of Algorithm \ref{alg:alpha_beta_sort}. There are two natural places to randomize: when deciding which jobs to test and the decision about the ordering of the jobs. These decisions directly correspond to the parameters $\alpha$ and $\beta$.
Making $\alpha$ randomized, for instance, could be achieved by defining $\alpha$ as a random variable with density function $f_\alpha: [1,\infty] \rightarrow \mathds{R}_0^+$ and testing $j$ if and only if $r_j := u_j/t_j \ge \alpha$. Then the probability for testing $j$ would be given by $p = \int_1^{r_j} f_\alpha(x) dx$. Using a random variable $\alpha$ like this would make the analysis unnecessarily complicated, therefore we directly consider the probability $p$ without defining a density, and let $p$ depend on $r_j$. This additionally allows us to compute the probability of testing \emph{independently} for each job.
Introducing randomness for $\beta$ is even harder. The choice of $\beta$ influences multiple jobs at the same time, therefore independence is hard to establish. Additionally, $\beta$ appears in the denominator of our analysis frequently, hindering computations using expected values. We therefore forgo using randomness for the $\beta$-parameter and focus on $\alpha$ in this paper. We encourage future research to try their hand at making $\beta$ random.
We give a short pseudo-code of our randomized algorithm in Algorithm \ref{alg:rand_sort}. It is given a parameter-function $p(r_j)$ and a parameter $\beta$, both of which are to be determined later.
\begin{algorithm}[ht]
$T \leftarrow \emptyset$, $N \leftarrow \emptyset$, $\sigma_j \equiv 0$\;
\ForEach{$j\in[m]$}{
add $j$ to $T$ with probability $p(r_j)$ and set $\sigma_j \leftarrow \beta t_j$\;
otherwise add it to $N$ and set $\sigma_j \leftarrow u_j$\;
}
\While{$N \cup T \neq \emptyset$}{
choose $j_{\min} \in \argmin_{j \in N \cup T} \sigma_j$\;
\uIf{$j_{\min} \in N$}{execute $j_{\min}$ untested\; remove $j_{\min}$ from $N$\;}
\uElseIf{$j_{\min} \in T$}{
\eIf{$j_{\min}$ not tested}
{test $j_{\min}$\; set $\sigma_{j_{\min}} \leftarrow p_{j_{\min}}$\;}
{execute $j_{\min}$\; remove $j_{\min}$ from $T$\;}
}
}
\caption{Randomized-SORT}
\label{alg:rand_sort}
\end{algorithm}
\begin{theorem}
\label{thm:rand_sort}
Randomized-SORT is $3.3794$-competitive for the objective of minimizing the sum of completion times.
\end{theorem}
\begin{proof}
Again, we let $\rho_1 \ge \hdots \ge \rho_n$ denote the ordered optimal running time of jobs $1,\dots,n$. The optimal objective value is given by \eqref{eq:opt_value}. Fix jobs $j$ and $k$. For easier readability, we write $p$ instead of $p(r_j)$. Since the testing decision is now done randomly, the algorithmic running time $p_j^A$ as well as the contribution $c(k,j)$ are now random variables. It holds
\begin{equation*}
p_j^A = \begin{cases}
t_j + p_j &\text{with probability } p\\
u_j &\text{with probability } 1-p\\
\end{cases}
\end{equation*}
For the values of $c(k,j)$ we consult the case distinctions from the proof of the Contribution Lemma \ref{lem:contribution_lemma}. If $j \in N$, one can easily determine that $c(k,j) \le (1+1/\beta)u_j$ for all cases. Note that for this we did not need to use the final estimates with parameter $\alpha$ from the case distinction. Therefore this upper bound holds deterministically as long as we assume $j \in N$. By extension it also trivially holds for the expectation of $c(k,j)$:
\begin{equation*}
E[c(k,j)\,|\, j \text{ untested}] \le (1+1/\beta)u_j.
\end{equation*}
Doing the same for the case distinction of $j \in T$, we get
\begin{equation*}
E[c(k,j)\,|\, j \text{ tested}] \le \max\left((1+\beta)t_j,\left(1+\frac{1}{\beta}\right)p_j,t_j+p_j\right).
\end{equation*}
For the expected value of the contribution we have by the law of total expectation:
\begin{equation*}
\begin{aligned}
E[c(k,j)] &= E[c(k,j)\,|\, j \text{ untested}] \cdot Pr[j \text{ untested}] \\
&\qquad + E[c(k,j)\,|\, j \text{ tested}] \cdot Pr[j \text{ tested}]\\
&\le \left(1+\frac{1}{\beta}\right) u_j \cdot (1-p) + \max\left((1+\beta)t_j,\left(1+\frac{1}{\beta}\right)p_j,t_j+p_j\right) \cdot p\\
\end{aligned}
\end{equation*}
Note that this estimation of the expected value is independent of any parameters of $k$. That means, for fixed $j$ we estimate the contribution to be the same for all jobs with small parameter $k \le j$. Of course, as before, for the jobs with large parameter $k > j$ we may also alternatively directly use the algorithmic runtime of $k$:
\begin{equation*}
E[c(k,j)] \le E[p_k^A].
\end{equation*}
Putting the above arguments together, we use the Contribution Lemma and linearity of expectation to estimate the completion time of $j$:
\begin{equation*}
\begin{aligned}
E[C_j] &= \sum_{j=1}^n E[c(k,j)]\\
&\le \sum_{k>j} E[p_k^A] + \sum_{k \le j} E[c(k,j)].\\
\end{aligned}
\end{equation*}
For the total objective value of the algorithm we receive again using linearity of expectation:
\begin{equation*}
\begin{aligned}
E\left[\sum_{j=1}^n C_j\right] &\le \sum_{j=1}^n (j-1) E[p_j^A] + \sum_{j=1}^n j \cdot E[c(k,j)]\\
&\le \sum_{j=1}^n (j-1) (u_j \cdot (1-p) + (t_j+p_j) \cdot p)\\
&\qquad + \sum_{j=1}^n j \Bigg( \left(1+\frac{1}{\beta}\right) u_j \cdot (1-p)\\
&\qquad + \max\left((1+\beta)t_j,\left(1+\frac{1}{\beta}\right)p_j,t_j+p_j\right) \cdot p \Bigg)\\
&\le \sum_{j=1}^n j \cdot \lambda_j(\beta,p),\\
\end{aligned}
\end{equation*}
where we define
\begin{equation*}
\begin{aligned}
\lambda_j(\beta,p) &:= \left(u_j+\left(1+\frac{1}{\beta}\right)u_j\right) \cdot (1-p)\\
&\qquad + \left(t_j+p_j + \max\left((1+\beta)t_j,\left(1+\frac{1}{\beta}\right)p_j,t_j+p_j\right)\right) \cdot p.\\
\end{aligned}
\end{equation*}
Having computed this first estimation for the objective of the algorithm, we now consider the ratio $\lambda_j(\beta,p) / \rho_j$ as a standalone. If we can prove an upper bound for this ratio, the same holds as competitive ratio for our algorithm.
Hence the goal is to choose parameters $\beta$ and $p$, where $p$ can depend on $j$, \text{s.t.\ } $\lambda_j(\beta,p) / \rho_j$ is as small as possible. In the best case, we want to compute
\begin{equation*}
\min_{\beta \ge 1, p \in [0,1]} \ \max_j \ \frac{\lambda_j (\beta, p)}{\rho_j}.
\end{equation*}
\begin{lemma}
\label{lem:min_max_rand}
There exist parameters $\hat{\beta} \ge 1$ and $\hat{p} \in [0,1]$ s.t.
\begin{equation*}
\max_j \frac{\lambda_j (\hat{\beta}, \hat{p})}{\rho_j} \le 3.3794.
\end{equation*}
\end{lemma}
The choice of parameters is given in the proof of the lemma, which can be found in Appendix \ref{append:min_max_rand}. During the proof we use computer-aided computations with Mathematica. The Mathematica code can be found in Appendix \ref{append:mathematica_code} and additionally on the webpage \cite{Albers2020} for download.
To conclude the proof of the theorem, we write
\begin{equation*}
E\left[\sum_{j=1}^n C_j\right] \le \sum_{j=1}^n j \cdot \lambda_j (\hat{\beta}, \hat{p}) \le 3.3794 \sum_{j=1}^n j \cdot \rho_j = 3.3794 \cdot \OPT.
\end{equation*}
\end{proof}
\section{Optimal Results for Minimizing the Makespan}
\label{sec:makespan}
In this section, we consider the objective of minimizing the makespan of our schedule. It turns out that we are able to prove the same tight algorithmic bounds for this objective function as D\"urr \text{et al.\ } in the unit-time testing case, both for deterministic and randomized algorithms. The decisions of the algorithms only depend on the ratio $r_j = u_j/t_j$. Refer to the appendix for the proofs.
\begin{theorem}
\label{thm:det_makespan}
The algorithm that tests job $j$ iff $r_j \ge \varphi$ is $\varphi$-competitive for the objective of minimizing the makespan. No deterministic algorithm can achieve a smaller competitive ratio.
\end{theorem}
\begin{theorem}
\label{thm:rand_makespan}
The randomized algorithm that tests job $j$ with probability $p = 1-1/(r_j^2-r_j+1)$ is $4/3$-competitive for the objective of minimizing the makespan. No randomized algorithm can achieve a smaller competitive ratio.
\end{theorem}
\section{Conclusion}
\label{sec:conclusion}
In this paper, we introduced the first algorithms for the problem of scheduling with testing on a single machine with general testing times that arises in the context of settings where a preliminary action can influence cost, duration or difficulty of a task. For the objective of minimizing the sum of completion times, we presented a $4$-approximation for the deterministic case, and a $3.3794$-approximation for the randomized case. If preemption is allowed, we can improve the deterministic result to $3.2361$. We also considered the objective of minimizing the makespan, for which we showed tight ratios of $1.618$ and $4/3$ for the deterministic and randomized cases, respectively.
Our results open promising avenues for future research, in particular tightening the gaps between our ratios and the lower bounds given by the unit case. Based on various experiments using different adversarial behaviour and multiple testing times it seems hard to force the algorithm to make mistakes that lead to worse ratios than those proven in \cite{Duerr2018} for the unit case. We conjecture that in order to achieve better lower bounds, the adversary must make live decisions based on previous choices of the algorithm, in particular depending on how much the algorithm has already tested, run or deferred jobs up to a certain point.
Further interesting directions for future work are the extension of the problem to multiple machines to consider scheduling problems like open shop, flow shop, or other parallel machine settings.
\bibliographystyle{splncs04}
|
1,108,101,564,549 | arxiv | \section{Introduction}
Gravitational-wave~(GW) echoes in the post-merger GW signal from a binary
coalescence might be a generic feature of quantum corrections at the horizon
scale~\cite{Cardoso:2016rao,Cardoso:2016oxy}, and might provide a smoking-gun
signature of exotic compact objects~(ECOs) and of exotic states of matter in
ultracompact stars~\cite{Ferrari:2000sr,Pani:2018flj} (for a review,
see~\cite{Cardoso:2017cqb,Cardoso:2017njb}).
In the last two years, tentative evidence for echoes in the combined LIGO/Virgo
binary black-hole~(BH) events have been
reported~\cite{Abedi:2016hgu,Conklin:2017lwb} with controversial
results~\cite{Ashton:2016xff,Abedi:2017isz,Westerweck:2017hus,Abedi:2018pst}.
Recently, a tentative detection of echoes in the post-merger signal of
neutron-star binary coalescence GW170817~\cite{TheLIGOScientific:2017qsa} has
been claimed at $4.2\sigma$ confidence level~\cite{Abedi:2018npz}, but
such a strong claim is yet to be confirmed/disproved by an independent
analysis. The
stochastic background produced by ``echoing remnants''~\cite{Du:2018cmp} and
spinning ECOs~\cite{Barausse:2018vdb} has been also studied recently.
While model-independent~\cite{Abedi:2018npz} and burst~\cite{Tsang:2018uie}
searches can be performed without knowing the details of the echo waveform, the
possibility of extracting as much information as possible from post-merger
events relies on one's ability to model the signal accurately. Furthermore,
using an accurate template is crucial for model selection and to discriminate
the origin of the echoes in case of a detection.
In the last year, there has been considerable progress in modeling the echo
waveform~\cite{Nakano:2017fvh,Mark:2017dnq,Maselli:2017tfq,Bueno:2017hyj,
Wang:2018mlp,Correia:2018apm,Wang:2018gin}, but the proposed approaches are
sub-optimal, because either they are based on analytical templates not
necessarily
related to the physical properties and parameters of the remnant, or they rely
on numerical waveforms which are inconvenient for direct searches through
matched filters.
In this work, we take the first step to overcome these limitations by building
an \emph{analytical} template directly anchored to the physical properties of a
given ECO model.
As we shall discuss, the template captures the rich phenomenology of the GW
echo signal, including amplitude and frequency modulation, which arise from the
physical origin of the echoes, namely radiation that bounces back and forth
between the object and the
photon-sphere~\cite{Cardoso:2014sna}, slowly
leaking to infinity through wave
tunneling~\cite{Cardoso:2016rao,Cardoso:2016oxy,Cardoso:2017cqb,Cardoso:2017njb}
.
As an illustration and anticipation of our results, in Fig.~\ref{fig:SNR} we
compare the ringdown$+$echo signal derived below against the power spectral
densities of current (aLIGO at design sensitivity~\cite{zerodet}) and future
(Einstein~Telescope~\cite{Punturo:2010zz},
Cosmic~Explorer~\cite{PhysRevD.91.082001}, LISA~\cite{LISA})
GW interferometers. Details on the template are provided in Sec.~\ref{sec:Setup}.
A preliminary parameter
estimation using current and future GW detectors is performed in
Sec.~\ref{sec:bounds}.
We conclude in Sec.~\ref{sec:Discussion} with future prospects.
Throughout this work, we use $G=c=1$ units.
\begin{figure*}[th]
\centering
\includegraphics[width=0.475\textwidth]{SNR_groundNC.pdf}
\includegraphics[width=0.5\textwidth]{SNR_LISANC.pdf}
\caption{Illustrative comparison between the ringdown$+$echo signal derived in
this work and the
power spectral densities of various interferometers as functions of the GW
frequency $f$. Left panel: we
considered an object with $M=30M_\odot$, at a distance of $D=400\,{\rm Mpc}$,
with compactness parameter $d=100M$ (roughly corresponding to near-horizon
quantum corrections, see below) and various values of the reflectivity
coefficient ${\cal R}$
(${\cal R}=0$ corresponds to the pure BH ringdown template). The sensitivity
curves
refer to aLIGO with its anticipated design-sensitivity
\texttt{ZERO\_DET\_high\_P} configuration~\cite{zerodet}, Cosmic~Explorer in
the
narrow band variant~\cite{Evans:2016mbw,Essick:2017wyl}, and Einstein Telescope
in its \texttt{ET-D} configuration~\cite{Hild:2010id}.
Right panels: the echo signal is compared to the recently proposed LISA's noise
spectral density~\cite{LISA}. We considered an object with $M=10^{6}M_\odot$,
at
a distance of $D=100\,{\rm Gpc}$ (corresponding to cosmological
redshift $\approx 10$), and with $d=100M$. Each small panel corresponds to a
different
value of ${\cal R}$.
For simplicity in all panels we neglected corrections due to the geometry of
the detector, sky averaging, and cosmological effects, and we assumed ${\cal
A}\sim M/D$ for the amplitude of the ringdown signal. Details are given in the
main text.} \label{fig:SNR}
\end{figure*}
\section{Setup}~\label{sec:Setup}
As a first step, we focus on nonspinning models, the extension to spinning
objects is underway.
Our approach is based on the analytical approximation of perturbations of the
Schwarzschild geometry in terms of the P\"oschl-Teller
potential~\cite{Poschl:1933zz,Ferrari:1984zz} and on the framework developed in
Ref.~\cite{Mark:2017dnq}, in which the echo signal is written in terms of a
transfer function that reprocesses the BH response at the horizon.
For the busy reader, our final template is provided in a ready-to-be-used form
in Eq.~\eqref{FINALTEMPLATE} and in a supplemental {\scshape
Mathematica}\textsuperscript{\textregistered} notebook~\cite{webpage}.
\subsection{An analytical template for GW echoes}
We model the stationary ECO (see Fig.~\ref{fig:setting}) with a background
geometry described, when $r>r_0$,
by the Schwarzschild metric,
\begin{equation}
ds^2=-Adt^2+A^{-1}dr^2+r^2d\Omega^2\,,
\end{equation}
with $A(r)=1-2M/r$, $M$ and $r_0$ being the total mass and the radius of the
object in Schwarzschild coordinates, respectively. At $r=r_0$, we assume the
presence of a membrane, the properties of which are parametrized by a complex
and (generically) frequency-dependent reflectivity
coefficient ${\cal R}$~\cite{Mark:2017dnq,Maggio:2017ivp}.
\begin{figure}[th]
\centering
\includegraphics[width=0.49\textwidth]{setting.pdf}
\caption{Schematic description of our ECO model (adapted from
Ref.~\cite{Mark:2017dnq}).}\label{fig:setting}
\end{figure}
This model is well suited to
describe near-horizon quantum structures (which belong to the \emph{ClePhO}
category introduced in Refs.~\cite{Cardoso:2017cqb,Cardoso:2017njb}).
After carrying out a Fourier transform and a spherical-harmonics decomposition,
various classes of perturbations of the background metric are described by a
master equation
\begin{equation}
\left[\frac{\partial^2}{\partial
x^2}+\omega^2-V_{sl}(r)\right]\tilde\Psi(\omega,x)=\tilde S(\omega,x)
\,,\label{master}
\end{equation}
where
\begin{equation}\label{eq:tortoise}
x = r + 2 M \log{ \left(\frac{r}{2 M} - 1 \right)}
\end{equation}
defines\footnote{We note that our definition of $x$ differs by a constant
term $-2M\log2$ relative to the one adopted in Ref.~\cite{Mark:2017dnq}.
\label{coordinate}} the tortoise coordinate $x$ of the Schwarzschild metric,
$l$
is the multipolar index, $s$ identifies the type of the perturbation, and
$\tilde S$ is a source term.
We assume that the surface of the object in tortoise coordinates is located near
the would-be horizon at $x_0=x(r_0)\ll - M$, as expected for near-horizon
quantum
corrections~\cite{Cardoso:2016rao,Cardoso:2016oxy,Cardoso:2017cqb,
Cardoso:2017njb}.
The potential reads
\begin{eqnarray}
V_{sl}(r)&=&A(r)\left(\frac{l(l+1)}{r^2}+\frac{1-s^2}{r}A'(r)\right)\,,
\label{potentialmaster}
\end{eqnarray}
where the prime denotes derivative with respect to the coordinate $r$. In the
above potential, $l\geq s$ with $s=0,1$ for test
Klein-Gordon and Maxwell fields, respectively, whereas $s=2$ for axial
gravitational perturbations (see Fig.~\ref{fig:PT}). Also polar gravitational
perturbations are described by Eq.~\eqref{master}, but in this case the
potential reads
\begin{eqnarray}
V_{2l}^{\rm P}(r)&=&2A\left[\frac{9 M^3+9 M^2 r \Lambda +3 M r^2 \Lambda ^2+r^3
\Lambda ^2
(1+\Lambda )}{r^3 (3 M+r \Lambda )^2}\right]\,,\nonumber\\\label{potentialPolar}
\end{eqnarray}
with $\Lambda=(l-1)(l+2)/2$ and $l\geq2$.
While axial and polar perturbations of a Schwarzschild BH are
isospectral~\cite{Chandra}, this property is generically broken for
ECOs~\cite{Cardoso:2017cqb,Cardoso:2017njb}.
\subsubsection{Transfer function}
By using Green's function techniques, Mark et al.~\cite{Mark:2017dnq} showed
that the solution of Eq.~\eqref{master} at infinity reads $\tilde
\Psi(\omega,x\to\infty)\sim \tilde Z^{+}(\omega) e^{i\omega x}$, with
\begin{equation}
\tilde Z^{+}(\omega) = \tilde Z_{\rm BH}^{+}(\omega)+ {\cal K}(\omega) \tilde
Z_{\rm BH}^{-}(\omega)\,. \label{signalomega}
\end{equation}
In the above equation, $\tilde Z_{\rm BH}^{\pm}$ are the responses of a
Schwarzschild BH (at infinity and near the horizon, for the plus and minus
signs, respectively) to the source $\tilde S$,
\begin{equation}
\tilde Z_{\rm BH}^{\pm}(\omega) = \int_{-\infty}^{+\infty}dx \frac{\tilde S
\tilde \Psi_{\mp} }{W_{\rm BH}}\,, \label{ZBH}
\end{equation}
where $W_{\rm BH}=\frac{d\tilde \Psi_+}{dx}\tilde \Psi_- -\tilde
\Psi_+ \frac{d\tilde \Psi_-}{dx}$ is the Wronskian, and $\tilde \Psi_\pm$ are
the solutions of the homogeneous equation associated
to Eq.~\eqref{master} such that
\begin{equation}\label{Psip}
\tilde \Psi_+(\omega, x) \sim \begin{cases}
\displaystyle
e^{+i \omega x} & \text{ as } x \to + \infty\\
\displaystyle
B_{\rm out}(\omega)e^{+i \omega x} + B_{\rm in}(\omega) e^{- i \omega x} & \text{ as } x \to - \infty
\end{cases} \,,\\
\end{equation}
\begin{equation}
\tilde \Psi_-(\omega, x) \sim \begin{cases}
\displaystyle
A_{\rm out}(\omega)e^{+i \omega x} + A_{\rm in}(\omega) e^{-i \omega x} & \text{ as } x \to + \infty \\
\displaystyle
e^{-i \omega x} & \text{ as } x \to - \infty \\
\end{cases} \, .
\end{equation}
The details of the ECO model are all contained in the transfer function
\begin{equation}
{\cal K}(\omega)=\frac{{\cal T}_{\rm BH} {\cal R} e^{-2 i \omega x_0}}{1-{\cal
R}_{\rm BH} {\cal R} e^{-2 i \omega x_0 }}\,, \label{transfer}
\end{equation}
where ${\cal T}_{\rm BH} = 1/B_{\rm out}$ and ${\cal R}_{\rm BH} = B_{\rm
in}/B_{\rm out}$ are the transmission and reflection coefficients for waves
coming from the \emph{left} of
the BH potential barrier~\cite{Mark:2017dnq,Chandra,NovikovFrolov}, whereas
${\cal R}(\omega)$ is the reflection coefficient at the surface of the object,
defined so that
\begin{equation}\label{eq:boundary}
\tilde \Psi\to e^{-i\omega (x- x_0)} + {\cal R}(\omega) e^{i\omega (x- x_0)}\,,
\end{equation}
near the surface at $x\sim x_0$, with $|x_0|\gg M$. In
Appendix~\ref{app:optics}, we provide a heuristic derivation of
Eq.~\eqref{transfer} in terms of a geometrical optics analogy.
The above equations are subject to the constraint that the time domain
waveforms
are real, which implies
\begin{equation}\label{eq:kreal}
{\cal K}(\omega) = {\cal K}^*(-\omega^*)\,
\end{equation}
and analogous relations for the other quantities.
We notice that the signal in the frequency domain is written as a sum of the BH
response plus an extra piece proportional to $\cal R$. The poles of this latter
piece are the quasinormal modes (QNMs) of the ECO which, in general, differ
from those of the BH. Since the BH QNMs scale with $1/M$ while the ECO QNMs
scale with $1/d$, when $d\gg M$ the signal at small times is
dominated by the poles of the BH (which are no longer QNMs since they do not
satisfy the right boundary conditions, see discussion below).
\begin{figure}[th]
\centering
\includegraphics[width=0.49\textwidth]{potential_PT_four.pdf}
\caption{Comparison between the exact potential governing perturbations in a
Schwarzschild geometry (red dashed curve) and its approximation given by the
P\"{o}schl-Teller potential (PT, blue continuous curve), as explained in the
text.}\label{fig:PT}
\end{figure}
\begin{table*}[th]
\begin{tabular}{c|ccc|cccc}
\hline
\hline
Potential &$s$ &$\omega_RM$&$\omega_IM$&
$\alpha M$ & $V_0 M^2$ & $x_m/M$ & $
{\Delta\omega_I}/{\omega_I}$\\
\hline
scalar &$0$ &0.4836& -0.09676& $0.2298$
& $0.2471$ & $1.466$ & $0.1876$ \\
electromagnetic &$1$ &0.4576& -0.09500& $0.2265$ &
$0.2222$ & $1.614$ & $0.1921$ \\
axial gravitational &$2$ &0.3737& -0.08896& $0.2159$ &
$0.1513$ & $2.389$ & $0.2136$\\
polar gravitational &$2$ &0.3737& -0.08896& $0.2161$ &
$0.1513$ & $1.901$ & $0.2148$\\
\hline
\hline
\end{tabular}
\caption{Numerical values of the fitting parameters of the P\"{o}schl-Teller
potential~\eqref{PT} used in this work to approximate the exact potential [see
Fig.~\ref{fig:PT}]. Scalar, electromagnetic and axial gravitational
perturbations are described by the potential~\eqref{potentialmaster}, whereas
polar gravitational perturbations are described by the
potential~\eqref{potentialPolar}. We restrict to $l=2$. As an indicator of the
quality of the analytical approximation, in the last column we show the
relative
difference $\frac{\Delta\omega_I}{\omega_I}$ between the exact imaginary part
of
the frequency (shown in the fourth column) and that given by the
P\"{o}schl-Teller potential. The parameter
$x_m$ is expressed in terms of the tortoise coordinate defined in
Eq.~\eqref{eq:tortoise}.
}\label{tab:PT}
\end{table*}
\subsubsection{P\"{o}schl-Teller potential}
The potential~\eqref{potentialmaster} [and~\eqref{potentialPolar}] can be
approximated by the P\"{o}schl-Teller
potential~\cite{Poschl:1933zz,Ferrari:1984zz,Bueno:2017hyj}
\begin{equation}
V_{{\rm PT}}(x) = \frac{V_0}{\cosh^2 [\alpha(x - x_m)]}\,, \label{PT}
\end{equation}
where $\alpha$, $V_0$ and $x_m$ are free parameters. We chose $V_0$ and $x_m$
such that the position of the maximum and its value coincide with those of the
corresponding $V_{sl}$. The remaining parameter $\alpha$ was found by
imposing that the real part of the fundamental QNM of the
P\"{o}schl-Teller potential,
\begin{equation}
\omega_R = \sqrt{V_0 - \alpha^2/4}\,,
\end{equation}
coincides with the exact one, as found numerically~\cite{Berti:2009kk}.
The values of $\alpha$, $V_0$ and $x_m$ obtained through this procedure for
different classes of potentials are given in Table~\ref{tab:PT}. The
P\"{o}schl-Teller potential provides an excellent approximation on the left of
the potential barrier and near the maximum (see Fig.~\ref{fig:PT}), which are
crucially the most relevant regions for our model. On the right of the potential
barrier the behavior is different, since the P\"{o}schl-Teller potential decays
exponentially as $x\gg M$, whereas the exact potential decays as $\sim 1/x^2$.
We have checked that this different behavior would affect only the reprocessing
of very low frequency signals, but it is negligible for the first echoes, which
are characterized by the reprocessing of the dominant QNMs with $\omega_R M\sim
1$.
Using the approximate potential, the homogeneous equation corresponding to
Eq.~\eqref{master} can be solved analytically. The general solution of the
homogeneous problem, $\tilde\Psi_0$, can be expressed in terms of associated
Legendre functions as
\begin{equation}
\tilde\Psi_0= c_1 P_{\frac{i \omega_R }{\alpha } - \frac{1}{2}}^{\frac{i \omega
}{\alpha }}(2 \xi-1)+c_2 Q_{\frac{i \omega_R }{\alpha } - \frac{1}{2}}^{\frac{i
\omega }{\alpha }}(2 \xi-1)\,, \label{tildePsi0}
\end{equation}
where $c_1$ and $c_2$ are integration constants and $\xi = [{1 + e^{-2
\alpha(x - x_m)}}]^{-1}$ is a new variable.
Taking $c_1 = e^{i \omega x_m} \Gamma(1 - \frac{i \omega}{\alpha})$ and $c_2 =
0$, $\tilde\Psi_0$ reduces to $\tilde \Psi_+$
and we obtain
\begin{eqnarray}
{\cal T}_{\rm BH} &=& -\frac{i}{\pi} \sinh \left(\frac{\pi \omega }{\alpha
}\right)\Upsilon\,, \label{TBH}\\
{\cal R}_{\rm BH} &=&-\frac{1}{\pi }\cosh\left(\frac{\pi
\omega_R}{\alpha}\right) \Upsilon e^{2 i \omega x_m} \,, \label{RBH}
\end{eqnarray}
where we defined
\begin{equation}
\Upsilon=\Gamma\left(\frac{1}{2} -
i\frac{\omega+\omega_R}{\alpha}\right)\Gamma\left(\frac{1}{2} -
i\frac{\omega-\omega_R}{\alpha}\right)\frac{\Gamma\left(1+\frac{i\omega}{\alpha}
\right)}{\Gamma\left(1-\frac{i\omega}{\alpha}\right)}\,.\label{gamma}
\end{equation}
By replacing the above expressions in Eq.~\eqref{transfer}, we finally obtain
\begin{equation}
{\cal K}(\omega)= -i\frac{e^{2i\omega d}{\cal
R}(\omega)\Upsilon\sinh{\left(\frac{\pi \omega}{\alpha
}\right)}}{\pi+e^{2i\omega d}{\cal R}(\omega)\cosh{(\frac{\pi
\omega_R}{\alpha})}\Upsilon}e^{-2i\omega x_m}\,, \label{Kappa}
\end{equation}
where we defined the width of the cavity\footnote{For ECOs with near-horizon
quantum structures, one expects $d\sim nM|\log(l_P/M)|$, where $l_P$ is the
Planck length and $n\sim {\cal O}(1)$ depends on the
model~\cite{Cardoso:2016oxy,Cardoso:2017cqb,Cardoso:2017njb}. This gives
$d/M\sim 100n$ roughly for both stellar-mass and supermassive objects.
In this
case, the redshift at the surface roughly reads $z\sim M/l_P$.}
$d=x_m-x_0>0$ (recall that $x_0<0$ and $x_m>0$), which is also related to the
redshift at the surface, $z\sim e^{d/(4M)}$ when $d\gg M$.
As we shall show later, the final signal depends only on the physical quantity
$d$; the dependence on $x_m$ in Eq.~\eqref{Kappa} will disappear from the final
result\footnote{This must be the case and, in fact, it is possible to add a
phase term in the definitions of the transfer function and of the BH response
at
the horizon so that $x_m$ never appears in the equations. We chose not to do
so in order to follow the notation of Ref.~\cite{Mark:2017dnq} more closely.
Equivalently, in a
coordinate system such that the maximum of the potential sits at the origin,
all results would depend only on the physical quantity $d$.}.
In summary, for a given choice of ${\cal R}(\omega)$, the above relations yield
an \emph{analytical} approximation to the transfer function ${\cal K}$. In
Appendix~\ref{app:comparison}, we compare some results for the
approximate analytical expressions of ${\cal T}_{\rm BH}$, ${\cal R}_{\rm BH}$
and ${\cal K}$, with their exact numerical counterparts as computed in
Ref.~\cite{Mark:2017dnq}.
\subsubsection{Modeling the BH response}
The inverse Fourier transform of the BH response $\tilde Z_{\rm
BH}^\pm(\omega)$
[see Eq.~\eqref{ZBH}] can be deformed in the complex frequency plane, yielding
three contributions~\cite{Leaver:1986gd,Berti:2009kk}: (i) the high-frequency
arcs that govern the prompt response; (ii) a sum-over-residues at the poles of
the complex frequency plane (defined by $W_{\rm BH}=0$), which correspond to
the
QNMs and that dominate the signal at intermediate times; (iii) a branch cut on
the negative half of the imaginary axis, giving rise to late-time tails due to
backscattering
off the background curvature.
The post-merger ringdown signal is very well approximated by the second
contribution only, so that for most astrophysical applications the BH response
at infinity can be written as a superposition of QNMs~\cite{Berti:2009kk}.
Considering for simplicity only the dominant mode, one gets
\begin{equation}
Z_{\rm BH}^{+} (t)\sim \mathcal{A}\, \theta(t - t_0) \cos(\omega_R t+\phi)
e^{-t/\tau}\,, \label{ZBHplus}
\end{equation}
where the complex QNM frequency reads $\omega_R+i\omega_I$, $\tau=-1/\omega_I$,
${\cal A}$ and $\phi$ are the amplitude and the phase, respectively, and $t_0$
parametrizes the starting time of the ringdown. In the above expression we have
defined ${\cal A}= M {\cal A}_{lmn} S_{lmn}/D$, where $M$ is the mass of the
object, $D$ is the distance of the source, $A_{lmn}$ is the amplitude of the
BH QNM with quantum numbers $(l,m,n)=(2,2,0)$, and $S_{lmn}$ are the
corresponding
spin-weighted spheroidal harmonics~\cite{Berti:2005ys}.
Given the BH response in the time domain, the frequency-domain waveform is
obtained through the Fourier transform,
\begin{equation}
\tilde Z_{\rm BH}^{\pm}(\omega) = \int_{- \infty}^{+ \infty} \frac{dt}{\sqrt{2
\pi}} Z_{\rm BH}^{\pm}(t) e^{i \omega t}.
\end{equation}
For the BH response at infinity, the Fourier transform yields
\begin{equation}\label{eq:bhtemplateINF}
\tilde Z_{\rm BH}^{+}(\omega) \sim \frac{{\cal A}}{2\sqrt{2\pi}} \left(
\frac{\alpha_1}{\omega - \Omega_+} + \frac{\alpha_2}{\omega - \Omega_-}
\right)e^{i \omega t_0},
\end{equation}
where $\Omega_\pm = \pm \omega_R + i \omega_I$, $\alpha_1=-ie^{-i(\phi+t_0
\Omega_+)}$ and $\alpha_2=-{\alpha_1}^*$.
It is worth noting that the complex poles are the same for $Z_{\rm BH}^+$ and
for $Z_{\rm BH}^-$, since those are defined by $W_{\rm BH}=0$. This suggests
that the BH response near the horizon might also be described by a
superposition
of QNMs\footnote{Note that, since the P\"oschl-Teller potential is symmetric
under reflections around its maximum, within our approximation the BH response
near the horizon would be equivalent to the response at infinity provided the
source term in Eq.~\eqref{ZBH} has the same symmetry of the potential. However,
this is generally not the case.\label{notasimmetria}}.
\begin{table}[th]
\begin{tabular}{ll}
\hline
\hline
$M$ & total mass of the object \\
${\cal A}$ & amplitude of the BH ringdown \\
$\phi$ & phase of the BH ringdown \\
$t_0$ & starting time of the BH ringdown \\
\hline
$d$ & width of the cavity ($z\sim e^{d/(4M)}$) \\
${\cal R}(\omega)$ & reflection coefficient at the surface \\
\hline
\hline
\end{tabular}
\caption{Parameters of the ringdown$+$echo template presented in this work. The
first four parameters characterize the ordinary BH ringdown. The parameter $z$
is the gravitational redshift at the ECO surface.} \label{tab:template}
\end{table}
More in general, we are interested in perturbations produced by sources
localized near the object, as expected for merger remnants. If we assume $\tilde
S(\omega,x)=C(\omega) \delta(x-x_s)$,
for any $x_s$ well inside the cavity (where $V(x_s)\approx0$), from
Eq.~\eqref{ZBH} we can derive a relation between $\tilde
Z_{\rm BH}^{+}$ and $\tilde Z_{\rm BH}^{-}$, namely:
\begin{eqnarray}
\tilde Z_{\rm BH}^- = e^{2i\omega x_s}\left(1+{\cal R}_{\rm BH}e^{-2i\omega
x_s} \right)\frac{\tilde Z_{\rm BH}^+}{{\cal T}_{\rm BH}}\,.
\label{eq:bhtemplateHOR}\\ \nonumber
\end{eqnarray}
Remarkably, the above relation is independent of the function $C(\omega)$
characterizing the source.
Thus, $\tilde Z_{\rm BH}^-$ can be computed analytically using the expressions
for ${\cal R}_{\rm BH}$, ${\cal T}_{\rm BH}$ and $\tilde Z_{\rm BH}^+$ derived
above. As
expected, the quantity ${\cal K} \tilde Z_{\rm BH}^-$ [appearing in
Eq.~\eqref{signalomega}] is independent of $x_m$ and depends only on the
physical width of the cavity $d$.
An expression similar to Eq.~\eqref{eq:bhtemplateHOR} can be obtained for a
source localized\footnote{Another particular case is
when the
source is
localized well outside the
light ring ($x_s\gg M$). In this case we obtain
\begin{equation}\label{greenout}
\tilde Z_{\rm BH}^+ = e^{-2i\omega x_s}\left(1+\hat{\cal R}_{\rm
BH}e^{2i\omega x_s} \right)\frac{\tilde Z_{\rm BH}^-}{\hat{\cal T}_{\rm BH}}\,,
\end{equation}
where $\hat{\cal R}_{\rm BH}$ and $\hat{\cal T}_{\rm BH}$ are respectively the
reflection and transmission coefficients for the scattering of left-moving
waves
\emph{from} infinity. Note that, since the P\"oschl-Teller
potential is symmetric around its maximum, within our framework these
coefficients coincide with ${\cal R}_{\rm BH}$ and ${\cal T}_{\rm BH}$,
respectively, modulo a phase difference.} at any point $x_s$ by
using the explicit form of
Eq.~\eqref{tildePsi0}~\cite{thesis}. The generic expression is given in
Appendix~\ref{app:genericsource}.
In what follows we will consider a source localized near the surface, $x_s
\approx x_0$, which should provide a model for post-merger excitations.
\subsubsection{Ringdown$+$echo template}
Putting together all the ingredients previously derived, the final expression
of
the full ECO response reads
\begin{equation}
\tilde Z^+(\omega) = Z_{\rm BH}^+(\omega) \left[1+{\cal R}\frac{\pi-e^{2 i
\omega d}\Upsilon\cosh\left(\frac{\pi \omega_R}{\alpha}\right)}{\pi+e^{2 i
\omega d}{\cal R}\Upsilon\cosh\left(\frac{\pi \omega_R}{\alpha}\right)} \right]
\,,
\label{FINALTEMPLATEGEN}
\end{equation}
where $\Upsilon$ is defined in Eq.~\eqref{gamma}.
In this form, the signal is written as the reprocessing of a generic BH response
$Z_{\rm BH}^+(\omega)$. Note that the BH response $Z_{\rm BH}^+(\omega)$ might
not necessarily be restricted to a ringdown signal in the frequency domain. In
general, if the remnant is an ECO, one might expect that the post-merger phase
can be obtained through the reprocessing of the ringdown part \emph{and} of the
late-merger phase, since already during the formation of the final ECO radiation
might be trapped by an effective photon-sphere.
If instead we model the BH response $Z_{\rm BH}^+(\omega)$ through a single
ringdown [Eq.~\eqref{eq:bhtemplateINF}], the explicit final form of the
ringdown$+$echo
signal reads
\begin{widetext}
\begin{equation}
\tilde Z^+(\omega) = \sqrt{\frac{\pi}{2}}{\cal A} \frac{
e^{i(\omega-\omega_I)t_0} (1+{\cal R}) \Gamma\left(1-\frac{i \omega }{\alpha
}\right) (\omega_R\sin(\omega_R t_0+\phi)+i (\omega +i \omega_I)
\cos(\omega_R t_0+\phi)}{\left[(\omega +i \omega_I)^2-\omega_R^2\right]
\left[\pi \Gamma\left(1-\frac{i \omega }{\alpha }\right)+e^{2 i d \omega }
{\cal R}
\cosh\left(\frac{\pi \omega_R}{\alpha}\right)\Gamma\left(\frac{1}{2} -
i\frac{\omega+\omega_R}{\alpha}\right) \Gamma\left(\frac{1}{2} -
i\frac{\omega-\omega_R}{\alpha}\right)
\Gamma\left(1+\frac{i \omega }{\alpha }\right)\right]} \,.
\label{FINALTEMPLATE}
\end{equation}
\end{widetext}
The above result is valid for any type of perturbation, provided the parameters
$\alpha$ and $V_0$ (or, equivalently, $\alpha$ and $\omega_R$) are chosen
appropriately (see Table~\ref{tab:PT}). In the next section we will use
Eq.~\eqref{FINALTEMPLATE} for polar gravitational perturbations.
For a linearly polarized wave, our template is defined by $4$ real parameters
(${\cal A}$, $\phi$, $t_0$ and $d$) plus the mass (which sets the scale for the
other dimensionful quantities) and the complex function ${\cal R}(\omega)$ that
is model dependent (see Table~\ref{tab:template}). Clearly, for ${\cal R}=0$
one
recovers a single-mode BH ringdown template in the frequency domain. The echo
contribution is fully determined only in terms of $d$ and ${\cal R}(\omega)$
once the parameters of the ordinary ringdown are known. If two polarizations
are included, the number of real parameters increases to $7$ (${\cal A}_+$,
${\cal A}_\times$, $\phi_+$, $\phi_\times$, $t_0$, $M$, $d$), plus the function
${\cal
R}(\omega)$.
The templates~\eqref{FINALTEMPLATEGEN} and~\eqref{FINALTEMPLATE} are also
publicly available in a ready-to-be-used supplemental {\scshape
Mathematica}\textsuperscript{\textregistered} notebook~\cite{webpage}.
\subsection{Properties of the template}
\subsubsection{Comparison with the numerical result}
Our final analytical template agrees remarkably well with the exact numerical
results. A representative example is shown in
Fig.~\ref{fig:comparison_signal_FD}. Here we compare the template
against the result of a
numerical integration of the Regge-Wheeler equation for a source localized
near the ECO surface. We show the second term of the right hand side of Eq.~\eqref{signalomega}, normalized by the standard BH response $Z_{\rm
BH}^+$ (since $Z_{\rm
BH}^-$ is proportional to $Z_{\rm BH}^+$, the final result is independent of the
specific BH response). The agreement is overall very good and also the
resonances are properly reproduced.
\begin{figure}[th]
\centering
\includegraphics[width=0.47\textwidth]{comparison_signal_FD.pdf}
\caption{Comparison between our analytical template and the result of a
numerical integration of the Regge-Wheeler equation. We show the
(absolute value of the) second term of the right hand side of Eq.~\eqref{signalomega}, normalized by
the standard BH response, $\tilde Z_{\rm BH}^+$, for $d\approx
20M$, $l=2$ axial perturbations, and a source localized near the ECO
surface.} \label{fig:comparison_signal_FD}
\end{figure}
\subsubsection{Time-domain echo signal}
The time-domain signal can be simply computed through an inverse Fourier
transform,
\begin{equation}
h(t) = \frac{1}{\sqrt{2 \pi}}\int_{- \infty}^{+ \infty} d\omega
\tilde{Z}^{+}(\omega) e^{-i \omega t}.
\end{equation}
In Figs.~\ref{fig:template0} and~\ref{fig:template2}, we present a
representative slideshow of our template for different values of $d$, ${\cal
R}(\omega)$, and for scalar and polar gravitational perturbations,
respectively.
For simplicity, we consider ${\cal R}(\omega)={\rm const}$.
The time-domain waveform contains all the features previously reported
for the echo signal, in particular amplitude and frequency
modulation
\cite{Cardoso:2016rao,Cardoso:2016oxy,Cardoso:2017cqb,Cardoso:2017njb}, and
phase inversion~\cite{Abedi:2016hgu,Wang:2018gin} of each echo relative to the
previous one due to the reflective boundary conditions.
\begin{figure*}[th]
\centering
\includegraphics[width=0.95\textwidth]{rdepen.pdf}
\caption{Examples of the ringdown$+$echo template in the time domain for
different values of $d$ and ${\cal R}(\omega)={\rm const}$ and for scalar
perturbations. From top to bottom: $d=100M$, $d=75 M$, $d= 50 M$; from left to
right: ${\cal R}=1$, ${\cal R}=0.75$, ${\cal R}=0.5$. The waveform is
normalized
to its peak value during the ringdown (not shown in the range of the $y$ axis
to better visualize the subsequent echoes).} \label{fig:template0}
\end{figure*}
\begin{figure*}[th]
\centering
\includegraphics[width=0.95\textwidth]{rdepen_gp.pdf}
\caption{Same as in Fig.~\ref{fig:template0}, but for polar gravitational
perturbations.}
\label{fig:template2}
\end{figure*}
\subsubsection{Decay of the echo amplitude in time}
Several qualitative features of the waveforms can be understood
with a simple geometrical-optics toy model presented in
Appendix~\ref{app:optics}. From this model, we expect the complex amplitude of
each echo (in the frequency domain) relative to the previous one to be
suppressed by a frequency-dependent factor ${\cal R}{\cal R}_{\rm BH}$, where
we dropped
the
phase term $e^{- 2 i \omega d}$ that accounts for the time delay between the
two.
The first echo has already a factor ${\cal R}_{\rm BH}(\omega)$, which is
essentially a low-pass filter. As shown in Fig.~\ref{fig:TBK}, ${\cal R}_{\rm
BH}(\omega) \approx 0$ for $\omega \gtrsim \omega_R$, whereas $|{\cal R}_{\rm
BH}(\omega)| \approx 1$ for $\omega \lesssim \omega_R$. Thus, for frequencies
$\omega<\omega_R$, we expect that the amplitude of the echoes in
the time domain should decrease as
\begin{equation}
A_{\rm echo}(t)\propto |{\cal R}{\cal R}_{\rm BH}|^\frac{t}{2d} \approx |{\cal
R}|^\frac{t}{2d} \,.\label{slope}
\end{equation}
As shown in Fig.~\ref{fig:slope},
Eq.~\eqref{slope} agrees almost perfectly with our numerical results in the time
domain (we expect the small departure of the line from the data to be a
consequence of the fact that ${\cal R}_{\rm BH}$ and ${\cal T}_{\rm BH}$
are not exactly constant).
\begin{figure}[th]
\centering
\includegraphics[width=0.52\textwidth]{slope.pdf}
\caption{Normalized absolute value of
$h(t)$ for scalar (top panel) and polar gravitational (bottom panel) echo
template. Continuous black lines show the slope $|h(t)|\sim 0.1|{\cal
R}|^{t/(2d)}$, see Eq.~\eqref{slope}. We set $d=75M$ and considered
${\cal R}=(1,0.75,0.5)$; different choices of the parameters give similar
results.}
\label{fig:slope}
\end{figure}
\subsubsection{Phase inversion of subsequent echoes}
The phase inversion between subsequent echoes shown in
Figs.~\ref{fig:template0} and \ref{fig:template2} can be understood by
considering the extra
factor ${\cal R}{\cal R}_{\rm BH}$ each echo has with respect to the previous
one. When ${\cal R}$ is real and positive, the phase is set by ${\cal R}_{\rm
BH}(\omega)$. Since ${\cal R}_{\rm BH}(\omega \sim 0) \approx -1$ for
low-frequency signals (which are the only ones that survive the first filtering
by ${\cal R}_{\rm BH}$), the $n$-th echo has a phase factor $\left[{\cal
R}_{\rm
BH}(\omega \sim 0)\right]^n \approx (-1)^n$. If the ECO is a wormhole, there is
no phase
inversion because in this case ${\cal R}={\cal R}_{\rm BH}
e^{-2i\omega
x_0}$~\cite{Mark:2017dnq}, so that ${\cal R}{\cal R}_{\rm BH} = {\cal R}_{\rm
BH}^2$, where again we dropped the time-shifting phase.
Likewise, if ${\cal R}$ is real and negative, the $n$-th echo has a phase
factor $\left[{\cal R}{\cal R}_{\rm
BH}\right]^n \approx 1$ for any $n$, so also in this case there is no phase
inversion. Although
not shown in Figs.~\ref{fig:template0} and \ref{fig:template2}, we have checked
that all these properties are reproduced by the time-domain templates.
Additional waveforms are provided online~\cite{webpage}.
\subsubsection{Dependence on the location of the source}
From the geometrical optics analogy of Appendix~\ref{app:optics}, we expect
that, for a source located inside the cavity at some $x_s = x_0 + {\ell}$,
the
effect of the surface will appear only after a (coordinate-time) delay of $2
{\ell}$ with respect to the RD, because of the extra time it takes for the
left-going perturbation to
reach the surface and come back.
Since the latter has a relative amplitude ${\cal R}$, the amplitude of the
prompt signal is
$\approx{\cal A}(1+{\cal R} e^{2i\omega {\ell}})$.
If $x_s \approx x_0$ (i.e., $\ell\approx0$), there is no delay between the
proper ringdown signal and the first reflection. This is consistent with the
behavior of the signal shown in Fig.~\ref{fig:SNR}: the full response at high
frequencies (i.e., those which are not reflected by the potential but only by
the surface) differs from the BH ringdown by a relative factor $1+{\cal R}$.
Note that, if $x_s=x_0$, in the Dirichlet case (${\cal R}=-1$) the prompt
signal and the reflected one interfere with opposite phase and the signal
vanishes, as clear from Eq.~\eqref{FINALTEMPLATE}.
When the source is located outside the light ring
[and consequently one needs to use Eq.~\eqref{greenout}], the frequency
content of the ECO response at infinity changes significantly. Indeed, in
this
case the low-frequency content of an incident packet would not be able to probe
the cavity and would be reflected regardless of the nature of the object and of
the boundary conditions at $x_0$. On the other hand, the very high frequency
component should pass through the light ring barrier unmodified and be
reflected only by the ECO surface.
The frequency-domain signal for a source localized at a generic location $x_s$
is given in Appendix~\ref{app:genericsource}.
\subsubsection{Energy of echo signal}
Finally, let us discuss the energy contained in the ringdown$+$echo signal.
Because of partial reflection, the energy contained in the full signal is
always
larger
than that of the ringdown itself (for ${\cal R} >0$). This is shown in
Fig.~\ref{fig:EvsR}, where
we
plot
the energy
\begin{equation}
E\propto \int_0^\infty d\omega \omega^2 |\hat Z^+|^2\,, \label{energy}
\end{equation}
normalized by the one
corresponding to the ringdown alone, $E_{\rm RD}=E({\cal R}=0)$, as a function
of the reflectivity ${\cal R}$. In the above equation, $\hat Z^+$ is the
frequency-domain full response obtained by using the Fourier transform of
\begin{equation}
Z_{\rm BH}^{+} (t)\sim \mathcal{A}\, \cos(\omega_R t+\phi) e^{-|t|/\tau}\,,
\label{ZBHplusB}
\end{equation}
rather than using Eq.~\eqref{ZBHplus}. This is the prescription used in
Ref.~\cite{Flanagan:1997sx} to compute the ringdown energy, and it circumvents
the fact that the Heaviside function in Eq.~\eqref{ZBHplus} produces a spurious
high-frequency behaviour of the energy flux, leading to infinite energy in the
ringdown signal. With the above prescription, the energy defined in
Eq.~\eqref{energy} is finite and reduces to the result of
Ref.~\cite{Flanagan:1997sx} for the BH ringdown
when ${\cal R}=0$.
\begin{figure}[th]
\centering
\includegraphics[width=0.49\textwidth]{EnergyvsRbis.pdf}
\caption{Total energy contained in the ringdown$+$echo signal normalized by
that
of the ringdown alone as a function of $\cal {R}$.
The inset shows the same quantity as a function of $1-{\cal R}$ in a
logarithmic
scale.
Note that the energy is much larger than the ringdown energy when ${\cal
R}\to1$. We set $t_0=0=\phi$ and $d=100M$; the result is independent of $d$ in
the large-$d$ limit.} \label{fig:EvsR}
\end{figure}
As shown in Fig.~\ref{fig:EvsR}, the energy contained in the ringdown$+$echo
signal depends strongly on ${\cal R}$ and can be $\approx 38$ times
larger than that of the ringdown
alone when ${\cal R}\to 1$ (the exact number depends also on $t_0$ and $\phi$).
This is due to the resonances\footnote{As shown in Fig.~\ref{fig:SNR}, when
${\cal R}=1$ the resonances are very narrow and high. However, the most
narrow resonances contribute little to the total energy. For example, in the
extreme case ${\cal R}=1$ and $d=100M$, the integral~\eqref{energy} converges
numerically when using fixed frequency steps $\Delta f\approx0.001/M$ or
smaller. This suggests that not resolving all resonances in the signal might
not
lead to a significant loss in the SNR. We are grateful to Jing Ren, whose
comment prompted this analysis.} corresponding to the low-frequency QNMs of the
ECO, that can be excited with large amplitude [see Fig.~\ref{fig:SNR}]. At high
frequency there are no resonances, but the energy flux $dE/d\omega$ is larger
than the ringdown energy flux by a factor $(1+{\cal R})^2$, in agreement with
our previous discussion.
\subsubsection{Connection with the QNMs}
The final expression~\eqref{FINALTEMPLATEGEN} also helps clarifying why the
``prompt'' ringdown signal of an ECO is identical to that of a BH even if the
BH QNMs are not part of the QNM spectrum of an
ECO~\cite{Cardoso:2016rao,Barausse:2014tra}. While this fact is easy to
understand in the time domain due to causality (in terms of time needed
for the perturbation to
probe the boundaries~\cite{Cardoso:2016rao}), it is less obvious in the
frequency domain.
The crucial point is that the BH QNMs are still poles of the ECO full
response, $\tilde Z^+(\omega)$, in the complex frequency plane, even if they
are not part of the ECO QNM spectrum. Indeed, Eq.~\eqref{FINALTEMPLATEGEN}
contains two types of complex poles: those associated with $\tilde Z_{\rm
BH}^+(\omega)$ which are the standard BH QNMs (but do not appear in the ECO QNM
spectrum), and those associated with the poles of the transfer function ${\cal
K}$, which correspond to the ECO QNMs.
The late-time signal is dominanted by the second type of poles, since the
latter have lower frequencies than the BH QNMs. On the other hand, the prompt
ringdown is dominanted by the first type of poles, i.e. by the dominant QNMs of
the corresponding BH spacetime.
\section{Projected constraints on ECOs} \label{sec:bounds}
In this section we use the template previously derived
[Eq.~\eqref{FINALTEMPLATE}] for a preliminary parameter estimation of the ECO
properties using current and future GW detectors.
We shall focus on polar gravitational perturbations with $l=2$, which are
typically the dominant ones.
As already illustrated in Fig.~\ref{fig:SNR}, the ringdown$+$echo signal
displays sharp peaks which originate from the resonances of the transfer
function ${\cal K}$ [see Fig.~\ref{fig:TBK}] and
correspond to the long-lived QNMs of the ECO. Indeed, they are very well
described by the harmonics of normal modes in a cavity of width $d$
[see Eq.~\eqref{modes}] (with a
finite resonance width given by the small imaginary part of the
mode~\cite{Cardoso:2014sna,Brito:2015oca}) and their frequency separation is
$\Delta \omega\propto 1/d$. The
relative amplitude of each resonance in the signal depends on the source, and
the dominant modes are not necessarily the fundamental
harmonics~\cite{Mark:2017dnq,Bueno:2017hyj}.
Although only indicative, Fig.~\ref{fig:SNR} already shows two important
features. First, the amplitude of the echo signal in the frequency domain can
be
\emph{larger} than that of the ringdown itself. This explains the
aforementioned
energy content of the echo signal with respect to the ordinary ringdown
(Fig.~\ref{fig:EvsR}) and suggests that GW echoes
might be detectable even when the ringdown is not (we note that this feature is
in agreement with some claims of
Refs.~\cite{Abedi:2016hgu,Conklin:2017lwb,Abedi:2018npz}).
Second, as shown in the right panels of Fig.~\ref{fig:SNR}, the amplitude of
the echo signal depends strongly on the value of ${\cal R}$ and changes by
several orders of magnitude between ${\cal R}=0.5$ and ${\cal R}=1$ (and
even more for smaller values of ${\cal R}$). This suggests that the
detectability of (or the constraints on) echoes will strongly depend on ${\cal
R}$ and would be much more feasible when ${\cal R}\approx 1$. Below we shall
quantify this expectation.
\subsection{Fisher analysis}
For simplicity, we employ a Fisher analysis, which is accurate at large
signal-to-noise ratios (SNRs) (see, e.g., Ref.~\cite{Vallisneri:2007ev}).
We shall assume that the signal
is linearly polarized; including two polarizations is a
straightforward extension.
Furthermore, since our ringdown$+$echo template was built for nonspinning
ECOs, in
principle we should also neglect the spin of the final object. However, since
the statistical errors obtained from the Fisher matrix depend on the number of
parameters, it is more realistic to assume that the template depends on the
spin
(through $\omega_R$ and $\omega_I$, taken to be those of a Kerr BH rather than
of
a Schwarzschild BH) and to perform the Fisher analysis by injecting a
vanishing value of the spin. This procedure introduces some systematics, since
we are ignoring the remaining spin dependence of the echo template.
Nonetheless,
it should provide a more reliable order-of-magnitude estimate of the
statistical errors on the ECO parameters in the spinning case.
The Fisher matrix $\Gamma$ of a template $\tilde h(f)$ for a detector with
noise
spectral density $S_n(f)$ is defined as
\begin{equation}\label{fisher}
\Gamma_{i j} = (\partial_i \tilde h, \partial_j \tilde h)= 4 \, \Re
\int_{0}^{\infty} \frac{\partial_i \tilde h^*(f) \partial_j \tilde
h(f)}{S_n(f)}
df\,,
\end{equation}
where $i,j=1,...,N$, with $N$ being the number of parameters in the template,
and $f=\omega/(2\pi)$ is the GW frequency.
The SNR $\rho$ is defined as
\begin{equation}
\rho^2 = 4 \int_{0}^{\infty} \frac{\tilde h^*(f) \tilde h(f)}{S_n(f)} df\,.
\label{SNR}
\end{equation}
The inverse of the Fisher matrix, $\Sigma_{ab}$, is the covariance matrix of
the
errors on the template's parameters: $\sigma_{i}=\sqrt{\Sigma_{ii}}$ gives the
statistical error associated with the measurement of $i$-th parameter.
We note that, to the leading order in the large SNR limit, the statistical
errors estimated through the Fisher matrix are independent of the systematic
errors arising from approximating the true signal with an imperfect theoretical
template~\cite{PhysRevD.89.104023}.
\subsubsection{Validation of the method: BH ringdown}
As a check of our computation, we have reproduced the results of the BH
ringdown
analysis performed in Ref.~\cite{Berti:2005ys}. This can be achieved by
neglecting the echo part of our template, i.e. by setting ${\cal R}=0$ or
simply considering only the first term in Eq.~\eqref{signalomega}.
We have first reproduced the analytical results presented in Appendix~B
of Ref.~\cite{Berti:2005ys} for what concerns the statistical errors on the
parameters of the ringdown waveform. We have then derived the same results
through a numerical integration of Eq.~\eqref{fisher}.
Note that Ref.~\cite{Berti:2005ys} adopted a $\delta$-approximation (i.e.,
white
noise,
$S_n(f) = {\rm const}$), which is equivalent to consider the signal as
monochromatic. In this limit, the quantity $\rho \sigma_i$ is by
construction
independent of the detector sensitivity. As a
further check, we have relaxed the $\delta$-approximation and repeated the
analysis using the recently proposed LISA's noise spectral
density~\cite{LISA}, obtaining similar results.
\subsubsection{Analysis for GW echoes}
After having validated our scheme, we computed numerically the Fisher
matrix~\eqref{fisher} with the template~\eqref{FINALTEMPLATE} using the
sensitivity curves presented in Fig.~\ref{fig:SNR} for current and future
detectors. As previously discussed, for linearly polarized waves the template
contains $5$ ringdown parameters (mass, spin, phase, amplitude, and starting
time), and two ECO parameters (the frequency-dependent reflection
coefficient ${\cal R}(\omega)$ and the width of the cavity $d$) which
fully characterize the echoes. The parameter $d$ is directly related to
physical quantities, such as the compactness of the ECO or the redshift at
the surface.
\subsection{Results}
\begin{figure*}[th]
\centering
\includegraphics[width=0.47\textwidth]{deltaRvsR.pdf}
\includegraphics[width=0.47\textwidth]{deltaDvsR.pdf}
\caption{Left panel: relative (percentage) error on the reflection coefficient,
$\Delta {\cal R} / {\cal R}$ multiplied by the SNR, as a function of ${\cal
R}$. The inset shows the
same quantity as a function of $1-{\cal R}$ in a logarithmic scale.
Right panel: same as
in the left panel but for the width of the cavity,
$\Delta (d/M)/(d/M)$. For simplicity, we assumed that ${\cal R}$
is real and positive.
In both panels we considered $M=30M_\odot$ for ground-based detectors and
$M=10^6 M_\odot$ for LISA. We assume $d=100 M$ but the errors are independent
of
$d$ in the large-$d$ limit, see Appendix~\ref{app:d}.}
\label{fig:errors}
\end{figure*}
Our main results for the statistical errors on the ECO parameters are
shown in Fig.~\ref{fig:errors}. In the large SNR limit, the errors scale as
$1/\rho$ so we present the quantity $\rho\Delta {\cal R}/{\cal R}$ (left panel)
and
$\rho\Delta (d/M)/(d/M)$ (right panel\footnote{We
adopted
dimensionless parameters to define the Fisher matrix, in particular $M/M_\odot$
and $d/M$. The
statistical error on $d$ can be easily obtained from the full
covariance matrix.}). Figure~\ref{fig:errors} shows a number of
interesting features:
\begin{itemize}
\item The relative errors are almost independent of the sensitivity curve
of the detector and only depend on the SNR. This suggests that the parameter
estimation of echoes will be only mildly sensitive to the details of future
detectors. Obviously, future interferometers on Earth and
in space will allow for very high SNR in the post-merger phase
$(\rho\approx 100$ for ET/Cosmic Explorer, and possibly even larger for LISA),
which
will put more stringent constraints on the ECO parameters.
This mild dependence of the relative errors on the sensitivity curve is valid
for signals located near the minimum of the sensitivity curve, as those adopted
in Fig.~\ref{fig:errors}. Less optimal choices of the injected parameters
would give a more pronounced
(although anyway small) dependence, which is due to the different behavior of
the various
sensitivity curves at low/high frequencies.
\item Although Fig.~\ref{fig:errors} was obtained by injecting the
value $d=100M$, the statistical absolute errors on ${\cal R}$ and $d$ do
not depend on
the injected value of $d$ in the $d\to \infty$ limit. We give an analytical
explanation of this seemingly counter-intuitive property in
Appendix~\ref{app:d}. The statistical errors for $d=100M$ are very similar to
those for $d=50M$ and saturate for larger values of $d/M$. Overall, our
analysis suggests that the detectability of the echoes is independent of $d$ in
the large-$d$ limit (i.e., for ultracompact objects).
\item A further important feature is
the strong dependence of the relative errors on the value of the reflection
coefficient ${\cal R}$. In particular, the relative errors for ${\cal R}=1$ are
smaller than those for ${\cal R}\approx0.5$ roughly by $4$ orders of
magnitude. The
reason for this is related to what is shown in Figs.~\ref{fig:SNR}
and~\ref{fig:EvsR}: the amplitude of individual
echoes and the total energy of the signal depend strongly on the reflection
coefficient. This feature suggests that it should be relatively straightforward
to rule out or detect models with ${\cal R}\approx 1$, whereas it is
increasingly more difficult to constrain models with smaller values of ${\cal
R}$.
\end{itemize}
As an example, let us consider the extremal case ${\cal R}=1$. Although this
case is ruled out by the ergoregion
instability~\cite{Cardoso:2008kj,Maggio:2017ivp} and by the
absence of GW stochastic background in LIGO O1~\cite{Barausse:2018vdb}, it is
interesting to explore the level of constraints achievable in this case. For
a reference event with $M=30 M_\odot$ and $d>50 M$ with aLIGO,
we obtain
\begin{equation}\label{estimateDeltaR}
\frac{\Delta {\cal R}}{{\cal R}}\approx
5\times 10^{-8}\left(\frac{8}{\rho_{\rm ringdown}}\right) \qquad {\rm for}\;\,
{\cal R}=1\,,
\end{equation}
where, as a reference, we normalized $\rho$ such that the value of the SNR
\emph{in the ringdown phase only}\footnote{When ${\cal R}\approx1$, the SNR in
the ringdown phase can differ significantly from the
SNR in the whole post-merger phase,
see~Fig.~\ref{fig:EvsR}. For ${\cal R}=1$, and for the parameters considered in
Eq.~\eqref{estimateDeltaR}, the total SNR is $\rho\approx 18\rho_{\rm
ringdown}$.} is that of
GW150914~\cite{GW150914}. This suggests that this model could be detected or
ruled out compared to the BH case (${\cal R}=0$) at more than $5\sigma$
confidence level with aLIGO/Virgo.
Figure~\ref{fig:errors} also shows a strong dependence of $\Delta{\cal
R}/{\cal R}$ when ${\cal R}<1$. It is therefore
interesting to calculate the SNR necessary to discriminate a
partially-absorbing ECO from a BH on the basis of a measurement of ${\cal R}$
at some confidence level. Clearly, if $\Delta {\cal R}/{\cal R}>100\%$, the
measurement would be compatible with the BH case (${\cal R}=0$). On the other
hand, if $\Delta {\cal R}/{\cal R}<(4.5,0.27,0.007,0.00006)\%$, one might be
able (by performing a more sophisticated analysis than the one presented here)
to detect or rule out a given model at $(2,3,4,5)\sigma$ confidence level,
respectively.
\begin{figure}[th]
\centering
\includegraphics[width=0.52\textwidth]{R_vs_SNR_new.pdf}
\caption{Projected exclusion plot for the ECO reflectivity ${\cal R}$ as a
function of the SNR in the ringdown phase and at different $\sigma$
confidence levels.
The shaded areas represent regions that can be excluded at a
given confidence level.
This plot is based on Fig.~\ref{fig:errors} and assumes $d\gg M$ and $M=
30M_\odot$ ($M=10^6M_\odot$) for ground- (space-) based detectors.
The red marker corresponds to $\rho_{\rm ringdown}=8$ and ${\cal
R}=0.9$, which is the value claimed in Ref.~\cite{Abedi:2016hgu} at
$(1.6\div2)\sigma$ level.} \label{fig:RvsSNR}
\end{figure}
The result of this preliminary analysis is shown in Fig.~\ref{fig:RvsSNR},
where we present the exclusion plot for the parameter ${\cal R}$ as a function
of the SNR in the ringdown phase.
Shaded areas represent regions which can be
excluded at some given confidence level. Obviously, larger SNRs would allow to
probe values of ${\cal R}$ close to the BH limit, ${\cal R}\approx 0$.
The extent of the constraints strongly depends on the confidence
level.
For example, ${\rm SNR\approx100}$ in the ringdown would allow to distinguish
ECOs with ${\cal R}\gtrsim 0.3$ from BHs at $2\sigma$ confidence level, but a
$3\sigma$ detection would require ${\cal R}\gtrsim0.85$. The reason for this is
again related to the strong dependence of the echo signal on ${\cal R}$ (see
Figs.~\ref{fig:SNR} and~\ref{fig:EvsR}).
Finally, the red marker in Fig.~\ref{fig:RvsSNR} corresponds to a
detection at $\rho=8$ in the ringdown phase (such as
GW150914~\cite{GW150914}) and an ECO with reflectivity ${\cal
R}=0.9$. These are roughly the values of the tentative detection claimed in
Ref.~\cite{Abedi:2016hgu} at $(1.6\div2)\sigma$ level (but see also
Refs.~\cite{Conklin:2017lwb,Ashton:2016xff,Abedi:2017isz,Westerweck:2017hus,
Abedi:2018pst}). Although our analysis is preliminary, it is interesting
to note that our results are not in tension with the claim of
Ref.~\cite{Abedi:2016hgu}, since Fig.~\ref{fig:RvsSNR} suggests that an ECO
with
${\cal R}=0.9$ could be detected at $\lesssim 2.5\sigma$ confidence level
through an event with $\rho_{\rm ringdown}\approx8$.
On the other hand, the Fisher analysis only gives an estimate of the
statistical errors based on a theoretical template, without using
real data. As such, it would also be in
agreement with a negative search or with smaller significance, like that
reported in
Ref.~\cite{Westerweck:2017hus}.
We consider the results
shown in
Fig.~\ref{fig:RvsSNR} as merely indicative that interesting constraints on (or
detection of) quantum-dressed ECOs are within reach of current and,
especially,
future detectors.
Indeed, Fig.~\ref{fig:RvsSNR} shows that to confirm the putative detection of
Ref.~\cite{Abedi:2016hgu} at $3\sigma$ ($4\sigma$) level would require a
single-event detection with $\rho_{\rm ringdown}\approx 50$ ($\rho_{\rm
ringdown}\approx1800$).
\section{Discussion and Outlook}~\label{sec:Discussion}
We have presented an analytical template that describes the ringdown and
subsequent echo signal of a ultracompact horizonless object motivated by
putative near-horizon quantum structures. This template
depends on the physical parameters of the echoing remnant, such as the
reflection coefficient ${\cal R}$ and the redshift at the surface of the object.
This study is the first step in the development of an accurate template to be
used in direct searches for GW echoes using matched filters and in parameter
estimation.
We have characterized some of the features of the template, which are anchored
to the physical properties of the ECO model. The time-domain
waveform contains all features previously reported for the echo signal, namely
amplitude and frequency modulation, and phase inversion of each echo relative
to
the previous one due to the reflective boundary conditions.
The amplitude of subsequent echoes (both in the frequency and in the time
domain)
depends strongly on the reflectivity ${\cal R}$. When ${\cal R}\approx
1$
the echo signal has amplitude and energy significantly larger than those
of the ordinary BH ringdown. This suggests that GW echoes in
certain models might be detectable even when the ringdown is not.
Using a Fisher analysis, we have then estimated the statistical errors on the
template parameters for a post-merger GW detection with current and future
interferometers. Interestingly, for signals in the optimal frequency window,
the statistical errors at a given SNR depend only mildly on the detector's
sensitivity curve.
Our analysis suggests that ECO models
with ${\cal R}\approx 1$ can be detected or ruled out even with aLIGO/Virgo
(for events with $\rho\gtrsim 8$ in the ringdown phase) at $5\sigma$
confidence level. The same event might allow us to probe values of
the reflectivity as small as ${\cal R}\approx 0.8$ roughly at $2\sigma$
confidence level.
Overall, the detectability of the echoes is independent of the parameter $d$ in
the large-$d$ limit (i.e., for ultracompact objects).
ECOs with ${\cal R}=1$ are ruled out by the ergoregion
instability~\cite{Cardoso:2008kj,Maggio:2017ivp} and by the
absence of GW stochastic background in LIGO O1~\cite{Barausse:2018vdb}.
Excluding/detecting echoes for models with smaller values of the reflectivity
(for which
the ergoregion instability is absent~\cite{Maggio:2017ivp})
will require
SNRs in the post-merger phase of ${\cal O}(100)$. This will be achievable only
with third-generation detectors (ET and Cosmic Explorer) and with the
space-mission LISA.
Although our analysis is preliminary, we believe that our results
already indicate that interesting constraints on (or
detection of) quantum-dressed ECOs are within reach with current (and
especially
future) interferometers.
Extensions of this work are manifold. A template valid for the spinning case
is underway. This case is particularly interesting, not only because merger
remnants are spinning, but also because of the rich
phenomenology of spinning horizonless objects, which might undergo various
types
of instabilities~\cite{1978CMaPh..63..243F,Moschidis:2016zjy,Brito:2015oca,
Cardoso:2007az,Cardoso:2008kj,Cardoso:2014sna,Maggio:2017ivp}. In particular,
due to superradiance~\cite{Brito:2015oca} and to the ergoregion
instability~\cite{Cardoso:2014sna,Vicente:2018mxl}, the echo signal will grow
in time over a timescale $\tau_{\rm ergoregion}$ which is generically much
longer than $\tau_{\rm echo}\sim d$.
Another natural extension concerns the role of the boundary conditions at the
surface for gravitational perturbations~\cite{Price:2017cjr} --~especially in
the spinning case~-- and on model-independent ways to describe the interior of
the object in case of partial absorption (cf., e.g.,
Ref.~\cite{Barausse:2018vdb}).
Furthermore, a more realistic model could be obtained by reprocessing the full
form of $Z_{\rm BH}^+$ containing both the ringdown and the late-merger signal
[i.e., using the template~\eqref{FINALTEMPLATEGEN}], or using a superposition
of QNMs. This extension will be particularly
important to compare our template (constructed within perturbation theory and
therefore strictly speaking valid only for weak sources) with the post-merger
signal of
coalescences forming an ``echoing'' ultracompact horizonless object.
Unfortunately, numerical simulations of these systems are
currently unavailable, but we envisage that a comparison between analytical
and numerical waveforms will eventually follow a path similar to what done in
the past for the matching of ringdown templates with numerical-relativity
waveforms (see, e.g., Ref.~\cite{Buonanno:2006ui}).
Another interesting prospect is to analyze the prompt-ringdown
signal and the late-time one separately. Since the prompt response is
universal~\cite{Cardoso:2016rao}, it could be used to
infer the mass and the spin of the final object, thus making it easier to
extract the echo parameters from the post-merger phase at late times.
Finally, it should be straightforward to extend our analysis to complex
and (possibly) frequency-dependent values of ${\cal R}$ and to include two
polarizations.
These analyses and other applications are left for future work.
\begin{acknowledgments}
The authors acknowledge interesting discussion with the members of the
GWIC-3G-XG/NRAR working group, the financial support provided under the
European
Union's H2020 ERC, Starting Grant agreement no.~DarkGRA--757480, and networking
support by the COST Action CA16104. PP acknowledges the kind hospitality of the
Universitat de les Illes Balears, where this work has been finalized.
\end{acknowledgments}
|
1,108,101,564,550 | arxiv | \section{Introduction}
One challenge in survey cosmology is to infer the true dark matter
density field from observables, i.e. galaxies and galaxy clusters
from galaxy surveys in the UV, visible, NIR, or radio,
or galaxy clusters detected in X-ray or Sunyaev-Zeldovich (SZ)
\citep{Sunyaev72} surveys. All these observable populations, as well
as their host dark matter halos, are biased tracers of their underlying
dark matter. It is well know that the bias depends on halo properties,
e.g. mass, formation time, etc. Moreover, the tracers are
not deterministic, even on large scales, i.e. there is randomness
in their clustering relative to that of the dark matter.
Understanding such randomness, or stochasticity,
is crucial for precise mass reconstruction and strengthening
constraints in cosmological parameters from observables. In this work,
we take a step back by assuming that we have perfect ``observations''
of dark matter halos
and we aim to understand the stochasticity between halos and dark matter,
and to develop an optimal mass estimator from halo catalogs.
The simplest assumption about the distribution of halos (or galaxies)
is that they are drawn from the mass distribution in a biased Poisson
process. This remains a common assumption, even though mass
conservation arguments strongly suggest that this model cannot be
correct \citep{Sheth99b}.
On the other hand, the consequences of the simple truism that the
mass {\em is} the mass-weighted sum of all halos are only just beginning
to be worked out. \citet{Abbas07} showed that the usual expressions for
halo bias \citep{MW96, ST02} -- which derive from the fact that halo
abundances are top-heavy (i.e., massive halos are overrepresented) in
denser regions---can also be derived from noting that a region which
happens to have a top-heavy mass function will also be overdense.
The two approaches provide rather different prescriptions for how
to construct an estimator of the mass field given a (subset of) the
halos. The estimator assuming halos are Poisson sampled from the
mass is considerably noisier than the one in which halos are mass
weighted \citep[H10]{SHD09, Hamaus10}. In what follows, we will compare
the differences between these two approaches, as well as provide a
prescription for finding the optimal weight when only a subset of the
halos are available.
In many respects, our approach is similar to that of H10.
However, while they concentrate on the problem of writing a weighted
halo field as a linear function of the mass field, we will focus on
the inverse problem -- that of writing the mass field as a weighted
sum over the halos, and minimizing the RMS residual $E$ of the mass
estimation.
This re-evaluation of the stochasticity in mass estimators is important
for observational programs to constrain cosmological parameters via
measurement of the mass power spectrum: the weighting scheme that
minimizes stochasticity may suggest a change in targeting strategies
for spectroscopic surveys, and will reduce the resources required for
a volume-limited measure of the mass power spectrum.
Other cosmological probes and tests of general relativity require
cross-correlation of the estimated mass field with other observables
such as gravitational lensing. These experiments should gain even more
through the use of optimal mass reconstruction.
Section~\ref{estimators} describes previous work on how to use
halos to estimate the mass distribution, when halos are a sampling
of the mass field, and then contrasts these with the prescription
which follows from assuming that the mass is mass-weighted halos.
Section~\ref{sims} presents various tests in simulations of this
approach.
Section~\ref{andso} discusses implications of our findings, and
a final section summarizes.
\section{Bias, stochasticity, and linear estimators}
\label{estimators}
Suppose we have two random variables, $m$ and $h$, both defined to
have zero mean. We will later be interested in cases where these are
the mass and halo density fluctuations in spatial cells, or Fourier
coefficients of the fluctuation fields. Their covariance matrix can
always be written as
\begin{equation}
\label{pvr}
\mC = \left(\begin{array}{cc}
P & rb_{\rm var} P \\
r b_{\rm var} P & b^2_{\rm var} P
\end{array}
\right).
\end{equation}
$r$ is the correlation coefficient between $m$ and $h$, and $b_{\rm
var}$ is called the ``variance bias'' by \citet{DL99}. Throughout
this paper, the variable $P$ without subscript will mean the power in
the mass distribution. We can quantify the error in any estimator
$\hat m$ of the mass variable via
\begin{equation}
E^2 \equiv \frac{\left\langle (m- \hat m)^2 \right\rangle}{\langle m\rangle^2}.
\label{esq1}
\end{equation}
We will refer to $E$ as the {\em stochasticity} between the two
variables. We can extend the definition to mean the RMS residual
error after subtraction of any estimator $\hat m$ of the mass field.
If we restrict the estimator $\hat m$ to be a linear function of $h$,
{\it i.e.\/} $\hat m = wh$ for some weight $w$, then substitution into
(\ref{esq1}) yields
\begin{equation}
E^2 = \left(1+w^2+b_{\rm var}^2-2wrb_{\rm var}\right).
\end{equation}
$E$ is minimized at $w=r/b_{\rm var}$,
which yields
\begin{equation}
E_{\rm opt}^2 = 1-r^2 = \frac{ |\mC|}{\langle m^2\rangle \langle h^2\rangle}.
\label{esq2}
\end{equation}
In many cases cosmological information is carried by the ratio $b$
between the two variables. If $m$ and $h$ are drawn from a bi-variate
Gaussian distribution, then we can use the Fisher matrix formalism
from \citet{TTH} to form the covariance matrix of the parameters
$\{P, r, b_{\rm var}\}$ in \mC. If we draw $N_m$ pairs $(m,h)$ from
the distribution, the uncertainties on $b_{\rm var}$ and $P$ after
marginalization over other parameters are
\begin{eqnarray}
\frac{\sigma_P}{P} & = & \sqrt\frac{2}{N_m}\\
\frac{\sigma_{b_{\rm var}}}{b_{\rm var}} & = & \sqrt\frac{1-r^2}{N_m}
= \frac{E_{\rm opt}}{\sqrt{N_m}}
= \sqrt{\frac{E_{\rm opt}^2}{2}}\, \frac{\sigma_P}{P}.
\end{eqnarray}
[See \citet{SCS05} for the real space version of this argument.]
Note that the same stochasticity $E$ appears in
$\sigma_b$.\footnote{\citet{Seljak04} and \citet{Bonoli09} write
$\sigma_b/b=S=\sqrt{2(1-r)}$ per mode. The difference between this
and the correct expression $\sqrt{1-r^2}$ is small when $r$ is
close to 1.}
The appeal of cosmological tests based on ratios like $b_{\rm var}$
instead of power variables like $P$ is that the former require
$E_{\rm opt}^2/2$ fewer mode measurements to reach the same accuracy and
can hence be more powerful for a given survey. We will show below that
this factor can reach $E_{\rm opt}^2/2 < 10^{-3}$ for estimation of the
mass distribution from halo distributions.
In this paper we will adopt a {\em covariance} definition of bias for
variable $h$ against $m$, in which the covariance matrix is written as
\begin{equation}
\label{bcov}
\mC = \left(\begin{array}{cc}
P & b P \\
b P & b^2P + \noise
\end{array}\right).
\end{equation}
The ``noise'' component is $\noise\ge 0$. Note that our $P$ is explicitly
the mean power spectrum in realizations of the field $m$.
We do not subtract shot noise.
From (\ref{esq2}), the stochasticity of $h$ and $m$ is
$E_{\rm opt}^2 = \noise/(b^2P + \noise) = (1+b^2P/\noise)^{-1}$.
If $h$ is a Fourier coefficient of the fluctuations in a biased
Poisson sampling of the field $m$, with mean space density $n$, then
$\noise=1/n$. For any field $h$, therefore, we can define an effective
space density via
\begin{equation}
\label{nb2eff}
(nb^2)_{\rm eff} P = E_{\rm opt}^{-2} -1.
\end{equation}
Recently, H10 concentrated on optimization of the quantity
\begin{equation}
\sigma_w^2 \equiv \left\langle (\delta_w - \hat \delta_w)^2 \right\rangle,
\end{equation}
where $\delta_w$ is a weighted halo map and $\hat \delta_w$ is a
linear function of the mass fluctuations.
Our approach (and motivation) is similar, although we focus on the
inverse problem -- that of writing the mass field as a weighted sum
over the halos, and minimizing the RMS residual $E$ of the mass
estimation.
\subsection{Multidimensional linear estimators}
Now consider attempting to estimate the mass fluctuation $\delta_m$
from the fluctuations $\delta_i$ of a collection of tracers, {\it e.g.}
the fluctuations in bins of halo mass. We wish to construct an
estimator $\hat\delta_m$ from a linearly weighted sum of the tracers:
\begin{equation}
\hat \delta_m \equiv \sum w_i\delta_i = \vw\cdot \vdelta.
\end{equation}
With complete generality we can define the power $P$, the covariance
bias vector ${\bf b}$ and the halo covariance matrix \mC\ via
\begin{eqnarray}
\langle \delta_m^2 \rangle & = & P, \\
\label{covbias}
\langle \delta_m \delta_i \rangle & = & b_iP, \\
\langle \delta_i \delta_j \rangle & = & C_{ij}.
\end{eqnarray}
Note that we have not subtracted a shot noise contribution from the
halo variance. We will never subtract a Poisson shot noise term from
the power spectra in this paper because it is our intention to test
the simple assumptions about the nature of shot noise.
For any choice of weight vector \vw, the stochasticity of the resulting
mass estimator is
\begin{equation}
E^2 = 1 - 2\vb^T \vw + \vw^T \mC \vw / P.
\end{equation}
The choice of \vw\ that minimizes the stochasticity is
\begin{equation}
\label{wopt}
\vw_{\rm opt} = \left(\mC/P\right)^{-1} \vb ,
\end{equation}
making
\begin{equation}
\label{eopt}
E^2_{\rm opt} = 1 - \vb^T \left(\mC/P\right)^{-1} \vb.
\end{equation}
Note that this is the stochasticity-minimizing linear estimator
quite generally, independent of any assumptions about Gaussianity or
any details of the process generating the halo distributions. Linear
estimators which minimize the RMS residual of a target are known as
Wiener filters, and our form
$\vw = \mC^{-1}\vb$ is typical for such cases.
Also it is generally true that the optimized mass estimator
$\hat \delta_{\rm opt}$ will satisfy
$\langle \hat \delta_{\rm opt}^2 \rangle
= \langle \hat \delta_{\rm opt} \delta_m \rangle$.
The weight Eq.~(\ref{wopt}) is proportional to that in eqn.~(19)
of H10, even though their derivation assumes Gaussianity and is not
based on minimizing the stochasticity $E$.
\subsection{Principal components}
\citet{Bonoli09} investigate the stochasticity of the halo field with
respect to the matter by taking the weight function to be the first
(or higher) principal component (PC) of the halo covariance matrix \mC.
In other words the weight \vw\ is the eigenvector of \mC\ having the
largest eigenvalue. If the $\delta_i$ are rotated into the principal
components, then the matrix \mC\ becomes diagonal. If principal
component $j$ has correlation coefficient $r_j$ with respect to the
matter, then it is easy to see that
\begin{equation}
E^2_{\rm opt} = 1 - \sum_j r_j^2.
\end{equation}
Hence a drawback of PC weighting is that the stochasticity of the
first principal component (PC1) achieves the optimally low value only
if no other PC's correlate with the mass. \citet{Bonoli09} show that
this is a good approximation only on the largest scales.
Another issue with PC weighting is that it is not stable to re-binning
of the halo population. For example, when the halos occupy the mass
distribution with a Poisson process, the bins must be chosen with
equal $n_i$ in order for the first PC to dominate the correlation with
the mass (as was done by H10). The optimal weighting Eq.~(\ref{wpoi})
shown later in our paper is recovered only in the limit of vanishing shot noise,
$n_iP\rightarrow\infty$. Since PC weighting is binning-dependent and
non-optimal, we will not focus on it.
H10 find that the optimal weight vector is very close to the weakest
principal component of the ``shot noise matrix'' $\mC - \vb P\vb^T$
when the halos are binned in equal numbers. This is intriguing since
there is no algebraic requirement for the correspondence.
\subsection{Mass as mass-weighted halos}
\subsubsection{Mass completeness relation}
If we partition {\em all} of the mass into halos, {\it i.e.} extend
the halo catalog to zero mass, then the mass distribution {\em is} the
mass-weighted sum of the halo distributions \citep[e.g.,][]{Abbas07}.
Hence we will obtain a perfect $E=0$ estimator of mass if we weight
each halo bin by the fraction of the total mass it holds:
\begin{equation}
\label{w=m}
\eta_i = \frac{n_i m_i}{\sum_i n_i m_i} = \frac{n_i m_i}{\bar\rho},
\end{equation}
where $m_i$ is the mean mass of halos in bin $i$ and $\bar\rho$ is the
overall mean density. Since $E=0$ is clearly the optimal result, it
is mass weighting which is optimal. If, on the other hand,
halos occupy the mass distribution via a biased Poisson process,
it is optimal to weight halos by their bias factors (we show this
explicitly below).
Therefore, the biased Poisson model cannot be correct in the limit
that the halo catalog includes all of the mass.
The simple truism that the mass {\em is} the mass-weighted sum of
all halos suggests that the optimal estimator will tend toward mass
weighting as we include lower-mass halos. \citet{Park09} note in N-body
simulations that mass-weighted halo catalogs attain lower stochasticity
than uniform weighting. \citet{SHD09} show that weighting by mass (or
other functions of mass) produces significantly lower stochasticity
than expected from Poisson shot noise. H10 derived that weight
function which optimizes $\sigma_w^2$. We derive the analogous
optimal weight for $E$.
\subsubsection{Mass estimation with incomplete halo catalogs}
Suppose we are given a list of halo masses and positions.
Suppose that this list is complete down to some limiting mass $m_d$.
We can assign the remainder of the mass to a ``dust bin,'' which
contains the fraction $\eta_d = 1 - \sum \eta_i$ of mass that is
not in the halos.
If $\delta_d$ is the relative density fluctuation field of the mass
in the dust bin, we can define the power of the dust field as
\begin{equation}
P_d = \langle \delta_d^2 \rangle
\end{equation}
and a bias vector $\vc_d$ between halos and dust by
\begin{equation}
c_{di} = \frac{C_{di}}{P_d} = \frac{\langle \delta_d \delta_i \rangle}{P_d} .
\end{equation}
Because
\begin{equation}
\delta_m \equiv \eta_d \delta_d + \sum \eta_i \delta_i,
\label{msum}
\end{equation}
the bias and optimal weighting for the halo bins against the mass are:
\begin{eqnarray}
P & = & \eta_d^2 P_d + 2 \eta_d P_d \veta^T \vc_d + \veta^T \mC \veta, \\
P \vb & = & \eta_d P_d \vc_d + \mC \veta, \\
\label{smoothw}
\vw_{\rm opt} & = & \veta + (\eta_d P_d) \mC^{-1} \vc_d, \\
\label{smoothE}
E^2_{\rm opt} & = & \frac{ \eta_d^2}{ P/P_d} \,
\left( 1 - \vc_d^T \mC^{-1} \vc_d P_d \right).
\end{eqnarray}
In this formulation it is apparent that as our halo catalog extends to
lower masses and $\eta_d \rightarrow 0$, the optimal weight
$\vw_{\rm opt}\rightarrow \veta$, {\it i.e.} mass weighting, and
$E^2\rightarrow 0$.
The further question of interest for finite $\eta_d$ is: How well do
the known halo
fluctuations $\delta_i$ predict the dust bin density $\delta_d$?
\subsection{Sampling Models}
\label{biasedPoisson}
So far the analysis has been completely general as to the generation
of the mass field and the designation of halos within it. Now we
examine models for the relation between halos and mass.
The most common assumption about the distribution of halos (or galaxies)
is that they are drawn from the mass distribution in a biased Poisson
process. If we are examining the Fourier coefficients of the mass and
halo distributions, the biased Poisson model can be broken down into
three assumptions:
\begin{enumerate}
\item The halos are a linearly biased sampling of some continuous ``halo
field'' $\delta_h$ with power $\langle \delta_h^2 \rangle = P_h$,
via some stochastic process that has no spatial correlations.
Then the covariance matrix of halo bins can be written as a rank-one
matrix plus a diagonal ``shot noise:''
\begin{eqnarray}
\label{vvt}
\mC & = & P_h \vv \vv^T + {\rm diag}(\vN), \\
\noise_i & = & f_i / n_i.
\end{eqnarray}
Here $f_i>0$ is a ``clump size'' factor relating the noise $\noise_i$ in bin $i$ to the
space density $n_i$ of sources in the bin. Halos in bin $i$ are drawn
with bias $v_i$ from the halo field.
\item The halos are placed by a Poisson process so that $f_i=1$.
\item We identify the halo field $\delta_h$ with the mass $\delta_m$
such that $P_h=P$ and $\vv=\vb$. In this case
$\mC = P\vb\vb^T + {\rm diag}(1/n_i)$.
\end{enumerate}
Assumption (iii) of the biased-Poisson model has been noted to be
inconsistent with the assumption that the halo catalog can be extended
to comprise all of the mass. We will therefore consider models in
which assumption (i) holds without (iii), such that the halos sample a
field that is distinct from the mass distribution. We will take care
therefore to distinguish \vv, the bias of the halos with respect to
the halo field $\delta_h$, from \vb, which we always define via the
covariance with mass as per (\ref{bcov}).
The assumptions that halos are linearly biased, and that the halo
generating process has no spatial correlations are idealizations:
in fact, halos in simulations do not overlap (almost by definition),
and their bias is non-linear. We will return to the limits of these
assumptions later.
When \mC\
takes the form (\ref{vvt}), two things are of note: first, this
description is stable under re-binning of the halos in the limit of
narrow bins. More specifically, if $v_i$ and $f_i$ are slowly-varying
functions of the mass $m_i$ of halos in bin $i$, then the
$v_i$ and $f_i$ do not change if two adjacent bins are merged. In
other words we can write functions $v(m)$ and $f(m)$, and all of our
linear-algebra formulations can be carried over into integrals over
halo mass $m$. The second useful fact about (\ref{vvt}) is that it
can be inverted analytically using the Sherman-Morrison formula:
\begin{eqnarray}
\label{vvtinv}
P_h\left(\mC^{-1}\right)_{ij} & = &
x_i \delta_{ij} - \frac{ x_i v_i x_j v_j}{1 + \sum x_i v_i^2} \\
x_i & \equiv & P_h/\noise_i = n_i P_h/f_i.
\end{eqnarray}
When all three conditions of the biased-Poisson model are met,
the optimal weight function and
stochasticity are found simply from (\ref{wopt}) and (\ref{vvtinv}):
\begin{eqnarray}
\label{wpoi}
\frac{w_{{\rm opt},i}}{n_i} & = & b_i \frac{P}{1+\sum n_i b_i^2 P}, \\
\label{epoi}
E^2_{\rm opt} & = & \left(1 + \sum n_i b_i^2 P \right)^{-1}, \\
\label{nbpoi}
(nb^2)_{\rm eff} & = & \sum n_i b_i^2.
\end{eqnarray}
This recovers the result from \citet{Percival04} that the optimal
linear mass estimator in a Poisson model weights each halo by its bias
$b_i$ (times a mass-independent factor), and the stochasticity of the
estimator is determined by $\sum n_i b_i^2 P$.
Conveniently the weights scale with the bias, independent of the range
of halo masses included in the estimator. This property does
{\em not} hold for more general forms of \mC\ and \vb,
{\it e.g.} it fails when $\vv \ne \vb$ and condition (iii) is violated.
In this paper we will {\em not} assume that the halos occupy the
matter distribution via a biased Poisson process. We will examine the
\mC\ matrix for halos in numerical simulations, examine what if any
of the three biased-Poisson conditions actually holds, and then use
the general formulae (\ref{wopt}) and (\ref{eopt}) to find how the
optimal stochasticity differs from the Poisson predictions.
\subsubsection{General sampling model}
\label{samplingmodel}
We now examine the case where all halos, and the dust, are indeed
placed by a local process that is biased relative to some halo field
$\delta_h$, so assumption (i) holds but (ii) and (iii) are not
assumed.
This model illustrates the difference between the halo field that
is {\em sampled} and the mass field that the halos {\em comprise}.
The two fields cannot be equivalent, even in the linear regime.
In \S\ref{testsampling}, we examine whether halo covariance matrices measured
in simulations are in fact consistent with this sampling model.
If all the halos and dust are placed in this halo field by
independent processes, and we define a weighted field
\begin{equation}
\delta_v \equiv \frac{\sum_i n_i v_i \delta_i}{\sum_i n_i v_i},
\end{equation}
then the covariances between the dust, the halo bins, and $\delta_v$
are
\begin{eqnarray}
\label{vvNmodel}
\mC & = & \vv \vv^T P_h + {\rm diag}(f_i/n_i) \\
P_d = C_{dd} & = & v_d^2 P_h + \noise_d \\
C_{di} & = & v_d v_i P_h \\
C_{dv} & = & v_d \frac{\sum_i n_i v_i^2}{\sum_i n_iv_i} P_h\\
C_{vv} & = & \left(\frac{\sum_i n_i v_i^2}{\sum_i n_iv_i}\right)^2 P_h
+ \frac{\sum_i n_i v_i^2 f_i}{(\sum_i n_i v_i)^2}
\end{eqnarray}
We have freedom in setting the normalization of $v_d$ and \vv\ which we
do by requiring
\begin{equation}
\label{normV}
\eta_d v_d + \veta \cdot \vv = 1.
\end{equation}
In this case the mass, which is the sum of halo and dust densities,
has power
\begin{eqnarray}
\label{PmPh}
P= C_{mm} & = & P_h + \eta_d^2\noise_d + \sum \eta_i^2 f_i / n_i \\
& = & P_h + \eta_d^2\noise_d
+ (1-\eta_d)^2 \langle fm^2\rangle / \langle m\rangle^2 \\
& \equiv & P_h + \noise_m.
\end{eqnarray}
The angle brackets denote number-weighted averages over the halo
population, and the final expression defines $\noise_m$.
We see that when the mass field is a sampled realization of the halo
field, then $P\equiv C_{mm}$ is larger than $P_h$ by terms representing
the sampling shot noise.
The bias \vb\ of the halo mass bins relative to the mass distribution
will not in general equal the bias \vv\ with respect to the halo
field. Since $b_i P = \langle \delta_i \delta_m\rangle,$ we can expand
$\delta_m$ using (\ref{msum}), and use the $\mC$ elements above, to
derive
\begin{equation}
\label{vbsample}
\vb = \frac{\vv + {\rm diag}(m_i f_i/P_h\bar\rho)}{P/P_h}.
\end{equation}
Formally, only $v_i \propto f_i m_i$ will yield $\vb \propto \vv$.
In general, $b_i>v_i$ at sufficiently large masses, and $b_i<v_i$ at
lower masses. Equality is at $m_i = b_i \noise_m\bar\rho$, which
occurs for $z=0$ at $\sim 10^{14} h^{-1}M_\odot$. In a $\Lambda$CDM
model, the $b=v$ crossover will occur for halos with $b\approx1.6$ for
a wide range of redshifts.
We can also solve for the weight vector of the optimal linear mass
estimator:
\begin{equation}
\label{woptsample}
\frac{w_{{\rm opt},i}}{n_i} = \frac{m_i}{\bar\rho}
+ \eta_d v_d\,(v_i/f_i)\, P_h\,E^2_{\rm pois},
\end{equation}
where we have set
$E^2_{\rm pois} \equiv (1 + \sum_j n_j v_j^2 P_h/f_j)^{-1}$;
the sum over $j$ is over the halo bins in the catalog. $E^2_{\rm
pois}$ describes the fidelity with which the halos can estimate the
{\em halo} field, as opposed to the {\em mass} field.
Notice that the first term is the mass weighting
$(w\propto m)$, and the second correction term has the same form
as that for the standard Poisson model $(w\propto v)$, but its
importance depends on the mass fraction in dust, and how it clusters.
Thus, equation~(\ref{woptsample}) is a weighted sum of
equations~(\ref{w=m}) and~(\ref{wpoi}).
As the halo catalog comprises more of the total mass,
$\eta_d\rightarrow 0$, and mass weighting becomes optimal.
If we define $\delta_{\rm opt} \equiv \sum_i w_{{\rm opt},i}\delta_i$, then
the optimized stochasticity is
\begin{eqnarray}
\label{E2opt}
E^2_{\rm opt} &=& 1 - \frac{\langle\delta_{\rm opt}\delta_m\rangle^2}
{\langle\delta_{\rm opt}^2\rangle\langle\delta_m^2\rangle}
= 1 - \frac{\langle\delta_{\rm opt}\delta_m\rangle}
{\langle\delta_m^2\rangle} \\
&=& \frac{\eta_d^2C_{dd}}{C_{mm}} \,
\left(1 - \frac{C_{dv}^2}{C_{dd}C_{vv}}\right)\\
&=& \frac{\eta_d^2}{P_m} \left( \noise_d + v_d^2 P_h\,E^2_{\rm pois}\right).
\label{E2sample}
\end{eqnarray}
We have written the second equality explicitly to show that $E_{\rm opt}$
makes physical sense: since $\eta_d^2 C_{dd}/C_{mm}$ is the fraction of
the total $C_{mm}$ that is in dust, the stochasticity is the yet
smaller portion
of this dust power that cannot be recovered via the correlation between
the dust and the bias weighted halos.
The final expression shows that $E_{\rm opt}\to 0$ as the mass fraction
in dust $\eta_d\to 0$, as it should.
When $P_h\to 0$, then $E_{\rm opt}^2\to \eta_d^2\noise_d/\noise_m$:
in this limit, the stochasticity is determined by the fraction of the
noise term $\noise_m$ which is contributed by the dust.
The opposite limit is when $P_h\gg \noise_m$, where $P\approx P_h$
and $E_{\rm opt}\to \eta_d v_d E_{\rm pois}$. Since $\eta_d v_d < 1$, this
is why the optimal stochasticity can be substantially smaller than
in the Poisson model.
The optimized weights and relations between $b$ and $v$ are similar to
what H10 found in their analysis of $\sigma_w$.
E.g., our equation~(\ref{vbsample}) reduces to their equation~(41)
upon taking $f_i\rightarrow 1$, $P_h\rightarrow P_{\rm lin}$, and
$v_i\rightarrow b_i$.
In the discussion following their Eq.~(36), H10 note that their optimal
weight is a linear combination of mass and bias weighting, a point they
make again with their Eq.~(49). But our expression for the relative
contributions of these two weights is substantially more transparent
than theirs. For instance, our formulation shows that this factor is,
in fact, the one associated with the usual Poisson-sampling model, times
a factor which accounts for the dust -- this is not obvious from their expressions.
\subsection{The halo model description}
\label{halomodel}
The halo model is a specific case of the sampling model in the
previous section. The halo model is particularly well-suited to
describing the effect of weighting halos \citep{Sheth05}; it
predicts not only \mC, but also the dust-bin quantities like
$\noise_d$ and $P_{d}$ which are not observable and were left
unspecified in the previous section. Hence the halo model allows
an estimate of the stochasticity $E_w$ associated with any weight
function $w$ applied to the halos, so it can be used to estimate
$E_{\rm opt}$.
(In the context of the optimal weight discussed earlier, it provides
a prescription for the effect of bias weighting halos.)
In what follows, we will explicitly set $f=1$; comparison of the
predictions of this calculation with the measurements in simulations
provides a measure of the accuracy of this assumption.
In particular, if we define
\begin{eqnarray}
\label{nw}
n_w &=& \int_{m_d}^\infty dm\, \frac{dn}{dm}\, w(m) ,\\
\noise_w &=& \int_{m_d}^\infty dm\, \frac{dn}{dm}\, \frac{w^2(m)}{n_w^2}, \\
\noise_\times &=& \int_{m_d}^\infty dm\, \frac{dn}{dm}\,
\frac{m\,u(k|m)}{\bar\rho}\, \frac{w(m)}{n_w}, \\
\noise_m &=& \int_0^\infty dm\, \frac{dn}{dm}\,
\frac{m^2\,|u(k|m)|^2}{\bar\rho^2}
\label{noisem}
\end{eqnarray}
then
\begin{eqnarray}
\label{pk}
C_{ww} &=& v_w^2\, P_h(k) + \noise_w ,\\
C_{wm} &=& v_w\, P_h(k) + \noise_\times ,
\end{eqnarray}
where
\begin{equation}
\label{bw}
v_w = \int_{m_d}^\infty dm\, \frac{dn}{dm}\, \frac{w(m)}{n_w}\, v(m).
\end{equation}
Here $v(m)$ is the bias with respect to $P_h(k)$; it is related to
the bias $b(m)$ with respect to the mass field by equation~(\ref{vbsample}).
The factor $u(k|m)$ in the expressions above represents the fact
that halo catalog is treated as if all mass is concentrated at the
center of mass, but real halo mass is smeared in a density profile: $u$ is the
Fourier transform of the density profile, normalized so that $u\to1$
as $k\to 0$.
Strictly speaking, this writing of the halo model is not quite
correct, because halos of a given mass may have a range of density
profiles, i.e. $u$ is not the same for all halos, and stochasticity
in $u$ will contribute to $E$. If we use the mean $u$ for a given $k$
in the expressions above, and define
$\sigma_u^2(k|m) \equiv \langle |u^2| \rangle - \langle u\rangle^2$,
then then the scatter in profile shapes will contribute an
additional term
\begin{equation}
\label{structurenoise}
\noise_m \rightarrow \noise_m + \int dm \frac{dn}{dm} \sigma_u^2(k|m)
\left(\frac{m}{\bar \rho}\right)^2.
\end{equation}
We expect this additional term to be unimportant at the scales
$k<0.1h\,{\rm Mpc}^{-1}$ that are of most interest for cosmology
\cite{ShethJain03}. Section~\ref{depart} shows that this is indeed
the case. However, this term grows as $k^4$, so it could dominate
the stochasticity at higher $k$.
The results of the previous section suggest that an optimal linear
reconstruction of the mass can be obtained if we use
equation~(\ref{woptsample}) for the weight function.
If we set $f_i=1$, then
\begin{equation}
\label{halomodelw}
w_{\rm opt}(m) = \frac{m\,u(k|m)}{\bar\rho}
+ F_v\, \frac{v(m)\, P_h(k)}{1 + (nv^2)_h P_h(k)},
\end{equation}
where
\begin{eqnarray}
F_v &=& 1 -
\int_{m_d}^\infty dm\,\frac{dn}{dm}\frac{m\,u(k|m)}{\bar\rho}\,v(m) \\
(nv^2)_h &=& \int_{m_d}^\infty dm\,\frac{dn}{dm}\,v^2(m).
\end{eqnarray}
This makes the stochasticity
\begin{equation}
\label{halomodelE}
E_{w_{\rm opt}}^2 = 1 - \frac{C_{wm}^2}{C_{mm}C_{ww}} = 1 - \frac{n_w\,C_{wm}}{C_{mm}}
\end{equation}
equal the expression given in equation~(\ref{E2opt}).
(Note that, for these choices of $w$ and $v$, $n_w^2\,C_{ww} = n_w C_{wm}$,
and $E_{w_{\rm opt}}^2$ is independent of $u$.)
Thus, we can use the halo model to estimate the stochasticity $E_w^2$
associated with the weight $w$ of equation~(\ref{halomodelw}) as follows.
We can measure the mass power spectrum $P$ and the covariance bias
$b(m)$ in the simulations, and then use equation~(\ref{vbsample})
to infer $v(m)P_h$. We then use the halo model to estimate $\noise_m$,
which we subtract from the measured $P$ to get $P_h$, and hence $v(m)$.
These can then be used to estimate $F_\nu$ and $(nv^2)_h$ by summing over
the halos. The halo model can also be used to estimate $\noise_m$;
by directly measuring the component of this that comes from the halos
in the sample, one can determine $\noise_d$ and so estimate $E_w$.
In the following section, we will compare this Poisson sampled and
mass-weighted halo model for $E_w$ with the optimal stochasticity in
simulations.
\subsubsection{Halo exclusion and other subtleties}
We could have made heavier use of the halo model as follows.
The usual implementation \citep{Sheth05} replaces
$v(m)\,P_h \to b_{\rm pbs}(m)\,P_{m}$, where
$b_{\rm pbs}(m)$ is the peak background split bias \citep[][e.g.]{Bardeen86, Cole89, MW96,Sheth99},
and $P_m$ is the power spectrum of the mass, usually approximated
by $P_{\rm lin}$ at small $k$.
H10 make this same assumption in their halo model of $\sigma_w$.
We will show later that $b_{\rm pbs}(m)$ appears to be closer to
$v(m) = (C_{hm}-\noise_\times)/(C_{mm}-\noise_m)$ than it is to
$b(m) = C_{hm}/C_{mm}$.
However, note that our discussion indicates that $P_h$ is {\em not}
to be identified with the mass power spectrum and $v(m)$ is not the
same as the linear bias factor between the halo and mass fields.
This is one reason why our construction of the halo model above was slightly
different from standard. In particular, we did not begin from the mass field
$\delta_m$, and immediately set $P_h = P_m$, as is usually done. Rather, we
framed our discussion in terms of the field $\delta_h$, and weighted samplings
of it. Because we assumed that halos were linearly biased Poisson samplings of
this field, we explicitly ignored the fact that, in reality halos are spatially
exclusive, and the sampling function is not just a linear function of the mass.
The exclusion property means that the assumption that the halos are obtained from
independent sampling processes for every mass bin cannot be correct. Indeed, in
the sampling algorithm described in \citep{Sheth99b}, this lack of independence
appears explicitly -- and it also contributes to the non-linearity of the bias
relation (see their equation 17). [For more recent discussion of the effects of
halo exclusion and non-linear bias on $P_m$, and another way of seeing why exclusion
alone can produce effects which appear as scale dependent non-linear bias, see
\citet{Smith07}.] As we shall see, our neglect of these effects sets a limit to
the accuracy of our approach (e.g., the contribution to the stochasticity which
comes from bias-weighting the halos may not be optimal).
\subsubsection{Halo-mass-dependent selection function}
The results above assume that all halos above a sharp threshold in
mass are observed. If the threshold is not sharp, but is a
function of mass, $0\le p(m)\le 1$, then it is straightforward to
verify that equation~(\ref{halomodelw}) remains the optimal weight,
provided that, in the previous expressions for $C_{ww}$, $C_{wm}$,
$F_v$ and $(nv^2)_h$ (but not $\noise_m$, of course), all occurrences
of $(dn/dm)$ are replaced by $(dn/dm)\,p(m)$. As a result, the
sub-sampling decreases $n_wC_{wm}$; since $C_{mm}$ does not change (of
course), the sub-sampling degrades (i.e., increases) $E$.
\begin{table*}
\centering
\caption{Basic parameters used in the Millennium and NYU simulations.
$\epsilon$ is the Plummer-equivalent comoving softening length of
the gravitational force; $N_{\rm re}$ is the number of realizations for
each simulation; $z_{\rm start}$ is the starting redshift for the simulation.}
\bigskip
\begin{tabular}{lcccccccccccc}
\hline
Name &$N_p$ &$M_p$ & $L_{box}$ & $\epsilon$& $\Omega_m$ & $\Omega_{\Lambda}$ & $\Omega_b$ & $\sigma_8$ & $n_s$ & $H_0$ & $z_{start}$ & $N_{re}$ \\
\hline
Millennium &$2160^3$&$8.6\times10^8~h^{-1}M_{\odot}$& 500~$h^{-1}$Mpc & 5$h^{-1}$kpc &0.25 & 0.75 & 0.045 & 0.9 & 1 & 73 & 127 & 1 \\
NYU &$640^3$ &$6\times10^{11}~h^{-1}M_{\odot}$ & 1280~$h^{-1}$Mpc &20$h^{-1}$kpc &0.27 & 0.73 & 0.046&0.9 & 1 & 72 & 50 & 49\\
\hline
\end{tabular}
\label{tab1}
\end{table*}
\section{Optimal weights and mass estimators from simulations}\label{sims}
\subsection{Simulations}
In this section we use N-body simulations to find optimal
mass-estimation weights for halos binned by mass (rather than,
e.g., angular momentum, axis ratio, formation time or concentration).
We show the stochasticity that results from applying the optimal
weights and compare to common sub-optimal choices.
For purposes of exploration, we wish to have a wide range of halo
masses, while still having good statistics for large halos and
large-scale modes. For the first purpose, we use the Millennium
simulation \citep{Springel05b}, which resolves halos down to
$\sim 10^{10} h^{-1}M_{\odot}$.
For the second, we use the suite of 49 cosmological dark matter
``NYU'' simulations described in \citet{Manera10}, which have a
total volume $800\times$ larger than the Millennium, but the minimum
resolved halo mass is $1000\times$ larger.
The simulations assume very similar $\Lambda$CDM cosmological models
and were carried on using the same {\sc GADGET-2} code \citep{Springel05a}.
Table~\ref{tab1} provides details of the basic simulation parameters.
The initial power spectrum was generated by {\sc CMBFAST} \citep{Seljak96}.
The initial density field of the Millennium simulation was realized by
perturbing a homogeneous, glass-like particle distribution with a
Gaussian random realization of the initial power spectrum \citep{white96},
while the NYU simulations use an algorithm motivated by Second Order
Lagrangian Perturbation Theory \citep{Scoccimarro98}.
Dark matter halos with at least 20 particles are identified in both
simulations using the friends-of-friends (FOF) group finder
with a linking length of $0.2$ times the mean particle separation
\citep{Davis85}. The lowest mass halo of the two
simulations that we use are $1.7\times10^{10}h^{-1}M_{\odot}$ and
$1.0\times10^{13}h^{-1}M_{\odot}$ respectively.
The Millennium simulation has $\approx 1.8\times10^7$ halos at
at $z=0$, $z=0.5$ and $z=1$ while the NYU simulations have
$\sim 4.3\times10^7$, $3.5\times10^7$ and $2.5\times10^7$ halos in
each of these redshift slices.
For a more detailed analysis of the halo mass function and bias factors
in these simulations, see \citet{Springel05b} and \citet{Manera10}.
\begin{figure*}
\begin{center}
\resizebox{\hsize}{!}{
\includegraphics[angle=0]{Optimal_W_NYU_k0.1_model3.eps}
\includegraphics[angle=0]{Optimal_W_Mill_k0.1_model3.eps}}
\end{center}
\caption{The weight function for optimal reconstruction of the mass
field on the scale $k=0.1h$Mpc$^{-1}$ at $z=0$ as a function of halo mass
for the NYU simulations (left) and the Millennium simulation (right).
The optimal weight depends on the minimum halo mass used in the
reconstruction; solid lines (with arbitrary vertical offset), which
are measured from simulations using Eq.~(14), show this dependence.
Dashed, dotted and blue curves show $w\propto M$,
equation~(\ref{halomodelw}), and $w\propto b$,
the latter being optimal if the Poisson model is correct.
The optimal weighting steepens as $M_{\rm min}$ decreases, approaching
$w\propto M$, although not exactly along the dotted curves predicted
by our halo model implementation of the sampling model.
Cyan and orange lines show the mean halo occupancy
distributions (HODs) for blue galaxies and luminous red
galaxies, respectively, in the low-$z$ SDSS spectroscopic sample.
}
\label{wvsm}
\end{figure*}
\begin{figure*}
\begin{center}
\resizebox{\hsize}{!}{
\includegraphics[angle=0]{E_M_All1.eps}
}
\end{center}
\caption{Stochasticity $E$ of the estimators of the mass field
derived from the weights shown in Figure~\ref{wvsm}, shown
as a function of the minimum mass of halos in the catalog.
The three panels show different $k$ values, all at $z=0$.
In each panel, data at higher $M_{\rm min}$ are from the NYU
simulations; lower $M_{\rm min}$ measurements are from
the Millennium simulation.
From the bottom up: black, purple, blue, and red solid curves show
optimal, mass, bias and uniform weighting of the halos.
Bias weighting would be optimal if the standard biased Poisson
model were correct; it clearly is not.
Dashed curve shows the halo model calculation of $E$ which
assumes the mass is sum of halos that are
Poisson-sampled from some halo field. The failure of the
model at $M<10^{12}h^{-1} M_\odot$ is discussed in the text.
}
\label{evsm}
\end{figure*}
\begin{figure*}
\begin{center}
\resizebox{\hsize}{!}{
\includegraphics[angle=0]{Scatter1.NoWeight.ps}
\includegraphics[angle=0]{Scatter1.Opt.ps}
}
\resizebox{\hsize}{!}{
\includegraphics[angle=0]{Scatter3.NoWeight.ps}
\includegraphics[angle=0]{Scatter3.Opt.ps}
}
\end{center}
\caption{Unweighted (left) and optimally weighted (right) real-space halo density
fluctuations $\delta_h$ versus dark matter density fluctuations $\delta_m$
smoothed by a spherical top-hat window function with the radius of $R=50 h^{-1} Mph$.
The three contour levels (0.1, 1, 10 shown in green, yellow, red) indicate the relative
number density of data points in the $\delta_m$-$\delta_h$ plane.
Diagnal black solid lines indicate $\delta_h=\delta_m$. Dash lines show the
fitting results of the function $\delta_h=b_0+b_1\delta_m+b_2\delta_m^2$, with
best-fit parameters shown in the figures.
Top panels are results from the Millennium simulation with the
mimimal halo mass of $M_{\rm min}=1.4\times10^{10}h^{-1}M_{\odot}$. Bottom
panels show results from the NYU simulations, with $M_{\rm min}=3.1\times 10^{13}h^{-1}M_{\odot}$.}
\label{Scatter}
\end{figure*}
\subsection{Measuring power spectra and covariance matrices in simulations}
We start our measurements by dividing halos into bins sorted by mass.
The clustering of halos depends weakly on the halo mass when halo mass
is low but increases rapidly with mass when $M>10^{13}~h^{-1}M_{\odot}$.
Furthermore halo abundances drop sharply at high mass. As we do not
want a wide range of masses within a single bin, we include fewer
halos per bin at high masses.
We choose bins so that the number of halos in each decreases exponentially
at high masses. The optimal weighting and stochasticity are robust to
changes in the binning, as long as the function $b(M)$ is well sampled.
We divide the halos into 10 bins for the NYU simulations, and use up
to 30 bins for the Millennium simulation. We have tested that using
more bins does not change our results.
Within each halo bin, we weight halos by their masses and assign them
to a $N_g^3=256^3$ 3D mesh of cubic grid cells using the cloud-in-cell
(CIC) assignment scheme \citep{Hockney81},
i.e. we take the Fourier transform of the mass distribution within
a halo bin. This is in anticipation of the result below that optimal
weighting is closer to mass-weighting than number-weighting of halos.
If the bins are narrow in mass, this choice of intra-bin weighting
should have little effect, which we have verified.
We separately Fourier transform the overdensity field of each halo bin
and the total mass distribution. We correct each Fourier mode for the
convolution with the CIC window function by the operation:
\begin{equation}
\delta(\vk)=\delta(\vk)\,
\left(\frac{\sin(x)}{x}\frac{\sin(y)}{y}\frac{\sin(z)}{z}\right)^{-2},
\end{equation}
where $\{x, y, z\}=\{k_xL_{box}/2N_g, k_yL_{box}/2N_g, k_zL_{box}/2N_g\}$,
and $N_g$ is the number of grid cells in each dimension.
For each bin in $k$ we construct the covariance matrix of Fourier
coefficients
$C_{ij}(k)=\langle\delta_i(k)\,\delta_j(k)\rangle$,
where $i$ and $j$ range over all halo bins,
as well as the mass power $P=\langle \delta_m^2\rangle$, and hence
the covariance biases \vb\ of the halos against the mass.
For the NYU simulations we average results from all 49 realizations to
produce a mean covariance matrix.
\begin{figure*}
\begin{center}
\resizebox{\hsize}{!}{
\includegraphics[angle=0]{nb2_vs_k.eps}
\includegraphics[angle=0]{nb2_vs_m.both.eps}
}
\end{center}
\caption{{\em Left:} The effective source density $(nb^2)_{\rm eff}$
of an optimally-weighted mass estimator is plotted (solid) vs $k$ for
catalogs with a different halo mass cutoffs $M_{\rm min}$ as
labeled. This measure of stochasticity is found to vary little
with scale in the linear regime $k<0.1h$Mpc$^{-1}$. The dashed lines show
the $nb^2$ expected via Equation~(\ref{nbpoi}) under a biased-Poisson
model of galaxy stochasticity. The dotted line shows
the $(nb^2)_{\rm eff}$ required to attain $E=0.5$, {\it i.e.} a
volume-limited power spectrum measurement.
{\em Right:}
For three different redshifts, we plot the optimal $(nb^2)_{\rm
eff}$ vs the minimum mass of the halo catalog used for mass
reconstruction (solid lines). The dashed lines plot the $nb^2$ that
we would expect if halos were a Poisson sampling of the mass
distribution, as per Equation~(\ref{nbpoi}). Because halos comprise
the mass rather than sampling the mass, the $(nb^2)_{\rm eff}$ is up
to $15\times$ higher, {\it i.e.} the mass estimator is less
stochastic, than the Poisson model predicts.
}
\label{nb2vsk}
\end{figure*}
\subsection{Measured stochasticity and optimal weights}
\label{measure}
Figures~\ref{wvsm} shows the optimal weights we derive from the
simulations (black solid lines), and compares them with various
functions of mass. Purple-dashed lines show $w\propto m$;
blue curves show the bias weighting $w\propto b$ that would
be appropriate if the standard biased-Poisson model were correct,
and dotted curves show the optimal weight in equation~(\ref{halomodelw})
derived from the halo model of Section~\ref{halomodel}.
Clearly, neither $M$ nor $b$ are optimal weights.
Indeed, the shape of $w_{\rm opt}(M)$ depends on the cut-off mass
$M_{\rm min}$ of the halo catalogue. (This dependence follows that
found by H10, in their study of $\sigma_w$.)
When $M_{\rm min}<10^{13}~h^{-1}M_{\odot}$, the massive end of the
$w_{\rm opt}(M)$ is close to mass weighting, as illustrated by the
right-hand plot of Figures~\ref{wvsm}.
When $M$ is close to $M_{\rm min}$, however, $w_{\rm opt}(M)$ is
flatter than mass weighting. Moreover, the slope of $w_{\rm opt}(M)$
gets shallower as $M_{\rm min}$ increases, as shown in the left hand
plot. Weighting halos by their masses is a poorer approximation to the
optimal weight when $M_{\rm min} >10^{13}~h^{-1}M_{\odot}$.
The halo model prediction of the optimal weight is generally in good
agreement with the measurements. The agreement is not perfect, however,
especially when $M$ approaches $M_{\rm min}$.
Figure~\ref{evsm} shows the stochasticity $E$ associated with these
linear estimators $\hat \delta_m$ of the mass distribution, as a
function of the minimum mass $M_{\rm min}$ of halos included in the
sample. Black, purple, blue, and red solid curves show $E$ derived from
optimal, mass, bias and uniform weighting of the halos.
Weighting halos by their masses yields lower $E$ than bias weighting
or equal weighting, but is significantly worse than the optimal when
the halo catalog has $M_{\rm min}\approx 10^{13}h^{-1}M_\odot$ or lower.
Bias weighting would be optimal if the standard biased Poisson model
were correct, but is clearly far from optimal for halos in $N$-body
simulations.
The dashed curves show the halo model description of $E_{\rm opt}$
(equation~\ref{halomodelE}). The model agrees with the measurements
at $M_{\rm min}>10^{12}h^{-1}M_{\odot}$, but it does not predict the
inflection we measure at smaller $M_{\rm min}$.
We discuss this further in Section~\ref{depart}.
As an additional check of our numerical methods, we have verified
that inclusion of the ``dust bin'' in the estimator leads to a
perfect mass estimator ($E=0$) with optimal weights directly
proportional to halo mass.
\subsection{The scatter between the halo field and the mass field}
To illustrate the gain from applying the optimal weights, we show in
Figure~\ref{Scatter} the scatter between the fluctuations of the
halo field $\delta_m$ and the mass density field $\delta_m$,
before and after applying the optimal weights. Notice that
$\delta_h$ and $\delta_m$ are both density contrasts in
configuration space that are smoothed by the same spherical
top-hat window function. We do the smoothing by multiplying
the density contrasts in Fourier space with the Fourier transform
of the window function, $\delta_{h,m}(k)=W_R(k)\delta_{h,m}(k)$, where
$W_R(k)=3[\sin (kR)-kR\cos (kR)]/(kR)^3$ and $R$ is the radius of the window function.
Then we Fourier transform back and get the smoothed $\delta_h$ and $\delta_m$.
We fit the scatter plots with the
polynomial function $\delta_h=b_0+b_1\delta_m+b_2\delta_m^2$,
to see if there is any indication of non-linear bias factor $b_2$.
We usually find very small fitted values of $b_2$, especially for the
optimal weighted cases. We also find an increase of $b_2$
value when increasing the low mass cut of the halo sample.
In general, we see a significant improvment of
applying the optimal weights, indicated by the shrinking of the scatter.
This shows that the optimal weights indeed work well, without any
higher-order bias correction.
\subsection{Scale dependence of stochasticity}
Figure~\ref{nb2vsk} illustrates that the optimal $(nb^2)_{\rm eff}$ is
nearly independent of $k$ at fixed $M_{\rm min}$ in the linear
regime, where $(nb^2)_{\rm eff}$ is related to $E_{\rm opt}$ by
equation~(\ref{nb2eff}). Since both $(nb^2)_{\rm eff}$ and the
Poisson prediction $(nb^2)$ (the simple bias weighted sum over
the halo population), are nearly constant across the linear regime,
we compare them in the right panel of Figure~\ref{nb2vsk}
as a scale-independent measure of stochasticity.
We find that at all redshifts and $M_{\rm min}$ values, the achievable
$(nb^2)_{\rm eff}$ is significantly better (higher) than would have
been expected in the model where halos are a Poisson sampling of the
mass. The ratio $(nb^2)_{\rm eff}/(nb^2)$ can be as high as $\approx 15$
for surveys of $M > 10^{12}h^{-1}M_\odot$ halos at $z=0$.
Even for surveys limited to massive clusters,
$M_{\rm min}\approx 10^{14}h^{-1}M_\odot$, the effective source
density of the optimal estimator is $\approx 2\times$ better than the
Poisson model predicts.
\subsection{Departures from the halo model}
\label{depart}
The dashed curve in Figure~\ref{evsm} plots the halo model prediction
of the optimal $E$, which assumes the mass is comprised of halos that
are Poisson-sampled from some halo field.
The $E_{\rm opt}$ from simulations becomes shallower for $M_{\rm
min}<10^{12}h^{-1}M_\odot$ where as the halo model does not.
Either the halo model is not accurate at
$M_{\rm min} < 10^{12}h^{-1}M_\odot$, or there is some bias in the
numerical estimation of $E_{\rm opt}$ from the simulation catalogs.
Calculating $E_{\rm opt}^2$ at $M_{\rm min} < 10^{12} h^{-1}M_\odot$ sets
heavy demands on the measurement of the covariances in the simulation,
because $\vb^T \mC^{-1} \vb$ must be calculated to a fractional
accuracy of $E^2_{\rm opt} < 10^{-3}$ in this regime.
We have considered the possibility that $E_{\rm opt}$ levels off at
small $M_{\rm min}$ because of discreteness effects. Specifically,
in the simulations, mass comes in units of $m_p$, so the ``dust'' is
not made of arbitrarily small halos. This makes
$\noise_d\to \noise_d + 1/n_d$ approximately. However, this
additional factor is too small to explain the flattening we see. We
have also verified that the plateau in $E_{\rm opt}$ is unaffected by
the size of the bins in mass or $k$.
As noted earlier, the stochasticity $E$ will be degraded if halos of a
given mass have varying internal structure, while we treat halos
of a given mass as being identical point masses in the analysis.
Within the halo model, the stochasticity adds to $\noise_m$ as per
(\ref{structurenoise}).
We test the magnitude of this effect by creating new halo overdensity
maps from the full sample of $N$-body particles belonging to halos in
each mass bin. These ``true'' halo mass maps are then used to create
an optimal mass estimator. Figure~\ref{fullHalos} shows
that the point-mass approximation has negligible impact on $E$ at
$k<0.15h\,{\rm Mpc}^{-1}$, but at higher $k$ the mass estimator is
increasingly degraded by the absence of information on
the variability of internal structure of massive halos.
\begin{figure}
\begin{center}
\resizebox{\hsize}{!}{
\includegraphics[angle=0]{E_k_ExtendedHalo.eps}
}
\end{center}
\caption{The optimal stochasticity $E$ as a function of
wave-number $k$ from using all halos in the Millennium simulation is
shown for two cases: the upper red curve treats each halo as a point
mass, while the lower black curve uses the full spatial distribution
of the particles comprising the mass of each halo. If the halo
catalog does not contain information on the variability of internal
halo structure at a given mass, it cannot fully map the mass
distribution. This significantly degrades $E$ for $k>0.15h\,{\rm
Mpc}^{-1}$, but does not explain the inflection in $E_{\rm opt}$
observed at $k\le 0.1h\,{\rm Mpc}^{-1}$ for $M<10^{12}h^{-1}M_\odot$
in Figure~\ref{evsm}.
}
\label{fullHalos}
\end{figure}
Even when we use the full halo mass distributions in our optimal
estimator, the performance for $M_{\rm min}<10^{12} M_\odot$ remains
worse than $E$ predicted by the halo model.
It is possible that noise in \vb\ and \mC\ from having
too few modes is compromising our measurements of $E_{\rm opt}$ in the
Millennium simulation. This would most strongly affect the
low-$k$ region where modes are scarcest, however the inflection
does not exhibit this behavior.
We conclude
that we have reached limits of the
assumption that halos are a Poisson sampling of an underlying halo
field. For $M \approx 10^{12}h^{-1}M_\odot$, $\eta_d \approx 0.66$
so the effects of exclusion/mass conservation are beginning to
matter, so it is perhaps not surprising that the optimal mass
reconstruction is poorly described by a model that presumes halos to
be independently sampled from the halo field. We will defer to future
work investigation of mass reconstruction in the presence of exclusion
and other non-linear effects.
\subsection{Explicit test of the sampling model}
\label{testsampling}
Are the observed covariance matrices of the halos consistent with
their being stochastic discrete realizations of an underlying ``halo
field'' $\delta_h$, the model of \S\ref{samplingmodel}? We answer
this question by asking how well the elements $C_{ij}$ of the
simulations' covariance matrices can be fit by appropriate values of
the $v_i$ and $f_i$. We find the $\{v_i,f_i\}$ which minimize the L2
norm of the residual to the model (\ref{vvNmodel}):
\begin{equation}
\lVert\delta \mC\rVert_2 \equiv \sum_{ij} \left( v_i v_j +
\delta_{ij}f_i/n_iP - C_{ij}/P\right)^2.
\end{equation}
(We use the mass power $P$ instead of $P_h$ in the fit, which slightly
changes the values of $v_i$ and $f_i$ without affecting the quality of
the fit.)
Figure~\ref{bvn} plots the best-fitting values of $f_i$ and $v_i$ for
each halo mass bin in the NYU simulations, along with the bias $b_i$.
The fitted values of $f_i$ slightly exceed unity, but this is consistent
with halos being a Poisson sampling ($f_i=1$) of a ``halo field''
because the mass-weighting within each halo bin will cause $f_i$ to
rise slightly above unity, particularly in the most massive bin.
\begin{figure}
\resizebox{\hsize}{!}{
\includegraphics[angle=0]{bvn.z0.0.nyu.eps}
}
\caption{The best fit of the sampling model
$C_{ij}=v_iv_jP + \delta_{ij} f_i/n_i$ to the $z=0$ halo catalog
of the NYU simulation is shown as a function of $k$. The solid
lines in the upper panel plot the bias $b_i$ of halos in each mass
bin, while the dashed lines are the best-fit $v_i$ for each bin.
Higher-mass bins have higher bias. Note that $v_i < b_i$ for
massive halos, by an amount that grows at $k>0.1h\,{\rm Mpc}^{-1}$,
but that $v_i > b_i$ at lower masses, as predicted.
The lower panel shows the best-fit $f_i$. Poisson sampling would
induce values slightly above $f_i=1$ because of the mass weighting
within a halo bin, particularly for the most massive bin, as observed.
}
\label{bvn}
\end{figure}
The fit to the sampling model does confirm a departure from the
simple biased-Poisson model, however, in that the biases $v_i$ of the
halos with respect to the parent halo field are not equal to the
covariance biases $b_i$ with respect to the mass field. There is a
divergence of \vb\ from \vv\ toward the trans-linear regime.
The peak-background split model yields an analytic prediction
$b_{\rm pbs}$ for the bias of halos vs the underlying mass distribution.
Figure~\ref{pbs} shows that for the NYU simulations, it is the bias
$v$ of the halos vs the {\em halo field} $\delta_h$ that is best
described by $b_{\rm pbs}$, not the bias $b$ of the halos vs the
{\em mass distribution.}
This observation should lead to a deeper theoretical understanding
of the halo distribution. In particular, it may help resolve the
discrepancy reported by \cite{Manera10} between $b_{\rm pbs}$ and
their measurements of halo bias, which were effectively what we
call $b$, rather than $v$.
\begin{figure}
\resizebox{\hsize}{!}{
\includegraphics[angle=0]{bvm.eps}
}
\caption{For the full NYU halo sample at $z=0$, we plot three
types of bias. The open symbols are the covariance bias $b$ vs the
mass distribution. The solid symbols are the bias $v$ vs the
underlying continuum ``halo field'' $\delta_h$ in the sampling model
that best fits the halo-mass covariance matrix. The curve is the
analytic bias prediction of the peak-background split model.
}
\label{pbs}
\end{figure}
The quality of the fit of the sampling model to the covariance matrix
can be gauged by the ratio
\begin{equation}
R \equiv \frac{\sqrt{\lVert\delta \mC\rVert }}{\lVert \vv \rVert}.
\end{equation}
This quantity measures the ratio of the RMS error in elements of \mC\ to
the RMS value of the model---excluding the diagonal elements, which
are always perfectly fit by adjusting the $f_i$.
For the NYU
simulations at $z=0$ we find \mC\ well fit by this model:
$0.01<R<0.035$ for all $k<0.25~h$Mpc$^{-1}$.
For the bin at
$k=0.07~h$Mpc$^{-1}$ in the NYU simulations, we would expect
$R\approx0.02$ from Gaussian sample variance in the estimation of the
$C_{ij}$. At higher $k$, the sample variance in the estimation of
\mC\ should drop, so it is likely that by $k=0.2$ the residuals of
\mC\ to the sampling model are in excess of statistical errors. We
expect effects such as halo exclusion and halo internal structure
to induce departures from the simplest sampling models at scales
approaching the halo sizes.
The sampling model is not able to fit the $\mC$ matrix across the full
mass range of the $z=0$ Millennium halo catalog, yielding $R>0.1$.
Excluding the two most massive of 30 halo bins from $\mC$ permits a
solution with $R\approx0.07$ in the linear regime, however this
solution requires values of $f_i<1$ or even negative $f_i$ for the
least massive halos. We take this as an additional sign that low-mass
halos cannot be considered to populate the halo field independently,
e.g. exclusion may be important. The much smaller volume of the
Millennium simulation leads to larger statistical
fluctuations in the elements of \mC, so at this time we refrain from
detailed analysis of departures from the sampling model for
less-massive halos.
\section{Practical consequences}
\label{andso}
With an optimal (linear) mass reconstruction algorithm in hand, we can
examine strategies for conducting cosmological measurements with
maximal efficiency.
\subsection{Power spectrum measurement with halo mass estimates}
We first posit a survey attempting to measure the shape of the matter
power spectrum near $k=0.2~h$Mpc$^{-1}$, {\it e.g.} a baryon acoustic
oscillation measurement. Since the error in a power spectrum
determination from $N_m$ modes of a stochastic estimator is
$\sigma_P/P = [(1-E^2)N_m/2]^{-1/2}$, it is typical to consider a
survey with $E\le 0.5$ as sample-variance limited. This is equivalent
to the criterion $(nb^2)_{\rm eff}P = 3$. [Here we assume that the
bias of the estimator is known or immaterial.]
If the redshift of a halo can be obtained with a single spectrum of
its central galaxy, then clearly the strategy for attaining a mass
estimator with a given stochasticity with the fewest redshift
measurements is to measure redshifts for all halos above a chosen mass limit.
In practice one could identify candidate halos from multicolor imaging
data using a variant of red-sequence detection or cluster-finding with
photometric redshifts \citep[e.g.][]{MaxBCG,RCS}.
If imaging data can be used successfully to identify clusters and
estimate their host halo masses, then the more expensive spectroscopic
redshift survey need target only the brightest member of each cluster
or group to determine a redshift for the presumed dark matter halo.
X-ray or Sunyaev-Zeldovich survey data could also potentially contribute to
halo finding and mass estimation.
What is the minimum number of spectra that one must obtain to make the
volume-limited power spectrum measurement described above?
We answer this question for the case when halos are
optimally weighted, and also for comparison the (incorrect) prediction
for halos that occupy the mass by the biased-Poisson model.
We list in Table~\ref{tab2}
the minimal mass of halos and minimal number density of halos for the
above two cases in three different redshifts. The optimally weighted
case needs a factor 4--12 fewer spectra than predicted by the
biased-Poisson model to achieve the sample-variance limit $E\le0.5$.
To complete a volume-limited survey for $0<z<1$
that achieve $(nb^2)_{\rm eff}P = 3$ at the BAO scale
$k\sim 0.2~h$Mpc$^{-1}$, the total number
of spectra one needs is $\approx6\times 10^6 f_{\rm sky}$ if halos are optimal weighted.
If the biased-Poisson model were correct, one would require
$\approx 33\times10^6 f_{\rm sky}$, a factor of 5 more.
For a redshift survey to $z<0.7$ such as the ongoing SDSS-III
Baryon Oscillation Spectroscopic Survey (BOSS)\footnote{http://www.sdss3.org/cosmology.php},
the two cases yield $2\times10^6 f_{\rm sky}$ and $14\times10^6 f_{\rm
sky}$, respectively. Hence BOSS at $f_{\rm sky}=0.25$ would require
only 500,000 optimally targeted and weighted redshifts to achieve
$(nb^2)_{\rm eff}P = 3$, while the survey plans to obtain 1.5~million
redshifts. Given that precise halo masses are not
easily accessible, the weighting for a real survey may be sub-optimal.
We will show in Section~\ref{LRG} that if one weights halos by the
numbers of LRGs, a factor of 3 more redshifts are required to attain $E<0.5$.
\begin{table*}
\centering
\caption{The number density of halos $n^{\rm opt}$ and the corresponding minimal halo
mass $M^{\rm opt}_{\rm min}$ needed to achieve $(nb)_{\rm eff}^2P=3$
($E=0.5$) at BAO scales ($k=0.2h\,{\rm Mpc}^{-1}$) when applying the optimal weighting.
For comparison, the minimal halo mass $M^{\rm P}_{\rm min}$ and number density $n^{\rm P}$ needed to achieve
the same accuracy in a Poisson model are also listed. The last column
gives the ratio of required redshift measurements in the two models.}
\bigskip
\begin{tabular}{lccccccc}
\hline
\hline
$z$ &$M^{\rm opt}_{\rm min}$ & $M^{\rm P}_{\rm min}$ & $n^{\rm opt}$ & $n^{\rm P}$ & $n^{P}/n^{opt}$ \\
&$h^{-1}M_{\odot}$&$h^{-1}M_{\odot}$&($h~$Mpc)$^{-3}$& ($h~$Mpc)$^{-3}$ & \\
\hline
0.0 & $6.2\times 10^{13}$ &$6.5\times 10^{12}$& $4.2\times10^{-5}$ & $5.7\times10^{-4}$ & 13 \\
0.5 & $2.6\times 10^{13}$ &$5.2\times 10^{12}$& $8.0\times10^{-5}$ & $5.7\times10^{-4}$ & 7 \\
1.0 & $1.1\times 10^{13}$ &$3.8\times 10^{12}$& $1.4\times10^{-4}$ & $5.8\times10^{-4}$ & 4 \\
\hline
\hline
\end{tabular}
\label{tab2}
\end{table*}
\subsection{Bias calibration with weak lensing}
A more ambitious measurement is to cross-correlate the mass distribution
estimated from a galaxy redshift survey with a weak gravitational
lensing shear map, thereby calibrating the bias of the estimator
\citep{Pen04}. Measures of the redshift dependence of the
cross-correlation between lensing and matter can also strongly
constrain the curvature and $D(z)$ function of the Universe
\citep{Bernstein04}. A simplified analysis of these problems considers
the covariance matrix between the gravitational convergence field
$\kappa$ and the galaxy-based mass estimator $g$ to be
\begin{equation}
\label{ckg}
\mC = \left(\begin{array}{cc}
P + \noise_\kappa & bP \\
bP & b^2P + \noise_g
\end{array} \right).
\end{equation}
where $\noise_\kappa$ is the noise in the weak lensing mass estimation.
To fully exploit the lensing data, the mass estimator should attain
$E^{-2}=1+b^2P/\noise_g \ge P/\noise_\kappa$ such that the $S/N$ ratio per mode
of the mass estimator exceeds the $S/N$ ratio per mode of the
lensing map.
If the lensing noise level $\noise_\kappa$ is known and one is
inferring $P, b,$ and $\noise_g$ from the values of the lensing and mass
estimators in $N_m$ modes of the sky, then the marginalized error in
the estimate of $b$ becomes
\begin{equation}
\label{sigbxc}
\frac{\sigma_b}{b} = \sqrt\frac{ (1+\noise_\kappa/P) \left( 1/(nb^2)_{\rm eff}P + \noise_\kappa/P\right) +
(\noise_\kappa/P)^2}{N_m}.
\end{equation}
When the lensing and mass reconstruction both have high $S/N$ per
mode, this becomes
\begin{equation}
\frac{\sigma_b}{b}
\approx \sqrt\frac{\noise_\kappa/P + E^2}{N_m}.
\end{equation}
As an example, consider a lensing source plane consisting of $n=30$
galaxies arcmin$^{-2}$ as $z_s=1$ being cross-correlated with a
transverse mass mode at $k=0.1h/$Mpc at $z=0.5$, near the peak of the
lensing kernel. The shear signal will appear at multipole
$\ell=kD\approx130$ where the total power in the shear signal is
$C_\ell \approx 8\times10^{-8}$. The shape noise power is
$\sigma^2_\gamma / n \approx 2\times10^{-10}$, giving
$P/\noise_\kappa\approx400$, or a $S/N$ per mode
of $\approx 20$. The cosmological
measurements will hence continue to
benefit from higher halo survey density until $E\ll 0.05$. Even with
optimal halo weighting this does not occur until
$M_{\rm min}<10^{12}M_\odot$. In most of the relevant regime, the
optimal mass weighting require 10 or more times fewer surveyed
redshifts than the Poisson formulae would have suggested.
\begin{figure}
\resizebox{\hsize}{!}{
\includegraphics[angle=0]{modes_vs_n.both.eps}
}
\caption{The vertical axis is the number of modes $N_m$ that must be
measured for a 1\% measure of the bias of an optimal mass estimator using
the cross-correlation of lensing and galaxy redshift survey. It is
plotted vs the density of sources in the redshift survey. The solid
lines are for an optimal survey which targets one galaxy in each
halo above some mass limit. The dashed lines assume that target
galaxies are predominantly selected from halos with
$b\approx1$. The dashed blue line follows $N_m\propto n^{-5/6}$,
illustrating that the total number of galaxies $\propto nN_m$ required in
the optimal survey is a weak function of survey depth.
}
\label{mvsn}
\end{figure}
Given the high source density required to saturate the accuracy in
$b$, we ask whether it is more efficient to conduct a deep survey or
a shallower survey covering more sky and hence more modes.
Figure~\ref{mvsn} plots the number of modes that must be observed to
determine $b$ to 1\% accuracy, vs the space density $n$ of halos
surveyed, as we lower $M_{\rm min}$ of the survey. We find
$N_m\propto n^{-5/6}$ describes the results well. The total
number of redshifts to be obtained in the survey is $nN_m\propto
n^{1/6}$, hence this measure of the survey's expense depends very
weakly on depth. We also note that the source density required for
the bias measurement is nearly independent of redshift using the
optimal survey strategy. In contrast, a survey of number-weighted
emission-line galaxies, plotted with the dashed lines,
requires $>15\times$ higher $n$ than the optimal strategy at $z=0$,
and galaxy densities that increase to $z=1$.
\subsection{Galaxies as weights}
\label{LRG}
Most measurements of large-scale structure to date have used galaxy
densities as mass estimators. A spectroscopic survey of galaxies can
be thought of as weighting halos by the number of galaxies per halo
\cite{Scoccimarro01}.
The halo occupancy distribution (HOD) gives the probability $p(N|M)$
of finding $N$ galaxies in a halo of mass $M$.
The use of galaxies as mass tracers will be inferior to an optimal
halo-weighting scheme, in the sense of having higher stochasticity $E$
for a given number of redshifts, for three reasons:
\begin{enumerate}
\item Ideally only one redshift per halo is needed, so a galaxy survey
is in some sense wasting spectroscopic resources if more than one
galaxy is targeted per halo.
\item The mean HOD $g(M)\equiv \langle p(N|M) \rangle $ may
not match the optimal halo weighting.
\item The occupancy of a given halo is an integer drawn from the
HOD, which adds a source of stochasticity to the weight assignment
and can propagate into increased stochasticity in the mass estimation,
even if the mean HOD is a close to the optimal weight.
\end{enumerate}
The first penalty is typically not large: the HOD is typically divided
into the probability $f_c(M)$ of having a single central galaxy in the
halo, plus a distribution $p(N_s|M)$ of the number $N_s$ of satellite
galaxies. The latter is typically taken to be a Poisson distribution
with a mean $\bar N_s(M)$. For most galaxy samples, the fraction of
satellites is 10--20\%, a small perturbation to the number of redshift
targets.
This also implies, however, that $\approx 90\%$ of the halos detected
in a survey are occupied by a single target galaxy, and hence given
equal weight. We are then drawn to the second issue: does the mean
HOD $g(M)$ serve as a nearly-optimal weight function?
In Figures~\ref{wvsm}, we plot the mean number of blue galaxies
(in cyan lines) and LRGs (in orange lines) in each halo as a function
of halo mass, as determined from SDSS data.
For a simple HOD in which $f_c$ is a step function, the mean HOD is
\begin{equation}
\label{wsdssxi}
g(M) =
\begin{cases}
1 + \left(\frac{M}{M_1}\right)^{\alpha} & M\ge M_0 \\
0 & M<M_0.
\end{cases}
\end{equation}
$M_1$, $\alpha$, and the cutoff $M_0$ are dependent upon the
luminosity cut or other criteria used to define the galaxy sample.
For luminosity-limited samples, typical values are
$M_1 \approx 20 M_0$ and $\alpha\approx 1$ \citep{Zehavi05, Zehavi10}.
But note the similarity of this mean HOD functional form to
Equations~(\ref{woptsample}) and (\ref{halomodelw})
for the optimal weight.
The optimal weight for a catalog of halos with mass $M>M_{\rm min}$
is very well approximated by a function of the form
$w(M)= 1 + (M/\beta M_{\rm min})^{\alpha}$,
where $\alpha$ and $\beta$ depend on $M_{\rm min}$.
For low $M_{\rm min}$, $\alpha \approx 1$, with $\alpha$ decreasing
at higher $M_{\rm min}$. For $M_{\rm min}$ in the range
$10^{11}$--$10^{13}M_\odot$ at $z=0$, values of $3<\beta<9$ yield
the least stochasticity, with $E$ values indistinguishable from
the optimal weighting.
Hence by a useful coincidence, the mean HOD for luminosity-selected
galaxies bears close resemblance to an optimal halo weighting function,
except that the HOD tends to have a longer flat low-mass plateau
($\beta\approx 20$) than the optimal weight ($\beta\approx 4$).
H10 noted that the optimal weight function they derived is
well approximated by equation~(\ref{wsdssxi}), with $\beta\sim 3$,
but they did not make the connection to galaxy HODs.
How far from optimal are the mean HODs?
We examine the case of luminous red galaxies (LRGs) first.
We obtain the mean number of LRGs from the HOD fitting results of
\citet{Zheng09}, using equation~(B3) in their paper to model the
dependence of model parameter on $\sigma_8$. The mean HOD is
shown in orange in Figure~(\ref{wvsm}).
Figure~\ref{evsn} plots the stochasticity $E$ vs the space density
$n$ of target halos at $k=0.1h$Mpc$^{-1}$ and $z=0$, with the solid
black line showing the best possible result from optimal targeting
and weighting of halos ($M_{\rm min}$ is an implicit parameter for
the black curve). The dashed red line shows the result of using
the mean LRG HOD as a halo weight. A subtlety is that the LRG HOD
does not have a step-function cutoff---the probability $f_c$ of a
central galaxy follows an error function and hence has no single
well-defined $M_{\rm min}$.
The red dashed line shows the result of
varying a low-mass cutoff applied to the LRG HOD.
We find that the $E$ vs $n$ behavior is within 10--20\% of the optimal
result as long as we cut out halos with $f_c<0.1$.
For LRGs, at least, the mean HOD is therefore a good
choice of weight function. We will examine other galaxy
classes below.
To examine the impact of item (iii), the stochasticity of the HOD, on
mass-estimator performance, we populate the halos in the Millennium
simulation with LRGs as per the HOD prescription. We place central
galaxies in the specified fraction of halos, plus a Poisson-distributed
number of satellite galaxies in the halos which host a central galaxy.
The red triangle in Figure~\ref{evsn} shows the stochasticity $E$ vs
the number density of LRGs. We find stochastically occupied halos
achieve a factor of two higher (worse) $E$ than the deterministic
weighting by the mean HOD. While the stochastic LRG HOD requires
$\approx 3\times$ as many redshifts to reach $E=0.5$ as an ideal
survey would require, it is similar to what one would predict from an
optimal survey if the biased-Poisson model were a correct description
of halo stochasticity (dashed black line).
The green solid curve in Figure~\ref{evsn} shows the result of
weighting the halos by the number of galaxies drawn from HODs with
varying minimum galaxy luminosities, from \citet{Zehavi10}. The
behavior is similar to the LRG HOD: even though the mean number of
galaxies per halo looks like the optimal weight, using the actual
number of galaxies as weight results in larger $E$.
The randomness in the number of galaxies in each halo introduces
additional stochasticity that degrades the mass estimator.
If galaxies are to be used to provide optimal reconstructions of
the mass, then they must themselves be weighted in some way so as
to reduce the stochasticity in the weight applied to halos of a
given mass. Determining the optimal mark is an interesting problem
for the future. Formalism for treating this more general problem
has been developed in \citet{Sheth05}, and can be used directly,
but is beyond the scope of this work.
Our results suggest that it is interesting to study how best to supplement
the spectroscopic galaxies with a larger, deeper photometric-redshift
sample. The spectroscopic galaxies can be weighted by the number of
photo-z galaxies consistent with sharing the same halo. The deeper
photo-z catalog potentially has lower stochasticity in halo mass
estimates.
\begin{figure}
\resizebox{\hsize}{!}{
\includegraphics[angle=0]{Evsn.eps}}
\caption{Stochasticity $E$ as a function of the number density of
redshifts obtained in the survey.. The black lines plot
the result of an ideal mass-reconstruction strategy, in
which one redshift is obtained for each halo above a cutoff
mass, and an optimal $w(M)$ is applied to each surveyed
halo. The dotted black curve plots the stochasticity one
would expect from this strategy if halos were a biased
Poisson sampling of the mass---the optimal survey requires
significantly fewer redshifts than predicted by the Poisson
model for the same $E$. The green line gives $E$
resulting from weighting each halo by the number of
galaxies above a luminosity cut---assuming galaxies occupy
halos as per \citet{Zehavi10}. The blue and red triangles
result from galaxy-weighting using the HODs for blue and
luminous red galaxies, respectively, in the SDSS. The LRG
and luminosity-cut surveys are worse than optimal primarily
because of randomness in the halo occupancy; the dashed red
line shows the result of eliminating this randomness by
weighting each halo with the {\em mean} HOD. Blue galaxies
are very inefficient for reconstructing the mass; if
emission-line spectroscopic galaxies are a random 10\% or
1\% subsample of the blue population, they are also
inefficient, not even attaining the $E=0.5$ required for a
volume-limited power spectrum measurement.
}
\label{evsn}
\end{figure}
\begin{figure}
\resizebox{\hsize}{!}{
\includegraphics[angle=0]{E_Mass_up_40bins2.eps}
}
\caption{We plot the stochasticity of an optimal mass estimator
created from a catalog of halos in the Millennium simulation in the
mass range
$10^{12} h^{-1} M_\odot < M < M_{\rm max}$ vs the {\em
upper} limit $M_{\rm max}$ of the halo catalog. This
demonstrates that one cannot improve the stochasticity of the
estimator below some floor unless one detects (and heavily weights)
the rare halos above $\approx 10^{14}h^{-1} M_\odot$.
}
\label{rollup}
\end{figure}
\subsubsection{Surveys with blue or emission-line galaxies}
We have seen that the mean HODs for LRGs and luminosity-selected
galaxies are good approximations to the optimal weight, but the
stochasticity in halo occupation degrades $E$ for a given source
density $n$. Galaxy redshift surveys based on emission line detection
will likely result in substantially different halo weightings, so we
investigate the mass-reconstruction performance of such a survey
relative to an LRG survey or optimal weighting.
We model the emission-line sample by starting with the mean HOD for
blue galaxies
given by equations~(10) and~(11) and Table~4 in \citep{Zehavi05}:
\begin{equation}
g_{blue}(m) = \left(\frac{m}{M_{B}}\right)^{0.8}
+ 0.7\,{\rm e}^{-[2\log(M/10^{12} h^{-1}M_\odot)]^2}
\end{equation}
where $M_B = 7\times10^{13} h^{-1}M_{\odot}$ and $\alpha_B=0.8$
\citep[following][]{Sheth01}. This is plotted as the cyan line in
Figure~\ref{wvsm}.
There is a bump in the
number of blue galaxies between $\sim M^{11}~h^{-1}M_{\odot}$ to
$\sim M^{12}~h^{-1}M_{\odot}$, which is very different from
the optimal weight. The outcome of $E$ from weighting halos in the
Millennium simulation according to galaxy counts drawn from this HOD
is shown in the
blue triangle of Figure~(\ref{evsn}). Although the blue galaxy sample
achieves lower $E$ than the LRG does, notice that it requires $100\times$
more redshifts, i.e., is $100\times$ more costly than an optimally weighted
sample. If we were to under-sample the blue galaxies---e.g. by
obtaining redshifts for ten percent, or one percent of the sample
with the brightest emission lines---then $E$ would rise to 0.54 and
0.86, respectively, if the sub-sampling rate is independent of halo
properties. Clearly, emission line samples are a very inefficient
way to reconstruct the mass, although this disadvantage is countered
by the fact that emission lines can be much stronger and easily
detected relative to LRG absorption features. Optimization of a
survey would need to weigh these effects.
\subsubsection{Surveys that under-weight massive clusters}
Galaxies in massive clusters tend to be strongly deficient in 21-cm
emission and other gas-phase emission lines. A redshift survey
selected by such criteria will tend to miss or under-weight the most
massive clusters. In the Figure~\ref{rollup}, we show that the
absence of high-mass halos from a catalog sets a floor on the
attainable stochasticity even with optimal mass estimation. When
halos with $\sim 10^{12}h^{-1}M_{\odot}<M<M_{\rm max}$ are detected by
the survey, we find that $E \approx 0.5$ is achievable at
$z=0$, $k=0.2h$Mpc$^{-1}$ without detecting massive halos.
However, cosmological inferences requiring lower values of $E$
require a means of detecting and appropriately weighting clusters
above $10^{14}h^{-1}M_\odot$. Surveys without the ability to identify
massive clusters reach a limit in $E$ that cannot be improved by
addition of more low-mass halo detections.
The floor on $E$ can be estimated using equation~(\ref{E2sample}).
Consider the case when our halo catalog only contains objects
{\em below} some $m_d$, so the ``dust'' is the mass in massive objects.
Then $\sum_j n_jv_j^2$ can be large, so $E_{\rm pois}\to 0$.
$\noise_m$ is dominated by the most massive objects: at $z=0$, about
90\% of $\noise_m$ is contributed by objects with masses above
$10^{14}h^{-1}M_\odot$ (we have assumed $\sigma_8 = 0.9$).
If these are missing from the catalog, then $\noise_d\to \noise_m$,
making $E_{\rm opt}\approx (1 + P/\noise_m)^{-1/2}$; as a result,
on scales of $k\sim 0.1h$Mpc$^{-1}$, $E_{\rm opt}$ cannot be made
smaller than about 0.25.
\section{Conclusion and Discussion}\label{discuss}
We have determined the weighting scheme that minimizes the
stochasticity between the linearly weighted halo field and the
associated mass density field. The optimal weight function depends
on the mass range of the halo catalog, how much mass is missing from
the halo catalogue, and how the halos cluster. (I.e., we showed how
the weight is modified by a halo-mass dependent selection function.)
We show that neither mass weighting nor bias weighting of the halos
is optimal. The first principal component of the halo covariance
matrix $\mC$ is also
usually not the optimal, although H10 show that the weakest PC of the
altered noise matrix $\mC-\vb P \vb^T$ is a good approximation to an optimal
weight
when the numbers of halos in all bins are equal.
Rather, we demonstrate that, under very general circumstances,
the optimal weight will be a mix of bias weighting and mass weighting,
simply because the mass is comprised of the mass-weighted halo catalog
plus the mass in the ``dust bin'' of structures below the halo detection
threshold.
The halo model can generally give a reasonably good description for
the optimal weight function and its associated stochasticity with
two important alterations: first, it is necessary to treat the
halos as though they sample a continuous ``halo field'' that is
distinct from the mass distribution, and that they have a bias
$v(M)$ with respect to the halo field that differs from the bias
$b(M)$ with respect to mass.
Second, halo catalogs extending below $10^{12}h^{-1}M_{\odot}$ do not
reconstruct the mass as well as the halo model predicts.
However, we find that the model generally overestimates the optimal
stochasticity, even on the large scales where one might have expected
to find good agreement. We suspect this is due to the combined effects
of non-linear bias and halo exclusion, which our treatment currently
ignores.
We also note that the randomness in the halo shapes at fixed mass---i.e.
ellipticity, concentration, and/or substructure---introduces stochasticity
into a mass estimator built from a halo catalog in which the halo profiles
are reduced to points, setting a lower limit on the attainable $E$.
In the Millennium catalog, this structure stochasticity substantially
degrades the mass estimation for $k>0.2h\,{\rm Mpc}^{-1}$.
Since information about halo shapes is difficult to obtain in
observations, the lower limit on $E$ from point-like halos in simulations
is also the best one can achieve in real observations -- although,
because galaxies are expected to be reasonably faithful tracers of
halo profiles, it may be that they can be used to further reduce $E$
into the non-linear regime.
An optimally weighted halo catalog can have an effective number
density $(nb^2)_{\rm eff}$ up to $15\times$ better (higher) than one
would have predicted for the same halo catalog in a biased-Poisson
model of halo stochasticity. This gain means that a volume-limited
measurement of the linear-regime power spectrum of matter for the
entire observable $z<1$ universe could in principal be accomplished
with only 6 million spectroscopic redshift measurements. Such a
program would require outside information, perhaps a deep imaging
photo-z survey, to identify halos and provide reliable mass estimates
or marks to apply to spectroscopic targets.
(See H10 for an estimate of the effect of mass-estimator
degradation due to a generic log-normal error distribution in the
estimation of halo masses.)
We use halo occupancy distribution models to estimate the
stochasticity $E$ resulting from more traditional surveys which
apply uniform weights to targeted galaxies.
The mean HODs for luminosity-thresholded samples and LRGs are
remarkably useful approximations to
the optimal weight functions. However, additional
stochasticity is introduced into the mass estimator by the random
variations in halo occupancy about the mean.
Hence luminosity-selected or LRG catalogs require $\approx 3\times$
more redshifts to reach a given $E$ than a survey with perfect
knowledge of halo masses for which the optimal weighting can be
applied. In contrast, HODs for blue or emission-line galaxies do
not resemble the optimal weights, and hence require
$\approx 100\times$ more redshift measurements than an
optimally-weighted survey to obtain a given $E$. Random sub-sampling
such galaxies to the same space density as LRGs yields $E$ values
that are $2\times$ larger than those for LRGs.
We also find that low-mass halos cannot reconstruct the shot noise
contributed by the massive halos, setting an upper limit on the
fidelity of the mass reconstruction for surveys that fail to identify
the most massive clusters.
Application of optimal halo weighting can be even more beneficial
for studies of cross-correlation between gravitational potential
(i.e. mass) and other cosmological signals, since these experiments
gain rapidly as the stochasticity $E$ drops below the $E=0.5$ needed
to make volume-limited power-spectrum measurements.
\citet{Park10}, find, for example, that mass weighting halos can greatly
accuracy in estimation of gravitational potential if the halo catalog
extends down to $\approx10^{13}h^{-1}M_\odot$ or lower.
Applying the optimal weight is obviously efficient in reducing noise in
the estimation of BAO from power spectra and in cross-correlation
cosmological tests. In future work we will extend this study to the
use of redshift space distortions to measure the growth rate of
structure \citep[e.g.][]{Okumura10}. We are also investigating the
potential for galaxy marking and non-linear mass estimators to further
improve the ability to trace large-scale structure with observational
data.
\section*{ACKNOWLEDGMENT}
We thank Roman Scoccimarro for providing the NYU simulations. This work is supported by DOE grant DE-FG02-95ER40893, and grants AST-0908027 and AST-0908241 from the National Science Foundation. The Millennium simulation used in this paper was carried out as part of the programme of the Virgo Consortium on the Regatta supercomputer of the Computing Centre of the Max-Planck-Society in Garching. We thanks John Helly for helping accessing the Millennium database. YC thanks the hospitality of the Institute for Computational Cosmology in Durham University when this work was finishing.
\bibliographystyle{mn2e}
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1,108,101,564,551 | arxiv |
\section{Introduction}
\label{sec:Intro}
Accretion disks in black hole X-ray binaries (BHXRBs) and active galactic nuclei (AGN) are typically described by axisymmetric, time-independent models. A key factor that determines the most suitable model is the accretion rate with respect to the Eddington limit. At the Eddington limit, radiation pressure rivals gravity and plays a dynamically important role. The most well-known models are the geometrically thin standard accretion disk model \citep{ss73}, which describes sources accreting at an appreciable fraction of their Eddington limit ($L \gtrsim 0.01-0.1\,L_{\rm Edd}$), and the geometrically thick advection dominated accretion flow model \citep{Narayan1994, Narayan2003}, which describes highly sub-Eddington sources ($L \lesssim 0.01 L_{\rm Edd}$). Slim disk models, in which the advection of trapped photons dominates, work well near and above the Eddington limit (e.g. \citealt{Abramowicz1978}). While these models can provide a reasonable description of the multi-wavelength emission, they -- by construction -- do not address the variability of their light curves and spectra. Such variability can encode important information about the structure of the accretion disk and corona (e.g. \citealt{Kara2019,UttleyCackett2014}).
For example, recent observational evidence indicates that accretion disks in changing-look AGN can undergo drastic changes in luminosity and spectral shape over the course of less than a year (e.g. \citealt{Maclead2016, Yang2018}) or during quasi-periodic eruptions (QPEs) over the course of hours (e.g. \citealt{Qian2018}). In these AGN, the luminosity can change by up to two orders of magnitude while the spectrum significantly hardens or softens. This suggest a dramatic change in the accretion flow on timescales which are too short to be explained by the viscous timescales inferred from the Maxwell stresses created by magnetorotational instability (MRI, \citealt{Balbus1991, Balbus1998}) induced turbulence (e.g. \citealt{Lawrence2018, Dexter2019}). Various physical mechanisms were proposed to reconcile theory with observations, including instabilities in the accretion disk \citep{Sniegowska2020}, magnetically elevated accretion disks \citep{Begelman2007}, reprocessing of point-source radiation by the outer accretion disk (e.g. \citealt{Clavel1992}), disk tearing \citep{Raj2021_AGN} and an orbiting compact object (e.g. \citealt{King2020, Arcodia2021}).
Similar to their changing-look AGN counterparts, the power spectra of BHXRBs display a wide range of mysterious features ranging from broad-spectrum variability to narrow spectral peaks known as quasi-periodic oscillations (QPOs). Such features can encode unique information about the structure and dynamics of the accretion disk in addition to the spin and mass of the black hole. QPOs are typically divided into low- and high-frequency QPOs, which can sometimes be observed together (e.g. \citealt{IngramMotta2020}). Various physical mechanisms have been proposed to explain QPOs. These include geometric effects such as precession of a misaligned disk around the black hole spin axis (e.g. \citealt{Stella1998, Ingram2009}) and intrinsic effects such as parametric resonances (e.g. \citealt{Kluzniak2002, Rezzolla2003, Abramowicz2003}) and disco-seismic modes (e.g. \citealt{Kato2004, Dewberry2019, Dewberry2020}). All of these models make simplifying assumptions about the underlying physics. Most significantly, they typically do not include magnetized turbulence (though see \citealt{Dewberry2020, Wagoner2021}), which can potentially dampen oscillatory modes. Thus, general relativistic magnetohydrodynamic (GRMHD) simulations, which can simulate accretion from first-principles, are very attractive for addressing the origin of QPOs.
\citet{Musoke2022} recently demonstrated using GRMHD simulations (first presented in \citealt{Liska2020}) that accretion disks, which are misaligned with the black hole spin axis, can be torn apart by the Lense-Thirring torques \citep{LT1918} due to the frame dragging of a spinning black hole. Figure~\ref{fig:contourplot_3d} shows that the inner part of an accretion disk tears off the parent disk, leading to two independently precessing sub-disks. Such tearing events lead to spikes in the accretion rate (see also \citealt{Raj2021}). Disk tearing occurs in very thin disks when the viscosity cannot communicate the differential Lense-Thirring torques. This is expected to occur when a disk accretes at an appreciable fraction of the Eddington limit (see e.g. \citealt{Esin1997} for a review) and was observed in hydrodynamic simulations \citep{Nixon2012B, Nixon2012counter, Nealon2015, Raj2021}. Disk tearing can lead to a rapid burst of accretion on timescales shorter than the viscous time and is a promising mechanism to explain large luminosity and spectral swings observed in accreting black holes (e.g. \citealt{Nixon2014, Raj2021_AGN}).
Based on the simulations, \citet{Musoke2022} argued that low-frequency and high-frequency QPOs can be explained by disk tearing. In their model, a disk tears off at a near-constant radius ($r \sim 10^1 r_g$) and precesses for $\lesssim 10$ periods before falling into the black hole. While precessing at $\nu=2.5$~Hz (for $M_{BH}=10M_{\odot}$), the disk emits a periodically modulated light curve. \citet{Musoke2022} also found strong radial epicyclic oscillations at $\nu \sim 56$~Hz in the inner disk, which they argue can potentially explain some high-frequency QPOs. The geometric origin of low-frequency QPOs is supported by recent observations, which suggest that that iron line centroid frequency moves from blue-to-redshifted with the same phase as the low-frequency QPO \citep{Ingram2016_method, Ingram2016_QPO}. On the other hand, observations have been able to constrain neither an intrinsic nor geometric origin of the much rarer high-frequency QPOs (e.g. \citealt{IngramMotta2020}).
Thus, the results of \citet{Musoke2022} suggest that disk tearing is a very promising mechanism to explain the multi-wavelength variability in many accreting black holes. However, all GRMHD simulations that found tearing \citep{Liska2019b, Liska2020} have relied on a cooling function \citep{Noble2009} to artificially set the temperature and scale height of the disk. In reality, the temperature of the disk is determined by complex physics involving viscous heating, radiative cooling and the advection of energy. Such physics can modify the radial and vertical structure of the disk, such as its scale height and vertical temperature profile. This makes it extremely challenging to benchmark such GRMHD models against multi-wavelength observations because they might not capture the important physics that sets the disk thermodynamic state. For example, in radiation pressure supported disks imbalance between the local dissipation rate and cooling rate can lead to thermal and viscous instabilities that are not captured by a cooling function (e.g. \citealt{Lightman1974, Shakura1976, Sadowski2016_mag, Jiang2019, Liska2022}). In addition, when the inner disk precesses, it will beam its radiation field periodically towards the outer disk. This radiation field will scatter off the outer disk, and the radiation back-reaction force can cause additional warping and dissipation (e.g. \citealt{Pringle1997, Wijers1999}).
Here, we present the first radiation-transport two-temperature GRMHD simulation of an accretion disk tilted by $65^{\circ}$ relative to a rapidly spinning $M_{BH}=10M_{\odot}$ black hole of dimensionless spin of $a = 0.9375$ accreting at $35 \%$ the Eddington luminosity. We show that the disk undergoes tearing. The accretion disk does not contain any large scale vertical magnetic flux to begin with and does not launch a jet: this makes the simulation applicable to the soft(-intermediate) states of BHXRBs (see \citealt{FenderBelloni2004}). In Sections~\ref{sec:Numerics} and \ref{sec:Physical_Setup} we describe the numerical setup and initial conditions, before presenting and discussing our results in Sections~\ref{sec:Results} and \ref{sec:discussion}, and concluding in Section~\ref{sec:conclusion}.
\section{Numerical Setup}
\label{sec:Numerics}
In this work we utilize the radiative version of our GPU accelerated GRMHD code H-AMR{} \citep{Liska2018A, Liska2020}. H-AMR evolves the ideal GRMHD equations and radiation transfer equations in addition to the electron and ion entropies. We use a (modified) spherical grid in Kerr-Shild foliation with coordinates $r$, $\theta$ and $\phi$. Spatial reconstruction of primitive variables is performed using a 1-dimensional PPM method, which guarantees second order convergence in 3 dimensions. Magnetic fields are evolved on a staggered grid as described in \citet{Gardiner2005,White2016}. Inversion of conserved to primitive variables is performed using a 2-dimensional Newton-Raphson routine \citep{Noble2006} or Aitken Acceleration scheme \citep{Newman2014} for the energy equation and a 1-dimensional Newton-Raphson routine for the entropy equation. The energy based inversion is attempted first, and, if it fails, the entropy based inversion is used as a backup. This dual energy formulation is now standard in many GRMHD codes \citep{Porth2019}.
The radiative transfer equations are closed with a two-moment M1-closure \citep{Levermore1984} whose specific implementation is described in \citet{Sadowski2013, Mckinney2013}. We additionally evolve (see e.g. \citealt{Noble2009, Ressler2015}) the electron and ion entropy tracers ($\kappa_{e,i}=p_{e,i}/\rho^{\gamma_{e,i}}$) and include Coulomb collisions \citep{Stepney1983}. We use adiabatic indices $\gamma_e=\gamma_i=5/3$ for the electrons and ions (and $\gamma=5/3$ for the gas). This is appropriate for the ion and electron temperatures in the accretion disk. We use a reconnection heating model \citep{Rowan2017} to distribute the dissipative heating between ions and electrons. This leads to roughly $20-40\%$ of the dissipation going into the electrons with the remainder heating the ions.
We include Planck-averaged bound-free, free-free, and cyclo-Synchrotron absorption ($\kappa_{abs}$), emission ($\kappa_{em}$) and electron scattering ($\kappa_{es}$) opacities as given in \citet{Mckinney2017} for solar abundances $X=0.7$, $Y=0.28$, and $Z=0.02$. In addition, we account for Comptonization through a local blackbody approximation \citep{Sadowski2015}. This works reasonably well when emission, absorption and scattering are localized, but becomes inaccurate when the radiation field is anisotropic and/or the energy spectrum deviates from a blackbody. Namely, the radiation temperature used to calculate the absorption opacity is approximated as a blackbody with $T_{r}=(\hat{E}_{rad}/a)^{0.25}$ with $\hat{E}_{rad}$ the fluid frame radiation energy density and $a$ the radiation density constant. If the radiation deviates from a blackbody this approximation can severely under or overestimate the radiation temperature. We assume that only electrons can absorb and emit radiation and set the ion opacity to $\kappa_i=0$. This is a reasonable approximation in the accretion disk where bound-free processes, which involve energy exchange between ions/electrons/radiation, only become important when the density is high enough for Coulomb collisions to equilibrate the ion and electron temperature. The limitations of the M1 closure relevant to our work are further addressed in Section \ref{sec:discussion}.
To resolve the accretion disk in our radiative GRMHD models we use a base grid resolution of $N_r \times N_{\theta} \times N_{\phi} = 840 \times 432 \times 288$. We then use beyond $r \gtrsim 4 r_g$ up to 3 layers of adaptive mesh refinement (AMR) to progressively increase this resolution in the disk from the base resolution at $r \sim 4 r_g$ to $N_r \times N_{\theta} \times N_{\phi} = 6720 \times 2304 \times 4096$ at $r \sim 10 r_g$. (In addition, we use static mesh refinement \citealt{Liska2020} to reduce the $\phi$-resolution to $N_{\phi}=16$ at the polar axis.) This is exactly half of the resolution used in \citet{Liska2020} and \citet{Musoke2022}. Nevertheless, the turbulence is still very well resolved with the number of cells per MRI-wavelength, $Q^{i}=\lambda_{MRI}^{i}/N^{i}$, exceeding $(Q^{\theta}, Q^{\phi}) \gtrsim (20,125)$ within the entire disk. We place the radial boundaries at $R_{in}=1.13 r_g\approx 0.84r_{\rm H}$ and $R_{out}=10^5 r_g$, which ensures that both boundaries are causally disconnected from the flow; here $r_{\rm H} = 1+(1-a^2)^{1/2}r_g$ is the event horizon radius. The local adaptive time-stepping routine in H-AMR{} increases the timestep of each mesh block independent of the refinement level in factors of 2 (up to a factor of 16) based on the local Courant condition \citep{Courant1928}. This increases the effective speed of the simulations several-fold while reducing the numerical noise \citep{Chatterjee2019}.
Since ideal GRMHD is unable to describe the physical processes responsible for injecting gas in the jet funnel (such as pair creation) we use density floors in the drift frame \citep{Ressler2017} to ensure that ${\rho c^2}\ge{p_{mag}}/12.5$. However, due to the absence of any poloidal magnetic flux, no jets are launched in this work and thus the corresponding density floors are never activated. Instead, the ambient medium is floored at the density $\rho=10^{-7}\times r^{-2}$ and internal energy $u_{tot}=10^{-9} \times r^{-2.5}$, where $u_{tot}=u_{gas}+u_{rad}$ is the sum of the gas, $u_{gas}$, and radiation, $u_{rad}$, internal energies.
\begin{figure*}
\centering
\includegraphics[width=1.0\linewidth,trim=0mm 0mm 0mm 0,clip]{contourplot_3d.png}
\vspace{-10pt}
\caption{The first demonstration of disk tearing in a radiation pressure supported accretion disk, as seen through iso-countour 3D renderings of density at three different times. \textbf{(Left panel)} The disk at $t=28,955 r_g/c$ forms a rigid body that is warped, but shows no differential precession. \textbf{(Middle panel)} The disk tears off a precessing inner disk at $t=31,000 r_g$ that precesses for two periods between $t=35,000 r_g/c$ and $t=37,000 r_g/c$. While precessing, the inner disk aligns with the BH, shrinking in size until it fully disappears. \textbf{(Right panel)} The disk forms a new tear at $r \sim 25 r_g/c$ around $t \sim 45,000 r_g/c$, and the tearing process repeats. }
\label{fig:contourplot_3d}
\end{figure*}
\section{Physical Setup}
\label{sec:Physical_Setup}
To study the effect of disk tearing in a highly warped accretion disk we evolve our model in two stages. In both stages we assume a rapidly spinning black hole of mass $M_{BH}=10M_{\odot}$ and dimensionless spin $a=0.9375$. The initial radial ($r$) and vertical ($z$) density profile of the disk is $\rho(r,z) \propto r^{-1}\exp{(-{z^2}/{2h^2})}$. The disk extends from an inner radius, $r_{in}=6 r_g$, to the outer radius, $r_{out}=76 r_g$ at a constant scale height $h/r = 0.02$. The covariant magnetic vector potential is $A_{\theta} \propto (\rho-0.0005)r^{2}$ and normalized to give an approximately uniform $\beta=p_{b}/p_{tot} \sim 7$ in the initial conditions. Here $p_b$ is the magnetic pressure and $p_{tot}$ is the sum of the electron/ion pressures, $p_{e,i}$, and the radiation pressure, $p_{rad}$. The whole setup is subsequently rotated by 65 degrees with respect to the black hole spin axis, which itself is aligned with polar axis of the grid. Physical equivalence between tilting the disk (e.g. \citealt{Liska2018A, White2019A}) and tilting the black hole (e.g. \citealt{Fragile2005, Fragile2007}) is demonstrated in the Appendix of \citet{Liska2018A}.
In the first stage (model T65), which is described in \citet{Liska2020}, we evolve the disk in ideal GRMHD for $\Delta t = 150,000 r_g/c$ with a preset scale height of $h/r=0.02$ set by a cooling function \citep{Noble2009}. The disk undergoes a large tearing event around $t \sim 45,000 r_g/c$, which is analysed in \citet{Musoke2022}. During this tearing event a sub-disk of size $\Delta r \sim 10-15 r_g$ tears off and precesses for $\sim 6$ periods between $t \sim 45,000r_g/c$ and $t \sim 80,000 r_g/c$. While precessing, the inner disk exhibits a very prominent radial epicyclic oscillation at $\nu = 56$~Hz for a 10 solar mass BH. Another tearing event between $t \sim 120,000 r_g/c$ and $t \sim 145,000 r_g/c$ produces a similar oscillation albeit at a slightly higher frequency of $\nu = 69$~Hz. These frequencies are consistent with observed high frequency QPOs (e.g. \citealt{IngramMotta2020}).
In the second stage (model RADT65), which is the focus of this work, we restart this simulation in full radiative GRMHD at $t=29,889 r_g/c$ as described above (after downscaling the resolution by a factor $2$). This is well before the occurrence of any large tearing event in stage 1. We assume thermodynamic equilibrium between gas and radiation by setting $T_r=T_e=T_i$ in the initial conditions and run this model (RADT65) for $\Delta t \sim 14,000 r_g/c$ until $t \sim 45,000 rg/c$. The density scaling is set such that the average accretion rate corresponds to approximately $\dot{M}/\dot{M}_{\rm Edd} \sim 0.35$ with $M_{\rm Edd}=\frac{1}{\eta_{NT}} L_{\rm Edd}/c^2$ the Eddington accretion rate, $L_{\rm Edd}=\frac{4 \pi G M_{BH}}{c k_{es}}$ the Eddington luminosity, and $\eta_{NT}=0.179$ the \citet{Novikov1973} efficiency. According to \citet{Piran2015} this choice of parameters is roughly consistent with a thin disk solution \citep{Sadowski2011} of scale height of $h/r=0.02-0.06$ between $r=r_{isco}$ and $r=100 r_g$.
\begin{figure*}
\centering
\includegraphics[width=1.0\linewidth,trim=0mm 0mm 0mm 0,clip]{space_time.png}
\caption{ \textbf{Panels (a, b)} Space-time diagrams of density, $\rho$, and warp amplitude, $\psi$. The warp amplitude peaks where the disk is about to tear or has already torn. Since dissipation is enhanced in warps, maxima in warp amplitude correspond to minima in density. The main tearing event is denoted by black dotted lines. \textbf{Panels (c, d)} Space-time diagrams of the electron, $T_e$, and ion, $T_i$, temperatures. The ion and electron temperature correlate with the warp amplitude and anti-correlate with the density. Close to the black hole and in the tear the ions are significantly hotter than the electrons since Coulomb collisions are unable to equilibrate the ion and eletron temperatures. \textbf{Panels (e, f)} Space-time diagrams of the tilt, $\mathcal{T}$, and precession, $\mathcal{P}$, angle of the disk. Around $t \sim 36,500 r_g/c$ a disk tears off and precesses for $\lesssim 2$ periods, during which it gradually aligns with the black hole spin axis. }
\label{fig:ST}
\end{figure*}
\section{Results}
\label{sec:Results}
\subsection{Tearing Process}
\label{sec:Tearing}
The disk in RADT65 undergoes a tearing event between $t=31,000 r_g/c$ and $t=40,000 r_g/c$, as visualised using a 3D density isocontour rendering at three different times in Fig.~\ref{fig:contourplot_3d}). Figure~\ref{fig:ST} shows the space-time diagram, in which we calculate the average density of the disk $\bar{\rho}=\frac{\int^{2 \pi}_{0} \int^{\pi}_{0} \rho^{2} \sqrt{-g} d\theta d\phi}{\int^{2 \pi}_{0} \int^{\pi}_{0} \rho \sqrt{-g} d\theta d\phi}$ (Fig.~\ref{fig:ST}a), the warp amplitude, $\Psi=r \frac{\partial \vec{l}}{\partial r}$ with $\vec{l}$ being the angular momentum (Fig.~\ref{fig:ST}b), the average electron and ion temperatures, $T_{e,i}=\frac{\int^{2 \pi}_{0} \int_{0}^{\pi} p_{e,i} \sqrt{-g} d\theta d\phi}{\int^{2 \pi}_{0} \int^{\pi}_{0} \rho \sqrt{-g} d\theta d\phi}$ (Fig.~\ref{fig:ST}c,d), the tilt and precession angles, $T_{disk}$ and $P_{disk}$ (Fig.~\ref{fig:ST}e,f as calculated in \citealt{Fragile2005}).
During this tearing event the disk precesses for $\lesssim 2$ periods between $36,000<t<38,000 r_g/c$ (Fig.~\ref{fig:ST}e) before it aligns with the spin axis of the black hole (Fig.~\ref{fig:ST}f). Alignment happens through warp driven dissipation of misaligned angular momentum on the accretion time of the inner disk, which differs from the Bardeen-Petterson alignment \citep{bp75} mechanism that manifests itself on much shorter timescales during which the structure and density of the disk remains in a steady state (see discussion in \citealt{Liska2018C}). In addition to this `main' tearing event there are indications, such as the formation of an inwards moving ring-like structure and an increase in the warp amplitude around $r \sim 30 r_g$, that a much larger tearing event will occur after the end of this simulation. Due to the large computational expense associated with extending the duration of RADT65, we leave the analysis of this (potentially) much longer tearing event to future work.
Whether a disk is able to tear depends on how the internal torques in the disk react to differential Lense-Thirring precession \citep[e.g.][]{Nixon2012}. The Lense-Thirring precession rate of a point particle around a rapidly spinning black hole follows a steep radial dependence of $\nu_{LT} \propto a/r^3$. However, an accretion disk will precess with a single frequency set by the integrated LT torque in the disk if the viscous torques are able to counteract the differential precession rate (e.g. \citealt{Fragile2005, Fragile2007}). Naively, one expects that the warp amplitude decreases monotonically with the increasing distance from the black hole. This is (roughly) the case in the inner and outer disk of RADT65 for $t \lesssim 32,000 r_g/c$.
As Fig.~\ref{fig:ST}(a),(b) shows, the warp amplitude and surface density in RADT65 are anti-correlated for the entire runtime. At $r \lesssim 20 r_g$ the warp amplitude is rather high and a cavity of low density gas forms at both early and late times. In this cavity the plasma thermally decouples into hot electrons with $T_e \sim 10^8-10^9K$ and extremely hot ions with $T_i \sim 10^9-10^{10} K$. The strong anti-correlation between warp amplitude and density suggests that accretion is driven by dissipation in warps (e.g. \citealt{Papaloizou1983, Ogilvie1999, Nelson2000}). In a companion paper (Kaaz et al 2022, submitted) we demonstrate that warp-driven dissipation is responsible for the bulk of the dissipation and that magneto-rotational instability driven turbulence (e.g. \citealt{Balbus1991, Balbus1998}) plays only a very subdominant role.
However, after $t \gtrsim 32,000 r_g/c$ the warp amplitude is not a monotonically decreasing function of radius. Instead, the warp amplitude forms local minima and maxima, and concentric rings of higher density gas centered at the (local) minima of the warp amplitude form. It is known that when the warp amplitude exceeds a critical value, it can become unstable \citep{Dogan2018, Dogan2020}. When this happens, the viscous torques in the warp drop to zero and the disk tears. In RADT65 the tearing process starts at $r \sim 19 r_g$ around $33,000 r_g/c$ and the disk fully detaches at $r \sim 10 r_g$ around $t \sim 36,500 r_g/c$. We see later that during this process warp-driven dissipation leads to a temporary burst in the BH accretion rate and luminosity around $t \sim 36,500 r_g/c$ (Sec.~\ref{sec:variability}). The disk even (briefly) enters the slim disk regime where $\dot{M}\sim \dot{M}_{edd}$ and $L \sim 0.5 L_{edd}$, suggesting the luminosity is slightly suppressed due to photon trapping (e.g. \citealt{Abramowicz1978}).
\begin{figure}
\centering
\includegraphics[width=\columnwidth,trim=0mm 0mm 0mm 0,clip]{streamers.png}
\vspace{-10pt}
\caption{ A volume rendering of the density during the main tearing event at $t=37,026 r_g/c$. The red plane cuts the disk at the $\phi=0$ surface. Streamers from the outer disk (green) deposit low angular momentum gas onto the inner disk (blue). This causes the inner accretion disk to shrink in size (Kaaz et al. 2022, submitted).}
\label{fig:streamers}
\end{figure}
As the gas from the inner disk falls into the black hole it transports angular momentum outwards. Angular momentum conservation dictates that the radius of the inner disk should increase. This process, called viscous spreading, affects all finite size accretion disks that are not resupplied externally with gas (e.g. \citealt{Liska2018A, Porth2019}). However, the inner disk in RADT65 does not spread viscously, but, in fact, decreases in size. This discrepancy could potentially be explained by cancellation of misaligned angular momentum (e.g. \citealt{Nixon2012, Hawley2019}). Namely, when gas from the outer disk falls onto the inner disk the misaligned components of angular momentum cancel out leading to a net decrease in the inner disk radius. In addition, it is possible that the inner disk transfers some of its excess angular momentum to the outer disk. In a companion paper (Kaaz et al. 2022, submitted) we quantify the relative contributions of the angular momentum cancellation and outward angular momentum transport to the evolution of the tearing radius.
\begin{figure*}
\centering
\includegraphics[width=\linewidth,trim=0mm 0mm 0mm 0,clip]{vertical_plot.png}
\vspace{-10pt}
\caption{A top down view of the accretion disk during the main tearing event at $t=37,026 r_g/c$ corresponding to the volume rendering in Figure \ref{fig:streamers}. \textbf{(Panel a)} The surface density, $\Sigma$, drops sharply in the tear due to additional dissipation as the gas undergoes a radical plane change. \textbf{(Panel b)} The vertically integrated scattering optical depth ($\tau_{es}$) drops below unity in the tear suggesting optically thin emission. \textbf{(Panels c, d)} The electron, $T_e$, and ion, $T_i$, temperatures peak perpendicular to the line of nodes (due to a `nozzle' shock, see Sec.~\ref{sec:rad_sign}) and in the tear at $r \sim 10 r_g$. \textbf{(Panel e)} The density scale height, $\theta_d$, of the accretion disk oscillates in azimuth. It peaks along the x-axis and reaches a minimum along the y-axis. \textbf{(Panel f)} The bolometric luminosity, $\lambda$, forms a non-axisymmetric pattern ranging from $\lambda \sim 0.15 L_{\rm Edd}$ to $\lambda \sim 0.5 L_{\rm Edd}$. \textbf{(Panels g, h)} The gas entropy entropy, $s_{gas}$, does not increase substantially when gas crosses the scaleheight compression along the vertical y-axis, but the radiation entropy, $s_{rad}$, does. This suggests the shock heating is radiatively efficient.}
\label{fig:vertical_plot}
\end{figure*}
\subsection{Dissipation and Radiative Signatures}
\label{sec:rad_sign}
To better understand the structure of the disk during the tearing event at $t=37,026 r_g/c$ we show a volume rendering of the density in Figure~\ref{fig:streamers} and a vertical projection of several quantities in Figure \ref{fig:vertical_plot}. In this projection the line of nodes, where the disk crosses the equatorial plane, is aligned with the horizontal x-axis. The projected quantities include the surface density $\Sigma=\int_{0}^{\pi} \rho \sqrt{g_{\theta \theta}} d\theta$ (panel a), the scattering optical depth $\tau_{es} \sim 0.34 \Sigma$ (panel b), electron and ion temperature $T_{e,i}=\frac{\int_{0}^{\pi}p_{e,i}\sqrt{g_{\theta \theta}}d\theta}{\Sigma}$ (panels c and d), density scale height $\theta_{d}=\frac{\int_{0}^{\pi} \rho(\theta - \pi/2) \sqrt{g_{\theta \theta}}d\theta}{\Sigma}$ (panel e), effective bolometric luminosity $\lambda=\frac{\int_{0}^{\pi} R^{r}_{t} \sqrt{g_{\theta \theta}} 2 \pi d\theta}{L_{edd}}$ (panel f), gas entropy per unit mass $s_{gas}=\frac{\int_{0}^{\pi}p_{e,i} /\rho^{\gamma} \rho u^{t} \sqrt{g_{\theta \theta}}d\theta}{\Sigma}$ (panel e), and radiation entropy per unit mass $s_{rad}=\frac{\int_{0}^{\pi} E_{rad}^{3/4} \sqrt{g_{\theta \theta}}d\theta}{\Sigma}$ (panel h). Integration is performed in a spherical coordinate system aligned with the local angular momentum vector of the disk. An animation of Figure \ref{fig:streamers} and panels a, c and f of Figure \ref{fig:vertical_plot} can be found on our YouTube channel (\href{https://www.youtube.com/watch?v=wJwBLtvHLT8}{movie}).
Interestingly, the electron and ion temperatures do not decline smoothly with distance from the black hole (Fig.~\ref{fig:vertical_plot}b,c) as predicted by idealized models of thin accretion disks \citet{Novikov1973}. At the tearing radius of $10 r_g$, where the warp amplitude reaches a maximum, the electron temperature reaches $T_e \gtrsim 10^8K$ and the plasma becomes optically thin. Here streamers of gas get thrown onto highly eccentric orbits and subsequently rain down on the inner disk (Fig.~\ref{fig:streamers}). The rapid rise in temperature in the tear suggests that gas crossing the tear is subject to additional dissipation as it undergoes a rapid orbital plane change. If we assume the gas has a temperature of $T_i=0K$ before crossing the tear at $r \sim 7 r_g$, and the gas dissipates $\epsilon=10^{-2}$ of its orbital kinetic energy $U_{kin}\approx\frac{1}{2r}$, we find that dissipation can easily heat the gas to $T_i \sim 10^{10} K$. In reality, $\epsilon$ can be much higher if radiative cooling is efficient.
The scale height of the disk exhibits a prominent $m=2$ azimuthal oscillation (Fig.~\ref{fig:vertical_plot}e). This can also be observed in Figure \ref{fig:phiplot} where we show a $\theta-\phi$ slice at $r=25 r_g$ of the density, electron temperature, and gas entropy. At the line of nodes the scale height reaches a maximum of $h/r \sim 0.1$ while perpendicular to the line of nodes the scale height drops to $h/r \sim 0.005$. Around this nozzle-like `compression', the gas reaches a temperature ranging from $T_e \sim 2 \times 10^7 K$ at $r \sim 20 r_g$ to $T_e \gtrsim 10^8K$ within $r \lesssim 5 r_g$. At select times $T_e$ can even reach $T_e \sim 10^9K$ closer to the black hole (see YouTube \href{https://www.youtube.com/watch?v=wJwBLtvHLT8}{movie}). The radial and azimuthal fluctuations in the temperature lead to a highly non-axisymmetric emission pattern (Fig.~\ref{fig:vertical_plot}f). At this specific snapshot ($t=37,026 r_g/c$) the effective bolometric luminosity varies azimuthally between $\lambda \sim 0.15 L_{edd}$ and $\lambda \sim 0.5 L_{edd}$ at $r=200 r_g$.
\begin{figure}
\centering
\includegraphics[clip,trim=0.0cm 0.0cm 0.0cm 0.0cm,width=\columnwidth]{phiplot.png}
\vspace{-10pt}
\caption{A $\theta-\phi$ slice through the disk at $r=15 r_g$ in tilted coordinates with the ascending node at $\phi=1/2\pi$ and the descending node at $\phi=3/2\pi$. The pink and black lines correspond to the $\tau_{es}=1$ and $\tau_{bf}=1$ surfaces respectively. \textbf{(Upper panel)} The gas density, $\rho$, peaks in the `nozzle', which is located perpendicular to the local line of nodes at $\phi=0$ and $\phi=\pi$. \textbf{(Middle panel)} The electron temperature, $T_e$, increases by a factor $\sim 5$ in the nozzle, suggesting the radiative emission will be hardened. \textbf{(Lower panel)} The gas entropy per unit mass, $\kappa_{gas}$, does not increase in the nozzle, suggesting the rise in gas temperature is adiabatic (Sec. \ref{sec:rad_sign})}
\label{fig:phiplot}
\end{figure}
The specific gas entropy does not increase in the nozzle (Fig.~\ref{fig:vertical_plot}g), suggesting heating of gas in the nozzle is adiabatic. This might seem surprising since nozzles in tidal disruption events (TDEs) are typically associated with shock heating and steep gradients in the gas entropy (e.g. \citealt{Rees1988, Kochanek1994, Andalman2020}). However, if the radiative cooling timescale is very short, shocks do not automatically lead to an increase of the gas entropy. Instead, they will lead to an increase in the radiation entropy. This increase in radiation entropy is visible along the vertical y-axis in Fig.~\ref{fig:vertical_plot}(g). In a companion paper (Kaaz et al 2022, submitted), we demonstrate that a shock forms in the nozzle that dissipates roughly $\sim 1.8\%$ of the orbital energy each time the gas passes through it.
\subsection{Outflows and Variability}
\label{sec:variability}
To understand the variability associated with disk tearing and the energetics of the outflows we plot in Figure~\ref{fig:timeplot} the mass accretion rate (panel a), radiative emission rate (panel b) and (radiative) outflow efficiencies (panel c). The mass accretion rate measured at the event horizon ($\dot{M}=\int_{0}^{2\pi} \int_{0}^{\pi} \rho u^r \sqrt{-g} d\theta d\phi$ with $g$ the metric determinant and $u^{\mu}$ the fluid 4-velocity) exhibits a factor $\sim 40$ variation in time, ranging from $\dot{M}=2 \times 10^{-2} \dot{M}_{\rm Edd}$ to $\dot{M}=\dot{M}_{\rm Edd}$ (defined in Sec.~\ref{sec:Physical_Setup}). However, the luminosity ($L=\int_{0}^{2\pi} \int_{0}^{\pi} R^{r}_{t} \sqrt{-g} d\theta d\phi$ with $R^{r}_{t}$ the $r$-$t$ component of the radiation stress energy tensor) only exhibits a factor $\sim 5$ variation and peaks at $L \sim 0.5 L_{edd}$.
The outflow efficiencies with respect to the BH mass accretion rate are defined at the event horizon as $\eta_{wind}=\frac{\dot{M}-\dot{E}}{|\dot{M}|_{t}}$, $\eta_{rad}=\frac{L_{r=r_{200 r_g}}}{|\dot{M}|_t}$, and $\eta_{adv}=\frac{-L_{r=r_{BH}}}{|\dot{M}|_t}$. Here $\dot{E}=\int_{0}^{\pi} \int_{0}^{2\pi} T^{r}_{t} \sqrt{-g} d\theta d\phi$ is the energy accretion rate with $T^{r}_{t}$ the stress energy tensor, and, $|\dot{M}|_t$ is a running average of $\dot{M}$ over a time interval of $500 r_g/c$. Unless stated otherwise, we measure $\dot{M}$ and $\dot{E}$ at the event horizon, $L$ at the event horizon for $\eta_{adv}$ and at $r=200 r_g$ for $\eta_{rad}$. As Fig.~\ref{fig:timeplot}(c) shows, when the accretion rate reaches the Eddington limit the advective efficiency, $\eta_{adv}$, is of comparable magnitude to the radiative efficiency, $\eta_{rad}$ (Fig.~\ref{fig:timeplot}d). This implies that a significant fraction of the emitted radiation will fall into the black hole before it reaches the observer suggesting the disk might have entered the slim disk regime at the peak of luminosity. Nevertheless, the time-averaged radiative efficiency reaches $|\eta_{rad}|=14.7 \%$, which is only slightly below the canonical \citet{Novikov1973} efficiency of $\eta_{NT}=17.8\%$.
We do not observe any evidence for a radiation or magnetic pressure driven wind. While the outflow efficiency measured at the event horizon exceeds $\eta_{wind}\gtrsim 15 \%$ this drops to $\eta_{wind} \lesssim 0.1 \%$ when measured at $r=200 r_g$, which demonstrates that both the magnetic pressure (e.g. \citealt{Liska2018C}) and radiation pressure (e.g. \citealt{Kitaki2021}) are insufficient to accelerate the wind to escape velocities. This suggests that poloidal magnetic fields are a key ingredient to accelerate sub-relativistic outflows, even when the accretion rate approaches the Eddington limit.
\begin{figure}
\centering
\includegraphics[clip,trim=0.0cm 0.0cm 0.0cm 0.0cm,width=\columnwidth]{timeplot.png}
\vspace{-10pt}
\caption{Time evolution of RADT65 with the main tearing event denoted by black dotted lines. \textbf{(Panel a)} The mass accretion rate increases from $\dot{M} \sim 10^{-2} \dot{M}_{\rm Edd}$ to $\dot{M} \sim 1 \dot{M}_{\rm Edd}$. \textbf{(Panel b)} The luminosity increases from $L \sim 0.1 L_{\rm Edd}$ to $L \sim 0.5 L_{\rm Edd}$. \textbf{(Panel c)} The tilt $\mathcal{T}$ of the accretion disk. The inner accretion disk ($r \lesssim 10 r_g$) aligns with the black hole just after the main tearing event while the outer accretion disk remains misaligned. \textbf{(Panel d)} The wind efficiency, $\eta_{wind}$, radiative efficiency, $\eta_{rad}$, advective radiative efficiency, $\eta_{adv}$, and \citet{Novikov1973} efficiency, $\eta_{NT}$, and total efficiency, $\eta_{tot}$, as defined in Sec.~\ref{sec:variability}. When the accretion rate peaks, advection of radiation becomes important and suppresses the total luminosity of the system. \textbf{(Panel e)} The thermal scale height of the disk, $\theta_{t}$, stays stable or increases throughout the disk, which suggests the disk is thermally stable (Sec.~\ref{sec:stability}).}
\label{fig:timeplot}
\end{figure}
\subsection{Radial Structure}
\label{sec:rad_structure}
To better understand the structure of the accretion disk we show in Figure~\ref{fig:radplot} time-averaged (between $t=[43,000, 44,000] r_g/c$) radial profiles of the density, $\bar{\rho}$ (panel a), luminosity, $L$ (panel b), and radiation, ion, electron, and magnetic pressure, $p_{rad}$, $p_i$, $p_e$, and $p_{b}$, respectively (panel c). During this time interval a new tear starts to form around $r \sim 30 r_g$, which manifests itself as a drop in density and increase in the radiative flux. While the magnetic and radiation pressure are in approximate equipartition within $r \lesssim 20 r_g$, the accretion disk becomes fully radiation pressure dominated between $r=[20,40] r_g$. The sum of the ion and electron pressure is a factor $\sim 10-100$ lower than the radiation pressure, suggesting that gas pressure does not play a dynamically important role.
We have already mentioned in Section \ref{sec:Tearing} that warp driven dissipation plays an important role in RADT65. To illustrate how much warp-driven dissipation accelerates the inflow of gas in RADT65, we translate the radial infall speed of the gas into an effective viscosity, $\alpha_{\rm eff}=\frac{\langle v^{r}v^{\phi}\rangle_{\rho}}{\langle c_{s}^{2}\rangle_{\rho}}$ with $v^{r}=u^{r}/u^{t}$ and $v^{\phi}=u^{\phi}/u^{t}$. Here, $\langle x\rangle_y=\frac{\int_{0}^{2\pi}\int_{0}^{\pi} x y \sqrt{-g} d\theta d\phi d\phi}{\int_{0}^{2\pi}\int_{0}^{\pi} y \sqrt{-g} d\theta d\phi}$ gives the $y-$weighted average of quantity $x$. In Fig.~\ref{fig:radplot}(c), we find $\alpha_{\rm eff} \sim 1-5 \times 10^1$ throughout the disk. The Maxwell stresses $\alpha_{\rm m}=\frac{\langle b^{r}b^{\phi}\rangle_{\rho}}{\langle p_{tot}\rangle_{\rho}} \sim 10^{-2}-10^{-1}$ induced by magnetized turbulence cannot account for this effective viscosity, which suggests that other physical processes such as shocks (e.g. \citealt{Fragile2008}) drive accretion in warped accretion disks.
\subsection{Thermal Stability}
\label{sec:stability}
Interestingly, \citet{Liska2022} found that an aligned version of the accretion disk in RADT65 collapsed within $t \lesssim 5,000 r_g/c$ to an exceedingly thin slab that could not be resolved numerically. This runaway cooling (see also \citealt{Sadowski2016_mag,Fragile2018, Jiang2019, Mishra2022}) is a manifestation of the thermal instability \citep{Shakura1976}. However, we do not observe any evidence for the thermal instability in RADT65. In fact, as Fig.~\ref{fig:timeplot}(e) demonstrates, the thermal scale height, $\theta_{t}=\frac{\langle c_s\rangle_{\rho}}{\langle v^{\phi}\rangle_{\rho}}$, remains stable or even increases depending on the radius.
This unexpected result could potentially be explained within $r\lesssim 20 r_g$ by magnetic fields, which can stabilize a thermally unstable accretion disk if they reach equipartition with $p_b \sim p_{rad}$ (e.g. \citealt{Begelman2007, Sadowski2016_mag, Jiang2019, Liska2022, Mishra2022}). However, magnetic fields will not be able to stabilize the disk between $r=[20,40] r_g$ where the disk is strongly radiation pressure dominated with $p_{rad}/p_{b} \sim 10^1-10^2$.
We found in Sec.~\ref{sec:variability} that advection of radiation becomes significant at the event horizon. If the radiation diffusion timescale becomes longer than the accretion timescale, photon trapping might stabilize the disk against runaway cooling (e.g. \citealt{Abramowicz1978}). To test if photon trapping can explain the apparent thermal stability of the disk we plot the radiation diffusion time, $t_{\rm diff}\sim \frac{h}{v_{rad}} \sim 3 h \tau_{es}$ and the accretion time, $t_{\rm acc}=\frac{r}{\langle v_r\rangle_{\rho}}$, in Fig.~\ref{fig:radplot}(e). Here we use for $v_{rad}$ the optically thick radiation diffusion speed $v_{\rm rad} \sim \frac{1}{3 \tau_{es}} c$ (e.g. \citealt{Mckinney2013}). As is evident from Fig.~\ref{fig:radplot}(e), while the radiation diffusion and accretion timescales are comparable in the inner disk, the radiation diffusion timescale becomes much shorter than the accretion timescale in the outer disk. This suggests that photon trapping will not be able to stabilize the outer disk against runaway cooling or heating.
Instead, we propose in Section \ref{sec:discussion} that nozzle shock driven accretion disks are inherently thermally stable.
\begin{figure}
\centering
\includegraphics[clip,trim=0.0cm 0.0cm 0.0cm 0.0cm,width=\columnwidth]{radplot.png}
\vspace{-10pt}
\caption{Radial structure of the disk time-averaged between $t=[43,000; 44,000] r_g/c$. This is the end state of our simulation at which point a new tear starts to form. \textbf{(Panel a)} The gas density, $\bar{\rho}$, peaks around $r \sim 25 r_g$. We expect the new tear to form just behind the peak in density. \textbf{(Panel b)} The cumulative luminosity of the disk increases until $35 r_g$, partially driven by enhanced dissipation in the forming tear between $r=20 r_g$ and $r=30 r_g$. \textbf{(Panel c)} The disk is radiation pressure, $p_{rad}$, dominated between $r=[15, 40] r_g$ and magnetic pressure, $p_b$, dominated everywhere else. The ion, $p_i$), and electron, $p_e$, pressures are very weak compared to either the radiation or magnetic pressure and thus do not play an dynamically important role. \textbf{(Panel d)} The effective viscosity, $\alpha_{\rm eff}$, is much larger than the Maxwell, $\alpha_{\rm m}$, stress. This suggests that dissipation is not driven by magnetized turbulence induced by magnetic stresses. Instead, we demonstrate in a companion paper that shocks are driving accretion (Kaaz et al 2022, submitted). \textbf{(Panel e)} The photon diffusion timescale, $t_{\rm diff}$, is shorter than the accretion timescale, $t_{\rm acc}$, except very close to the black hole. This suggest advection of radiation internal energy is only dynamically important very close to the black hole. }
\label{fig:radplot}
\end{figure}
\section{Discussion}
\label{sec:discussion}
\subsection{Applicability to Spectral States}
The accretion disk in RADT65 does not launch any relativistic jet. This suggests RADT65 is applicable to the high-soft and potentially soft-intermediate spectral states. In these spectral states the spectrum is dominated by thermal blackbody radiation and no radio jets are detected (though see \citealt{Russel2020} for detection of compact jets in the infrared).
Nevertheless, the emission profile of RADT65 presents a radical departure from the \citet{Novikov1973} model of a thin accretion disk. Instead of an axisymmetric power-law emission profile, emission in warped accretion disks can become hardened around the nozzle shock or at the tearing radius. Here the temperature of plasma can reach $T_e \sim 10^8-10^9 K$, which might lead to Comptonization of cold accretion disk photons. This could potentially contribute to the high-energy power law emissions tail observed in the spectra of soft state BHXRBS (e.g. \citealt{Remillard2006}) in addition to the soft X-Ray excess in the spectra of high-luminosity AGN (e.g. \citealt{Gierlinski2004, Gierlinski2006}). Note that part of this high-energy emission can come from within the ISCO as suggested by recent GRMHD simulations \citep{Zhu2012} and semi-analytical models \citep{Hankla2022}. Future general relativistic ray-tracing (GRRT) calculations will need to quantify how much the spectrum deviates from a blackbody spectrum and in which wavelength bands this optically thin shock-heated gas can be detected. The energy spectrum of the accelerated electrons will depend on the microphysics of the shocks, which will need to be addressed by particle-in-cell simulations.
In the hard-intermediate state (HIMS) jetted ejections imply that the accretion disks might be saturated by vertical magnetic flux (e.g. \citealt{Mckinney2012, Tchekhovskoy2011, Chatterjee2020}). When an accretion disk is saturated by magnetic flux it enters the magnetically arrested disk (MAD) regime \citep{Narayan2003} in which magnetic pressure overcomes gravity and accretion proceeds through non-axisymmetric instabilities. Recent two-temperature radiative GRMHD simulations \citep{Liska2022} have demonstrated that magnetic flux saturation can lead to `magnetic' truncation during which the inner disk decouples into a magnetic pressure supported `corona' of hot electrons ($T_e \gtrsim 5 \times 10^8 K$) and extremely hot ions ($T_i \gtrsim 10^{10}K$). This happens within the magnetic truncation radius, which is determined by the amount and location of the excess poloidal magnetic flux in the system. It is conceivable that the combined effects of magnetic reconnection in current sheets and shocks in warps can heat the plasma to temperatures far above what was observed in this work or in \citet{Liska2022}. Future work will need to address if, where and how such misaligned truncated disks tear, and, if their spectral and timing signatures are consistent with sources in the HIMS.
\subsection{Quasi-periodic Variability}
This work and \citet{Musoke2022} have demonstrated that disk tearing can lead to short luminosity outbursts of length $\Delta t \sim 1000-3000 r_g/c$ every $\Delta t_{tear} \sim 10,000-50,000 r_g/c$. Coincident with a tearing event the disk precesses on a timescale of $\Delta t_{prec} \sim 1000-6000 r_g/c$ suggesting tearing events might be accompanied by a sinusoidial modulation of the light curve. In parallel, the mass-weighted radius of the precessing inner disk oscillates on a timescale of $\Delta t_{osc} \sim 150-400 r_g/c$ suggesting another sinusoidal modulation of the lightcurve albeit at a higher frequency (Fig. \ref{fig:radiusplot} and \citealt{Musoke2022}). We summarize the timescales of these variability phenomena in Table \ref{table:variability} for a BHXRB ($M_{BH}=10 M_{\odot}$), a low-mass AGN ($M_{BH} = 10^6 M_{\odot}$), and a high-mass AGN ($M_{BH} = 10^9 M_{\odot}$). Please note that these timescales might vary by at least an order of magnitude depending the location of the tearing radius, which we expect will depend on e.g. the accretion rate, black hole spin, misalignment angle, and amount of large scale magnetic flux.
\begin{figure}
\centering
\includegraphics[clip,trim=0.0cm 0.0cm 0.0cm 0.0cm,width=\columnwidth]{radiusplot.png}
\vspace{-10pt}
\caption{The mass weighted radius, $r_{M}=\frac{\int_{0}^{2 \pi} \int_{0}^{\pi} \Sigma r^3 \sqrt{-g} d\theta d\phi}{\int_{0}^{2 \pi} \int_{0}^{\pi} \Sigma r^2 \sqrt{-g} d\theta d\phi}$, of the inner disk (after tearing) in RADT65 (blue) features a radial epicyclic oscillation consistent with the radial epicyclic frequency of an oscillating ring of gas at $r=6.9 r_g$ (orange). Such oscillations may be associated with HFQPOs detected in BHXRB lightcurves.}
\label{fig:radiusplot}
\end{figure}
Table \ref{table:variability} suggests that the timescales associated with disk tearing are consistent with variety of astrophysical phenomena. The tearing events themselves might be associated with the heartbeat modes (on $1-100$~s timescales) detected in the BHXRBs GRS 1915+105 and IGR J17091–3624 (e.g. \citealt{Belloni2000, Neilsen2011, Weng2018, Altamirano2011}) in addition to QPEs detected in various AGN (e.g. \citealt{Miniutti2019, Giustini2020, Arcodia2021, Miniutti2022}). If the tearing events in some spectral states are spaced further apart, they could potentially also explain some changing-look phenomena observed in AGN with masses of $10^7-10^9 M_{\odot}$ that occur on timescales of years to decades (e.g. \citealt{Sniegowska2020}). In all of these cases the luminosity fluctuates by a factor $10^{0}-10^{2}$ suggesting some significant flaring event in the accretion disk reminiscent of a tearing event. The precession of the disk that follows each tearing event can potentially explain low frequency QPOs detected in numerous BHXRB with $\nu \sim 10^{-1}-10^{1}$~Hz (e.g. \citealt{IngramMotta2020}). Precession is also consistent with the $44$ day LFQPO detected in the $2 \times 10^8 M_{\odot}$ AGN KIC 9650712 \citep{Smith2018} and the $3.8H$ LFQPO detected in the $10^5-10^6 M_{\odot}$ AGN GJ1231+1106 \citep{Lin2013}. In addition, the oscillation of the mass-weighted radius might be associated with HFQPOs of $\nu \sim 10^1-10^2$~Hz in BHXRBs and some of the short duration QPOs detected in various AGN. These include the $\sim 1H$ QPO in the lightcurve of the $10^6-10^7 M_{\odot}$ AGN RE J1034+396 \citep{Gierlinski2008, Czerny2016}, the $1.8H$ QPO in the $5 \times 10^6 M_{\odot}$ MRK 766 \citep{Zhang2017}, and the $2H$ QPO in the $4 \times 10^6 M_{\odot}$ AGN MS 2254.9-3712 \citep{Alston2015}.
The heated gas at the tearing radius suggests the QPO emission will be substantially hardened through Comptonization. This is consistent with frequency-resolved spectroscopy of BHXRBs which has demonstrated that the Type-C QPO waveform is dominated by hardened emission \citep{Sobolweska2006} and recent observations of AGN that suggest an increase in the soft-X-ray excess during QPEs \citep{Giustini2020}. Interestingly, our simulations suggest that tearing events are accompanied by a low-frequency sinusoidal signal caused by precession of the inner disk and a high-frequency sinusoidial signal caused by a radial epicyclic oscillation at the tearing radius. This can potentially be tested using observational data for both BHXRBs and AGN.
\begin{table}
\begin{center}
\caption{Time period between large tearing events, $\Delta t_{tear}$, disk precession period, $\Delta t_{prec}$, and period of radial epicyclic oscillations, $\Delta t_{osc}$, for various black hole masses. These timescales can be associated with various variability phenomena in BHXRBs and AGN such as QPEs, QPOs, and heartbeat oscillations (Sec.~\ref{sec:discussion}). }
\label{table:variability}
\begin{tabular}{|c|c|c|c|}
\hline
\textbf{$M_{BH}$} & \textbf{$\Delta t_{tear}$} & \textbf{$\Delta t_{prec}$} & $\Delta t_{osc}$\\
\hline
N/A & $50,000 r_g/c$ & $6000 r_g/c$ & $400 r_g/c$ \\
\hline
$10M_{\odot}$ & $2.5s$ & $0.3s$ & $20 ms$\\
\hline
$10^6 M_{\odot}$ & $70h$ & $9h$ & $0.5h$\\
\hline
$10^9 M_{\odot}$ & $8yr$ & $1yr$ & $20d$\\
\hline
\end{tabular}
\end{center}
\end{table}
\subsection{Comparison to Semi-analytical Models}
Semi-analytical models typically divide warped accretion disks into two categories (e.g. \citealt{Papaloizou1983}) depending on the scale height of the disk relative to the $\alpha-$viscosity parameter. In the first category bending waves excited by oscillating radial and vertical pressure gradients in the warp are damped and the warp is propagated through viscous diffusion only. This happens when the scaleheight is smaller than the viscosity parameter, e.g. $h/r<\alpha$. In this regime instability criteria were derived that predict the tearing of accretion disks (e.g. \citealt{Ogilvie2000, Dogan2018, Dogan2020}). In the second category the warp is propagated through bending waves that travel at a fraction of the sound speed \citep{Papaloizou1995}. These bending waves are excited by pressure gradients in a warp. Recent work has made substantial progress in describing warp propagation in the bending wave regime (e.g. \citealt{Ogilvie2013_parametric, Paardekooper2018, Deng2020, Ogilvie2022}), but has not yet come up with criteria for disk tearing.
Interestingly, \citet{Deng2022} predicted using the affine model of \citet{Ogilvie2018} that oscillations in the vertical pressure gradient lead to an azimuthal $m=2$ oscillation in the scale height of the disk similar to what is observed in our work. These authors also demonstrated that a parametric instability can induce turbulence and create ring-like structures in warped accretion disks. Since $h/r \sim \alpha_m$ (Fig.~\ref{fig:radplot}d) and the warp is highly non-linear it is unclear in which regime of warp propagation our disk falls (see also \citealt{Sorathia2013a, Hawley2019}). We do note though that there is some evidence for wave-like structures in the disk (Fig.~\ref{fig:waves}), which suggests the disk might be in the bending wave regime of warp propagation. Nevertheless, future work will need to isolate wave-like from diffusive (misaligned) angular momentum transport to make more direct comparison to semi-analytical models possible.
\begin{figure}
\centering
\includegraphics[clip,trim=0.0cm 0.0cm 0.0cm 0.0cm,width=\columnwidth]{alphaplot.png}
\vspace{-10pt}
\caption{A scatterplot of the effective viscosity, $\alpha_{\rm eff}$, with respect to the thermal scaleheight, $\theta_{t}$ at $r=5 r_g$ in green, $r=10 r_g$ in black, and $r=20 r_g$ in blue. We show power-law fits to the data as solid lines with fitting function $\alpha_{\rm eff} \propto (h/r)^{-\gamma}$. The effective viscosity increases as the disk becomes thinner, which suggests warp-driven dissipation becomes more important for thinner disks. We argue in Section~\ref{sec:stability2} that warp-driven dissipation can stabilize the disk against runaway thermal collapse (e.g. \citealt{Shakura1976}).}
\label{fig:alphaplot}
\end{figure}
\subsection{Thermal and Viscous Stability of Warped Disks}
\label{sec:stability2}
In Section \ref{sec:stability} we demonstrated that warped, radiation pressure supported, accretion disks remain thermally stable for much longer than similar non-warped, radiation pressure supported, accretion disks considered in previous work \citep{Liska2022}. We concluded that neither magnetic pressure nor advection of radiation internal energy can explain this thermal stability. Here we propose an alternative explanation.
For a disk to remain thermally stable the derivative of the dissipation rate, $Q_{vis}$, with respect to the temperature needs to be smaller than the derivative of the cooling rate, $Q_{rad}$, with respect to the temperature such that $\frac{Q_{vis}}{dT} < \frac{Q_{rad}}{dT}$. For a radiation pressure supported $\alpha-$disk $Q_{vis} \propto T^{r}_{\phi} H \propto \alpha p_{tot} H \propto \alpha p_{tot}^2 \propto \alpha T^8$ and $Q_{rad} \propto T^4$ \citep{ss73, meier2012}. This suggests that with a constant $\alpha$-viscosity radiation pressure supported accretion disks are thermally unstable. However, if the dissipation rate follows $Q_{vis} \propto T^{\zeta}$ with $\zeta \lesssim 4$ due to warping, the disk will remain thermally stable.
To determine $\zeta$ in our simulation we present in Figure \ref{fig:alphaplot} a scatter plot of the effective viscosity, $\alpha_{\rm eff}$, with respect to the thermal scale height, $\theta_{t}$. We only use data after $32,000 r_g/c$ to allow the disk to adjust after seeding it with radiation and exclude data where the tilt angle is smaller than $\mathcal{T}<60^{\circ}$. We visually fit the data to find that $\alpha_{\rm eff} \propto H^{-\gamma}$ with $\gamma \sim 1.25 - 1.75$. From the relation $H \propto p_{tot} \propto T^4$ we find that $\zeta= 8 - 4 \gamma \sim 1-3$. This suggests that even though our accretion disk is radiation pressure dominated at $r=20 r_g$, warp-driven dissipation can potentially prevent the disk from becoming thermally unstable.
Following a similar argument, one can show that the warp-driven dissipation also avoids the viscous \cite{LightmanEardley1974} instability of radiation pressure dominated disks. If a disk is subject to the viscous inability, it will disintegrate into rings of high density gas (e.g. \citealt{Mishra2016}). Even though high-density rings do form in our simulation (Figs.~\ref{fig:ST} and \ref{fig:vertical_plot}), we conclude that this cannot be a manifestation of the viscous instability since similar rings form in the non-radiative analogue to RADT65 presented by \citet{Musoke2022}. This suggests that the ring formation in RADT65 is warp-induced.
\begin{figure}
\centering
\includegraphics[clip,trim=0.0cm 0.0cm 0.0cm 0.0cm,width=\columnwidth]{waves.png}
\vspace{-10pt}
\caption{A transverse slice through the density along the line of nodes at $50 r_g$. This snapshot is taken beyond the `official' duration of our simulation (Sec.~\ref{sec:Physical_Setup}), just after the onset of the next tearing event. Outwards propagating wave-like structures (visible as vertical `stripes') form in the outer disk suggesting that wave-like angular momentum transport might be present. The vertical extent of the outer disk is artificially inflated because the frame is sliced at an angle with respect to the disk.}
\label{fig:waves}
\end{figure}
\begin{figure*}
\centering
\includegraphics[width=1.0\linewidth,trim=0mm 0mm 0mm 0,clip]{rad_feedback.png}
\caption{Radiation field warps the inner disk, as seen in a transverse slice of density through the $x-z$ plane of our initial conditions (left) and after $\Delta t=2,000 r_g/c$ (right) with radiation streamlines shown in red. Even though our M1 radiation scheme is unable to properly model multi-beam radiation (Sec.~\ref{sec:warping}), these results suggest that radiation warping might be an important effect.}
\label{fig:rad_feedback}
\end{figure*}
\subsection{Radiation Warping}
\label{sec:warping}
When the inner disk precesses it will periodically beam radiation towards the outer disk. This can potentially warp the outer disk and induce additional dissipation (similar to \citealt{Pringle1997, Wijers1999}). In a similar context \citet{Liska2019b} has demonstrated that when a precessing jet collides with the outer accretion disk it can inject energy and angular momentum into the disk, leading to an increase of particle orbits in the outer disk. Since the energy efficiency of the radiative outflows in RADT65 is non-negligible ($|\eta_{rad}| \sim 14.7 \%$) we expect that radiation feedback can have a profound effect on the structure and dynamics of an accretion disk.
Radiative feedback is an interesting avenue of future research that is hard to address in this article due to limitations of our M1 radiation closure \citep{Levermore1984}. M1 works well in modeling radiative cooling and energy transport in optically thick regions, and in optically thin regions for a single radiation bundle. However, M1 is unable to properly model crossing radiation bundles. The reason for this is that M1 treats radiation as a highly collisional fluid and is thus unable to model crossing radiation bundles in optically thin media. Instead the energy and momentum of the two radiation bundles will be summed together. While the M1 closure conserves the total amount of energy and momentum in the photon-fluid, it can lead to a nonphysical redistribution of energy and angular momentum.
Despite these limitations we demonstrate in Figure \ref{fig:rad_feedback} that the structure of the inner accretion disk in RADT65 differs significantly after seeding it with radiation. In Figure \ref{fig:rad_feedback} the radiation streamlines get deflected by the inner disk presumably leading to an increase in the warp amplitude compared to the start of the simulation. In this snapshot the tearing process has not started yet, which suggests that radiation warping can be important even in the absence of disk tearing. While these preliminary results are encouraging, more advanced numerical simulations will be necessary to reliably address radiation warping.
\section{Conclusion}
\label{sec:conclusion}
In this article we have presented the first radiative two-temperature GRMHD simulation of an $65^{\circ}$ misaligned disk accreting at $\dot{M} \sim 0.35 \dot{M}_{edd}$. Similar to the idealized GRMHD models presented in \citet{Liska2019b, Liska2020, Musoke2022} the radiation pressure supported accretion disk tears apart and forms a precessing disk. During the tearing process the mass accretion rate increases by a factor $\sim 10-40$ and the luminosity increases by a factor $\sim 5$, almost exceeding the Eddington limit at the peak of the outburst. The timescales and amplitudes of these luminosity swings are roughly consistent with QPEs in AGN (e.g. \citealt{Miniutti2019, Giustini2020, Arcodia2021, Miniutti2022}) and several heartbeat modes in BHXRBs (e.g. \citealt{Belloni2000, Neilsen2011, Altamirano2011, Weng2018}). Following a tearing event, the inner disk precesses for several periods with a frequency that is consistent with low-frequency QPOs (e.g. \citealt{IngramMotta2020}). While precessing, the inner disk also exhibits a radial epicyclic oscillation of its center of mass whose frequency is consistent with high frequency QPOs (e.g. \citealt{IngramMotta2020}). Future long-duration GRMHD simulations combined with radiative transfer calculations will need to test if disk tearing indeed produces QPO-like variability. This will require the tearing radius to remain stable over many cycles of tearing.
Warping of the disk forms two nozzle shocks directed perpendicular to the line of nodes. At the nozzle shock, the scale height of the disk decreases by a factor of $\sim 5$ and the temperature increases up to $T_e \sim 10^8-10^9 K$ and $T_i \sim 10^9-10^{10}K$. In a companion paper (Kaaz et al 2022, submitted), we demonstrate that dissipation in the nozzle shock leads to a much shorter accretion time scale than predicted by $\alpha$-viscosity based models (e.g. \citealt{ss73}) seeded by magnetized turbulence (e.g. \citealt{Balbus1991}). This can potentially explain the timescales associated with some changing look phenomena in AGN including QPEs. In addition, we find during tearing events that gas crossing the tear heats up to $T_e \sim 10^{8} K$ and $T_i \sim 10^{10} K$. This is caused by an increase in the dissipation rate as the gas undergoes a rapid orbital plane change. These results imply that the emission profile from warped accretion disks will deviate substantially from the \citet{Novikov1973} model of a geometrically thin accretion disk. This can, pending future radiative transfer calculations, have far-reaching consequences for spectral fitting and black hole spin measurements (e.g. \citealt{Zhang1997, Kulkarni2011}).
While radiation pressure supported accretion disks in GRMHD simulations typically collapse (e.g. \citealt{Sadowski2016_mag, Fragile2018, Jiang2019, Mishra2020, Liska2022}) into an infinitely thin slab due to the thermal instability \citep{Shakura1976}, this did not happen in our simulation. Here we argue that the warp and associated nozzle shock stabilize the disk against thermal collapse (Sec.~\ref{sec:discussion}). These shocks differ from spiral density wave induced shocks (e.g. \citealt{Arzamasskiy2018}). In a companion paper (Kaaz et al 2022, submitted), we demonstrate that nozzle shock dissipation drives accretion in geometrically thin warped accretion disks within at least $r \lesssim 50 r_g$. This upends the standard paradigm that invokes magneto-rotational instability induced turbulence for the dissipation of orbital kinetic energy (e.g. \citealt{Balbus1991, Balbus1998}). Future work will need to address the range of spectral states, accretion rates, and misalignment angles where nozzle shock dissipation becomes important.
Past work almost exclusively invoked magnetic reconnection in coronae to explain high energy (non-)thermal emission from accretion disks (e.g. \citealt{Sironi_2014, Beloborodov2017, RipperdaLiskaEtAl2021}). However, this work suggests that heated plasma formed in the nozzle and at the tearing radius can potentially be another source of high energy emission. This is especially appealing to explain non-thermal emission in the high-soft and soft-intermediate states of BHXRBs. Those states do not launch any large-scale jets and are thus unlikely to be saturated by magnetic flux (e.g. \citealt{Mckinney2012}), which might be a key ingredient to form a corona where the prime source of dissipation comes from magnetic reconnection (e.g. \citealt{Liska2022, RipperdaLiskaEtAl2021}). Test-particle methods (e.g. \citealt{Bacchini2019}) informed by first principles particle-in-cell simulations are an interesting avenue for future research to quantify the contribution of magnetic reconnection and warp-driven shocks to the acceleration of non-thermal particles and corona-like emission.
\section{Acknowledgements}
We thank Michiel van der Klis, Adam Ingram, Sera Markoff and Ramesh Narayan for insightful discussions. An award of computer time was provided by the Innovative and Novel Computational Impact on Theory and Experiment (INCITE), OLCF Director's Discretionary Allocation, and ASCR Leadership Computing Challenge (ALCC) programs under award PHY129. This research used resources of the Oak Ridge Leadership Computing Facility, which is a DOE Office of Science User Facility supported under Contract DE-AC05-00OR22725.This research used resources of the National Energy Research
Scientific Computing Center, a DOE Office of Science User Facility
supported by the Office of Science of the U.S. Department of Energy
under Contract No. DE-AC02-05CH11231 using NERSC award
ALCC-ERCAP0022634. We acknowledge PRACE for awarding us access to JUWELS Booster at GCS@JSC, Germany. ML was supported by the John Harvard Distinguished Science Fellowship, GM is supported by a Netherlands Research School for Astronomy (NOVA), Virtual Institute of Accretion (VIA) postdoctoral fellowship, NK by an NSF Graduate Research Fellowship, and AT by the National Science Foundation grants
AST-2206471,
AST-2009884,
AST-2107839,
AST-1815304,
OAC-2031997,
and AST-1911080.
\newpage
|
1,108,101,564,552 | arxiv | \section{INTRODUCTION}\label{sec:intro}
Mixed-morphology supernova remnants (SNRs) are characterized a center-filled thermal X-ray morphology with a synchrotron radio shell, accounting for more than 25\% of Galactic SNRs \citep{1998ApJ...503L.167R}. Most of the mixed-morphology SNRs are interacting with the dense interstellar medium (ISM) evidenced by radio-line emission such as CO, \ion{H}{1}, and/or 1720~MHz OH masers \citep[e.g.,][]{1998ApJ...505..286S,2003ApJ...585..319Y,2018ApJ...864..161K}. In addition, some of them are associated with GeV/TeV gamma-ray sources, which likely arise from interactions between accelerated cosmic-ray (CR) protons and dense clouds in the vicinity of the SNRs \citep[e.g.,][]{2008A&A...481..401A,2016PASJ...68S...5B}. Moreover, shock-propagation into the clumpy ISM and/or dense circumstellar matter (CSM) have a potential to explain their mixed-morphology and thermal X-ray radiation \citep[e.g.,][]{2012PASJ...64...24S,2017ApJ...846...77S,2019ApJ...875...81Z}. Therefore, the shock-interacting ISM plays an important role in understanding their morphology, plasma conditions, and cosmic-ray acceleration \citep[see also reviews by][]{2012A&ARv..20...49V,2020AN....341..150Y,2021arXiv210600708S}. To unveil the physical processes and high-energy phenomena in the mixed-morphology SNRs, detailed comparative studies among the radio-line emission, X-rays, and gamma-rays are needed.
W49B (also known as G43.3$-$0.2) is a well-studied Galactic mixed-morphology SNR with the bright radio-continuum shell and thermal-dominated center-filled X-rays as shown in Figure \ref{fig1}. The SNR is thought to be lying on the far-side of the Galaxy from us \citep[e.g.,][]{1978A&A....67..355L,2001ApJ...550..799B}. The small apparent diameter of $\sim$$3'$--$5'$ is consistent with the larger distance of $\sim$7.5--11.3~kpc \citep[][]{2014ApJ...793...95Z,2018AJ....155..204R,2020AJ....160..263L} and its young age \citep[5--6~kyr,][]{2018AA...615A.150Z}. W49B is also thought to be an efficient accelerator of CR protons owing to its bright GeV/TeV gamma-rays with the pion-decay bump \citep{2018A&A...612A...5H}. The total energy of CR protons was derived to be $\sim$10$^{49}$--10$^{51}$~erg, assuming the targeted gas density of 10--1000~cm$^{-3}$ \citep{2010ApJ...722.1303A}.
\begin{figure}[]
\begin{center}
\vspace*{0.2cm}
\includegraphics[width=\linewidth,clip]{w49b_fig01_v2_rgb.pdf}
\caption{Intensity map of {\it{Chandra}} broad-band X-rays \citep[$E$: 0.5--7.0~keV, e.g.,][]{2013ApJ...764...50L} superposed on the VLA radio continuum contours at 20~cm obtained from MAGPIS \citep{2006AJ....131.2525H}. The contour levels are 5.0, 7.6, 15.4, 28.4, 46.6, and 70.0 mJy beam$^{-1}$. The region enclosed by dashed rectangle corresponds to the observed region with ALMA ACA.}
\label{fig1}
\end{center}
\vspace*{0.5cm}
\end{figure}%
The X-ray radiation of W49B is characterized by three properties: the most luminous in Fe K-shell line emission \citep{2014ApJ...785L..27Y}, non-thermal Bremsstrahlung \citep{2018ApJ...866L..26T}, and recombining (overionized) plasma where the ionization temperature goes even higher than the electron temperature \citep{2009ApJ...706L..71O,2010A&A...514L...2M,2013ApJ...777..145L,2018AA...615A.150Z,2018ApJ...868L..35Y,2020ApJ...893...90S,2020ApJ...903..108H,2020ApJ...904..175S}. The elongated structure of Fe-rich ejecta is believed to be related to a bipolar/jet-driven Type Ib/Ic explosion and/or interactions between the shock and a surrounding interstellar cloud \citep{2007ApJ...654..938K,2013ApJ...764...50L,2017MNRAS.468..140B}. On the other hand, recent X-ray studies on metal abundances favor Type Ia models \citep{2018AA...615A.150Z,2020ApJ...904..175S}. Additionally, \cite{2020ApJ...893...90S} conclude that the SN type is unclear, with neither core-collapse or Ia models perfectly reproducing their best-fit abundances. The origin of recombining plasma---thermal conduction with cold-dense clouds and/or adiabatic cooling---is still being debated \citep[e.g.,][]{2018ApJ...868L..35Y,2020ApJ...893...90S,2020ApJ...903..108H}. If we detect decreasing the electron temperature toward the shocked clouds, we can confirm the thermal conduction scenario as a formation mechanism of the recombination plasma \citep[e.g.,][]{2017ApJ...851...73M,2018PASJ...70...35O,2020ApJ...890...62O}.
Although W49B is thought to be interacting with interstellar clouds, it is a perplexing question which clouds are physically associated. \cite{2007ApJ...654..938K} discovered 2.12~$\mu$m shocked H$_2$ emission toward the eastern and southwestern shells by near-infrared photometric observations. The total mass of shocked H$_2$ is estimated to 14--550~$M_{\sun}$. Subsequent radio observations using CO line emission revealed three molecular clouds at velocities of $\sim$10, $\sim$40, and $\sim$60~km~s$^{-1}$, which are possibly associated with the SNR \citep[e.g.,][]{2014ApJ...793...95Z,2016ApJ...816....1K,2020AJ....160..263L}. \cite{2014ApJ...793...95Z} argued that the molecular cloud at $\sim$40~km~s$^{-1}$ is interacting with W49B because of its wind-bubble like morphology. On the other hand, \cite{2016ApJ...816....1K} found a line-broadening of $^{12}$CO profile in a molecular cloud at $\sim$10~km~s$^{-1}$ located toward the western shell of W49B, and hence the authors claimed that the cloud at 10~km~s$^{-1}$ is interacting with the SNR. The velocity is roughly consistent with the \ion{H}{1} absorption studies \citep{2001ApJ...550..799B,2018AJ....155..204R}. Most recently, \cite{2020AJ....160..263L} performed near-infrared spectroscopy of shocked H$_2$ emission toward four strips on W49B. The authors found that a central velocity of shocked H$_2$ is $\sim$64~km~s$^{-1}$ and then concluded that the molecular cloud at $\sim$60~km~s$^{-1}$ located toward the center and the southwest shell of W49B is likely associated with the SNR. In either case, detailed spatial and kinematic studies as well as deriving cloud properties (e.g., mass, density, kinetic temperature) have not been performed due to the modest sensitivity and angular resolution of CO datasets up to $\sim$$20''$, corresponding to a spatial resolution of $\sim$1~pc at the distance of 10~kpc.
In the present paper, we report on results of new millimeter wavelength observations using CO($J$ = 2--1) line emission with the Atacama Compact Array (ACA, also known as Morita Array) which is a part of the Atacama Large Millimeter/submillimeter Array (ALMA). The unprecedented sensitivity and high-angular resolution of $\sim$$7''$ ($\sim$0.3~pc at the distance of 10~kpc) of the ALMA CO data enable us to identify the interacting molecular cloud and its physical relation to the high-energy phenomena in W49B. Section \ref{sec:obs} describes the observational datasets and reductions. Section \ref{sec:results} comprises five subsections: Sections \ref{subsec:overview}--\ref{subsec:pv} present overview distributions of X-ray, radio continuum, and CO; Section \ref{subsec:ratio}--\ref{subsec:lvg} show physical conditions of molecular clouds. Discussion and conclusions are given in Sections \ref{sec:discussion} and \ref{sec:conclusions}, respectively.
\section{OBSERVATIONS AND DATA REDUCTIONS}\label{sec:obs}
\subsection{CO}\label{subsec:co}
Observations of $^{12}$CO($J$ = 2--1) and $^{13}$CO($J$ = 2--1) line emission were conducted using ALMA ACA Band 6 (211--275 GHz) as a Cycle 6 project (proposal no. 2018.1.01780.S). We used the mosaic observation mode with 10--12 antennas of 7-m array and four antennas of 12-m total power (TP) array. The observed areas were $5\farcm1 \times 2\farcm7$ rectangular regions centered at ($\alpha_\mathrm{J2000}$, $\delta_\mathrm{J2000}$) $=$ ($19^\mathrm{h}11^\mathrm{m}09\fs00$, $+9\arcdeg06\arcmin24\farcs8$) and ($19^\mathrm{h}11^\mathrm{m}07\fs44$, $+9\arcdeg05\arcmin56\farcs2$). The actual observed area is shown in Figure \ref{fig1}. The combined baseline length of 7-m array data is from 8.85 to 48.95 m, corresponding to $u$--$v$ distances from 6.8 to 37.6 $k\lambda$ at 230.538 GHz. Two quasars, J1924$-$2914 and J1751$+$0939, were observed as bandpass and flux calibrators. We also observed four quasars, J1907$+$0127, J1922$+$1530, J1938$+$0448, and J1851$+$0035, as phase calibrators. We performed data reduction using the Common Astronomy Software Application \citep[CASA,][]{2007ASPC..376..127M} package version 5.5.0. We utilized ``tclean'' task with multi-scale deconvolver and natural weighting. The emission mask was also selected using the auto-multithresh procedure \citep{2020PASP..132b4505K}. We combined the cleaned 7-m array data and the calibrated TP array data using ``feather'' task to recover the missing flux and diffuse emission. The beam size of feathered data is $8\farcs23 \times 4\farcs77$ with a position angle of $-75.31\degr$ for the $^{12}$CO($J$ = 2--1) data, and $8\farcs28 \times 5\farcs04$ with a position angle of $-79.37\degr$ for the $^{13}$CO($J$ = 2--1) data. The typical noise fluctuations are $\sim$0.065 K for the $^{12}$CO($J$ = 2--1) data and $\sim$0.055 K for the $^{13}$CO($J$ = 2--1) data at the velocity resolution of 0.4 km~s$^{-1}$.
We also used archival datasets of $^{12}$CO($J$ = 1--0) and $^{12}$CO($J$ = 3--2) line emission for estimating physical properties of molecular clouds. The $^{12}$CO($J$ = 1--0) data are from the FOREST Unbiased Galactic Plane Imaging survey with the Nobeyama 45~m telescope \citep[FUGIN,][]{2017PASJ...69...78U}, and the $^{12}$CO($J$ = 3--2) data are from the CO High-Resolution Survey \citep[COHRS,][]{2013ApJS..209....8D} obtained with the James Clerk Maxwell Telescope (JCMT). The angular resolution is $\sim$$20''$ for the $^{12}$CO($J$ = 1--0) data and $\sim$16\farcs6 for the $^{12}$CO($J$ = 3--2) data. The velocity resolutions of $^{12}$CO($J$ = 1--0) and $^{12}$CO($J$ = 3--2) data are 1.3 and 1.0 km s$^{-1}$, respectively. To improve the signal-to-noise ratio of the $^{12}$CO($J$ = 3--2) data, we combined four spatial pixels and a rebind pixel size is to be $12''$. The typical noise fluctuations are $\sim$1.4 K for the $^{12}$CO($J$ = 1--0) data and $\sim$0.18 K for the $^{12}$CO($J$ = 3--2) data at each velocity resolution.
\subsection{Radio Continuum}\label{subsec:rc}
The radio continuum data at 20 cm wavelength are from the Multi-Array Galactic Plane Imaging Survey \citep[MAGPIS,][]{2006AJ....131.2525H} obtained with the Very Large Array (VLA) and the Effelsberg 100-m telescope. The angular resolution is $\sim$$6''$, which is compatible for the ALMA ACA resolution. The typical noise fluctuations are $\sim$1--2 mJy.
\subsection{X-rays}\label{subsec:xrays}
We utilized archival X-ray data obtained by {\it{Chandra}} (observation IDs are 117, 13440, and 13441), which have been published in numerous papers \citep[e.g.,][]{2005ApJ...631..935K,2009ApJ...691..875L,2009ApJ...706L.106L,2011ApJ...732..114L,2013ApJ...777..145L,2013ApJ...764...50L,2009ApJ...692..894Y,2016ApJ...821...20K,2018AA...615A.150Z}. The X-ray datasets were taken with the Advanced CCD Imaging Spectrometer S-array (ACIS-S3). We used Chandra Interactive Analysis of Observations \citep[CIAO,][]{2006SPIE.6270E..1VF} software version 4.12 with CALDB 4.9.1 \citep[][]{2007ChNew..14...33G} for data reduction and imaging. After reprocessing for all datasets using the ``chandra\_repro'' procedure, we created an energy-filtered, exposure-corrected image using the ``fluximage'' procedure in the energy bands of 0.5--7.0~keV (broad-band, see Figure \ref{fig1}), 0.5--1.2 keV (soft-band), 1.2--2.0 keV (medium band), 2.0--7.0 keV (hard-band), and 4.2--5.5 keV (continuum band). Because the soft- and medium-band images are heavily affected by interstellar absorption \cite[e.g.,][]{2018AA...615A.150Z}, in this paper we focus on the X-ray images at energies greater than 2.0 keV. We also created an exposure-corrected, continuum-subtracted image of Fe He$\alpha$ line emission (6.4--6.9 keV) following the method presented by \cite{2013ApJ...764...50L}. The typical angular resolution of {\it{Chandra}} images is $\sim$0\farcs5.
\begin{figure*}[]
\begin{center}
\includegraphics[width=\linewidth,clip]{w49b_fig02_v8_rgb.pdf}
\caption{Integrated intensity maps of ALMA ACA $^{12}$CO($J$ = 2--1) ({\it{upper panels}}) and $^{13}$CO($J$ = 2--1) ({\it{lower panels}}) for (a, b) the 10~km~s$^{-1}$ cloud, (c, d) the 40~km~s$^{-1}$ cloud, and (e, f) the 60~km~s$^{-1}$ cloud. The integration velocity range is from 1 to 15 km~s$^{-1}$ for the 10~km~s$^{-1}$ cloud; from 38 to 47 km~s$^{-1}$ for the 40~km~s$^{-1}$ cloud; and is from 57 to 67 km~s$^{-1}$ for the 60~km~s$^{-1}$ cloud. Superposed contours are the same as shown in Figure \ref{fig1}.}
\label{fig2}
\end{center}
\end{figure*}%
\section{RESULTS}\label{sec:results}
\subsection{Overview of X-ray, Radio Continuum, and CO Distributions}\label{subsec:overview}
Figure \ref{fig1} shows the {\it{Chandra}} broad-band X-ray image of W49B superposed on the VLA radio continuum contours at 20 cm wavelength. As presented in previous studies, a barrel-shaped radio-continuum shell with several co-axis filaments and the center-filled X-rays are seen \citep[e.g.,][]{2007ApJ...654..938K,2009ApJ...691..875L,2011ApJ...732..114L}. The X-ray elongated feature brighter than $\sim5 \times 10^{-7}$ photon cm$^{-2}$ s$^{-1}$ is roughly consistent with the spatial distribution of Fe He$\alpha$ emission (see Figure 3 in \citeauthor{2013ApJ...764...50L}~\citeyear{2013ApJ...764...50L}). We note that overall distributions of X-rays and radio continuum are quite different between the northeastern and southwestern halves: the shell boundary of the northeastern half roughly coincides with each other, whereas the southwestern shell of X-rays is significantly deformed compared to that of radio continuum. In particular, the X-rays are dim around positions at ($\alpha_\mathrm{J2000}$, $\delta_\mathrm{J2000}$) $\sim$ ($19^\mathrm{h}11^\mathrm{m}00\fs0$, $+09\arcdeg06\arcmin00''$), ($19^\mathrm{h}11^\mathrm{m}03\fs0$, $+09\arcdeg05\arcmin00''$), and ($19^\mathrm{h}11^\mathrm{m}08\fs0$, $+09\arcdeg06\arcmin06''$): the first two correspond to the bright peaks of radio-continuum and the other is partially surrounded by radio filaments. This trend is also seen in the 4.2--5.5 keV band image which is mostly free from the interstellar absorption as well as line-emission.
\begin{figure*}[]
\begin{center}
\includegraphics[width=\linewidth,clip]{w49b_fig03ab_v3_rgb.pdf}
\caption{Velocity channel maps of ALMA ACA $^{12}$CO($J$ = 2--1) ({\it{upper panels}}) and $^{13}$CO($J$ = 2--1) ({\it{lower panels}}) for each cloud. Each panel shows CO integrated intensity distribution integrated over the velocity range from 1 to 15 km s$^{-1}$ every 2.8 km s$^{-1}$ for the 10~km~s$^{-1}$ cloud; from 38 to 48 km s$^{-1}$ every 2 km s$^{-1}$ for the 40~km~s$^{-1}$ cloud; and 57 to 67 km s$^{-1}$ every 2 km s$^{-1}$ for the 60~km~s$^{-1}$ cloud. Superposed contours are the same as shown in Figure \ref{fig1}.}
\label{fig3}
\end{center}
\end{figure*}%
\setcounter{figure}{2}
\begin{figure*}[]
\begin{center}
\includegraphics[width=\linewidth,clip]{w49b_fig03c_v3_rgb.pdf}
\caption{{\it{Continued.}}}
\end{center}
\end{figure*}%
Figure \ref{fig2} shows integrated intensity maps of $^{12}$CO($J$ = 2--1) and $^{13}$CO($J$ = 2--1) for three velocity ranges of 1--15 km s$^{-1}$ (hereafter ``10 km s$^{-1}$ cloud''), 38--47 km s$^{-1}$ (hereafter ``40 km s$^{-1}$ cloud''), and 57--67 km s$^{-1}$ (hereafter ``60 km s$^{-1}$ cloud'') as previously mentioned in several papers \citep[e.g.,][]{2014ApJ...793...95Z,2016ApJ...816....1K,2020AJ....160..263L}. The kinematic distance of molecular cloud is $\sim$11 kpc for the 10 km s$^{-1}$ cloud; $\sim$9 kpc for the 40 km s$^{-1}$ cloud \footnote{Although \cite{2014ApJ...793...95Z} suggested the distance of 40 km s$^{-1}$ cloud to be $\sim$10~kpc using a Galactic rotation curve model with $R_0 = 8.5$~kpc and $\Theta_0 = 220$~km s$^{-1}$ \citep{1986MNRAS.221.1023K}, we adopt its distance to be $\sim$9~kpc using the latest Galactic parameters of $R_0 = 7.92$ kpc and $\Theta_0 = 227$ km s$^{-1}$ \citep{2020PASJ...72...50V}. Here $R_0$ is the distance from the Sun to the Galactic center and $\Theta_0$ is the rotation velocity of the local standard of rest. We use the latter values throughout the paper.}; and $\sim$7 kpc for the 60 km s$^{-1}$ cloud \citep{2020arXiv201111916S}. Note that there are no other CO clouds within the velocity range from $-15.0$ to 92.6 km s$^{-1}$, and hence we focus on the three molecular clouds in the present paper.
In the 10 km s$^{-1}$ cloud (Figures \ref{fig2}a and \ref{fig2}b), there is an intensity gradient increasing from southeast to northwest. The radio continuum shows fairly good spatial correspondence with molecular clouds in the northern inward protrusion and along the north, northwest, and southwest rims. On the other hand, both the $^{12}$CO and $^{13}$CO emission lines are faint in the southeastern shell where shocked H$_2$ emission is strongly detected \citep[e.g.,][]{2007ApJ...654..938K}. Dense clouds traced by $^{13}$CO emission are located not only outside the shell boundary, but also inside the radio continuum shell.
In the 40 km s$^{-1}$ cloud (Figures \ref{fig2}c and \ref{fig2}d), the $^{12}$CO emission has a relatively uniform distribution rather than that of the 10 km s$^{-1}$ cloud, but the $^{13}$CO emission shows an intensity gradient increasing from northwest to southeast. We note that the radio brightest shell in west shows lack of both the $^{12}$CO and $^{13}$CO emission lines. The bright $^{12}$CO emission and $^{13}$CO clumps are located both toward the southwest of the SNR as well as inside the southeastern shell. The southwestern CO clumps seem to be along the sharp edge of radio shell, whereas the CO clumps inside the SNR show no significant spatial correlations with the radio shell morphology. At this spatial coverage, we could not find the bubble-like CO structure toward W49B as mentioned by \cite{2014ApJ...793...95Z}.
In the 60 km s$^{-1}$ cloud (Figures \ref{fig2}e and \ref{fig2}f), there are dense clouds across the SNR from northeast to southwest with two bright CO peaks at ($\alpha_\mathrm{J2000}$, $\delta_\mathrm{J2000}$) $\sim$ ($19^\mathrm{h}11^\mathrm{m}17\fs0$, $+09\arcdeg07\arcmin30''$) and ($19^\mathrm{h}10^\mathrm{m}57\fs5$, $+09\arcdeg05\arcmin07''$). The former contains two \ion{H}{2} regions cataloged by \cite{2009A&A...501..539U}, whereas the latter does not have any cataloged objects. The diffuse CO emission inside the SNR appears to be spatially anti-correlated with the radio continuum contours.
To derive masses of three molecular clouds, we used the following equations:
\begin{eqnarray}
M = m_{\mathrm{H}} \mu \Omega D^2 \sum_{i} N_i(\mathrm{H}_2),\\
N(\mathrm{H}_2) = X \cdot W(\mathrm{CO}),
\label{eq1}
\end{eqnarray}
where $m_\mathrm{H}$ is the mass of hydrogen, $\mu = 2.8$ is the mean molecular weight, $\Omega$ is the solid angle of each pixel, $D$ is the distance to W49B, $N_i(\mathrm{H}_2)$ is the column density of molecular hydrogen for each pixel, $X$ is CO-to-H$_2$ conversion factor of $2 \times 10^{20}$~cm$^{-2}$ (K~km~s$^{-1})^{-1}$ \citep{1993ApJ...416..587B}, and $W$(CO) is the velocity integrated intensity of $^{12}$CO($J$ = 1--0) emission line obtained from the FUGIN data \citep{2017PASJ...69...78U}. We estimated the mass of molecular cloud inside the radio shell to be $\sim$$4.1 \times 10^4$ $M_{\sun}$ for the 60 km s$^{-1}$ cloud and $\sim$$2.7 \times 10^4$ $M_{\sun}$ for the other two clouds, where we adopted the shell radius of 2\farcm5 centered at ($\alpha_\mathrm{J2000}$, $\delta_\mathrm{J2000}$) $=$ ($19^\mathrm{h}11^\mathrm{m}07\fs34$, $+09\arcdeg06\arcmin01\farcs1$). These values are roughly consistent with previously derived cloud masses using the $^{13}$CO($J$ = 1--0) emission and the $^{13}$CO-to-H$_2$ conversion factor \citep{2018A&A...612A...5H}.
\subsection{Detailed Spatial Comparison with the Radio Continuum Shell}\label{subsec:comprc}
Figure \ref{fig3} shows velocity channel maps for each molecular cloud superposed on the radio continuum contours. We find that dense $^{13}$CO clumps at a velocity rang from 3.8 to 9.4 km s$^{-1}$ are nicely along not only with the northern and southern shell, but also with radio filaments inside the SNR at positions of ($\alpha_\mathrm{J2000}$, $\delta_\mathrm{J2000}$) $\sim$ ($19^\mathrm{h}11^\mathrm{m}06\fs4$, $+09\arcdeg07\arcmin03''$) and ($19^\mathrm{h}11^\mathrm{m}05\fs5$, $+09\arcdeg05\arcmin56''$). We also find that both the $^{12}$CO and $^{13}$CO clouds at the velocity range of 9.4--12.2 km s$^{-1}$ show global anti-correlation with the radio shell. In addition, $^{12}$CO emission at the velocity range of 12.2--15.0 km s$^{-1}$ shows a good spatial correspondence with the western shell especially for the sharp edge of the southwestern rim. This velocity range is consistent with that inferred from Kilpatrick et al. (2016). The southwestern CO clumps at 42.0--44.0 km s$^{-1}$ seem to be along the sharp edge of radio shell especially prominent in the $^{13}$CO line emission. Moreover, the $^{13}$CO clouds at 38.0--40.0 km s$^{-1}$ shows the good spatial correspondence with the southeastern half of the radio continuum shell. Furthermore, the 63--65 km s$^{-1}$ $^{12}$CO map shows a lack of CO emission along the eastern shell and shows bright CO emission in the gaps between the western and northern radio contours. Although the other CO clouds also appear to be overlapped with the radio continuum shell and filaments, their spatial correspondence is not clear.
\begin{figure*}[]
\begin{center}
\includegraphics[width=\linewidth,clip]{w49b_fig04_v8_rgb.pdf}
\caption{Position-velocity diagram of ALMA ACA $^{12}$CO($J$ = 2--1) ({\it{upper panels}}) and $^{13}$CO($J$ = 2--1) ({\it{lower panels}}) for (a, b) the 10~km~s$^{-1}$ cloud, (c, d) the 40~km~s$^{-1}$ cloud, and (e, f) the 60~km~s$^{-1}$ cloud. The integration range of Right Ascension is from 283\fdg76 to 287\fdg79. Dashed curves and circles delineate expanding gas motion (see the text). Vertical dashed lines indicate integration velocity ranges for each cloud.}
\label{fig4}
\end{center}
\end{figure*}%
\subsection{Position--Velocity Diagrams}\label{subsec:pv}
Figure \ref{fig4} shows position--velocity diagrams for the three molecular clouds. The velocity distributions of $^{12}$CO and $^{13}$CO emission are similar to each other except for a velocity at $\sim$40 km s$^{-1}$, suggesting that the $^{12}$CO emission at $\sim$40 km s$^{-1}$ is subject to self-absorption due to optically thick component. In fact, the $^{12}$CO emission at $\sim$40 km s$^{-1}$ shows dip-like feature, whereas the $^{13}$CO spectrum at the same velocity has strong line emission. We also find incomplete and complete cavity-like structures in the 10 and 40 km s$^{-1}$ clouds, respectively (see dashed curves in Figures \ref{fig4}a--\ref{fig4}d). The incomplete cavity (or arc-like distribution) in the 10 km s$^{-1}$ cloud lies from 5 km s$^{-1}$ to 11 km s$^{-1}$ with a velocity dispersion of a few km s$^{-1}$. On the other hand, the complete cavity in the 40 km s$^{-1}$ cloud is clearly seen especially for $^{12}$CO emission, whose velocity range is from 41 km s$^{-1}$ to 47 km s$^{-1}$. Note that spatial extents of these cavities are roughly consistent with the diameter of the radio continuum shell. By contrast, there is no clear evidence for such cavity-like structure in the position--velocity diagram of the 60 km s$^{-1}$ cloud (see Figures \ref{fig4}e and \ref{fig4}f).
\subsection{Intensity Ratio Maps}\label{subsec:ratio}
Figure \ref{fig5} shows intensity ratio maps of $^{12}$CO($J$ = 3--2) / $^{13}$CO($J$ = 2--1) (hereafter $R_\mathrm{12CO32/13CO21}$) toward the three molecular clouds. A higher value of $R_\mathrm{12CO32/13CO21}$ tends to be observed in diffuse warm gas thermalized by supernova shocks and/or stellar radiation assuming that the abundance ratio of $^{12}$CO/$^{13}$CO is constant within a molecular cloud complex \citep[cf.][]{2011ApJS..196...18B,2020A&A...644A..64D}. We find that high-intensity ratios of $R_\mathrm{12CO32/13CO21}$ $\sim$10 are distributed toward the shell of W49B in the 10 km s$^{-1}$ cloud. The southwestern edge of the shell (hereafter SW-edge) shows the highest value of $R_\mathrm{12CO32/13CO21}$ $\sim$20, where the radio continuum shell is strongly deformed. On the other hand, the 40 km s$^{-1}$ cloud shows no significant enhancement of $R_\mathrm{12CO32/13CO21}$ toward the SNR shell. The southeast shell in the 60 km s$^{-1}$ cloud also shows higher values of $R_\mathrm{12CO32/13CO21}$, while the regions with high-intensity ratios continuously extend beyond the radio-shell boundary and may not be related to W49B.
\setcounter{figure}{4}
\begin{figure*}[]
\begin{center}
\includegraphics[width=\linewidth,clip]{w49b_fig05_v7_rgb.pdf}
\caption{Intensity ratio maps of $^{12}$CO($J$ = 3--2) / $^{13}$CO($J$ = 2--1) for (a) the 10~km~s$^{-1}$ cloud, (b) the 40~km~s$^{-1}$ cloud, and (c) the 60~km~s$^{-1}$ cloud. Each data was smoothed to match the beam size 16\farcs6. The velocity range of each cloud and superposed solid contours are the same as shown in Figure \ref{fig1}. Superposed dashed contours indicate $^{13}$CO($J$ = 2--1) integrated intensities, whose lowest contour levels are 3 K km s$^{-1}$ and contour intervals are 1.0 K km s$^{-1}$ for the 10~km~s$^{-1}$ cloud; 0.5 K km s$^{-1}$ for the 40~km~s$^{-1}$ cloud; and 1.5 K km s$^{-1}$ for the 60~km~s$^{-1}$ cloud. The gray areas represent that the $^{12}$CO($J$ = 3--2) and/or $^{13}$CO($J$ = 2--1) data show the lower significance than $\sim$7$\sigma$. Yellow crosses (Reference SW/NW, SW/NE-shell, central-filament) and a dashed circle (SW-edge) discussed in Section \ref{subsec:lvg} are indicated. Green squares and triangles also indicate the positions of \ion{H}{2} regions and IRAS point sources, respectively \citep{1988iras....1.....B,2009A&A...501..539U}.}
\label{fig5}
\end{center}
\end{figure*}%
\subsection{Physical conditions of the molecular clouds}\label{subsec:lvg}
To reveal physical conditions for each cloud in detail, we performed the Large Velocity Gradient (LVG) analysis \citep[e.g.,][]{1974ApJ...189..441G,1974ApJ...187L..67S}. The LVG analysis can calculate the radiative transfer of molecular line emission, assuming a spherically isotropic cloud with uniform photon escape probability, temperature, and radial velocity gradient of $dv/dr$. Here, $dv$ is the half-width half-maximum of CO line profiles and $dr$ is a cloud radius. We selected six individual CO peaks for each cloud, which are significantly detected both the $^{12}$CO and $^{13}$CO emission lines. Four of them appear to be along the radio continuum shell or filaments (hereafter refer to as ``shell clouds''), and the others were selected as reference which are located outside of the shell (hereafter refer to as ``reference clouds''). CO spectra toward each position are shown in Figure \ref{fig6}. We adopt $dv/dr = {2.5}$ km s$^{-1}$ pc$^{-1}$ for the 10 km s$^{-1}$ SW-shell and the 40 km s$^{-1}$ SE-shell; {1.5} km s$^{-1}$ pc$^{-1}$ for the 60 km s$^{-1}$ SW-{edge}, NW-shell, and Reference-{SW}; and {1.0} km s$^{-1}$ pc$^{-1}$ for the others. We also utilized the abundance ratio of [$^{12}$CO/H$_2$] = $5 \times 10^{-5}$ \citep{1987ApJ...315..621B} and the isotope abundance ratio of [$^{12}$CO/$^{13}$CO] = 49 \citep{1990ApJ...357..477L}.
\begin{figure*}[]
\begin{center}
\includegraphics[width=\linewidth,clip]{w49b_fig06_v5_rgb.pdf}
\caption{CO intensity profiles toward individual CO peaks in (a--f) the 10~km~s$^{-1}$ cloud, (g--l) the 40~km~s$^{-1}$ cloud, and (m--r) the 60~km~s$^{-1}$ cloud. The CO spectra enclosed by vertical lines represent individual CO peaks that are focused on the present study. Each CO data was smoothed to match the beam size 16\farcs6 and the velocity resolution of 1 km s$^{-1}$. In SW-edge spectra of the 10~km~s$^{-1}$ cloud, we combined four pixels around the highest intensity ratio of $^{12}$CO($J$ = 3--2) / $^{13}$CO($J$ = 2--1) as shown in Figure \ref{fig5}a.}
\label{fig6}
\end{center}
\end{figure*}%
\begin{figure*}[]
\begin{center}
\includegraphics[width=\linewidth,clip]{w49b_fig07_v11_rgb.pdf}
\caption{LVG results on the number density of molecular hydrogen, $n$(H$_2$), and the kinetic temperature, $T_{\mathrm{kin}}$ for each cloud as shown in Figure \ref{fig6}. The red lines and blue dash-dotted lines indicate the intensity ratios of $^{12}$CO($J$ = 3--2) / $^{12}$CO($J$ = 2--1) and $^{12}$CO($J$ = 3--2) / $^{13}$CO($J$ = 2--1), respectively. The shaded areas in red and blue represent $1\sigma$ error ranges of each intensity ratio. Yellow crosses indicate the best-fit values of $n$(H$_2$) and $T_{\mathrm{kin}}$ for each cloud. The results are summarized in Table \ref{tab1}.}
\label{fig7}
\end{center}
\end{figure*}%
Figure \ref{fig7} shows the LVG results on the number density of molecular hydrogen, $n$(H$_2$), and the kinematic temperature, $T_\mathrm{kin}$, toward the six positions for each cloud. The best-fit values of $n$(H$_2$) and $T_\mathrm{kin}$ are summarized in Table \ref{tab1}. In the 10 km s$^{-1}$ cloud, we find that the shell clouds show $T_\mathrm{kin}$ $\sim$20--{60} K, which are significantly higher than that of the reference clouds ($T_\mathrm{kin} = {15}$ K). By contrast, all shell clouds in both the 40 km s$^{-1}$ and 60 km s$^{-1}$ components show $T_\mathrm{kin}$ $\sim$10 K which are roughly consistent with their reference clouds except for Reference-SE in the 60 km s$^{-1}$ ($T_\mathrm{kin} = {21}$ K). We also note that there is no relation between the number density of molecular hydrogen and kinetic temperature for each cloud.
\section{DISCUSSION}\label{sec:discussion}
\subsection{Molecular Clouds Associated with W49B}\label{subsec:mc}
Previous studies proposed three candidates of molecular clouds which are interacting with the SNR W49B: namely the 10 km s$^{-1}$ cloud, 40 km s$^{-1}$ cloud, and the 60 km s$^{-1}$ cloud \citep{2014ApJ...793...95Z,2016ApJ...816....1K,2020AJ....160..263L}. Their claim is mainly based on three elements: (1) a line-broadening feature of CO emission at $\sim$10 km s$^{-1}$, (2) a wind-bubble like morphology of CO cloud at $\sim$40 km s$^{-1}$, and (3) a central velocity of shocked H$_2$ line emission at $\sim$64 km s$^{-1}$. In this section, we discuss which cloud is the most likely to be associated with W49B in terms of spatial distributions, presence of the expanding gas motion, and physical conditions of CO clouds.
\subsubsection{Spatial Distributions of CO Clouds}
We first emphasize that the 10 km s$^{-1}$ cloud shows a clear spatial correspondence with the radio continuum shell and filaments (see Figure \ref{fig3} and Section \ref{subsec:comprc}). In particular, a majority of CO clouds at $\sim$10 km s$^{-1}$ are located along the outer boundary of the radio continuum shell: $^{13}$CO clouds in the northern shell at 6.6--9.4 km s$^{-1}$, and arc-like $^{12}$CO clouds in the southwestern shell at 12.2--15.0 km s$^{-1}$. Moreover, $^{13}$CO clumps at 3.8--9.4 km s$^{-1}$ spatially coincide well with radio filaments inside the shell. Such spatial correspondence is naturally expected as a result of shock--cloud interactions. According to magneto-hydrodynamical (MHD) simulations, interactions between supernova shocks and clumpy clouds enhance turbulent magnetic field up to $\sim$1 mG on the surface of shocked clouds, where the synchrotron radio/X-ray radiation becomes brighter \citep{2009ApJ...695..825I,2012ApJ...744...71I,2019MNRAS.487.3199C}. This was further supported by several observations toward the Galactic and Magellanic SNRs \citep[][]{2013ApJ...778...59S,2017ApJ...843...61S,2017JHEAp..15....1S,2019ApJ...873...40S,2020ApJ...904L..24S,2018ApJ...863...55Y,2018ApJ...864..161K}, and hence it should not be surprising that shock--cloud interactions with the magnetic field amplification occurred in W49B as well.
However, we cannot rule out the possibility of shock-interaction with the 40 km s$^{-1}$ and 60 km s$^{-1}$ clouds from the spatial comparative studies alone. In fact, molecular clouds at 38.0--40.0 km s$^{-1}$ and 42.0--44.0 km s$^{-1}$ show good spatial correspondences with the southeastern half and southwestern shell of the SNR, respectively (see Figure \ref{fig3}). The 63.0--65.0 km s$^{-1}$ CO map also shows a good anti-correlation with the radio shell, which is not inconsistent with the picture of magnetic field amplification via the shock--cloud interaction.
\subsubsection{Shock and Wind Induced Expanding Gas Motion}
We argue that the cavity-like structures in the position-velocity diagrams at the 10 km s$^{-1}$ and 40 km s$^{-1}$ clouds provide further supports for the shock interactions (see Figures \ref{fig4}a--\ref{fig4}d). Because such cavity-like structure toward a SNR indicates an expanding gas motion, which is thought to be formed by a combination of shock acceleration and strong gas winds from the progenitor system: stellar winds from a high-mass progenitor or disk winds from a progenitor system of post single-degenerate explosion. In the present study, an expanding velocity $\Delta V$ is derived to be $\sim$6 km s$^{-1}$ for the 10 km s$^{-1}$ cloud; and $\sim$3 km s$^{-1}$ for the 40 km s$^{-1}$ cloud. These values are roughly consistent with other Galactic/Magellanic SNRs \citep[e.g.,][]{1990ApJ...364..178K,1991ApJ...382..204K,2017JHEAp..15....1S,2019ApJ...881...85S,2018ApJ...864..161K}.
It is noteworthy that the two expanding cavities are independent because their $\Delta V$ values are much smaller than the velocity difference of the 10 km s$^{-1}$ and 40 km s$^{-1}$ clouds. Therefore, either expanding shell is located at the same distance with W49B, and the forward shock has been impacted the wind-cavity wall where the shock--cloud interaction occurred. The other expanding shell is likely not associated with W49B. According to \cite{2020PASJ...72L..11S}, there are many relics of fully evolved SNRs in the Galactic plane that cannot be observed by radio continuum, optical, infrared, and X-rays. Because thermal radiation from SNRs has shorter cooling time (below $\sim$10 kyr) than the lifetime of giant molecular clouds \citep[$\sim$10 Myr, e.g.,][]{1999PASJ...51..745F,2009ApJS..184....1K}. The expanding gas motion of the 10 km s$^{-1}$ or 40 km s$^{-1}$ cloud is therefore likely one of such objects that happen to be located along the line of sight.
\subsubsection{Kinetic Temperature of Molecular Clouds}
According to the LVG analysis in Section \ref{subsec:lvg}, the higher kinetic temperature $T_{\mathrm{kin}} \sim$20--{60} K of the shell clouds are seen in the 10 km s$^{-1}$ cloud, suggesting that shock-heating likely occurred. Because the $T_{\mathrm{kin}}$ values are roughly consistent with the previous studies of shock-heated molecular clouds in the vicinity of middle-aged SNRs \citep[e.g.,][]{1998ApJ...505..286S,2012A&A...542L..19G,2013ApJ...768..179Y,2014A&A...569A..81A,2020A&A...644A..64D}. In addition, the presence of high temperature dust components of $45 \pm 4$ K and $151 \pm 20$ K also support the shock-heating scenario \citep{2014ApJ...793...95Z}. Moreover, the bright 24-micron emission is detected in the southwestern shell where SW-edge of the 10 km s$^{-1}$ cloud shows the highest kinetic temperature of $\sim${60} K. It is noteworthy that there are no other extra heating sources such as IRAS point sources or \ion{H}{2} regions toward the shell clouds (see also Figure \ref{fig5}).
By contrast, all shell clouds in the 40 km s$^{-1}$ and 60 km s$^{-1}$ components show $T_{\mathrm{kin}} \sim$10 K, implying quiescent molecular clouds without any extra-heating processes such as shock heating and stellar radiation. Interestingly, Reference-SE in the 60 km s$^{-1}$ cloud shows warmer temperature of $\sim$20 K, despite the reference cloud is far from the SNR shell. A possible scenario is that a part of the 60 km s$^{-1}$ cloud is located at the tangent point of the Galaxy, and hence the velocity crowding would accumulate diffuse gas and increase the ambient gas temperature \cite[e.g.,][]{2019ApJS..240...14L}. In any case, there is no shock-heated gas in both the 40 km s$^{-1}$ and 60 km s$^{-1}$ clouds.
\begin{deluxetable*}{lccccccc}
\tablecaption{Results of LVG analysis at the 10 km s$^{-1}$, 40 km s$^{-1}$, and 60 km s$^{-1}$ clouds}
\tablehead{
& \multicolumn{2}{c}{$\mathrm{^{12}CO}$} & &$\mathrm{^{13}CO}$ & & \\
\cline{2-3}\cline{5-5}
\multicolumn{1}{c}{Name} & $J$~=~3--2 & $J$~=~2--1 &&$J$~=~2--1 & $n$(H${_2}$) & \phantom{0}$T_{\mathrm{kin}}$\phantom{0} \\
& (K) & (K) &&(K) & \phantom{0}($\times10^3$ cm$^{-3}$)\phantom{0} & (K) \\
\multicolumn{1}{c}{(1)} & (2) & (3) && (4) & (5) & (6)}
\startdata
\uline{10 km s$^{-1}$ cloud}\phantom{0}\phantom{0}\phantom{0}&&&&&&\\
SW-edge & 1.02 & 1.35 &\phantom{0}& 0.14 & {$0.83^{+0.10}_{-0.05}$} & {$60^{+47}_{-26}$} \\
SW-shell & 3.12 & 3.97 && 1.12 & {$2.45^{+0.50}_{-0.16}$} & ${18}^{{+7}}_{-3\phantom{0}}$ \\
Inner-filament & 2.06 & 3.25 && 0.37 & {$0.78^{+0.03}_{-0.02}$} & {$23^{+9}_{-5\phantom{0}}$} \\
NE-shell & 2.16 & 2.96 && 0.60 & {$1.26^{+0.15}_{-0.09}$} & ${20}^{{+8}}_{-4\phantom{0}}$ \\
Reference-SW & 3.02 & 4.03 && 1.43 & {$1.86^{+0.14}_{-0.08}$} & ${15}^{{+3}}_{-2\phantom{0}}$ \\
Reference-NW & 2.49 & 3.19 && 1.41 & {$2.29^{+0.46}_{-0.20}$} & ${15}^{+6}_{{-3}\phantom{0}}$ \\
\hline
\uline{40 km s$^{-1}$ cloud}\phantom{0}\phantom{0}&&&&&&\\
SE-shell & 1.81 & 3.32 && 0.68 & {$1.78^{+0.08}_{-0.00}$} & $10^{{+2}}_{{-1}\phantom{0}}$ \\
SW-edge & 1.78 & 3.13 && 2.40 & {$1.26^{+0.03}_{-0.03}$} & $\phantom{0}9^{+2}_{-1\phantom{0}}$ \\
Inner-filament & 0.64 & 1.91 && 0.22 & {$0.69^{+0.02}_{-0.00}$} & $\phantom{0}8^{+3}_{-2\phantom{0}}$ \\
N-shell & 0.93 & 5.22 && 0.23 & {$0.76^{+0.05}_{-0.02}$} & $13^{{+6}}_{{-3}\phantom{0}}$ \\
Reference-SW & 1.65 & 2.83 && 0.77 & {$1.32^{+0.09}_{-0.09}$} & $10^{+2}_{-2\phantom{0}}$ \\
Reference-N & 1.05 & 2.40 && 0.42 & $0.93^{+0.03}_{-0.00}$ & $\phantom{0}9^{+2}_{-2\phantom{0}}$ \\
\hline
\uline{60 km s$^{-1}$ cloud}\phantom{0}\phantom{0}&&&&&&\\
SW-edge & 2.78 & 4.61 && 1.62 & $2.04^{+0.05}_{-0.04}$ & $\phantom{0}9^{+1}_{-1\phantom{0}}$ \\
SW-shell & 2.09 & 3.66 && 1.19 & $1.48^{+0.03}_{-0.03}$ & $\phantom{0}9^{+1}_{-1\phantom{0}}$ \\
Inner-filament & 5.22 & 7.93 && 2.82 & {$1.66^{+0.08}_{-0.04}$} & $11^{{+1}}_{-1\phantom{0}}$ \\
NW-shell & 3.98 & 6.13 && 1.31 & {$1.51^{+0.04}_{-0.03}$} & ${14}^{+1}_{{-2}\phantom{0}}$ \\
Reference-SW & 1.76 & 3.64 && 0.42 & {$0.91^{+0.02}_{-0.00}$} & $12^{{+3}}_{{-1}\phantom{0}}$ \\
Reference-SE & 5.20 & 7.15 && 1.33 & $1.17^{{+0.06}}_{{-0.02}}$ & {$21^{+3}_{-2\phantom{0}}$} \\
\enddata
\label{tab1}
\tablecomments{Col. (1): Region name for each cloud. Cols. (2)--(3): Radiation temperature for each line emission derived by the least-squares fitting using a single Gaussian function. Col. (4): Number density of molecular hydrogen. Col. (5): Kinetic temperature.}
\vspace*{-0.5cm}
\end{deluxetable*}
\subsubsection{Final Decision and Consistency with Previous Studies}\label{sec:final}
In conclusion, we claim that the 10 km s$^{-1}$ cloud is the one most likely associated with W49B in terms of its spatial distribution, kinetics, and physical conditions. This velocity is consistent not only with the line-broadening measurements by \cite{2016ApJ...816....1K}, but also with the latest \ion{H}{1} absorption measurement toward W49B by \cite{2018AJ....155..204R}. In this case, the kinematic distance of W49B is slightly revised to $11.0 \pm 0.4$ kpc assuming the Galactic rotation curve model of \cite{1993A&A...275...67B} and the latest Galactic parameters of $R_0 = 7.92$ kpc and $\Theta_0 = 227$ km s$^{-1}$ \citep{2020PASJ...72...50V}. This value is also roughly consistent with the previous distance to W49B of $11.3 \pm 0.4$ kpc derived by \ion{H}{1} absorption \citep{2018AJ....155..204R}.
On the other hand, there is large gap in radial velocities between the 10 km s$^{-1}$ cloud and shocked H$_2$ line emission at $\sim$64 km s$^{-1}$ \citep{2020AJ....160..263L}. We argue that this inconsistency should not be a problem considering the excitation condition for each line emission. In general, the CO line emission at 2.6 mm (also known as $^{12}$CO $J$ = 1--0 transition line) can trace a bulk mass of molecular cloud with low kinetic temperature of $\sim$10 K. On the other hand, the supernova-shocked H$_2$ line emission traces only a small portion of molecular cloud which is highly excited into $\sim$2000--3000 K \citep[e.g.,][]{1994ApJ...427..777M,2020AJ....160..263L}. In W49B, the CO-traced molecular cloud mass is $\sim$$2.7 \times 10^4$ $M_{\sun}$ for the 10 km s$^{-1}$ cloud, whereas the mass of shocked H$_2$ is only 14--550 $M_{\sun}$ \citep{2007ApJ...654..938K}. This indicates that the shocked H$_2$ mass is only $\sim$2\% of the CO-traced molecular cloud mass at most. Considering the momentum conservation between the SNR shocks and interacting clouds, the shocked H$_2$ component is more easily accelerated than the CO-traced molecular cloud. We therefore argue that the velocity inconsistency between the CO cloud and shocked H$_2$ component is caused by the difference of their excitation conditions and masses.
In any case, the physical interaction of the 10 km s$^{-1}$ cloud with W49B means that the bulk mass of molecular clouds is concentrated in the northwestern half of W49B, not in the southwest shell. This is consistent with the hydrogen density maps derived by \cite{2018AA...615A.150Z}, who found higher plasma densities in the west of W49B by the X-ray spectral modeling. The inhomogeneous gas distribution will significantly affect to understand the origins of recombining plasma and gamma-rays from W49B. We will discuss later them the latter Sections \ref{subsec:rp} and \ref{subsec:wp}.
\subsection{A Detailed Comparison with X-Rays}\label{subsec:xcomp}
To reveal a physical relation between the 10 km s$^{-1}$ cloud and X-ray radiation, we here compare the CO distributions with {\it{Chandra}} X-ray images\footnote{Because the shell boundary of X-rays is almost similar to that of radio continuum except for the western half, we here only present a spatial comparison with the CO map of 12.2--15.0 km s$^{-1}$ which is bright in the western part of the shell.}. Figure \ref{fig8}a shows the ALMA ACA $^{12}$CO($J$ = 2--1) integrated intensity overlaid with the {\it{Chandra}} X-ray contours at the energy band of 2--7 keV. In the integration velocity range of 12.2--15.0 km s$^{-1}$, CO clouds are perfectly along with the zigzag pattern of the western X-ray shell, indicating that the shock ionization occurred. Note that the spatial correspondence is also seen in the X-ray image at 4.2--5.5 keV, which are mostly free from the interstellar absorption. We thus suggest that the shockwave was strongly decelerated and deformed in the western shell along the dense clouds, whereas the eastern shell was freely expanded with a smooth shape of the forward shock. This also indicates that the shock velocity of the eastern shell is faster than that of the western shell. Further proper motion studies might be able to reveal the velocity difference in the east--west direction.
\begin{figure}[]
\begin{center}
\includegraphics[width=\linewidth,clip]{w49b_fig08_v7_rgb.pdf}
\caption{(a) Integrated intensity maps of ALMA ACA $^{12}$CO($J$ = 2--1) superposed on the {\it{Chandra}} X-ray intensity contours in the energy band of 2--7 keV. The integration velocity range of CO is from 12.2 to 15.0 km s$^{-1}$. The contour levels are 0.3, 0.4, 0.7, 1.2, 1.9, 2.8, 3.9, and $5.2 \times 10^{-7}$ photon cm$^{-2}$ s$^{-1}$. (b) Integrated intensity maps of ALMA ACA $^{13}$CO($J$ = 2--1) superposed on the continuum-subtracted Fe He$\alpha$ emission. The integration velocity range of CO is from 1.0 to 15.0 km s$^{-1}$. The contour levels are 0.5, 0.8, 1.1, 1.4, 1.7, and $2.0 \times 10^{-7}$ photon cm$^{-2}$ s$^{-1}$.}
\label{fig8}
\end{center}
\end{figure}%
Figure \ref{fig8}b shows an overlay map of the $^{13}$CO($J$ = 2--1) intensity image and the continuum-subtracted Fe He$\alpha$ emission in white contours. To compare the Fe-rich ejecta with the total amount of dense clouds, we use $^{13}$CO with the whole velocity range of 1.0--15.0 km s$^{-1}$. Although the elongated structure of Fe-rich ejecta is believed to be related to a bipolar/jet-driven Type Ib/Ic explosion \citep[][]{2007ApJ...654..938K,2013ApJ...777..145L}, the Fe-rich ejecta is mainly located on the void of dense molecular clouds. Moreover, almost Fe-rich ejecta is surrounded by dense molecular clumps. We argue that this situation is consistent with the supernova explosion inside a barrel-shaped cavity which was proposed by \cite{2018AA...615A.150Z}. The authors revealed that an enhancement of cool plasma component along the Fe-rich ejecta (or the void of dense clouds) was observed by a spatially resolved X-ray spectroscopy (see Figure 4 in \citeauthor{2018AA...615A.150Z}~\citeyear{2018AA...615A.150Z}). Following the proposed scenario, the forward shock was freely expanded in the low-density medium at the beginning, and then suddenly encountered with the dense gaseous materials traced by $^{13}$CO line emission and/or cool plasma component. Since the shock--cloud interaction generates multiple reflected (or inward) shocks, the Fe-rich ejecta is efficiently heated up at higher densities toward the center of the SNR \citep[see also][]{2019ApJ...873...40S}. The X-Ray Imaging and Spectroscopy Mission \citep[XRISM,][]{2020arXiv200304962X} will provide us with further understanding of shock-interactions through a detailed spatial comparison between the X-ray derived ionic properties and CO clouds.
\begin{figure*}[]
\begin{center}
\includegraphics[width=\linewidth,clip]{w49b_fig09_v6_rgb.pdf}
\caption{(a) Integrated intensity maps of ALMA ACA $^{12}$CO($J$ = 2--1) for the 10~km~s$^{-1}$ cloud superposed on {\it{NuSTAR}} Fe He$\alpha$ flux contours \citep{2018ApJ...868L..35Y}. The integration velocity range and contour levels are the same as shown in Figures \ref{fig2}a and \ref{fig6}, respectively. The bashed 12 boxes indicate $1' \times 1'$ regions used for the spatially resolved spectral analysis in \cite{2018ApJ...868L..35Y} and deriving CO averaged integrated intensities in Figure \ref{fig9}b. (b) Scatter plot between the electron temperature $kT_\mathrm{e}$ \citep{2018ApJ...868L..35Y} and peak integrated intensities of $^{12}$CO($J$ = 2--1) for box regions A1--3, B1--3, C1--3, and D1--3 as shown in Figure \ref{fig9}a. Error bars of CO and $kT_\mathrm{e}$ represent standard division of CO integrated intensity and 1$\sigma$ confidence level for each box region. The dashed line indicates the linear regression applying the least squares method.}
\label{fig9}
\end{center}
\end{figure*}%
\subsection{Origin of the High-Temperature Recombining Plasma in W49B}\label{subsec:rp}
It is a long-standing question how the recombining (overionized) plasma is formed in SNRs since its discovery in 2002 \citep[IC443,][]{2002ApJ...572..897K}. Subsequent detailed X-ray spectroscopic observations revealed that nearly 20 SNRs show the overionized state (e.g., W49B, \citeauthor{2009ApJ...706L..71O}~\citeyear{2009ApJ...706L..71O}; G359.1$-$0.5, \citeauthor{2011PASJ...63..527O}~\citeyear{2011PASJ...63..527O}; W28, \citeauthor{2012PASJ...64...81S}~\citeyear{2012PASJ...64...81S}; W44, \citeauthor{2012PASJ...64..141U}~\citeyear{2012PASJ...64..141U}; G346.6$-$0.2, \citeauthor{2013PASJ...65....6Y}~\citeyear{2013PASJ...65....6Y}; 3C 391, \citeauthor{2014ApJ...790...65E}~\citeyear{2014ApJ...790...65E}; CTB 37A, \citeauthor{2014PASJ...66....2Y}~\citeyear{2014PASJ...66....2Y}; G290.1$-$0.8, \citeauthor{2015PASJ...67...16K}~\citeyear{2015PASJ...67...16K}; LMC N49, \citeauthor{2015ApJ...808...77U}~\citeyear{2015ApJ...808...77U}; Kes 17, \citeauthor{2016PASJ...68S...4W}~\citeyear{2016PASJ...68S...4W}; G166.0$+$4.3, \citeauthor{2017PASJ...69...30M}~\citeyear{2017PASJ...69...30M}; 3C400.2, \citeauthor{2017ApJ...842...22E}~\citeyear{2017ApJ...842...22E}; LMC N132D, \citeauthor{2018ApJ...854...71B}~\citeyear{2018ApJ...854...71B}; HB21, \citeauthor{2018PASJ...70...75S}~\citeyear{2018PASJ...70...75S}; CTB1, \citeauthor{2018PASJ...70..110K}~\citeyear{2018PASJ...70..110K}; Sagittarius A East, \citeauthor{2019PASJ...71...52O}~\citeyear{2019PASJ...71...52O}; G189.6+3.3, \citeauthor{2020PASJ...72...81Y}~\citeyear{2020PASJ...72...81Y}, see also a review by \citeauthor{2020AN....341..150Y}~\citeyear{2020AN....341..150Y}). However, the physical origin of recombining plasmas is still under debate.
Since the recombining plasma is characterized by higher ionization temperature $kT_\mathrm{i}$ than the electron temperature $kT_\mathrm{e}$, rapid electron cooling or increasing ionization state is needed to produce the plasma state. Three scenarios have been proposed to explain the origin of recombining plasmas in SNRs, called adiabatic cooling, thermal conduction, and photoionization scenarios. In the adiabatic cooling (a.k.a. rarefaction) scenario, rapid electron cooling occurs when the shockwaves breakout from a dense ISM (e.g., CSM) into a much less dense medium \citep[e.g.,][]{1989MNRAS.236..885I,1994ApJ...437..770M,2018ApJ...868L..35Y}. In the thermal conduction scenario, such rapid electron cooling is caused by interactions between the shockwaves and cold dense clouds through thermal conduction \citep[e.g.,][]{2002ApJ...572..897K,2017PASJ...69...30M,2017ApJ...851...73M,2018PASJ...70...35O,2020ApJ...890...62O}. On the other hand, the photoionization scenario proposes that an external X-ray radiation or low-energy CRs) increase the ionization state via photoionization \citep[e.g.,][]{2013ApJ...773...20N,2019PASJ...71...52O,2019PASJ...71...37H}. Because photoionization can be seen in limited environments such as near the Galactic center or a SNR with strong \ion{Fe}{1} K$\alpha$ emission, the adiabatic cooling and thermal conduction scenarios are thought to be the formation mechanisms of recombining plasmas in most SNRs.
The origin of recombining plasma in W49B has been discussed in the past decade. The thermal conduction scenario was initially proposed by \cite{2005ApJ...631..935K}, whereas the adiabatic cooling scenario is more favored in the subsequent studies \citep{2010A&A...514L...2M,2013ApJ...777..145L,2011MNRAS.415..244Z,2018ApJ...868L..35Y}. Because the recombining plasma in W49B shows a positive correlation between the ionization timescale $n_\mathrm{e}t$ and $kT_\mathrm{e}$. Further, there is no correlation between the
plasma condition and ambient clouds traced by near infrared emission \citep{2018ApJ...868L..35Y}. This trend is in contrast to what is observed in W44 (see also \citeauthor{2020ApJ...890...62O}~\citeyear{2020ApJ...890...62O} and a review in \citeauthor{2020AN....341..150Y}~\citeyear{2020AN....341..150Y}). On the other hand, most recent X-ray studies presented that the X-ray spectra from W49B are reproduced by two ejecta components (low- and high-temperature plasma). The authors proposed thermal conduction scenarios especially for the high-temperature recombining plasma in W49B, considering the conduction timescale \citep{2020ApJ...893...90S,2020ApJ...903..108H}. Note that \cite{2020ApJ...903..108H} argued that thermal conduction is a possible origin of recombining plasma in the eastern regions of W49B because dense molecular clouds are thought to be associated in the southwestern shell \citep{2007ApJ...654..938K,2014ApJ...793...95Z}. In this section, we argue that the origin of the high-temperature recombining plasma in W49B can be understood as the thermal conduction scenario considering the CO-traced interacting molecular clouds in W49B.
\begin{deluxetable*}{lcccccl}[]
\tablecaption{Comparison of Physical Properties in Eleven Gamma-Ray SNRs}
\tablehead{
\multicolumn{1}{c}{Name} & Distance & Diameter & Age & $n_\mathrm{p}$ & $W_\mathrm{p}$ & \multicolumn{1}{c}{References} \\
& (kpc) & (pc) & (kyr) & (cm$^{-3}$) & ($10^{49}$~erg) & \\
\multicolumn{1}{c}{(1)} & (2) & (3) & (4) & (5) & (6) & \multicolumn{1}{c}{(7)}}
\startdata
RX~J1713.7$-3946$ & 1.0 & 18 & 1.6 & 130 & \phantom{00}$0.16^{+0.07}_{-0.08}$ & \cite{2012ApJ...746...82F} \\
RX~J0852.0$-$4622 & \phantom{0z}0.75$^\mathrm{a}$ & 24 & \phantom{z}1.7$^\mathrm{a}$ & 100 & \phantom{00}$0.07^{+0.02}_{-0.02}$ & \cite{2017ApJ...850...71F} \\
RCW~86 & 2.5 & 30 & 1.8 & \phantom{0}75 & \phantom{00}$0.11^{+0.01}_{-0.01}$ & \cite{2019ApJ...876...37S} \\
HESS~J1731$-$347 & 5.7 & 44 & 4.0& \phantom{0}60 & \phantom{00}$0.66^{+0.22}_{-0.22}$ & \cite{2014ApJ...788...94F} \\
G39.2$-$0.3 & 6.2 & 14 & \phantom{0}\phantom{0zw}$5.0^{+2.0}_{-2.0}$$^\mathrm{b}$ & 400 & $3.2^{+1.1}_{-0.8}$ & \cite{2020MNRAS.497.3581D} \\
W49B & 11.0\phantom{0} & 16 & \phantom{0}\phantom{0zw}$6.0^{+1.0}_{-1.0}$$^\mathrm{c}$ & 650 & $2.1^{+1.1}_{-0.6}$ & This work \\
Kes~79 & 5.5 & 16 & \phantom{0zw}$8.3^{+0.5}_{-0.5}$ & 360 & 0.5\phantom{0zw} & \cite{2018ApJ...864..161K} \\
W44 & \phantom{0}3.0$^\mathrm{d}$ & 27 & 20.0$^\mathrm{e}$& 200 & 1.0\phantom{0zw} & \cite{2013ApJ...768..179Y} \\
IC443 & \phantom{i}1.5$^\mathrm{f}$ & 20 & \phantom{0iiiii}$25.0^{+5.0}_{-5.0}$$^\mathrm{g}$ & 680 & 0.09\phantom{ccz} & \cite{2021MNRASsubmitted} \\
\hline
LMC N132D & 50.0\phantom{0} & 25 & \phantom{00zw}$2.5^{+0.2}_{-0.2}$$^\mathrm{h}$ & $< 2000$& $> 0.5$ & \cite{2020ApJ...902...53S} \\
LMC N63A & 50.0\phantom{0} & 18 & \phantom{00zii}$3.5^{+1.5}_{-1.5}$$^\mathrm{i}$ & 190 & $0.9^{+0.5}_{-0.6}$ & \cite{2019ApJ...873...40S} \\
\enddata
\label{tab2}
\tablecomments{Col. (1): Name of SNRs. Col. (2): Distance to SNRs in units of kpc. Col. (3): Diameter of SNRs in units of pc. Col. (4): Age of SNRs in units of kyr. Col. (5): Averaged number density of total interstellar protons $n_\mathrm{p}$ in units of cm$^{-3}$. Col. (6): Total energy of cosmic-ray protons $W_\mathrm{p}$ in units of 10$^{49}$ erg. Col. (7): References for CO/\ion{H}{1} derived $n_\mathrm{p}$ and $W_\mathrm{p}$ for each SNR. Other specific references are also shown as follows: $^\mathrm{a}$\cite{2008ApJ...678L..35K}, $^\mathrm{b}$\cite{2011ApJ...727...43S}, $^\mathrm{c}$\cite{2018AA...615A.150Z}, $^\mathrm{d}$\cite{1975AA....45..239C}, $^\mathrm{e}$\cite{1991ApJ...372L..99W}, $^\mathrm{f}$\cite{2003AA...408..545W}, $^\mathrm{g}$\cite{2008AJ....135..796L}; \cite{2001ApJ...554L.205O}, $^\mathrm{h}$\cite{2020ApJ...894...73L}, and $^\mathrm{i}$\cite{1998ApJ...505..732H}.}
\end{deluxetable*}
Figure \ref{fig9}a shows the $^{12}$CO($J$ = 2--1) integrated intensity map of the 10 km s$^{-1}$ cloud superposed on the {\it{NuSTAR}} Fe He$\alpha$ flux contours \citep{2018ApJ...868L..35Y}. The twelve $1' \times 1'$ boxes indicate regions used for spatially resolved spectral analysis using {\it{NuSTAR}} by \cite{2018ApJ...868L..35Y}. The authors fitted X-ray spectra above 3~keV using a single temperature model, because the energy band is dominated by the high-temperature plasma. We note that the CO integrated intensities are significantly changed region to region, which shows an intensity gradient from the southeast to the northwest as mentioned in Section \ref{sec:results}.
Figure \ref{fig9}b shows a scatter plot between $kT_\mathrm{e}$ of the high-temperature recombining plasma \citep{2018ApJ...868L..35Y} and peak integrated intensity of $^{12}$CO($J$ = 2--1) line emission for each box. We find a clear negative correlation between the two. More precisely, $kT_\mathrm{e}$ values in high-temperature plasma are increasing from the west (cloud rich) regions to the east (cloud poor) regions. This is consistent with the thermal conduction scenario: rapid electron cooling occurred in cold/dense cloud rich regions. Note that this finding will not rule out the adiabatic cooling scenario in W49B. In fact, the X-ray spectra from W49B are reproduced two ejecta components: the low-temperature recombining plasma favors the adiabatic cooling scenario whereas the high-temperature component is likely produced by the thermal conduction \citep{2020ApJ...893...90S,2020ApJ...903..108H}. In other words, both the thermal conduction and adiabatic cooling processes coexist in W49B.
Finally, we discuss the reason why our conclusion---the thermal conduction origin of the high-temperature plasma---is different from some previous studies. One of the most important issues is the previous evaluation of the ISM interacting with W49B. Almost all previous studies used the shocked H$_2$ distribution as the bulk mass of the ISM. However, as discussed in Section \ref{sec:final}, the shocked H$_2$ mass is only $\sim$2\% of the CO-traced molecular cloud mass. Because the shocked H$_2$ map is bright in the southeast, most of researchers believed that the southeast shell is interacting with dense molecular clouds and the ISM mass of east is higher than that of west. Some previous studies therefore concluded that the lower plasma temperature in west was caused by the adiabatic cooling process \citep[e.g.,][]{2020ApJ...903..108H,2018ApJ...868L..35Y}. By contrast, it is noteworthy that \cite{2018AA...615A.150Z} suggested that molecular cloud density is higher in the west than the east, which is compatible with our ALMA results. We also note there are different interpretations for the $n_\mathrm{e}t$ variation in the high-temperature plasma. \cite{2018ApJ...868L..35Y} used $n_\mathrm{e}t$ as a proxy for electron density, assuming a uniform time since heating and uniform initial temperature. The former contrasts with the results of \cite{2020ApJ...903..108H} and \cite{2018AA...615A.150Z}, who found higher recombination ages in the east than the west. The positive correlation between $n_\mathrm{e}t$ and $kT_\mathrm{e}$ in W49B may have to be reconsidered. In any case, we emphasize that the proper evaluation of the ISM surrounding an SNR is essential to understand the origin of recombining plasma correctly. Further detailed comparative studies of CO based molecular cloud properties and X-ray spectroscopic results are needed to better understand the origin of recombining plasma in SNRs.
\begin{figure*}
\begin{center}
\includegraphics[width=130mm,clip]{w49b_fig10_v6_rgb.pdf}
\caption{Scatter plot between the age of SNRs and the total energy of cosmic-ray protons $W_\mathrm{p}$. The data points and references are summarized in Table \ref{tab2}. The green solid line indicates the linear regression of the double-logarithmic plot applying the least squares method.}
\label{fig10}
\end{center}
\end{figure*}%
\subsection{Total Energy of Cosmic-Ray Protons}\label{subsec:wp}
It is a hundred-year problem how CRs, mainly comprising relativistic protons, are accelerated in interstellar space. SNRs are believed to be acceleration sites for Galactic CRs below $\sim$3~PeV through the diffusive shock acceleration \citep[DSA, e.g.,][]{1978MNRAS.182..147B,1978ApJ...221L..29B}. A conventional value of the total energy of CRs accelerated in an SNR is thought to be $\sim$$10^{49}$--$10^{50}$~erg, corresponding to $\sim$1--10\% of typical kinematic energy released by a supernova \citep[$10^{51}$~erg, e.g.,][]{2013ASSP...34..221G,2019AJ....158..149L}. One of the foremost challenges is to validate these predictions experimentally.
Gamma-ray and radio-line observations hold a key to understand the acceleration of CRs in SNRs. Gamma-rays from SNRs are produced by two different mechanisms: hadronic and leptonic processes \citep[e.g.,][]{1994A&A...285..645A,1994A&A...287..959D}. For the hadronic process, CR proton--interstellar proton interaction creates a neutral pion that quickly decays into two gamma-ray photons (hadronic gamma-ray). For the leptonic scenario, a CR electron energizes a low-energy photon into gamma-ray energy via inverse Compton scattering, in addition to produce gamma-rays through non-thermal Bremsstrahlung (leptonic gamma-ray). To confirm the acceleration of relativistic protons, the main component of CRs, it is crucial to detect the characteristic spectral feature of hadronic gamma-rays with a cut-off at a few GeV known as pion-decay bump \citep[e.g.,][]{2011ApJ...742L..30G,2013Sci...339..807A}. In addition, a good spatial correspondence between gamma-rays and interstellar protons provides an alternative support for the CR proton acceleration \citep[e.g.,][]{2003PASJ...55L..61F,2012ApJ...746...82F,2017ApJ...850...71F,2008A&A...481..401A,2013ApJ...768..179Y,2019ApJ...876...37S}, because the hadronic gamma-ray flux is proportional to the total energy of CR protons and the number density of interstellar protons. Note that the interstellar protons are mainly either neutral molecular and atomic hydrogen traced by CO and \ion{H}{1} radio-lines, respectively. Adopting the number density of targeted interstellar protons, the total energy of CR protons was estimated to be $W_\mathrm{p}\sim$10$^{48}$--$10^{49}$~erg toward a dozen gamma-ray SNRs. However, it is still under debate which parameters are important to understand the variety of observed (or in-situ) $W_\mathrm{p}$ values. To better understand the origins of CR protons and variety of $W_\mathrm{p}$, we need more samples as well as detailed gamma-ray and radio-line studies for SNRs.
W49B is thought to be one of the CR proton accelerators because of the detection of hadron-dominant gamma-rays with a pion-decay bump \citep{2018A&A...612A...5H}. In fact, the best-fit position of GeV gamma-ray detected by {\it{Fermi}} Large Area Telescope (LAT) is the edge of the SE shell, where the bright radio continuum, shocked H$_2$ emission, and the dense molecular clouds are located. To obtain the total energy of CR protons in W49B, we first estimate the number density of interstellar protons interacting with the SNR. Using the equations (1) and (2), the averaged number density of interstellar protons in molecular form is estimated to be $\sim$$650 \pm 200$ cm$^{-3}$ assuming a shell radius of 8 pc and a thickness of 3 pc \citep[e.g.,][]{1994ApJ...437..705M}. The error is derived as the typical uncertainty of the CO-to-H$_2$ conversion factor of $\sim$30\% \cite[cf.][]{2013ARA&A..51..207B}. Additionally, the interstellar protons in atomic form are neglectable in W49B because the derived column density of atomic hydrogen is significantly lower than that of molecular hydrogen \citep[see][]{2001ApJ...550..799B}. The similar situation is also seen in other middle-aged SNRs \citep[e.g., W44,][]{2013ApJ...768..179Y}. We therefore adopt the number density of interstellar protons $n_\mathrm{p}$ to be $\sim$$650 \pm 200$ cm$^{-3}$.
According to \cite{2018A&A...612A...5H}, the total energy of CR protons $W_\mathrm{p}$ is written as
\begin{eqnarray}
W_\mathrm{p} \sim2.0\mathrm{-}2.2 \times 10^{49} (n_\mathrm{p} / 650\; \mathrm{cm^{-3}})^{-1}\; (d / 11\; \mathrm{kpc})^{-2} \;\; \mathrm{erg},\;\;\;\;\;
\label{eq3}
\end{eqnarray}
where $d$ is the distance to the SNR. Adopting $n_\mathrm{p} = 650$ cm$^{-3}$ and $d = 11$ kpc, we than obtain $W_\mathrm{p} \sim$$2 \times 10^{49}$ erg, corresponding to $\sim$2\% of the typical kinematic energy released by a supernova explosion. Table \ref{tab2} compares physical properties of eleven gamma-ray SNRs including W49B. Here, all values of $n_\mathrm{p}$ and $W_\mathrm{p}$ were derived from CO/\ion{H}{1} radio-line observations. We find that $W_\mathrm{p}$ in W49B is roughly consistent with that in other gamma-ray bright SNRs located in our Galaxy or the Large Magellanic Cloud (LMC). In addition, it is noteworthy that young SNRs RX~J1713.7$-$3946, RX~J0852.0$-$4622 (a.k.a. Vela Jr.), and RCW~86 as well as an evolved SNR IC~443 show the lowest values of $W_\mathrm{p} \sim$$10^{48}$ erg, while the others hold higher values of $W_\mathrm{p} \sim$$10^{49}$ erg.
To better understand the trend, we plot $W_\mathrm{p}$ values as a function of the age of SNRs. Figure \ref{fig10} shows a scatter plot between the age of SNRs and $W_\mathrm{p}$. We find a positive correlation between two parameters in the SNRs with a young age less than $\sim$6000 yrs, suggesting that in-situ values of $W_\mathrm{p}$ are strongly limited by short duration time of acceleration also known as age-limited acceleration \citep[cf.][]{2010A&A...513A..17O}. On the other hand, other SNRs with an older age more than $\sim$8000 yrs show a steady decrease of $W_\mathrm{p}$ as SNRs get older. This trend could be understood considering the energy dependent diffusion of CRs \citep[e.g.,][]{1996A&A...309..917A,2007Ap&SS.309..365G}. In other words, in-situ values of $W_\mathrm{p}$ have been decreased due to CR escape from the SNR. In fact, hadron-dominant gamma-rays have been detected in nearby giant molecular clouds of W44, suggesting that the molecular clouds are illuminated by CR protons escaped from W44 \citep[e.g.,][]{2012ApJ...749L..35U,2020ApJ...896L..23P}. The authors suggested an actual value of $W_\mathrm{p}$ including escaped CRs is $\sim$$10^{50}$ erg, corresponding to 10\% of the typical kinematic energy released by a supernova explosion. In any case, W49B shows one of the highest in-situ values of $W_\mathrm{p}$ in the gamma-ray bright SNRs, which imply that the escape (diffusion) of CRs is not significant at the moment. Further gamma-ray observations using the Cherenkov Telescope Array (CTA) will unveil a transition phase from the age-limited acceleration to escape dominant stage in detail.
\section{CONCLUSIONS}\label{sec:conclusions}
We summarize the primary conclusions as follows:
\begin{enumerate}
\item New ALMA ACA CO($J$ = 2--1) observations at $\sim$$7''$ resolution have revealed the spatial and kinematic distributions of three candidates of interacting molecular clouds with the mixed-morphorogy SNR W49B, velocities of which are $\sim$10 km s$^{-1}$, $\sim$40 km s$^{-1}$, and $\sim$60 km s$^{-1}$. We found that western molecular clouds at $\sim$10 km s$^{-1}$ are obviously along with both the radio continuum boundary and inside filaments as well as the deformed X-ray shell, suggesting that shock-cloud interactions occurred. The 10 km s$^{-1}$ cloud also shows higher kinetic temperature of $\sim$20--{60} K than the reference clouds at {15} K, indicating that modest shock heating also occurred. The presence of a wind-bubble with an expanding velocity of $\sim$6 km s$^{-1}$ provides further evidence for the association of the 10 km s$^{-1}$ cloud.
\item The barrel-like structure of Fe-rich ejecta is mainly located on the void of dense molecular clouds, where a cool plasma component is enhanced. We propose a possible scenario that the barrel-like structure of Fe-rich ejecta was formed not only by the asymmetric supernova explosion, but also by interactions with dense molecular clouds. A supernova explosion occurred within the cylinder-like gaseous medium and then Fe-rich ejecta was efficiently heated-up at higher densities by multiple-reflected shocks formed by shock-cloud interactions.
\item The electron temperature $kT_\mathrm{e}$ of recombining plasma from Fe He$\alpha$ shows a negative correlation with the peak integrated intensity of CO line emission in the 10 km s$^{-1}$ cloud. More precisely, $kT_\mathrm{e}$ values in high-temperature recombining plasma are increasing from the west (cloud rich) regions to the east (cloud poor) regions, suggesting the thermal conduction origin. Note that this finding does not rule out the adiabatic cooling scenario in the low-temperature recombining plasma in W49B which was previously discussed \citep[][]{2020ApJ...893...90S,2020ApJ...903..108H}.
\item The total energy of CR protons $W_\mathrm{p}$ is estimated to be $\sim$$2 \times 10^{49}$ erg, which is one of the highest values in gamma-ray bright SNRs. We found that in-situ values of $W_\mathrm{p}$ in gamma-ray SNRs increase with age for the young group (with the age less than $\sim$6000 yr). On the other hand, other older SNRs show a steady decrease of $W_\mathrm{p}$ as SNRs get older due to the escapes/diffusion effect of CRs. We frame a hypothesis that W49B is undergoing an age-limited acceleration without a significant escape or diffusion of CRs from the SNR.
\end{enumerate}
\section*{Acknowledgements}
We are grateful Hiroya Yamaguchi for providing us with the {\it{NuSTAR}} data points used in this paper. This paper makes use of the following ALMA data: ADS/JAO.ALMA\#2018.1.01780.S. ALMA is a partnership of ESO (representing its member states), NSF (USA) and NINS (Japan), together with NRC (Canada), MOST and ASIAA (Taiwan), and KASI (Republic of Korea), in cooperation with the Republic of Chile. The Joint ALMA Observatory is operated by ESO, AUI/NRAO and NAOJ. The scientific results reported in this article are based on data obtained from the {\it{Chandra}} Data Archive (Obs IDs: 117, 13440, and 13441). This research has made use of software provided by the {\it{Chandra}} X-ray Center (CXC) in the application packages CIAO (v 4.12). This work was supported by JSPS KAKENHI Grant Numbers JP19H05075 (H. Sano), and JP21H01136 (H. Sano). K. Tokuda was supported by NAOJ ALMA Scientific Research Grant Number of 2016-03B. We are also grateful to the anonymous referee for useful comments which helped the authors to improve the paper significantly.
\software{CASA \citep[v 5.6.0.:][]{2007ASPC..376..127M}, CIAO \citep[v 4.12:][]{2006SPIE.6270E..1VF}, CALDB \citep[v 4.9.1][]{2007ChNew..14...33G}.}
\facilities{ALMA, Chandra, NuSTAR, Very Large Array (VLA), Nobeyama 45-m Telescope, James Clerk Maxwell Telescope (JCMT).}
|
1,108,101,564,553 | arxiv | \section{Introduction}
Clusters of galaxies are attracting considerable attention for their cosmological
applications. A conceptually simple observation, such as the number of clusters per
unit volume, is able to put strong constraints on the cosmological parameters
(or their combinations), for example on the equation of state of the dark energy
(e.g. Albrecht et al., 2006, i.e. the Dark Energy Report and references
therein). In essence, both
analytic predictions and gravitational N body simulations give the halo mass
function, $dN/dM/dV$, i.e. the number of halos of mass $M$ per unit halo mass and
universe volume. The number of halos is sensitive to the cosmological parameters
in two ways, linearly (with the cosmic volume) and exponentially (via the growth
function, i.e. how the cluster mass increases with time). Since one can in
principle measure the abundance of the clusters in the Universe, the
comparison of the observed number of clusters to the expected
(cosmologically-dependent) number of halos allows one to constrain the cosmological
parameters.
This is one of the drivers of many on-going cluster surveys, such
as the South Pole Telescope Survey\footnote{PI Carlstrom,
http://pole.uchicago.edu/} using clusters detected by the Sunayev-Zeldovich
effect, the XMM-Large Scale Survey\footnote{PI Pierre,
http://vela.astro.ulg.ac.be/themes/spatial/xmm/LSS/}, the XMM-Cluster Survey\footnote{PI
Romer, http://xcs-home.org/} using clusters detected by their X-ray
emission, MaxBCG (Koester et al. 2007) and the Red Sequence Cluster
Survey\footnote{PI Yee, http://www.rcs2.org/} using clusters detected by optical
data. More recently, lensing cluster surveys have started (e.g. Berg\'e et al.
2008).
As is known, each experiment measures a combination of cosmological
parameters, rather than the parameters per se. Only the combination of several
measures from different kinds of experiments is able to break this degeneracy
in the parameter space, also showing the absence of systematic effects. In this
sense, cluster counting is complementary to other experiments such as the
observations of SNIa, or the measurements of Baryon Acoustic Oscillations and
CMB, etc. This last aspect is very
important in order to test the idea that dark energy is
indeed a new source in Einstein equations rather than
e.g. the manifestation of a different theory of gravity; by
comparing observables which are mainly sensitive to the
growth of structures with tests of the redshift-distance
relation, we can look for inconsistencies that cannot be
explained by dark energy in the form of a new fluid (e.g.
Trotta \& Bower 2006).
The main obstacle to using clusters for cosmological tests is that no
technique is able to yield a direct measure of their masses, but instead they measure
proxies such as the X-ray flux, temperature or Yx (Kravtsov et al. 2006),
$n_{200}$ (a sort of galaxy richness, see below)
or the Sunayev-Zeldovich decrement.
The calibration between mass and mass proxy (average relation and
intrinsic scatter) can be achieved either by
specific follow-up observations (more direct, or at least independent,
measures of mass), or by a Bayesian technique called in the astronomical
context self-calibration (Majumdar \& Mohr, 2004; Gladders et
al. 2007), i.e. basically modelling the relation with generic functions
and marginalising over their parameters.
However, cosmological constraints are much
less tight when determined in the absence of an external measure of
the mass-scaling of the mass proxy. In particular, recent work by Wu
et al. (2008) has emphasised how self-calibration is
hampered by secondary parameters (i.e. the halo formation time and
concentration). Therefore, a direct measurement of the scaling relation
is essential to test the assumption of the self-calibration technique,
namely to determine the shape of the scatter (currently
Gaussian) and of the scaling (currently linear in log units) and
this is a valuable aim {\em per se.}
The caustic method (Diaferio \& Geller 1997; Diaferio 1999) offers
a robust path to estimating cluster mass.
It relies on the identification in projected phase-space (i.e. in
the plane of line-of-sight velocities and projected cluster-centric
radii, $v,R$) of the envelope defining sharp density contrasts
(i.e. caustics) between the cluster and the field region. The
amplitude of such an envelope is a measure of the mass inside $R$.
Of course, there are other observables available for measuring
cluster masses, but these require additional hypotheses. X-ray-determined
masses require measurements of temperature and surface brightness
profiles and are based on the assumption that the cluster hot gas is
in hydrostatic equilibrium, an assumption that has been questioned in
recent years (e.g. Rasia et al. 2006). Masses derived
using Sunayev-Zeldovich (SZ) decrements additionally assume the
intra-cluster medium is isothermal (e.g. Muchovej et al. 2007).
In this paper, we use caustic masses, i.e. masses derived from the caustic
technique which assumes that galaxies trace the velocity
field. As opposed to the dynamical masses, derived
from the virial theorem (i.e. from the velocity dispersion)
or from the Jeans method, caustic
mass does not require that the cluster is in dynamical
equilibrium (see Rines \& Diaferio 2006 for a discussion).
On the other hand, the relative novelty of caustic masses make
them much less studied through numerical
simulations and by comparisons to other mass proxies. For
this reason, we look for systematic errors on caustic
masses and we calibrate the mass-richness scaling with
velocity dispersion and with an additional mass proxy based
on velocity dispersion fixed by numerical simulations.
In this paper we
aim to give the absolute calibration of the relation between
$n200$, the number of red galaxies (brighter
than a specified limit and within a given clustercentric distance)
and mass. We want also to measure the
scatter of the $n200$ mass proxy and compare
its performance to the $L_X$ mass proxy.
The mass-richness calibration was partially addressed
in the pioneering work of maxBCG
(Koester et al. 2007; Rozo et al. 2008 and references
therein). Because these works
lack clusters with known masses and $r_{200}$ and their
analysis suffers of circularity ($r_{200}$ is derived for
stack of clusters of a given $n_{200}=n(<r_{200})$, i.e.
of clusters with a known $r_{200}$),
their calibration is doubtful, and in fact, their
$r_{200}$, used to measure $n200$, is found in
later papers to be on average twice
as large as the assumed $r_{200}$ radius (e.g. Sheldon et al. 2009;
Becker et al. 2007; Johnston et al. 2007),
i.e. they counted galaxies in a radius too large by a factor of two.
Furthermore, they found a
redshift dependence when none is assumed to be there by definition
(Rykoff et al. 2008; Becker et al. 2007).
Our analysis does not share the problems they encountered.
Throughout this paper we assume $\Omega_M=0.3$, $\Omega_\Lambda=0.7$
and $H_0=70$ km s$^{-1}$ Mpc$^{-1}$. In this paper, velocity
dispersion, usually denoted as $\sigma_v$ in the literature, is
denoted with the symbol $s$. All quantities are measured
in the usual units: velocity dispersions in km s$^{-1}$,
cluster radii in kpc, X-ray luminosities in erg s$^{-1}$,
cluster masses
in solar mass units.
\section{Parameter estimation in Bayesian Inference}
The Bayesian approach to statistics has become increasingly popular
over the past few decades as computational and algorithmic advances have
permitted the analysis of more complex
data sets and the use of more flexible models.
For the theoretician, there are interesting philosophical differences
to be explored between the Bayesian and frequentist approaches.
For the practictioner, Bayesian data analysis provides an additional
valuable statistics tool.
A good introduction to the Bayesian framework can be found in many textbooks
(e.g. Mackay 2003, D'Agostini 2003 and Gelman et al. 2003).
In this section we will summarise a Bayesian approach to an applied problem.
Suppose one is interested in estimating the (log) mass of a galaxy cluster,
$lgM$.
In advance of collecting any data, we may have certain
beliefs and expectations about the values of $lgM$.
In fact, these thoughts are often used in deciding which
instrument will be used to gather data and how this
instrument may be configured. For example, if we are wanting to
measure the mass of a poor cluster via the virial theorem, a Jeans
analysis or the caustic technique, we will select a spectroscopic
set up with adequate resolution, in order to avoid that velocity
errors are comparable to, or larger than, the likely low
velocity dispersion of poor clusters.
Crystalising these thoughts in the form of a probability distribution for
$lgM$ provides the prior $p(lgM)$, used, as mentioned,
in the feasibility section of the telescope time proposal, where
instrument, configuration and exposure time are set.
For example one may believe (e.g. from the cluster being somewhat poor)
that the log of the cluster mass is probably not far from $13$, plus or minus 1;
this might be modelled by saying that the probability distribution of the
log mass, here denoted $lgM$ is a Gaussian centred on $13$ and with $\sigma$, the
standard deviation, equal to $0.5$, i.e. $lgM \sim \mathcal{N} (13,0.5^2)$.
Once the appropriate instrument and its set up have been selected,
data can be collected on the quantities of interest.
In our example, this means we record a measurement of log mass, say
$obslgM200$, via, for example, a caustic analysis, i.e. measuring
distances and velocities. The physics or, sometimes simulations,
of the measuring process may allow us to estimate the reliability
of such measurements. Repeated
measurements are also extremely useful for assessing it.
The likelihood is the model which we adopt for how the noisy observation
$obslgM200$ arises given a value of $lgM$.
For example, we may find that the measurement technique allows us to measure
masses in an unbiased way but with a standard error of 0.1 and
that the error structure is Gaussian, ie.
$obslgM200 \sim \mathcal{N} (lgM,0.1^2)$.
If we observe $obslgM200=13.3$ we usually summarise the above by writing
$lgM=13.3\pm 0.1$.
How do we update our beliefs about the unobserved log mass $lgM$ in light
of the observed measurement, $obslgM200$?
Expressing this probabilistically, what is the posterior distribution
of $lgM$ given $obslgM200$, i.e. $p(lgM \ | \ obslgM200)$?
Bayes Theorem (Bayes 1764 and Laplace 1812) tells us that
\begin{equation}
p(lgM \ | \ obslgM200) = \frac{ p(obslgM200 \ |\ lgM) p(lgM)} {p(obslgM200)}
\end{equation}
The denominator $p(obslgM200)$, known as the
(Bayesian) evidence, is equal to the integral of the numerator
\begin{equation}
p(obslgM200) = \int p(obslgM200 \ |\ lgM) p(lgM) dlgM
\end{equation}
Notice that, as with frequentist statistical approaches, assumptions have
been made which should be assessed; neither priors nor likelihoods (on which
frequentist methods such as maximum likelihood estimation is based) are
set in stone.
Simple algebra shows, that in our example the posterior distribution of
$lgM \ | \ obslgM200$ is
Gaussian, with mean
$\mu=\frac{13.0/0.5^2+13.3/0.1^2}{1/0.5^2+1/0.1^2}=13.29$
and $\sigma^2=\frac{1}{1/0.5^2+1/0.1^2}=0.0096$.
$\mu$ is just the usual weighted average of
two ``input" values, the prior and the observation, with weights
given by prior and observation $\sigma$'s.
In our example, the posterior mean and standard deviation
are numerically almost indistinguishable from the observed
value and its quoted error, however, this is not the rule for
complex data analysis, for example when biases are there
or in frontier measurements, like in
Butcher-Oemler studies, where one often finds observed values outside the
range of acceptable values (see, e.g. Andreon et al. 2006).
From a computational point of view, only simple examples
such as the one described above can generally be tackled analytically.
Markov Chain Monte Carlo (MCMC) methods are widely used for more complex
problems.
Although this might sound intimidating to the astronomical end-user,
the advent of BUGS-like programs (Spiegelhalter et al. 1996) such as
JAGS (Plummer 2008), allow scientists to apply these ideas
for quite complicated models using a simple syntax.
In our example, we just need to write in an ascii file the symbolic
expression of the prior, $lgM \sim \mathcal{N} (13,0.5^2)$ and
likelihood, $obslgM200 \sim \mathcal{N} (lgM,0.1^2)$ and nothing more.
For the work in this paper, the JAGS code
is given in the appendix.
\begin{figure*}
\psfig{figure=direct_inverse_fit.ps,width=12truecm,clip=}
\caption[h]{Left panel: 500 points drawn from a bivariate Gaussian,
overlaid by the line showing the expected value of $y$ given $x$.
The yellow vertical stripe captures those $y$ for which $x$ is
close to 2.
Central panel: Distribution of the $y$ values for $x$ values in a narrow band of $x$ centred on $2$, as shaded in the left panel.
Right panel: as the left panel, but we also add the lines joining the
expected $x$ values at a given $y$, and the $x=y$ line.
}
\end{figure*}
\section{Uncertainties of predicted values in Bayesian Inference}
Suppose we want to estimate the value of a quantity not yet measured (e.g.
the mass of a not yet weighted cluster).
Before data $lgM$ are collected (or even considered), the distribution of the
predicted values $\widetilde{lgM}$ can be expressed
\begin{equation}
p( \widetilde{lgM} ) = \int p( \widetilde{lgM}, \theta ) d \theta = \int p( \widetilde{lgM} | \theta ) p(\theta) d \theta
\end{equation}
These two equalities result from the application of probability
definitions, the first equality is simply that a marginal distribution results
from integrating over a joint distribution,
the second one is Bayes' rule.
If some data $lgM$ have been already collected for similar objects, we can
use these data to improve our prediction for $\widetilde{lgM}$.
For example, if mass and richness in clusters are highly correlated,
one may better predict
the cluster mass knowing its richness than in the absence of such
information, simply because
mass shows a lower scatter at a given richness than when
clusters of all richnesses are considered (except if the relationship
has slope exactly equal to $\tan k \pi/2$, with $k=0,1,2,3$).
In making explicit the presence of such data, $lgM$,
we rewrite Eq. 3 conditioning on $lgM$:
\begin{equation}
p( \widetilde{lgM} | lgM ) = \int p( \widetilde{lgM} | lgM, \theta ) p(\theta|lgM) d \theta
\end{equation}
The conditioning on $lgM$ in the first term in the integral simplifies out
because $lgM$ and $\widetilde{lgM}$ are considered conditionally independent given
$\theta$, so that this term becomes simply $p( \widetilde{lgM} | \theta )$. The left
hand side of the equation is called the posterior predictive distribution
for a
new unobserved $\widetilde{lgM}$ given observed data $lgM$ and model parameters
$\theta$. Its width is a measure of the uncertainty of the predicted value
$\widetilde{lgM}$, a narrower distribution indicating a more precise prediction.
Let us first consider a simple example. Suppose we do not know
the mass, $\widetilde{lgM}$, of a given cluster and we
are interested in predicting it from our knowledge of its richness.
In this didactical example we assume for simplicity that a) all
probability distributions are Gaussian,
b) that previous data $lgM$ for clusters of the same richness
allowed us to determine that clusters of that richness have on average
a mass of $lgM=13.3\pm0.1$,
i.e. $p(\theta|lgM) = \mathcal{N}(13.3,0.1^2)$,
c) that the scatter between the individual
and the average mass of the clusters
is $0.5$ dex, i.e. $p( \widetilde{lgM} | \theta ) = \mathcal{N}(\theta,0.5^2)$. Then,
Eq. 4 is easily analytically solvable and gives
the intuitive solution that
$p( \widetilde{lgM} | lgM )$ is a Gaussian centred on $lgM=13.3$ and with
a $\sigma$ given by the sum in quadrature of
$0.1$ and $0.5$ ($=0.51$ dex). Therefore, a not-yet
weighed cluster of the considered richness has
a predicted mass of $13.3$ with an uncertainty of $0.51$ dex.
The latter is the
performance of richness as a mass estimator in our didactical
example. A different proxy, say X-ray luminosity, may give a different
value for the uncertainty of the predicted mass and the comparison
of these values allows us to rank the performances of these different
mass proxies.
Later in this paper, we measure and compare the performance
of mass and X-ray luminosity.
The assumptions we use then go beyond the simplistic ones of
the pedagogical example,
starting with the assumption of
having a set of clusters with richness identical to that
of the cluster whose mass we want to estimate, the
(tacit) assumption of living in an observational error-free world, the lack
of modelling of a trend between richness and mass, the perfect
knowledge of the parameters of the sampling distribution, a perfect
matching of the richness of clusters with available mass and those
with to-be-estimated mass, etc.
Despite this apparent complexity, to account for all these factors,
we only need to state a richness-mass scaling model (the
same one used to analyse the scaling itself, detailed in Sec 5.1) and use
Eq. 4 to measure the performances of the mass proxies.
Although the above methodology might appear initially intimidating to the
astronomical
end-user, the use of predictive posterior distributions is generally pain free
since programs such as BUGS offer it as a standard feature.
In practice,
the integral in Eq. 4 is computed
quite simply using sampling; repeatedly values of $\theta$ are drawn
from the posterior $p(\theta|lgM)$ and for each of these, values of
$\widetilde{lgM}$ are drawn from $p(\widetilde{lgM}|\theta)$. The values of
$\widetilde{lgM}$ are stored. The width of
the distribution of these values gives the uncertainty of the predicted value,
i.e. the performance of the considered mass proxy. Therefore,
the quoted performance accounts for
all terms entering into the modelling of proxy and mass,
which include the uncertainty of the proxy value
(richness and X-ray luminosity), the uncertainty on the parameters
describing the regression between mass and mass proxy (slope, intercept,
intrinsic scatter and their covariance), as well as other modelled terms (we
also account for the noisiness of the error itself in our analysis).
Some factors are automatically accounted for without any additional input,
for example, where data are scarce, for example near or outside the sampled
richness or $L_X$ range, predictions are noisier (because the
regression is poorly determined here). As a consequence, proxy performances
are poorer (the posterior predictive distribution is wider) there.
\section{Prediction with errors on predictor variables}
It is important to distinguish between the prediction of a
variable $y$ which is assumed to be linearly related to
a {\bf non-random} predictor variable $x$ and the prediction
of a variable $y$ which is linearly related to a predictor
variable $x$ which is itself a random variable.
The latter situation is the one in which we find ourselves here,
given that we want to predict mass as a function of richness
and for both quantities we must collect observational
data.
Figure 1 shows a set of 500 points drawn from a
bivariate Gaussian where marginally both $x$ and $y$ are
standard Gaussian with mean 0 and variance 1 and $x$ and $y$
have correlation $1/2$.
Superimposed on the left hand panel of Figure 1 is the
line giving the theoretical conditional expectation of
$y$ given $x$ (this is known theoretically for this bivariate
Gaussian to be $y = 0.5 x$).
By eye, this line perhaps seems too shallow with respect to
the trend identified by the points, which perhaps might be
captured by the $x=y$ line shown in blue in the right-hand panel.
However, if what we want to do is to predict a $y$ given an $x$
value, this ``too shallow'' line is more appropriate.
To illustrate why this is the case, the middle panel of Figure 1
concentrates on those observed for $x$ close to 2.
It is clear from their histogram that their average is closer
to the value predicted by the red line (1 in this case) than the value predicted by the blue (2 in this case).
To emphasise that although we treat $x$ and $y$ symmetrically
in terms of both being random variables, we have an asymmetry
in terms of our predictive goals, the right hand panel also
shows the expected value of $x$ given a value of $y$.
Akritas \& Bershady (1996) give a related description of
the various types of fit from a non-Bayesian perspective.
\section{Data \& data reduction}
\input mytab2.tex
Our starting point is the CIRS (Cluster Infall
Regions in SDSS, Rines \& Diaferio 2006) cluster catalogue.
Fundamentally, clusters are: a) X-ray flux-selected
b) with an upper cut at redshift $z=0.1$ (to allow a good caustic
measurement) and c) are in the SDSS DR6 spectroscopic survey.
These catalogues give cluster centres,
virial radii $r_{200}$ and masses within $r_{200}$,
$M_{200}$, derived by the caustic technique.
CIRS also lists
the cluster velocity dispersion, computed using just those
galaxies inside the caustic, and the turnaround radius.
The velocity dispersions are
computed using, on average, 208 member galaxies per cluster.
We note that in CIRS velocity dispersions are quoted
with slightly asymmetric errors.
D'Agostini (2004) suggests adopting the average
of the asymmetric errors as a point value of the error and the
midpoint between the upper and lower values as a point value of the
measurement (velocity dispersion) itself. Masses, as quoted by
CIRS have more asymmetric errors and are
such that the lower error bar includes negative mass for
some clusters.
This is compatible with symmetric errors on a log scale being
transformed onto a linear scale and is
supported by the way in which Rines \& Diaferio (2008)
summarise in their introduction their previous (CIRS)
paper. Therefore, we convert errors back on the log scale.
Our statistical analysis accounts for noisiness
of mass and velocity dispersion estimated errors.
For each cluster, we extract the galaxy catalogues
from the Sloan Digital Sky Survey (hereafter
SDSS) $6^{th}$ data release (Adelman-McCarthy et al., 2008),
discarding both clusters at $z<0.03$ to avoid shredding problems (large
galaxies are split in many smaller sources)
and two cluster pairs (requiring a deblending algorithm for
estimating the richness of each cluster component).
We also discard clusters not
wholly enclosed inside the SDSS footprint and a few clusters with hierarchical
centres that have converged on a secondary galaxy clump, instead
of on the main cluster.
One further cluster, the NGC4325 group,
has been removed because it is of very low richness (it has only two galaxies
brighter than the adopted luminosity limit), far lower than the
other clusters in the sample.
The list of the 53 remaining clusters is given in Table 1.
We emphasise that only two cluster pairs
have been removed from the original sample because of their
morphology, all the other excluded clusters
have been removed because they are not fully enclosed
in the sky area observed by SDSS or have suspect masses because
the CIRS algorithm converged on a secondary clump.
Basically,
we want to count red members within a specified luminosity range and
colour and within a given cluster-centric radius, typically $r_{200}$,
as is already done for other clusters at similar redshift
(e.g. Andreon et al. 2006) or in the distant universe (Andreon 2006; 2008;
Andreon et al. 2008b).
We only consider red galaxies because these objects
are those whose luminosity evolution is better known and because
their star formation rate (and therefore luminosity) cannot be altered
by cluster merging, these objects having already exhausted the barionic
reservoir needed to form new stars.
Since we aim to replicate the present analysis to include
additional clusters in future papers, we
take a (passive evolving) limiting magnitude of $M_V=-20$ mag, which is
the approximate completeness of the SDSS at $z=0.3$ and of the
CFHTLS wide survey and CTIO imaging (e.g. Andreon et al. 2004a) of
the XMM-LSS field at $z\sim1$; it is also a widely
used magnitude cut (e.g. de Lucia et al. 2007, Andreon 2008, etc.).
Magnitudes are passively evolving, modelled with a simple
stellar population of solar metallicity, Salpeter IMF, from Bruzual \&
Charlot (2003), as in De Lucia et al. (2007), Andreon (2008) amongst others.
Such a correction is applied for consistency
with other (past and future) work, but is actually unnecessary for our clusters
because it is negligible given the small redshift range ($0.03<z<0.1$) probed
in this work.
We count only red galaxies, defined as those
within 0.1 redward and 0.2 blueward in $g-r$ of the colour--magnitude relation.
This definition of ``red" is quite simple because for our cluster
sample the resulting number hardly
depends on the details of the ``red" definition; the
determination of the precise location of the colour--magnitude relation
is irrelevant because the latter is much
narrower than 0.3 mag and because practically all
galaxies brighter than the adopted luminosity cut are red.
Colours are corrected for the colour--magnitude
slope, but again this is a negligible correction given the small
magnitude range explored (a couple of magnitudes).
For the colour centre, we took the peak of the colour distribution.
Some of the galaxies counted in the cluster line of sight,
are actually in the cluster fore/background.
The contribution from background galaxies is estimated, as usual, from
a reference direction (e.g. Zwicky 1957; Oemler 1974; Andreon, Punzi \& Grado
2005). The reference direction
is taken outside the turnaround radius, or for the few
clusters too close or near to a SDSS border, near the turnaround radius.
Since richness is based on galaxy counts, it is computed
within a cylinder of radius $r_{200}$. Masses are instead calculated (by
Rines \& Diaferio 2006) within spheres of radius $r_{200}$.
Table 1 gives for our 53 clusters: 1) the cluster id; 2) the observed number of
galaxies in the cluster line of sight within $r_{200}$, $obstot_i$; 3) the
observed number of galaxies in the reference line of sight, $obsbkg_i$; 4) the
ratio between the cluster and reference solid angles, $C_i$. Columns 5 and 6 list
$obstot_i$ and $C_i$, but for the radius inferred using eq. 18, introduced in sec
5.1, based on the observed number of galaxies, within an aperture of 1 $h^{-1}$
Mpc, $obsn(<1.43)$. Column 7 lists $obsn(<1.43)$.
\section{Results}
\subsection{Richness-mass model}
The aim of this section is to present a Bayesian analysis of the
richness-mass model.
In particular, we wish to acknowledge the uncertainty in all the
measurements, including in error estimates. Most previous approaches
assume that errors are perfectly known, which is seldom the case for
astronomical measurements, in particular for complex
astronomical measurements such as caustic masses and velocity dispersions,
whose quoted errors come from a simplified analysis.
Furthermore, no regression method
for a Poisson quantity has been previously published in
astronomical journals and even
less so for a difference of Poisson deviates.
First of all, because of errors, observed and true values
are not identically equal.
The variables $n200_i$ and $nbkg_i$ represent the true richness
and the true background galaxy counts in the studied solid angles.
We measured the number of galaxies in both cluster and control field
regions, $obstot_i$ and $obsbkg_i$ respectively, for each of our 53
clusters (i.e. for $i=1,\ldots,53$).
We assume a Poisson likelihood for both and that
all measurements are conditionally independent.
The ratio between the cluster and control field solid angles,
$C_i$, is known exactly. In formulae:
\begin{eqnarray}
obsbkg_i &\sim& \mathcal{P}(nbkg_i) \\
obstot_i &\sim& \mathcal{P}(nbkg_i/C_i+n200_i)
\end{eqnarray}
For each cluster, we have a cluster
mass measurement and a measurement of the
error associated with this mass, $obslgM200_i$ and $obserrlgM200_i$
respectively.
We assume that the likelihood model is a Gaussian centred on the true
value of the cluster mass, $lgM200_i$, with a scatter given by the true
value of the mass error, $\sigma_i$:
\begin{equation}
obslgM200_i \sim \mathcal{N}(lgM200_i,\sigma^2_i)
\label{eqn:eq9}
\end{equation}
We now need to address the fact that we do not know the true
value of the mass error and that we only have an estimate of it
i.e. we need to model the relationship
between $\sigma_i$ and $obserrlgM200_i$.
We use a scaled $\chi^2$ distribution, chosen so that
$obserrlgM200_i^2$ will be unbiased for $\sigma^2_i$, with the
(welcome) additional property that positivity is enforced.
\begin{equation}
obserrlgM200_i^2 \sim \sigma^2_i \chi^2_\nu / \nu
\label{eqn:eqn10}
\end{equation}
Notice that for mathematical reasons we model the relationship between
variances rather than between standard deviations.
The degrees of freedom of the distribution, $\nu$, control
the spread of the distribution, with large $\nu$ meaning that
quoted errors will be close to true errors.
Our baseline analysis uses $\nu=6$ to quantify
that we are 95\% confident that quoted errors are correct
up to a factor of 2 (i.e. that
$\frac{1}{2}<\frac{obserrlgM200_i}{\sigma_i}<2$, derived via the equivalent
probability statement for $obserrlgM200_i^2$ and $\sigma^2_i$).
We note that when $\nu=6$, the $\chi^2$ distribution is quite skewed, and
most of the remaining 5\% probability lies below $1/2$.
We anticipate that results are relatively robust to the choice of $\nu$.
The shape of the adopted distribution, a $\chi^2$ distribution,
is for analogy
to the case in which the quoted error is derived as a result of
repeated observations; in such a case,
standard sampling theory for Gaussian data would have made
our choice extremely natural.
We now turn to the unobserved quantities in our model.
for which we will
specify independent prior distributions.
We assume a linear relation between the unobserved mass and $n200$ on the
log scale, with intercept $\alpha+14.5$, slope $\beta$ and
intrinsic scatter $\sigma_{scat}$:
\begin{equation}
lgM200_i \sim
\mathcal{N}(\alpha+14.5+\beta (\log(n200_i)-1.5), \sigma_{scat}^2)
\label{eqn:eqn12}
\end{equation}
Note that $\log(n200)$ is centred at an average value of 1.5 and
$\alpha$ is centred at -14.5, purely for computational advantages in
the MCMC algorithm used to fit the model (it speeds up
convergence, improves chain mixing, etc.).
Please note that the relation is between true values, not
between observed values, which may be biased,
as we will show in Appendix A for an astronomical sample
affected by Malmquist bias.
The priors on the slope and the intercept of the regression line in
Equation 9
are taken to be quite flat,
a zero mean Gaussian with very large variance for $\alpha$ and a
Students $t$ distribution with 1 degree of freedom for $\beta$.
The latter choice is made to avoid that properties of galaxy clusters
depend on astronomers rules to measure angles (from the x or from the y axis).
This agrees with the model choices in
Andreon (2006 and later works) but differs from some previous works
(e.g. Kelly 2007) that instead assume a uniform prior on the
slope $\beta=\tan b$ and, as a consequence, favour some angles
over others, depending on
the adopted convention on the way angles are
measured (i.e. from the x axis counterclockwise as in mathematics, or from the
y axis clockwise as in astronomy). Our $t$ distribution on $\beta$ is
mathematically equivalent to an uniform prior on the angle $b$.
\begin{eqnarray}
\alpha &\sim& \mathcal{N}(0.0,10^4) \\
\beta &\sim& t_1
\label{eqn:eqn11}
\end{eqnarray}
For the true values of the cluster richness and background,
we have tried not to impose strong a-priori values, only enforcing
positivity.
Both are given independent improper uniform priors.
\begin{eqnarray}
n200_i &\sim& \mathcal{U}(0,\infty) \\
nbkg_i &\sim& \mathcal{U}(0,\infty)
\label{eqn:eqn8}
\end{eqnarray}
Finally we need to specify the prior on the mass error, $\sigma_i$,
and on the intrinsic scatter of the mass-richness scaling,
$\sigma_{scat}$. These are positively defined (by definition),
but otherwise we impose quite weak prior information.
For mathematical reasons, we parameterise these priors on the variance rather
than on the standard deviations as might seem more natural (for astronomers).
An extremely common choice is the Gamma distribution:
\begin{eqnarray}
1/\sigma_i^2 &\sim& \Gamma(\epsilon,\epsilon)\\
1/\sigma_{scat}^2 &\sim& \Gamma(\epsilon,\epsilon)
\label{eqn:eqn13}
\end{eqnarray}
with $\epsilon$ taken to be a very small number.
The above equations translate almost literally into the
JAGS code given in Appendix B. The code is only about 15 lines
long in total, about two orders of magnitude
shorter than any previous implementation of a regression model
(e.g. Kelly 2007, Andreon 2006), none of which address the noisiness of the
quoted error.
Our model seems quite complex with a lot of assumptions,
more than other models
adopted in previous analysis, but actually it makes weaker
assumptions, plainly states what is actually also assumed
by other models (e.g. the conditional independence and
Poisson nature of $obsbkg_i$ and $obstot_i$, the positivity
of the intrinsic scatter, etc.) and
removes approximations adopted in other approaches.
For example, it is common to ignore the uncertainty in the count data and
to take $n200$ to be the observed $obsn200=obstot - obsbkg/C$.
However, doing so does not respect the fact that $n200$ must be non-negative
and in the low count regions $obstot - obsbkg/C$ can be
found to be negative (see Appendix B of Andreon et al. 2006).
Instead, we account for the difference and we will show in
the Appendix an example of the danger of ignoring
the difference between $obsn200$ and $n200$. Eq. 5 and 6 also capture
the Poisson nature of galaxy counts that, for small values, is fairly
different from the usual Gaussian approximation widely adopted in
regression models published in astronomical journals.
Furthermore, it is common to ignore the uncertainty in
the mass error. Our model may easily recover this case, by
letting $\nu$ take a large value (formally, to go to infinity).
Our model replaces this strong assumption with a weaker one, namely that
the quoted squared error is an unbiased measure of the true squared error.
Finally, the remaining ingredients are just uniform (or nearly so)
distributions in the appropriate space.
Essentially, our model assumes that the true richness and true mass are
linearly
related (with some intrinsic scatter) but rather than having these true
values we have noisy measurements of both richness and scatter,
with noise amplitude different from point to point.
In the statistics literature, such a model is know as an
``errors-in-variables regression'' (Dellaportas \& Stephens, 1995).
Our model goes one step beyond
the works of D'Agostini
(2004), Andreon (2006) and Kelly (2007), which all assume
errors to be perfectly known (and none of which
deal with Poisson processes as galaxy counts).
These works were, in turn, less approximate approaches
than previous fitting methods used in astronomy to regress two quantities
(for example, simple least-squares, bivariate correlated error and intrinsic
scatter, etc.).
To summarise, the novelty of the present approach is to treat in a symmetric
way measurements and estimates of errors.
The parameters of primary importance are those of the linear relationship
between true mass and richness, with associated intrinsic scatter
$\sigma_{scat}$ being of particular interest.
\begin{figure}
\psfig{figure=fig1.ps,width=8truecm,clip=}
\caption[h]{Richness-mass scaling.
The solid line marks the mean fitted regression line of $lgM200$ on $log(n200)$, while the dashed line
shows this mean plus or minus the intrinsic scatter $\sigma_{scat}$. The shaded region marks the 68\%
highest posterior credible interval for the regression. Error bars on the data points represent observed
errors for both variables. The distances between the data and the regression line is due in part to the
measurement error and in part to the intrinsic scatter.}
\label{fig:fig1}
\end{figure}
\begin{figure*}
\psfig{figure=fig2.ps,width=18truecm,clip=}
\caption[h]{Posterior probability distribution for the
parameters of the richness-mass scaling.
The black jagged histogram shows the posterior as computed
by MCMC, marginalised over the other parameters. The red curve
is a Gauss
approximation of it. The shaded (yellow) range shows
the 95\% highest posterior credible interval.
}
\label{fig:fig2}
\end{figure*}
\subsection{Richness-mass result}
Using the model above, we found, for our sample of 53 clusters:
\begin{equation}
lgM200 = (0.96\pm0.15) \ (\log n200 -1.5) +14.36\pm0.04
\end{equation}
(Unless otherwise stated, results of the statistical computations
are quoted in the form $x\pm y$ where $x$
is the posterior mean and $y$ is the posterior standard deviation.)
Figure 2
shows the scaling between richness and mass, observed
data, the mean scaling (solid line) and its 68\% uncertainty (shaded yellow
region) and the mean intrinsic scatter (dashed lines) around
the mean relation. The $\pm 1$ intrinsic scatter band is
not expected to contain 68\% of the data points, because of
the presence of measurement errors.
Figure 3
shows the posterior marginals for
the key parameters; for the intercept, slope and intrinsic
scatter $\sigma_{scat}$. These marginals
are reasonably well approximated by Gaussians.
The intrinsic mass scatter at a given richness,
$\sigma_{scat}=\sigma_{lgM200|\log n200}$, is small, $0.19\pm0.03$.
The small scatter and its small uncertainty is promising from the point
of view of
using $n200$ for cosmological aims, for example to estimate
the mass distribution, given the $obsn200$ distribution.
The slope between richness and mass
is very near to one (within one third of the estimated standard deviation), i.e.
clusters which have twice as many galaxies are twice as massive.
\begin{figure}
\psfig{figure=r200_sigmav.ps,width=8truecm,clip=}
\caption[h]{$r_{200}-s$ (velocity dispersion) scaling.
The line marks the expected scaling, $r_{200} \propto s$. The
good agreement between the trend identified by the data and the
expected scaling implies that there is no velocity dispersion (mass) dependent
systematic bias on the adopted $r_{200}$.}
\end{figure}
\subsection{Checks}
Firstly our results are robust to the choice
of $\nu$ (we tested $\nu=6$ vs $\nu=3,30,300$ or $\nu=3000$).
Second, the determination of the slope of the richness scaling requires setting
two (astronomical) parameters, a radius within which galaxies should be counted
and a limiting (reference) magnitude.
To investigate the dependence of the richness-mass slope on which
limiting magnitude is adopted, we recompute $n200$
using two different limiting magnitudes, one and two mag deeper than
our reference mag, $M_V=-20$ mag.
The resulting slopes of the mass-richness scaling are $0.98\pm0.15$
and $0.95\pm0.16$, both very close to the original slope
derived using the reference mag ($0.96\pm0.15$).
The intrinsic scatter changes insignificantly, by 0.01 dex, with
the limiting magnitude.
We now check whether the scaling of richness found with mass
may be biased (tilted) by having
hypothetically taken a systematically incorrect $r_{200}$ (for example,
too small an $r_{200}$ at large masses, or too big a one at small masses).
Figure 4 plots $r_{200}$ as a function of
cluster velocity dispersion.
The superimposed straight line comes from assuming that
$r_{200}$ is the virial radius (i.e. $M_{200}=M_{virial}$),
$r_{200} \propto s$ (e.g. eq. 1
in Andreon et al. 2005, eq. 1 of Carlberg et al. 1997, eq. 3.1 in
Muzzin et al.
2007) rather than as a fit to these points.
As the points are scattered roughly around the slope of
the expected relation, we reject the possibility that the slope
between richness and mass (or velocity dispersion) is biased because of
a bad choice of the reference radius in which
galaxies are counted (one that does not correctly scale with mass).
In summary, $n200$ tightly correlates with
mass, with 0.19 dex intrinsic scatter. The
slope is fairly robust to the choice of the reference magnitude,
the uncertainty of error terms ($\nu$) and the a-priori range of
mass errors. Furthermore, it is unbiased with respect to a (hypothetical)
bad choice of the reference radius.
\begin{figure}
\psfig{figure=richvsdisp.ps,width=8truecm,clip=}
\caption[h]{Velocity dispersion -- richness scaling.
Symbols are as in Fig. 2.
}
\end{figure}
\subsection{Richness-velocity dispersion scaling and results}
Velocity dispersions, $s$, are observationally more expensive
than $n200$ but less expensive
than caustic masses. They are also good tracers of
the cluster mass (e.g. Biviano et al. 2006; Mandelbaum \& Seljak
2007; Evrard et al. 2008). Since at high redshifts
caustic masses are observationally prohibitive to calculate,
from the perspective of testing the evolution
of the richness scaling
it is useful to
calibrate the scaling between richness and velocity dispersion.
The statistical model employed is very similar to that described for
the richness-mass scaling, essentially we
only need to read ``velocity dispersion" where mass was
written. Because velocity dispersion errors are easier to measure
than mass errors, we adopt $\nu=50$, i.e.
we are 68\% confident that quoted errors are correct up to a
factor 1.1 (i.e. within 10\%).
Because of the different measurement units, the intercept $\alpha$
is now centred at $2.8$ (for computational purposes in
JAGS).
For our sample, we found
\begin{equation}
\log s = (0.30\pm0.04) \ (\log n200 -1.5) +2.77\pm0.01
\end{equation}
Figure 5 shows the fitted scaling between richness and velocity dispersion, the
observed data, the posterior mean scaling (solid line) and its uncertainty
(shaded yellow region) and the mean intrinsic scatter (dashed lines).
Similarly to the richness-mass scaling, the
intercept, slope and intrinsic scatter
have posterior marginals which are close to Gaussian.
The intrinsic velocity dispersion scatter at a given richness,
$\sigma_{scatt}=\sigma_{\log s |\log n200}$, is small, $0.05\pm0.01$.
As in the case of the richness-mass scaling, these results
are robust to the choice of $\nu$, for $\nu \ga 10$.
The fitted slope of the richness -- velocity dispersion scaling is one third
of the slope of the richness -- mass scaling, as it should be, given that
velocity dispersion scales with mass with power 0.33 (e.g. Evrard et al.
2008).
\begin{figure}
\psfig{figure=plotmasses.ps,width=8.5truecm,clip=}
\caption[h]{Caustic masses (ordinate) vs masses derived from
the cluster velocity
dispersion using relations calibrated with numerical simulations
(left panel: Evrard et al. 2008, right panel: Biviano et al. 2006).
The solid, slanted, line mark the equality and it is not a fit
to the data.
}
\end{figure}
\subsection{Caustic mass systematic errors}
In the previous sections we have not accounted for possible systematic
error
in the caustic mass, except indirectly in a couple of locations:
a) in Figure 4, when comparing
the cluster velocity dispersion with $r_{200}$: if a systematic
error
on M200 were present, then the data would not scatter around
the expected relation; b) in section 5.4, where we found
that the slope of richness-velocity dispersion is one third of
the slope of the richness-mass, as expected for a mass that
scales with the cube of velocity dispersion.
In order to further investigate the lack of gross systematic errors on
caustic masses we plot in Figure 6 caustic masses
against two masses, derived from velocity
dispersion using relations calibrated with numerical simulations
(left panel: Evrard et al. 2008, right panel: Biviano et al. 2006).
The solid line is the one-to-one relation
rather than a fit to the points. If caustic masses were
systematically larger or smaller than masses derived
from velocity dispersion, then these points might well be
systematically above or below the solid line. If instead
caustic masses were too big at high masses and too small at
low masses, or vice versa,
points should have a different (tilted) slope from the plotted
line. Figure 6 clearly shows that neither of the two cases occurs.
A 30 \% offset error or a 30 \% tilt would be obvious to the eye.
A second obvious conclusions coming from this figure is
that the two panels are virtually indistinguishable.
This is because the two calibrations of the velocity
dispersion-mass relation, although independent, are
actually almost identical.
To summarise, this section shows the lack of
an obvious gross systematic error in caustic masses.
``Statistical" errors on caustic mass and noisiness
of errors are built-in in our model.
\section{Richness as mass proxy}
The richness-mass scaling derived in previous sections
needs a known $r_{200}$,
the radius within which galaxies have to be counted.
If we want to use $n200$ as a mass proxy, $r_{200}$ should
be instead considered as unknown. Lopes et al. (2009)
disagree with this reasoning because in their
work they measured the performance of
mass proxies assuming $r_{200}$ (or $r_{500}$) known, when
instead it is unknown for clusters with unknown masses.
We now measure the performances of a richness estimate
that does not require the knowledge of $r_{200}$, counting
galaxies within some reference radius, $\widehat{r_{200}}$,
that can be measured from imaging data\footnote{The hat above symbols
is introduced to distinguish these values, derived from
eq. 18, from values used in previous sections which were taken from
CIRS.}. Since
there are a number of ways $\widehat{r_{200}}$ may be
estimated, we consider some of them.
In principle, we may be interested in:
a) $\sigma_{scat}$, i.e. the intrinsic scatter in mass at a given richness. This may be
of interest to those who want to known which part of the
observed scatter is intrinsic.
b) the uncertainty of the mass estimated
from the cluster richness. This is,
for example, the case when one has one or few clusters with a measurement of
richness and we would like to know their estimated mass.
With real data, cluster richness is known with a finite precision
which induces a minimal floor in the performances
of richness as mass proxy.
To this end, we first need to find a way to estimate $\widehat{r_{200}}$
from galaxy counts, because clusters for which we want an estimate of
mass will not have a known $r_{200}$. Then we will
calibrate the measured $\widehat{n_{200}}$
($n_{200}$ values within $\widehat{r_{200}}$) with mass and
estimate the uncertainty of the predicted
$lgM200$ for a cluster sample, the latter using Eq. 4.
Recall that the
performance of richness as a mass predictor
accounts for
all terms entering into the modelling of proxy and mass,
which include the uncertainty of the proxy value and
the uncertainty on the parameters
describing the regression between mass and mass proxy (slope, intercept,
intrinsic scatter and their covariance).
As in some literature approaches, we use the same sample
both to establish the scaling between regressed
quantities and to measure the proxy performance.
However, these literature
approaches compute the proxy performances
from a single regression (usually
named the best fit, i.e. for a single value of $\theta$), ignoring that
other fits are similarly acceptable and that the best fit itself
is uncertain (i.e. ignoring
uncertainties on slope, intercept and intrinsic scatter).
When the best fit scaling is defined as the
one minimising the scatter (and this is not our case),
the measured scatter
underestimates the true scatter, by definition.
Our approach supersedes these previous approaches,
allowing for errors other approaches neglect and also
including their covariance.
\begin{figure}
\psfig{figure=n200hat_mass.ps,width=8truecm,clip=}
\caption[h]{Richness--mass scaling
for a richness measured within $\widehat{r_{200}}$, an $r_{200}$ radius
estimated from optical data. Symbols are as in Fig 2.
}
\end{figure}
\subsection{Reference case}
Since we do not known a priori which approach is the optimal
way to estimate $r_{200}$ from imaging data alone, in this
section we consider a reference case and in the following section we
make a number of tests to see how robust are our conclusions to
the assumptions made in the reference case.
We simply compute the number of cluster galaxies (i.e. $obstot-obsbkg/C$)
within a radius of 1.43 Mpc, $obsn(r<1.43)$ and then we estimate
$\widehat{r_{200}}$ as
\begin{equation}
\widehat{\log r_{200}} = 0.6 \ (\log obsn(r<1.43)-1.5)
\end{equation}
The slope, $0.6$ and the radius, $1.43$ Mpc are taken for
consistency with Koester et al. (2007). The intercept
is chosen to reproduce the trend between known $obsn(r<1.43)$
and $r_{200}$ radii. Therefore, our $\widehat{r_{200}}$ has no
bias (or at most a negligible one) with respect to
$r_{200}$ by construction. Instead,
the normalisation (intercept) adopted in Koester et al. (2007) has
been later discovered (Becker et al. 2007; Johnston et al. 2007)
to give radii too large by a factor of two.
Having adopted the radius above, we need to count the galaxies
within this radius and recompute the
solid angle ratio.
Asymptotically, $\widehat{n200}$ is given by
$\widehat{obstot}-obsbkg/\widehat{C}$, but our analysis does not
assume that this asymptotic behaviour holds within our finite
sample\footnote{Background counts do not need to be
recomputed, which is why there is no hat on $obsbkg$.}.
Using our fitting model, we found
\begin{equation}
lgM200 = (0.57\pm0.15) \ (\log \widehat{n200} -1.5) +14.40\pm0.05
\end{equation}
The data and fit are depicted in Figure 7.
The major difference with respect to our fit performed
on measurements using knowledge of $r_{200}$ (i.e. sec 6.2) is
the shallower slope, which is 1.9 combined $\sigma$ shallower
than it was.
Intercept, slope and intrinsic scatter have posteriors close to Gaussian.
The intrinsic scatter
is small, it has mean $0.27\pm0.03$ dex.
With respect
to the case where $r_{200}$ is known, the intrinsic scatter
is larger ($0.27\pm0.03$ vs $0.19\pm0.03$), as
expected because we are not using
our knowledge about $r_{200}$. We emphasise
that this is the uncertainty on the mass inferred from $\widehat{n200}$
if we were able to measure the latter quantity with very large precision,
being $0.27$ dex the part of the mass scatter not associated to
measurement errors.
Since $\widehat{n200}$ is not better known than allowed
by the observed data, the mass error inferred
from a (noisy) estimate of the cluster richness is larger and is given by
the average uncertainty of predicted $lgM200$, which is
found to be $0.29\pm0.01$ dex. Therefore, we can predict the mass
of a cluster within $0.29$ dex by measuring
its richness.
Since the uncertainty on the predicted mass is only
slightly larger than the intrinsic scatter, the uncertainty
on the mass-richness scaling (regression) and proxy uncertainty
only account for
small amounts of the variability. Therefore the performance of richness as mass
proxy is dominated by the mass scatter at a given richness.
In comparison to caustic cluster masses, which have, on average,
a $0.14$ dex error, masses estimated from $\widehat{n200}$ have
twice worse accuracy ($0.29$ vs $0.14$ dex).
Although $\widehat{n200}$ is noisier mass proxy than caustic
masses, the former requires far less expensive observations
than the latter and as a consequence
is available for almost a two hundred times larger sample.
The last number is computed
as the ratio of the number of clusters with available richness
from SDSS (e.g. maxBCG clusters, about
13000, Koester et al. 2007)
with those with caustic masses
in the same sky region (74, listed in Rines \& Diaferio 2006).
\subsection{Other paths to $\widehat{n200}$}
\begin{figure*}
\psfig{figure= Lxn200_mass.ps,width=12truecm,clip=}
\caption[h]{Comparison of the performances, as mass predictor, of
X-ray luminosity and richness. The solid line mark the mean model,
the dashed lines delimit the mean model plus and minus the average
uncertainty of predicted masses. Equal ranges (3.5 dex) are adopted
for richness and X-ray luminosity. See section 4 for a discussion of
the slopes of prediction lines.
}
\end{figure*}
To check the resulting robustness of our results to the few parameters
involved in the computation, we make some tests.
First, we take as reference radius a value near
to the average of our $r_{200}$, 1.25 Mpc and use $obsn(<1.25)$ as pivot
value for estimating $\widehat{r_{200}}$. Second, we changed the
slope to $0.55$, because some maxBCG papers (Koester et al. 2007;
Hansen et al. 2005; Becker et al. 2007) disagree on the
slope value ($0.55$ or $0.6$) adopted in Koester et al. (2007).
Third, we decide to count galaxies in a radius twice larger than
$\widehat{r_{200}}$, to check the sensitivity to the adopted
reference radius. The factor two is adopted to follow the maxBCG papers,
which adopted an $r_{200}$ radius later discovered to be too large by a
factor of two.
Fourth, we consider the simplest case, we adopt a fixed
aperture, 1.43 Mpc, for all clusters, irrespective of their
size or mass. In all these cases we found similar
slopes, intrinsic mass scatter and average uncertainty of predicted $lgM200$
as in our reference case. This is expected,
given that the intrinsic scatter alone accounts for most of
the uncertainty of predicted masses.
In summary, if $r_{200}$ has to be estimated from a scaling relation based
on counting red galaxies within in aperture in imaging
data, it seems that we have reached a floor on the quality of
mass determination, $0.27$ dex of intrinsic scatter,
and $0.29$ dex of average
uncertainty of predicted $lgM200$,
no matter how precisely $\widehat{r_{200}}$ is defined.
\subsection{Comparison with other mass proxies}
In this section we want to compare the performances of the X-ray
luminosity and richness as mass proxies. Among all possible
proxies, we choose X-ray luminosity because it is measurable from survey
data, as is richness. Other mass proxies, such as $Y_X$, do require
follow-up observations and it would be unwise to compare them to
(optical) mass estimates derived from survey data.
Of course, in this comparison, both mass proxies
are measured without using knowledge of mass or
linked quantities, such as $r_{200}$, because they are
unknown for clusters with unknown masses. Lopes et al. (2009)
disagree with this reasoning
because they measured and compared the performance of
mass proxies assuming knowledge of $r_{200}$ (or $r_{500}$).
Richness and its performance
as a mass predictor have been measured by us in the previous section.
In short, richness offers a mass with a $0.29$ dex uncertainty.
X-ray luminosities are collected by Rines \& Diaferio (2006) and
come, in order of preference, from the ROSAT-ESO Flux-Limited X-Ray
(REFLEX), the Northern ROSAT All-Sky galaxy cluster
survey (NORAS), the Bright
Cluster Survey (BCS) and its extension (eBCS) and, finally, from the
X-Ray Brightest Abell Cluster Survey (XBACS).
Rines \& Diaferio (2006) do not list errors for
X-ray luminosities,
therefore we repeat our analysis with
a 5\% and a 30\% error and we found that
results are robust to the adopted error.
The model
for the logarithm of the X-ray flux is assumed to be Gaussian and
the following equations
\begin{eqnarray}
obslgLx_i & \sim & \mathcal{N}(lgLx_i, err^2) \\
lgLx_i & \sim & \mathcal{U}(0,\infty) \\
lgM200_i & \sim & \mathcal{N}(\alpha+14.5+\beta (lgLx_i-42.5), \sigma_{scat}^2)
\end{eqnarray}
replace eq. 5, 6 and 9. Before proceeding further, we emphasise
that our analysis involving $L_X$ ignores the Malmquist bias due
to the X-ray selection of the cluster sample (e.g. Stanek et al. 2006),
i.e. clusters brighter than average for their mass
are over-represented (easier
to detect and thus more likely to be in the sample).
Figure 8 shows richness vs mass and X-ray luminosity vs mass,
the fitted scaling (posterior mean, solid line) and the mean
model plus and minus the uncertainty of predicted masses (dashed lines).
By eye, our fit seems shallower than the data suggest.
Our derived slopes match those derived by other fitting algorithms,
for example, the $L_X$ vs mass regression has
a slope of $0.30\pm0.10$ using our fitting
algorithm, a slope of $0.29\pm0.10$ neglecting the uncertainty
on the error (i.e. following Andreon 2006 and
Kelly 2007) and a BCES$(Y|X)$ (Akritas \& Bershady 1996)
slope of $0.31\pm0.07$. This slope is
theoretically different from the slope of the underlying
relation between these quantities
because we are interested here in something
different, namely prediction as explained in Sec 4.
As a further cautionary check,
we verified that the uncertainty of predicted masses, the
quantity of interest here, is robust, in particular
we forced a steeper slope (e.g. we keep the $L_X$-mass slope to $0.5$),
getting an identical value for the uncertainty.
For the richness, we found (sec 7.1)
a mass uncertainty of $0.29\pm0.01$ dex.
For the $L_X$ proxy, we found an identical value for the
mass uncertainty,
$0.30\pm0.01$ dex.
Therefore,
masses predicted by $L_X$ or richness are comparably precise, to
about $0.30$ dex. Qualitatively, one may reach the same
conclusion by inspecting Figure 8 and performing an approximate
analysis requiring a number of assumptions
that are unnecessary in our statistical analysis; the precision
of a mass proxy is, in our case, dominated by the intrinsic scatter
in mass at a given proxy value, which in turn is not
too dissimilar from the vertical scatter in Figure 8 because observational
mass errors are not large. The two data point clouds display similar widths
at a given value of the proxy (see Figure 8)
and therefore the two proxies display similar
performances as mass predictors. Our statistical analysis removes
approximations and holds when the qualitative
analysis does not, for example if the regression is poorly determined,
or the mass errors are large, or the richness is poorly determined,
or in the presence of a mismatch in proxy value between clusters with
known and to-be-estimated mass.
A
plot similar to our Figure 8 by Borgani \& Guzzo (2001)
seems to show a better $L_X$ performance,
but only when compared to an optical richness estimated by eye (Abell 1958;
Abell et al. 1989).
\begin{figure*}
\psfig{figure= Lxn200_masssigma.ps,width=12truecm,clip=}
\caption[h]{Comparison of the performances, as mass predictors, of
X-ray luminosity and richness. In this figure we use masses inferred
from cluster velocity dispersion, but the basic result does not change,
X-ray luminosity and richness score similarly as mass proxies.
Lines and symbols as in previous
figure.
}
\end{figure*}
The fact that the studied sample is mainly (but not exactly),
an X-ray flux-limited sample gives an advantage
to $L_X$ as a mass proxy; had we taken a volume complete mass-limited
sample of the same cardinality in the same Universe volume
(i.e. unbiased with mass) instead of the
adopted (almost) flux-limited sample, some clusters would not be X-ray
detected and thus would have a very loose mass constraint, lacking an
$L_X$ detection. A richness-selected sample formed by
all clusters with $n200$ above a threshold would also have favoured
the $n200$ mass proxy, because the scatter between
$n200$ and $L_X$ would have included in the sample
clusters undetected in X-ray. Therefore,
in spite of the selection favouring the X-ray proxy, richness
performs as $L_X$ in predicting cluster masses.
Richness has a further advantage, it is available for a larger
number of clusters per unit Universe volume. Let us consider for example
$z<0.3$, the number of clusters with optical estimates of mass (i.e.
with $\widehat{n200}$) outnumber the one with an X-ray based proxy
(i.e. with $L_X$) by a factor 58; there are 5800 ster$^{-1}$ optically
detected clusters (maxBCG clusters, Koester et al. 2007) and only 100
ster$^{-1}$ X-ray detected (REFLEX clusters, Bohringer et al. 2001).
The optically-selected cluster sample is a quasi volume limited sample.
If, mimicking what has been done
for X-ray measurements,
a cut on $\widehat{n200}$ signal to noise is used, instead of adopting
a quasi volume limited sample,
the number of optically detected clusters grows significantly.
Similarly, in about four degrees squares, there are about
106 clusters with $obsn200>6$ (Andreon et
al., in preparation) and $0.32<z<0.8$. In the very same
area there are 9 C1 clusters (Pacaud et al. 2007), i.e.
ten times fewer. If, as for X-ray data, a
cut on $obsn200$ signal to noise is adopted, the number
of optically detected clusters would be larger. Schuecker, Bohringer and
Voges (2004) claim SDSS being deeper than the Rosat All Sky Survey
(RASS) ``can thus be used to guide a cluster detection in RASS
down to lower X-ray flux limits". Unsurprisingly, the fact that
COSMOS (Scoville et al. 2007) and many X-ray cluster surveys keep
X-ray detections that match with optical clusters (e.g.
Finoguenov et al. 2007, for COSMOS) {\it assumes} that in current data
sets X-ray clusters are a sub-set of optically selected clusters, i.e.
a smaller sample. Finally, while X-ray
selected clusters are almost always optically detected, the reverse
has proved much more difficult, which clarifies that optical cluster detection
is observationally
cheaper. Therefore, studies that require large samples of
clusters or a denser sampling of the universe volume may adopt
optically-selected cluster samples because they offer a
mass estimates of comparable quality for larger samples.
Some words of caution are in order. The good performance of richness as
a mass estimator holds for our sample and should be confirmed on a sample of
clusters optically selected. Of particular relevance is
the frequency of catalogued optically selected clusters
being instead line of sight superpositions of smaller systems.
Such points will be addressed by our X-ray follow-up of
all (53) clusters optically selected
with $59<n_{200}<70$ and $0.1<z<0.3$ in the maxBCG catalogue
(Koester et al. 2007a).
Similarly, the same caution is in order for other
mass estimators. For example $L_X$ has
been proposed as a mass estimator by Maughan (2007).
Its performances as a mass predictor, however, has been measured on
data having $L_X$
based on hundreds or thousands of photons and therefore the
noisiness of $L_X$ itself in establishing his
performances as a mass estimator has been largely underestimated.
Furthermore, point sources are identified and
removed through (high-resolution) Chandra observations,
making the identification and
flagging of point sources easy and studied
clusters have preferentially large count rates and are
little affected by residual, unrecognised as such,
point sources. To summarise, the good
performances of $L_X$ cannot be immediately extrapolated to common
cluster samples, dominated by objects with noisy $L_X$/count rate
(because of the steep cluster number counts) and
for which residual point source contamination is
more important and which are perhaps observed by survey
instruments as XMM, having a lower resolutions and
therefore a more difficult identification and of
contaminating point sources.
\section{A third mass calibration}
Our approach can be used to calibrate richness
against mass, no matter which mass we are talking about
(e.g. lensing, caustic, Jean, etc).
In section 6.4 we use velocity
dispersion (uncorrected with any numerical simulation) to
calibrate the richness scaling, recycling the
same model already used for caustic masses.
As a further example,
we recycle our model to calibrate richness against $M_s$, the
mass derived
from velocity dispersion, $s$, fixed with a mass-$s$
relation derived by numerical simulations. We
adopt the mass-$s$ relation in Biviano et al. (2006).
As shown in Figure 6, had we used the mass-$s$ relation
in Evrard et al. (2007) we would have found near indistinguishable
results.
To use the masses $M_s$ in place of the caustic ones,
we need only write their values (and their errors) in the data file and
run our same model. Mass errors are derived
by combining in quadrature velocity dispersion errors (converted
in mass) and
the intrinsic noisiness of $M_s$ (12 \%, from Biviano et al. 2006).
We adopt, as for
velocity dispersions, $\nu=50$ (but results do not
depend on $\nu$, if $\nu\ga 30$).
We found, for our sample of 53 clusters:
$ lgM_s = (0.92\pm0.11) \ (\log n200 -1.5) +14.35\pm0.03 $
with an intrinsic scatter of $0.12\pm0.04$ dex.
The intercept, slope and intrinsic scatter
have posterior marginals which are close to Gaussian,
as for the scaling with caustic masses.
Unsurprisingly, we found a near identical slope and intercept
to those using caustic masses; to find different values we would
need
that caustic masses be tilted or offset from
velocity-dispersion derived masses, whereas Figure 6 shows the
lack of a gross tilt or offset.
We also found compatible values for intrinsic scatter
($0.12\pm0.04$ vs $0.19\pm0.03$). The similarity of the
two intrinsic scatters testifies that
errors on caustic masses and on $M_s$ are on a consistent
scale, i.e. similarly correct (or incorrect); for example, if
the caustic mass error is overestimated, the intrinsic
scatter of the caustic mass-richness scaling would be lower
than the one using $M_s$, because the intrinsic scatter
is the part of the scatter not accounted for the measurement
errors.
We now move on to consider $\widehat{n200}$, the
cluster richness estimated without knowledge of the cluster mass
and linked quantities, as $r_{200}$. How does it perform
as a mass proxy, when $lgM_s$ is used as mass? At the minimal
effort
of listing the data in the data file, we find:
$lgM_s = (0.62\pm0.12) \ (\log \widehat{n200} -1.5) +14.40\pm0.04 $
\noindent
ie, indistinguishable from the scaling with caustic masses,
$lgM200 = (0.57\pm0.15) \ (\log \widehat{n200} -1.5) +14.40\pm0.05 $
The intercept, slope and intrinsic scatter have posteriors close to Gaussian,
as with the scaling with caustic masses. The intrinsic scatter
is small, it has mean $0.21\pm0.03$ dex, very similar to the one
obtained using caustic masses, $0.27\pm0.03$ dex.
The average uncertainty of predicted $lgM_s$, i.e. the quality of
richness as mass proxy, is
found to be $0.23\pm0.01$, similar to the one obtained using
caustic masses, $0.29\pm0.01$ dex.
To explore the quality of $L_X$ as a mass proxy when mass is measured
by $lgM_s$, we only need to
list the data and run our model, with no change. We found
that mass, predicted from $L_X$ has an average uncertainty of
$0.22\pm0.01$. When caustic masses are used,
the average uncertainty of predicted masses was $0.30\pm0.01$.
Figure 9 shows the richness vs mass
and X-ray luminosity vs mass, for velocity-dispersion derived masses.
It is the equivalent of Figure 8.
The bottom line is hardly different
from that derived in sec 7.3;
richness and X-ray luminosity show comparable performances in
predicting cluster mass (either caustic or derived from cluster
velocity dispersion). If anything, there is some tentative evidence
that both X-ray luminosity and richness better predict
masses derived from velocity dispersions
than caustic masses ($\sim0.21$ vs $\sim0.29$). We will defer to a
future paper an in-dept examination of the significance of this
possible effect.
\section{Discussion and Conclusions}
In order to exploit clusters as cosmological probes, it is important
to know the mass-proxy scaling. Although self-solving for
the scaling itself is feasible, an independent
calibration of the scaling
is a safety check and allows us to improve cosmological
constraints.
In this paper we computed the richness (number of red galaxies
brighter than $M_V=-20$ mag) of 53 clusters with
available caustic masses, the latter
having the advantage that, unlike other masses,
they do not require the cluster to be in
dynamical or hydrostatic equilibrium. We investigated the
possibility of systematic biases by comparing caustic masses
to masses derived from velocity dispersions and found
no gross offsets or tilt.
Richness is computed from SDSS imaging data both with and without
knowledge of the reference radius $r_{200}$ from SDSS imaging
data. We then measure the scaling between richness and caustic mass.
Our richness-mass calibration is solid, both from an astrophysical
perspective, because we the adopted masses are amongst
the most hypothesis-parsimonious estimates of cluster
mass and statistically, because we account
for terms usually neglected,
such as the Poisson nature of galaxy counts, the
intrinsic scatter and uncertain
errors. Our cluster sample is
larger, by a factor of a few, than previous samples used
in comparable works. The data and code used for the stochastic computation
are distributed with this
paper. This code is quite general, we used it to derive two
alternative richness-mass calibrations, using as a mass proxy
the cluster velocity dispersion or a mass calibrated from the
velocity dispersion via numerical simulations.
We found a slope between richness and (caustic) mass of $0.96\pm0.15$ with
knowledge of $r_{200}$, i.e. clusters which have twice the number of galaxies are twice as massive.
The intrinsic scatter is small, $0.19$ dex.
An identical result is found using masses calibrated from the
velocity dispersion via numerical simulations.
When the reference radius in which galaxies should be
counted has to be estimated from optical data, the slope
decreases to $0.57\pm0.15$ and masses inferred
by the cluster richness are good to within $0.29\pm0.02$ dex,
largely independently of the way the radius itself is
estimated.
The uncertainty of predicted masses
is twice the average uncertainty of caustic masses (0.14 dex), but
observationally less expensive to obtain and for this reason
available for a two hundred
times larger sample. Richness is a
mass proxy of quality comparable to X-ray luminosity, both
showing a $0.29$ dex mass uncertainty,
but is less observationally expensive
than the latter, as testified by the larger number
density of optically-detected clusters with respect to X-ray
detected clusters in current catalogues.
This has important applications in the
estimation of cosmological parameters from optical cluster surveys, because in
current surveys clusters detected in the optical range outnumber, by at least one
order of magnitude, those detected in X-ray.
In particular, we note
that our richness is computed
from the shallowest data ever used
by us, 54 s exposures at a 2m telescope, taken
under mediocre seeing conditions (1.5 arcsec FWHM), i.e. SDSS
imaging data. Similar or better data should be available for
every cluster; we are unaware of a cluster of galaxy claimed to be
so without some optical imaging of it.
The similar performances of X-ray luminosity and richness
in predicting cluster masses has been confirmed using cluster
masses derived from velocity dispersion fixed by numerical
simulations.
People wanting to estimate the mass of one or two clusters
have to measure galaxy counts brighter than $M_V=-20$ mag
within the $r_{200}$ radius estimated from eq. 18 plus a similar measurement
in an area devoid of cluster galaxies to account for background galaxies,
list these values with our measurements
and run the JAGS code listed in the appendix. Those in a hurry
and accepting a reduced quality of the mass estimation and of its uncertainty
may simply insert the measured $n200$ in eq. 19 and take a
$\pm 0.29$ dex mass error.
In the appendix we present an individual comparison with
the literature addressing the richness-mass scaling.
Here we emphasise that our measurement
of the performances of mass proxies conceptually differs from
some other published works; a) we quote the posterior predictive uncertainty
and not the scatter. The former accounts for the uncertainty
in the richness-mass scaling, while the latter does not. Since the
scaling between mass and richness is not known perfectly, we
prefer posterior predictive uncertainty to the scatter. b) Our
own measurement of the scatter is not biased low, whereas literature
values are sometimes biased low as a result of the way the best fit
model is found, minimising the scatter. The best fit relation
is preferred (by other authors) to the true relation if this leads
to a lower scatter.
The effect is intuitively obvious (and quantitatively important)
for small samples. We prefer, instead, not to be optimistic.
c) Some works (e.g. Lopes et al. 2009) evaluate the performances
of a mass proxy assuming that mass-linked quantities, such as $r_{200}$
are known, while they are unknown for clusters with unknown
masses. This logical inconsistency has an important impact on
the final result. Had we followed Lopes et al. (2009), we should have
concluded that richness returns masses with a 0.19 dex precision, instead
of with a 0.27 dex precision, almost a 50 \% underestimate. d) Some works, instead,
forget important items, such as Malmquist bias.
As detailed in the Appendix where individual works are considered, generally
speaking, authors tend to be more optimistic about
the quality of the richness-mass calibration and
the proxy performances than their data allow.
As mentioned, in order to use richness for cosmological
studies, we need to check that our results hold for
an optically selected cluster and, if a large redshift
range is considered, we need to measure the evolution of
the scaling, similarly to what is necessary for calibrating
every other mass proxy. The first issue will be
attacked by our (running) X-ray observations of an optically
selected cluster sample, the second one by a lensing analysis
of an intermediate redshift ($0.3<z<0.8$) cluster sample.
From this perspective, in this paper we also calibrated richness against
cluster velocity dispersion, which are easier to measure than caustic
masses.
Evolution of red galaxies is now well
understood (Stanford et al. 1998; Kodama et
al. 1998; De Propris et al. 1999; Andreon 2006; Andreon et
al. 2008b), quite differently
from another
widely used mass tracer, the X-ray emission from the intracluster
medium. For the latter
one is forced to assume self-similar evolution for lack of
better knowledge (e.g. Vikhlinin et al. 2009) even if available X-ray
observations argue against
this scenario (e.g. Markevitch 1998, Pratt et al. 2008).
\section*{Acknowledgements}
We thank Vincent Eke and the anonymous referee
for their useful comments
which prompted us to insert Sec 3, 6.5 and 8. We also thank
D. Johnston, A. Moretti, R. Trotta, L. Aguillar and E. Meyer
for useful suggestions.
For the standard SDSS acknowledgement see,
http://www.sdss.org/dr6/coverage/credits.html
|
1,108,101,564,554 | arxiv | \section{Introduction}
\label{sec:intro}
Chaotic dynamical systems are characterized by the existence of a
\textit{predictability horizon} in time, beyond which the information
on the state available from the initial conditions is not enough for
meaningful predictions. Thus it appears a difficult task to perform
an orbit determination for a chaotic dynamical system, at least when
the observations are spread over a time-span longer than the
predictability horizon.
Nevertheless there are practical problems of orbit determination in
which the system is chaotic and the time-span of the
observations is very long. It is important to understand the
behaviour of the solutions, with their estimated uncertainties, in
particular when the variables to be solved for include not
just the initial conditions but also some dynamical parameters. If the
number of available observations grows, but simultaneously
the time interval over which they are spread grows up to
values comparable to the predictability horizon, does the solution
become more accurate, and is the iterative procedure of
differential corrections \citep[Chap. 5]{orbit_det} to find the least
squares solution still possible?
In this paper we use a model problem, namely the discrete dynamical
system defined by the standard map of the pendulum, with just one
dynamical parameter, the $\mu$ coefficient appearing in
equation~(\ref{stmap}). We also set up an observation process
in which both coordinates of the standard map are observed after
each iteration. In the observations we include a simulated random
noise with a normal distribution. Then, each experiment of
orbit determination is also a concrete computation of a segment
limited to $n$ iterations (of the map and of its inverse) of a
\textit{$\varepsilon$ shadowing orbit} for the \textit{$\delta$ pseudo
trajectory} defined by the observation process. The
\textit{Shadowing Lemma} (see Section~\ref{sec:SL}) provides a
mathematically rigorous result on the availability of shadowing
orbits, but thanks to the orbit determination process we make explicit
the relationship between $\varepsilon$ and $\delta$ (see
Section~\ref{sec:SL_OD}), and we explicitly compute the
$\varepsilon$-shadowing orbit.
At the same time, each experiment provides an estimate of the standard
deviation of each of the variables, including initial
conditions and the parameter. These estimates can be plotted as a
function of $n$, thus showing the relationship between accuracy,
number of observations and time interval, measured in Lyapounov times
(see Section~\ref{sec:numerical}).
Of course the numerical experiments are limited to a finite number of
iterations, while the Shadowing Lemma refers to an infinite
orbit. However, the maximum number of iterations is
controlled by another time limit, the \textit{computability horizon}
due to round off error. This limit can be estimated approximately by a
simple formula, and it is found in numerical experiments as a function
of both the initial conditions and the numeric precision used in the
computations.
\subsection{Wisdom hypothesis}
\label{sec:wisdom}
In 1987 J. Wisdom was discussing the chaotic rotation state of
Hyperion, when he claimed that numerical experiments indicated that
\textit{the knowledge gained from measurements on a chaotic dynamical
system grows exponentially with the timespan covered by the
observations} \citep{wisdom87}. This pertained in particular
to the information on dynamical parameters like the moments of inertia
ratios for Hyperion, as well as the rotation state at the midpoint of
the time interval covered by the observations, which he proposed would
be determined with exponentially decreasing uncertainty.
Therefore Wisdom suggests that the orbit determination for a
chaotic system might be in fact more effective than for a non-chaotic
one. It is clear from the context that he was referring to
numerical results, thus his statement can only be verified with finite
computations as close as possible to a realistic data processing of
observations of a chaotic system with dynamical parameters to be
determined.
We have set as a goal in this paper to test the behavior of
the uncertainty in the dynamical parameter of our model
problem. We shall discuss the implications for Wisdom's claim in
Section~\ref{sec:discwisd}.
\subsection{Application to planet-crossing asteroids}
In our solar system there are \textit{planet-crossing minor bodies},
including asteroids and comets, by definition such that their
orbits can, at some times, intersect the orbit of the major
planets. In particular many of the \textit{Near Earth Asteroids
(NEAs)} can intersect the orbit of the Earth. These orbits are
necessarily chaotic, at least over the timespan accessible to accurate
numerical computations.
Unfortunately, these orbits are especially important and necessary to
be studied because of the very reason of chaos, namely close
approaches to the major planets including the Earth: these approaches
may, in some cases, be actual impacts on a finite size planet.
The attempt to predict possibility of impacts by NEAs, in particular
on our planet, is called \textit{Impact Monitoring}, and it is indeed
a form of orbit determination for chaotic orbits. There is a subset of
cases of NEAs for which non-gravitational perturbations, such as the
ones resulting from the Yarkovsky effect, are not negligible
in terms of Impact Monitoring because of the exponential
divergence of nearby orbits which amplifies initially very small
perturbations\\ \citep{apophis,bennu,1950DA,2009FD}.
Thus the Impact Monitoring for these especially difficult cases is an
instance of orbit determination of a chaotic system, with as
parameters the 6 initial conditions and at least one dynamical
parameter, such as a Yarkovsky effect coefficient to be solved for.
We shall show in Section~\ref{sec:impacts} that the weak determination
of the dynamical parameter is a key feature of these cases.
\section{Orbit determination for the standard map}
\label{sec:OD_stmap}
The simplest example of a conservative dynamical system which has both
chaotic and ordered orbits can be built by means of an area preserving
map of a two dimensional manifold:
\begin{equation}\label{stmap}
S_{\mu}(x_k,y_k)=\left\{
\begin{array}{ccl}
x_{k+1} & = & x_k+y_{k+1}\\
y_{k+1} & = & y_k-\mu \sin{x_k}
\end{array}
\right.
\end{equation}
where $\mu$ is the perturbation parameter, and $S$ is the standard
map. The system has more regular orbits for small $\mu$, and
more chaotic orbits for large $\mu$. We choose an
intermediate value of $\mu$, e.g. $\mu=0.5$, in such a way that
ordered and chaotic orbits are both
present. Figure~\ref{fig:smap_05_zoom} shows the strongly chaotic
region around the hyperbolic fixed point, and a few regular orbits on
both sides.
\begin{figure}[h!]
\figfig{8cm}{smap_05_zoom}{Orbits of the standard map for the
perturbation parameter $\mu=0.5$. Plotted is a blow up of
the central region around the hyperbolic fixed point, showing the
strongly chaotic region and a few regular orbits on both sides.}
\label{smap_zoom}
\end{figure}
The advantage of such example is that the least square parameter
estimation process can be performed by means of an explicit formula.
First we compute the linearized map
\begin{displaymath}
DS = \left(
\begin{array}{cc}
\frac{\partial x_{k+1}}{x_{k}} & \frac{\partial x_{k+1}}{y_{k}} \\
\frac{\partial y_{k+1}}{x_{k}} & \frac{\partial y_{k+1}}{y_{k}}
\end{array}
\right)
= \left(
\begin{array}{cc}
1-\mu \cos(x_{k}) & 1 \\
-\mu \cos(x_{k}) & 1
\end{array}
\right)
\end{displaymath}
and from this the linearized state transition matrix
\begin{displaymath}\label{transition_matrix}
A_{k}=\frac{\partial(x_{k}, \ y_{k})}{\partial (x_{0}, \ y_{0})}
\end{displaymath}
which is the solution of the variational equation (for infinitesimal
displacement in the initial conditions), and given
by the recursion:
\begin{displaymath}
A_{k+1}=DS \ A_{k}\ \ ;\ \ A_0=I\ .
\end{displaymath}
The variational equation for the derivatives with respect to the model
parameter $\mu$ is:
\begin{eqnarray*}
\frac{\partial (x_{k+1}, \ y_{k+1})}{\partial \mu} & = & DS \ \frac{\partial (x_k, \ y_k)}{\partial \mu}+\frac{\partial S_\mu}{\partial \mu}\\
& = & DS \ \frac{\partial (x_k, \ y_k)}{\partial \mu} +
\left(
\begin{array}{c}
-\sin(x_{k}) \\
-\sin(x_{k})
\end{array}
\right)
\end{eqnarray*}
Then we set up an observation process, in which both coordinates $x$
and $y$ are observed at each iteration, and the observations are
Gaussian random variables with mean $x_k$ ($y_k$ respectively) and
standard deviation $\sigma$: we use the notation $x_k(\mu_0,\sigma)$
to indicate that the probability density function of the observation
$x_k$ is the normal $\mathcal{N}(x_k(\mu_0), \sigma^2)$ one,
and similarly for $y_k$. The residuals are:
\begin{eqnarray}\label{residuals}
\Bigg\{
\begin{array}{ccl}
\xi_k & = & x_k(\mu_0,\sigma)-x_k(\mu_1)\\
\bar{\xi}_{k} & = & y_k(\mu_{0},\sigma)-y_{k}(\mu_1).
\end{array}
\end{eqnarray}
for $k=-n,\dots,n$. In (\ref{residuals}) $x_k(\mu_0,\sigma)$ and
$y_k(\mu_0,\sigma)$ are the observations at each iteration, $\mu_0$ is
the ``true'' value and $\mu_1=\mu_0+d\mu$ is the current guess.
Then the least squares fit is obtained from the normal equations
\citep{orbit_det}:
\begin{equation}\label{eq:normaleq}
C = \sum_{k=-n}^{n}B_{k}^{T}B_{k} \ \ ; \ \
D = -\sum_{k=-n}^{n}B_{k}^{T} \left(
\begin{array}{c}
\xi_{k} \\
\bar{\xi_{k}}
\end{array}
\right)
\end{equation}
\[
B_{k}=\frac{\partial({\xi_{k},\bar{\xi}_{k}})}{\partial(x_{0},y_{0},\mu)}=-\left(A_{k}|\frac{\partial(x_{k},y_{k})}{\partial
\mu}\right).
\]
The least squares solution for both, the parameter $\mu$ and the initial conditions, is:
\[
\left(
\begin{array}{c}
\delta x \\
\delta y \\
\delta \mu
\end{array}
\right)=\Gamma D, \quad \Gamma=C^{-1}
\]
with $\Gamma$ the covariance matrix of the result. This is enough to
find the least squares solution by iteration of the
above \textit{differential correction}. However, to assess
the uncertainty of the result, weights should be assigned to
the residuals consistently with the probabilistic model, in this case
each residual needs to be divided by its standard deviation $\sigma$;
then the distribution of the result $(x,y,\mu)$ is a normal
distribution with covariance matrix $\sigma^2\,\Gamma$.
\section{Shadowing Lemma}
\label{sec:SL}
The shadowing problem is that of finding a deterministic orbit as
close as possible to a given noisy orbit. The so-called Shadowing
Lemma is the main result about shadowing near a hyperbolic set of a
diffeomorphism. Anosov~\citep{anosov} and Bowen ~\citep{bowen}
proved the existence of arbitrarily long shadowing solutions
for invertible hyperbolic maps. Here we give an overview of
these classical results, as in \citep{pilyugin}.
Let $(X,d)$ be a metric space and let $\Phi$ be a homeomorphism
mapping $X$ onto itself. A $\delta$-pseudotrajectory of the dynamical
system $\Phi$ is a sequence of points $\zeta=\{x_k \in X: \ k \in
\mathbf{Z}\}$ such that the following inequalities
\begin{eqnarray}\label{pseudotrajectory}
d(\Phi(x_k),x_{k+1})<\delta.
\end{eqnarray}
hold. For a graphical description of a $\delta$-pseudotrajectory, see
Fig.~\ref{fig:delta_pseudo}.
\begin{figure}[h!]
\figfig{2cm}{delta_pseudo}{A $\delta$-pseudotrajectory.}
\label{delta_pseudo}
\end{figure}
Usually, a $\delta$-pseudotrajectory is considered as the result of
using a numerical method to compute orbits of the dynamical system
$\Phi$, e.g., because of round off error. We say that a point $x \in
X$ \emph{$(\varepsilon, \Phi)$-shadows} a pseudotrajectory
$\zeta=\{x_k\}$ if the inequalities
\begin{eqnarray}\label{shadows}
d(\Phi^k(x),x_k)<\varepsilon
\end{eqnarray}
hold (see Figure~\ref{fig:epsi_shadowing}).
\begin{figure}[h!]
\figfig{2cm}{epsi_shadowing}{An $\varepsilon$-shadowing.}
\label{epsi_shadowing}
\end{figure}
If only one dynamical system $\Phi$ is considered, we will
usually write $\varepsilon$-shadows $\zeta$. The existence of a
shadowing point for a pseudotrajectory $\zeta$ means that $\zeta$ is close
to a real trajectory of $\Phi$.
The following statement is usually called the Shadowing Lemma.
\begin{theorem}
If $\Lambda$ is a hyperbolic set for a diffeomorphism $\Phi$, then
there exists a neighborhood $W$ of $\Lambda$ such that for all
$\varepsilon>0$ there exists $\delta >0$ such that for any
$\delta$-pseudotrajectory with initial conditions $\zeta\in W$ there
is a point $x$ that $\varepsilon$-shadows $\zeta$.
\end{theorem}
The Anosov shadowing requires the existence of a hyperbolic set. It
means that at each point there are two directions where the motion is
either exponentially expanding (stable manifold) or exponentially
contracting (unstable manifold).
\begin{definition}
We say that a set $\Lambda$ is hyperbolic for a diffeomorphism $\Phi
\in C^1(\mathbf{R}^n)$ if:
\begin{itemize}
\item[(a)] $\Lambda$ is compact and $\Phi$-invariant;
\item[(b)] there exist constants $C>0$, $\lambda_0 \in (0,1)$, and families
of linear subspaces $S(p)$, $U(p)$ of $\mathbf{R}^n$, $p \in \Lambda$, such
that
\begin{itemize}
\item[(b.1)] $S(p) \oplus U(p)=\mathbf{R}^n$;
\item[(b.2)] $D\Phi(p)T(p)=T(\Phi(p))$, $p \in \Lambda$, $T=S,U$;
\item[(b.3)]
\begin{eqnarray*}
|D\Phi^m(p)v| \leq C \lambda_0^m|v| \ for \ v \in S(p), \ m \geq 0 ;\\
|D\Phi^{-m}(p)v| \leq C \lambda_0^m|v| \ for \ v \in U(p), \ m \geq 0 ;
\end{eqnarray*}
\end{itemize}
\end{itemize}
The definition of a hyperbolic set is strictly related to the one of
Lyapounov exponent: for each orbit inside a hyperbolic set, the
Lyapounov exponents must be either $>\log(\lambda_0)$ or
$<-\log(\lambda_0)$.
\end{definition}
\subsection{Shadowing Lemma and orbit determination}
\label{sec:SL_OD}
Our goal is to connect the Shadowing Lemma with the chaotic orbit
determination, involving the least squares fit and the differential
corrections.
First of all we build a $\delta$-pseudotrajectory by using a simulated
observations process. In Section~\ref{sec:OD_stmap} we have
set up such an observations process, with observations
$(x_k(\mu_0,\sigma),y_k(\mu_0, \sigma))$. We claim that a sequence
$\zeta=\left\{(x_k(\mu_0, \sigma),y_k(\mu_0, \sigma))\right\}$ is a
$\delta$-pseudotrajectory for the dynamical system $S_{\mu^*}(x_0,
y_0)$, with $\delta=\sqrt{2}|\mu^*-\mu_0|+\mathcal{K}\sigma$,
$\mathcal{K} \in \mathbf{R}$. To obtain this result we compute the Euclidean
distance:
\begin{equation}
\label{delta_dist}
d(S_{\mu^*}(x_{k}(\mu_0,\sigma),y_{k}(\mu_0,\sigma)),
(x_{k+1}(\mu_0,\sigma),y_{k+1}(\mu_0,\sigma)))
\end{equation}
For the sake of simplicity $(\bar{x}_{k+1},\bar{y}_{k+1})$ are the
observations, i.e. Gaussian random variables with mean $x_{k+1}$
($y_{k+1}$, respectively), and standard deviation $\sigma$, as in
Sec.~\ref{sec:OD_stmap}, and
$S_{\mu^*}(\bar{x}_{k},\bar{y}_{k})=(\tilde{x}_{k+1},\tilde{y}_{k+1})$.
Using these notations, (\ref{delta_dist}) turns into
\begin{equation*}
d(S_{\mu*}(\bar{x}_{k},\bar{y}_{k}),
(\bar{x}_{k+1},\bar{y}_{k+1})) =
\sqrt{(\tilde{x}_{k+1}-\bar{x}_{k+1})^2+(\tilde{y}_{k+1}-\bar{y}_{k+1})^2}
\end{equation*}
We compute separately the two differences.
\begin{eqnarray}
\label{eq:diff_y}
|\tilde{y}_{k+1}-\bar{y}_{k+1}|&=&|\bar{y}_{k+1}-\mu^*\sin{\bar{x}_k}-y_{k+1}-\mathcal{N}(0,\sigma^2)|\nonumber\\
&=&|\mathcal{N}(0,2\sigma^2)-\mu^*\sin{x_k}\cos({\mathcal{N}(0,\sigma^2)})+\mu^*\sin({\mathcal{N}(0,\sigma^2)})\cos{x_k}+\mu_0\sin{x_k}|\nonumber\\
&<&\mathcal{N}(0,2\sigma^2)+|\mu_0-\mu^*|
\end{eqnarray}
To allow the last estimate, we need to solve a technical problem: the
Shadowing Lemma uses a uniform norm, that is the maximum of the
distance between the $\delta$-pseudotrajectory and the
$\varepsilon$-shadowing. On the contrary, the natural norm for the
residuals of the fit is the Euclidean norm with the square root of the
sum of the squares. However, since the number of residuals is not only
finite but sharply limited by the numerical phenomena discussed in
Section~\ref{sec:numerical}, in a given experiment we can just take
the maximum absolute value of the residuals and it is going to be
${\cal K}\sigma$, with ${\cal K}$ a number which in practice cannot be
too large, e.g., in our experiment ${\cal K}=5.9$.
Then we can approximate the quantities ${\cal O}(\sigma)$ and smaller,
e.g., $\cos({\mathcal{N}(0,\sigma^2)}) \sim 1$ and
$\sin({\mathcal{N}(0,\sigma^2)}) \sim 0$.
The $x$ coordinate gives a similar result:
\[
|\tilde{x}_{k+1}-\bar{x}_{k+1}|=|\bar{x}_{k+1}+\tilde{y}_{k+1}-x_k-y_{k+1}
-\mathcal{N}(0,\sigma^2)|
\]
\begin{equation}
\label{eq:diff_x}
< \mathcal{N}(0,\sigma^2)+\mathcal{N}(0,2\sigma^2)+|\mu_0-\mu^*|=
\mathcal{N}(0,3\sigma^2)+|\mu_0-\mu^*|
\end{equation}
Putting together (\ref{eq:diff_y}) and (\ref{eq:diff_x}) we obtain
\begin{gather}
\begin{aligned}
\sqrt{(\tilde{x}_{k+1}-\bar{x}_{k+1})^2+(\tilde{y}_{k+1}-\bar{y}_{k+1})^2}&< \\
< \sqrt{(\mathcal{N}(0,3\sigma^2)+|\mu_0-\mu^*|)^2+(\mathcal{N}(0,2\sigma^2)+|\mu_0-\mu^*|)^2}&<\\
< \sqrt{2}|\mu_0-\mu^*| + \sqrt{(\mathcal{N}(0,3\sigma^2))^2+(\mathcal{N}(0,2\sigma^2))^2}< \sqrt{2}|\mu_0-\mu^*| +\mathcal{K}\sigma\
\end{aligned}\label{eq:deltaps}
\end{gather}
with $\mathcal{K} \in \mathbf{R}$.
Therefore the sequence generated by the observations is a
$\delta$-pseudotrajectory for the dynamical system $S_{\mu^*}$ with
$\delta=\sqrt{2}|\mu_0-\mu^*|+\mathcal{K}\sigma$.
Figure~\ref{fig:deltapseudo_matlab} is an example of observations as a
$\delta$-pseudotrajectory. The observations are built with
$\sigma=10^{-3}$ and $\mu_0=0.5$, and the dynamical system is
$S_{\mu^*}$, with $|\mu^*-\mu| = 10^{-1}$; the circles have radius $\delta$.
\begin{figure}[h!]
\figfig{8cm}{deltapseudo_matlab}{An example of a
$\delta$-pseudotrajectory. Initial conditions are $x_0=3$,
$y_0=0$, $\mu_0=0.5$. $\delta \mu=10^{-1}$, and $\sigma=10^{-3}$.}
\end{figure}
The solution of the least squares fit (to the observations from $-n$
to $n$), obtained by convergent differential corrections, is a finite
$\varepsilon$-shadowing, valid for the iterations from $-n$ to $n$.
We choose a value $\varepsilon>0$, that is a boundary on the maximum
of the residuals. Then we choose $\sigma<\varepsilon/{\cal K}$ and we set up
the observations process. Next, we create a first guess: a new orbit
obtained with a small change of the initial conditions and of the
dynamical parameter $\mu$:
$\{(x_k(\mu_g),y_k(\mu_g))\}=S_{\mu_g}(x_g,y_g)$, with $x_g=x_0+dx$,
$y_g=y_0+dy$, and $\mu_g=\mu_0+d\mu$. Then we apply the differential
corrections to the orbit. If the iterations converge, that is the last
correction is very small, the maximum of the norm of the residuals is
less than $\varepsilon$ (because the individual residuals are less
than $3\sigma$).
At convergence, we obtain an initial condition
$(x^*,y^*)$ and a value of the dynamical parameter $\mu^*$, such that
$(x^*,y^*)$ is the $(\varepsilon,S_{\mu^*})$-shadowing for the
$\delta$-pseudotrajectory with
$\delta=\sqrt{2}|\mu^*-\mu_0|+\mathcal{K}\sigma$, for all the points
used in the fit.
The most important requirement is the convergence of the differential
corrections, otherwise we cannot obtain the
$\varepsilon$-shadowing. This is far from trivial, because the chaotic
divergence of the orbits introduces enormous nonlinear effects, for
which the linearized approach of differential corrections may
fail. To guarantee convergence, first we select the initial
conditions $x_0,y_0$ to be at the center of the observed interval,
otherwise the initial conditions would be essentially undetermined
along the stable manifold of the initial conditions. Second, we use a
\textit{progressive solution} approach, namely, given the solution
with $2n+1$ observations with indexes between $-n$ and $n$, we use the
convergent solution $x_0^*, y_0^*, \mu^*$ for $n$ as first guess for
the solution with $2n+3$ observations (between $-n-1$ and $n+1$).
Then the initial guess is actually used only for the solution with $3$
observations, for which the nonlinearity is negligible. Still the
convergence of the differential corrections depends critically upon
the number $n$ of iterations of the map, as explained in
Section~\ref{sec:numerical}.
\section{Numerical results}
\label{sec:numerical}
The experiment was performed with the initial conditions at $x_{0}=3$
and $y_{0}=0$, and the value of the dynamical parameter
$\mu_0=0.5$. The dynamical context for this orbit can be appreciated
from Figure~\ref{smap_zoom}, showing that the initial conditions
are indeed in a portion of the initial conditions space
containing mostly chaotic orbits. For the observation noise we have
used a standard deviation $\sigma=10^{-10}$.
\subsection{Computability horizon}
\label{sec:horizon}
\begin{figure}[h!]
\figfig{10cm}{det_eigenvalues_double}{The eigenvalues and the
determinant of the state transition matrix in a semilogarithmic
scale, as a function of the number of iterations. Also shown is
the linear fit to the large eigenvalue based on the first $180$
iteration, with slope $+0.091$. The computation is in double
precision and the number of iterations of the standard map $n$ is
$300$ with the map and $300$ with its inverse. The determinant of
the state transition matrix would be $1$, for all $n$, in an exact
computation. The numerical instability occurs when the eigenvalues
reach the critical values $\sqrt{\varepsilon_d},
\sqrt{1/\varepsilon_d}$ marked by the dotted lines.}
\end{figure}
\begin{figure}[h!]
\figfig{10cm}{det_eigenvalues}{The eigenvalues and the determinant
of the state transition matrix in a semilogarithmic scale, as a
function of the number of iterations. Also shown is the
linear fit to the large eigenvalue based on the first
$300$ iteration, with slope $+0.086$. The computation is in
quadruple precision and the number of iterations of the standard
map $n$ is $800$ with the map and $800$ with its inverse. The
numerical instability occurs when the eigenvalues reach the
critical values $\sqrt{\varepsilon_q}, \sqrt{1/\varepsilon_q}$
marked by the dotted lines.}
\end{figure}
Figures~\ref{fig:det_eigenvalues_double} and \ref{fig:det_eigenvalues}
show the absolute value of the eigenvalues of the state transition
matrix forward and backward. The product of two eigenvalues should be
$1$ in exact arithmetic. When the condition number of the matrix
becomes larger than the inverse of the machine rounding off error, the
computation of the matrix becomes numerically impossible, and the
computed value of the determinant is far from $1$.
In Figure~\ref{fig:det_eigenvalues_double} the computations are
performed in the standard double precision, that is with a mantissa of
$52$ binary digits and a round off relative error of
$\varepsilon_d=2^{-53}=1.1\times 10^{-16}$. We observe a numerical
instability after $\simeq 180$ iterations: the determinant
deviates from the exact value of $1$ and the small
eigenvalue starts increasing; the large eigenvalue keeps increasing,
but there is a slight change of slope. Then we fit the slope
of the large eigenvalue curve for the first $180$ iterations, and get
a Lyapounov indicator $+0.091$: it approximates the maximum Lyapounov
exponent $\chi$ for the orbit to which our differential corrections
converge\footnote{There is no way to rigorously compute the Lyapounov
exponents: in practice \textit{Lyapounov indicators} extracted from
finite propagations are used to assess, but not rigorously prove,
the chaotic nature of the orbits. Note that it is a numerically well
documented phenomenon that the indicators are not constant, but
actually depend upon the time interval over which they are computed,
although in most cases these changes are not very large and the
conclusion that an orbit is chaotic is reliable.}.
The Lyapounov time is $T_L=1/\chi$, in this example $T_L\simeq 11$. To
reach a ratio of eigenvalues of the state transition matrix of
$1/\varepsilon_d$ we need a number of Lyapounov times
$\log(1/\sqrt{\varepsilon_d})$, in this case $\simeq 18.4\,T_L\simeq
202$ iterations of the map. At about this number of iterations the
maximum and minimum eigenvalues of $A_n$ are so widely apart in size
that a bad conditioning horizon is reached, and the computation of the
state transition matrix becomes numerically inaccurate. Hence near
$\pm 18.4\, T_L$ we observe the numerical instability in the
computation of the determinant and of the eigenvalues of the state
transition matrix.
Figure~\ref{fig:det_eigenvalues} shows the same computations, with the
same initial condition, but in quadruple precision, with a $112$ bit
mantissa and $\varepsilon_q=2^{-113}=9.6\times 10^{-35}$. The change
of slope in the eigenvalues curves occurs after $\simeq 300$
iterations, while a full blown numerical instability occurs after
$\simeq 550$ iterations. The fit to the large eigenvalue for the first
$300$ iterations gives a Lyapounov indicator $+0.086$, not very
different from the one obtained in double precision. Thus we would
expect the numerical instability to occur after
$\log(1/\sqrt{\varepsilon_q})\,T_L\simeq 39.2\, T_L=455$
iterations. It appears that the rate of divergence decreases after
$300$ iterations, as shown by the change in slope, allowing to
maintain at least the determinant near the exact value for about $100$
more iterations beyond the value predicted above.
The \textit{computability horizon} represents the maximum number of
iterations we can reach, before the computation becomes numerically
unstable. The computability horizon strongly depends on the
chaoticity of the system: more chaos, that is larger $\chi$, more
instability; but also upon the precision of the computations.
Thus, in the following we perform the numerical experiments in
quadruple precision, in order to mitigate the problem of the numerical
instability. We compute $500$ iterations forwards and backwards, but
we use only the first $300$ iterations for the linear fits, to avoid
the possibility that changes in slope, such as the ones apparent in
Figure~\ref{fig:det_eigenvalues}, contaminate our experimental
results.
The compatibly horizon is a hard limit in that it is not
practically possible to increase the number of iterations to a much
higher value. E.g., to push the horizon by a factor $10$
above the value for double precision, we would need computations
performed with real numbers represented with $800$
bytes\footnote{Software to perform arithmetic computations with an
arbitrary number of digits is available, but the algorithms are too
slow to be used even for our simple example.}.
The conclusion is that the practical problem of chaotic orbit
determination is meaningful only for a finite number of iterations,
and the accuracy of the results can be tested only within the boundary
of the computability horizon.
\subsection{Chaotic case}
\label{sec:chaos}
Figure~\ref{fig:chaos_unc_3par} shows the results in quadruple
precision of the full 3-parameter fit: the 3 parameters are the
initial conditions and the dynamical parameter $\mu$. The
determination of $\mu$ is indeed not possible without simultaneous
determination of the initial conditions.
Even in quadruple precision we find a maximum value of $n$ beyond
which the iterative solution of the nonlinear least squares problem is
divergent. This maximum turns out to be $599$ in this experiment: it
is close to what we have called the computability horizon, that
is this limitation is due to the difficulty of computing the state
transition matrix when the condition number is too large.
\begin{figure}[h!]
\figfig{10cm}{chaos_unc_3par}{Standard deviation of the solutions
for the initial conditions and for the dynamical parameter $\mu$
(continuous lines), and actual error (nominal solution of
the fit minus real value used in the simulation, dashed lines),
as a function of the number of iterations.}
\end{figure}
The curves in Fig.~\ref{fig:chaos_unc_3par} represent both the formal
standard deviation and the actual error of the solutions of the least
squares fit, as a function of $n$ in a semilogarithmic plot. Both the
formal standard deviation and the actual error of $\mu$ do not
decrease exponentially. Indeed, in Fig.~\ref{fig:chaos_unc_3par_fit}
we have the same behavior of the curves that we have already seen in
Fig.~\ref{fig:chaos_unc_3par}, but in a log-log plot, in
which a constant slope $a$ would imply a power law proportional to
$n^a$. The slopes of the lines that fit the uncertainties are:
$-0.675$ for the dynamical parameter $\mu$, $-0.833$ and $-12.030$ for
the initial conditions $x$ and $y$, respectively. This plot in
logarithmic scale shows that the uncertainty for $\mu$ and $x$ does
not decrease exponentially. It is also apparent that one of the
initial conditions ($y$) is better determined than the other one
($x$), with an improvement as a function of $n$ which could be
exponential. This is a property of the specific initial condition we
have used, for other choices we can get three parameters determined
with comparable accuracy, none of them with exponential
improvement\footnote{This depends upon the orientation of the stable
and unstable directions at the initial condition.}.
\begin{figure}[h!]
\figfig{10cm}{chaos_unc_3par_fit}{Uncertainty of the solution of the
least squares fit for the initial conditions and for the dynamical
parameter $\mu$ in a logarithmic scale. }
\end{figure}
Figure~\ref{fig:chaos_unc_2par} shows the results for the standard
deviation and the actual error when solving only for the initial
condition. The 2x2 portion of the normal matrix which refers only to
the initial conditions is not badly conditioned. Also as a result of
this, we are able to get convergence of the differential corrections
up to $\pm 742$ iterations, which is even beyond the numerical
stability boundary. If the fit is done by using only up to $300$
iterates, to avoid the apparent slope change, the slopes shown in this
Figure are $-0.084$ for $x$ and $-0.083$ for $y$; note that the
Lyapounov indicator for the same interval is $+0.086$. Thus
exponentially improving determination of the initial conditions only
is possible, and the exponent appears to be very close to the opposite
of the Lyapounov exponent.
\begin{figure}[h!]
\figfig{10cm}{chaos_unc_2par}{Standard deviation of the solutions
for the initial conditions (continuous lines), true errors for the
same 2 parameters (dashed lines).}
\end{figure}
\subsection{Ordered case}
\label{sec:ord}
An ordered case can be obtained with a change of the initial
conditions. For the numerical experiments we have chosen $x_0=2$,
$y_0=0$ and $\mu_0=0.5$. In the ordered case we have not the problem
of the computability horizon and the Lyapounov exponent is very
small: actually, it could be zero if we are on a Moser invariant curve.
Thus we have computed $5000$ iterations.
Figure~\ref{fig:ordered} gives a summary of our numerical
experiment in the ordered case. The Lyapounov indicator is very small
($\simeq 10^{-4}$), and can be made even smaller by continuing the
experiment for larger values of $n$. As a consequence, the state
transition matrix is not badly conditioned, and the computability
horizon is much beyond the number of iterations we have used (if it
exists at all). Thus the lack of chaoticity implies the practical
absence of the computability horizon, and we can determine all the
parameters with very good accuracy, even if we are not in exact
arithmetic. The values of the slopes of the fit to the uncertainty
are $-0.504$ for $\mu$, $-0.504$ and $-0.488$ for $x$ and $y$
respectively, the corresponding regression lines are shown in the
log-log plot on the bottom right. As it is clear by comparing the top
right and the bottom left plot, the standard deviation for the
solution with only 2 parameters have very much the same behavior,
indeed in a log-log plot (not shown) we can get slopes $-0.511$ for
$x$, $-0.481$ for $y$.
All these power laws are close to the inverse square root of the
number of iterations, namely the same rule as the standard deviation
in the computation of a mean. We do not have a formal proof of this,
but we conjecture that for an orbit on a Moser invariant curve (for
which the Lyapounov exponents are exactly zero) the standard
deviations for all the parameters decrease as $1/\sqrt{n}$.
\begin{figure}[ht]
\begin{minipage}[b]{0.6\linewidth}
\centering
\includegraphics[width=.8\linewidth]{det_eigenvalues_ord}
\vspace{4ex}
\end{minipage
\begin{minipage}[b]{0.6\linewidth}
\centering
\includegraphics[width=.8\linewidth]{ord_unc_2par}
\vspace{4ex}
\end{minipage}
\begin{minipage}[b]{0.6\linewidth}
\centering
\includegraphics[width=.8\linewidth]{ord_unc_3par}
\vspace{4ex}
\end{minipage
\begin{minipage}[b]{0.6\linewidth}
\centering
\includegraphics[width=.8\linewidth]{ord_unc_3par_fit}
\vspace{4ex}
\end{minipage}
\caption{Top left: eigenvalues of the state transition matrices, for
the chosen regular initial conditions and for $\pm 5000$
iterations. Top right: solutions for the initial condition only,
from the top condition number of the normal matrix,
standard deviation of $y$ and for $x$. Bottom left: solutions for
three parameters, from the top condition number, standard
deviation of $x$, of $y$, of $\mu$. Bottom right: log-log plot of
the 3 standard deviations, with very similar slopes.}
\label{fig:ordered}
\end{figure}
\section{Conclusions}
\label{sec:conclusions}
We have understood the concept of \textit{computability horizon} as a
consequence of numerical instability in the computation of the state
transition matrices, thus providing a comparatively simple empirical
formula to approximately predict the horizon. This is a practical
limitation which applies to any attempt at orbit determination of
chaotic orbits.
We have used numerical experiments in quadruple precision, but
nevertheless limited by the computability horizon to few hundreds of
iterations for the very chaotic orbit used in our test. From these we
have found the following three empirical facts:
\begin{enumerate}
\item If only initial conditions are determined for a chaotic orbit,
the uncertainty can decrease exponentially with the number $n$ of
iterations of the map, and the exponent of this decrease is close to
a Lyapounov exponent.
\item If a dynamical parameter $\mu$ is determined together with the
initial conditions of a chaotic orbit, the decrease in uncertainty
is polynomial in $n$ for $\mu$ and for at least one of the initial
coordinates.
\item If the initial conditions belong to an apparently ordered orbit,
that is such that there is no evidence of a positive Lyapounov
exponent, it is possible to determine simultaneously $\mu$ and the
initial conditions with uncertainty decreasing polynomially with
$n$. The case in which only the initial conditions are determined
gives the same result. Moreover, all the power laws $n^a$ for these
uncertainties appear numerically to have $a\simeq -1/2$.
\end{enumerate}
\subsection{Discussion on the Wisdom hypothesis}
\label{sec:discwisd}
The statement by Wisdom, as a practical rule for concrete orbit
determination, appears to be first limited by the computability
horizon. Second, the actual decrease of the uncertainty, going as far
as it can be done numerically, is not exponential, but polynomial, as
$n^a$, with $a$ negative and rather small, although we have found that
the value of $a$ depends upon the initial conditions\footnote{We are
showing figures and giving data only for one initial condition, but
of course we have run many tests.}. Note that the orbit
determinations in which the only parameters to be solved are the 2
initial coordinates show an exponential decrease as $\exp(-\alpha\,
n)$, where $\alpha$ appears to be close to the Lyapounov exponent
$\chi$, but the strong correlations appearing when 3 parameters are
solved degrade the result in a very substantial way.
This needs to be compared to the regular case, shown in
Figure~\ref{fig:ordered}, where the standard deviations for each of
the 3 fit parameters decrease approximately according to an
$1/\sqrt{n}$ law, as prescribed by the standard rule for the estimate
of the mean with errors having a normal distribution. Indeed it is
possible that the determination of $\mu$ for some chaotic cases,
including the example shown in Figure~\ref{fig:chaos_unc_3par_fit},
decreases faster than for an ordered case, but the decrease is anyway
polynomial, proportional to $n^a$ with some different negative $a$,
thus the difference is not very large, given the tight constraint on
the maximum possible value of $n$.
\subsection{Examples from Impact Monitoring}
\label{sec:impacts}
One feature of our results is that adding a dynamical parameter to the
list of parameters to be determined results in degradation in the
normal matrix, thus in much slower decrease of the uncertainties as
the number of observations grows. The problems of orbit determination
for NEA undergoing several close approaches to the Earth (or other
planets) is more complex than our simple model, but we have found that
the phenomenon described above does occur in a remarkably similar way.
\begin{figure}[h!]
\figfig{10cm}{pdf_yarko_noyarko}{Two different Probability Density
Functions (PDF) for the trace of possible solutions on the Target
Plane of the close approach of asteroid (410777) 2009 FD to the
Earth in the year 2185. Superimposed and on a different
vertical scale are the keyholes relative to impacts in
different years between 2185 and 2196; the height of the bar is
proportional to the width of the keyhole, thus the Impact
Probability can be computed as product of the width and the PDF.}
\end{figure}
In Figure~\ref{fig:pdf_yarko_noyarko} we show two probability
distributions, as derived from the orbit determination of the
asteroid (410777) 2009 FD. The narrow peaked distribution corresponds
to an orbit determination with 6 parameters, the initial conditions
only: the standard deviation is $6\times 10^4$ km. The much wider
distribution corresponds to a fit with 7 parameters, including the
constant $A_2$ appearing in the transverse acceleration due to the
Yarkovsky effect: the STD is $\simeq 2.3\times 10^6$ km. The Yarkovsky
effect is a form of non-gravitational perturbation due to thermal
radiation emitted anisotropically by the asteroid, and is indeed very
small. However, when the uncertainty resulting from the covariance
matrix of the orbit determination is propagated for $\sim 170$ years
after the last observation available, not only the Yarkovsky effect
has a long enough time to accumulate but it is also enhanced
by the exponential divergence of nearby orbits, the Lyapounov time
being about $15.3$ years \citep{2009FD}[Figure 5].
The practical consequence of this increase of the uncertainty
arises from the fact that the Target Plane of 2009 FD for 2815
includes some \textit{keyholes}, small portions corresponding to
impacts with the Earth (either at that time or a few years later,
until 2196). With the 7 parameters solutions these keyholes are
within the range of outcomes with a significant value of the
Probability Density Function, thus the impacts have a non-negligible
probability, the largest being an \textit{Impact Probability} of
$\simeq 1/370$ for 2185. If on the contrary the orbit was estimated
with 6 parameters only, then the probability would appear to be even
larger for an impact in 2190, and all the other keyholes
(including the one for 2185) would correspond to negligible
impact probabilities. Given that the impact, if it was to occur, would
release an energy equivalent to $3,700$ MegaTons of TNT, this
difference is practically relevant. In fact, the solution including
the Yarkovsky effect leads to a more reliable estimate of the Impact
Probabilities, because the Yarkovsky effect exists and needs to be
taken into account.
Is the discrepancy in the uncertainties with and without the dynamical
parameter in the fit essentially the same phenomenon we have found in
our simple model? We do not know the answer to this question, but we
shall investigate this issue in the future.
\begin{acknowledgements}
This work has been partially supported by the Marie Curie Initial Training
Network Stardust, FP7-PEOPLE-2012-ITN, Grant Agreement 317185. The
authors were also sponsored by an internal research fund of the
Department of Mathematics of the University of Pisa.
\end{acknowledgements}
|
1,108,101,564,555 | arxiv | \section{Introduction}\label{Section1}
In Bayesian statistics, it is a common problem to collect and compute random samples from a probability distribution. Markov Chain Monte Carlo (MCMC) is an intensive technique commonly used to address this problem when direct sampling is often arduous or impossible. MCMC using Bayesian inference is often used to solve problems in biology \cite{valderrama2019mcmc}, forensics \cite{taylor2014interpreting}, education \cite{drousiotis2021early}, and chemistry \cite{dumont2021quantification}, among other areas making it one of the most widely used algorithms when a collection of samples from a probability distribution is needed.
Monte Carlo applications are generally considered embarrassingly parallel since each chain can run independently on two or more independent machines or cores. Despite that, the main problem is that each chain is not embarrassingly parallel, and when the feature space and the proposal are computationally expensive, we can not do much to improve the running time and get results faster. When we have to handle huge state-spaces and complex compound states, it takes significant time for an MCMC simulation to converge on an adequate model not only in terms of the number of iterations required but also the complexity of the calculations occurring in each iteration(such as searching for the best features and tree shape of a decision tree). For example, running an MCMC on a single chain Decision tree for a dataset of $400 000$ datapoints and $15$ features took upwards of 6 hours to converge when run on a $2.3 - 5.10 GHz$ Intel Core i7-10875H. In \cite{byrd2008speculative}, an approach aiming to parallelise a single chain is presented, and the improvement achieved is at its best 2.2 times faster. The functionality of this kind of solution is therefore limited as in real-time, and life applications run time is critical. The work presented in this paper aims to find methods to significantly reduce the MCMC Decision tree's runtime by emphasising on the implementation of MCMC rather than the statistical algorithm itself. We aim to reduce significantly and up to an order of magnitude the run time of the MCMC Decision Tree on a single laptop or personal computer which is going to make the algorithm widely applicable and suitable for non tecinacl users. The remainder of this paper is organised as follows. Section 2 explains the MCMC in General and the Most Recent Work. Section 3 presents the MCMC in Decision trees. Our method is outlined in section 4, with the possible theoretical improvements. We introduce the case study in which we applied our method and reviewed results in section 5. Section 6 concludes the paper.
\section{Markov Chain Monte Carlo in General and Most Recent Work}\label{Section2}
One of the most widely used algorithms is the Metropolis \cite{metropolis1953equation} and its generalisation (see algorithm\ref{Metropolis Hashting}), the Metropolis-Hastings sampler (MH) \cite{hastings1970monte}. Given a partition of the state
vector into components, i.e., $x = (x_1, . . ., x_k )$, and that we wish to update the $i_th$ component, the Metropolis-Hastings update proceeds as follows. We first have a density for producing candidate observation $x'$, such that $x'_{i} = x_{i}$, which is denoted by $q(x,x')$. Given the chains ergodic condition, the definition of $q$ is arbitrary, and it has a stationary distribution $\pi$ which is selected so that the observations may be generated relatively easily. After the new state generation $ x' = (x_1, . . ., x_{i-1},
x_i, x_{i+1}, . . ., x_k )$ from density $q(x,x')$, the new state is accepted or rejected using the Rejections Sampling principle with acceptance probability $\alpha(x,x')$ given by equation \ref{chain1}. If the proposed state is rejected, the chain remains in the current state.
It is worth mentioning that acceptance probability in this form is not unique, considering there are many acceptance functions that supplies a chain with the required properties. Nevertheless, Peskun(1973) \cite{peskun1973optimum} proved that MH is the optimal one where the proper states are rejected least often, which maximises the statistical efficiency meaning that more samples are collected with fewer iterations.
\begin{equation}\label{chain1}
a(\chi, \chi') = \min(1, \frac{\pi(x')}{\pi(x')} \frac{q(\chi|\chi') }{q(\chi'|\chi) })
\end{equation}
On a Markov process, the next step depends on the current state, which makes it hard for a single Markov chain to be processed contemporaneously by several processing elements.
Byrd \cite{byrd2008reducing}. proposed a method to parallelise a single Markov chain(Multithreading on SMP Architectures), where we consider backup move “B” in a separate thread of execution as it is not possible to determine whether move “A” will be accepted. If “A” is accepted, the backup move ‘B’ - whether accepted or rejected - must be discarded as it was based upon a now supplanted chain state. If “A” is rejected, control will pass to “B”, saving much of the real-time spent considering “A” had “A” and “B” been evaluated sequentially. Of course, we may have as many concurrent threads as desired.
At this point, it is worth mentioning that the single chain parallelisation can become quickly problematic as the efficiency of the parallelisation is not guaranteed, especially for computationally cheap proposal distributions. Also, we need to consider that nowadays, computers make serial computations much faster than in 2008, when the single parallelisable chain was proposed.
Another way of making faster MCMC applications is to reduce the convergence rate by requiring fewer iterations. Metropolis-Coupled MCMC($(MC)^{3}$) utilised multiple MCMC \cite{altekar2004parallel} chains to run at the same time, while one chain is treated as the "cold" where its parameters are set to normal while the other chains are treated as "hot", which are expected to accept the proposed moves. The space will be explored faster through the "hot" chains than the "cold" as they are more possible to make disadvantageous transitions and not to remain at near-optimal solutions. The speedup increased when more chains and cores were added.
Our work is focused on achieving a faster execution time of the MCMC algorithm on Decision trees through multiprocessor architectures. We aim to reduce the number of iterations while the number of samples collected is not affected. Multi-threading on SMP Architectures and $(MC)^{3}$ differs from our work as the former targets rejected moves as a place for optimisation, and the latter requires communication between the chains. Moreover, the aims are different as $(MC)^{3}$ expands the combination of the chain, enhancing the possibilities of discovering different solutions and assisting avoid the simulation getting stuck in local optima.
\subsection{Probabilistic trees packages and level of parallelism}
Most of the existing probabilistic tree packages are only supported by the R programming language.
BART \cite{chipman2010bart} software included in the CRAN package \footnote{https://cran.r-project.org/web/packages/BART/index.html} supports multi threading based on OpenMP, where there are numerous exceptions for operating systems, so it is difficult to generalise. Generally, Microsoft Windows lacks OpenMP detection since the GNU autotools do not natively exist on this platform and Apple macOS since the standard Xcode toolkit is also not provided. The parallel package provides multi-threading via forking, only available in Unix. BART under CRAN uses parallelisation for the predict function and running concurrent chains.
BartMachine \footnote{https://cran.r-project.org/web/packages/bartMachine/index.html}, which is written in Java and its interface is provided by rJava package, which requires Java Development Kit(JDK), provides multi-threading features similar to BART. BartMachine is recommended only for those users who have a firm grounding in the java language and its tools to upgrade the package and get the best performance out of it. Similar to BART, its parallelisation is based on running concurrent chains.
The rest of the available packages, BayesTree\footnote{https://cran.r-project.org/web/packages/BayesTree/BayesTree.pdf},dbarts\footnote{https://cran.r-project.org/web/packages/dbarts/index.html},Bartpy\footnote{https://pypi.org/project/bartpy/},XBART\footnote{https://jingyuhe.com/xbart.html} and imptree\footnote{https://cran.r-project.org/web/packages/imptree/index.html} does not support any kind of parallelisation.
Concurrent chains can not solve the problem of long hours of execution time. For example, if a single chain needs 50 hours to execute, 5 chains will still need 50 hours if run concurrently. In contrast, in our case, a chain that serially needs 50 hours now takes approximately 2 hours for each chain. Moreover, we can to run concurrent chains where each chain is parallelised. If our implementation is compared to a package like BartMachine and BART, the runtime improvement we achieved is around 18 times faster, and if we compare it with a package that does not offer any parallelisation like most of the existing ones, the run time improvement for 5 chains is around 85 times faster.
\section{Markov Chain Monte Carlo in Decision Tree}\label{Section2}
A decision tree typically starts with a root node, which branches into possible outcomes. Each of those outcomes leads to additional decision nodes, which branch off into other possibilities ending up in leaf nodes. This gives it a treelike shape.
Our model describes the conditional distribution of $y$ given $x$, where $x$ is a vector of predictors $[x = (x_1,x_2,...,x_p)]$.
The main components of the $tree(T)$ includes the depth of the tree$(d(T))$, the features$(k(T))$ and the thresholds$(c(T))$ for each node where $k(T),c(T) \in \theta$, and the possibilities $p(Y|T,\theta,x)$ for each leaf node$(L(T))$.
If $x$ lies in the region
corresponding to the $i_th$ terminal node, then $y|x$ has distribution $f(y|\theta_i)$, where f represents a parametric family
indexed by $\theta_i$. The model is called a probabilistic classification tree, according to the quantitative response y.
As Decision Trees are identified by $(\theta, T)$, a
Bayesian analysis of the problem proceeds by specifying
a prior probability distribution $p(\theta ,T)$. Because $\theta$ indexes the parametric model for each $T$, it will usually be convenient to use the relationship
\begin{equation}\label{fullformula}
p(Y_1:_N,T,\theta|x_1:_N) = p(Y|T,\theta,x)p(\theta|T)p(T)
\end{equation}
In our case it is possible to analytically obtain eq \ref{fullformula} and calculate the posterior of $T$ as follows:
\begin{equation}\label{labels probabiliteis}
p(Y|T,\theta,x) = \prod_{i = 1}^{N} p(Y_i|x_i,T,\theta)
\end{equation}
\begin{equation}\label{features and thresholds}
p(\theta|T) = \prod_{j\in(T)}p(\theta_j|T) = \prod_{j\in(T)} p(k_j|T)p(c_j|k_j,T)
\end{equation}
\begin{equation}\label{prior}
p(T) = \frac{a}{(1+d)^\beta}
\end{equation}
Equation \ref{labels probabiliteis} describes the product of the probabilities of every data point($Y_i$) classified correctly given the datapoints features($x_i$), the tree structure($T$), and the features/thresholds($\theta$) on each node on the tree.
Equation \ref{features and thresholds} describes the product of possibilities of picking the specific feature($k$) and threshold($c$) on every node given the tree structure($T$).
Equation \ref{prior} is used as the prior for tree $T_i$. This formula is recommended by \cite{chipman2010bart} and three aspects specify it: the probability that a node at depth $d (=0.1.2. . . .)$ is nonterminal, the parameter $a\in {0,1}$ which controls how likely a node would split, with larger $\alpha$ values increasing the probability of split, and the parameter ${\beta > 0}$ which controls the number of terminal nodes, with larger values of $\beta$ reducing the number of terminal nodes. This feature is crucial as this is the penalizing feature of our probabilistic tree which prevents it from overfitting and allowing convergence to the target function $f(X)$ \cite{rovckova2019theory}, and it puts higher probability on "bushy" trees, those whose terminal nodes do not vary too much in depth.
An exhaustive evaluation of equation \ref{fullformula} over all trees $T$ will not be feasible, except in trivially small problems, because of the sheer
number of possible trees, which makes it nearly impossible to determine precisely which trees have the largest posterior probability.
Despite these limitations, Metropolis-Hastings algorithms can still be used to explore the posterior. Such algorithms simulate a Markov chain sequence of trees such as:
\begin{equation}\label{chain}
T_0, T_1, T_2,....,T_n
\end{equation}
which are converging in distribution to the posterior
$p(Y|T,\theta,x)p(\theta|T)p(T)$ in equtaion \ref{fullformula}.
Because such a simulated sequence will tend to gravitate toward regions of higher posterior probability, the simulation can be used to search for high-posterior probability trees stochastically. We next describe the details of such algorithms and their implementation.
\subsection{Specification of the Metropolis-Hastings
Search Algorithm on Decision Trees}
The Metropolis-Hastings(MH) algorithm for simulating
the Markov chain in Decision trees (see equation \ref{chain}) is defined as follows. Starting with an initial tree $T_0$, iteratively simulate the transitions from $T_i$ to $T_i+1$ by these two steps:
\begin{enumerate}
\item Generate a candidate value $T'$ with probability distribution $q(T_i, T')$.
\item Set $T_{i+1} = T'$ with probability
\begin{equation}\label{chain}
a(T_i, T') = \min(1, \frac{\pi(Y_1:_N,T',\theta'|x_1:_N)}{\pi(Y_1:_N,T,\theta|x_1:_N)} \frac{q(T,\theta|T',\theta') }{q(T',\theta'|T,\theta) })
\end{equation}
Otherwise set $T_{i+1} = T_i$.
\end{enumerate}
To implement the algorithm, we need to specify the transition kernel $q$. We consider kernels $q(T, T')$, which generate $T'$ from $T$ by randomly choosing among four steps:
\begin{itemize}
\item Grow(G) : add a new $D(T)$ and choose uniformly a $k(T)$ and a $c(T)$
\item Prune(P) : choose uniformly a $D(T)$ to become a leaf
\item Change(C) = choose uniformly a $D(T)$ and change randomly a $k(T)$ and a $c(T)$
\item Swap(S) = choose uniformly two $D(T)$ and swap their $k(T)$ and $c(T)$
\end{itemize}
The rules are chosen by picking a number uniformly between 0 and 1 and each action have its own interval. For example, $p(G)=0.3 ,p(P)=0.3 ,p(C)=0.2 ,p(S)=0.2, [0, 0.3, 0.6, 0.8, 1]$
The probabilities (see equation \ref{Actions}) represent the sum of the probabilities of every accepted forward move. P(G), p(P), p(C), p(S) are set by the user who chooses how often each move wants to be proposed.
\begin{equation}\label{Actions}
q(T',\theta'|T,\theta) = q(T'|T)q(\theta'|T')=\sum_a q(a) q(T'|T,a)q(\theta'| T',\theta,a)
\end{equation}
where :
\begin{equation}\label{Grow}
q(G)q(T'|T,G) q(\theta'|T',\theta,G) = p(G) \times \frac{1}{c} \times \frac{1}{k} \times \frac{1}{|L(T)|}
\end{equation}
\begin{equation}\label{Prune}
q(P)q(T'|T,P) q(\theta'|T',\theta,P) =p(P) \times \frac{1}{|D(T)| - 1}
\end{equation}
\begin{equation}\label{Change}
q(C)q(T'|T,C) q(\theta'|T',\theta,c) = p(C) \times \frac{1}{ |D(T)|}\times \frac{1}{c} \times \frac{1}{k}
\end{equation}
\begin{equation}\label{Swap}
q(S)q(T'|T,S) q(\theta'|T',\theta,S) =p(S) \times \frac{1}{ (|D(T)|(|D(T)| - 1)) / 2}
\end{equation}
Equation \ref{Grow} can be described as the possibility of proposing the grow move including the probability of choosing the specific feature($k$), threshold($c$) and leaf node($|L(T)|$) to grow. P(G) is multiplied by the number of features($k$), the unique number of datapoints($c$) and the number of leaf nodes($|L(T)|$). For example, given a dataset with 100 unique datapoints($c$), 5 features($k$), a tree structure($T$) with 7 leaf nodes($|L(T)|$) and a $p(G)=0.3$ eq\ref{Grow} will be $0.3\times \frac{1}{100}\times \frac{1}{5}\times \frac{1}{7}$
Equation \ref{Prune} is the possibility of proposing the prune move, where p(P) is multiplied by the number of decision nodes subtracting one($(|D(T)| - 1)$ we are not allowed to prune the root node). In practise, given a $p(P)=0.3$ and a tree structure($T$) with 7 decision nodes($|D(T)|$) eq\ref{Prune} will be $0.3\times \frac{1}{10-1}$
Equation \ref{Change} is the possibility of proposing the change move where p(C) is multiplied by the number of decision nodes($|D(T)|$), the number of features($k$), and the number of unique datapoints($c$). For example, given a dataset with with 100 unique datapoints($c$), 5 features($k$), a tree structure($T$) with 12 decision nodes($|D(T)|$) and a $p(G)=0.2$ eq\ref{Grow} will be $0.2\times \frac{1}{100}\times \frac{1}{5}\times \frac{1}{12}$
Equation \ref{Swap} is the possibility of proposing the swap move where p(S) is multiplied by the number of paired decision nodes($|D(T)|$).In practise, given a tree structure($T$) with 12 decision nodes($|D(T)|$) and a $p(S)=0.2$ eq\ref{Swap} will be $0.2\times \frac{1}{((12)(12-1))/2}$
\begin{theorem}\label{testenv-theorem}Transition kernel(see equation \ref{reverseActions}) yields a reversible Markov chain, as every step from $T$ to $T'$ has a counterpart that can move from $T'$ to $T$.
\end{theorem}
\begin{equation}\label{reverseActions}
q(T',\theta'|T,\theta)
\end{equation}
\begin{proof}
Assume a Markov chain, starting from its unique invariant distribution $\pi$. Now, take into consideration that for every sample $T_0,T_1,...,T_n$ have the same joint probability mass function(p.m.f) as their time reversal $T_n,T_{n-1},...,T_0$, so as we can call the Markov chain reversible, as well as its invariant distribution $\pi$ is reversible. This can be explained as a simulation of a reversible chain that looks the same if it runs backward.
The first thing we have to look for is if the Markov chain starts at $\pi$, and it can be checked by equation\ref{reversiblejump}
\begin{equation}\label{reversiblejump}
\begin{split}
&P(T_{k} = i|T_{k+1}=j, T_{k+2} = i_{k+2}, ...., T_n=i_n )\\
&
= \frac{P(T_{k} = i,T_{k+1}=j, T_{k+2} = i_{k+2}, ...., T_n=i_n )}{P(T_{k+1}=j, T_{k+2} = i_{k+2}, ...., T_n=i_n )}\\
&
=\frac{\pi P_{ij}P{ji_{k+2}} .... P_{i_{n-1}i_n}}{\pi P{ji_{k+2}} .... P_{i_{n-1}i_n}}\\
&
=\frac{\pi P_{ij}}{\pi_j}
\end{split}
\end{equation}
Equation\ref{reversiblejump} is only dependent on $i$ and $j$ where this expression for reversibility must be the same as the forward transition probability $P(X=T_{k+1} = i|X=T_k = j) = P_{ji}$. If, both original and the reverse Markov chains have the same transition probabilities, then their p.f.m must be the same as well.
\end{proof}
An example for our probabilistic tree is the following:
Assume a tree structure($T$) with 5 leaf nodes($|L(T)|$) and 11 decision nodes($|D(T)|$) sampling from a given dataset with 4 features($c$) and 100 unique datapoints($k$) for each feature.
If for example the forward proposal($q(T',\theta'|T,\theta)$) = ($"change"$) with p(C) = 0.2, we end up with the following equation:
$0.2\times \frac{1}{11}\times\frac{1}{4}\times\frac{1}{100}$.
At the same time the reverse proposal(going from the current position to the previous) ($q(T,\theta|T',\theta')$) equation looks exactly the same as the forward proposal. Given the above practical example we have strengthen our proof which shows that ($q(T',\theta'|T,\theta)$) = ($q(T,\theta|T',\theta')$), which shows in practise the reverse transition kernel nature of our model.
\begin{algorithm}
\caption{The General Metropolis-Hashting Algorithm}\label{Metropolis Hashting}
\begin{algorithmic}
\State Initialize $X_0$
\For{$i = 1$ to $N$}
\State sample $\chi'$ from $q(\chi'|\chi_{t-1})$
\State Calculate $\alpha(\chi_i,\chi_{t-1}) = \min(1, \frac{\pi(x')}{\pi(x')} \frac{q(\chi|\chi') }{q(\chi'|\chi) })$
\State Draw $u$ from $u[0,1]$
\If{$u < \alpha(\chi_i|\chi_{t-1})$ }
\State $\chi_i = \chi'$
\Else
\State $\chi_i = \chi_{t-1}$
\EndIf
\EndFor
\end{algorithmic}
\end{algorithm}
\section{Parallelising a single Decision Tree MCMC Chain}
Given a Decision's Tree MCMC chain with $N$ iterations, we propose a method that utilises $C$ number of cores aiming to enhance the running time of a single chain by at least an order of magnitude. As stated in Section\ref{Section1} and Section\ref{Section2}, at each iteration, a new sample $\chi'$ is drawn from the proposal distribution. Our method requires sampling from $C$ number of cores, $S (C=S)$ number of samples in parallel. We then accept the sample with the greatest $a(T_i, T')$ and repeat the same method until the Markov Chain converges to a stationary distribution.
In our method, we check the Markov chain convergence when the F1-score fluctuates less than $\pm 3\%$ for at least 100 iterations. Once the Chain has converged, we proceed to the second phase of our method. We now keep producing samples using $C$ cores, but we can now collect more than one sample which satisfies $a(T_i, T') >= u$. ($u$ is a random uniform number$[0,1]$), otherwise we collect $T_i$
From this point, we will propose new samples from the sample with the greatest $a(T_i, T')$ until we are happy with the number of samples collected. Using this method, we can collect the same number of samples and explore the feature space as effectively as the serial implementation, but 18 times faster using an average laptop or personal computer.
Our algorithm reduces the number of iterations and explores the feature space faster as we use more cores. This provides us with a significant run time improvement up to 18 times faster when the feature space is big and the proposal is expensive. The following sections will evaluate the running time improvement and the quality of the samples produced.
\begin{algorithm}
\caption{Single Chain parallelisation on MCMC Decision Trees}\label{DecisionTreeMetropolisParallel}
\begin{algorithmic}
\State Initialize $T_0$
\For{$i = 1$ to $N$}
\State sample $C$ number of $T'$ from $Q(T{^j}'|T{^j}_{t-1})$
\State Calculate in parallel $\alpha(T{^j}_i|T{^j}_{t-1}) = \min(1,\frac{p(Y_1:_N,T',\theta'|x_1:_N)}{p(Y_1:_N,T,\theta|x_1:} \frac{q(T,\theta|T',\theta') }{q(T',\theta'|T,\theta) })$ for each sample
\State Store every sampled posterior $\alpha(T{^j}_t|T{^j}_{t-1})$ value
\State Until converge $T'$ = $\max{\alpha(T{^j}_i|T{^j}_{t-1})}$
\If{Markov Chain converged}
\State Draw $u$ from $uniform[0,1]$
\For {$j= 1$ to $j$} \algorithmiccomment{For loop run in parallel}
\If{$\alpha(T{^j}_i|T{^j}_{t-1}) > u$}
\State Collect Sample $T'$
\State $T' = \max{\alpha(T{^j}_i|T{^j}_{t-1}})$
\Else
\State Collect Sample $T$
\EndIf
\EndFor
\EndIf
\EndFor
\end{algorithmic}
\end{algorithm}
\subsubsection{Theoretical gains}
Using $C$ cores simultaneously, the programme cycle consists of repeated "steps," each performing the equivalent of between 1 and $n$ iterations. We need to calculate the number of iterations based on the acceptance rate to produce the same number of samples($S$) when we increase $C$. The moves are considered in parallel, where they are accepted or rejected. Given that the average probability of a single arbitrary move being rejected is $p_r$, the probability of the $i_{th}$ in every single concurrent core is $pr$. This step continues for $i$ iterations where equations \ref{iterations}, \ref{Runtime}, and \ref{speedup} show the iterations needed, run time, and speedup improvement, respectively, given a time($t$) in minutes for each iteration. Theoretical speedup (see figure\ref{speedupgraph}) were plotted for varying cores.
\begin{equation}\label{iterations}
i = \frac{S/C}{pr}
\end{equation}
\begin{equation}\label{Runtime}
Runtime = \frac{i}{t}
\end{equation}
\begin{equation}\label{speedup}
Speedup = \frac{Runtime}{C}
\end{equation}
For example, given a single MCMC Decision Tree chain running for $10000$ iterations and an acceptance rate of $70\%$, after $3000$ burn-in iterations, we end up with $4900$ samples.
For the parallel MCMC chain, given the same settings as the serial one, with a $30\%$ burn-in period and 25 cores, we will collect the same number of samples with 500 iterations. This provides us with a 25 times faster execution time. Algorithm \ref{DecisionTreeMetropolisParallel} indicates the part implemented on a parallel environment, and figure 3 the maximum theoretical benefits from utilising our method.
Considering the communication overhead, the parts of the algorithm that are not parallelised, and the fact that the cores does not receive constant utilisation, in practise the speedups of this order are not achievable. Therefore, we will test it in practise and find out how it performs on real-life scenarios respect to the accuracy and the runtime improvement
\section{Results}
\subsection{Quality of the samples between serial and parallel implementation}
We have used the Wine dataset from scikit learn datasets\footnote{https://scikit-learn.org/stable/datasets/toydataset.html} repository as well as Pima Indians Diabetes and Dermatology from UCI machine learning repository \footnote{https://archive.ics.uci.edu/ml/index.php}, which are publicly available, to examine the quality of the samples on several testing hypotheses, including the different number of cores per iteration given the average F1-Score. Precision and Recall were also calculated for more depth and detailed insights about the performance and quality of the samples. The results, including the F1-Score, Precision, Recall, and Accuracy(see table2, table3 and table4) produced through 25-Fold Cross-Validation, ensure that every observation from the original dataset has the possibility of appearing in the training and test sets and also reduce any statistical error. Before any performance comparison, we need to examine whether the samples produced for each test case(25 cores and 40 cores) have any statistical difference from the serial implementation. We next examine if extracted samples by utilising 25 and 40 cores are representative of the family of the data they come. We use as ground truth data the F1-scores on each sample collected on every fold of the serial implementation, and we compare them with the corresponding collected samples from the other two test cases. In order to check any statistical difference between the samples, we performed the two-sample t-test for unpaired data\cite{fisher19783}, which is defined as follows:
\begin{equation}\label{t_test}
T = \frac{Y_1 - Y_2}{\sqrt{\frac{s_{1}^{2}}{N_1}+\frac{s_{2}^{2}}{N_2}}}
\end{equation}
\begin{equation}\label{degrees of freedom}
|T| > t_{1-a/2,\nu}
\newline
where:
\newline
\nu =
\frac{
\left(\dfrac{s_{1}^{2}}{n_1}+\dfrac{s_{2}^{2}}{n_2}\right)^{\!2}
}{
\dfrac{(s_{1}^{2}/n^{}_1)^2}{n_1-1}+
\dfrac{(s_{2}^{2}/n^{}_2)^2}{n_2-1}
}
\end{equation}
Formula\ref{t_test} is used to calculate the t-Test statistic equation where $N_1$ and $N_2$ are the sample sizes, $Y_1$ and $Y_2$ are the sample means, and $s_{1}^{2}$ and $s_{2}^{2}$ are the sample variances. The null hypothesis is rejected when equation \ref{degrees of freedom} holds, which is the critical value of the $t$ distribution with $\nu$ degrees of freedom.
For our first dataset(Wine), we first examine the serial implementation with the parallel using 25 cores. The absolute value of the t-Test, 0.62, is less than the critical value of 1.964, so we prove the null hypothesis and conclude that the samples drawn by using 25 cores have not any statistical difference at the 0.05 significance level.
We then compare the serial implementation with the parallel using 40 cores. In this case, the absolute value of the t-Test, 0.63, is less than the critical value of 1.964, so we prove the null hypothesis and conclude that the samples drawn by using 40 cores have not any statistical difference at the 0.05 significance level.
We also examine the parallel implementations between them(using 25 and 40 cores accordingly). In this case, the absolute value of t-Test, 0.64, is less than the critical value of 1.964, so we reject the null hypothesis and conclude that the samples drawn using 40 cores(in comparison with the samples drawn with 25 cores) have not a statistical difference at the 0.05 significance level.
The results for the rest of the datasets are presented explicitly in table 1
\begin{table}[htbp]\label{T}
\begin{tabular}{|l|c|c|c|c|}
\hline
Datasets & 1 vs 25 cores & 1 vs 40 cores & \multicolumn{1}{l|}{25 vs 40 cores} & \multicolumn{1}{l|}{Critical T value} \\ \hline
Pima Indians Diabetes & 0.51 & 0.63 & 0.63 & 1.97 \\
Dermatology & 0.73 & 0.77 & 0.77 & 1.97 \\
Wine & 0.62 & 0.64 & 0.65 & 1.97 \\ \hline
\end{tabular}\caption{Datasets Critical Values}
\end{table}
T-test proves that if we use up to 40 cores for sampling(rarely laptops and personal computers have more than 40cores), the quality of the samples are the same, ending up with statistically same samples as the serial implementation.Table 1, table 2, and table 3 shows that when we sample in parallel using up to 40 cores, the accuracy and the F1-score remain on the same levels as the serial implementation.
Tables 1, 2, 3, 4 indicate that a single chain on MCMC Decision Trees can not be an embarrassingly parallel algorithm as we can only improve the running time of a single chain by utilising a specific number of cores. The running time improvement we achieved($\times18$ faster) is the maximum run-time enhancement we can achieve on an MCMC decision tree to maintain the high metrics produced by the serial implementation. If we try to extract up to 40 samples per iteration, it is highly probable to get samples that does not affect the final results negatively. According to our results, the maximum number of cores can that be used is 40. Furthermore, Precision, Recall, and F1-score metrics indicate that no overfit is observed even when more than 3 labels exist, proving the samples' quality even in multi class classification problems.
\begin{table}[htbp]\label{tablewine}
\begin{tabular}{|l|cll|lll|lll|}
\hline
Labels & \multicolumn{3}{c|}{Precision} & \multicolumn{3}{c|}{Recall} & \multicolumn{3}{l|}{F1-score} \\ \hline
& 1 core & 20cores & 40cores & \multicolumn{1}{c}{1 core} & 20cores & 40cores & \multicolumn{1}{c}{1 core} & 20cores & 40cores \\ \cline{2-10}
0 & 0.85 & 0.88 & 0.88 & 1.00 & 1.00 & 1.00 & 0.92 & 0.94 & 0.94 \\
1 & 1.00 & 1.00 & 1.00 & 0.78 & 0.78 & 0.78 & 0.88 & 0.88 & 0.88 \\
2 & 1.00 & 0.93 & 0.93 & 1.00 & 1.00 & 1.00 & 1.00 & 0.96 & 0.96 \\ \hline
accuracy & \multicolumn{1}{l}{} & & & & & & \textbf{0.93} & \textbf{0.93} & \textbf{0.93} \\ \hline
\end{tabular}\caption{Results for Wine Dataset}
\end{table}
\begin{table}[htbp]\label{tablepimaindiandiabetes}
\begin{tabular}{|l|cll|lll|lll|}
\hline
Labels & \multicolumn{3}{c|}{Precision} & \multicolumn{3}{c|}{Recall} & \multicolumn{3}{l|}{F1-score} \\ \hline
& 1 core & 20cores & 40cores & \multicolumn{1}{c}{1 core} & 20cores & 40cores & \multicolumn{1}{c}{1 core} & 20cores & 40cores \\ \cline{2-10}
0 & 0.83 & 0.84 & 0.84 & 0.86 & 0.83 & 0.83 & 0.84 & 0.84 & 0.84 \\
1 & 0.65 & 0.63 & 0.63 & 0.61 & 0.65 & 0.86 & 0.63 & 0.64 & 0.64 \\ \hline
accuracy & \multicolumn{1}{l}{} & & & & & & \textbf{0.78} & \textbf{0.78} & \textbf{0.78} \\ \hline
\end{tabular}\caption{Results for Pima Indian Diabetes Datasets}
\end{table}
\begin{table}[htbp]\label{tabledermatology}
\begin{tabular}{|l|lll|lll|lll|}
\hline
Labels & \multicolumn{3}{c|}{Precision} & \multicolumn{3}{c|}{Recall} & \multicolumn{3}{l|}{F1-score} \\ \hline
& \multicolumn{1}{c}{1 core} & 20cores & 40cores & \multicolumn{1}{c}{1 core} & 20cores & 40cores & \multicolumn{1}{c}{1 core} & 20cores & 40cores \\ \cline{2-10}
0 & \multicolumn{1}{c}{0.93} & \multicolumn{1}{c}{0.93} & 0.93 & 1.00 & 1.00 & 1.00 & 0.97 & 0.97 & 0.97 \\
1 & 0.94 & 0.94 & 0.93 & 1.00 & 1.00 & 1.00 & 0.97 & 0.97 & 0.96 \\
2 & 1.00 & 1.00 & 1.00 & 0.96 & 0.96 & 0.96 & 0.98 & 0.98 & 1.00 \\
3 & 0.94 & 0.94 & 0.92 & 0.94 & 0.94 & 0.95 & 0.94 & 0.94 & 0.95 \\
4 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 \\
5 & 1.00 & 1.00 & 1.00 & 0.67 & 0.67 & 0.67 & 0.80 & 0.80 & 0.80 \\ \hline
accuracy & & & & & & & \textbf{0.96} & \textbf{0.96} & \textbf{0.96} \\ \hline
\end{tabular}\caption{Results for Dermatology Datasets}
\end{table}
\subsection{Practical gains}
Figure 1 presents the theoretical and practical speedup achieved given the number of cores used demonstrating a remarkable runtime improvement, especially when the feature space is ample and the proposal is expensive. Figure 1 demonstrates that in practise, theoretical speedups of this order can not be achieved for various reasons, including communications overhead, and as well as the cores do not receive constant utilisation. The practical improvement achieved used the novel method we proposed, speeds up the process up to 18 times depending on the number of cores the user may choose to utilise. Moreover, figure 1 demonstrates that even if we use more than 25 cores, the speedup achieved is the same, because of the architecture of the cores. When we run on a local machine(laptop or personal computer) a medium size dataset(500,000 entries), the memory is not enough to run every single parallel tree on a separate core. Given that, we have to wait for a core to finish the task, in order to allocate its memory to another core. Given that, we end up that it is not always beneficial to use more cores, as faster execution time and speedup is not guaranteed. Scaling up to 25 cores is the ideal, having in mind that any number above that, might not benefit the run time.
To the best of our knowledge it is the first time where a single chain in general, specifically on decision trees, is parallelised with our proposed method.
\begin{figure}[htbp]
\centerline{\includegraphics[width=0.85\textwidth]{speedup.png}}
\caption{Speedup achieved by utilising different number of cores}
\label{speedupgraph}
\end{figure}
\section{Conclusion}
Our novel proposed method for parallelising a single MCMC Decision tree chain takes advantage of multicore machines without altering any properties of the Markov Chain. Moreover, our method can be easily and safely used in conjunction with other parallelisation strategies, i.e., where each parallel chain can be processed on a separate machine, each being sped up using our method.
Furthermore, our approach can be applied to any MCMC Decision tree algorithm which needs to process hundreds of thousands of data given an expensive proposal where an execution time of 18 times faster can be easily achieved. As multicore technology improves, CPUs with multiple processing cores will provide speed-ups closer to the theoretical limit calculated. By taking advantage of the improvements in modern processor designs our method can help make the use of MCMC Decision tree-based solutions more productive and increasingly applicable to a broader range of applications.
Future work includes on expanding our method on a High Performance Computer(HPC), servers, and cloud which are build for this kind of tasks to compare and demonstrate possible runtime improvements, and discover the merits of such technologies. Moreover, we plan to implement more MCMC single chain parallelisation techniques, including data partitioning, and conduct experiments with various size and shapes datasets, to find the most effective technique, given the type and shape of the dataset.
|
1,108,101,564,556 | arxiv | \section{Introduction}
The Calogero model and its various generalizations is one of the most studied integrable systems in physics and mathematics
\cite{Calogero-1969,Sutherland-1971,Moser}. In its most basic form it describes $N$ identical non-relativistic particles in one dimension interacting through two-body inverse-square potentials in the presence of an external harmonic potential.
The Hamiltonian of the rational Calogero model in a harmonic trap reads
\begin{eqnarray}
{\cal H} &=& \frac{1}{2}\sum_{j=1}^{N}\big(p_j^{2}+\omega^2 x_j^2 \big)
+\frac{1}{2} \sum_{j,k=1; j\neq k}^{N} \frac{g^{2}}{(x_{j}-x_{k})^{2}},
\label{hCM}
\label{Hsquare}
\label{Eq1}
\end{eqnarray}
where $x_{j}$ are the coordinates of the particles, $p_{j}$ are their canonical momenta, and $g$ is the coupling constant. We normalized the mass of the particles to be unity. This model appears in many branches of physics and mathematics and has connections and relevance to fractional statistics, fluid mechanics, spin chains, two-dimensional gravity, strings in low dimensions etc; As a result, it has been studied extensively (see \cite{OlshaPer,Perelomov-book, Sutherland-book, APreview} for reviews and a comprehensive list of references).
The above model is classically and quantum mechanically integrable, and can be obtained as a reduction of a matrix system.
Several generalizations are also integrable, involving hyperbolic, trigonometric or elliptic mutual potentials, and also general
external potentials of quartic type or corresponding trigonometric type \cite{i3,APnew1,APnew2}. Moreover, integrable generalizations where the particles carry internal degrees of freedom have been proposed \cite{Kawakami1,Kawakami2,HaHa,MinahanAP,HiWa}.
A particular scaling limit (the ``freezing trick") then leads to integrable spin chains with long-range interactions, including the
Haldane-Shastry spin model as well as the non-translation invariant harmonic spin chain \cite{APspinchain, APspinchain1}.
A remarkable fact is that the hydrodynamic limit $N\to\infty$ of the system (\ref{Eq1}) can be found exactly using the methods of collective field theory \cite{JevickiSakita,Sakita-book,Jevicki-1992} or using the methods of Ref. \cite{2005-AbanovWiegmann,2009-AbanovBettelheimWiegmann}. This is quite nontrivial, as writing classical microscopic Hamiltonians in collective fluid mechanical
variables usually involves several approximations. By contrast, the Calogero fluid theory reproduces the dynamics of the many-body
system to all orders in $1/N$, with any corrections being of nonperturbative nature.
The integrability and other rich properties of the underlying particle systems suggest that the corresponding fluid mechanical equations are also integrable and point to the existence of soliton solutions. This has, indeed, been verified for the free ($\omega =0$) Calogero model \cite{1995-Polychronakos}
as well as the periodic trigonometric (Sutherland) model \cite{1995-Polychronakos}. A remarkable method of finding multi-soliton solutions
was recently proposed \cite{kul1,kul2}, relying on a ``dual" representation of the model, where the dual particles play the role of
``soliton variables''. The existence of such a dual form is far from obvious, and the demonstration that it reduces to the usual
Calogero model is quite involved.
The main goal of this paper is to present a first-order formalism based on a generating function (``prepotential")
that makes the self-dual version of the model
apparent and greatly simplifies the derivation of its connection to the second-order Calogero equations of motion. Based on this formalism, generalizations are proposed that can, in principle, admit non-identical particles with different masses and particle-dependent
interactions.
Conditions of stability and reality, subsequently, reduce the system to the dual formulation of the usual Calogero model and
its trigonometric and hyperbolic generalizations, but with more general external potentials which, interestingly, turn out to be the integrable quartic or trigonometric potentials found before \cite{i3,APnew1,APnew2}. In this formulation, soliton solutions can be identified in these more general potentials. Remarkably, solitons behave as regular Calogero particles but with negative mass and complex coordinates.
From this starting point, a similar finite dimensional reduction can also be performed in the hydrodynamic model, with the fluid motion parametrized in terms of a finite number of complex parameters representing soliton positions and speeds. Issues of stability and soliton condensation are also discussed
The paper is organized as follows:
In Section 2, we introduce our first-order formalism and derive the general form of two-body and external potentials that can be described this way. The solution of the functional equations that appear and derivation of the specific potentials are delegated to the appendices. We examine the stability and reality conditions and establish the appearance of the generalized quartic-type potentials and solitons.
In Section 3, we focus on systems with quartic external potentials and derive their dual formulation and soliton solutions. We comment on the mapping of the soliton problem to an electrostatic problem and present numerical solutions that demonstrate the integrable
nature of the soliton solutions.
In Section 4, we deal with the collective field and fluid mechanical formulation of these models and present the corresponding
soliton solutions in terms of meromorphic fields.
Finally, in Section 5 we state our conclusions and point to directions of future investigation.
\section{General First-Order Formalism of Interacting Systems}
In this section, we will formulate the first-order dual equations of motion for a dynamical system of particles
in terms of a generating function (prepotential). This formulation greatly simplifies the proof of
equivalence of the dual equations with the second-order equations of a Calogero-like system and allows for
generalizations involving more general two-body and external potentials.
The starting point is a system of $n$ particles on the line with coordinates $x_a$,
$a = 1, \dots , n$, obeying the {\it first-order} equations of motion
\begin{eqnarray}
\label{xdot1}
m_{a}\dot{x}_{a}=\partial_{a}\Phi
\end{eqnarray}
with $\Phi$ a function of the $x_a$, $\partial_a \equiv {\partial \over \partial x_a}$,
and $m_a$ a set of constant ``masses". Taking another time derivative
we obtain
\begin{eqnarray}
m_{a}\ddot{x}_{a}&=& \sum_b \partial_b \partial_a \Phi \, \dot{x}_b = \sum_b \partial_{a}\partial_{b}\Phi\frac{1}{m_{b}}\partial_{b}\Phi \nonumber \\&=&\partial_{a}\left[\sum_{b}\frac{1}{2 m_{b}}\left(\partial_{b}\Phi\right)^{2}\right]
\end{eqnarray}
This has the form of a standard equation of motion for particles of mass $m_a$ inside a potential
\begin{equation}
V = -\sum_{a}\frac{1}{2 m_{a}}\left(\partial_{a}\Phi\right)^{2}
\label{V}
\end{equation}
such that
\begin{eqnarray}
m_{a}\ddot{x}_{a}&=&-\frac{\partial V}{\partial x_a}
\end{eqnarray}
We wish the potential to contain only one-body and two-body terms. Further, the interactions (two-body terms)
should depend only on the relative particle distance. So we choose a $\Phi$ of the general form
\begin{eqnarray}
\Phi=\frac{1}{2} \sum_{a\neq b}F_{ab}(x_{ab})+ \sum_a W_a(x_a) ~,~~~ x_{ab} := x_a - x_b
\end{eqnarray}
By symmetrizing the sum, the $F_{ab}$ can be chosen to satisfy $F_{ab} (x) = F_{ba} (-x)$.
This symmetry ensures that the function $F_{ab}$ depends on the difference between $x_a$ and $x_b$ without caring about
the order of particles. This gives us
\begin{eqnarray}
\label{derphi}
\partial_{a}\Phi= \sum_{b}f_{ab}(x_{ab})+w_{a}(x_{a})
\end{eqnarray}
where we defined
\begin{eqnarray}
f_{ab} (x) = -f_{ba} (-x) = F'_{ab} (x) ~,~~~ w_a (x)= W'_a (x)
\end{eqnarray}
We look for conditions for $f_{ab}$ and $w_a$ such that the potential contains only
one- and two-body terms. We first examine the case of no external potential.
\subsection{The case $W_a (x)=0$}
At the moment, let us ignore $W_a (x_a)$, i.e, consider the case of no external potential. The expression
for the potential $V$ then is
\begin{eqnarray}
V =-\sum_{b\neq c,d}\frac{1}{2 m_{b}}f_{bc}(x_{bc})f_{bd}(x_{bd})
\end{eqnarray}
Let us define from here on the shorthand (and similarly for other functions)
\begin{eqnarray}
f_{ab} \equiv f_{ab} (x_{a}-x_{b}) ~,~~{\rm satisfying}~~ f_{ab} = - f_{ba}
\end{eqnarray}
We also define renormalized functions $\tilde{f}_{ab}$,
\begin{eqnarray}
f_{ab}=m_{a}m_{b}\tilde{f}_{ab}
\end{eqnarray}
Then the above potential becomes
\begin{eqnarray}
V &=&-\frac{1}{2}\sum_{b\neq c,d} m_{b}m_{c}
m_{d}\tilde{f}_{cb}\tilde{f}_{db} \nonumber \\
&=&-\frac{1}{2} \sum_{b\neq c, c=d} m_b m_c^2 \tilde{f}_{bc}^2 -
\frac{1}{2}
\sum_{b\neq c\neq d}m_{b}m_{c}m_{d} \tilde{f}_{bc}\tilde{f}_{bd}
\end{eqnarray}
and symmetrizing over the summation indices
\begin{eqnarray}
V
&=&-\frac{1}{4} \sum_{b\neq c} m_b m_c (m_b + m_c ) \tilde{f}_{bc}^2 \nonumber \\
&& -\frac{1}{6}
\sum_{b\neq c\neq d}m_{b}m_{c}m_{d}\left[\tilde{f}_{bc}\tilde{f}_{bd}
+\tilde{f}_{cb}\tilde{f}_{cd}+\tilde{f}_{db}\tilde{f}_{dc}\right]
\label{Vsymm}
\end{eqnarray}
The term in the last bracket is, in general, a three-body term. We demand that it be, instead, a sum of
two-body terms. That is, we impose the condition
\begin{eqnarray}
\tilde{f}_{bc}\tilde{f}_{bd}
+\tilde{f}_{cb}\tilde{f}_{cd}+\tilde{f}_{db}\tilde{f}_{dc}= g_{bc} + g_{bd} + g_{cd}
\end{eqnarray}
for some functions $g_{ab} (x)$, for all distinct $b$, $c$, $d$.
The above is a functional equation similar to equations encountered in the study of the Lax pair of identical Calogero
particles \cite{calfunc}
${\tilde f}_{ab}$ and $g_{ab}$ can depend on the particle indices. Its solution can be obtained with methods
similar to the ones for the indistinguishable case, and is derived in Appendix A. In general, the solution
involves elliptic Weierstrass functions. In this paper we will focus on the simpler case where the functions
$g_{ab}$ are constants, that is
\begin{eqnarray}
\tilde{f}_{cb}\tilde{f}_{db}
+\tilde{f}_{dc}\tilde{f}_{bc}+\tilde{f}_{bd}\tilde{f}_{cd}=C_{bcd}
\label{functf}
\end{eqnarray}
where $C_{bcd}$ is a constant independent of $x$. In this case, the three-body term in the potential (\ref{Vsymm})
becomes an irrelevant constant and the potential becomes a sum of two-body terms $V_{ab}$.
Up to rescalings of the coordinate $x$, we have the folowing three possibilities for
$\tilde{f}_{ab}(x_{ab})$, $F_{ab}(x_{ab})$ and $V_{ab} (x_{ab})$ for various values of $C_{bcd}$, described in Table \ref{tab1} with $g$ a constant.
\begin{table}
\caption {Solutions to functional equations without external potential} \label{tab1}
\begin{center}
\begin{tabular}{|c|c|c|c|}
\hline
$C_{abc}$ & $\tilde{f}_{ab}(x_{ab})$ & $F_{ab}(x_{ab})$ & $- V_{ab} ( x_{ab} )$ \tabularnewline
\hline
\hline
0 & $g/{x_{ab}}$ & $g \, m_a m_b \log\left|x_{ab}\right|$ &{\large ${g \,m_a m_b (m_a + m_b ) \over 4 \, x_{ab}^2}$}\tabularnewline
\hline
$-g^2$ & $g \,{\cot x_{ab}}$ & $g \, m_a m_b \log\left|\sin x_{ab}\right|$ & {\large ${g^2 m_a m_b (m_a + m_b ) \over 4 \sin^2 x_{ab}}$}\tabularnewline
\hline
$+g^2$ & $g\,{\coth x_{ab}}$ & $g \, m_a m_b \log\left|\sinh x_{ab}\right|$ &{\large ${g^2 m_a m_b (m_a + m_b ) \over 4 \sinh^2 x_{ab}}$}\tabularnewline
\hline
\end{tabular}
\end{center}
\end{table}
The above are essentially the rational, periodic and hyperbolic integrable models of
Calogero type but with particle-dependent two-body couplings. The integrablity of this generalized version
of family of Calogero models (unequal coupling constants and masses) remains unexplored.
\subsection{The case $W_a (x ) \neq 0$ (External Potentials)}
We now consider the case where the prepotential includes one-body terms $W_a (x_a )$, leading to
a term $w_a = W'_a (x_a )$ in (\ref{derphi}). In analogy with $f_{ab}$ we define
\begin{equation}
w_a = m_a {\tilde w}_a
\end{equation}
The expression for the potential, using ${\tilde f}_{ab} = - {\tilde f}_{ba}$ and symmetrizing in the indices,
becomes
\begin{eqnarray}
-V
&=&\frac{1}{4} \sum_{b\neq c} m_b m_c (m_b + m_c ) \tilde{f}_{bc}^2 \nonumber \\
&+& \frac{1}{6}
\sum_{b\neq c\neq d}m_{b}m_{c}m_{d}\left[\tilde{f}_{bc}\tilde{f}_{bd}
+\tilde{f}_{cb}\tilde{f}_{cd}+\tilde{f}_{db}\tilde{f}_{dc}\right] \nonumber \\
&+& \frac{1}{2} \sum_{b\neq c} m_b m_c ({\tilde w}_b - {\tilde w}_c ) {\tilde f}_{bc} + \frac{1}{2} \sum_b m_b {\tilde w}_b^2
\label{VWsymm}
\end{eqnarray}
The terms in the first two lines are the same as for the $W_a = 0$ case, and the requirement that they reduce to
two-body terms gives the same solutions for ${\tilde f}_{ab}$ as before. The last line contains two-body and one-body
potentials. Demanding that two-body terms depend only on particle distance imposes the condition
\begin{equation}
({\tilde w}_b - {\tilde w}_c ) {\tilde f}_{bc} = u_{bc} + v_b + v_c
\label{bc}
\end{equation}
for some particle-dependent functions $u_{ab} (x)$ and $v_a (x)$.
The above is a functional equation for the functions ${\tilde w}_a$ whose solutions depend on the functions
$ {\tilde f}_{ab}$. Its treatment is given in Appendix B, and we state in Table \ref{tab2} the solutions in each case,
up to rescalings of $x$, with $m_{tot} = \sum_a m_a$ the total mass, $C_a, c_1 , c_2 , c_3$ constants, and as usual $C_{ab} = C_a - C_b$.
\begin{table}
\caption {Solutions to functional equations with external potential} \label{tab2}
\begin{center}
{\hspace*{-1.1cm}
\begin{tabular}{|p{1.5cm}|p{2.9cm}|p{2.5cm}|p{8.4cm}|}
\hline
$\tilde{f}_{ab}(x_{ab})$ & $~~~~{\tilde w}_a ( x_a )$ & $~~~u_{ab} ( x_{ab} )$ & $~~~~~~~~~~~~~~~~~
-2 V_a (x_a )$\tabularnewline
\hline
\hline
{\vspace*{-0.1cm}$g/{x_{ab}}$ }& $C_a + c_1 x_a + c_2 \, x_a^2 + c_3 \, x_a^3$ & { {\vspace*{-0.15cm}{\large$\frac{g C_{ab}}{x_{ab}}$}$-\frac{1}{2}{g c_3} x_{ab}^2$}}&
{\vspace*{-0.1cm} $m_a {\tilde w}_a^2 +
g (m_{tot}-m_a) m_a (c_2 \, x_a + {3 \over 2} c_3 \, x_a^2)$}\tabularnewline
\hline
{\vspace*{-0.1cm}$g \cot x_{ab}$}& $C_a + c_1 \cos 2x_a + c_2 \sin 2x_a + c_3 x_a $&{$g C_{ab}\cot x_{bc}+g c_3 x_{ab} \cot x_{ab}$ }& \vspace*{-0.1cm}{$m_a {\tilde w}_a^2 + g
(m_{tot}-m_a) m_a ( c_2 \cos 2x_a - c_1 \sin 2x_a) $}\tabularnewline
\hline
{\vspace*{-0.1cm}$g \coth x_{ab}$} & \small{$C_a + c_1 \cosh 2x_a + c_2 \sinh 2x_a + c_3 x_a$}&$g C_{ab} \coth x_{bc}+g c_3 x_{ab} \coth x_{ab}$ &{\vspace*{-0.1cm} $m_a {\tilde w}_a^2 + g
(m_{tot}-m_a) m_a (c_2 \cosh 2x_a + c_1 \sinh 2x_a )$}\tabularnewline
\hline
\end{tabular}
}
\end{center}
\end{table}
In the case that we restrict our solutions to $u_{bc} = 0$ (a justification of why this may be relevant is given later), that is, we only allow one-body terms to appear
in the right hand side of (\ref{bc}), then $c_3$ must vanish and all $C_a$ have to be equal. The acceptable forms for ${\tilde w}_a$ and coresponding one-body potentials are given in Table \ref{tab3}. Interestingly, we recover the restricted form of the integrable potentials found in \cite{i3,APnew1, APnew2}, for which the proof of integrability simplifies considerably. (The most general class of integrable potentials, derived in \cite{APnew1,APnew2},
depends on one additional parameter.)
\begin{table}
\caption {Solutions to functional equations with one-body external potentials} \label{tab3}
\begin{center}
{\hspace*{-1cm}
\begin{tabular}{|c|c|c|}
\hline
$\tilde{f}_{ab}(x_{ab})$ & ${\tilde w}_a ( x_a )$ & $-2 V_a (x_a )$\tabularnewline
\hline
\hline
$g/{x_{ab}}$ & $c_0 + c_1 x_a + c_2 x_a^2$ & { $m_a {\tilde w}_a^2 + g
(m_{tot}-m_a) m_a c_2 \, x_a$}\tabularnewline
\hline
{$g \cot x_{ab}$}&{$c_0 + c_1 \cos 2x_a + c_2 \sin 2x_a$ }&{ $m_a {\tilde w}_a^2 + g
(m_{tot}-m_a) m_a ( c_2 \cos 2x_a - c_1 \sin 2x_a ) $}\tabularnewline
\hline
{$g \coth x_{ab}$} & \small{$c_0 + c_1 \cosh 2x_a + c_2 \sinh 2x_a$} &{ $m_a{\tilde w}_a^2 +
g (m_{tot}-m_a) m_a ( c_2 \cosh 2x_a + c_1 \sinh 2x_a )$ }\tabularnewline
\hline
\end{tabular}
}
\end{center}
\end{table}
It would seem peculiar that the eliminated term is the cubic term in the rational case, while it is the linear term in the
trigonometric and hyperbolic cases. We can check, however, that the small-$x$ limit of the above trigonometric or hyperbolic
potentials, upon proper scaling
of the coefficients, reduces to the rational case. In particular, the presence of the linear term
in the trigonometric and hyperbolic cases introduces an extra parameter which can be tuned to make
the coefficient of $x^3$ finite and nonzero in the small-$x$ limit.
\subsection{Reality Conditions and Solitons}
The potential of the above dynamical system, as given by (\ref{V}), is negative definite and thus, in principle,
unstable. To obtain more interesting stable systems, we extend the variables and parameters to the complex
plane. The goal is to find a set of parameters for which the particles, or at least a subset of them, will
remain on the real axis and will constitute a stable real dynamical system.
With an appropriate choice of parameters, we can ensure that the potential will turn positive and thus be
stable when the $x_a$ are on the real axis. This is simply achieved by taking $\Phi \to i \Phi$, that is,
\begin{equation}
{\tilde f}_{ab} \to i {\tilde f}_{ab} ~,~~{\tilde w}_a \to i {\tilde w}_a ~,~~ V \to -V
\end{equation}
The first-order equations of motion in terms of $i {\tilde f}_{ab}$ and $i {\tilde w}_a$, however, become
\begin{equation}
{\dot x}_a = i \sum_{b \neq a} m_b {\tilde f}_{ab} + i {\tilde w}_a
\label{xadot}
\end{equation}
We observe that even if the initial positions of the particles are real, their velocities will be imaginary and
therefore will escape into the complex plane. It is thus impossible to obtain a stable system with
all the particle coordinates real.
We demand, therefore, that a subset of the particle coordinates, say $x_1 , \dots x_N$, to remain real,
while the rest of them, $x_{N+1} , \dots x_n$, will become complex. For clarity, we call the real coordinates
$x_j$, $j=1,\dots N$, and the remaining complex ones $z_\alpha$, $\alpha = 1, \dots M$ with $M=n-N$.
For reasons that will become apparent later on, we call the $z_\alpha$ ``solitons".
The first basic requirement is that the potential should contain no couplings between $x_j$ and $z_\alpha$,
else such mutual forces would drive the $x_j$ into the complex plane. This means, in particular, that the
top line in (\ref{VWsymm}) should contain no mixed terms; that is
\begin{equation}
m_j m_\alpha (m_j + m_\alpha ) = 0 ~~{\rm for~all}~~j,\alpha
\end{equation}
The only possibility (other than $m_j =0$ or $m_\alpha =0$) is
\begin{equation}
m_j = - m_\alpha = m
\end{equation}
That is, all particles have the same (positive) mass, while solitons have the same negative mass. The second
line in (\ref{VWsymm}) is a constant in our case, so it is of no concern. The first term in the third line, however,
in principle couples particles and solitons, as its mixed terms are
\begin{equation}
m_j m_\alpha u_{j\alpha}
\end{equation}
We thus demand $u_{j\alpha} = 0$, which leaves the possibilities found before for vanishing $u_{ab}$.
The above conditions will secure that the potential does not couple (real) particles and (complex) solitons
and thus their second-order equations of motion decouple.
Particles will obey their own Calogero-like equation of motion and solitons will
obey their own similar equations.
To make sure that the motion of particles will remain real, we must further ensure that their initial velocities are
real. Their expression (for $m_j =m$ and $m_\alpha = -m$) is
\begin{equation}
{\dot x}_j = i m \sum_{k (\neq j)} {\tilde f}_{jk} - i m \sum_{\alpha} {\tilde f}_{j\alpha} + i {\tilde w}_j
\label{xri}
\end{equation}
while the corresponding expression for solitons is
\begin{equation}
{\dot z}_{\alpha} = - i m \sum_{k } {\tilde f}_{\alpha k} + i m \sum_{\beta (\neq \alpha)} {\tilde f}_{\alpha \beta} + i {\tilde w}_{\alpha}
\label{zjs2}
\end{equation}
For real $x_j$, the first and last terms in the right hand side of (\ref{xri}) are purely imaginary. Reality of the ${\dot x}_j$
implies
\begin{eqnarray}
&& m \sum_{k (\neq j)} {\tilde f}_{jk} - m~ {\rm Re} \left( \sum_{\alpha} {\tilde f}_{j\alpha} \right)
+ {\tilde w}_j = 0 \label{Imz} \\
&&{\dot x}_j = m ~{\rm Im} \left(\sum_{\alpha} {\tilde f}_{j\alpha} \right)
\label{Rez}
\end{eqnarray}
The above are, altogether, $2N$ real constraints for $2N+ 2M$ real variables (the $N$ initial positions $x_j$,
$N$ initial velocities ${\dot x}_j$ and the $M$ complex initial values of complex $z_\alpha$). Overall, there are
$2M$ free parameters in this model, which can be chosen as the initial values of $z_\alpha$. (We caution that
the parameters $z_\alpha$
may not actually uniquely determine the state of the system, as equation (\ref{Imz}) may have none or more than one
solutions for the $x_j$.)
We see that, in general, the real particle system $x_j$ follows a motion that is parametrized by the initial values
of the $M$ complex soliton parameters $z_\alpha$, which can, then, be viewed as phase space variables for
the system. The number of solitons $M$ determines the degrees of freedom of the system. In particular,
for $M=0$ all velocities are zero and particles are in their equilibrium position determined by (\ref{Imz})
for ${\tilde f}_{j\alpha} =0$.
For the case $M=1$, $z_1 (t)$ moves as a solitary particle inside the one-body potential $V (z_1 )$
given in Table 3. Its presence deforms the distribution of $x_j$, creating a coalescence of
particles near Re$(z_1 (t) )$, that is, a solitary wave. The imaginary part of $z_1 (t)$ determines the
width of the wave, while also determining the velocity of the soliton. For $M>1$ the various $z_\alpha$
move as particles in the potential $V ( z)$ interacting through Calogero-type potentials, while the real
particles $x_j$ perform a motion ``guided" by the $z_\alpha$. The above features justify the identification
of $z_\alpha$ as soliton parameters, and the corresponding motion of $x_j$ as solitonic many-body ``waves".
\subsection{Stability}
In the previous section we ensured that the dynamical system described by the $x_j$ obeys second-order equations
of motion in a stable potential. We still need to ensure, however, that the initial value problem as determined
by the choice of soliton parameters $z_\alpha$ has solutions. As we shall see, this imposes nontrivial constraints.
We start with the simplest case of no solitons. As stated, this corresponds to static $x_j$ obeying the condition
\begin{equation}
m \sum_{k (\neq j)} {\tilde f}_{jk} + {\tilde w}_j = 0
\end{equation}
or, using the definitions of ${\tilde f}_{jk}$ and ${\tilde w}_j$ as well as $m_j = m_k = m$,
\begin{equation}
{\partial \Phi \over\partial x_j} = 0 ~,~~~ \Phi = \sum_{j<k} F_{jk} (x_{jk}) + \sum_j W(x_j )
\end{equation}
These are the equilibrium conditions for particles $x_j$ inside the (real) prepotential $\Phi$ defined above. In order
for this to have solutions it must be of an appropriate form.
Let us examine the rational case. The prepotential and real potential in this case become
\begin{eqnarray}
\Phi = \sum_j \left( c_0 x_j + {c_1 \over 2} x_j^2 + {c_2 \over 3} x_j^3 \right) + \sum_{j<k} g \, m \ln | x_{jk} |
~~~~~~~~~~\cr
V= \sum_j \left[ {m \over 2} \left( c_0 + c_1 x_j + {c_2} x_j^2 \right)^2 + g (N-1) m^2 c_2 x_j
\right] + \sum_{j<k} {g \, m^2 \over x_{ab}^2}
\label{Vquar}
\end{eqnarray}
The prepotential consists of a {\it cubic} one-body potential and a logarithmic two-body interaction. We can always
choose the sign
of $g$ that makes the interaction term repulsive, $g <0$ (in the opposite case we can flip the sign of
$g$ as well as $c_0 , c_1 , c_2$ which leads to the same potential $V$).
So this becomes a standard problem of finding the
equilibrium position of particles in a potential with repulsive logarithmic interactions.
Generically, this problem may not have solutions since the cubic potential is unbounded from below. This happens,
in particular, for the simplest nontrivial purely cubic case ($c_0 = c_1 = 0$). To guarantee the existence of solutions
the prepotential must have a ``well" such that it can trap particles. The existence of such a well
requires the condition
\begin{equation}
c_1^2 > 4 c_0 c_2
\end{equation}
This condition is necessary but not sufficient. The well must also be deep enought to hold $N$ particles repelling
each other with a logarithmic interaction of strength $g m$. We have no explicit expressions for this restriction in
terms of the constants of the problem. We stress that the above restrictions are necessary so that the equilibrum
problem can be addressed in the first-order formalism. The real potential $V$, being a quartic expression in the coordinate,
always has an equilibrium solution.
For a nonzero number of solitons the situation becomes more complicated, as now the set of equations
(\ref{Imz}, \ref{Rez}) must admit solutions. In general this will imply restrictions to both the form of the
potential and the range of ``phase space" parameters $z_\alpha$. We point out, however, that the interaction
between particles and solitons is attractive, so in general the presence of solitons improves the situation.
In fact, it may be that the unstable prepotentials for which the zero-soliton case has no static solutions
become stable in the presence of a large number of solitons, akin to a ``broken" symmetry phase with a soliton condensate. We postpone the investigation of this issue for a future publication.
\subsection{A Note on Integrability}
Solitons are a hallmark of integrability and, indeed, Calogero-like systems are one of the most celebrated
classes of integrable models. The systems derived here, however, differ from standard Calogero models in the fact that
(i) the particles are not identical, with distinct masses and corresponding two-body interactions, and (ii) in the presence
of a more general one-body potential.
In the above analysis, the issue of integrability remains unaddressed. There are, nevertheless, some intriguing
indications that these systems are integrable. First, the reality conditions necessary to have stable potentials
naturally restrict the system to identical particles ($m_j = m$) which is integrable. Further, the extended
one-body potentials that we obtained (quartic in the rational case, harmonic with two nontrivial modes in the
trigonometric and hyperbolic cases) are exactly the potentials that are know to be integrable \cite{APnew1,APnew2}.
In fact, the obtained potentials belong to a somewhat restricted class.
Specifically, a generic quartic potential has 4 nontrivial parameters (ignoring the constant term). Our potential,
however, has only 3 parameters ($c_0, c_1 , c_2$), being essentially a complete square. Although arbitrary quartic
potentials are integrable, for the above restricted potentials of ``Bogomolny" type the proof of integrability simplifies
considerably \cite{APnew1, APnew2, i3}.
Finally, an inspection of the known integrals of motion for the standard quadratic Calogero model ($c_3 = 0$)
reveals that they are all zero. (This is not a contradiction, since particles and solitons contribute with opposite
signs.) Although this is not yet a proof of integrability, since the value of higher integrals could fluctuate
between particles and solitons, it is a tantalizing clue that a general proof of integrability may well be
possible within the first-order formalism. This and other issues are saved for future investigation.
\section{Dual representation and solitons of the rational model in external quartic potentials }
\label{dualCM}
In this section we focus on the particular concrete example of rational Calogero models in external quartic potentials. Using the general formalism of the previous section, we demonstrate the existence of a dual version involving soliton variables $z_\alpha$ and derive analytical and numerical solutions.
\subsection{First order equations for the rational Calogero case}
\label{sec:dcm}
The general formalism discussed above greatly helps in writing down the first order equations for the rational Calogero case. If we take the first row of Table 2 and impose that particles (indexed by $j$) have mass $m_j =1$ and solitons (indexed by $\alpha$) have mass $m_\alpha =-1$, then (\ref{xri}) and (\ref{zjs2}) give
\begin{eqnarray}
\dot{x}_{j} -iw(x_{j}) & = & -ig\sum_{k=1 (k\neq j)}^{N}\frac{1}{x_{j}-x_{k}}
+ig\sum_{\alpha=1}^{M}\frac{1}{x_{j}-z_{\alpha}},
\label{xjdot} \\
\dot{z}_{\alpha} -iw(z_{\alpha})& = & ig\sum_{\alpha=1(\alpha \neq \beta)}^{M}\frac{1}{z_{\alpha}-z_{\beta}}
-ig\sum_{j=1}^{N}\frac{1}{z_{\alpha}-x_{j}},
\label{zjdot}
\end{eqnarray}
for $x_{j}(t)$ with $j=1,2,\ldots, N$ and $z_{\alpha}(t)$ with $\alpha=1,2,\ldots,M$. Here, the function $w$ is given in Table 3 as
\begin{equation}
w (x) = c_0 + c_1 x + c_2 x^2
\end{equation}
Since both $w_a$ and $V_a$ are independent of the particle index $a$ (see Table 3), we dropped the subscripts.
Equations (\ref{xjdot},\ref{zjdot}) are first order in time and, for complex coordinates, the dynamics is fully defined by the initial values of $x_{j}$, $z_{\alpha}$, i.e., by $N+M$ complex numbers ($2N+2M$ real variables). If the particle coordinates $x_j$ are real (with real velocities), as discussed in section 2.3, the dynamics is fully determined by the $M$ complex initial values of
$z_\alpha$. Applying Eq. (\ref{V}) of the general formalism, the corresponding second order equations are given by
\begin{eqnarray}
\ddot{x}_{j} & = & -\frac{g^{2}}{2}\frac{\partial}{\partial x_{i}}\sum_{i\neq j}^{N}
\frac{1}{\left(x_{i}-x_{j}\right)^{2}}-V^{\prime}(x_{j}), \qquad j=1,\dots, N
\label{pmotx}
\\
\ddot{z}_{\alpha} & = & -\frac{g^{2}}{2}\frac{\partial}{\partial z_{\alpha}}\sum_{\alpha=1 (\alpha \neq \beta)}^{M}\frac{1}{\left(z_{\alpha}-z_{\beta}\right)^{2}}-V^{\prime}(z_{\alpha}), \qquad \alpha=1,\dots, M
\label{pmotz}
\end{eqnarray}
We refer to the system (\ref{xjdot},\ref{zjdot}) as the \textit{dual Calogero system in external potential $V$}. We emphasize that the general formalism greatly simplifies the transition from first order to second order equations. Lack of such a general formalism would have involved a laborious algebra to arrive at the second order equations (\ref{pmotx},\ref{pmotz}). We also point out that the derivation of (\ref{pmotx},\ref{pmotz}) from (\ref{xjdot},\ref{zjdot}) holds for
arbitrary $N$ and $M$, although we are mostly interested in the case $M<N$.
\subsection{Multi-Soliton solutions}
\label{sec:Msolutions}
From the first order equations derived above for the rational Calogero model, we get (separating real and imaginary parts of (\ref{xjdot})),
\begin{eqnarray}
w(x_{j}) &=& g\sum_{k=1 (k\neq j)}^{N}\frac{1}{x_{j}-x_{k}}
-\frac{g}{2}\sum_{\alpha=1}^{M}\left(\frac{1}{x_{j}-z_{\alpha}}+\frac{1}{x_{j}-\bar{z}_{\alpha}}\right),
\label{xjreal} \\
{\dot x}_j = p_{j}& =&
i\frac{g}{2}\,\sum_{\alpha=1}^{M}\left(\frac{1}{x_{j}-z_{\alpha}}-\frac{1}{x_{j}-\bar{z}_{\alpha}}\right).
\label{pjreal}
\end{eqnarray}
As we noted in section 2.3, if we specify the $M$ complex positions $z_{\alpha}$ at any time we can find both the $N$ real positions $x_{j}$ and the corresponding real momenta $p_{j}$. If we are given the initial values of $x_{j}$ and $p_j$, their evolution is fully determined by (\ref{pmotx}). However, these initial values are not, in general, independent, as they are related by (\ref{xjreal},\ref{pjreal}) through the values of the $M$ complex parameters ($2M$ real parameters) $z_{\alpha}$ . (Only when $M \ge N$ we can choose them independently.) The initial values of ${\dot z}_\alpha$
are always restricted, as they need to satisfy (\ref{zjdot}) and the $x_j$ are fully fixed by the $z_\alpha$.
\subsection{Solution for zero solitons}
\label{background}
In this section we discuss the case of zero solitons. This gives rise to a spacially inhomogeneous static background configuration where all particles are at rest. (We also note that this is the same as the limit where all $z_\alpha$
go to infinity.) Taking $M=0$ we have $p_{j}=0$ for all $j$ and the particle coordinates settle in the equilibrium
positions
\begin{equation}
\label{Hermite}
w (x _j ) = c_0 + c_1 x_j + c_2 x_j^2 = g \sum_{k=1 (k\neq j)}^{N} \frac{1}{x_j-x_k}.
\end{equation}
If we just had a harmonic trap as in \cite{kul1,kul2}, in which case $w^{harm}(x)=\omega x$,
then it is known that the solution of this system of algebraic equations is given by the roots of the $N$-th Hermite polynomial (Stiltjes formula \cite{Szego-1975,mehta}). We are unaware of a generalization of the Stiltjes formula when $w(x)$ has a quadratic form as above.
\subsection{The single-soliton solution}
\label{1soliton}
The single soliton case is essentially equivalent to one complex $z$ coordinate moving freely in a quartic polynomial potential. For $M=1$ equations (\ref{xjreal},\ref{pjreal}) become
\begin{eqnarray}
c_0 &+& c_1 x_j + c_2 x_j^2 = g\sum_{k=1 (k\neq j)}^{N} \frac{1}{x_j-x_k}
-\frac{g}{2} \left( \frac{1}{x_j -z} + \frac{1}{x_j-\bar{z}}\right),
\label{M1xj} \\
p_j &=& i\frac{g}{2} \left( \frac{1}{x_j-z} - \frac{1}{x_j-\bar{z}}\right).
\label{M1pj}
\end{eqnarray}
Equation (\ref{M1xj}) is a further generalization of the Stieltjes problem (\ref{Hermite}) (see Refs. \cite{Szego-1975,forrester,orive}). To our understanding, exact solutions of (\ref{M1xj}) are not known (not even in the case of harmonic potential), and the solution may not be unique. Equation (\ref{M1xj}) is essentially the equilibrium position of $N$ particles repelling each other but held in an external potential and also attracted to an additional particle of opposite ``charge", as will be further elaborated in section 3.6.
The soliton is a single particle moving in an external quartic polynomial potential. That is,
equation (\ref{pmotz}) in the case $M=1$ takes the simple form
\begin{equation}
\ddot{z} = -V^{\prime}(z) = -k_3 z^3 - k_2 z^2 - k_1 z - k_0
\label{oscillator}
\end{equation}
where $k_0 , \dots , k_3$ are related to the parameters of the model (see Table 3 and eq.\ (\ref{Vquar})).
This is a single anharmonic complex oscillator. Typically, analytical solutions for the above equation are not available for quartic polynomials $V(z)$, but solving a single particle problem is numerically easy. Knowing the value of $z(t)$, we then use (\ref{M1xj}) and (\ref{M1pj}) to find the $x_j$ and $p_j$. The upper left panel of Fig.\ 1 demonstrates the
particle evolution profile for the case of a single solition.
We see that the worldline of particles clearly shows a robust soliton. The motion of the corresponding variable $z(t)$ is given in the lower left panel of the same Fig.\ 1.
\subsection{The two-soliton solution}
The case when the system has two solitons is nontrivial even with respect to the $z_1 , z_2$ variables. In this case, we have two complex soliton coordinates in a quartic potential interecting with Calogero forces. Putingt $M=2$ in (\ref{xjreal},\ref{pjreal}) we
obtain
\begin{eqnarray}
w(x_{j}) &=& g\sum_{k=1 (k\neq j)}^{N}\frac{1}{x_{j}-x_{k}}
-\frac{g}{2}\Big(\frac{1}{x_{j}-z_{1}}+\frac{1}{x_{j}-\bar{z}_{1}} \label{xjreal2}\\
&+& \frac{1}{x_{j}-z_{2}}+\frac{1}{x_{j}-\bar{z}_{2}}\Big),
\nonumber\\
p_{j}& =&
i\frac{g}{2}\left(\frac{1}{x_{j}-z_{1}}-\frac{1}{x_{j}-\bar{z}_{1}} + \frac{1}{x_{j}-z_{2}}-\frac{1}{x_{j}-\bar{z}_{2}} \right)
\label{pjreal2}
\end{eqnarray}
while solitons obey the coupled second-order equations
\begin{eqnarray}
\ddot{z}_{1} & = & -\frac{g^{2}}{2}\frac{\partial}{\partial z_{1}}\sum_{i\neq j}^{M}\frac{1}{\left(z_{1}-z_{2}\right)^{2}}-V^{\prime}(z_{1}) \\
\ddot{z}_{2} & = & -\frac{g^{2}}{2}\frac{\partial}{\partial z_{2}}\sum_{i\neq j}^{M}\frac{1}{\left(z_{2}-z_{1}\right)^{2}}-V^{\prime}(z_{2})
\end{eqnarray}
Initial conditions for ${\dot z}_{1,2}$ can be set by specifying $z_{1,2}$ and using (\ref{xjreal2})
to find $x_j$ and then (\ref{zjdot}) to find ${\dot z}_{1,2}$. Then the evolution of $z_{1,2}$ can be found
by solving the above Calogero equations.
The upper right panel of Fig. 1 shows the
particle evolution profile in the case of two solitions.
We see that the worldline of particles clearly shows two robust solitons, one moving left and another moving right. The motion of the corresponding complex soliton variables is given in the lower right panel of the same Fig. 1.
\subsection{Mapping solitons to an electrostatic problem and the numerical protocol}
Let us take a close look at equation (\ref{xjreal}). It is essentially the derivative of the real part of the
prepotential $\Phi$:
\begin{eqnarray}
U = Real(\Phi) &=& \sum_{j=1}^{N}W(x_j) -\sum_{j<k}\ln|x_{j}-x_{k}| \nonumber\\&+&\frac{1}{2}\sum_{j=1}^{N}\sum_{n=1}^{M}\Big[\ln|x_{j}-z_{\alpha}|+\ln|x_{j}-\bar{z}_{\alpha}|\Big].
\label{estatic}
\end{eqnarray}
where we remind the reader that the function $W(x)$ is related to $w(x)$ as
\begin{eqnarray}
W(x)=\int_{0}^{x}w(x^{\prime})dx^{\prime}
\end{eqnarray}
Eq. (\ref{xjreal}) is then the extremum condition ${\partial U \over \partial x_j} = 0 $ for the above function.
The function $U$ can be thought of as the ``electrostatic energy'' of $N$ particles with unit charges interacting through a logarithmic potential (2d Coulomb potential), restricted to move along a straight line (the real axis) and in the presence of $2M$ external charges $-1/2$ placed at $z_{\alpha},\bar{z}_{\alpha}$ and of an external potential $W(x)$.
The solution of (\ref{xjreal}) is not necessarily a minimum of (\ref{estatic}), but may correspond to any fixed point (maximum, minimum or saddle point) of (\ref{estatic}), and there may be several such points.
The above observation forms the basis for a numerical procedure for solving Eq. (\ref{xjreal}), at least for solutions
corresponding to a local minimum. The basic idea is to let the particles slide towards the minimum of the
above potential by introducing a ``viscous" force that allows them
to move towards their equilibrium positions.
That is, we introduce the following $N$ coupled ODEs
\begin{eqnarray}
\dot{x}_j = -\gamma \frac{\partial{U}}{\partial x_j}
\end{eqnarray}
It is clear that the above drives the system to the minimum of the potential $U$. In fact, the above equation
implies
\begin{eqnarray}
\label{ueq}
\frac{d U}{dt}
=-\frac{\gamma}{2} \sum_{j=1} ^{N} \left( \frac{\partial{U}}{\partial x_j} \right)^2
\end{eqnarray}
so the potential decreases until it reaches a fixed point. Local maxima can also be dealt this way by flipping
the sign of $U$ and ttuning them into minima. Saddle points, on the other hand, will be missed.
The above first-order equation can be integrated numerically. Once we find the solutions for $x_j$, we then use Eq. (\ref{M1pj}) to find the initial momenta. These form the special initial conditions for the particles that correspond to
a set of solitons, and we can evolve them according to Eq. \ref{pmotx} without further reference to the soliton
variables.Therefore, we have mapped the problem of finding soliton configurations to an electrostatic problem of a function $U$.
\begin{figure}
\includegraphics[scale=0.48]{onesol.pdf}
\includegraphics[scale=0.55]{twosol.pdf}
\includegraphics[scale=0.55]{Zonesol.pdf}
\includegraphics[scale=0.30]{Ztwosol.pdf}
\caption{\textbf{(Upper Panel Left)} One soliton solution for rational Calogero model in external quartic polynomial potential. Here we take N=31 particles and the special initial condition is dictated by a single $z = 0.0239 i $. We notice, in the worldine picture, a remarkable evidence of a robust soliton We use the prepotential $w(x) = x +0.06 x^2$ which makes the potential
$V(x)= 0.87 x + 0.5 x^2 + 0.06 x^3 + 0.0018 x^4$ using first row of Table 3.
\textbf{(Lower Panel Left)} We show the motion of one dual variable ``z" corresponding to one soliton solution.
The initial condition is $z(t=0) = 0.0239 i $ and $\dot{z}(t=0)= -43.0768 - 0.00103378 i$
(which is fixed by Eq. \ref{zjdot}) and the equation it satifies is $\ddot{z} = -V^{\prime}(z)$ where
$V(z) = 0.93 z +0.5z^2 + 0.06 z^3 + 0.0018 z^4$ again using first row of Table 3.
\textbf{(Upper Panel Right)} Two soliton solution where initial conditions are dictated by two values of dual variable, $z_1 = + 0.0239 i$ and $z_2 = - 0.0239 i$. We use the prepotential $w(x) = x +0.06 x^2$ which makes the potential
$V(x)= 0.84 x + 0.5 x^2 + 0.06 x^3 + 0.0018 x^4$ using first row of Table 3. We see clear evidence of two solitons in the system.
\textbf{(Lower Panel Right)} Motion of two coupled dual variables $z_1$ (black) and $z_2$ (blue) is shown here. They satisfy the equation $\ddot{z}_1 = g^2 \frac{1}{(z_1-z_2)^3} -V^{\prime}(z_1)$ where
$V(z) =0.9 z_1+0.5 z_1^2 + 0.06 z_1^3 + 0.0018 z_1^4$
(similarly for the other dual variable $z_2$). The initial conditions are $z_1(t=0) = + 0.0239 i $, $z_2(t=0) = - 0.0239 i $, $\dot{z}_1(t=0) = -64.297 - 0.00129161 i $, $\dot{z}_2(t=0) = 64.297 - 0.00129161 i$.
}
\label{scalingplot}
\end{figure}
\section{Hydrodynamic Limit and Meromorphic Fields}
\label{sec:mero}
\subsection{General formalism}
In this section we consider the generalized Calogero models with external potentials and take the hydrodynamic limit to derive soliton solutions for the corresponding fluid mechanical density and velocity of the particles. We do this by introducing specific meromorphic functions with poles on the position of particles and solitons and taking their many-particle limit. The approach is related to the one of \cite{2005-AbanovWiegmann,2009-AbanovBettelheimWiegmann},
but we will give an independent simplified exposition, directly following from our first-order formulation.
We consider a system with $N$ (real) particle coordinates $x_j$ and $M$ (complex) solitons $z_\alpha$. We will
take the prepotential to be of the form that ensures a stable potential and absence of 3-body forces, as found in
section 2, leading to the first-order equations (\ref{xadot})
\begin{equation}
{\dot x}_a = i \sum_b m_b {\tilde f}_{ab} + i {\tilde w}_a
\label{xdagain}\end{equation}
and corresponding second-order equations
\begin{equation}
\hspace*{-0.28cm}
{\ddot x}_a = -{1\over m_a} \partial_a V = -\partial_a \left[ \sum_{b (\neq a)} \frac{1}{2} m_b (m_a + m_b) {\tilde f}_{ab}^2
+ (m_{tot} - m_a ) v_a + \frac{1}{2} {\tilde w}_a^2 \right]
\label{xdd}
\end{equation}
with $v_a = v (x_a )$ and $ {\tilde w}_a= {\tilde w}(x_a)$ as found in section 2.2.
Coordinates $x_a$ run over particles (for $a = j$) and solitons ($a=\alpha$), and we take the masses of particles to be
$m_j =1$ and the masses of solitons $m_\alpha = -1$, so $m_{tot} = N-M$.
To this system we add one more ``spectator" particle $a=0$ with coordinate $x_0 =x$ and mass $m_0$. The full system
of $N+M+1$ particles and total mass $N-M+m_0$ retains its Calogero-like form.
Using Eq. \ref{xdagain} for this spectator particle, the velocity $u$ of the spectator particle is, in particular,
\begin{eqnarray}
u = {\dot x} &=& i\sum_{a \neq 0} m_a {\tilde f} (x - x_a ) + i {\tilde w} (x)\nonumber \\
&=& i\sum_{j=1}^N {\tilde f} (x - x_j ) - i \sum_{\alpha=1}^M {\tilde f} (x - z_\alpha ) + i {\tilde w} (x)
\label{u(x)}
\end{eqnarray}
and its corresponding acceleration (using Eq. \ref{xdd}) is
\begin{eqnarray}
{du \over dt} = {\ddot x} &=& -\partial_x \left[ \sum_{j=1}^N {1+m_0 \over 2} {\tilde f} (x - x_j )^2
+ \sum_{\alpha=1}^M {1 -m_0 \over 2} {\tilde f} (x - z_\alpha )^2 \right. \nonumber\\
&& ~~~~~~~~~+ \left. (N-M)\, v (x) + \frac{1}{2} {\tilde w} (x)^2 \right]
\end{eqnarray}
The additional particle $x$ creates an additional term $m_0 {\tilde f} (x_a - x )$ in the equation for
${\dot x}_a$ of the remaining particles, and a corresponding term in the potential. We wish this particle to be a spectator, that is, not to modify
the motion of the remaining particles (while itself being influenced by them). So we take the limit $m_0 \to 0$, which leaves the
$N$ particles and solitons the same as in the original $N+M$-particle Calogero-like system. Equation (\ref{u(x)}) for $u = {\dot x}$
remains unchanged, while the equation for ${\ddot x}$ becomes
\begin{eqnarray}
{du \over dt} &=& -\partial_x \Bigg[ \sum_{j} \frac{1}{2} {\tilde f} (x - x_j )^2
\nonumber \\ &+& \sum_{\alpha} \frac{1}{2} {\tilde f} (x - z_\alpha )^2 + (N-M) v (x) + \frac{1}{2} {\tilde w} (x)^2 \Bigg]
\label{udd}
\end{eqnarray}
The role of the spectator particle is that it monitors and essentially determines both the position and the velocity of the remaining
particles. To this end, we consider $u$ as defined in (\ref{u(x)}) as a function of the spectator particle position $x$ and promote
$x$ to an independent variable, defining a field $u(x)$. The time derivative of $u(x)$, written as ${\partial u \over \partial t}$, is thus
the time variation of $u$ arising from its dependence on $x_j (t)$ and $z_\alpha (t)$, but {\it not} on $x$.
In other words, we define,
\begin{eqnarray}
{\partial u \over \partial t} \equiv \sum_j {\partial u \over \partial x_j} {\dot x}_j
+ \sum_\alpha {\partial u \over \partial z_\alpha} {\dot z}_\alpha
\label{ttdef}
\end{eqnarray}
The total time derivative entering (\ref{udd}) is, therefore,
\begin{eqnarray}
{du \over dt} &=& {\partial u \over \partial x} {\dot x} + \sum_j {\partial u \over \partial x_j} {\dot x}_j
+ \sum_\alpha {\partial u \over \partial z_\alpha} {\dot z}_\alpha \nonumber \\
&=& u \partial_x u + {\partial u \over \partial t} = \partial_x \left(\frac{1}{2} u^2 \right) + {\partial u \over \partial t}
\label{ttd}
\end{eqnarray}
where we used ${\dot x} = u$ and the definition Eq. \ref{ttdef}.
The above relation and (\ref{udd}) allow us to find the equation of motion of
the field $u(x,t)$. To do this, we need to express the terms involving $x_j$ and $z_\alpha$ in (\ref{udd}) in terms of
$u$. This can be achieved by noting that all prepotentials ${\tilde f} (x)$ found in section 2.2
(rational, trigonometric or hyperbolic) satisfy the relation
\begin{equation}
{\tilde f}(x)^2 = g \, \partial_x {\tilde f} (x) + C ~,~~~ f(x) = -{g \over x}, ~ -g\cot x , ~ -g \coth x
\label{der}
\end{equation}
where $C$ is a constant
(zero, $+g^2$ and $-g^2$ for rational, hyperbolic and trigonometric prepotential respectively). Therefore
the terms involving sums of ${\tilde f}^2$ in (\ref{udd}) can be expressed as derivatives of terms
in $u(x)$. We note, however, that particle and soliton terms come with opposite sign in $u(x)$, while they
have the same sign in (\ref{udd}). This necessitates splitting $u(x)$ into two parts:
\begin{eqnarray}
u^+ (x) &=& - i \sum_{\alpha} {\tilde f} (x - z_\alpha ) + i \lambda {\tilde w} (x) \label{up}\\
u^- (x) &=& i\sum_{j} {\tilde f} (x - x_j ) + i (1-\lambda) {\tilde w} (x) \label{um}\\
u(x) &=& u^+ (x) + u^- (x) \nonumber
\end{eqnarray}
where $\lambda$ is an arbitrary parameter that splits the term ${\tilde w} (x)$ between the two
functions. Using (\ref{udd},\ref{ttd}, \ref{der}) and (\ref{up},\ref{um}) we arrive at the equation of motion for $u$
\begin{eqnarray}
{\partial u \over \partial t} &+& \partial_x \Bigg[ \frac{1}{2} u^2 + {i g \over 2} \partial_x ( u^+ - u^- ) \nonumber \\ &+& \frac{1}{2} {\tilde w}^2
+ (N-M) v +(\lambda - {\textstyle \frac{1}{2}}) \, g \,\partial_x {\tilde w}\Bigg] = 0
\end{eqnarray}
The terms independent of $u$ above are the one-body potential $V(x)$ entering the equation of motion of
particles $x_j$, with the difference that $v(x)$ is multiplied by $N-M$ (rather than $N-M-m_j = N-M-1$
and the extra term
involving $\partial_x {\tilde w} (x)$. In fact, for all cases of $\tilde f$ (rational, trigonometric and hyperbolic),
$\partial_x {\tilde w} (x)$ is proportional to $v(x)$:
\begin{equation}
g \, \partial_x {\tilde w} (x) = -2 v(x)
\end{equation}
The equation for $u(x,t)$ therefore takes the form:
\begin{equation}
{\partial u \over \partial t} + \partial_x \left[ \frac{1}{2} u^2 + {i g \over 2} \partial_x ( u^+ - u^- ) +V
+ (\lambda -1)g \, \partial_x {\tilde w} \right] = 0
\label{ut}\end{equation}
If we want the equation of motion of $u$ to involve the same one-body potential $V(x)$ as that of
particles, we must choose $\lambda =1$, assigning the full prepotential ${\tilde w} (x)$ to $u^+$.
This is in fact the opposite convention than the one in \cite{2009-AbanovBettelheimWiegmann}, where
$u^-$ contains the term ${\tilde w} (x)$. We stress that any choice of $\lambda$ is allowed, leading to
different definitions of $u^+$, $u^-$ and an extra term in the evolution equation for $u$.
We also note that $\lambda$ does not appear in the equation of $u$ (\ref{ut}) in the quadratic (harmonic)
rational Calogero case studied in
Ref. \cite{2009-AbanovBettelheimWiegmann}, since $w(x)$ is
linear and $v(x)$ is a constant that drops from the equation.
\subsection{The rational case and derivation of the hydrodynamic limit}
The above construction holds for all three types of Calogero potentials. We first examine the rational case.
We pick $\lambda=1$ as the most natural choice and define
\begin{eqnarray}
u^+ (x) &=& ig\sum_{\alpha=1}^{M}\frac{1}{x-z_{\alpha}} + i {\tilde w} (x) \label{upp}\\
u^- (x) &=& -ig\sum_{j=1}^{N}\frac{1}{x-x_j}\label{umm}
\end{eqnarray}
A priori, it looks like we have a single equation of motion (\ref{ut}) for two fields $u^+$ and $u^-$.
As we stressed before, however, the particle system is actually fully determined by the values of $z_\alpha$,
so in principle $u^+$ is enough to fully fix the system.
$u^+$ is a meromorphic function of $x$ with $M$
simple poles at $z_\alpha$, so it fixes the number of solitons.
Using the equations of motion (\ref{xjdot}) we see that $u^+$ satisfies
\begin{equation}
u^+(x_j) = {\dot x}_j + ig\sum_{k=1 (k\neq j)}^{N} \frac{1}{x_j-x_k}.
\label{u+j}
\end{equation}
Therefore, {\it if we also know that there are $N$ particles}, the function $u^+ (x)$ fully determines the system, as:
\begin{eqnarray}
&& Im ~u^{+}(x_j ) = g\sum_{k=1 (k\neq j)}^{N} \frac{1}{x_j-x_k} \label{Iu}
\\
&& Re ~ u^{+}(x_j ) = {\dot x}_j = v_j
\label{Ru}
\end{eqnarray}
The $N$ equations (\ref{Iu}) in principle determine the $N$ real variables $x_j$, and subsequently
(\ref{Ru}) determines ${\dot x}_j$. The known values of $x_j$, then, determine the function $u^- (x)$ through
equation (\ref{umm}).
The definition of $u^+ (x)$, (see Eq. \ref{upp}) however, does not involve $N$, and the {\it same} function $u^+ (x)$
can describe systems of an arbitrary number of particles (see Eq. \ref{Iu}, \ref{Ru}). The real part of $u^+ (x)$ for real $x$, in particular,
defines a continuous velocity field
$v(x)$ that is the actual particle velocity on the position of particles. Similarly, its imaginary part defines
a field that can be related to the position of particles, for any number of them. It is, therefore, a good tool to
deal with the hydrodynamic limit $N \to \infty$ where the interparticle distance goes to zero and the system is
described by a continuous density $\rho (x)$ and velocity $v(x)$.
From the above discussion it follows that, in the $N \to \infty$ limit, the real part of $u^+ (x)$ straightforwardly
goes over to the fluid velocity field $v(x)$. To express the imaginary part in terms of the fluid particle density
requires a bit more work. In particular, we need to express the sum in (\ref{Iu}) in terms of the particle
density $\rho (x)$, including all perturbative corrections in $1/N$. This is nontrivial because of the singularity
as $x_k$ approaches $x_j$ and has to be evaluated carefully. This has been done in \cite{Stone}. Here we will
follow a slightly different approach that will allow us to separate the perturbative and non-perturbative parts.
For a large number of particles we define the continuous position function $x(s)$ such that $x(j) = x_j$.
For finite $N$ any smooth interpolation between the $x_j$ will do, while in the
$N \to \infty$ limit this function becomes unique. It is related to the continuous density $\rho (x)$ by
\begin{equation}
x' (s) = {1 \over \rho (x(s))}
\end{equation}
The sum of interest is
\begin{equation}
\sum_{k(\neq j )=1}^N {1 \over x_j - x_k} = \sum_{k(\neq j)=1}^N {1 \over x(j) - x(k)}
\label{sum}
\end{equation}
which needs to be expressed in terms of $x(s)$ or $\rho(x)$.
Our starting point is the identity
\begin{equation}
\sum_{n=-\infty}^\infty f(n) = \sum_{n=-\infty}^\infty {\tilde f} (2\pi n)
\end{equation}
where $f(s)$ is any function of $s$ and ${\tilde f} (q)$ is its Fourier transform, defined as
\begin{equation}
{\tilde f} (q) = \int_{-\infty}^\infty ds~ e^{-i q s} f(s)
\end{equation}
For a function smooth at the scale of $\Delta s \sim 1$, the Fourier transform ${\tilde f} (2\pi n)$ for
$n \neq 0$ will be negligibly small. In fact, terms with $n\neq 0$ are nonperturbative in $1/N$
(instanton corrections), as we will explain in the sequel. So, up to nonperturbative corrections
\begin{equation}
\sum_{n=-\infty}^\infty f(n) = {\tilde f} (0) = \int_{-\infty}^\infty ds ~f(s)
\label{sumint}
\end{equation}
To apply this formula to the sum (\ref{sum}) we view the summand as a function of $k$ and define it to be zero for $k$
outside its range, extending the summation to all integers. Still, a straightforward application of (\ref{sumint})
is hindered by
the fact that the summed function $1/[x(j) - x(s)]$ is not smooth, due to the singular behavior near $s=j$, and thus
higher Fourier modes contribute substantially. We can proceed in two different ways. The first is to evaluate
the higher Fourier modes and sum their contribution. The second is to regularize the integrand in a way that
renders it smooth, as was done in \cite{Stone}. Both methods lead to the
same result. Following the second method, we define the function
\begin{equation}
f(s) = {1 \over x(j) - x(s)} + {1 \over x' (j) (s-j) + \epsilon (s-j)^3} ~,~~ f(j) = {x'' (j) \over 2 x' (j)^2}
\end{equation}
for $\epsilon >0$. The above function is continuous everywhere and smooth at the scale of $\Delta s \sim 1$,
since the
singularity at $s=j$ is subtracted by the second (regulator) term. Summing it over integer values of $s$ we have
\begin{eqnarray}
\sum_{k=-\infty}^\infty f(k) &=& f(j) + \sum_{k(\neq j)=-\infty}^\infty f(k) \nonumber\\
&=&{x'' (j) \over 2 x' (j)^2} + \sum_{k(\neq j)=-\infty}^\infty {1 \over x(j) - x(k)}
\end{eqnarray}
since the sum of the regulator term is absolutely convergent (due to $\epsilon$) and
vanishes due to antisymmetry in $k-j$. Applying (\ref{sumint}) we have the result
\begin{eqnarray}
\sum_{k(\neq j)=-\infty}^\infty {1 \over x(j) - x(k)} &=& -{x'' (j) \over 2 x' (j)^2} + \int_{-\infty}^\infty
f(s) ds \nonumber \\
&=& -{x'' (j) \over 2 x' (j)^2} + P.V. \int_{-\infty}^\infty {ds \over x(j) - x(s)}
\end{eqnarray}
where $P.V.$ stands for principal value. Finally, changing integration variable from $s$ to $y= x(s)$
and using $x' = 1/\rho$ and thus $x'' = -\partial_x \rho / \rho^3$ we obtain
\begin{eqnarray}
\sum_{k(\neq j)=-\infty}^\infty {1 \over x - x(k)} &=& {\partial_x \rho (x) \over 2 \rho (x)}
+ P.V. \int_{-\infty}^\infty {\rho (y) dy \over x - y} \nonumber \\
&=& \frac{1}{2} \partial_x \ln { \rho(x)} - \pi \rho^H (x)
\end{eqnarray}
where we put $x(j) = x$, and $\rho^H (x)$ is the Hilbert transform of $\rho(x)$.
The above result is perturbatively exact in $1/N$. Indeed, the higher Fourier modes of $f(s)$ expressed
in terms of $y = x(s)$ are
\begin{equation}
{\tilde f} (q) = \int_{-\infty}^\infty dy \, \rho(y) ~ e^{-i q \int_{-\infty}^y dw \, \rho(w)} \, f(s(y))
\end{equation}
For a continuous particle distribution, $x(j+1) - x(j) \sim 1/N$, $\rho(x) \sim N$, and thus the
oscillating exponent in the above expression for $q \neq 0$ is of order $N$. The integral is thus of order
$e^{-N}$, which is nonperturbative in $1/N$. So the $q=0$ term captures the full perturbative contribution.
Overall, from (\ref{Iu},\ref{Ru}) we obtain for $u^+ (x)$
\begin{equation}
\label{u+td}
u^+ (x)=v(x)-i\pi g \rho^H (x) +ig\, \partial_x\ln {\sqrt{\rho (x)}}
\label{u+rv}
\end{equation}
Nonperturbative contributions are in general negligible in the fluid limit ($N \to \infty$). The one instance
in which they become relevant is when the distribution of particles breaks into two or more disjoint components.
In this case the distance $x(K) - x(K +1)$ between the last particle $K$ in one component and the first particle
$K+1$ in the next is large, and thus the function $f(s)$ is not smooth at $s=K$.
The appearance of multiple fluid components signals the onset of nonperturbative effects and needs to
be described in terms of multiple functions $\rho_a (x)$, one for each component, with compact disjoint supports. In our paper, we do not encounter this scenario and hence, emergence of relavant perturbative corrections is not a concern.
The continuous version of $u^- (x)$ can similarly be found. Its definition (\ref{umm}) involves the parameter
$x$ and a sum over the full set of particles. By taking the variable $x$ to be complex and off the real axis the
issue of singularities is avoided and the summand becomes a smooth function of $x(s)$. So, up to nonperturbative
contributions,
\begin{equation}
u^- (x) = -ig\sum_j \frac{1}{x-x(j)} = -ig \int ds \, \frac{1}{x-x(s)} = -ig \int dy\, {\rho(y) \over x- y}
\end{equation}
So $u^- (x)$ is the Cauchy transform of $\rho(x)$. As $x$ approaches the real axis the above expression
has a discontinuity. We obtain
\begin{equation}
\label{u-td}
u^-(x\pm i0)=\mp \pi g \rho + i\pi g \rho^H
\label{u-r}
\end{equation}
with the discontinuity
\begin{equation}
u^{-}(x+i0)-u^{-}(x-i0)=-2\pi g \rho(x).
\label{u-jump}
\end{equation}
Expressions (\ref{u+rv}) and (\ref{u-r}) determine $u^\pm (x)$ in terms of fluid quantities. Note that with our choice of $\lambda=1$ in the definition (\ref{up},\ref{um}) of $u^\pm (x)$
their expression involves only the fluid density $\rho(x)$ and velocity $v(x)$ and not the prepotential ${\tilde w} (x)$.
Substituting these expressions into the equation (\ref{ut}) for $u(x)$ we obtain in principle 4 real equations
(2 for the real part and 2 for the imaginary part at $x\pm i0$) for the two real fields $\rho(x)$ and $v(x)$.
These equations are compatible and reduce to the fluid equations
\begin{eqnarray}
\partial_t \rho &+& \partial_x (\rho v ) = 0 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\
\partial_t v &+& \partial_x \Bigg[ \frac{1}{2} v^2 + {\pi^2 g^2 \over 2} \rho^2 + \pi g^2 \partial_x \rho^H
\\ \nonumber &-& {g^2 \over 8} \left( \partial_x \ln { \rho}\right)^2 - {g^2 \over 4}
\partial_x^2 \ln {\rho} + V \Bigg]= 0
\end{eqnarray}
The above equations can be seen to arise from the Hamiltonian
\begin{equation}
H = \int dx \left[ \frac{1}{2} \rho v^2 +{\pi^2 g^2\over 6} \rho^3 +{\pi g^2\over 2} \rho \, \partial_x \rho^H
+ {g^2} {(\partial_x \rho)^2 \over 8\rho} + \rho V \right]
\end{equation}
using the standard fluid mechanical Poisson structure
\begin{equation}
\{ \rho (x) , v (y) \} = \delta' (x-y)
\end{equation}
\subsection{One soliton solution of Calogero model with quartic potentials in terms of meromorphic fields}
The one-soliton solution is given by
\begin{equation}
u^{+}(x) = \frac{ig}{x-z_{1}(t)} + i {\tilde w} (x),
\label{u+1sol}
\end{equation}
The single soliton solution for $u^+ (x)$ is a meromorphic function with a single pole. $z_{1}(t)$ above satisfies (\ref{zjdot}) for $M=1$ which is simply
\begin{equation}
\dot{z}_1 -i{\tilde w} (z_1 ) = -ig\sum_{j=1}^{N}\frac{1}{z_1-x_{j}}
\end{equation}
and the second-order equation
\begin{equation}
{\ddot z}_1 = - V' (z_1 )
\label{onezquart}
\end{equation}
From the expression of $u^+ (x)$ in terms of hydrodynamic quantities
\begin{equation}
\label{TDred1}
v - ig(\pi\rho^H-\partial_x\log\sqrt{\rho})=\frac{ ig}{x-z_{1}} + i {\tilde w} (x)
\end{equation}
we can, in principle, find the density and velocity fields from the position $z_{1}$. Writing $z_1 (t) = a(t) + i b(t)$
and taking the real and imaginary parts of (\ref{TDred1}) we obtain
\begin{eqnarray}
v&=&-g \, \frac{b}{(x-a)^2 + b^2} \label{1solTDv}\\
\pi\rho^H &-& \partial_x\log\sqrt{\rho}
= -\frac{x-a}{(x-a)^2 + b^2} - {1 \over g} {\tilde w} (x)
\label{1solTD}
\end{eqnarray}
So $a(t)$ parametrizes the position of the soliton while $b(t)$ parametrizes its width.
The second equation above needs to be solved for $\rho (x)$. This is nontrivial,
although the solution can be found analytically in the limit of thin solitons. In this limit, the width $b \rightarrow 0$, and the soliton solution (upto $O(1/N)$ corrections), can be written as
\begin{eqnarray}
\rho_{sol} (x,t)= \rho_{0} + \delta(x - a(t))
\end{eqnarray}
where the background density $\rho_{0} $ satisfies, $\pi\rho_0^H - \partial_x\log\sqrt{\rho_0}
+ {1 \over g} {\tilde w} (x)=0
$
The soliton parameter $z_{1}(t)$ is moving in the complex plane along a non-trivial curve guided by its quartic polynomial
as in (\ref{onezquart}). Therefore, (\ref{1solTDv},\ref{1solTD}) give a one-dimensional reduction of an infinite dimensional rational Calogero system with quartic potential in the hydrodynamic limit. The procedure to go to the hydrodynamic limit can similarly be extended for the two-soliton and multi-soliton case.
\vskip 0.1in
\section{Conclusions and Outlook}
\label{sec:conclusion}
To summarize, in this paper we introduced a first order formalism based on a prepotential and derived its general form that gives rise to two-body and external potentials. Imposing the requirements of stability and reality conditions,
we demonstrated the natural emergence of the Calogero family of models in generalized quartic and trigonometric external potentials. Our general formalism provides a relatively straightforward route to finding soliton solutions,
a task otherwise considered to be an enormous challenge. Using the more common version of the Calogero family of models, namely, the rational Calogero model (in quartic polynomial external potentials), we demonstrate the existence of soliton solutions. We derived the particle time evolution for the case when the system has one and two solitons and we showed that our method can be easily extended to $M$ solitons. We showed that finding soliton solutions can be achieved via a mapping to an electrostatic problem. Using a fluid formalism involving meromorphic fields, we have also identified soliton solutions in the hydrodynamic limit.
One of the the main lesson from the work presented in this paper is that there may exist further extensions of the Calogero
family of models beyond the known systems, and that they may admit dual formulations that identify their collective
degrees of freedom and provide solutions to their fluid mechanical versions. Clearly, there are many open issues and
directions of possible future research.
The most immediate questions are the ones on stability and integrability.
It is puzzling that the dual formulation of the quartic potential model is stable only within a subset of its parameters, which
actually exclude the purely quartic case. Although we conjecture that models outside the stability regime correspond to a
soliton condensate, an explicit demostration of this fact, and derivation of the soliton solutions, would be desirable.
Similarly, our approach does not deal with integrability. Again, it is remarkable that the systems that can be
dealt with this formalism do fall eventually within a subclass of the generalized Calogero models that were known to
be integrable.
A direct derivation of integrability seems to be possible within this formalism and, if there, has yet to be uncovered.
Extension of our results to other members of the Calogero family is also an open issue. We restricted our derivation to
the rational, trigonometric and elliptic models and their external potential generalizations, mainly for reasons of
mathematical clarity and simplicity. An extension to the elliptic (Weierstrass) model is well within the reach of the
formalism. In this context, the identification of elliptic models with external potentials would be a very interesting
advance. Similar remarks hold for models of particles with internal degrees
of freedom. Clearly an extension of the formalism is needed to incorporate internal particle coordinates, and this is a topic
of further research.
Finally, there exist several intriguing similarities of the present formalism with quantum mechanical features of the
Calogero model, although our treatment is purely classical. The generating function clearly alludes to a quantum
mechanical wavefunction, at least in the equilibrium semiclassical limit. Similarly, the stable and unstable domains of
quartic dual systems are in direct analogy with the broken and unbroken phases of supersymmetric quantum mechanical systems,
the ``unstable" broken phase leading to a soliton condensate. Aspects of our formalism also bear similarities with
techniques from matrix models and the exchange operator formulation. These and related issues are left for future investigation.
\section{Acknowledgments}
M. K. gratefully acknowledges the Ramanujan Fellowship SB/S2/RJN-114/2016 from the Science and Engineering Research Board (SERB), Department of Science and Technology, Government of India. A.P.'s research is supported by NSF grant 1213380 and by a PSC-CUNY grant.
\section{Appendix A: Solutions to the functional equations without external potentials}
\label{app-hilbert}
In this Appendix we provide the solutions to the functional equations for ${\tilde f}_{ab}$, thereby yielding Table 1.
We start with the equation (\ref{functf}) for a fixed triplet of indices $b,c,d$:
\begin{eqnarray}
\tilde{f}_{bc}\tilde{f}_{bd}
-\tilde{f}_{bc}\tilde{f}_{cd}+\tilde{f}_{bd}\tilde{f}_{cd}=C_{bcd}
\end{eqnarray}
where we used the antisymmetry of ${\tilde f}_{ab} = - {\tilde f}_{ba}$ to put the indices $b,c,d$ in order.
Let us define, $x\equiv x_b - x_c$, $y\equiv x_c - x_d$ such that $x+y = x_b-x_d$.
For convenience, we also rename the doublets of indices $bc \equiv 1$, $cd \equiv 2$, $ bd \equiv 3$ and call
$C_{bcd}$ simply $C$. The above equation, then, reads
\begin{eqnarray}
\tilde{f}_{1} (x)\tilde{f}_{3}(x+y) - \tilde{f}_{1}(x) \tilde{f}_{2}(y)+\tilde{f}_{3}(x+y) \tilde{f}_{2}(y)=C
\end{eqnarray}
In terms of the reciprocal functions $g_j (x)=\frac{1}{{\tilde f}_j (x)}$ the above becomes
\begin{eqnarray}
g_1 (x) + g_2 (y)- g_3 (x+y) = C g_1 (x) \, g_2 (y) \, g_3 (x+y)
\end{eqnarray}
and solving for $g_3$ we obtain
\begin{eqnarray}
g_3 (x+y) = \frac{g_1 (x) + g_2 (y)}{1 + C g_1 (x) g_2 (y)}
\label{g3}
\end{eqnarray}
Taking derivatives with respect to $x$ and $y$ and equating the results
(since $\partial_x g_3 (x+y) = \partial_y g_3 (x+y)$) we obtain
\begin{eqnarray}
\frac{g_1^{\prime} (x) }{1-C g_1^2 (x)} = \frac{g_2^{\prime} (y)}{1-C g_2^2 (y)} = k =
{\rm constant}
\label{g12}
\end{eqnarray}
The above differential equations determine $g_1$ and $g_2$ and, through (\ref{g3}),
they also determine $g_3$. Their solution depends on the sign of the constant $C$. Specifically
\begin{eqnarray}
C=0:& &g_j (x) = k x + b_j \nonumber\\
C= -g^2 < 0 :& &g_j (x) = \frac{1}{g} \sin (kg x +g \, b_j ) \nonumber \\
C= g^2 > 0 :& &g_j (x) = \frac{1}{g} \sinh (kg x + g \, b_j ) \nonumber \\
j=1,2,3,& &{\rm with}~~ b_1 + b_2 = b_3
\end{eqnarray}
Repeating the above analysis for a triplet involving one new particle, say, $b,c,e$, leads to the
same equations (\ref{g12}), with the same constants $C$ and $k$ (fixed by the form of $g_{bc} = g_1$
found above). Inductively, we conclude that the constants $C=C_{abc}$ and $k$ are common
for the full set of particles, while the constants $b_{ab}$ satisfy $b_{ab} + b_{bc} = b_{ac}$
for all $a,b,c$. These constants can actually be absorbed by a shift in the position of particle
coordinates:
\begin{equation}
x_a \to x_a - \frac{b_{1a}}{k} ~,~~ b_{11} \equiv 0
\end{equation}
where we chose arbitrarily particle $1$ as a reference, so we can take all $b_a =0$.
Finally, the constant $k$ can be set to $1/g$ through the rescaling of coordinates
$x_a \to (kg)^{-1} x_a$. Overall, we recover the ${\tilde f}_{ab} = 1/g_{ab}$ as given
in Table 1.
\section{Appendix B: Solutions to functional equations for external potentials}
In this Appendix we derive the solutions to the functional equation (\ref{bc}), written explicitly as
\begin{equation}
\big[{\tilde w}_b (x_b ) - {\tilde w}_c (x_c )\big] \,{\tilde f}_{bc} (x_{bc} )= u_{bc} (x_{bc} ) + v_b (x_b )+ v_c (x_c )
\end{equation}
We define $x_b = t+s$ and $x_c = t-s$, which implies $x_{bc} = 2s$. Using also the inverse functions
$g_{bc} = 1/{\tilde f}_{bc}$ defined in the previous Appendix, the above equation becomes
\begin{eqnarray}
{\tilde w}_b (t+s) - {\tilde w}_c (t-s) &=& g_{bc} (2s) \bigg[ u_{bc} (2s)+ v_b (t+s)+ v_c(t-s) \bigg]\nonumber \\
&=& h_{bc} (s) + g_{bc} (2s) \bigg[v_b (t+s)+ v_c(t-s) \bigg]
\label{bcapp}
\end{eqnarray}
where we defined $h_{bc} (s) = g_{bc} (2s) u_{bc} (2s)$.
The solution of this equation depends on the form of $g_{bc}$ and we treat it in a case-by-case basis for
the solutions derived in the previous Appendix.
\subsection{The rational case $g_{bc}={1 \over {\tilde f}_{bc} } = \frac{1}{g} x_{bc}$}
In the rational case (\ref{bcapp}) becomes
\begin{eqnarray}
{\tilde w}_b (t+s) - {\tilde w}_c (t-s) &=& {2s \over g} \bigg[ u_{bc} (2s)+ v_b (t+s)+ v_c(t-s) \bigg]
\nonumber \\
&=& h_{bc} (s) + {2s \over g} \bigg[v_b (t+s)+ v_c(t-s) \bigg]
\label{wrat}
\end{eqnarray}
with $h_{bc} (s) = 2s \, u_{bc} (2s)/g$.
The left hand side is regular around $s=0$ and has a well-defined Taylor expansion in $s$, therefore so must be
$h_{bc} (s)$. Expanding in powers of $s$ we obtain
\begin{eqnarray}
s^0 :& &{\tilde w}_b (t) - {\tilde w}_c (t) = h_{bc}(0) \label{Taylor.0} \\
s^1 :& &{\tilde w}_b' (t) + {\tilde w}_c' (t) = h_{bc}'(0) + {2\over g} \big[ v_b (t) + v_c(t) \big]\label{Taylor.1} \\
s^2 :& &{\tilde w}_b'' (t) - {\tilde w}_c'' (t) = h_{bc}''(0) + {4\over g} \big[ v_b' (t) - v_c' (t) \big]\label{Taylor.2}\\
s^3 :& &{\tilde w}_b''' (t) + {\tilde w}_c''' (t) = h_{bc}'''(0) + {6\over g} \big[ v_b'' (t) + v_c'' (t) \big] \label{Taylor.3}
\end{eqnarray}
Eq.\ (\ref{Taylor.0}) states that ${\tilde w}_b (t)$ and ${\tilde w}_c (t)$ differ by a constant.
Differentiating (\ref{Taylor.1}) twice with respect to $t$ and combining with (\ref{Taylor.3}) we obtain
\begin{equation}
{\tilde w}_b''' (t) = -{1\over 4} h_{bc}''' (0) = {\rm constant}
\end{equation}
which means that ${\tilde w}_{b}$ and ${\tilde w}_{c}$ must be of the form
\begin{equation}
{\tilde w}_{b,c} (t) = C_{b,c} + c_1 t + c_2 t^2 + c_3 t^3
\end{equation}
The constants $C_b$ and $C_c$ can differ, but $c_1 , c_2 , c_3$ are the same for any two particles $b$ and $c$,
therefore they are common to the system. Substituting this form for ${\tilde w}_{b,c} (t)$ in (\ref{Taylor.0}-\ref{Taylor.3}),
or directly in (\ref{wrat}), we obtain $v_b (x)$, $v_c (x)$, $h_{bc}(x)$ and $u_{bc} (x)$. In doing this, we note that
the constant terms of $v_b$, $v_c$ and $u_{bc}$ can be combined together; similarly, $v_b' (0) - v_c' (0)$ and
$u_{bc}' (0)$, contributing a term proportional to $x_b - x_c = x_{bc}$, can also be combined. We use this to choose
$v_b (0) = v_c (0) = v_b' (0) - v_c' (0) = 0$. We eventually obtain
\begin{eqnarray}
{\tilde w}_{b,c} (x) &=& C_{b,c} + c_1 x + c_2 x^2 + c_3 x^3 \\
v_{b,c} (x) &=& g c_2 x + {3g \over 2} c_3 x^2 \\
u_{bc} (x) &=& g {C_b - C_c \over x_{bc}} + g c_1 -{g \over 2} c_3 x^2
\end{eqnarray}
The above recover the potentials presented in Table 2. Note that the constant $g c_1$ in $u_{bc}$ is
dynamically irrelevant and can be omitted. Note also that the contribution of $v_a (x_a )$ in the full
potential involves a sum $\sum_{a\neq b} m_a m_b v_a = \sum_a (m_{tot} - m_a ) m_a v_a$, which explains the
coefficient of the corresponding terms in $V_a (x_a )$.
\subsection{The trignometric case $g_{bc} = {1 \over {\tilde f}_{bc} } = {1\over g} \tan x_{bc}$}
In the trignometric or hyperbolic case we proceed in a similar way, Taylor expanding (\ref{bcapp}) in $s$. The equations
are the same as
in the previous section for orders $s^0$, $s^1$ and $s^2$, since $\tan x$ is the same as
$x$ up to quadratic order. At order $s^3$, however, we get instead of (\ref{Taylor.3})
\begin{equation}
{\tilde w}_b''' (t) + {\tilde w}_c''' (t) = h_{ab}'''(0) + {6\over g} \big[ v_b'' (t) + v_c'' (t) \big]
+{16 \over g} \big[ v_b (t) + v_c (t) \big]
\end{equation}
Combining this with the other equations yields
\begin{eqnarray}
{\tilde w}_b''' (t) + 4 {\tilde w}_b' (t) &=& -{1 \over 4} h_{bc}''' (0) + 2 h_{bc}' (0) \equiv {\rm C}\nonumber\\
{\rm or} ~~ {\tilde w}_b'' (t) + 4 {\tilde w}_b (t) &=& {\rm C} t + {\rm C}'
\end{eqnarray}
This is like a driven harmonic oscillator of frequency $2$, with general solution of the form
\begin{equation}
w_{b,c} (t) = C_{b,c} + c_1 \cos 2t + c_2 \sin 2t + c_3 t
\end{equation}
Putting this form in the remaining equations, or in the original functional equation, we finally find
\begin{eqnarray}
{\tilde w}_{b,c} (x) &=& C_{b,c} + c_1 \cos 2x + c_2 \sin 2x + c_3 x \\
v_{b,c} (x) &=& g \, c_2 \cos 2x - g \, c_1 \sin 2x \\
u_{bc} (x) &=& g (C_b - C_c) \cot x + g \, c_3 \, x \cot x
\end{eqnarray}
The hyperbolic case is treated in exactly the same way, or can be obtained by simple analytic continuation
$x \to i x$, $g \to ig$, $c_2 \to -i c_2$. Altogether, we recover the potentials presented in Table 2.
\section*{References}
|
1,108,101,564,557 | arxiv | \section{Introduction}
Quantum discrimination of two optical quantum states plays a crucial rule in quantum information processing tasks, e.g., continuous-variable quantum key distributions and optical quantum communications \cite{silberhorn2002continuous,grosshans2003quantum,ghorai2019asymptotic,yuan2020free}. Due to the good compatibility with classical optical infrastructures, coherent states \cite{glauber1963coherent} generated by lasers are usually employed as the information carriers in quantum communication systems. However, the minimum discrimination error probability (MDEP) of discriminating two coherent states cannot be zero because of the non-orthogonal property of two coherent states \cite{glauber1963coherent}. To improve the performance of the quantum discrimination, new information carriers using non-coherent states draw much attention recently \cite{guerrini2019quantum,guerrini2020quantum}.
For example, the problem of discriminating between two noisy photon-added coherent states (PACSs) was addressed in \cite{guerrini2020quantum}. The PACS is generated by sequentially applying the displacement operator and the creation operator on a vacuum state. It was demonstrated that the error probability can be significantly reduced when PACSs instead of coherent states are employed in pulse position modulations \cite{guerrini2019quantum}. Inspired by \cite{guerrini2020quantum}, we focus on the quantum discrimination between two noisy displaced number states (DNSs). The DNS is generated by sequentially applying the creation operator and the displacement operator on a vacuum state. The properties of noiseless DNS were discussed in \cite{de1990properties,tanas1992phase,mo1996displaced}. However, the thermal noise is inevitable in preparing a DNS and the property of a noisy DNS has not been studied yet. Besides, to the best of the authors' knowledge, the problem of discriminating between two noisy DNSs has not been addressed.
In this letter, we first address the problem of discriminating between two noiseless DNSs. Then we derive the Fock representation of noisy DNSs and address the problem of discriminating between two noisy DNSs. Using the Fock representation of noisy DNSs, we further prove that the optimal quantum discrimination of two noisy DNSs can be achieved using a Kennedy receiver with a threshold detection. The simulation results verify our theoretical derivations. We also explore the possibility of employing DNSs instead of coherent states in on-off keying (OOK) modulations; and find that the error probability of OOK modulation using a DNS can be significantly reduced compared with the error probability of OOK modulation using a coherent state with the same average energy.
\section{Quantum Discrimination of Two Noiseless DNSs}\label{DiscriminateNoiselessDNS}
\subsection{Displaced Number State}
The DNS is generated by sequentially applying the creation operator and the displacement operator on a vacuum state; and it can be written as \cite{de1990properties,tanas1992phase}
\begin{equation}
\ket{\mu,k}= \hat{D}(\mu)\ket{k}
\end{equation}
\noindent where $\hat{D}(\mu)$ is the displacement operator; and $\ket{k}$ is the number state containing $k$ photons. The number state decomposition of a DNS can be obtained as \cite{tanas1992phase}
\begin{equation}
\ket{\mu,k}=\sum_{n=0}^{\infty}b_n\ket{n}
\end{equation}
\noindent where the coefficient $b_n$ is given by \cite{tanas1992phase}
\begin{equation}
b_n=
\left\{
\begin{array}{ll}
\sqrt{\frac{n!}{k!}}(-\mu^*)^{k-n}e^{-\frac{|\mu|^2}{2}}L_n^{(k-n)}(|\mu|^2), \text{ for } n<k\\
\sqrt{\frac{k!}{n!}}\mu^{n-k}e^{-\frac{|\mu|^2}{2}}L_n^{(n-k)}(|\mu|^2), \text{ for } n\geq k\\
\end{array}
\right.
\end{equation}
\noindent and where $L_n^{(a)}(x)$ is the generalized Laguerre polynomial of order $n$ with parameter $a$.
Using the number state decomposition, we can obtain the inner product of two DNSs $\ket{\mu,k}$ and $\ket{\xi,h}$, where we let $h\geq k$ without loss of generality, as
\begin{equation}\label{Superposition}
\begin{aligned}
\langle \xi,h|\mu,k \rangle=&\bra{h}\hat{D}(\mu-\xi)\ket{k}\\
=&\sum_{n=0}^{\infty}\bra{h}b_n(\mu-\xi,k)\ket{n}\\
=&\sqrt{\frac{k!}{h!}}(\mu-\xi)^{h-k}e^{-\frac{|\mu-\xi|^2}{2}}L_k^{(h-k)}(|\mu-\xi|^2)
\end{aligned}
\end{equation}
\noindent where in the first step we have used the properties of displacement operator: $\hat{D}(\alpha)=\hat{D}^{\dagger}(-\alpha)$ and $\hat{D}(\alpha)\hat{D}(\beta)=\hat{D}(\alpha+\beta)$.
\subsection{Discriminate Two Noiseless DNSs}
The key of discriminating between any two quantum states $\{\hat{\rho}_0, \hat{\rho}_1\}$ with prior probabilities $\{p_0, p_1\}$ is to find two positive operator-valued measure (POVM) operators $\{\hat{\Pi}_0,\hat{\Pi}_1\}$ that can minimize the discrimination error probability. According to the Helstrom's theory \cite{helstrom1969quantum}, the optimal POVM operators can be obtained as
$\hat{\Pi}_0=\sum_{\lambda_n<0}\ket{\lambda_n}\bra{\lambda_n}$ and $\hat{\Pi}_1=\hat{\mathbb{I}}-\hat{\Pi}_0$, where $\lambda_n$ and $\ket{\lambda_n}$ are the eigenvalue and the eigenvector of the decision operator $\hat{\Delta}=p_1\hat{\rho}_1-p_0\hat{\rho}_0$; $\hat{\mathbb{I}}$ is the identity operator. The MDEP of discriminating $\{\hat{\rho}_0, \hat{\rho}_1\}$ is obtained as the Helstrom bound \cite{helstrom1969quantum}
\begin{equation}\label{HelstromBound_1}
\begin{aligned}
P_e&=\frac{1}{2}(1-\|\hat{\Delta}\|_1)\\
&=p_1-\sum_{\lambda_n>0}\lambda_n
\end{aligned}
\end{equation}
\noindent where $\|\hat{A}\|_1=\text{tr}\{\sqrt{\hat{A}^{\dagger}\hat{A}}\}$ denotes the trace norm of the operator $\hat{A}$.
For discriminating between two pure states $\hat{\rho}_0=\ket{\psi_0}\bra{\psi_0}$ and $\hat{\rho}_1=\ket{\psi_1}\bra{\psi_1}$, the Helstrom bound \eqref{HelstromBound_1} can be rewritten as
\begin{equation}\label{HelstromBound_2}
P_e=\frac{1}{2}-\frac{1}{2}\sqrt{1-4p_0p_1|\left \langle \psi_0|\psi_1 \right \rangle|^2}.
\end{equation}
Therefore, the MDEP for discriminating between two noiseless DNSs $\ket{\xi,h}$ and $\ket{\mu,k}$ is determined by the inner product $\langle \xi,h|\mu,k \rangle$. Substituting \eqref{Superposition} into \eqref{HelstromBound_2}, we can obtain the MDEP as
\begin{equation}\label{P_e_noiseless}
\begin{aligned}
P_e=\frac{1}{2}-\frac{1}{2}&\left\{1-4p_0p_1\frac{k!}{h!}|\mu-\xi|^{2(h-k)}e^{-|\mu-\xi|^2}\right.\\
&\quad \times\left.\left[L_k^{(h-k)}(|\mu-\xi|^2)\right]^2\right\}^{\frac{1}{2}}.
\end{aligned}
\end{equation}
From \eqref{P_e_noiseless}, we can observe that the perfect discrimination with zero error probability happens in the following two situations: (i) $\mu=\xi$ and $h\neq k$; (ii) $L_k(|\mu-\xi|^2)=0$ and $h=k\neq 0$. Notice that when $h=k=0$, the two DNSs becomes two non-orthogonal coherent states $\ket{\xi}$ and $\ket{\mu}$. Then the MDEP approaches zero when $|\mu-\xi|^2$ approaches $\infty$.
\section{Quantum Discrimination of Two Noisy DNSs}
\subsection{Noisy DNSs}\label{NoisyDNS}
A noisy number state $\hat{\rho}(k)$ is obtained by applying creation operators on a thermal state $\hat{\rho}_{th}$, which results in
\begin{equation}
\hat{\rho}(k)=\frac{(\hat{A}^{\dagger})^k\hat{\rho}_{th}\hat{A}^k}{\text{tr}\{(\hat{A}^{\dagger})^k\hat{\rho}_{th}\hat{A}^k\}}
\end{equation}
\noindent where $\text{tr}\{\cdot\}$ denotes the trace operation.
Then the noisy DNS $\hat{\rho}(\mu,k)$ is defined as
\begin{equation}\label{noisy DNS}
\hat{\rho}(\mu,k)\triangleq\frac{\hat{D}(\mu)(\hat{A}^{\dagger})^k\hat{\rho}_{th}\hat{A}^k\hat{D}^{\dagger}(\mu)}{N_k}
\end{equation}
\noindent where $N_k$ can be obtained as
\begin{equation}
\begin{aligned}
N_k=\text{tr}\{\hat{D}(\mu)(\hat{A}^{\dagger})^k\hat{\rho}_{th}\hat{A}^k\hat{D}^{\dagger}(\mu)\}=k!(n_t+1)^k
\end{aligned}
\end{equation}
\noindent and where $n_t$ is the average number of thermal photons due to the presence of thermal noise. The following theorem presents the Fock representation for a noisy DNS.
\begin{theorem}{(Fock representation)}\label{theorem_Fock_representation}
The Fock representation of a noisy DNS $\hat{\rho}(\mu,k)$ is found to be
\begin{equation}
\begin{aligned}
\label{Fock representation}
\bra{n}&\hat{\rho}(\mu,k)\ket{m}\\
&=\sum_{i=0}^{k}\sum_{j=0}^{k}I(n\geq i;m\geq j)\frac{(-1)^{i+j}\binom{m}{j}\binom{k}{i}}{(k-j)!}\sqrt{\frac{n!}{m!}}e^{-\frac{|\mu|^2}{n_t+1}}\\
&\quad \times \frac{|\mu|^{2(k-j)}(\mu^*)^{m-n} n_t^{n-i}}{(n_t+1)^{m+k-j+1}}L_{n-i}^{(m-n+i-j)}\left(-\frac{|\mu|^2}{n_t(n_t+1)}\right)
\end{aligned}
\end{equation}
\noindent where $I(n\geq i;m\geq j)$ is an indicator function defined as
\begin{equation}
\begin{aligned}
I(n\geq i;m\geq j)\triangleq\left\{
\begin{array}{ll}
1,\text{ for } n\geq i \text{ and } m\geq j \\
0,\text{ otherwise.}
\end{array}\right.
\end{aligned}
\end{equation}
\end{theorem}
\begin{proof}
See Appendix \ref{A1}.
\end{proof}
Using the Fock representation in \eqref{Fock representation}, we can obtain the photon statistics of a noisy DNS as
\begin{equation}
\begin{aligned}
p(n)&= \bra{n}\hat{\rho}(\mu,k)\ket{n}\\
&=\sum_{i=0}^{k}\sum_{j=0}^{k}I(n\geq i;n\geq j)\frac{(-1)^{i+j}\binom{n}{j}\binom{k}{i}}{(k-j)!}e^{-\frac{|\mu|^2}{n_t+1}}\\
&\quad \times \frac{|\mu|^{2(k-j)} n_t^{n-i}}{(n_t+1)^{n+k-j+1}}L_{n-i}^{(i-j)}\left(-\frac{|\mu|^2}{n_t(n_t+1)}\right).
\end{aligned}
\end{equation}
Then the average number of photons $n_p$ contained in a noisy DNS can be obtained in the following lemma.
\begin{lemma}\label{Lemma_1}
The average number of photons $n_p(\mu,k)$ contained in a noisy DNS $\hat{\rho}(\mu,k)$ is found to be
\begin{equation}\label{n_p}
n_p(\mu,k)=|\mu|^2+k(n_t+1)+n_t.
\end{equation}
\end{lemma}
\begin{proof}
See Appendix \ref{A2}.
\end{proof}
\subsection{Discriminate Two Noisy DNSs}\label{DiscriminateNoisyDNS}
Now we consider the discrimination of two noisy DNSs $\hat{\rho}_0=\hat{\rho}(\xi,h)$ and $\hat{\rho}_1=\hat{\rho}(\mu,k)$ with prior probabilities $p_0$ and $p_1$, respectively. Here we let $h\geq k$ without loss of generality. According to the Helstrom bound \eqref{HelstromBound_1}, the MDEP is determined by all the positive eigenvalues $\lambda_n$ of the decision operator
$\hat{\Delta}=p_1\hat{\rho}(\mu,k)-p_0\hat{\rho}(\xi,h)$. Because the displacement operator is an unitary operator, $\hat{\Delta}$ and $\hat{D}(-\xi)\hat{\Delta}\hat{D}^{\dagger}(-\xi)$ share the same eigenvalues. Then it is readable to show that
\begin{equation}\label{tracenorm_2}
\|p_1\hat{\rho}(\mu,k)-p_0\hat{\rho}(\xi,h)\|_1=\|p_1\hat{\rho}(\mu-\xi,k)-p_0\hat{\rho}(0,h)\|_1.
\end{equation}
It is challenging to obtain an analytical form of the eigenvalues $\lambda_n$ for an arbitrary $\hat{\Delta}$. However, eq. \eqref{tracenorm_2} indicates that when $\mu=\xi$, we only need to obtain the eigenvalues of $p_1\hat{\rho}(0,k)-p_0\hat{\rho}(0,h)$. Using the Fock representation \eqref{Fock representation}, we can obtain the eigenvalues of $p_1\hat{\rho}(0,k)-p_0\hat{\rho}(0,h)$ as
\begin{equation}\label{Lambda_n}
\begin{aligned}
\lambda_n&=p_1\bra{n}\hat{\rho}(0,k)\ket{n}-p_0\bra{n}\hat{\rho}(0,h)\ket{n}\\
&=
\left\{
\begin{array}{ll}
0, \text{ for } n<k\\
p_1\binom{n}{k}\frac{n_t^{n-k}}{(n_t+1)^{n+1}}, \text{ for } k\leq n<h\\
p_1\binom{n}{k}\frac{n_t^{n-k}}{(n_t+1)^{n+1}}-p_0\binom{n}{h}\frac{n_t^{n-h}}{(n_t+1)^{n+1}}, \text{ for } n\geq h.
\end{array}
\right.
\end{aligned}
\end{equation}
Now the key of obtaining a tractable MDEP is to find all the positive eigenvalues. To achieve this, in the following we first introduce the Kennedy receiver with threshold detection \cite{yuan2020kennedy}, and then we prove that the Kennedy receiver with threshold detection can achieve the optimal discrimination and provide a tractable MDEP.
\subsection{Kennedy Receiver with Threshold Detection}\label{KennedyDetection}
A Kennedy receiver with threshold detection \cite{yuan2020kennedy} consists of a displacement operator $\hat{D}(\beta)$ and a photon counting process followed by a threshold detection based on the counted photons. The threshold detection is characterized by two POVM operators \begin{equation}\label{ThreholdOperators}
\hat{M}_0=\hat{\mathbb{I}}-\sum_{n=0}^{n_{th}}\ket{n}\bra{n}; \quad \quad \hat{M}_1=\sum_{n=0}^{n_{th}}\ket{n}\bra{n}
\end{equation}
\noindent where $n_{th}$ is the detection threshold of the counting photons. These two POVM operators correspond to the following threshold detection rule
\begin{equation}
\label{Threshold test}
n \mathop{\lesseqgtr} \limits_{\hat{\rho}_0}^{\hat{\rho}_1} n_{th}.
\end{equation}
If we set the displacement operator as $\hat{D}(\beta)=\hat{D}(-\mu)$, then the input states $\hat{\rho}(\mu,h)$ and $\hat{\rho}(\mu,k)$ are displaced as $\hat{\rho}(0,h)$ and $\hat{\rho}(0,k)$, respectively. Then the error probability of the receiver can be calculated by
\begin{equation}\label{P_e_2}
\begin{aligned}
P_e&=p_0\text{tr}\{\hat{M}_1\hat{\rho}(0,h)\}+p_1\text{tr}\{\hat{M}_0\hat{\rho}(0,k)\}.
\end{aligned}
\end{equation}
\noindent Substituting \eqref{ThreholdOperators} into \eqref{P_e_2}, we can obtain
\begin{equation}\label{P_e_3}
\begin{aligned}
P_e&=p_1-\sum_{n=0}^{n_{th}}\left(p_1\bra{n}\hat{\rho}(0,k)\ket{n}-p_0\bra{n}\hat{\rho}(0,h)\ket{n}\right)\\
&=p_1-\sum_{n=0}^{n_{th}}\lambda_n.
\end{aligned}
\end{equation}
The optimal threshold $n_{th}$ is obtained by minimizing the error probability in \eqref{P_e_3}. The following optimal discrimination theorem guarantees that the Kennedy receiver with optimal threshold $n_{th}$ can always achieve the MDEP.
\begin{theorem}{(Optimal discrimination)}\label{Theorem_optimal_discrimination}
The optimal discrimination of two noisy DNSs $\hat{\rho}(\mu,h)$ and $\hat{\rho}(\mu,k)$ can be achieved by the Kennedy receiver with threshold detection, where the displacement operator is $\hat{D}(-\mu)$; and the MDEP can be obtained as
\begin{equation}\label{P_e}
P_e=p_1-\sum_{n=0}^{n_{th}}\lambda_n
\end{equation}
\noindent where the optimal threshold $n_{th}$ is the maximum $n$ satisfying
\begin{equation}\label{optimalthreshold}
\begin{aligned}
\binom{n}{k}n_t^{h-k}\geq \binom{n}{h}\frac{p_0}{p_1}.
\end{aligned}
\end{equation}
\end{theorem}
\begin{proof}
See Appendix \ref{A3}.
\end{proof}
Although the POVM operators of optimal quantum discrimination for two quantum states can be obtained from the Helstrom's theory, the realization of the optical quantum discrimination is usually intractable. However, Theorem \ref{Theorem_optimal_discrimination} indicates that the optimal quantum discrimination of two noisy DNSs $\hat{\rho}(\mu,h)$ and $\hat{\rho}(\mu,k)$ is realizable by the Kennedy receiver with threshold detection \footnote{Note that the Kennedy receiver with threshold detection is a near-optimum receiver for discriminating between two coherent states.}.
\section{Numerical Results}
\label{sect:NumericalResults}
\begin{figure}
\begin{center}
\includegraphics[width=0.48\textwidth, draft=false]{TwoNoiselessDNS.eps}
\caption{MDEP for discriminating two noiseless DNSs ($n_t=0$)}
\vspace{-0.4cm}
\label{Fig:TwoNoiselessDNS}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=0.48\textwidth, draft=false]{TwoNoisyDNS.eps}
\caption{MDEP for discriminating two noisy DNSs ($n_t=0.2$)}
\vspace{-0.4cm}
\label{Fig:TwoNoisyDNS}
\end{center}
\end{figure}
The prior probabilities are set as $p_0=p_1=0.5$ in this section. Figs. \ref{Fig:TwoNoiselessDNS} and \ref{Fig:TwoNoisyDNS} present the MDEP for discriminating two noiseless DNSs and two noisy DNSs, respectively. From Fig. \ref{Fig:TwoNoiselessDNS}, we can observe that when $k=h$, the MDEP decreases as $|\mu-\xi|$ increases; and when $k\neq h$, the MDEP achieves zero when $\mu=\xi$. Besides, when $k\neq h$, the MDEP decreases as the gap $h-k$ increases. Comparing Fig. \ref{Fig:TwoNoisyDNS} with Fig. \ref{Fig:TwoNoiselessDNS}, we can see that the MDEP for discriminating two noisy DNSs demonstrates similar properties to that for discriminating two noiseless DNSs. Besides, the MDEP for discriminating two noisy DNSs is always larger than that for discriminating two noiseless DNSs with the same parameters.
\begin{figure}
\begin{center}
\includegraphics[width=0.48\textwidth, draft=false]{P_e_vs_gap.eps}
\caption{Error probabilities under different gap ($\mu=\xi=1, n_t=0.2$)}
\vspace{-0.4cm}
\label{Fig:P_e_vs_gap}
\end{center}
\end{figure}
Next we check the error probability for discriminating between two noisy DNSs obtained by the Kennedy receiver with threshold detection with $\mu=\xi=1$ under different gaps, shown in Fig. \ref{Fig:P_e_vs_gap}. We also plot the MDEP results of Helstrom bound obtained by the optimal quantum discrimination. We can see that the error probabilities obtained by the Kennedy receiver with threshold detection coincide with the MDEP results of Helstrom bound obtained by the optimal quantum discrimination. This verifies the result of Theorem \ref{Theorem_optimal_discrimination}. The MDEP decreases as the gap increases for a given $k$, which is as expected. Besides, when the gap is fixed, the MDEP decreases as $k$ decreases. For a given gap, a smaller $k$ implies a smaller energy requirement. This indicates that if we use DNSs as the information carriers in intensity modulations, a smaller $k$ can achieve a better performance in not only error probability but also energy efficiency. Therefore, the OOK modulation with $k=0$ for a given energy gap is the optimal intensity modulation in terms of both the error probability and the energy efficiency.
\begin{figure}
\begin{center}
\includegraphics[width=0.48\textwidth, draft=false]{DNS_vs_Coherent.eps}
\caption{Error probabilities of OOK modulation under different thermal noises}
\vspace{-0.4cm}
\label{Fig:DNS_vs_Coherent}
\end{center}
\end{figure}
At last, we consider a special case of discriminating two noisy DNSs with $\mu=\xi=0$ and $k=0$ under different gaps $h$, which corresponds to an OOK modulation in communication systems. The error probabilities under different $h$ with different thermal noises are shown in Fig. \ref{Fig:DNS_vs_Coherent}. We also plot the error probabilities of the OOK modulation employing a coherent state with the same average energy per information bit. We can see that the error probability decreases as $h$ increases, which is as expected. Besides, we can also see that the error probability of OOK modulations employing a DNS can be significantly reduced compared with that of employing a coherent state with the same average energy.
\section{Conclusion}
\label{sect:Conclusion}
We addressed the problem of discriminating between two noisy DNSs. We first considered the quantum discrimination of two noiseless DNSs. Then we derived the Fock representation of a noisy DNS, and then used the Fock representation to derive the MDEP of discriminating two noisy DNSs. We further proved that the optimal quantum discrimination of two noisy DNSs can be achieved by the Kennedy receiver with threshold detection. The simulation results verified our theoretical derivations. Besides, we found that the error probability of OOK modulation employing a DNS is significantly less than that of OOK modulation employing a coherent state with the same average energy.
\bibliographystyle{IEEEtran}
|
1,108,101,564,558 | arxiv |
\section{Conclusion}
\label{sec:conclude}
We believe that the quantum computing field can learn from classical computing design techniques. The PRK are a powerful design tool,
and we posit that there is a need for a similar approach for hardware/software co-design for quantum computing. We have introduced the concept
of QRK and have provided a few examples of QRK. This paper is a call to the quantum computing user community to work
together to develop a complete set of QRK that can guide the design and development of quantum computing.
\section{Introduction}
\label{sec:intro}
Quantum computers have the potential to transform computing. Their advent promises to open up new classes of applications and enable the solution of problems that are intractable for
classical computers today. To reach that potential, however, requires great technological innovations, including research advances in fundamental physics, the
development of new quantum programming models that ``normal'' humans can use, and solutions to a host of issues in both software and hardware concerning building quantum
systems.
Quantum computing has a long way to go, but we believe we can leverage best practices from classical digital computing to advance through the stages of development
at a faster rate. In this paper, we focus on one lesson learned from the era of digital computing: that the best systems emerge from a hardware/software co-design
process. All too often hardware is designed and then ``thrown over the fence'' to software developers to figure out how to
make the hardware useful. It is more effective to have the software and hardware teams working together, so
the programming systems are ready as soon as the hardware is available and the hardware includes features
specifically needed to make the software both more efficient to run and easier to write.
However, it is challenging to do hardware/software codesign when you don't know what future applications will look like. Hypothesis: Even though we
may not know what future applications will look like, we do know the features of a system that will limit these applications.
If we can define these ``bottlenecks'' and build a system that collectively minimizes their impact, we can be confident that the
system will be effective for these future applications.
This was the basic principle behind Parallel Research Kernels (PRK) ~\cite{PRK2014} in the parallel computing field. The PRK are a collection of simple kernels
that expose the features of a system that limits parallel performance. They are small, generate their own data, do a computation (so
systems can't ``cheat''), and test their results. In essence, they are a way for application programmers to precisely define what they need
from a system; to guide hardware developers to build systems that will work well for applications.
They have proven useful for designing systems~\cite{SCC10} and to explore the suitability of extreme scale programming models~\cite{prkexa:16}.
The goal of this paper is to launch a conversation about using an approach similar to the PRK for quantum computing. We call these the
Quantum Research Kernels (QRK). Can we define features of quantum computing systems that will limit applications for these systems?
Can we produce well defined kernels to expose these features? Can we anticipate the breadth of application design patterns so we can
be confident that the set of QRK are complete? These are challenging questions. In this paper, we define the problem
and propose a process to answer these questions.
\section{Parallel Research Kernels}
\label{sec:prk}
The Parallel Research Kernels stress a system in ways parallel applications in high performance computing would.
They are listed in Table~\ref{table:PRK} where we provide the name of each kernel, a brief definition, and
the features of a parallel system stressed by the kernel. The kernels are defined mathematically and with a
reference implementation using C, OpenMP, and MPI. They are available in a github repository~\cite{prkRepo}.
\begin{table}[!htbp]
\centering
\caption{\textbf{Parallel Research Kernels:}
-- \small
A set of basic kernels designed to stress features of a system that limit the performance of HPC applications.}
\label{table:PRK}
\begin{tabular}{|l|l|l|}
\hline
\emph{Name} & \emph{Definition} & \emph{Exposed system feature} \\
\hline
Transpose & Transpose a dense matrix & Bisection Bandwidth \\
\hline
Reduce & Elementwise sum of & Message passing, local \\
& multiple private vectors & memory bandwidth \\
\hline
Sparse & Sparse-martix vector product & scatter/gather operations \\
\hline
Random & Random update to a table & Bandwidth to memory \\
& & with random updates \\
\hline
Synch\_global & Global synchronization & Collective synch. \\
\hline
Synch\_p2p & point-to-point & message passing latency, \\
& synchronization & remote atomics \\
\hline
Stencil & Stencil method & nearest neighbor and \\
& & asynchronous comm. \\
\hline
Refcount & Update shared or private & Mutual exclusion locks \\
& counters & \\
\hline
Nstream & daxpy over large vectors & Peak memory bandwidth \\
\hline
DGEMM & Dense Matrix Product & Peak floating point perf. \\
\hline
Branch & Inner loop with branches & Misses to the instruction \\
& & cache, branch prediction. \\
\hline
PIC & Particle in cell & unstructured asynch. \\
& &multitasking \\
\hline
AMR & Adaptive mesh refinement & hierarchical asynch. \\
& &multitasking \\
\hline
\end{tabular}
\end{table}
We submit that the Parallel Research Kernels are complete. If a system is built that does well with all of them, then
it is likely that system will be effective for running parallel HPC applications. The PRK were selected by an ad hoc committee
of parallel application programmers. When we started the project (in 2005) parallel computing in various forms had been around
over 25 years. We were able to convene a committee of experts with decades of experience. Over the course of several meetings
and long email chains, the committee came up with the list of kernels.
\section{Quantum Research Kernels}
\label{sec:qrk}
The quantum research kernels (QRK), inspired by the PRK, will define a set of kernels designed to
expose features of a quantum computing system that will constrain the success of applications
written for quantum computers. The QRK are intended to be a sufficiently complete set such that
if a system were constructed that did well for each of the QRK, that system would most likely be
a successful system for supporting key applications.
As with the PRK, the QRK will be produced through a community-driven process by programmers interested in
writing applications for quantum computers. To help nurture this conversation, we provide an example
of a potential QRK. One of the bottlenecks for scalable quantum computing, is the
difficulty of loading the data (particularly classical data) into the quantum
machine. State preparation can be an exponentially hard problem itself, leading
to the conundrum that loading of the data into the quantum machine can completely
negate the speed-up gained by doing the problem on a quantum computer \cite{aronson}.
Hence, the first QRK follows.
\begin{itemize}
\item {\bf Name}: Encode
\item{\bf Definition}: Create a sequence of classical values from $i=0$ to $i=N$ equal to $4i{\pi}/N$
\item{\bf Action}: Encode qubits with the values from the previous step. Rotate each qubit by ${\pi}/6$.
\item{\bf Test}: Read qubits and confirm that they have the correct rotated value
\end{itemize}
Note this has all the features we would expect in a QRK. It defines problem input that is generated so the QRK can
scale to any number of qubits. A specific operation is defined so at the end of the QRK, the result can be validated.
Finally, it exposes a specific feature of a quantum computer that will limit the ability of applications to run on the system
expressed in terms that architects can understand and use to guide the design of a quantum computer.
We have the beginnings of two additional QRK. The first is one we call the \emph{Computational Area}.
This is the product of a number of qubits that can be entangled times the number of operations that can be
carried out before the entangled state can no longer be maintained. A high Computational Area can be produced by a small number of
qubits that remain entangled over a large number of operations or by having a large number of qubits that remain entangled
for a small number of operations. There are advantages to both cases so this measure supports both.
The second additional QRK is called \emph{Parallel Streams}. This measures the ability of a quantum computer
to execute multiple independent streams of operations at the same time and in parallel. There are a number of options on how
to define the work that must be carried out in parallel. Initially, we would run the Computational Area QRK in each stream, though
for a wider range of applications for quantum computers, we may find a better case for each stream.
These three QRK are just a start. We need a larger set that covers the full range of features needed from a successful
quantum computer, hopefully, on the order of 10. The system features each QRK stresses overlap, but taken together
they need to cover the full range of features need by application programmers from a quantum computer so those designing
parallel systems can be confident a system that does well on the full set of QRK will meet the needs of applications programmers.
This is the primary goal of the QRK. However,
once established, an application designer can model the needs of an application in terms of a linear combination of QRK
thereby using them to help select the quantum computer best suited to a particular application.
|
1,108,101,564,559 | arxiv | \section{Introduction}
The inflationary paradigm \cite{inflation} provides a simple framework to generate primordial nearly scale-invariant perturbations with a slightly red tilt, as well as small non-gaussianties which are in good agreement with observations \cite{Planck1,Planck3,Planck4}. Despite its successes, the inflationary scenario has certain issues \cite{problems-ijjas-inflation,trans-Planckian} within its models and there is much discussion within the early universe cosmology community around attempts to fix these issues within the paradigm. The other approach is to try to construct an alternative paradigm \cite{Gasperini2002,brand} to see if this can also explain the observations of the late-time Universe that we have today. The bouncing cosmology paradigm is one such attempt \cite{review,review2}.
\paragraph*{}
We will be interested in model realisations within this paradigm that are non-singular. This means that these models do not suffer from the initial Big Bang singularity that is ubiquitous in the case of cosmologies that have expanded for its entire history \cite{vilenkin-borde-guth}. The trade-off in the case of a bouncing model is that there has to be some new physics at the expansion minimum (which coincides with the `bounce') to allow the Universe to re-expand \cite{Lehners:2007ac,Lehners:2008vx,Lehners2008}. This new physics is in the form of a null energy condition violation in the case of spatially flat models. The inclusion of positive curvature does satisfy one of the conditions on the vanishing of the Hubble rate without the need of violating the null-energy condition \cite{BarrDab}. However a second condition on the successful fulfillment of the bounce is to have the first derivative of the Hubble rate be negative and some new physics would be needed to have the Universe re-expand from the minimum. In a previous work, we have parametrised this `new physics' by hypothesising a non-linear equation of state \cite{me4}. This non-linear equation of state was used in \cite{bruni1,bruni2} to model a dynamical dark energy scenario. This model has the advantage that there exists a high energy cosmological constant which represents the maximum value the energy density can take without causing the model to exhibit phantom behaviour. At this energy density, the null energy condition is not violated at the level of phenomenology. This means that in the presence of positive spatial curvature it is still possible to have a bounce with no violation of the null-energy condition, and only a violation of the strong energy condition.
\paragraph*{}
An additional problem that plagues contracting universe scenarios, such as one that must be incorporated in a bouncing cosmology, is the problem of growing anisotropies and inhomogeneities. The issue with growing anisotropies is not only that this is in direct contradiction to the very tight constraints on anisotropy \cite{saadeh} and the time of isotropisation in the late Universe - but also that a successful bounce must necessarily have very low anisotropy going into it from the contracting phase \cite{ekpyrosis_mixmaster,ekpyrosis_numerics}. For example, it has been shown \cite{review} that at the level of perturbations if the mechanism for example is a scalar field $\chi$ that is causing the bounce to occur, then the perturbative anisotropy energy density $\sigma ^2$ must obey the ratio $\frac{\dot{\chi}^2}{\sigma ^2} \gg 1$ for the bounce to occur successfully and lead to re-expansion. This can be understood non-perturbatively as well - a contracting closed anisotropic Universe suffers an instability known as a BKL instability by which the Universe undergoes infinite chaotic oscillations on a finite time interval until it finally collapses to a singularity \cite{mixmaster_numerics_1,mixmaster_numerics_2,bkl,belinski_henneaux_2017,Belinskii1972}. The anisotropies thus need to be suppressed to avoid this instability from being manifested in a contracting Universe.
\paragraph*{}
A way of dealing with the unbounded growth of anisotropies has been to incorporate an ekpyrotic phase \cite{Lehners:2007ac,Lehners:2008vx,Lehners2008,review,review2}. This is described by an `ultra-stiff' equation of state i.e.\ an equation of state that is given by $P\gg \rho$ where $P$ and $\rho$ denote the pressure and energy densities of the fluid. In the ekpyrotic scenario, this is incorporated with the aid of a slow contraction mediated by a fast-rolling scalar field rolling down a negative exponential potential. While this model has been proved to be very successful in dealing with the BKL instability \cite{mixmaster_numerics_1,mixmaster_numerics_2,ekpyrosis_mixmaster}, it still has some issues. For example, it seems to be a fine-tuned scenario when one incorporates the effects of anisotropic pressures even if these follow the condition of ultra-stiffness on average \cite{me1}. It also has a problem of perturbations being blue-tilted without the incorporation of an additional scalar field \cite{review,review2}. There is also the probability of super-luminal propagation of the kinetic energy modes of the scalar field that is an issue in any scenario that has $p>\rho$. All of these reasons have prompted us to search for an alternative mechanism for isotropising a contracting Universe. It has been shown in \cite{me4,misner_aniso} that the incorporation of a negative anisotropic stress in the form of shear viscosity such that the anisotropic stress is proportional to the shear anisotropy with a negative proportionality constant leads to the isotropisation of a Bianchi IX Universe.
\paragraph*{}
In this work we shall study the global stability of the isotropic Friedmann-Lemaitre point in the absence of the ultra-stiff fluid equation of state and in the presence of this shear viscous stress. We shall show that a contracting Universe sourced by a non-linear equation of state and with the incorporation of shear viscosity demonstrates an attractor behaviour on approach to this isotropic Friedmann-Lemaitre point. In doing so, we are drawing on previous work on no-hair theorems in contracting universes \cite{Lidsey2005} to derive a new no-hair theorem which shows that isotropisation with the aid of shear viscous anisotropic stress is possible in the most general spatially homogeneous Universe. We do not model the bounce in this work. In order to do this, we would need a specific geometry as it would depend on whether the model in question was flat or had non-zero spatial curvature. Our aim remains to derive a general result for the process of isotropisation in the presence of shear viscosity in a contracting Universe.
\paragraph*{}
In the next section, we shall provide a brief background on Bianchi cosmologies which shall serve as the framework for the remainder of the paper. In the subsequent sections, we shall use the expansion normalised variables of the orthonormal formalism as done in \cite{WEllis,Lidsey2005} to prove a general cosmic no-hair theorem for the isotropisation of the Bianchi cosmologies in the presence of shear viscous anisotropic stresses for Bianchi Types I-VIII, Type IX and separately Bianchi Class B.
\paragraph*{}
For the entirety of this work we have used natural units where $c = \hbar = G =1$.
\section{A background on Bianchi cosmologies}
We are, in this work, interested in studying the stability of the isotropic Friedmann Lemaitre point. Thus we employ the class of the most general spatially homogeneous, anisotropic cosmologies which are given by the Bianchi cosmologies. The summary in this section will largely follow the formalism as laid out in \cite{WEllis}.
These admit a local group $G_3$ group of isometries that act simply transitively on a spacelike hypersurface. Then the line element can be written in the form $ds^2 = -dt^2 +h_{ab}\omega^a \omega ^b$ where $\omega ^a$ are the $1$-forms. These $1$-forms follow the Maurer-Cartan equations where $d\omega ^c =\frac{1}{2}C^c_{ab} \omega ^a \wedge \omega ^b$ where $C^c_{ab}$ are the structure constants of the Lie algebra. They are anti-symmetric so $C^c_{(ab)}=0$. This means that the independent components of the structure constants can be expressed as a symmetric $3\times 3$ matrix $n_{ab}$ and the components of a $3\times 1$ vector $A_b =C^a_{ab}$. So the structure constants can be written as,
\begin{equation}
C^c_{ab} = n^{cd}\epsilon_{dab} + \delta^c_{[a}A_{b]}
\end{equation}
where $\epsilon_{abc}$ is the antisymmetric tensor. Using the Jacobi identity,
\begin{equation}
C^e_{d[a}C^d_{bc]}=0
\end{equation}
we find that,
\begin{equation}
n_{ab}A^a =0
\end{equation}
We use the freedom of choice of orthonormal frame to rotate the frame vectors and diagonalise the tensor $n_{ab}$ as
\begin{equation}
n_{ab} = \mathrm{diag}\left\{n_1,n_2,n_3\right\}
\end{equation}
If $A^a \neq 0$ then we see that the above is an eigenvalue equation with the eigenvector $A^a$ always being able to be expressed as $A^a = (A,0,0)$. Therefore the eigenvalue equation simplifies to,
\begin{equation}\label{eq:paramAB}
n_1A =0
\end{equation}
\textit{For Bianchi Class A, $A=0$ and for Bianchi Class B $A\neq 0$}. We also assume that the direction of fluid flow is orthonormal to the group orbits. Hence we use the orthonormal frame approach and therefore we can write out the Einstein Field Equations in all generality for the Bianchi Classes A and B. The fluid velocity which is the unit timelike vector is given by $u_a$. This defines a projection tensor for the metric on the surface $g_{ab}$ given by,
\begin{equation}
h_{ab} = g_{ab} +u_a u_b
\end{equation}
which at each point projects onto the $3$-space orthogonal to the vector $u_a$ \cite{WEllis}.
The generalised expansion scalar is given by $\Theta$
\begin{equation}
\Theta = u^a_{;a}
\end{equation}
$\Theta= 3H$ where $H$ is the Hubble rate.
The evolution equation for the fluid velocity is given by,
\begin{equation}
\dot{u}_a = u_{a;b}u^b
\end{equation}
In this notation, the shear tensor is given by,
\begin{equation}
\sigma_{ab} = u_{(a;b)}-\frac{1}{3}\Theta h_{ab}-\dot{u}_a u_b
\end{equation}
The evolution equation for the shear is given by,
\begin{equation}\label{eq:evshear}
\dot{\sigma}_{ab } = -3 H\sigma _{ab} +2 \epsilon ^{m n}_{(a}\sigma_{b)m}\Omega_{n}- \phantom{p}^3 S_{ab} +\pi_{ab}
\end{equation}
The shear $\sigma _{\alpha \beta}$ can be diagonalised in the frame in which $n_{\alpha \beta}$ is diagonal and by the requirement of tracelessness has only $2$ independent components. We shall work with some linear combinations of these components as,
\begin{equation}
\sigma _+ =\frac{1}{2}\left(\sigma _{22}+\sigma_{33}\right),\;\;\; \sigma _- =\frac{1}{2\sqrt{3}}\left(\sigma _{22}-\sigma_{33}\right)
\end{equation}
The other unknown quantity in \eqref{eq:evshear} is the local angular velocity of the spatial frame specified by a set of $1$-forms $\{\textbf{e}_{\alpha}\}$ with respect to a Fermi-propagated spatial frame. We shall call this the Fermi rotation which is given by,
\begin{equation}
\Omega ^{a} = \frac{1}{2}\epsilon^{a m n}e^i_{m}e_{n i;j}u^j
\end{equation}
As mentioned before, we can choose a frame where $n_{\alpha \beta}$ are diagonalised and the $A_{a}$ can be expressed as $(A,0,0)$ via the Jacobi equation. We can show that the shear tensor can also be diagonalised in this frame as long as $\pi_{\alpha \beta} =0$ or $\pi_{\alpha \beta} \propto \sigma_{\alpha \beta}$. Details of this proof can be found in the appendix.
Thus the Einstein field equations in the orthonormal system formalism become,
\begin{align}
\dot{\sigma}_{\pm}&=-3H\sigma_{\pm} - ^{(3)}S_{\pm}\\
\dot{n}_1 &=\left(-H-4\sigma _+\right)n_1\\
\dot{n}_2 &= \left(-H+2\sigma_+ + 2\sqrt{3}\sigma _-\right)n_2\\
\dot{n}_3 &= \left(-H+2\sigma_+ -2\sqrt{3}\sigma _-\right)n_3
\end{align}
along with the continuity equation,
\begin{equation}
\dot{\rho}=-3H\left(\rho +p\right) -\left(\pi_+\sigma_+ + \pi_-\sigma_-
\right)
\end{equation}
where $\pi_{\pm}$ is given by,
\begin{equation}
\pi_{\pm} = -\kappa H \sigma_{\pm}
\end{equation}
In subsequent sections, we shall express these equations as a phase plane system through the process of expansion normalisation. In the case of Bianchi Types I-VIII we can normalise the gravitational field variables with respect to the Hubble expansion rate and in the case of Type IX which has a positive spatial curvature, this normalisation occurs with respect to a modified expansion variable.
\paragraph*{}
Before we begin our analysis of the stability of the isotropic Friedmann-Lemaitre point, we shall review the inclusion of dissipative effects in a cosmology.
\section{Shear and Bulk viscosity}
In standard cosmological analysis, dissipative effects are ignored. However their inclusion can give rise to many important effects - such as singularity resolution or novel isotropisation mechanisms.
\paragraph*{}
Dissipative effects can be modelled by including shear and bulk viscosity of the component fluids in the cosmology. This has been studied in several works \cite{gron-viscous}. The effect of bulk viscosity has been to increase the expansion rate i.e.\ modify the volume expansion. The manifestation of this is that the pressure of the component fluid gets modified as $\tilde{p} = p -3\xi H$ where $H$ is the Hubble expansion and $\xi$ is the coefficient of bulk viscosity. For bulk viscosity arising from radiation fluids, this effect vanishes when the equation of state of the radiation fluid is $p=(1/3)\rho$ i.e.\ the relativistic limit. However, viscosity effects can become important arbitrarily early in the history of the Universe if these effects were generated by particle collisions from gravitons, or processes like the evaporation of mini black holes. Bulk viscosity has also been studied as a mechanism of singularity resolution in cosmologies - for example for certain initial conditions, flat FLRW models with non-linear viscous terms are shown to be able to admit non-singular solutions. The non-linear equation of state that has been studied in \cite{bruni1,bruni2, me4} can be modelled as a bulk viscosity that is a function of the expansion rate $H$ - in the cases where the curvature is zero. Thus it is not surprising that non-singular solutions are found in the context of these non-linear fluids \cite{bruni1}.
\paragraph*{}Shear viscosity, on the other hand, usually manifests as an anisotropic pressure term and modifies the tensor modes, or the anisotropies of the cosmology. This has been studied with respect to neutrinos in \cite{misner_aniso} and also in the case of the early Universe for warm inflationary models \cite{gil}. It has been used to resolve the anisotropy problem in bouncing cosmologies in the contracting phase in \cite{me4}. For the case of small anisotropy, the form of shear viscosity is derived in \cite{weinberg_viscosity}. The anisotropic pressure term derived from shear viscosity there is given by $-\tilde{\kappa} \sigma_{ab}$ where $\tilde{\kappa}$ is the coefficient of viscosity. In \cite{belinski_henneaux_2017}, the coefficient of viscosity is dependent on density. When $\tilde{\kappa} \sim \rho ^n$, for values of $n < 1/2$, the Friedmann singularity for a contracting Universe is unstable and acausal. For $n>1/2$, the Friedmann singularity forms but is unstable. The only stable isotropisation on contraction occurs at $n=1/2$. Hence to achieve effective isotropisation, we choose the shear viscosity to take the form $-\kappa \rho ^{1/2} \sigma _{ab}$. For Universes that are flat and isotropic, this is the same as $-\kappa H \sigma_{ab}$. This is not the case in all of the general Bianchi types that we will be discussing in this work. However this choice makes the process of expansion normalisation and hence the phase plane analysis tractable. This form of the shear viscous term also makes the modified coefficient of viscosity $\kappa$ dimensionless. This simplification would not make a difference to the isotropising effect of shear viscosity. We can demonstrate this by considering the most anisotropic spatially homoegeneous geometry in a situation of anisotropy domination - the anisotropy dominated Kasner Universe. In this scenario, $H\sim \sigma ^2 \sim \sigma_{ab}\sigma^{ab}$. The evolution equation for the shear in the presence of viscous stress as modelled by $-\kappa H \sigma_{ab}$ is given by,
\begin{equation}
\dot{\sigma}_{ab} +3 H\sigma_{ab} = -\kappa H \sigma_{ab}
\end{equation}
Using the limit of anisotropy domination, we find the solution to the anisotropy energy density $\rho_{\sigma}$ which is proportional to $\sigma ^2 \sim \sigma_{ab}\sigma^{ab}$ given by,
\begin{equation}
\rho_{\sigma} \sim (6t +2 kt -c_1)^{-2}
\end{equation}
If the singularity which would turn into the bounce in the presence of positive curvature or new physics occurred at $t\rightarrow -\infty$ then $\rho_{\sigma}\rightarrow 0$ at this point. So the shear viscous term is successful in isotropising a Universe with maximum possible anisotropy i.e.\ the anisotropy-domanted Kasner Universe.
\section{Cosmic no-hair theorems}
In this section we follow the procedure laid out in \cite{Lidsey2005} to prove a global stability theorem for the most general, spatially homogeneous cosmologies. We consider a fluid with an equation of state given by,
\begin{equation}
p = \left(\gamma(\rho)-1\right)\rho
\end{equation}
Our results hold for a general $\gamma(\rho)$ - for the ideal equation of state, $\gamma$ is a constant. In \cite{Lidsey2005} they had explored the possibility of using an ultra-stiff equation of state i.e.\ $\gamma >2$ to prove a general cosmic no-hair theorem about the isotropic FL point being an attractor for initially contracting Universes of Bianchi Classes A and B. We use the same methods to prove a similar theorem - but this time in the absence of an ultra-stiff equation of state. Our ingredient is a shear viscous term - further cementing the idea that such a viscous effect can be used to isotropise Universes without necessarily having to take recourse to an `ekpyrosis'-like ultra-stiff fluid \cite{me4}. We have simply required that the Strong Energy Condition be obeyed, i.e.\ $\gamma >2/3$.
We define a new time variable $\tau$ which is defined as
\begin{equation}
\frac{dt}{d\tau}=\frac{1}{H}
\end{equation}
The time derivatives are taken with respect to this variable $\tau$ and is denoted by $'$. The Big Crunch singularity would take place at $\tau \rightarrow -\infty$. For a bounce model, this singularity would be replaced by a bounce and subsequent re-expansion. This would require new physics at the expansion minimum. For the case of a non-linear model \cite{bruni1,bruni2}, the presence of positive curvature and the non-linear equation of state fluid satisfies the conditions $H=0$ and the derivative of the Hubble rate with respect to cosmic time, $\dot{H}>0$ required for the bounce to occur. For the purposes of this work, we have not modelled the bounce but are instead interested in isotropisation in the contracting phase.
\paragraph*{}
As we are interested in the dynamics of the contracting phase alone, we can write out the Einstein's equations in expansion normalised form.
\begin{align}
\dot{\Sigma}_{\pm}' &=-(2-q)\Sigma_{\pm} -S_{\pm}\\
\dot{N}_1'&=\left(q-4\Sigma_+\right)N_1\\
\dot{N}_2'&=\left(q+2\Sigma_++2\sqrt{3}\Sigma_-\right)N_2\\
\dot{N}_3'&=\left(q+2\Sigma_+-2\sqrt{3}\Sigma_-\right)N_2\\
\end{align}
where
\begin{equation}\label{eq:defOfq}
q = 2\Sigma ^2 +\frac{1}{2}\left(3\gamma -2\right)\Omega
\end{equation}
and the expansion normalised variables are given by,
\begin{equation}\label{eq:normalisationI-VIII}
\left(\Sigma_+,\Sigma_-,N_1,N_2,N_3,\Omega\right)=\left(\frac{\sigma_+}{H},\frac{\sigma_-}{H},\frac{n_1}{H},\frac{n_2}{H},\frac{n_3}{H},\frac{\rho}{3H^2}\right)
\end{equation}
The Friedmann constraint is given by,
\begin{equation}\label{eq:Friedmann_constraint}
\Sigma ^2 +\Omega + K =1
\end{equation}
Here $K$ and $\Sigma ^2$ are respectively given by,
\begin{equation}\label{eq:defK}
K =\frac{1}{12}\left(N_1^2+N_2^2+N_3^3 -2N_1N_2-2N_2N_3-2N_3N_1\right)
\end{equation}
\begin{equation}\label{eq:defsigma}
\Sigma^2 = \Sigma _+^2+\Sigma _-^2
\end{equation}
which for Types I-VIII, is a positive definite quantity.
Let us look at the expansion normalised evolution equation for the density $\Omega$,
\begin{equation}
\Omega ^{\prime} = \left[ -(3\gamma -2)K +3(2-\gamma)\Sigma ^2\right]\Omega
\end{equation}
In the case of an ultra-stiff fluid $\gamma >2$, we can see that under the condition that $K>0$ , $\Omega ^{\prime} \leq 0$ for any initially contracting Bianchi $I-VIII$ Universe, where equality occurs iff $K=\Sigma ^2 =0$ for any non-vacuum orbit $\Omega\neq 0$. As $\Omega$ is a monotonic decreasing function of $\tau$ and it is bounded by the Friedmann constraint, we can infer that $\lim_{\tau \to -\infty}\Omega^{\prime}=0$. This further implies that,
\begin{equation}
\lim_{\tau \to -\infty}K=0,\; \lim_{\tau \to -\infty}\Sigma =0,\;\lim_{\tau \to -\infty}\Omega = 1
\end{equation}
Thus this means that for an initially contracting Bianchi $I-VIII$ Universe, isotropisation must have occurred as the bounce point is approached. If we now have a situation where the anisotropic stress is non zero and is of the form,
\begin{equation}
\pi _{ab} = -\kappa H \sigma_{ab}
\end{equation}
we get the evolution equation for $\Omega$ as,
\begin{equation}\label{eq:omegaEvEq}
\Omega ^{\prime} = -(3\gamma -2)K\Omega +\Sigma ^2 \left[3(2-\gamma)\Omega -\kappa\right]
\end{equation}
For our quadratic EOS, we have\cite{me4}
\begin{equation}
\gamma = \frac{P}{\rho} +1 = (\alpha+1) -\beta \frac{\rho}{\rho_C}
\end{equation}
We use the expansion normalised Friedmann constraint \eqref{eq:Friedmann_constraint} to write the evolution equation for $\Omega$ \eqref{eq:omegaEvEq} as follows,
\begin{equation}\label{eq:EvModOmega}
\Omega ^{\prime} = -(3\gamma -2)K \Omega +\left[\left\{3(2-\gamma)-\kappa\right\}\Omega -\kappa \Sigma ^2 -\kappa K\right]\Sigma^2
\end{equation}
If the strong energy condition i.e.\ $\rho + 3p\geq0$ is obeyed, then $(3\gamma -2)$ in the first term of \eqref{eq:EvModOmega} is positive definite. If $\kappa>0$, then using the fact that $\Sigma ^2 >0$ and $K>0$, we have the condition on $\Omega ^{\prime} \leq 0$ as
\begin{equation}\label{eq:condition}
\kappa > 3\left(2-\gamma\right)
\end{equation}
The case in which $\kappa =0$, for $\Omega ^{\prime} \leq 0$, we need the equation of state to be ultra-still effectively i.e.\ $\gamma(\rho)>2$.
If $\Omega ^{\prime} \leq 0$ for initially contracting Bianchi Types $I$-$VIII$ with equality when $K=\Sigma ^2 =0$, this means that $\Omega$ is a monotonically decreasing function of $\tau$ and as $\Omega$ is bounded by the Friedmann constraint \eqref{eq:Friedmann_constraint}.
we can conclude that $\lim_{\tau \rightarrow -\infty}\Omega ^{\prime} =0$. This implies that $\lim_{\tau \rightarrow -\infty}K=\lim_{\tau \rightarrow -\infty}\Sigma ^2 =0$ and by \eqref{eq:Friedmann_constraint}, $\lim_{\tau \rightarrow -\infty}\Omega =1$. Using $\Sigma^2 = \Sigma _+^2+\Sigma _-^2$, we can conclude that $\lim_{\tau \rightarrow -\infty}\Sigma _+ =\lim_{\tau \rightarrow -\infty}\Sigma _-=0$. This also implies that $\lim_{\tau \rightarrow -\infty}q=(3\gamma -2)/2>0$ for fluids obeying the strong energy condition. The evolution equations for the expansion normalised curvature variables are as follows,
\begin{align}
N_1 '&=\left(q-4\Sigma_+\right)N_1\\
N_2'&= \left(q+2\Sigma_+ +2\sqrt{3}\Sigma_-\right)N_2\\
N_3'&= \left(q+2\Sigma_+ -2\sqrt{3}\Sigma_-\right)N_3\\
\end{align}
Using the value of $q$ at $\tau \rightarrow -\infty$ we see that there exists a parameter $\epsilon >0$ such that $N'_i/N_i >\epsilon$ for a sufficiently negative $\tau$ so that $\lim _{\tau \rightarrow -\infty}N_i =0$. So following the same reasoning as in \cite{Lidsey2005}, we can conclude that for the values of $\kappa$ obeying \eqref{eq:condition}, the isotropic solution is a global attractor for a contracting Universe.
\section{Bianchi IX Cosmic No Hair Theorem}
The analysis for Bianchi IX has to be done slightly differently to the Bianchi Types I-VIII above. This is because the curvature variables $n_1$, $n_2$, $n_3$ are all positive in this Bianchi type and hence the spatial curvature is also positive. The Bianchi Type IX is an anisotropic generalisation of the closed Friedmann Universe. This also means that the expansion is no longer monotonic and the Hubble rate $H$ can be zero. Thus expansion normalisation of the state vector $\textbf{x} = \left \{ H,\sigma_+,\sigma_-,n_1,n_2,n_3\right\}$ must be done with respect to a modified expansion variable,
\begin{equation}\label{eq:defOfD}
D = \sqrt{H^2 + \frac{1}{4}\left(n_1 n_2 n_3\right)^{2/3}}
\end{equation}
The normalisation then becomes,
\begin{align}\label{eq:normalisationIX}
\left(\bar{H},\bar{\Sigma}_+,\bar{\Sigma}_-,\bar{N}_1,\bar{N}_2,\bar{N}_3,\bar{\Omega}\right)\equiv\\ \nonumber
\left(\frac{H}{D},\frac{\sigma_+}{D},\frac{\sigma_-}{D},\frac{n_1}{D},\frac{n_2}{D},\frac{n_3}{D},\frac{\rho}{3D^2}\right)
\end{align}
From \eqref{eq:defOfD} and \eqref{eq:normalisationIX}, we see that $\bar{H}$ is constrained by,
\begin{equation}\label{eq:defHbar}
\bar{H}^2 +\frac{1}{4}\left(\bar{N}_1 \bar{N}_2 \bar{N}_3\right)^{2/3} =1
\end{equation}
In correspondence to the Types I-VIII, we must define a new time variable $\bar{\tau}$,
\begin{equation}
\frac{dt}{d\bar{\tau}}=\frac{1}{D}
\end{equation}
The evolution equations then become
\begin{align}
D^{\star}&=-(1+\bar{q})\bar{H}D\\
\bar{\Sigma}_{\pm}^{\star} &= -(2-\bar{q})\bar{H}\bar{\Sigma}_{\pm}-\bar{S}_{\pm}\\
\bar{N}_1^{\star}&=\left(\bar{H}\bar{q}-4\bar{\Sigma}_+\right)\bar{N}_1\\
\bar{N}_2^{\star}&=\left(\bar{H}\bar{q}+2\bar{\Sigma}_++2\sqrt{3}\bar{\Sigma}_-\right)\bar{N}_2\\
\bar{N}_3^{\star}&=\left(\bar{H}\bar{q}+2\bar{\Sigma}_+-2\sqrt{3}\bar{\Sigma}_-\right)\bar{N}_3
\end{align}
Here the quantity $\bar{q}$ is defined by
\begin{equation}\label{eq:defOfqBIX}
\bar{q}= \frac{1}{2}(3\gamma -2)(1-\bar{V})+\frac{3}{2}(2-\gamma)\bar{\Sigma}^2
\end{equation}
The normalised Friedmann constraint is given by,
\begin{equation}\label{eq:TypeIXFriedConst}
\bar{\Sigma}^2 +\bar{\Omega} +\bar{V}=1
\end{equation}
where
\begin{equation*}
\bar{\Sigma}^2 = \bar{\Sigma}_+^2 + \bar{\Sigma}_-^2
\end{equation*}
and
\begin{align*}
\bar{V} = \frac{1}{12}\left[
\bar{N}_1^2 +\bar{N}_2 ^2 \bar{N}_3^2 -2\bar{N}_1\bar{N}_2-2\bar{N}_2\bar{N}_3-2\bar{N}_1\bar{N}_3\right.\\
\left. +3\left(\bar{N}_1\bar{N}_2\bar{N}_3\right)^{2/3}
\right]
\end{align*}
which implies that $\bar{V}\geq 0$. The corresponding evolution equation for the expansion normalised density parameter is given by,
\begin{equation}
\bar{\Omega}^{\star} = \left[-(3\gamma -2)\bar{V}+ 3(2-\gamma)\bar{\Sigma}^2\right]-\kappa \bar{H}\bar{\Sigma}^2
\end{equation}
Now using the Friedmann constraint $\bar{q}$ can also be written as,
\begin{equation}\label{eq:defofqbar}
\bar{q} = 2 \bar{\Sigma}^2 +\frac{1}{2}(3\gamma -2)\bar{\Omega}
\end{equation}
and $\bar{q} \geq 0$ being equal to $0$ when $\bar{\Sigma}^2 =\bar{\Omega}=0$. Using \eqref{eq:defHbar}, we find that $\bar{H}$ is bounded as $-1<\bar{H}<1$. The evolution equation for $\bar{H}$ is given as,
\begin{equation}
\bar{H}^{\star} = -(1-\bar{H}^2)\bar{q}
\end{equation}
Using the fact that $\bar{q} \geq 0$, and $\vert \bar{H}\vert <1$, we can assume that $ \bar{H}^{\star} <0$. This implies that $\lim _{\bar{\tau}\rightarrow -\infty}\bar{H} = 1$ and $\lim _{\bar{\tau}\rightarrow +\infty}\bar{H} =-1$. This implies that an initially expanding model will undergo a recollapse when $\bar{H}$ will become zero. After the recollapse point i.e.\ $\bar{\tau}\rightarrow \infty$, the condition
\begin{equation}
\kappa > 3\left(2-\gamma\right)
\end{equation}
along with the positivity of $\bar{\Sigma}^2$ and $\bar{V}$ as well as requiring that the Strong Energy Condition $(\rho +3p)>0$ ensure that $ \bar{\Omega}^{\star} \geq 0$. Thus $\lim _{\bar{\tau}\rightarrow +\infty}\bar{\Omega} =1$ which also implies that $\lim _{\bar{\tau}\rightarrow +\infty}\bar{\Sigma}^2=0$ and $\lim _{\bar{\tau}\rightarrow +\infty}\bar{V}=0$.
From \eqref{eq:defofqbar} then implies that $\lim _{\bar{\tau}\rightarrow +\infty}\bar{q} = (3\gamma -2)/2$. From the evolution equations for the $\bar{N}_i$ for $i=1,2,3$, there exists a parameter $\epsilon >0$ so that, $d\mathrm{ln}\bar{N}_a/d\bar{\tau}<-\epsilon$ and so $\lim _{\bar{\tau}\rightarrow +\infty}\bar{N}_a =0$. Thus we can arrive at a similar conclusion to types I-VIII, that the inclusion of a shear viscous term with the coefficient of viscosity obeying the condition \eqref{eq:condition} will act as an isotropisation mechanism to prevent the growth of anisotropies on approach to the expansion minima or `bounce'.
\section{Bianchi Class B}
For Class B models the parameter A defined in \eqref{eq:paramAB} is non-zero. Following \cite{hwain}, we define the shear variables $\tilde{\sigma}\equiv\frac{1}{2}\tilde{\sigma}_{ab}\tilde{\sigma}^{ab}$ where $\tilde{\sigma}_{ab}$ is the traceless part of the shear tensor $\sigma_{ab}$. The trace for Class A is zero but for Class B is given by $\sigma_+ \equiv \frac{1}{2}\sigma_a^a$. Similarly we can define $\tilde{n}\equiv \frac{1}{6}\tilde{n}_{ab}\tilde{n}^{ab}$ where $\tilde{n}_{ab}$ is the traceless part of $n_{ab}$ and the trace is given by $n_+ =\frac{1}{2}n^a_a$. The quantities $n_+$ and $\tilde{n}$ are related through the non-zero $A$ and the group parameter $h=\tilde{h}^{-1}$ (for some constant $\tilde{h}$) as,
\begin{equation}\label{eq:tildeN}
\tilde{n} = \frac{1}{3}\left(n_+^2 - \tilde{h}A^2\right)
\end{equation}
We proceed with the expansion normalisation as in the case of Types I-VIII in Class A,
\begin{equation}
\left(\Sigma_+,\tilde{\Sigma},N_+,\tilde{N},\Omega,\tilde{A}\right)=\left(\frac{\sigma_+}{H},\frac{\tilde{\sigma}}{H},\frac{n_+}{H},\frac{\tilde{n}}{H},\frac{\rho}{3H^2},\frac{A^2}{H^2}\right)
\end{equation}
Of course following \eqref{eq:tildeN}, we have,
\begin{equation}
\tilde{N} = \frac{1}{3}\left(N_+^2 - \tilde{h}\tilde{A}\right)
\end{equation}
Following \cite{Lidsey2005,hwain}, we use the fact that the evolution equation for the expansion normalised variable $\Omega$ for Bianchi Class B remains the same as \eqref{eq:EvModOmega} and the definition of $q$ in \eqref{eq:defOfq} also doesn't change. However now the variables $K$ and $\Sigma ^2$ defined previously in \eqref{eq:defK} and \eqref{eq:defsigma} are still positive definite, and are given by,
\begin{align}
K= \tilde{N}+\tilde{A}\\
\Sigma ^2 = \tilde{\Sigma} + \Sigma_+^2
\end{align}
In correspondence with the condition drawn on $\Omega'\leq0$ in Types I-VIII, we have $\lim_{\tau \rightarrow -\infty}K=\lim_{\tau \rightarrow -\infty}\Sigma ^2 =0$ and by \eqref{eq:Friedmann_constraint}, $\lim_{\tau \rightarrow -\infty}\Omega =1$.
This further implies that $\lim_{\tau \rightarrow -\infty}\Sigma =0$ and $\lim_{\tau \rightarrow -\infty}K =0$, that $\lim_{\tau \rightarrow -\infty}\tilde{\Sigma} =0$ and $\lim_{\tau \rightarrow -\infty}\Sigma_+ =0$. And that $\lim_{\tau \rightarrow -\infty}\tilde{N} =0$ and $\lim_{\tau \rightarrow -\infty}\tilde{A} =0$. Thus the spatial curvature variables given by,
\begin{equation}
\mathcal{S}_+ = 2\tilde{N},\;\;\;\;\;\;\mathcal{S}^{ab}\mathcal{S}_{ab} = 24\left(\tilde{A}+\tilde{N}^2\right)\tilde{N}
\end{equation}
vanish at the bounce. The conclusions drawn about the stability of the isotropic FL point in an initially contracting Universe with an equation of state $p=\left(\gamma(\rho)-1\right)\rho$ such that $\rho +3p>0$ drawn in Bianchi Types I-VIII remain unchanged.
\newline
\section{Conclusion}
The problem of isotropisation in contracting Universes has been a key challenge in the formulation of bouncing alternatives to inflation. In \cite{me4}, we have shown that the inclusion of shear viscous stresses leads to isotropisation of a Bianchi Type IX Universe. In this work we have used the orthonormal frame formalism used by \cite{Belinskii1972,WEllis} and subsequent expansion normalisation to write the Einstein Field Equations as a phase plane system. This has allowed us to derive global stability theorems for the Bianchi Class I-VIII and separately Type IX as well as the Bianchi Class B, which show that the isotropic, spatially homoegenous solution becomes the attractor in an initially contracting Universe on approach to the expansion minima.
\paragraph*{}
Our results hold for an equation of state following a form given by $p=\left(\gamma(\rho)-1\right)\rho$ and hence the case of non-linear equation of state is encapsulated in our proof. In the case of positive spatial curvature, with a Universe sourced with the non-linear equation of state fluid \cite{bruni1,bruni2,me4}, the expansion minima results in a `bounce' and a subsequent re-expansion. In this work we have demonstrated only the isotropisation in the contracting phase of the most general spatially homogeneous cosmology and have not modelled the physics of a bounce.
\paragraph*{}
The inclusion of shear viscosity in the contracting phase thus appears to be an effective isotropisation mechanism for all types of spatially homogeneous cosmologies - without having to use an ekpyrotic fluid $p\gg\rho$ as is done in \cite{Lidsey2005} . As the form of shear viscosity used here and in \cite{me4} is dependent on the Hubble parameter $H$ through $\pi_{\alpha\beta}=-\kappa H\sigma_{\alpha\beta}$, this viscous effect will vanish at the bounce point $H=0$. This should have an effect on the damping of perturbative anisotropies in the contracting phase but should not produce significant gravitational waves at the bounce.
\paragraph*{}
The effect of fully non-linear inhomogeneities is yet to be studied in these models. There have been some numerical studies of its evolution in the contracting phase with an ekpyrotic phase \cite{will1,will2}. At the linear level, the scalar perturbations are dependent on the shear tensor through the electric part of the Weyl Tensor and hence should remain small in the case of effective isotropisation. Furthermore, if we follow the assumptions made in \cite{bkl,belinski_henneaux_2017}, then on approach to a singularity, each point of an inhomogeneous Universe behaves as a Bianchi cosmology and hence these results should hold. However full numerical GR should be used to test whether this is actually the case.
|
1,108,101,564,560 | arxiv | \section{Introduction}
Relation extraction (RE) is a task of identifying typed relations between known entity mentions in a sentence.
Most existing RE models treat each relation in a sentence individually~\cite{miwa2016end,nguyen2015perspective}.
However, a sentence typically contains multiple relations between entity mentions.
RE models need to consider these pairs simultaneously to model the dependencies among them.
The relation between a pair of interest (namely ``target" pair) can be influenced by other pairs in the same sentence.
The example illustrated in Figure~\ref{fig:ex} explains this phenomenon.
The relation between the pair of interest \textit{Toefting} and \textit{capital}, can be extracted directly from the target entities or indirectly by incorporating information from other related pairs in the sentence.
The person entity (PER) \textit{Toefting} is directly related with \textit{teammates} through the preposition \textit{with}.
Similarly, \textit{teammates} is directly related with the geopolitical entity (GPE) \textit{capital} through the preposition \textit{in}.
\textit{Toefting} and \textit{capital} can be directly related through \textit{in} or indirectly related through \textit{teammates}.
Substantially, the path from \textit{Toefting} to \textit{teammates} to \textit{capital} can additionally support the relation between \textit{Toefting} and \textit{capital}.
\begin{figure}[t!]
\centering
\includegraphics[scale=0.53]{example2}
\caption{Relation examples from ACE (Automatic Content Extraction) 2005 dataset~\cite{doddington2004automatic}.}
\label{fig:ex}
\end{figure}
Multiple relations in a sentence between entity mentions can be represented as a graph.
Neural graph-based models have shown significant improvement in modelling graphs
over traditional feature-based approaches in several tasks.
They are most commonly applied on knowledge graphs (KG) for knowledge graph completion~\cite{jiang2017attentive} and the creation of knowledge graph embeddings~\cite{wang2017knowledge,shi2017proje}.
These models rely on paths between existing relations in order to infer new associations between entities in KGs.
However, for relation extraction from a sentence, related pairs are not predefined and consequently all entity pairs need to be considered to extract relations.
In addition, state-of-the-art RE models sometimes depend on external syntactic tools to build the shortest dependency path (SDP) between two entities in a sentence~\cite{xu2015neg,miwa2016end}.
This dependence on external tools leads to domain dependent models.
In this study, we propose a neural relation extraction model based on an entity graph, where entity mentions constitute the nodes and directed edges correspond to ordered pairs of entity mentions.
The overview of the model is shown in Figure~\ref{fig:model}.
We initialize the representation of an edge (an ordered pair of entity mentions) from the representations of the entity mentions and their context.
The context representation is achieved by employing an attention mechanism on context words.
We then use an iterative process to aggregate up-to $l$-length walk representations between two entities into a single representation, which corresponds to the final representation of the edge.
The contributions of our model can be summarized as follows:
\begin{itemize}[nolistsep]
\item We propose a graph walk based neural model that considers multiple entity pairs in relation extraction from a sentence.
\item We propose an iterative algorithm to form a single representation for up-to $l$-length walks between the entities of a pair.
\item We show that our model performs comparably to the state-of-the-art without the use of external syntactic tools.
\end{itemize}
\section{Proposed Walk-based Model}
The goal of the RE task is given a sentence, entity mentions and their semantic types, to extract and classify all related entity pairs (target pairs) in the sentence.
The proposed model consists of five stacked layers: embedding layer, BLSTM Layer, edge representation layer, walk aggregation layer and finally a classification layer.
As shown in Figure~\ref{fig:model}, the model receives word representations and produces simultaneously a representation for each pair in the sentence.
These representations combine the target pair, its context words, their relative positions to the pair entities and walks between them.
During classification they are used to predict the relation type of each pair.
\begin{figure}[t!]
\includegraphics[width=\linewidth]{graph_model.pdf}
\caption{Overview of the walk-based model.}
\label{fig:model}
\end{figure}
\subsection{Embedding Layer}
The embedding layer involves the creation of $n_w$, $n_t$, $n_p$-dimensional vectors which are assigned to words, semantic entity types and relative positions to the target pairs.
We map all words and semantic types into real-valued vectors $\mathbf{w}$ and $\mathbf{t}$ respectively.
Relative positions to target entities are created based on the position of words in the sentence.
In the example of Figure~\ref{fig:ex}, the relative position of \textit{teammates} to \textit{capital} is $-3$ and the relative position of \textit{teammates} to \textit{Toefting} is $+16$.
We embed real-valued vectors $\mathbf{p}$ to these positions.
\subsection{Bidirectional LSTM Layer}
The word representations of each sentence are fed into a Bidirectional Long-short Term Memory (BLSTM) layer, which encodes the context representation for every word.
The BLSTM outputs new word-level representations $\mathbf{h}$~\cite{hochreiter1997long} that consider the sequence of words.
We avoid encoding target pair-dependent information in this BLSTM layer.
This has two advantages:
(i) the computational cost is reduced as this computation is repeated based on the number of sentences instead of the number of pairs,
(ii) we can share the sequence layer among the pairs of a sentence.
The second advantage is particularly important as it enables the model to indirectly learn hidden dependencies between the related pairs in the same sentence.
For each word $t$ in the sentence, we concatenate the two representations from left-to-right and right-to-left pass of the LSTM into a $n_e$-dimensional vector, $\mathbf{e}_t = {[ \overrightarrow{\mathbf{h}_t}; \overleftarrow{\mathbf{h}_t} ]}$.
\subsection{Edge Representation Layer}
The output word representations of the BLSTM are further divided into two parts: (i) target pair representations and (ii) target pair-specific context representations.
The context of a target pair can be expressed as all words in the sentence that are not part of the entity mentions.
We represent a related pair as described below.
A target pair contains two entities $e_i$ and $e_j$.
If an entity consists of $N$ words, we create its BLSTM representation as the average of the BLSTM representations of the corresponding words,
$\mathbf{e} = \frac{1}{|I|}\sum_{i \in I} {\mathbf{e}_i}$, where $I$ is a set with the word indices inside entity $e$.
We first create a representation for each pair entity and then we construct the representation for the context of the pair.
The representation of an entity $e_i$ is the concatenation of its BLSTM representation $\mathbf{e}_i$, the representation of its entity type $\mathbf{t}_i$ and the representation of its relative position to entity $e_j$, $\mathbf{p}_{ij}$.
Similarly, for entity $e_j$ we use its relative position to entity $e_i$, $\mathbf{p}_{ji}$.
Finally, the representations of the pair entities are as follows:
$\mathbf{v}_i = {[ \mathbf{e}_i ; \mathbf{t}_i ; \mathbf{p}_{ij} ]}$ and
$\mathbf{v}_j = {[ \mathbf{e}_j ; \mathbf{t}_j ; \mathbf{p}_{ji} ]}$.
The next step involves the construction of the representation of the context for this pair.
For each context word $w_z$ of the target pair $e_i$, $e_j$, we concatenate its BLSTM representation $\mathbf{e}_z$, its semantic type representation $\mathbf{t}_z$ and two relative position representations: to target entity $e_i$, $\mathbf{p}_{zi}$ and to target entity $e_j$, $\mathbf{p}_{zj}$.
The final representation for a context word $w_z$ of a target pair is,
$\mathbf{v}_{ijz} = [ \mathbf{e}_z ; \mathbf{t}_z ; \mathbf{p}_{zi} ; \mathbf{p}_{zj} ]$.
For a sentence, the context representations for all entity pairs can be expressed as a three-dimensional matrix $\mathbf{C}$, where rows and columns correspond to entities and the depth corresponds to the context words.
The context words representations of each target pair are then compiled into a single representation with an attention mechanism.
Following the method proposed in \citet{zhou2016attention}, we calculate weights for the context words of the target-pair and compute their weighted average,
\begin{equation}
\begin{aligned}
\mathbf{u} &= \mathbf{q}^{\top} \; \tanh(\mathbf{C}_{ij}) , \\
\mathbf{\alpha} &= \softmax(\mathbf{u}) , \\
\mathbf{c}_{ij} &= \mathbf{C}_{ij} \; \mathbf{\alpha}^{\top},
\end{aligned}
\label{eq:att}
\end{equation}
where $\mathbf{q} \in \mathbb{R}^{n_d}, n_d = n_e + n_t + 2n_p $ denotes a trainable attention vector,
$\alpha$ is the attended weights vector
and
$\mathbf{c}_{ij} \in \mathbb{R}^{n_d}$ is the context representation of the pair as resulted by the weighted average.
This attention mechanism is independent of the relation type. We leave relation-dependent attention as future work.
Finally, we concatenate the representations of the target entities and their context ($ \in \mathbb{R}^{n_m}$). We use a fully connected linear layer, $\mathbf{W}_s \in \mathbb{R}^{n_m \times n_s}$ with $n_s < n_m$ to reduce the dimensionality of the resulting vector.
This corresponds to the representation of an edge or a one-length walk between nodes $i$ and $j$:
$\mathbf{v}^{(1)}_{ij} = \mathbf{W}_s \; [ \mathbf{v}_{i} ; \mathbf{v}_{j} ; \mathbf{c}_{ij} ] \in \mathbb{R}^{n_s}$.
\subsection{Walk Aggregation Layer}
Our main aim is to support the relation between an entity pair by using chains of intermediate relations between the pair entities.
Thus, the goal of this layer is to generate a single representation for a finite number of different lengths walks between two target entities.
To achieve this, we represent a sentence as a directed graph, where the entities constitute the graph nodes and edges correspond to the representation of the relation between the two nodes.
The representation of one-length walk between a target pair $\mathbf{v}^{(1)}_{ij}$, serves as a building block in order to create and aggregate representations for one-to-$l$-length walks between the pair.
The walk-based algorithm can be seen as a two-step process: walk construction and walk aggregation.
During the first step, two consecutive edges in the graph are combined using a modified bilinear transformation,
\begin{equation}
f(\mathbf{v}^{(\lambda)}_{ik}, \mathbf{v}^{(\lambda)}_{kj}) = \sigma \left( \mathbf{v}^{(\lambda)}_{ik} \; \odot \; ( \mathbf{W}_b \; \mathbf{v}^{(\lambda)}_{kj} ) \right),
\label{eq:w_construct}
\end{equation}
where $\mathbf{v}^{(\lambda)}_{ij} \in \mathbb{R}^{n_b}$ corresponds to walks representation of lengths one-to-$\lambda$ between entities $e_i$ and $e_j$,
$\odot$ represents element-wise multiplication,
$\sigma$ is the sigmoid non-linear function and
$\mathbf{W}_b \in \mathbb{R}^{n_b \times n_b}$ is a trainable weight matrix.
This equation results in walks of lengths two-to-2$\lambda$.
In the walk aggregation step, we linearly combine the initial walks (length one-to-$\lambda$) and the extended walks (length two-to-$2\lambda$),
\begin{equation}
\mathbf{v}_{ij}^{(2\lambda)} = \beta \mathbf{v}_{ij}^{(\lambda)} + (1 - \beta) \sum_{k \neq i,j} f(\mathbf{v}_{ik}^{(\lambda)}, \mathbf{v}_{kj}^{(\lambda)}),
\label{eq:w_aggregate}
\end{equation}
where
$\beta$ is a weight that indicates the importance of the shorter walks.
Overall, we create a representation for walks of length one-to-two using Equation (\ref{eq:w_aggregate}) and $\lambda=1$. We then create a representation for walks of length one-to-four by re-applying the equation with $\lambda=2$.
We repeat this process until the desired maximum walk length is reached, which is equivalent to $2\lambda = l$.
\subsection{Classification Layer}
For the final layer of the network, we pass the resulted pair representation into a fully connected layer with a softmax function,
\begin{equation}
\mathbf{y} = \softmax(\mathbf{W}_r \mathbf{v}_{ij}^{(l)} + \mathbf{b}_r),
\end{equation}
where $\mathbf{W}_r \in \mathbb{R}^{n_b \times n_r}$ is the weight matrix,
$n_r$ is the total number of relation types
and $b_r$ is the bias vector.
We use in total $2r + 1$ classes in order to consider both directions for every pair, i.e., left-to-right and right-to-left. The first argument appears first in a sentence in a left-to-right relation while the second argument appears first in a right-to-left relation. The additional class corresponds to non-related pairs, namely ``no relation" class.
We choose the most confident prediction for each direction and choose the positive and most confident prediction when the predictions contradict each other.
\section{Experiments}
\subsection{Dataset}
We evaluate the performance of our model on ACE 2005\footnote{\url{https://catalog.ldc.upenn.edu/ldc2006t06}} for the task of relation extraction.
ACE 2005 includes $7$ entity types and $6$ relation types between named entities.
We follow the preprocessing described in \citet{miwa2016end}.
\subsection{Experimental Settings}
We implemented our model using the Chainer library~\citep{tokui2015chainer}.\footnote{\url{https://chainer.org/}}
The model was trained with Adam optimizer~\citep{kingma2014adam}.
We initialized the word representations with existing pre-trained embeddings with dimensionality of $200$.\footnote{\url{https://github.com/tticoin/LSTM-ER}}
Our model did not use any external tools except these embeddings.
The forget bias of the LSTM layer was initialized with a value equal to one following the work of \citet{jozefowicz2015empirical}.
We use a batchsize of $10$ sentences and fix the pair representation dimensionality to $100$. We use gradient clipping, dropout on the embedding and output layers and L2 regularization without regularizing the biases, to avoid overfitting. We also incorporate early stopping with patience equal to five, to chose the number of training epochs and parameter averaging.
We tune the model hyper-parameters on the respective development set using the RoBO Toolkit~\cite{klein-bayesopt17}.
Please refer to the supplementary material for the values.
We extract all possible pairs in a sentence based on the number of entities it contains.
If a pair is not found in the corpus, it is assigned the ``no relation'' class.
We report the micro precision, recall and F1 score following~\citet{miwa2016end} and~\citet{nguyen2015perspective}.
\section{Results}
Table~\ref{tab:res} illustrates the performance of our proposed model in comparison with \textit{SPTree} system~\citet{miwa2016end} on ACE 2005.
We use the same data split with \textit{SPTree} to compare with their model. We retrained their model with gold entities in order to compare the performances on the relation extraction task.
The \textit{Baseline} corresponds to a model that classifies relations by using only the representations of entities in a target pair.
\begin{table}[t!]
\centering
\renewcommand*{\arraystretch}{1.1}
\begin{tabular}{|lccc|}
\hline
Model & P & R & F1 (\%) \\
\hline \hline
SPTree & 70.1 & 61.2 & 65.3 \\
\hline
Baseline & 72.5 & 53.3 & 61.4\rlap{$^*$} \\
No walks $l$ = 1 & 71.9 & 55.6 & 62.7 \\
\ \ + Walks $l$ = 2 & 69.9 & 58.4 & 63.6\rlap{$^\diamond$} \\
\ \ + Walks $l$ = 4 & 69.7 & 59.5 & 64.2\rlap{$^\diamond$} \\
\ \ + Walks $l$ = 8 & 71.5 & 55.3 & 62.4 \\
\hline
\end{tabular}
\caption{Relation extraction performance on ACE 2005 test dataset. * denotes significance at p $<$ 0.05 compared to SPTree, $\diamond$ denotes significance at p $<$ 0.05 compared to the Baseline. }
\label{tab:res}
\end{table}
As it can be observed from the table, the \textit{Baseline} model achieves the lowest F1 score between the proposed models.
By incorporating attention we can further improve the performance by 1.3 percent point (pp).
The addition of $2$-length walks further improves performance (0.9 pp).
The best results among the proposed models are achieved for maximum $4$-length walks.
By using up-to $8$-length walks the performance drops almost by 2 pp.
We also compared our performance with \citet{nguyen2015perspective} (\textit{CNN}) using their data split.\footnote{The authors kindly provided us with the data split.} For the comparison, we applied our best performing model ($l$ = 4).\footnote{We kept the same parameters when we apply our model to the this data split. We did not remove any negative examples unlike the \textit{CNN} model.}
The obtained performance is 65.8 / 58.4 / 61.9 in terms of P / R / F1 (\%) respectively.
In comparison with the performance of the \textit{CNN} model, 71.5 / 53.9 / 61.3, we observe a large improvement in recall which results in 0.6 pp F1 increase.
We performed the Approximate Randomization test~\cite{noreen1989computer} on the results. The best walks model has no statistically significant difference with the state-of-the-art \textit{SPTree} model as in Table~\ref{tab:res}. This indicates that the proposed model can achieve comparable performance without any external syntactic tools.
\begin{table}[t!]
\centering
\renewcommand*{\arraystretch}{1.1}
\begin{tabular}{|lcccc|}
\hline
\# Entities & $l = 1$ & $l$ = 2 & $l$ = 4 & $l$ = 8 \\
\hline \hline
$2$ & 71.2 & 69.8 & 72.9 & 71.0 \\
$3$ & 70.1 & 67.5 & 67.8 & 63.5\rlap{$^*$} \\
$[4, 6)$ & 56.5 & 59.7 & 59.3 & 59.9 \\
$[6, 12)$ & 59.2 & 64.2\rlap{$^*$} & 62.2 & 60.4 \\
$[12, 23)$ & 54.7 & 59.3 & 62.3\rlap{$^*$} & 55.0 \\
\hline
\end{tabular}
\caption{Relation extraction performance (F1 \%) on ACE 2005 development set for different number of entities. * denotes significance at p $<$ 0.05 compared to $l = 1$.}
\label{tab:anal}
\end{table}
Finally, we show the performance of the proposed model as a function of the number of entities in a sentence. Results in Table~\ref{tab:anal} reveal that for multi-pair sentences the model performs significantly better compared to the no-walks models, proving the effectiveness of the method. Additionally, it is observed that for more entity pairs, longer walks seem to be required. However, very long walks result to reduced performance ($l$ = 8).
\section{Related Work}
Traditionally, relation extraction approaches have incorporated a large variety of hand-crafted features to represent related entity pairs \cite{hermann2013role,miwa2014table,nguyen2014employing,gormley2015embedding}.
Recent models instead employ neural network architectures and achieve state-of-the-art results without heavy feature engineering.
Neural network techniques can be categorized into recurrent neural networks (RNNs) and convolutional neural networks (CNNs). The former is able to encode linguistic and syntactic properties of long word sequences, making them preferable for sequence-related tasks, e.g. natural language generation~\cite{goyal2016natural}, machine translation~\cite{sutskever2014sequence}.
State-of-the-art systems have proved to achieve good performance on relation extraction using RNNs \cite{cai2016bidirectional,miwa2016end,xu2016improved,liu2015dependency}. Nevertheless, most approaches do not take into consideration the dependencies between relations in a single sentence \cite{santos2015ranking,nguyen2015perspective} and treat each pair separately.
Current graph-based models are applied on knowledge graphs for distantly supervised relation extraction~\cite{zeng2017paths}. Graphs are defined on semantic types in their method, whereas we built entity-based graphs in sentences.
Other approaches also treat multiple relations in a sentence~\cite{gupta2016table,miwa2014table,li2014incremental}, but they fail to model long walks between entity mentions.
\section{Conclusions}
We proposed a novel neural network model for simultaneous sentence-level extraction of related pairs.
Our model exploits target and context pair-specific representations and creates pair representations that encode up-to $l$-length walks between the entities of the pair.
We compared our model with the state-of-the-art models and observed comparable performance on the ACE2005 dataset without any external syntactic tools.
The characteristics of the proposed approach are summarized in three factors: the encoding of dependencies between relations, the ability to represent multiple walks in the form of vectors and the independence from external tools.
Future work will aim at the construction of an end-to-end relation extraction system as well as application to different types of datasets.
\section*{Acknowledgments}
This research has been carried out with funding from AIRC/AIST, the James Elson Studentship Award, BBSRC grant BB/P025684/1 and MRC MR/N00583X/1. Results were obtained from a project commissioned by the New Energy and Industrial Technology Development Organization (NEDO). We would finally like to thank the anonymous reviewers for their helpful comments.
|
1,108,101,564,561 | arxiv | \section{Introduction}
In this paper we investigate further the deformed T-dual (DTD) supercoset sigma models introduced in \cite{Borsato:2016pas}, and we find results that are of interest also when considering the undeformed case, i.e. when applying just non-abelian T-duality (NATD).
The construction of DTD models is equivalent to applying NATD on a centrally extended subalgebra as first suggested in \cite{Hoare:2016wsk}.\footnote{The first hint of the relation of YB models to NATD appeared in~\cite{Orlando:2016qqu} for the case of Jordanian deformations.} The models are constructed by picking a subalgebra of the (super)isometry algebra $\tilde{\alg{g}}\subset \alg g$---the canonical example is the $AdS_5\times S^5$ superstring where $\alg g=\alg{psu}(2,2|4)$---and a 2-cocycle, i.e. an anti-symmetric linear map $\omega:\,\tilde{\mathfrak g}\otimes\tilde{\mathfrak g}\rightarrow\mathbbm{R}$ satisfying
\begin{equation}
\omega(X,[Y,Z])
+\omega(Z,[X,Y])
+\omega(Y,[Z,X])
=0\,,\qquad\forall X,Y,Z\in\tilde{\mathfrak g}\,.
\label{eq:2-cocycle}
\end{equation}
Together with an element of the corresponding group $\tilde g\in\tilde G$, the 2-cocycle defines a 2-form $B=\omega(\tilde g^{-1}d\tilde g,\tilde g^{-1}d\tilde g)$ which is closed, i.e. $dB=0$, thanks to the 2-cocycle condition. The idea behind the construction is to add this topological term to the supercoset sigma model Lagrangian and then perform NATD on $\tilde G$. If $\zeta B$ is added to the Lagrangian, with $\zeta$ a parameter, the resulting model can be thought of as a deformation of the non-abelian T-dual of the original model with deformation parameter $\zeta$. The classical integrability of the original sigma model is preserved by the deformation, since both adding a topological term and performing NATD preserve integrability. We refer to \cite{Borsato:2016pas} for more details on how this procedure relates to the construction of \cite{Hoare:2016wsk}.
Let us remark that DTD models may be constructed starting from a generic $\sigma$-model, for example the principal chiral model as in~\cite{Borsato:2016pas}, and the starting model does not have to be (classically) integrable. In this paper we will only consider the supercoset case.
It was proven in \cite{Borsato:2016pas} that the so-called Yang-Baxter (YB) sigma models \cite{Klimcik:2002zj,Klimcik:2008eq,Delduc:2013qra,Kawaguchi:2014qwa}, defined by an R-matrix solving the classical Yang-Baxter equation (CYBE), are equivalent to DTD models with invertible $\omega$. This relation was first conjectured and checked for many examples---in the language of T-duality on a centrally extended subalgebra---in \cite{Hoare:2016wsk}. See also \cite{Hoare:2016wca} for a more detailed discussion of some of the examples. In \cite{Borsato:2016pas} we used the fact that when $\omega$ is invertible its inverse $R=\omega^{-1}$ solves the CYBE, and therefore defines a corresponding YB model; by means of a field redefinition and relating the deformation parameters as $\eta=\zeta^{-1}$ we could prove the equivalence of the two sigma model actions~\cite{Borsato:2016pas}.
Note that simply by setting the deformation parameter to zero, DTD models include all non-abelian and abelian T-duals of the original supercoset model, including fermionic T-dualities. Therefore all the statements we prove for DTD models apply also to (non-abelian) T-duals of supercoset models. They are also easily seen to describe all so-called TsT-transformations of the underlying supercoset model. In fact we will argue here that the class of DTD models is closed under the action of NATD, as well as certain deformations, meaning that applying these operations yields a new DTD model. They therefore represent a very broad class of integrable string sigma models.
\vspace{12pt}
It was shown in \cite{Borsato:2016pas} that these models are invariant under kappa symmetry, which is needed to interpret them as Green-Schwarz superstrings. From the results of \cite{Wulff:2016tju} it follows that their target spaces must solve the generalised supergravity equations of \cite{Arutyunov:2015mqj,Wulff:2016tju} that ensure the one-loop scale invariance of the string sigma model. To have a fully consistent superstring, however, we must require the stronger condition of Weyl invariance, which implies that the target space should be a solution of the more stringent standard supergravity equations. Here we show that Weyl invariance of the DTD model is equivalent to the Lie algebra $\tilde{\mathfrak g}$ being unimodular, i.e. its structure constants should satisfy $f_{ij}^j=0$. In fact, this condition is precisely the one found in \cite{Alvarez:1994np,Elitzur:1994ri} when analysing the Weyl invariance of bosonic sigma models under NATD by path integral considerations. The presence of $\omega$ and the deformation does not modify the supergravity condition.
When $\omega$ is invertible the condition is also equivalent to unimodularity of the R-matrix $R=\omega^{-1}$, as defined in \cite{Borsato:2016ose}, which was shown there to be the condition for Weyl invariance of YB models. The fact that these conditions are the same was in fact an important hint that the latter should have an interpretation involving NATD \cite{Hoare:2016wsk}.
Here we give the detailed proof of kappa symmetry for DTD models and extract the target space superfields from components of the torsion as was done for $\eta$ (i.e. YB) and $\lambda$ models in \cite{Borsato:2016ose}. In particular, the RR fields and dilaton are difficult to extract by other means but we find that they are given by the simple expressions
\begin{equation}
e^{-2\phi}=\mathrm{sdet}'\widetilde{\mathcal{O}}\,,\qquad
\mathcal S^{\alpha1\beta2}=
-8i[\mathrm{Ad}_h(1+4\mathrm{Ad}_f^{-1}\widetilde{\mathcal{O}}^{-T}\mathrm{Ad}_f)]^{\alpha1}{}_{\gamma1}\widehat{\mathcal K}^{\gamma1\beta2}\,,
\end{equation}
with $\widetilde{\mathcal{O}}$ defined in (\ref{eq:def-op}) and $\mathcal S$ defined in (\ref{eq:calS})---for definitions of the remaining quantities see sections \ref{sec:DTD} and \ref{sec:target}. A by-product of these expressions is a formula for the transformation of RR fields under NATD for the case of supercosets. As we show in section \ref{sec:target} it agrees, for bosonic T-dualities, with the formula conjectured in \cite{Sfetsos:2010uq}, see also~\cite{Lozano:2011kb}, but our formula is valid also when doing fermionic T-dualities.
An advantage of the formulation of DTD models is that many statements about the sigma model boil down to simple algebraic statements about the Lie algebra $\tilde{\mathfrak g}$. One example is the Weyl invariance condition already mentioned, while another concerns their transformation under NATD---possibly including additional deformation. The advantages are clear also when discussing the isometries of these models. We show that they fall into two classes; in fact, besides the standard ones, i.e. the unbroken part of the $G$ isometries, there are also certain (abelian) shift isometries. We prove that T-dualising on either type of isometry we get back a DTD model; in particular, T-dualising on the first type of isometries is equivalent to the simple operation of enlarging $\tilde{\mathfrak g}$ by the corresponding generators, while T-dualising on the shift isometries removes generators from $\tilde{\mathfrak g}$. The latter operation can be used to prove, in this context, that solutions of the generalised supergravity equations are (formally) T-dual to solutions of the standard supergravity equations \cite{Arutyunov:2015mqj}. For more general NATD, where one applies T-duality on both types of isometries at the same time, we propose that the resulting model is still obtained in a similar way, namely simply by adding to $\tilde{\mathfrak g}$ the isometry generators that lie outside of it and removing from it the generators that are inside. We show that this conjecture is indeed consistent, i.e. the resulting model is a well-defined DTD model, which turns out to be quite non-trivial. As already mentioned this suggests that the class of DTD models is closed under (bosonic and fermionic) NATD, including also the deformations considered here.
It was suggested in \cite{Borsato:2016pas} that it might be possible to think of all DTD models as non-abelian T-duals of YB models. Here we show that this is in fact not true by providing an example of a DTD model which cannot be obtained from a YB model by NATD.
The outline of the paper is as follows. In section \ref{sec:DTD} we introduce the DTD models based on supercosets, discuss their gauge invariances and the equivalence to YB models when $\omega$ is invertible. Section \ref{sec:symm} describes the two classes of global symmetries, or isometries, of these models. We also address the question of what happens if one performs NATD and deformation of a DTD model and argue that this gives a new DTD model, proving this in simpler cases. Models which cannot be obtained by NATD of YB models are also discussed. In section \ref{sec:kappa} we demonstrate the kappa symmetry of DTD models and write the DTD model as a Green-Schwarz superstring. Given these results it is then straightforward to derive the target space fields of the DTD model from components of the superspace torsion, which we do in section \ref{sec:target}. This includes a derivation of the Weyl-invariance condition for these models. In section \ref{sec:ex} we work out the supergravity background for two examples of DTD models. The first is equivalent to a well known TsT-background but is useful to demonstrate the procedure. The second example is one of the new examples which cannot be obtained from a YB model by NATD. We finish with some conclusions and open problems. Three appendices contain some useful algebraic identities, a derivation of the DTD model action and a proof of integrability.
\section{The Deformed T-dual models}\label{sec:DTD}
As described in the introduction the deformed T-dual (DTD) models are constructed as follows. We start with a supercoset sigma model, e.g. the $AdS_5\times S^5$ superstring \cite{Metsaev:1998it} or one of the other examples in \cite{Zarembo:2010sg,Wulff:2014kja}. We single out a subalgebra $\tilde{\mathfrak g}\subset\mathfrak g$ of the ($\mathbbm Z_4$-graded) superisometry algebra and write the group element as $g=\tilde gf$ with $\tilde g\in\tilde G$ and $f\in G$. This parametrization is of course redundant and introduces a corresponding $\tilde G$ gauge symmetry $\tilde g\rightarrow\tilde g\tilde h^{-1}$ and $f\rightarrow \tilde hf$ on which we will comment below. The second ingredient, which is responsible for the deformation, is a Lie algebra 2-cocycle $\omega$ on $\tilde{\mathfrak g}$ satisfying (\ref{eq:2-cocycle}). We add to the original supercoset sigma model action the term
\begin{equation}
S_\omega=\tfrac{T}{4}\int_\Sigma\,\zeta\omega(\tilde g^{-1}d\tilde g,\tilde g^{-1}d\tilde g)\,,
\end{equation}
where $\zeta$ is a parameter introduced to keep track of the deformation---if there exist many 2-cocycles we could introduce a parameter for each.\footnote{If $\omega$ has mixed Grassmann even-odd components the corresponding deformation parameter $\zeta$ would be fermionic. Since the interpretation of such a fermionic deformation is not so clear we will generally assume that $\omega$ has only even-even and odd-odd components and that $\zeta$ is real.} As explained already, this is equivalent to adding a B-field to the action, which is closed by virtue of the 2-cocycle condition. This term is therefore topological and has no effect on local properties of the theory---issues with boundary conditions are more subtle and will not be considered here.
The final step is to perform NATD on $\tilde{\mathfrak g}$. This is done in the usual way by gauging the global $\tilde{\mathfrak g}$ symmetry and integrating out the gauge field. This procedure guarantees that properties like integrability are preserved, see appendix \ref{sec:integr} for an explicit proof. However, since T-duality is a non-local transformation of the fields of the sigma model, $\omega$ will now affect \emph{local} properties of the deformed model.
If $\omega$ is a coboundary, meaning that $\omega(X,Y)=f([X,Y])$ for some function $f:\,\tilde{\mathfrak g}\rightarrow\mathbbm{R}$, the B-field is exact; this is equivalent to no deformation at all since $B$ is pure gauge---alternatively a field redefinition can remove the $\zeta$ dependent contributions in the deformed model. Therefore non-trivial deformations are classified by the second (Lie algebra) cohomology group $H^2(\tilde{\mathfrak g})$. The same group also classifies non-trivial central extensions of $\tilde{\mathfrak g}$, consistent with the interpretation of these models as arising from NATD on a centrally extended subalgebra of the isometry algebra \cite{Hoare:2016wsk}.
Performing the above procedure one obtains the DTD supercoset model action
\begin{equation}
S=-\tfrac{T}{2}\int d^2\sigma\,\tfrac{\gamma^{ij}-\epsilon^{ij}}{2}\mathrm{Str}\big(J_i\hat d_fJ_j+(\partial_i\nu-\hat d_f^TJ_i)\widetilde{\mathcal{O}}^{-1}(\partial_j\nu+\hat d_fJ_j)\big)\,,\qquad\gamma^{ij}=\sqrt{-h}h^{ij}\,,
\label{eq:S-DTD}
\end{equation}
and we refer to appendix~\ref{app:action} for the details of its derivation.
Here $J=dff^{-1}$ encodes the degrees of freedom in $f$, while $\nu\in\tilde{\mathfrak g}^*$ denotes the dualised degrees of freedom coming from $\tilde g$. We have further defined
\begin{equation}
\hat d_f=\mathrm{Ad}_f\hat d\mathrm{Ad}_f^{-1}\,,\qquad\hat d=P^{(1)}+2P^{(2)}-P^{(3)}\,,\quad\hat d^T=-P^{(1)}+2P^{(2)}+P^{(3)}\,,
\label{eq:dhat}
\end{equation}
where $P^{(i)}$ project onto the corresponding $\mathbbm{Z}_4$-graded component of $\mathfrak g=\sum_{i=0}^3\mathfrak g^{(i)}$ and $\widetilde{\mathcal{O}}^{-1}$ is the inverse\footnote{Notice that $\widetilde{\mathcal{O}}\op^{-1}=\tilde P^T$ and $\widetilde{\mathcal{O}}^{-1}\widetilde{\mathcal{O}}=\tilde P$ rather than $1$.} of the linear operator $\widetilde{\mathcal{O}}:\tilde{\alg{g}}\to \tilde{\alg{g}}^*$
\begin{equation}\label{eq:def-op}
\widetilde{\mathcal{O}}=\tilde P^T(\hat d_f-\mathrm{ad}_\nu-\zeta\omega)\tilde P\,.
\end{equation}
Given a basis $\{T_i\}$ of $\tilde{\mathfrak g}$ and using the fact that $\mathfrak g$ has a non-degenerate metric given by the supertrace, we define the Lie algebra $\tilde{\mathfrak g}^*\subset\mathfrak g$ dual to $\tilde{\mathfrak g}$ by taking as dual basis $\{T^i\}$, where $\mathrm{Str}(T^jT_i)=\delta^j_i$. Then we have $\tilde P$ and $\tilde P^T$ which are projectors onto $\tilde{\mathfrak g}$ and $\tilde{\mathfrak g}^*$ respectively. At the same time we are thinking of the 2-cocycle $\omega$ as a map $\omega:\,\tilde{\mathfrak g}\rightarrow\tilde{\mathfrak g}^*$ so that the cocycle condition takes the form
\begin{equation}
\omega[x,y]=\tilde P^T\left([\omega x,y]+[x,\omega y]\right)\,,\qquad\forall x,y\in\tilde{\mathfrak g}\,.
\label{eq:cocycle2}
\end{equation}
Therefore, modulo the projector on the right-hand-side, $\omega$ acts as a derivation with respect to the Lie bracket, similarly to $\mathrm{ad}_\nu$ which is a derivation thanks to the Jacobi identity.
In general one needs to make sure that the inverse $\widetilde{\mathcal{O}}^{-1}$ exists in order to be able to define the model, and this puts some restrictions on the subalgebra $\tilde{\mathfrak g}$. By expanding in the parameter $\zeta$ we can think of the DTD model as a deformation of the non-abelian T-dual of the original model, since taking $\zeta=0$ reduces to ordinary NATD. Therefore, at least for a small deformation parameter the invertibility is guaranteed if one can apply NATD with respect to $\tilde{\mathfrak g}$.
There may also be cases in which NATD cannot be implemented but the operator is invertible for finite values of $\zeta$, i.e. the cocycle removes the 0-eigenvalues of $\widetilde{\mathcal{O}}$.
We now want to turn to the discussion of the gauge invariances of the action~\eqref{eq:S-DTD} of DTD models. Besides the fermionic kappa symmetry, which will be discussed separately in section~\ref{sec:kappa}, the action has two types of gauge invariances:
\begin{itemize}
\item[1.] \emph{Local Lorentz invariance}:
\begin{equation}
f\rightarrow fh\,,\qquad h\in H=G^{(0)}\,.
\label{eq:LocalLorentz}
\end{equation}
\item[2.] \emph{Local $\tilde{G}$ invariance}:
\begin{equation}
f\rightarrow\tilde hf\,,\quad\nu\rightarrow\tilde P^T\left(\mathrm{Ad}_{\tilde h}\nu+\zeta\frac{1-e^{\mathrm{ad}_x}}{\mathrm{ad}_x}\omega x\right)\,,\qquad \tilde h=e^x\in\tilde G\subset G\,.
\label{eq:LocalGtilde}
\end{equation}
\end{itemize}
The former is obvious and, as in the case of supercosets, it boils down to the fact that $P^{(0)}$ is missing in $\hat d$.
As mentioned at the beginning of this section, the latter comes about from the decomposition of the original group element as $g=\tilde gf$ where multiplication of $\tilde g$ from the right by an element of $\tilde G$ can be compensated for by multiplying $f$ on the left by the inverse group element. To verify that the action is indeed invariant under the second type of symmetry we use the identities~\eqref{eq:transfo-op} and~\eqref{eq:transfo-dnu} that say how the transformations of $\widetilde{\mathcal{O}}$ and $d\nu$ can be rewritten. Then the difference of the actions after and before the transformation~\eqref{eq:LocalGtilde} is proportional to
\begin{equation}
\int d^2\sigma\epsilon^{ij}\mathrm{Str}\Big(
2\partial_i\nu\tilde h^{-1}\partial_j\tilde h
+\tilde h^{-1}\partial_i\tilde h(\mathrm{ad}_\nu+\zeta\omega)(\tilde h^{-1}\partial_j\tilde h)
\Big)\,.
\end{equation}
The terms involving $\nu$ combine to a total derivative, and the one with $\omega$ is closed as already remarked, meaning that it is also a total derivative at least locally. This establishes the invariance of the action under the local transformation (\ref{eq:LocalGtilde}). This gauge invariance is obviously present also in the case of NATD, where the shift of $\nu$ is absent since $\zeta=0$.
\vspace{12pt}
The classical integrability of DTD models may be argued by the fact that they are obtained by adding a closed $B$-field and then applying NATD to the action of a supercoset, since neither of these operations breaks classical integrability, see e.g. \cite{Sfetsos:2013wia} for the argument in the case of NATD.
In appendix~\ref{sec:integr} we give a direct proof of the classical integrability of these models by showing that, similarly to what was shown in the case of DTD of PCM in~\cite{Borsato:2016pas}, the on-shell equations can be recast into the flatness condition
\begin{equation}
\epsilon^{ij}(\partial_i\mathcal L_j+\mathcal L_i\mathcal L_j)=0\,,
\end{equation}
for the Lax connection
\begin{equation}
\mathcal L_i=A^{(0)}_i+z A^{(1)}_i+\frac{1}{2}\left(z^2+z^{-2}\right)A^{(2)}_i+\frac{1}{2}\gamma_{ij}\epsilon^{jk}\left(z^{-2}-z^2\right)A^{(2)}_i+z^{-1}A^{(3)}_i\,,
\end{equation}
where $z$ is the spectral parameter, $A^i=A^i_++A^i_-$ and $A_\pm^i\equiv \mathrm{Ad}_f^{-1}(\tilde A_\pm^i+J_\pm^i)$, with $\tilde A^i_\pm$ given in~\eqref{eq:solApm}. See appendix~\ref{app:action} for our notation.
Notice that the presence of the Lax connection still implies that we have conserved charges corresponding to the full original $\alg g$ symmetry. However, in contrast to the case of supercosets, for DTD models one cannot argue any more that they are all local, see appendix~\ref{sec:integr}.
\subsection{Relation to Yang-Baxter sigma models}\label{sec:YB}
Given a DTD model with a cocycle $\omega$ which is non-degenerate on $\tilde{\alg{g}}$, we can show that the action can be recast into the one of a YB model via a field redefinition. This result was first presented in~\cite{Borsato:2016pas} and we collect here more details of the proof.
Given a non-degenerate $\omega$ we denote its inverse by $R=\omega^{-1}$.
From the cocycle condition for $\omega$ it follows that $R$ solves the CYBE on $\tilde{\alg{g}}^*$.
Conversely any solution of the CYBE on $\alg g$ defines an invertible 2-cocycle on a subalgebra\footnote{This follows from the fact that the subspace on which $R$ is invertible must be a subalgebra due to the CYBE \cite{MR674005}. Since $\omega=R^{-1}$ is a 2-cocycle on this subalgebra the subalgebra is quasi-Frobenius. Note that these results are true also for non-semisimple algebras and superalgebras.} $\tilde{\alg{g}}$, which demonstrates the one-to-one correspondence between DTD models with invertible $\omega$ and YB sigma models based on an R-matrix solving the CYBE.
The field redefinition that relates the two models is
\begin{equation}\label{eq:field-red-DTD-YB}
\nu=\zeta\tilde P^T\frac{1-\mathrm{Ad}_{\bar g}}{\mathrm{ad}_{Rx}}\omega Rx\,,\qquad\bar g=e^{Rx}\in\tilde G\,,
\end{equation}
with $x\in\tilde{\mathfrak g}^*$ so that $Rx\in\tilde{\mathfrak g}$. In fact, using the identities in (\ref{eq:dmu}) and (\ref{eq:admu}) we find
\begin{equation}
d\nu=\tilde P^T(\mathrm{ad}_\nu+\zeta\omega)(\bar g^{-1}d\bar g)\,,\qquad
\tilde P^T\mathrm{ad}_\nu\tilde P=\zeta\tilde P^T\mathrm{Ad}_{\bar g}^{-1}\omega \mathrm{Ad}_{\bar g}\tilde P-\zeta\omega\,,
\end{equation}
and the action (\ref{eq:S-DTD}) becomes, after a bit of algebra,
\begin{equation}
S=-\tfrac{T}{2}\int d^2\sigma\,\tfrac{\gamma^{ij}-\epsilon^{ij}}{2}\mathrm{Str}\Bigg(
g^{-1}\partial_ig\hat d\left(1-\frac{R_g\hat d}{R_g\hat d-\zeta}\right) g^{-1}\partial_jg
+\bar g^{-1}\partial_i\bar g(\mathrm{ad}_\nu+\zeta\omega)\bar g^{-1}\partial_j\bar g
\Bigg)\,,
\end{equation}
where we have defined $g=\bar gf$ and $R_g=\mathrm{Ad}_g^{-1}R\mathrm{Ad}_g$. The last term vanishes up to a total derivative and we are left precisely with the action of the YB sigma model~\cite{Delduc:2013qra,Kawaguchi:2014qwa}
\begin{equation}
S=-\tfrac{T}{2}\int d^2\sigma\,\tfrac{\gamma^{ij}-\epsilon^{ij}}{2}\mathrm{Str}\left(g^{-1}\partial_ig\ \hat d\ (1-\eta R_g\hat d)^{-1}(g^{-1}\partial_jg)\right)\,,
\end{equation}
with deformation parameter $\eta=\zeta^{-1}$.
In the special case when $\tilde{\mathfrak g}$ is abelian the DTD model is equivalent to a TsT transformation of the original supercoset sigma model, in agreement with the YB side for abelian $R$~\cite{Osten:2016dvf,Hoare:2016wsk}.
Let us mention that one can also construct a YB model for an R-matrix solving the modified CYBE, whose action takes essentially the same form as the above one \cite{Delduc:2013qra}; however, in that case it is not clear how to define the operator corresponding to $\omega$, and the relation to DTD models remains unclear. This case should be related by Poisson-Lie T-duality to the $\lambda$-model of~\cite{Sfetsos:2013wia,Hollowood:2014qma}.
We will argue in the next section that all (bosonic and fermionic) non-abelian T-duals of YB sigma models can be described as DTD models with certain degenerate $\omega$.
The converse is not true, in fact it is possible to identify DTD models which are not related to YB models by NATD; we refer to section~\ref{sec:not-YB} for an example and a discussion on this.
\section{Global symmetries}\label{sec:symm}
We will now describe the global symmetries, i.e. superisometries, of DTD models. Setting $\zeta=0$ and ignoring the presence of $\omega$ this discussion reduces to what one would have in the case of NATD.
In order to identify the global symmetries of these models we study the global transformations that leave the action invariant, \emph{modulo} gauge transformations with a global parameter, since the latter would not produce any Noether charge.
We find two types of global symmetries:\footnote{The two sets of transformations do not commute and their commutator is a transformation of the second type.}
\begin{itemize}
\item[1.] \emph{Unbroken global $G$-transformations}:
\begin{equation}\label{eq:GlobalG}
\begin{aligned}
&f\rightarrow g_0f\,,\quad\nu\rightarrow\tilde P^T\mathrm{Ad}_{g_0}\nu\,,\qquad g_0\in G \mbox{ and } g_0\notin\tilde G,\\
&\mbox{such that}\quad (1-\tilde P)\mathrm{Ad}_{g_0}\tilde P=0\,,\quad {\tilde P^T\mathrm{Ad}_{g_0}^{-1}\omega\mathrm{Ad}_{g_0}\tilde P=\omega}\,.
\end{aligned}
\end{equation}
The requirement $g_0\notin\tilde G$ comes from the fact that for $g_0\in\tilde G$ a combination of this isometry and the shift isometries described below is equivalent to a global $\tilde G$ gauge transformation.
\item[2.] \emph{Global shifts of $\nu$}:
\begin{equation}
\nu\rightarrow\nu+\lambda\,,\qquad\lambda\in\tilde{\mathfrak g}^*\quad\mbox{such that}\quad\tilde P^T\mathrm{ad}_\lambda\tilde P=0
\,.
\label{eq:GlobalShift}
\end{equation}
Note that the set of such $\lambda$'s will in general \emph{not} close into a subalgebra, although the corresponding isometry transformations of course commute since they are just shifts of $\nu$.
\end{itemize}
In the case when $\omega$ is invertible, which is equivalent to a YB sigma model with $R=\omega^{-1}$, it is not hard to show that these isometries coincide with the ones of the YB model which are normally written as $t\in\mathfrak g$ such that $R\mathrm{ad}_t=\mathrm{ad}_tR$.
Having global symmetries at our disposal means that we can gauge them and implement further NATD. Before discussing the details of this in the next subsection, we would like to exploit this possibility to make a comment regarding Weyl invariance of DTD models. As we prove in section~\ref{sec:target}, the target spaces of DTD models solve the standard supergravity equations if and only if the Lie algebra $\tilde{\alg{g}}$ is unimodular, i.e. $f_{ab}{}^b=0$. The standard supergravity equations are equivalent to the Weyl invariance at one-loop for the sigma-model, as opposed to just the scale invariance implied by the generalised supergravity equations \cite{Arutyunov:2015mqj,Wulff:2016tju}. In the non-unimodular case $f_{ab}{}^b\neq0$, and this defines a distinguished element of $\tilde{\mathfrak g}$; we can rotate the basis so that this element is $T_1$, i.e. $f_{1b}{}^b\neq0$ and $f_{ab}{}^b=0$ for $a\neq1$. The important observation is that the dual of the generator $T_1$ corresponds to an isometry. In fact, taking the trace of the Jacobi identity we find $f_{ab}{}^1=0$ and therefore
\begin{equation}
\mathrm{Str}(T_b\mathrm{ad}_{T^1}T_a)=f_{ab}{}^1=0\,,
\end{equation}
where $T^a \in\tilde{\mathfrak g}^*$. This confirms that $T^1$ satisfies (\ref{eq:GlobalShift}) and can be used to generate a shift isometry. Using the results of the next subsection, applying T-duality along the isometry direction $T^1$ one obtains a DTD model where $T_1$ is removed from $\tilde{\mathfrak g}$, so that the subalgebra that is left is now unimodular. Therefore, to each DTD model which is not Weyl invariant we can associate a Weyl invariant one obtained by (formal\footnote{Our discussion of isometries is at the level of the classical sigma model action, where the dilaton only appears in the combination $\mathcal F=e^\phi F$---together with RR fields---and in derivatives $\partial \phi$. When performing the T-duality we ignore the Fradkin-Tseytlin term, which will break the isometry referred to here.}) T-duality along a particular isometry direction. Obviously this possibility fails if there are obstructions to carrying out the T-duality, e.g. if the isometry in question is a null isometry. More generally, solutions of the \emph{generalised} supergravity equations are formally T-dual to solutions of the \emph{standard} supergravity equations ~\cite{Arutyunov:2015mqj,Wulff:2016tju}, and the above argument shows this relation in the specific context of DTD models.
\subsection{DTD of DTD models}
It is interesting to start from a DTD model as in~\eqref{eq:S-DTD} and further perform NATD, possibly including a deformation by a cocycle. We do this on the one hand to show that the application of these transformations on the sigma model does not require to start from a supercoset formulation, on the other hand to show that after these transformations we obtain a new DTD model. We will also use these results to argue that the example of the next subsection is not related to a YB model by NATD.
We can apply NATD by gauging the global isometries discussed above and dualising the corresponding directions.
Obviously, the choice of the type of isometries that we want to dualise will produce qualitative differences.
In fact, if we consider isometries of the first type~\eqref{eq:GlobalG} and dualise a subalgebra $\hat{\mathfrak g}$, we essentially enlarge the subalgebra $\tilde{\mathfrak g}$.
If instead we consider isometries of the shift type~\eqref{eq:GlobalShift} and dualise a subspace $\bar V^*\subset\tilde{\mathfrak g}^*$, then we remove generators from the subalgebra $\tilde{\mathfrak g}$. The combination of isometry transformations that we consider here is therefore
\begin{equation}
f=\hat gf'\,,\quad\nu=\tilde P^T(\mathrm{Ad}_{\hat g}\nu'+\bar\lambda)\,,\qquad\mbox{with}\qquad\hat g\in\hat G\,,\quad\bar\lambda\in \bar V^*\,.
\end{equation}
After gauging them in the usual way we obtain a sigma model action which is just the one in~\eqref{eq:S-DTD}, where we replace\footnote{We will now use the notation $\check \nu\in\check{\alg{g}}$ for the field and the subalgebra of the DTD model from which we start. Similarly, we will denote the corresponding operators as $\check P,\ \check{\mathcal{O}}$, etc. We do this because we want to reserve the usual notation for the DTD model that is obtained at the end, after applying the further deformation of NATD.}
\begin{equation}
f\rightarrow f'\,,\qquad
J\rightarrow J'+\hat A\,,\qquad
d\nu\rightarrow d\check\nu+\check P^T[\hat A,\check\nu]+\bar a\,,
\end{equation}
where $\hat A\in \hat{\mathfrak g}$ is the non-abelian gauge field corresponding to the $\hat G$ isometries and $\bar a\in \bar V^*$ is the abelian gauge field corresponding to the shift isometries.
We add to the action the terms\footnote{For the sake of the discussion here we fix conformal gauge $\gamma^{+-}=\gamma^{-+}=\epsilon^{-+}=-\epsilon^{+-}=2$ where $\sigma^\pm= \tau\pm \sigma$. In principle it is also possible to add a deformation for the second type of isometry by adding a term $\bar a\bar\omega'\bar a$, but we will not consider this possibility further here.}
\begin{equation}
-T\int d^2\sigma\ \mathrm{Str}(\hat\nu \hat{F}_{+-}+\bar\rho \bar f_{+-}-\hat\zeta\hat A_+\hat\omega\hat A_-)\,,
\end{equation}
where $\hat{F}_{+-}=\partial_+\hat A_--\partial_-\hat A_++[\hat A_+,\hat A_-]$ and $\bar f_{+-}=\partial_+\bar a_--\partial_-\bar a_+$, $\hat\nu$ and $\bar \rho$ are two new Lagrange multipliers, and $\hat\omega$ is a cocycle on $\hat{\alg{g}}$.
Integrating out $\hat\nu$ and $\bar\rho$ one obtains the action from which we started; to apply NATD we integrate out $\hat A$ and $\bar a$ instead.
{We will now describe what happens when we dualise either $\hat{\alg{g}}$ or $\bar V^*$, and then use it to argue what should happen in the most general case where one dualises on both at the same time.}\footnote{In the rest of this section we absorb the parameter $\zeta$ into $\omega$ to simplify the expressions.}
\paragraph{Dualising type 1 isometries}
Consider first isometries of type 1 above, where we have $\hat P+\check P=\tilde P$ and $\hat P\check P=0$. After a bit of algebra and dropping primes, we find that the new action takes the form $S=-T\int d^2\sigma \mathrm{Str}(J_+\hat d_fJ_-+(\partial_+\nu-\hat d_f^TJ_+)\mathcal{Q}(\partial_-\nu+\hat d_fJ_-))$ where $\nu=\check\nu+\hat\nu$ and $\mathcal Q$ is an operator acting on $\tilde{\mathfrak g}=\check{\mathfrak g}\oplus\hat{\mathfrak g}$ which can be written in a $2\times 2$ block form as
\begin{equation}
\mathcal{Q}=
\left(
\begin{array}{cc}
\check{\mathcal O}^{-1}+\check{\mathcal O}^{-1}(\hat d_f-\mathrm{ad}_{\check\nu})U^{-1}(\hat d_f-\mathrm{ad}_{\check\nu})\check{\mathcal O}^{-1} & -\check{\mathcal O}^{-1}(\hat d_f-\mathrm{ad}_{\check\nu})U^{-1}\\
-U^{-1}(\hat d_f-\mathrm{ad}_{\check\nu})\check{\mathcal O}^{-1} & U^{-1}
\end{array}
\right),
\end{equation}
where\footnote{The operators $\check{\mathcal{O}},\hat{\mathcal{O}}$ are obtained from $\widetilde{\mathcal{O}}$ by dressing $\nu, \omega$ and the projectors with checks or hats.} $U=\hat{\mathcal O}-\hat P^T(\hat d_f-\mathrm{ad}_{\check\nu})\check{\mathcal O}^{-1}(\hat d_f-\mathrm{ad}_{\check\nu})\hat P$.
It is straightforward to check that if we take $\omega=\check\omega+\hat\omega$ and define $\widetilde{\mathcal{O}}$ as in~\eqref{eq:def-op}, then its decomposition in block form is
\begin{equation}
\widetilde{\mathcal{O}}
=
\left(
\begin{array}{cc}
\check{\mathcal O} & \check P^T(\hat d_f-\mathrm{ad}_{\check\nu})\hat P\\
\hat P^T(\hat d_f-\mathrm{ad}_{\check\nu})\check P & \hat{\mathcal O}
\end{array}
\right),
\end{equation}
and that $\mathcal Q=\widetilde{\mathcal{O}}^{-1}$.
Therefore performing DTD by exploiting the unbroken isometries of the first type is equivalent to the simple operation of enlarging the dualised subalgebra as $\tilde{\mathfrak g}=\check{\mathfrak g}\oplus\hat{\mathfrak g}$, which is a Lie algebra due to the isometry condition $[\hat{\mathfrak g},\check{\mathfrak g}]\subset\check{\mathfrak g}$. As for the deformation, we are just adding new contributions, and $\omega=\check\omega+\hat\omega$ is a 2-cocycle on $\tilde{\mathfrak g}$ due to the isometry conditions in~\eqref{eq:GlobalG}.
\paragraph{Dualising type 2 isometries}
For isometries of type 2 we have $\bar P^T$ that projects on the space $\bar V^*$, so that $\bar P\check P=\check P\bar P=\bar P$ and $\tilde P=\check P-\bar P$.
When integrating out $\bar a_\pm$ we get equations where $\bar P \check{\mathcal{O}}^{-1}$ appears, so that it is convenient to use the block decomposition on the space $\tilde{\alg{g}}\oplus \bar V$
\begin{align}
\check{\mathcal O}^{-1}
\equiv&
\left(
\begin{array}{cc}
\widetilde{\mathcal{O}} & \tilde P^T(\hat d_f-\mathrm{ad}_{\tilde\nu}-\check\omega)\bar P\\
\bar P^T(\hat d_f-\mathrm{ad}_{\tilde\nu}-\check\omega)\tilde P & \bar P^T(\hat d_f-\mathrm{ad}_{\tilde\nu}-\check\omega)\bar P
\end{array}
\right)^{-1}
\\
=&
\left(
\begin{array}{cc}
\widetilde{\mathcal{O}}^{-1}+\widetilde{\mathcal{O}}^{-1}(\hat d_f-\mathrm{ad}_{\tilde\nu}-\check\omega)U^{-1}(\hat d_f-\mathrm{ad}_{\tilde\nu}-\check\omega)\widetilde{\mathcal{O}}^{-1} & -\widetilde{\mathcal{O}}^{-1}(\hat d_f-\mathrm{ad}_{\tilde\nu}-\check\omega)U^{-1}\\
-U^{-1}(\hat d_f-\mathrm{ad}_{\tilde\nu}-\check\omega)\widetilde{\mathcal{O}}^{-1} & U^{-1}
\end{array}
\right), \nonumber
\end{align}
where $U=\bar P^T(\hat d_f-\mathrm{ad}_{\tilde\nu}-\check\omega)\bar P-\bar P^T(\hat d_f-\mathrm{ad}_{\tilde\nu}-\check\omega)\widetilde{\mathcal{O}}^{-1}(\hat d_f-\mathrm{ad}_{\tilde\nu}-\check\omega)\bar P$.
Note that $\tilde{\mathfrak g}=\{x\in\check{\mathfrak g}\,|\,\mathrm{Str}(x\lambda)=0\,,\,\forall\lambda\in\bar{V}^*\}$ is indeed a subalgebra since for $x,y\in\tilde{\mathfrak g}$ we have $\mathrm{Str}([x,y]\lambda)=-\mathrm{Str}(x\mathrm{ad}_\lambda y)=0$ as a consequence of (\ref{eq:GlobalShift}). In fact for $x,y\in\check{\mathfrak g}$ we have in the same way $[x,y]\in\tilde{\mathfrak g}$. This means in particular that if $\bar{V}$ closes into a subalgebra it must be abelian. Clearly $\check\omega$ reduces to a 2-cocycle $\tilde\omega=\tilde P^T\check\omega\tilde P$ on $\tilde{\mathfrak g}$.
After some algebra and dropping a total derivative $d\nu d\bar\rho$-term, the dualised action becomes
\begin{align}
-T\int & d^2\sigma\mathrm{Str}\Big(
(J_++\partial_+\bar\rho)\hat d_f(J_-+\partial_-\bar\rho)
+(\partial_+\tilde\nu-\hat d_f^TJ_+)\widetilde{\mathcal{O}}^{-1}(\partial_-\tilde\nu+\hat d_fJ_-)
\nonumber\\
&{}
+(\partial_+\tilde\nu-\hat d_f^TJ_+)\widetilde{\mathcal{O}}^{-1}(\hat d_f-\mathrm{ad}_{\tilde\nu}-\check\omega)\partial_-\bar\rho
-\partial_+\bar\rho(\hat d_f-\mathrm{ad}_{\tilde\nu}-\check\omega)\widetilde{\mathcal{O}}^{-1}(\partial_-\tilde\nu+\hat d_fJ_-)
\nonumber\\
&{}
-\partial_+\bar\rho(\hat d_f-\mathrm{ad}_{\tilde\nu}-\check\omega)\widetilde{\mathcal{O}}^{-1}(\hat d_f-\mathrm{ad}_{\tilde\nu}-\check\omega)\partial_-\bar\rho
-\partial_+\bar\rho(\mathrm{ad}_{\tilde\nu}+\check\omega)\partial_-\bar\rho
\Big)\,.
\end{align}
As expected $\bar\nu=\check\nu-\tilde\nu$ has dropped out, since we have dualised the corresponding directions. Finally $\bar\rho$ can be removed by the field redefinition
\begin{equation}
f\rightarrow \bar hf\,,\quad\tilde\nu\rightarrow\tilde P^T\left(\mathrm{Ad}_{\bar h}\nu+\frac{1-\mathrm{Ad}_{\bar h}}{\mathrm{ad}_{\bar\rho}}\check\omega\bar\rho\right)\,,\qquad \bar h=e^{-\bar\rho}\,,
\end{equation}
which resembles a $\tilde G$ gauge transformation except for the fact that $\bar h\notin\tilde{G}$.
To check that we match with the DTD action in~\eqref{eq:S-DTD} we use the fact that under the above redefinition $\widetilde{\mathcal{O}}\rightarrow\check P^T\mathrm{Ad}_{\bar h}\widetilde{\mathcal{O}}\mathrm{Ad}_{\bar h}^{-1}\check P$ which follows from\footnote{These are proved using~\eqref{eq:admu},~\eqref{eq:dmu} and $\tilde P\mathrm{Ad}_{\bar h}\check P=\mathrm{Ad}_{\bar h}\tilde P$, the last being a consequence of $[x,y]\in\tilde{\mathfrak g}$ for any $x,y\in\check{\mathfrak g}$.}
\begin{equation}
\begin{aligned}
\check P^T\mathrm{ad}_{\tilde\nu}\check P&\rightarrow
\check P^T\mathrm{Ad}_{\bar h}\check P^T\mathrm{ad}_\nu\check P\mathrm{Ad}_{\bar h}^{-1}\check P
+\check P^T\mathrm{Ad}_{\bar h}\check\omega\mathrm{Ad}_{\bar h}^{-1}\check P
-\check\omega\,,
\\
d\tilde\nu&\rightarrow\check P^T\mathrm{Ad}_{\bar h}(d\nu-\mathrm{ad}_\nu(\bar h^{-1}d\bar h)-\check\omega(\bar h^{-1}d\bar h))\,.
\end{aligned}
\end{equation}
The calculations are simple when $\bar{V}$ is a (abelian) subalgebra since in that case $\bar h^{-1}d\bar h=-\mathrm{Ad}_{\bar h}^{-1}d\bar\rho$ and the last $d\bar\rho d\bar\rho$ term vanishes up to a total derivative.
When $\bar{V}$ is not a subalgebra it is clear that it must still work since these are abelian isometries and we can just T-dualise one at a time. It is nevertheless instructive to show this explicitly. To do this we use the fact that $\bar h^{-1}d\bar h+\mathrm{Ad}_{\bar h}^{-1}d\bar\rho$ is in $\tilde{\mathfrak g}$ since it involves commutators of elements from $\bar{V}$. This simplifies the left-over terms to $\int d\sigma^2\epsilon^{ij}\mathrm{Str}(\bar h^{-1}\partial_i\bar h\ \check\omega(\bar h^{-1}\partial_j\bar h))$
which indeed is a total derivative term and can be dropped.
As anticipated, we get that T-dualising on the shift isometries is equivalent to shrinking $\tilde{\mathfrak g}$ by removing the generators in $\bar V$.
\iffalse
\begin{align}
&\int_\Sigma\mathrm{Str}\Big(
2d\nu\wedge(\mathrm{Ad}_{\bar h}^{-1}d\bar\rho+\bar h^{-1}d\bar h)
+\mathrm{Ad}_{\bar h}^{-1}d\bar\rho\wedge(\mathrm{ad}_{\tilde\nu}+\check\omega)(\bar h^{-1}d\bar h)
\nonumber\\
&{}\qquad
+\bar h^{-1}d\bar h\wedge(\mathrm{ad}_{\tilde\nu}+\check\omega)(\mathrm{Ad}_{\bar h}^{-1}d\bar\rho+\bar h^{-1}d\bar h)
\Big)
\nonumber\\
=&
\int_\Sigma\mathrm{Str}\Big(
2\nu\{\bar h^{-1}d\bar h,\mathrm{Ad}_{\bar h}^{-1}d\bar\rho\}
+\nu\{\bar h^{-1}d\bar h,\bar h^{-1}d\bar h\}
\nonumber\\
&{}\qquad
+\mathrm{Ad}_{\bar h}^{-1}d\bar\rho\wedge(\mathrm{ad}_{\tilde\nu}+\check\omega)(\bar h^{-1}d\bar h)
+\bar h^{-1}d\bar h\wedge(\mathrm{ad}_{\tilde\nu}+\check\omega)(\mathrm{Ad}_{\bar h}^{-1}d\bar\rho+\bar h^{-1}d\bar h)
\Big)
\nonumber\\
=&
\int_\Sigma\mathrm{Str}(\bar h^{-1}d\bar h\wedge\check\omega(\bar h^{-1}d\bar h))
\,,
\end{align}
\fi
\paragraph{Dualising type 1 and 2 isometries}
We have seen that dualising on the isometries outside of $\tilde{\mathfrak g}$ has the effect of adding the corresponding generators to $\tilde{\mathfrak g}$. Similarly dualising on isometries inside $\tilde{\mathfrak g}$ effectively removes the corresponding generators. The natural conjecture is then that dualising on both types of isometries at the same time again just adds/removes the generators outside/inside $\tilde{\mathfrak g}$ to give the $\tilde{\mathfrak g}$ of the resulting model.
To be more specific, start from a DTD model with a cocycle on the subalgebra\footnote{Also here we prefer to change notation and call $\check{\mathfrak g}$ the original subalgebra, so that $\tilde{\mathfrak g}$ will be used for the algebra obtained after applying NATD.} $\check{\alg{g}}$ and imagine the most general NATD of this DTD model where we dualise isometries $t_i\notin\check{\mathfrak g}$ of type 1 as in (\ref{eq:GlobalG}) and $\lambda_I\in\check{\mathfrak g}^*$ of type 2 as in (\ref{eq:GlobalShift}). Our conjecture is that this results in a new DTD model where now
\begin{equation}
\tilde{\mathfrak g}=\{x=\check y+a_it_i\,,\,\check y\in\check{\mathfrak g}\,|\,\mathrm{Str}(\lambda_I\check y)=0\,,\,\forall\lambda_I\,\,\,\mbox{such that}\,\,\,\mathrm{Str}(\lambda_I[t_i,t_j])=0\,\,,\forall t_i,t_j\}\,.
\label{eq:dual-gtilde}
\end{equation}
In other words, $\tilde{\mathfrak g}$ is obtained by adding to $\check{\mathfrak g}$ all generators $t_i$ and by removing all elements which are dual to $\lambda_I$, except when these are generated in commutators $[t_i,t_j]$. In fact, we want the last condition on $\lambda_I$ because the commutator of two isometries of type 1 can generate an isometry of type 2, and if we are adding the $t_i$ we want to make sure that they close into an algebra. Here we will not work out explicitly the transformation of the action under this NATD {since this is quite involved}, we will rather just check that this expectation makes sense and such a DTD model is well-defined.
To start, we must assume that the isometries on which we dualise form a subalgebra of the isometry algebra. This implies the conditions
\begin{equation}
[t_i,t_j]=c_{ij}{}^kt_k+\check c_{ij}{}^{K'}\check t_{K'}\,,\qquad\check\omega(\check t_{I'})=\delta_{I'}^I\lambda_I\,,\qquad\check P^T\mathrm{ad}_{t_i}\lambda_I=c_{iI}{}^J\lambda_J\,,
\label{eq:iso-subalg}
\end{equation}
with some coefficients $c_{ij}{}^k$, $\check c_{ij}{}^k$ and $c_{iI}{}^J$. {The generators $\check t_{K'}\in\check{\mathfrak g}$ appear because, as already mentioned, the commutators of two $t_i$ can generate an element in $\check{\mathfrak g}$. These must still satisfy the second condition in (\ref{eq:GlobalG}) which translates to the second condition above.} The first consistency check is to show that $\tilde{\mathfrak g}$ defined above indeed forms a subalgebra of $\mathfrak g$ so that the corresponding DTD model can be defined. Commuting two elements of $\tilde{\mathfrak g}$ we get
\begin{equation}
[\check y+a_it_i,\check z+b_jt_j]=
[\check y,\check z]
-b_i\mathrm{ad}_{t_i}\check y
+a_i\mathrm{ad}_{t_i}\check z
+a_ib_j[t_i,t_j]\,.
\end{equation}
The isometry conditions in (\ref{eq:GlobalG}) indeed imply that the second and third term are in $\check{\mathfrak g}$. Taking the supertrace with $\lambda_I$ satisfying $\mathrm{Str}(\lambda_I[t_i,t_j])=0$ we get
\begin{equation}
\mathrm{Str}([\check y,\check z]\lambda_I)
+b_ic_{iI}{}^J\mathrm{Str}(\check y\lambda_J)
-a_ic_{iI}{}^J\mathrm{Str}(\check z\lambda_J)
=
-\mathrm{Str}(\check y\mathrm{ad}_{\lambda_I}\check z)
=0\,,
\end{equation}
where we used the conditions (\ref{eq:iso-subalg}) and the fact that $\check y,\check z\in\tilde{\mathfrak g}$ and, in the last step, the isometry condition (\ref{eq:GlobalShift}) for $\lambda_I$. This proves that indeed $\tilde{\mathfrak g}$ in (\ref{eq:dual-gtilde}) defines a subalgebra of $\mathfrak g$.
To define a 2-cocycle on $\tilde{\mathfrak g}$ we take $\omega=\tilde P^T\check\omega\tilde P$---we could also add an additional deformation in the $t_i$ directions but we will not do so here--- and we find
\begin{align}
\omega[\check y+a_it_i,\check z+b_jt_j]
=&\,
\tilde P^T
\Big(
[\check\omega\check y,\check z+b_it_i]
+[\check y+a_it_i,\check\omega\check z]
+a_ib_j\check\omega[t_i,t_j]
\Big)
\nonumber\\
=&\,
\tilde P^T[\omega\check y,\check z+b_it_i]
+\tilde P^T[\check y+a_it_i,\omega\check z]
+a_ib_j\tilde P^T\check\omega[t_i,t_j]\,,
\end{align}
where we used the cocycle condition for $\check{\omega}$, the fact that $\mathrm{ad}_{t_i}$ commutes with $\check\omega$ (\ref{eq:GlobalG}), and in the last step we used (\ref{eq:PPT-rel}). The first two terms are precisely what we want, it remains to show that the last one vanishes. By the conditions (\ref{eq:iso-subalg}) this term is proportional to a combination of $\lambda_I$ and therefore the $\tilde P^T$ projection means that this term vanishes unless $\mathrm{Str}([t_k,t_l]\check\omega[t_i,t_j])\neq0$ for some $k,l$. However
\begin{align}
\mathrm{Str}([t_k,t_l]\check\omega[t_i,t_j])
=&\,
\tfrac12\mathrm{Str}(\check\omega[[t_i,t_j],[t_k,t_l]])
=
\tfrac12\mathrm{Str}(\check P^T[\check\omega[t_i,t_j],[t_k,t_l]])
+\tfrac12\mathrm{Str}(\check P^T[[t_i,t_j],\check\omega[t_k,t_l]])
\nonumber\\
=&\,
\tfrac12\check c_{ij}{}^I\mathrm{Str}(\check P^T\mathrm{ad}_{\lambda_I}[t_k,t_l])
-\tfrac12\check c_{kl}{}^I\mathrm{Str}(\check P^T\mathrm{ad}_{\lambda_I}[t_i,t_j])
=0\,,
\end{align}
where we used the cocycle condition and the isometry condition in (\ref{eq:GlobalShift}). Therefore $\omega$ is indeed a 2-cocycle on $\tilde{\mathfrak g}$ and the corresponding DTD model is well-defined.
\subsection{DTD models not related to YB models by NATD}\label{sec:not-YB}
Here we want to present an example of a DTD model which is not related to a YB model by NATD.\footnote{{Let us mention that it is possible to find examples where $\omega$---as well as any 2-cocycle in its equivalence class---is non-degenerate on a space which does not close into an algebra. This corrects a statement in the first version of~\cite{Borsato:2016pas}.}}
To argue that this is the case we use two important facts concerning the dualisation of the two types of isometries discussed above.
First, when dualising isometries of type 1, thanks to the condition~\eqref{eq:GlobalG} the original $\check{\mathfrak g}$ will become an ideal of the larger algebra $\tilde{\mathfrak g}$ that is obtained by adding the generators $t_i$, i.e. by applying NATD.
That means that starting from a YB model---or, rather, its corresponding DTD model with non-degenerate $\omega$---NATD on isometries of type 1 will produce a DTD model with a cocycle \emph{non-degenerate on an ideal} of $\tilde{\mathfrak g}$. When we include also isometries of type 2 it remains true that what is left of $\check{\mathfrak g}$ forms a proper ideal inside $\tilde{\mathfrak g}$, on which, however, $\omega$ does not have to be non-degenerate.
We also remark that, since they are realised as linear shifts, isometries of type 2 are {commuting} and are therefore still present even after applying {abelian} T-duality along them. After the dualisation the corresponding symmetry will be realised as an isometry of type 1.
Consider the following algebra and corresponding 2-cocycle
\begin{equation}
\tilde{\mathfrak g}=\mathrm{span}\{p_1,\,p_2,\,p_3,\,J_{12}\}\,,\qquad\omega=k_3\wedge J_{12}\,,
\label{eq:ex1}
\end{equation}
where we refer to \cite{Borsato:2016ose} for our definitions and conventions on the generators of the conformal algebra {$\mathfrak{so}(2,4)$}.
The above 2-cocycle is defined on a space which is not an ideal of $\tilde{\mathfrak g}$, and it is clear that {adding an exact term to $\omega$ cannot change this}, since the only terms that we could add are $k_1\wedge J_{12}$ and $k_2\wedge J_{12}$.
According to the above discussion, this rules out the possibility of this example coming from dualising isometries of type 1 of a YB model. {In fact, since there is no proper ideal in $\tilde{\mathfrak g}$ that contains the subspace $\{p_3,J_{12}\}$ where $\omega$ is defined, a combination of isometries of type 1 and type 2 is also ruled out. This leaves only the possibility that this example is generated by T-dualising isometries of type 2 only. If} it were true that it comes from a YB model by dualising isometries of type 2, these should be realised here as isometries of type 1 and we {would be able to dualise them back to find a YB model (in DTD form)}.
However, in this example the only isometry of type 1 corresponds to $p_0$, and adding $p_0$ to $\tilde{\mathfrak g}$ does not help in making the cocycle non-degenerate on the dualised algebra.
We therefore conclude that the above example is not related to a YB model by NATD,\footnote{It would be interesting to understand whether this or similar examples are related to YB models in other ways, e.g. contractions.} and we refer to section~\ref{sec:ex-not-YB} {for the corresponding supergravity background}.
The above example may be obtained by dropping one of the two terms in $R_{11}$ {in table 2} of \cite{Borsato:2016ose}, and similar examples coming from dropping a term in other rank 4 R-matrices of \cite{Borsato:2016ose} are e.g.
{
\begin{equation}
\begin{aligned}
&\tilde{\mathfrak g}=\mathrm{span}\{p_1,\,p_2,\,p_3,\,p_0+J_{12}\}\,,\qquad&&\omega=k_3\wedge({k_0+}\, J_{12})\,,&&&\text{ from } R_{10}\,.\\
&\tilde{\mathfrak g}=\mathrm{span}\{p_0,\,p_1,\,p_2,\,J_{12}\}\,,\qquad&&\omega=k_0\wedge J_{12}\,,&&&\text{ from } R_{13}\,.\\
&\tilde{\mathfrak g}=\mathrm{span}\{p_1,\,p_2,\,J_{12},\,J_{03}\}\,,\qquad&&\omega=J_{12}\wedge J_{03}\,,&&&\text{ from } R_{14}\,.
\end{aligned}
\end{equation}
In each case it is easy to see that $\omega$ cannot be defined on an ideal in $\tilde{\mathfrak g}$ even if we add exact terms---in the first case the only terms that we could add are $k_1\wedge({k}_0+J_{12})$ and $k_2\wedge({k}_0+J_{12})$, in the second and third case they are $k_1\wedge J_{12}$ and $k_2\wedge J_{12}$. In the first case the only isometry of type 1 corresponds to $p_0$, while in the second and third there is no isometry of type 1. Note that the second case can be embedded into $\mathfrak{so}(2,3)$ and therefore gives a deformation also of $AdS_4$.
}
\iffalse
\alert{Other DTD models which are not related to YB are those where $\omega$ is non-degenerate on a space which does not even close into an algebra\footnote{\alert{It is possible to find examples where no cocycle in the equivalence class is non-degenerate on a space which does not close into an algebra. This corrects a statement in the first version of~\cite{Borsato:2016pas}.}}. Take for example the 6-dimensional algebra $\tilde{\alg{g}}=\text{span}\{T_i|i=1,\ldots,6\}$ where the non-vanishing commutators are
\begin{equation}
[T_1,T_2]=T_5,\qquad
[T_3,T_4]=T_5,
\end{equation}
and take the 2-cocycle $\omega=T^1\wedge T^2$.
This $\omega$ is degenerate, and the vector space spanned by $\{T_1,T_2\}$ on which it is non-degenerate does not close into a subalgebra of $\tilde{\alg{g}}$. We also see that it is not possible to find an exact piece so that the new cocycle is non-degenerate on a subalgebra of $\tilde{\alg{g}}$, since we could only add $T^1\wedge T^2+T^3\wedge T^4$.
However, the above example does not seem to produce a DTD model, since $\widetilde{\mathcal{O}}(T_5)=0$ and the operator is not invertible.
We can be embedded $\tilde{\alg{g}}$ into $\alg{so}(2,4)\oplus\alg{so}(6)$ by taking e.g.
\begin{equation}
\begin{aligned}
&T_1=\frac{1}{2}\left(p_1-J_{02}+J_{23}\right),\qquad
&&T_2=p_2+J_{01}-J_{13},\qquad
&&&T_5=p_0+p_3,\\
&T_3=\frac{1}{2}\left(p_1+J_{02}-J_{23}\right),\qquad
&&T_4=-p_2+J_{01}-J_{13},\qquad
&&&T_6\in\alg{so}(6),
\end{aligned}
\end{equation}
where we refer to~\cite{Borsato:2016ose} for our conventions on the conformal generators.
The dual generators may be taken as
\begin{equation}
\begin{aligned}
&T^1=\frac{1}{2}\left(-k_1-J_{02}-J_{23}\right),\qquad
&&T^2=\frac{1}{4}\left(-k_2+J_{01}+J_{13}\right),\qquad
&&&T^5=\frac{1}{4}\left(k_0-k_3\right),\\
&T^3=\frac{1}{2}\left(-k_1+J_{02}+J_{23}\right),\qquad
&&T^4=\frac{1}{4}\left(k_2+J_{01}+J_{13}\right).
\end{aligned}
\end{equation}
We have $\omega(T_5)=0$ by construction\footnote{The cocycle condition for this $\tilde{\alg{g}}$ fixes the components $\omega_{i5}=0$.} and $\tilde P^T\mathrm{ad}_\nu T_5=0$ because $T_5$ is central in this case. Moreover it can be checked that $\tilde P^T\hat d_f T_5=0$ for a generic element $f\in SO(2,4)$. The non-invertibility of $\widetilde{\mathcal{O}}$ should be related to the impossibility of applying T-duality along the null direction $T_5$. We were not able to find examples of this type---i.e. \alert{where $\omega$ is non-degenerate on a space which does not even close into an algebra---that have invertible $\widetilde{\mathcal{O}}$}.
}
\fi
\section{Kappa symmetry and Green-Schwarz form}\label{sec:kappa}
As we will show in a moment the action of DTD models is invariant under kappa symmetry variations, and this will allow us to put it into the Green-Schwarz form. To show invariance under kappa symmetry we need to consider the variation of the action under the fields $\nu$ and $f$, as well as the worldsheet metric $\gamma^{ij}$.
The variation of the action with respect to the fields is computed in~\eqref{eq:delta-fnu-S}.
To define a kappa symmetry variation we should also say how $\delta f$ and $\delta \nu$ are expressed in terms of the kappa symmetry parameters $\tilde\kappa^{(j)}_i$, each of them being a local Grassmann parameter of grading $j$. We define $A_\pm^i\equiv \mathrm{Ad}_f^{-1}(\tilde A_\pm^i+J_\pm^i)$, where subscripts $\pm$ indicate that we act with the worldsheet projectors in~\eqref{eq:ws-proj} and $\tilde A^i_\pm$ is given in~\eqref{eq:solApm}; we take\footnote{We write the kappa symmetry transformation in this way rather than the one in~\cite{Borsato:2016pas} because we want $P^{(0)}\mathrm{Ad}_f^{-1}\delta_\kappa\nu=0$.}
\begin{equation}
\hat d^T(f^{-1}\delta_\kappa f)=\mathrm{Ad}_f^{-1}\delta_\kappa \nu=
-\{i\tilde\kappa^{(1)}_i,A_-^{(2)i}\}+\{i\tilde\kappa^{(3)}_i,A_+^{(2)i}\}\,.
\end{equation}
This relation is fixed by noticing that after we impose it the total variation of the action with respect to the fields simplifies considerably, and we find
\begin{equation}
\begin{aligned}
(\delta_f+\delta_\nu )S&=-\tfrac{T}{2}\int d^2\sigma \ 4\operatorname{Str}\left(
A_-^{(2)i}A_-^{(2)j}[A_{+i}^{(1)},i\tilde\kappa_j^{(1)}]
+A_+^{(2)i}A_+^{(2)j}[A_{-i}^{(3)},i\tilde\kappa_j^{(3)}]
\right)\\
&=-\tfrac{T}{2}\int d^2\sigma \ \tfrac{1}{2}\Big[
\operatorname{Str}\left(A_-^{(2)i}A_-^{(2)j}\right)\operatorname{Str}\left(W[A_{+i}^{(1)},i\tilde\kappa_j^{(1)}]\right)\\
&\qquad\quad\qquad\qquad+\operatorname{Str}\left(A_+^{(2)i}A_+^{(2)j}\right)\operatorname{Str}\left(W[A_{-i}^{(3)},i\tilde\kappa_j^{(3)}]\right)
\Big]\,.
\end{aligned}
\end{equation}
Here we used the property $A_\pm^i B_\pm^j=A_\pm^j B_\pm^i$, which follows from the identity $ P^{ij}_\pm P^{kl}_\pm= P^{il}_\pm P^{kj}_\pm$, as well as the identity
\begin{equation}
A_\pm^{(2)i}A_\pm^{(2)j} = \tfrac{1}{8}W\operatorname{Str}(A_\pm^{(2)i}A_\pm^{(2)j})+c^{ij}\mathbbm1_8\,,
\end{equation}
where $c^{ij}$ is an expression which is not interesting for this calculation, and $W=\text{diag}(1_4,-1_4)$ is the hypercharge.
The above variation does not vanish but it can be compensated by the contribution coming from varying the worldsheet metric.
In fact, we first notice that the contribution of the terms involving the worldsheet metric to the action may be written as
\begin{equation}\label{eq:Sgamma}
S_\gamma=-\tfrac{T}{2}\int d^2\sigma \gamma^{ij}\ \operatorname{Str}\left(E^{(2)}_i E^{(2)}_j\right)\,,
\end{equation}
where we have two possible choices for the bosonic vielbein which are related by a local Lorentz transformation, either $E^{(2)}=A_+^{(2)}$ or $E^{(2)}=A_-^{(2)}$, where
\begin{equation}\label{eq:def-Apm}
A_+=\mathrm{Ad}_f^{-1}(J+\widetilde{\mathcal{O}}^{-T}(d\nu-\hat d_f^TJ))\,,\qquad
A_-=\mathrm{Ad}_f^{-1}(J-\widetilde{\mathcal{O}}^{-1}(d\nu+\hat d_fJ))\,.
\end{equation}
The subscript on $A_\pm$ is here used only to distinguish the two fields and should not be confused with the $\pm$ used to denote the worldsheet projections; however, we choose this notation since projecting on $A_\pm$ with $ P_\pm^{ij}$ after reintroducing worldsheet indices we obtain in fact the $A_\pm^{i}$ used above.\footnote{A caveat is that the projections of $A_\pm$ in~\eqref{eq:def-Apm} with $P_\mp^{ij}$ do not vanish, while $P_\mp^{ij}A_{\pm j}=0$. We trust that this will not create confusion, since the notation has clear advantages and those projections will never be needed.}
We declare the kappa symmetry variation of the worldsheet metric to be
\begin{equation}
\delta_\kappa\gamma^{ij}=-\tfrac{1}{2}\left[\operatorname{Str}\left(W[A_{+}^{(1)i},i\tilde\kappa_+^{(1)j}]\right)
+\operatorname{Str}\left(W[A_{-}^{(3)i},i\tilde\kappa_-^{(3)j}]\right)\right]\,,
\end{equation}
so that the total variation of the action under kappa symmetry transformations vanishes $(\delta_f+\delta_\nu +\delta_\gamma ) S=0$.
The kappa symmetry transformations for the fields may be also recast into the form
\begin{equation}\label{eq:deltakappa}
i_{\delta_\kappa z}E^{(2)}=0\,,\qquad
i_{\delta_\kappa z}E^{(1)}=P^{ij}_-\{i\kappa^{(1)}_i,E_j^{(2)}\}\,,\qquad
i_{\delta_\kappa z}E^{(3)}=P^{ij}_+\{i\kappa^{(3)}_i,E_j^{(2)}\}\,,
\end{equation}
where $\kappa^{(1)}=\mathrm{Ad}_h\tilde\kappa^{(1)}$ and $\kappa^{(3)}=\tilde\kappa^{(3)}$ and where we made a choice for the bosonic and fermionic components of the supervielbeins
\begin{equation}\label{eq:E-A}
E^{(2)}=A_+^{(2)}=\mathrm{Ad}_hA_-^{(2)}\,,\qquad E^{(1)}=\mathrm{Ad}_hA_+^{(1)}\,,\qquad E^{(3)}=A_-^{(3)}\,.
\end{equation}
The above transformations are the standard ones for kappa symmetry, and the action also takes the standard Green-Schwarz form
\begin{equation}\label{eq:S-GS}
S=-\tfrac{T}{2}\int d^2\sigma\,\gamma^{ij}\mathrm{Str}(E_i^{(2)}E_j^{(2)})-T\int B\,,
\end{equation}
where the $B$-field is
\begin{equation}\label{eq:B-field}
B=\tfrac14\mathrm{Str}(J\wedge\hat d_fJ+(d\nu-\hat d_f^TJ)\wedge\widetilde{\mathcal{O}}^{-1}(d\nu+\hat d_fJ))\,.
\end{equation}
As already noticed, $A_+^{(2)}$ and $A_-^{(2)}$ are related by a local Lorentz transformation, $A_+^{(2)}=\mathrm{Ad}_hA_-^{(2)}$ for some $h\in G^{(0)}$. For later convenience we can also relate other components of $A_+$ and $A_-$ as follows\footnote{As a consequence of this we have for example $A_+^{(3)}=E^{(3)}-P^{(3)}ME^{(2)}$.}
\begin{align}\label{eq:M}
&A_-=MA_+\,,\qquad P^{(2)}M=\mathrm{Ad}_h^{-1}P^{(2)}
\,,
\\
&M=\mathrm{Ad}_f^{-1}[1-\tilde P-\widetilde{\mathcal{O}}^{-1}\widetilde{\mathcal{O}}^T-4\widetilde{\mathcal{O}}^{-1}\mathrm{Ad}_fP^{(2)}\mathrm{Ad}_f^{-1}(1-\tilde P)]\mathrm{Ad}_f=
1-4\mathrm{Ad}_f^{-1}\widetilde{\mathcal{O}}^{-1}\mathrm{Ad}_fP^{(2)}\,,
\nonumber
\end{align}
while $M^{-1}$ is given by the same expression as $M$ but with $\widetilde{\mathcal{O}}$ replaced by its transpose $\widetilde{\mathcal{O}}^T=\tilde P^T(\hat d_f^T+\mathrm{ad}_\nu+\zeta\omega)\tilde P$. From this we can derive the useful relation
\begin{equation}
M^{-1}-1=-(M-1)\mathrm{Ad}_h\,.
\end{equation}
\section{Target space superfields}\label{sec:target}
In this section we will derive the form of the target space supergravity superfields for the DTD model. The calculations are very similar to the ones performed in \cite{Borsato:2016ose} for the $\eta$-model and $\lambda$-model.
Once the action and kappa symmetry transformations are written in Green-Schwarz form as in~\eqref{eq:S-GS} and~\eqref{eq:deltakappa}, the easiest way to extract the background fields is by computing the torsion $T^a=dE^a+E^b\wedge\Omega_b{}^a$ and $T^\alpha=dE^\alpha-\frac14(\Gamma_{ab}E)^\alpha\wedge\Omega^{ab}$ where $\Omega^{ab}$ is the spin connection superfield. It was shown in \cite{Wulff:2016tju} that the constraints on the torsion implied by kappa symmetry take the form\footnote{This is valid only for a suitable choice of the spin connection, which can however be extracted from the same equations. We have dropped the $\wedge$'s for readability.}
\begin{align}
T^a=-\tfrac{i}{2}E\gamma^aE\,,\quad
T^{\alpha I}
=&\,
\tfrac12E^{\alpha I}\,E\chi
+\tfrac12(\sigma^3E)^{\alpha I}\,E\sigma^3\chi
-\tfrac14E\gamma_aE\,(\gamma^a\chi)^{\alpha I}
-\tfrac14E\gamma_a\sigma^3E\,(\gamma^a\sigma^3\chi)^{\alpha I}
\nonumber\\
&\qquad{}
-\tfrac18E^a\,(E\sigma^3\gamma^{bc})^{\alpha I} H_{abc}
-\tfrac18E^a\,(E\gamma_a\mathcal S)^{\alpha I}
+\tfrac12E^bE^a\,\psi^{\alpha I}_{ab}\,,
\label{eq:torsion}
\end{align}
for the type IIB case.\footnote{Essentially identical expressions hold for type IIA, cf. \cite{Wulff:2013kga}.} The target space superfields contained here are the dilatino superfields $\chi_{\alpha I}$, the gravitino field strengths $\psi_{ab}^{\alpha I}$, where $I=1,2$ denotes the two Majorana-Weyl spinors of type IIB, as well as the NSNS three-form field strength $H=dB$ and ``RR field strengths'' encoded in the anti-symmetric $32\times32$ bispinor
\begin{equation}
\mathcal S=-i\sigma^2\gamma^a\mathcal F_a-\tfrac{1}{3!}\sigma^1\gamma^{abc}\mathcal F_{abc}-\tfrac{1}{2\cdot5!}i\sigma^2\gamma^{abcde}\mathcal F_{abcde}\,.
\label{eq:calS}
\end{equation}
Kappa symmetry implies that the target space is generically only a solution of the \emph{generalised} type II supergravity equations defined in \cite{Wulff:2016tju} and first written down, for the bosonic sector, in~\cite{Arutyunov:2015mqj}. However, when the (Killing) vector
\begin{equation}
K^a=-\tfrac{i}{16}(\gamma^a\sigma^3)^{\alpha I\beta J}\nabla_{\alpha I}\chi_{\beta J}
\label{eq:Ka}
\end{equation}
vanishes one gets a solution of \emph{standard} type II supergravity, and a one-loop Weyl invariant string sigma model. In that case there exists a dilaton superfield $\phi$ such that $\chi_{\alpha I}=\nabla_{\alpha I}\phi$ and the RR field strengths are defined in terms of potentials in the standard way $\mathcal F=e^\phi dC+\cdots$ \cite{Tseytlin:1996hs,Wulff:2013kga}.
Given that the supervielbeins for the DTD model are defined in terms of $A_\pm$ as in (\ref{eq:E-A}) we need to compute the exterior derivative of $A_\pm$ defined in (\ref{eq:def-Apm}) to find the torsion. With a bit of work one finds the deformed ``Maurer-Cartan'' equations\footnote{We use anti-commutators rather than commutators because the objects that appear are one-forms, and therefore naturally anti-commute.}
\begin{align}
dA_+=&\,
\tfrac12\{A_+,A_+\}
-\tfrac12\mathrm{Ad}_f^{-1}\widetilde{\mathcal{O}}^{-T}\mathrm{Ad}_f\big(\hat d^T\{A_+,A_+\}-2\{A_+,\hat d^TA_+\}\big)\,,
\label{eq:dA+}
\\
dA_-=&\,
\tfrac12\{A_-,A_-\}
-\tfrac12\mathrm{Ad}_f^{-1}\widetilde{\mathcal{O}}^{-1}\mathrm{Ad}_f\big(\hat d\{A_-,A_-\}-2\{A_-,\hat dA_-\}\big)\,,
\label{eq:dA-}
\end{align}
where we have used the identity~\eqref{eq:PPT-rel} and the fact that, due to the Jacobi identity and the 2-cocycle condition~\eqref{eq:cocycle2}, both $\mathrm{ad}_\nu$ and $\omega$ effectively act as derivations on the Lie bracket.
Projecting the first equation with $P^{(2)}$ and using~\eqref{eq:E-A} and (\ref{eq:M}) we get
\begin{align}
dE^{(2)}=&
\{A_+^{(0)},E^{(2)}\}
+\tfrac12\{E^{(1)},E^{(1)}\}
+\tfrac12\{E^{(3)},E^{(3)}\}
-\{E^{(3)},P^{(3)}ME^{(2)}\}
-P^{(2)}M^T\{E^{(2)},E^{(3)}\}
\nonumber\\
&{}
+\tfrac12\{P^{(3)}ME^{(2)},P^{(3)}ME^{(2)}\}
+P^{(2)}M^T\{E^{(2)},P^{(3)}ME^{(2)}\}
-\tfrac12P^{(2)}M^T\{E^{(2)},E^{(2)}\}\,.
\end{align}
Using $A_+^{(0)}=\frac12A_+^{ab}J_{ab}$, $E^{(2)}=E^aP_a$ etc. and the algebra in appendix A of \cite{Borsato:2016ose} this gives the form for the bosonic torsion $T^a$ in~\eqref{eq:torsion} provided that we identify the spin connection with\footnote{The components of $M$ are defined as $MT_A=T_BM^B{}_A$.}
\begin{equation}
\Omega_{ab}
=
(A_+)_{ab}
+2i(E^2\gamma_{[a})_{\beta}M^{\beta2}{}_{b]}
+\tfrac{3i}{2}E^cM^{\alpha2}{}_{[a}(\gamma_b)_{\alpha\beta}M^{\beta2}{}_{c]}
+\tfrac12E^c(M_{ab,c}-2M_{c[a,b]})\,.
\end{equation}
In a similar way, using (\ref{eq:E-A}) and (\ref{eq:dA-}) we find that
\begin{align}
dE^{(3)}=&
\{A_+^{(0)},E^{(3)}\}
+\{P^{(0)}ME^{(2)},E^{(3)}\}
+\mathrm{Ad}_h^{-1}\{E^{(1)}+P^{(1)}\mathrm{Ad}_hME^{(2)},E^{(2)}\}
+\tfrac12P^{(3)}M\{E^{(3)},E^{(3)}\}
\nonumber\\
&{}
+2P^{(3)}\mathrm{Ad}_f^{-1}\widetilde{\mathcal{O}}^{-1}\mathrm{Ad}_f\Big(
2\mathrm{Ad}_h^{-1}\{E^{(1)}+P^{(1)}\mathrm{Ad}_hME^{(2)},E^{(2)}\}
+\mathrm{Ad}_h^{-1}\{E^{(2)},E^{(2)}\}
\Big),
\end{align}
which leads to the torsion $T^{\alpha 2}$ taking the form in (\ref{eq:torsion}) with the background fields given by\footnote{These expressions have obvious close analogies with the ones found for the $\eta$-model in \cite{Borsato:2016ose}.}
\begin{align}
H_{abc}=&3M_{[ab,c]}-3iM^{\alpha2}{}_{[a}(\gamma_b)_{\alpha\beta}M^{\beta2}{}_{c]}
\,,\quad
\mathcal S^{\alpha1\beta2}=
-8i[\mathrm{Ad}_h(1+4\mathrm{Ad}_f^{-1}\widetilde{\mathcal{O}}^{-T}\mathrm{Ad}_f)]^{\alpha1}{}_{\gamma1}\widehat{\mathcal K}^{\gamma1\beta2},
\label{eq:HandS}
\\
\chi^2_\alpha=&-\tfrac{i}{2}\gamma^a_{\alpha\beta}M^{\beta2}{}_a
\,,\quad
\psi_{ab}^{\alpha2}
=
2[\mathrm{Ad}_f^{-1}\widetilde{\mathcal{O}}^{-1}\mathrm{Ad}_f\mathrm{Ad}_h^{-1}]^{\alpha2}{}_{cd}\widehat{\mathcal K}_{ab}{}^{cd}
+\tfrac14[\mathrm{Ad}_hM]^{\beta1}{}_{[a}(\gamma_{b]}\mathcal S^{12})_\beta{}^\alpha\,.
\nonumber
\end{align}
Here $\widehat{\mathcal K}^{AB}$ denotes the inverse of the metric defined by the supertrace $\mathrm{Str}(T_AT_B)=\mathcal K_{AB}$, see appendix A of \cite{Borsato:2016ose} for more details on our conventions.
Since the DTD model contains NATD as a special case we obtain as a by-product the transformation rules for RR fields under NATD---starting from a supercoset model. As a check we can compare this to the formula conjectured in \cite{Sfetsos:2010uq} based on analogy to the abelian case \cite{Hassan:1999bv}---consistency of that formula was checked in some particular cases {also} in \cite{Hoare:2016wca}. Setting $\zeta=0$, which removes the deformation, and restricting to a bosonic $\tilde{\mathfrak g}$, so that $\tilde P=\tilde P(P^{(0)}+P^{(2)})=(P^{(0)}+P^{(2)})\tilde P$, we find\footnote{Note that $(P^{(0)}+P^{(2)})\mathrm{Ad}_fP^{(1)}=0+$fermions.}
\begin{equation}
\mathcal S^{\alpha1\beta2}=
-8i[\mathrm{Ad}_h|_{\theta=0}]^{\alpha1}{}_{\gamma1}\widehat{\mathcal K}^{\gamma1\beta2}
+\mbox{fermions}\,,
\end{equation}
which agrees with the transformations conjectured in \cite{Sfetsos:2010uq}. Note that our result generalises this to the case where also fermionic T-dualities are involved.
Finally we must compute $T^{\alpha1}$ to extract the other dilatino superfield $\chi^1$. We find
\begin{align}
dE^{(1)}
=&
\{\mathrm{Ad}_hA_+^{(0)}-dhh^{-1},E^{(1)}\}
+\mathrm{Ad}_h\{E^{(2)},E^{(3)}-P^{(3)}ME^{(2)}\}
+\tfrac12P^{(1)}\mathrm{Ad}_hM^{-1}\mathrm{Ad}_h^{-1}\{E^{(1)},E^{(1)}\}
\nonumber\\
&{}
+2P^{(1)}\mathrm{Ad}_h\mathrm{Ad}_f^{-1}\widetilde{\mathcal{O}}^{-T}\mathrm{Ad}_f\Big(
2\{E^{(2)},E^{(3)}-P^{(3)}ME^{(2)}\}
+\{E^{(2)},E^{(2)}\}
\Big)\,.
\end{align}
Taking the exterior derivative of the equation $A_+^{(2)}=\mathrm{Ad}_hA_-^{(2)}$, cf. (\ref{eq:M}), we find the relation
\begin{equation}
[\mathrm{Ad}_hA_+^{(0)}-dhh^{-1}]_{ab}
=
\Omega_{ab}
-\tfrac12E^cH_{abc}
+2i(E^1\gamma_{[a})_\alpha[\mathrm{Ad}_hM]^{\alpha1}{}_{b]},
\end{equation}
which can be used to show that the torsion again takes the form in (\ref{eq:torsion}), where the remaining components of the background fields are\footnote{Just as in \cite{Borsato:2016ose}, one finds a superficially different expression for $H_{abc}$ namely
$$
H_{abc}=3[\mathrm{Ad}_hM]_{[ab,c]}+3i[\mathrm{Ad}_hM]^{\alpha1}{}_{[a}(\gamma_b)_{\alpha\beta}[\mathrm{Ad}_hM]^{\beta1}{}_{c]}\,.
$$
However consistency requires this to be the same as the expression in (\ref{eq:HandS}) and this can also be verified explicitly similarly to \cite{Borsato:2016ose}.
}
\begin{equation}
\chi^1_\alpha=\tfrac{i}{2}(\gamma^a)_{\alpha\beta}[\mathrm{Ad}_hM]^{\beta1}{}_a\,,\quad
\psi^{\alpha1}_{ab}
=
2[\mathrm{Ad}_h\mathrm{Ad}_f^{-1}\widetilde{\mathcal{O}}^{-T}\mathrm{Ad}_f]^{\alpha1}{}_{cd}\widehat{\mathcal K}_{ab}{}^{cd}
-\tfrac14(\mathcal S^{12}\gamma_{[a})^\alpha{}_\beta M^{\beta2}{}_{b]}\,.
\label{eq:chi1}
\end{equation}
It remains only to analyse the question of when this is a solution to the standard or the generalised type II supergravity equations, in other words to identify the conditions under which $K^a$ defined in (\ref{eq:Ka}) vanishes. We do this in the next subsection.
\subsection{Supergravity condition and dilaton}
By analogy with the calculations performed in \cite{Borsato:2016ose} there is a natural candidate for the dilaton superfield for the DTD model namely\footnote{The prime on the superdeterminant denotes the fact that we must restrict to the subspace where $\widetilde{\mathcal{O}}$ is defined, i.e. the subalgebra $\tilde{\mathfrak g}$.}
\begin{equation}
e^{-2\phi}=\mathrm{sdet}'\widetilde{\mathcal{O}}\,.
\end{equation}
We will now show that this guess is indeed correct by verifying that its spinor derivatives reproduces the dilatini found above.
Using the formula for the supertrace $\mathrm{Str}\mathcal M=\widehat{\mathcal K}^{AB}\mathrm{Str}(T_A\mathcal MT_B)$ we find
\begin{align}
d\phi
=&
-\tfrac12\mathrm{Str}(d\widetilde{\mathcal{O}}\op^{-1})
=
-\tfrac12\widehat{\mathcal K}^{AB}\mathrm{Str}\big\{([J,\hat d^T_fT_A]-\hat d^T_f[J,T_A]+[d\nu,T_A])\widetilde{\mathcal{O}}^{-1}T_B\big\}
\nonumber\\
=&
-\tfrac12\widehat{\mathcal K}^{AB}\mathrm{Str}\big\{\big(
[J,\hat d^T_fT_A]
-\hat d^T_f[J,T_A]
+[\mathrm{Ad}_f\hat d^TA_+,T_A]
+[(\mathrm{ad}_\nu+\zeta\omega)(\mathrm{Ad}_fA_+-J),T_A]
\big)\widetilde{\mathcal{O}}^{-1}T_B
\big\}
\nonumber\\
=&
\tfrac12\widehat{\mathcal K}^{AB}\mathrm{Str}\big\{
T_A\big(
\hat d[A_+,\mathrm{Ad}_f^{-1}\widetilde{\mathcal{O}}^{-1}\mathrm{Ad}_fT_B]
+[\hat d^TA_+,\mathrm{Ad}_f^{-1}\widetilde{\mathcal{O}}^{-1}\mathrm{Ad}_fT_B]
-[A_+,\hat d\mathrm{Ad}_f^{-1}\widetilde{\mathcal{O}}^{-1}\mathrm{Ad}_fT_B]
\big)
\big\}
\nonumber\\
&{}
+\widehat{\mathcal K}^{AB}\mathrm{Str}\big\{[(\mathrm{Ad}_fA_+-J),T_A]\tilde PT_B\big\}\,.
\label{eq:dphi}
\end{align}
If the last term vanishes, then using (\ref{eq:E-A}), (\ref{eq:chi1}), (\ref{eq:HandS}) and (\ref{eq:M}) one may check that the $E^{(1,3)}$-terms are indeed equal to
\begin{equation}
E^{\alpha1}\chi^1_\alpha+E^{\alpha2}\chi^2_\alpha\,.
\end{equation}
Therefore $\chi_{\alpha I}=\nabla_{\alpha I}\phi$ which implies that $K^a$ in (\ref{eq:Ka}) vanishes and we have a solution to standard type II supergravity.
Since $(\mathrm{Ad}_fA_+-J)\in\tilde{\mathfrak g}$ can be regarded as an arbitrary element of the Lie algebra, the vanishing of the last term in (\ref{eq:dphi}) is equivalent to $f_{AB}{}^A=0$ for the structure constants of $\tilde{\mathfrak{g}}$, i.e. $\tilde{\mathfrak{g}}$ must be unimodular. This condition is therefore sufficient to get a standard supergravity solution. Following a calculation similar to the one done in \cite{Borsato:2016ose}, computing $K^a$ in (\ref{eq:Ka}) and requiring it to vanish one finds that this condition is also necessary.\footnote{{In very special cases it is possible for $K^a$ to decouple from the remaining generalized supergravity equations. One then obtains a background solving both the generalised and standard supergravity equations depending on if $K^a$ is included or not. One such example is the pp-wave solution discussed in Appendix B of \cite{Hoare:2016hwh}.} We thank B. Hoare and S. van Tongeren for pointing this out.}
Our results imply that the DTD model gives a one-loop Weyl invariant string sigma model precisely\footnote{{This is modulo possible subtleties with the special cases mentioned in the previous footnote. One should also note that this condition is true provided one only allows a \emph{local} (Fradkin-Tseytlin) counter-term. If one relaxes this condition one can find a \emph{non-local} counter-term also when $K^a$ is non-zero, since solutions of the generalised supergravity equations are formally T-dual to solutions of the standard ones; see also \cite{Sakamoto:2017wor}. This being said, cases where $K^a$ is null may be subtle and deserve further study.}} when the subalgebra $\tilde{\mathfrak{g}}$ is unimodular. This is in fact the same condition that was found long ago for NATD on bosonic sigma models by path integral considerations \cite{Alvarez:1994np,Elitzur:1994ri}. Since the DTD model includes NATD as a special case, the analysis here coupled with the results of \cite{Arutyunov:2015mqj,Wulff:2016tju}, gives an alternative derivation of the Weyl anomaly for NATD of supercosets.
A nice fact is that we do not have to impose extra conditions on the cocycle $\omega$ used to construct the deformation. When $\omega$ is non-degenerate unimodularity of $\tilde{\mathfrak{g}}$ is equivalent to unimodularity of $R=\omega^{-1}$ as defined in \cite{Borsato:2016ose}, see the discussion there; this is consistent with the fact that the YB models are a special case of the DTD models.
\section{Some explicit examples}\label{sec:ex}
Here we would like to collect some formulas that are useful when deriving the explicit background for a given DTD model, and then work out {two} examples in detail.
We denote the generators of $\tilde{\alg{g}}\subset \alg g$ by $ T_i$, $i=1,\ldots,N=\text{dim}(\tilde{\alg{g}})$, and those of the dual $\tilde{\alg{g}}^*$ by $ T^i$. They satisfy $\operatorname{Str}(T^iT_j)=\delta^i_j$. The action of the projectors on a generic element $x\in\alg g$ may be written as
\begin{equation}
\tilde P(x)=\operatorname{Str}(T^ix)T_i,\qquad
\tilde P^T(x)=\operatorname{Str}(T_ix)T^i,
\end{equation}
where summation of repeated indices is assumed. Given a cocycle $\omega=\tfrac12\omega_{ij}T^i\wedge T^j$ with $\omega_{ji}=-\omega_{ij}$, its action on an element of the algebra is
\begin{equation}
\omega(x) = \omega_{ij}T^i\operatorname{Str}(T^jx),
\end{equation}
and it must satisfy the cocycle condition, which may be written as
\begin{equation}
\operatorname{Str}\Big(T_k(\omega[{T_i},T_j]-[{T_i},\omega T_j]+[{T_j},\omega T_i])\Big)=0,
\qquad\quad \forall T_i,T_j,T_k\in\tilde{\alg{g}}.
\end{equation}
With the above definitions one may easily construct the operator $\widetilde{\mathcal{O}}:\tilde{\alg{g}}\to\tilde{\alg{g}}^*$ defined in (\ref{eq:def-op}), that can be encoded in an explicit $N\times N$ matrix
\begin{equation}
\tilde O_{ij}=\operatorname{Str}(\widetilde{\mathcal{O}}(T_i)T_j),
\end{equation}
so that $\widetilde{\mathcal{O}}(T_i)=\tilde O_{ij}T^j$.
The matrix $\tilde O$ can be inverted with standard methods and used to construct the action of the inverse operator as $ \widetilde{\mathcal{O}}^{-1}(x)=\operatorname{Str}(xT_i)(\tilde O^{-1})^{ij}T_j$, so that on the basis generators $\widetilde{\mathcal{O}}^{-1}(T^i)=(\tilde O^{-1})^{ij}T_j$.
Obviously, when choosing a parametrisation for the group element $f$, one should make sure that the corresponding degrees of freedom cannot be gauged away by applying the local transformations discussed in section~\ref{sec:DTD}.
To obtain the background fields we use the results of section~\ref{sec:target}.
The metric reads as $ds^2=\eta^{ab}E_aE_b$, where the components of the bosonic supervielbein are obtained by $E_a=\operatorname{Str}(A_+ P_a)$, and the $B$-field is given by equation~\eqref{eq:B-field}.
From the superdeterminant of the matrix $\tilde O$ it is also straightforward to compute the (exponential of the) dilaton $e^\phi=(\mathrm{sdet}\,\tilde O)^{-\frac{1}{2}}$.
In order to determine the RR fields one first identifies the components of the matrix $M_{ab}=\operatorname{Str}((M{P}_a){P}_b)$ and then one constructs the local Lorentz transformation on spinorial indices
\begin{equation}
{(\mathrm{Ad}_h)^{\beta}}_{\alpha} = \exp[- \tfrac{1}{4} (\log M)_{ab} \Gamma^{ab} ]{^{\beta}}_{\alpha}\,,
\end{equation}
so that $\mathrm{Ad}_h\Gamma_a\mathrm{Ad}_h^{-1} =M_a^{\ b}\Gamma_b$, where $\Gamma_a$ are $32\times 32$ Gamma-matrices\footnote{Alternatively one can use the $16\times16$ gamma matrices used in the previous section.}. From (\ref{eq:calS}) and (\ref{eq:HandS}) one finds that the expression for RR fields is obtained by solving the equation
\begin{equation}
(\Gamma^a F_a + \tfrac{1}{3!} \Gamma^{abc} F_{abc} + \tfrac{1}{2\cdot 5!} \Gamma^{abcde} F_{abcde})\Pi =
e^{-\phi} \ [\mathrm{Ad}_h(1+4\mathrm{Ad}_f^{-1}\widetilde{\mathcal{O}}^{-T}\mathrm{Ad}_f)](4 \Gamma_{01234})\Pi,
\end{equation}
where $\Pi = \tfrac{1}{2} (1-\Gamma_{11})$ is a projector\footnote{With these conventions the self-duality for the 5-form is $F^{(5)}=*F^{(5)}$.} and $(-4 \Gamma_{01234})\Pi$ corresponds to the 5-form flux of AdS$_5\times$S$^5$.
In order to find the component $F_{a_1\ldots a_{2m+1}}$ it is then enough to multiply the above equation by $\Gamma_{a_1\ldots a_{2m+1}}$ and take the trace. As already explained, when the subalgebra $\tilde{\alg{g}}$ is bosonic the above result simplifies considerably, and only $\mathrm{Ad}_h$ remains inside square brackets.
After obtaining the components in tangent indices we translate them into form language using
$F^{(2m+1)}=\frac{1}{(2m+1)!}E^{a_{2m+1}}\wedge\ldots\wedge E^{a_1} F_{a_1\ldots a_{2m+1}}$.
\subsection{A TsT example}
{First we will work out a simple example where we dualise} a two-dimensional abelian subalgebra of the isometry of the sphere $\alg{so}(6)$, so that the deformation is equivalent to doing a TsT there~\cite{Lunin:2005jy,Frolov:2005ty,Frolov:2005dj}. This example was worked out already in~\cite{Hoare:2016wsk} for the NSNS sector, and the RR fields were taken into account in~\cite{Hoare:2016wca} by following the T-duality rules of~\cite{Sfetsos:2010uq}.
Here we will use the matrix realisation of the $\alg{psu}(2,2|4)$ superalgebra used in~\cite{Borsato:2016ose}, see also~\cite{Arutyunov:2015qva}.
We take $\tilde{\alg{g}}$ to be the abelian algebra spanned by two Cartans of $\alg{so}(6)$, $T_1\equiv J_{68},T_2\equiv J_{79}$, and for the dual generators we may just take $T^1= J_{68},T^2= J_{79}$.
We parametrise the bosonic fields as\footnote{The group elements parametrised by $\varphi$, $\xi$ and $r$ coincide with those in (A.1) of~\cite{Arutyunov:2013ega}.}
\begin{equation}
\nu = \tilde \varphi_iT^i,\qquad\qquad
f=f_{\alg{a}} \cdot \exp(\varphi P_5)\exp(-\xi J_{89})\exp(-\arcsin rP_9),
\end{equation}
where $f_{\alg{a}}$ is a coset group element parametrised by fields in AdS$_5$.
We take $\omega=T^1\wedge T^2$ which obviously satisfies the cocycle condition. The matrix corresponding to $\widetilde{\mathcal{O}}$ is very simple
\begin{equation}
\tilde O_{ij}=\left(
\begin{array}{cc}
2 r^2 \sin ^2\xi & \zeta \\
-\zeta & 2 r^2 \cos ^2\xi \\
\end{array}
\right),
\end{equation}
and it is easily inverted. Following the above discussion we immediately find the fields of the NSNS sector
\begin{equation}
\begin{aligned}
&ds^2=ds_{\alg{a}}^2+
\frac{r^2 }{\zeta ^2+r^4 \sin ^2(2 \xi )}(\cos ^2\xi\, d\tilde\varphi_1^2+\sin ^2\xi\, d\tilde\varphi_2^2)
+(1-r^2)d\varphi^2+r^2d\xi^2+\frac{dr^2}{1-r^2}\,,
\\
&e^\phi=(\zeta ^2+r^4 \sin ^2(2 \xi ))^{-\frac{1}{2}}\,,\qquad B=\frac{\zeta}{2}\ \frac{d\tilde \varphi_1\wedge d\tilde \varphi_2}{\zeta ^2+r^4 \sin ^2(2 \xi )}\,,
\end{aligned}
\end{equation}
where $ds_{\alg{a}}^2 $ is the metric of AdS$_5$.
After computing the matrix $M_{ab}$ and the local Lorentz transformation\footnote{For $32\times 32$ Gamma matrices we find convenient the basis used in~\cite{Arutyunov:2015qva}.} we get that only $F^{(3)}$ and $F^{(5)}$ are non-vanishing
\begin{equation}
\begin{aligned}
&F^{(3)}=4 r^3 \sin (2 \xi ) d\varphi \wedge d\xi \wedge dr,\\
&F^{(5)}=-2 \zeta(1+*)\left(\frac{ r^3 \sin (2 \xi )\, d\tilde \varphi_1\wedge d\tilde \varphi_2\wedge d\varphi \wedge d\xi \wedge dr}{\zeta ^2+r^4 \sin ^2(2 \xi )}\right).
\end{aligned}
\end{equation}
Since $\omega$ is non-degenerate on $\tilde{\alg{g}}$ we can relate the above background to a YB deformation of AdS$_5\times$S$^5$, see also section~\ref{sec:YB}. In this particularly simple example the $R$-matrix of the YB model is abelian, and therefore it corresponds just to a TsT transformation on the sphere, see also~\cite{Osten:2016dvf}.
In fact, consider the following TsT transformation on AdS$_5\times$S$^5$
\begin{equation}
\varphi_1\to T(\varphi_1),\qquad
\varphi_2\to \varphi_2-2\eta T(\varphi_1),\qquad
T(\varphi_1)\to \varphi_1,
\end{equation}
which produces the following background\footnote{As a starting point we take the undeformed AdS$_5\times$S$^5$ background as written in~\cite{Arutyunov:2015qva}.}
\begin{equation}
\begin{aligned}
&ds^2=ds_{\alg{a}}^2+
\frac{r^2 }{1+\eta^2r^4 \sin ^2(2 \xi )}(\cos ^2\xi\, d\varphi_2^2+\sin ^2\xi\, d\varphi_1^2)
+(1-r^2)d\varphi^2+r^2d\xi^2+\frac{dr^2}{1-r^2}\,,\\
&e^\phi=(1+\eta ^2r^4 \sin ^2(2 \xi ))^{-\frac{1}{2}}\,,\qquad
B= -\frac{\eta r^4 \sin ^2(2 \xi ) d \varphi_1\wedge d \varphi_2}{1+\eta^2r^4 \sin ^2(2 \xi )}\,
\end{aligned}
\end{equation}
for the NSNS sector and
\begin{equation}
\begin{aligned}
&F^{(3)}=4\eta r^3 \sin (2 \xi ) d\varphi \wedge d\xi \wedge dr,\\
&F^{(5)}=-2 (1+*)\left(\frac{ r^3 \sin (2 \xi )\, d \varphi_1\wedge d \varphi_2\wedge d\varphi \wedge d\xi \wedge dr}{1+\eta^2r^4 \sin ^2(2 \xi )}\right),
\end{aligned}
\end{equation}
for the RR sector.
To match with the above TsT background we need to implement the field redefinition~\eqref{eq:field-red-DTD-YB} at the level of the DTD background, which in this case just reduces to $\tilde \varphi_1=\eta^{-1}\varphi_2, \tilde \varphi_2=-\eta^{-1}\varphi_1$ since $\tilde{\alg{g}}$ is abelian.
We find agreement only if we also use the gauge freedom for $B$ to subtract the exact term $\frac{1}{2\eta}d \varphi_1\wedge d \varphi_2$; moreover we also need to redefine the constant part of the dilaton to reabsorb a factor of $\eta$, which then appears in front of the RR fields.
\subsection{A new example}\label{sec:ex-not-YB}
{Let us now consider the example in (\ref{eq:ex1})}
\begin{equation}
\tilde{\mathfrak g}=\mathrm{span}\{p_1,\,p_2,\,p_3,\,J_{12}\}\,,
\qquad
\tilde{\mathfrak g}^*=\mathrm{span}\{-\tfrac{1}{2}k_1,\,-\tfrac{1}{2}k_2,\,-\tfrac{1}{2}k_3,\,-J_{12}\}
\qquad\omega=k_3\wedge J_{12}\,.
\end{equation}
In this case we have just one isometry of type 1 corresponding to $p_0$, and the isometries of type 2 are $k_3$ and $J_{12}$. Inspired by the parametrisation used in (6.19) of~\cite{Borsato:2016ose} we parametrise\footnote{Even if present, one could remove $k_2$ in $\nu$ by means of a gauge transformation.}
\begin{equation}
\nu=\tilde \xi\ J_{12}+\tilde r\ k_1+ \tilde x^3\ k_3\,,
\qquad
f=\text{exp}(x^0p_0)\, \text{exp}(\log z D)\,.
\end{equation}
The above is a good parametrisation because it is not possible to remove degrees of freedom by applying gauge transformations. This will be confirmed e.g. by the fact that we get a non-degenerate metric in target space. We find that the (matrix corresponding to the) operator $\widetilde{\mathcal{O}}$ is
\begin{equation}
\tilde O_{ij}=\left(
\begin{array}{cccc}
\frac{2}{z^2} & 0 & 0 & 0 \\
0 & \frac{2}{z^2} & 0 & 2 \tilde{r} \\
0 & 0 & \frac{2}{z^2} & 2 \zeta \\
0 & -2 \tilde{r} & -2 \zeta & 0 \\
\end{array}
\right)\,,
\end{equation}
which is clearly invertible. We find the following NSNS sector fields
\begin{equation}
\begin{aligned}
&ds^2=\frac{-(dx^0)^2+dz^2}{z^2}+d\tilde r^2z^2
+\frac{d\tilde \xi^2}{4 z^2 \left(\zeta
^2+\tilde{r}^2\right)}+\frac{\tilde{r}^2 z^2(d\tilde x^3)^2}{\zeta
^2+\tilde{r}^2}+ds_{\alg{s}}^2
\,,\\
&e^\phi=\left(\frac{16 \left(\zeta ^2+\tilde{r}^2\right)}{z^4}\right)^{-\frac12}\,,\qquad
B= -\frac{\zeta d\tilde{\xi}\wedge d\tilde x^3 }{2 \left(\zeta
^2+\tilde{r}^2\right)}\,,
\end{aligned}
\end{equation}
where $ds_{\alg{s}}^2$ is the metric on S$^5$.
In the RR sector we have only three-form flux
\begin{equation}
F^{(3)}=-\frac{8 (dx^0\wedge d\tilde{\xi}\wedge dz)}{z^5}\,.
\end{equation}
According to the discussion in section~\ref{sec:YB} the above background is not related to a YB model by NATD.
\section{Conclusions}
We have argued that DTD models based on supercosets represent a large class of integrable string models which is closed under NATD as well as (certain) deformations. Besides being a useful tool to generate new integrable supergravity backgrounds it would be very interesting if these deformations could be understood on the dual field theory side. In the case when the 2-cocycle is invertible these models are equivalent to YB sigma models, which have been argued to correspond to non-commutative deformations, e.g. \cite{Hashimoto:1999ut,Maldacena:1999mh}, of the field theory {\cite{Matsumoto:2014gwa,vanTongeren:2015uha,vanTongeren:2016eeb}} (see also \cite{Araujo:2017jkb}). This interpretation is consistent with the fact that TsT transformations are special cases of these models \cite{Matsumoto:2014nra,Osten:2016dvf} and this includes the so-called $\beta$ and $\gamma$-deformations which have a known interpretation in $\mathcal N=4$ super Yang-Mills \cite{Leigh:1995ep,Lunin:2005jy,Frolov:2005ty,Frolov:2005iq}. Recently a certain limit of the $\gamma$-deformation has been used to construct a simplified integrable scalar field theory \cite{Gurdogan:2015csr,Gromov:2017cja} and it would be very interesting to explore similar limits of the more general class of deformations considered here to see whether one can learn more about the AdS/CFT duality for those cases.
Another important question is how the DTD model relates to the other known deformations of the $AdS_5\times S^5$ string, i.e. the $\eta$-model with R-matrix solving the modified CYBE \cite{Delduc:2013fga} and the $\lambda$-model \cite{Hollowood:2014qma}. These two deformations are related by Poisson-Lie T-duality and the fact that the latter is Weyl-invariant \cite{Borsato:2016ose} while the former is not \cite{Arutyunov:2015qva,Arutyunov:2015mqj} is explained by the fact that the obstruction to the duality at the quantum level again involves the trace of the structure constants \cite{Tyurin:1995bu}.\footnote{We thank A. Tseytlin for this comment.} The fact that NATD is used also in the construction of the $\lambda$-model suggests that there might be a bigger picture relating it to the DTD construction considered here. In fact this seems to be part of an even bigger picture of general integrable deformations of sigma models where T-duality and its generalizations play a central role, see for example the recent paper \cite{Klimcik:2017ken}.
\section*{Acknowledgements}
{We thank B. Hoare, S. van Tongeren and A. Torrielli for interesting and useful discussions, and A. Tseytlin for illuminating discussions and comments on the manuscript.}
The work of R.B. was supported by the ERC advanced grant No 341222.
We also thank the Galileo Galilei Institute for Theoretical Physics (GGI) for the hospitality and INFN for partial support during the completion of this work at the program {\it New Developments in $AdS_3/CFT_2$ Holography}.
\vspace{2cm}
|
1,108,101,564,562 | arxiv | \section{Introduction}
A classical-quantum channel
$W:\,{\mathcal X}\to{\mathcal S}({\mathcal H})$ models a device
with a set of possible inputs ${\mathcal X}$, which, on an input $x\in{\mathcal X}$, outputs
a quantum system with finite-dimensional Hilbert space ${\mathcal H}$
in state $W(x)$.
For every such channel $W:\,{\mathcal X}\to{\mathcal S}({\mathcal H})$, we define the lifted channel
\begin{align*}
\ext{W}:\,{\mathcal X}\to{\mathcal S}({\mathcal H}_{\mathcal X}\otimes{\mathcal H}),\mbox{ }\mbox{ }\ds\mbox{ }\mbox{ }
\ext{W}(x):=\pr{x}\otimes W(x).
\end{align*}
Here, ${\mathcal H}_{{\mathcal X}}$ is an auxiliary Hilbert space, and $\{\ket{x}:\,x\in{\mathcal X}\}$ is an orthonormal basis in it.
As a canonical choice, one can use ${\mathcal H}_{{\mathcal X}}=l^2({\mathcal X})$, the $L^2$-space on ${\mathcal X}$ with respect to the counting measure,
and choose $\ket{x}:=\mathbf 1_{\{x\}}$ to be the characteristic function (indicator function) of the singleton $\{x\}$.
Note that this is well-defined irrespectively of the cardinality of ${\mathcal X}$.
The classical-quantum state $\ext{W}(P):=\sum_{x\in{\mathcal X}}P(x)\pr{x}\otimes W(x)$ plays the role of the joint distribution of the input and the output of the channel for a fixed finitely supported input probability distribution $P\in{\mathcal P}_f({\mathcal X})$.
Given a quantum divergence $\divv$, i.e., some sort of generalized distance of quantum states, there are various natural-looking but inequivalent ways to define the
corresponding capacity of the channel
for a fixed input probability distribution $P$.
One possibility is a mutual information-type quantity
\begin{align}
I_{\divv}(W,P)&:=\inf_{\sigma\in{\mathcal S}({\mathcal H})}\divv\left(\ext{W}(P)\|P\otimes\sigma\right),\label{cap1}
\end{align}
where one measures the $\divv$-distance of the joint input-output state of the channel from the set of uncorrelated states, while the first marginal is kept fixed. This can be interpreted as a measure of the
maximal amount of correlation that can be created between the input and the output of the channel with a fixed input distribution. The idea is that the more correlated the input and the output can be made, the more useful the channel is for information transmission.
Another option is to use the $P$-weighted $\divv$-radius of the image of the channel, defined as
\begin{align}
\chi_{\divv}(W,P)&:=\inf_{\sigma\in{\mathcal S}({\mathcal H})}\sum_{x\in{\mathcal X}}P(x)\divv(W(x)\|\sigma).\label{cap2}
\end{align}
This approach is geometrically motivated, and the idea behind is that the further away some states are in $\divv$-distance (weighted by the input distribution $P$), the more distinguishable they are, and the information transmission capacity of the channel is related to the number of far away states
among the output states of the channel.
In the case where $W$ is a classical channel, and $\divv$ is a R\'enyi divergence, these quantities were
studied by Sibson \cite{Sibson} and Augustin \cite{Augustin}, respectively; see \cite{Csiszar},
and the recent works
\cite{NakibogluRenyi,NakibogluAugustin,Cheng-Li-Hsieh2018} for more references on the history and applications of these quantities.
It was shown in \cite{Csiszar} that for classical channels,
both quantities yield the same channel capacity when $\divv=D_{\alpha}$ is a R\'enyi $\alpha$-divergence, i.e.,
\begin{align}\label{capacity equality}
C_{\divv}(W):=\sup_{P\in{\mathcal P}_f({\mathcal X})}\chi_{\divv}(W,P)
=
\sup_{P\in{\mathcal P}_f({\mathcal X})}I_{\divv}(W,P)
=
\inf_{\sigma\in{\mathcal S}({\mathcal H})}\sup_{x\in{\mathcal X}}\divv(W(x)\|\sigma),
\end{align}
where the last quantity is the $\divv$-radius of the image of the channel.
This was extended to the case of classical-quantum channels and a variety of quantum R\'enyi divergences
in a series of work \cite{KW,MH,MO-cqconv,WWY}.
Moreover, it was shown in \cite{MO-cqconv} (extending the corresponding classical result of \cite{DK,CsiszarKorner,Csiszar}) that
the strong converse exponent $\mathrm{sc}(W,R)$ of a classical-quantum channel $W$
for coding rate $R$ can be expressed as
\begin{align*}
\mathrm{sc}(W,R)=\sup_{\alpha>1}\frac{\alpha-1}{\alpha}\left[R-\chi_{\alpha}^{*}(W)\right],
\end{align*}
where $\chi_{\alpha}^{*}(W)=\chi_{D_{\alpha}^{*}}(W)$ is the R\'enyi capacity of $W$ corresponding to the
sandwiched R\'enyi divergences $D_{\alpha}^{*}$ \cite{Renyi_new,WWY}, thus giving an operational interpretation
to the sandwiched R\'enyi capacities with parameter $\alpha>1$.
After this, it is natural to ask which of the two quantities presented in \eqref{cap1} and \eqref{cap2} has an operational interpretation, and for what divergence. Note that the standard channel coding problem does not
yield an answer to this question, essentially due to \eqref{capacity equality}, so to settle this problem, one needs to consider a refinement of the channel coding problem where the input distribution $P$ appears on the operational side. This can be achieved by considering constant composition coding, where the codewords are required to have the same empirical distribution for each message, and these empirical distributions are required to converge to a fixed distribution $P$ on ${\mathcal X}$ as the number of channel uses goes to infinity.
It was shown in \cite{Csiszar} that in this setting
\begin{align}\label{Csiszar cc sc}
\mathrm{sc}(W,R,P)=\sup_{\alpha>1}\frac{\alpha-1}{\alpha}\left[R-\chi_{\alpha}(W,P)\right],
\end{align}
for any classical channel $W$ and input distribution $P$,
where $\mathrm{sc}(W,R,P)$ is the strong converse exponent for coding rate $R$, and
$\chi_{\alpha}(W,P)=\chi_{D_{\alpha}}(W,P)$.
This shows that, maybe somewhat surprisingly, it is not the perhaps more intuitive-looking concept of mutual information
\eqref{cap1} but the geometric quantity \eqref{cap2} that correctly captures the information transmission capacity
of a classical channel.
Our main result is an exact analogue of \eqref{Csiszar cc sc} for classical-quantum channels, as given in
\eqref{main result abstract}. Thus we establish that the operationally relevant notion of R\'enyi capacity with fixed input distribution $P$ for a
classical-quantum channel in the strong converse domain is the sandwiched R\'enyi divergence radius of the channel.
The structure of the paper is as follows.
After collecting some technical preliminaries in Section \ref{sec:Preliminaries}, we study the concepts of
divergence radius and center for general divergences in Section \ref{sec:gendivrad} and for R\'enyi divergences
in Section \ref{sec:Renyi divrad}.
One of our main results is the additivity of the
weighted R\'enyi divergence radius for classical-quantum channels, given in Section \ref{sec:additivity}.
We prove it using a representation of the minimizing state in \eqref{cap2} when $\divv=D_{\alpha,z}$
is a quantum $\alpha$-$z$ R\'enyi divergence \cite{AD},
as the fixed point of a certain map on
the state space. Analogous results have been derived very recently by Nakibo\u glu in \cite{NakibogluAugustin}
for classical channels, and by Cheng, Li and Hsieh in \cite{Cheng-Li-Hsieh2018} for classical-quantum channels
and the Petz-type R\'enyi divergences. Our results extend these with a different proof method, which in turn is
closely related to the approach of Hayashi and Tomamichel for proving the additivity of the sandwiched
R\'enyi mutual information \cite{HT14}.
In Section \ref{sec:sc}, we prove our main result, \eqref{main result abstract}.
The non-trivial part of this is the inequality LHS$\le$RHS, which we prove using a refinement of the arguments
in \cite{MO-cqconv}. First, in Proposition \ref{prop:sc upper} we employ a suitable adaptation of the techniques
of Dueck and K\"orner \cite{DK} and
obtain the inequality in terms of the log-Euclidean R\'enyi divergence, which gives a suboptimal bound. Then in Proposition \ref{prop:upper reg} we use the
asymptotic pinching technique developed in \cite{MO-cqconv} to arrive at an upper bound
in terms of the regularized sandwiched R\'enyi divergence radii, and finally we use the previously established
additivity
property of these quantities to arrive at the desired bound. In the proof of Proposition \ref{prop:sc upper} we need a constant
composition version of the classical-quantum channel coding theorem.
Such a result was established, for instance, by Hayashi in \cite{universalcq}, and very recently by
Cheng, Hanson, Datta and Hsieh in \cite{ChengHansonDattaHsieh2018}, with a different exponent, by refining another random coding argument by
Hayashi \cite{Hayashicq}. We give a slightly modified proof in Appendix \ref{sec:random coding exponent}.
Further appendices contain various technical ingredients of the proofs, and in Appendix \ref{sec:divrad further}
we give a more detailed discussion of the concepts of divergence radius and mutual information for general divergences and $\alpha$-$z$ R\'enyi divergences, which may be of independent interest.
\section{Preliminaries}
\label{sec:Preliminaries}
For a finite-dimensional Hilbert space ${\mathcal H}$, let ${\mathcal B}({\mathcal H})$ denote the set of all linear operators on ${\mathcal H}$,
and let ${\mathcal B}({\mathcal H})_{\mathrm{sa}}$, ${\mathcal B}({\mathcal H})_+$, and ${\mathcal B}({\mathcal H})_{++}$ denote the set of
self-adjoint, non-zero positive semi-definite (PSD), and positive definite operators, respectively.
For an interval $J\subseteq\mathbb{R}$, let
${\mathcal B}({\mathcal H})_{\mathrm{sa},J}:=\{A\in{\mathcal B}({\mathcal H})_{\mathrm{sa}}:\spec(A)\subseteq J\}$, i.e.,
the set of self-adjoint operators on ${\mathcal H}$ with all their eigenvalues in $J$.
Let ${\mathcal S}({\mathcal H}):=\{\varrho\in{\mathcal B}({\mathcal H})_+,\,\Tr\varrho=1\}$ denote the set of \ki{density operators}, or \ki{states}, on ${\mathcal H}$.
For a self-adjoint operator $A$, let $P^A_a:=\mathbf 1_{\{a\}}(A)$ denote the spectral projection of $A$
corresponding to the singleton $\{a\}$.
The projection onto the support of $A$ is $\sum_{a\ne 0}P^A_a$; in particular, if $A$ is positive semi-definite, it is equal to $\lim_{\alpha\searrow 0}A^{\alpha}=:A^0$. In general, we follow the convention that real powers
of a positive semi-definite operator $A$ are taken only on its support, i.e., for any $x\in\mathbb{R}$,
$A^x:=\sum_{a>0}a^x P^A_a$.
Given a self-adjoint operator $A\in{\mathcal B}({\mathcal H})_{\mathrm{sa}}$, the \ki{pinching} by $A$ is the operator
${\mathcal F}_A:\,{\mathcal B}({\mathcal H})\to{\mathcal B}({\mathcal H})$, ${\mathcal F}_A(.):=\sum_a P^A_a(.)P^A_a$, i.e., the block-diagonalization
with the eigen-projectors of $A$. By the pinching inequality \cite{H:pinching},
\begin{align*}
X\le|\spec(A)|{\mathcal F}_A(X),\mbox{ }\mbox{ }\ds\mbox{ }\mbox{ } X\in{\mathcal B}({\mathcal H})_+.
\end{align*}
Since ${\mathcal F}_A$ can be written as a convex combination of unitary conjugations,
\begin{align}\label{pinching mon}
f\left({\mathcal F}_A(B)\right)\le {\mathcal F}_A (f(B)),\mbox{ }\mbox{ }\ds\mbox{ }\mbox{ }
\Tr g\left({\mathcal F}_A(B)\right)\le \Tr g(B),
\end{align}
for any operator convex function $f$, and
any convex function $g$ on an interval $J$, and any $B\in{\mathcal B}({\mathcal H})_{\mathrm{sa},J}$. The second inequality above is due to the following well-known fact:
\begin{lemma}\label{lemma:trace function properties}
Let $J\subseteq\mathbb{R}$ be an interval and $f:\,J\to\mathbb{R}$ be a function.
\begin{enumerate}
\item
If $f$ is monotone increasing then $\Tr f(.)$ is monotone increasing on ${\mathcal B}({\mathcal H})_{\mathrm{sa},J}$.
\item
If $f$ is convex then $\Tr f(.)$ is convex on ${\mathcal B}({\mathcal H})_{\mathrm{sa},J}$.
\end{enumerate}
\end{lemma}
For a differentiable function $f$ defined on an interval $J\subseteq\mathbb{R}$, let
$f^{[1]}:\,J\times J\to\mathbb{R}$ be its \ki{first divided difference function}, defined as
\begin{align*}
f^{[1]}(a,b):=\begin{cases}
\frac{f(a)-f(b)}{a-b},&a\ne b,\\
f'(a),&a=b,
\end{cases}\mbox{ }\mbox{ }\ds\mbox{ }\mbox{ } a,b\in J.
\end{align*}
The proof of the following can be found, e.g., in \cite[Theorem V.3.3]{Bhatia} or \cite[Theorem 2.3.1]{Hiai_book}:
\begin{lemma}\label{lemma:op function derivative}
If $f$ is a continuously differentiable function on an open interval $J\subseteq\mathbb{R}$ then for any finite-dimensional Hilbert space ${\mathcal H}$, $A\mapsto f(A)$ is Fr\'echet differentiable on ${\mathcal B}({\mathcal H})_{\mathrm{sa},J}$, and its
Fr\'echet derivative $Df(A)$ at a point $A$ is given by
\begin{align*}
Df(A)(Y)=\sum_{a,b}f^{[1]}(a,b)P^A_aYP^A_b,\mbox{ }\mbox{ }\ds\mbox{ }\mbox{ } Y\in{\mathcal B}({\mathcal H})_{\mathrm{sa}}.
\end{align*}
\end{lemma}
It is straightforward to verify that in the setting of Lemma \ref{lemma:op function derivative}, the function
$A\mapsto \Tr f(A)$ is also Fr\'echet differentiable on ${\mathcal B}({\mathcal H})_{\mathrm{sa},J}$, and its
Fr\'echet derivative $D(\Tr \circ f)(A)$ at a point $A$ is given by
\begin{align}\label{Tr derivative}
D(\Tr \circ f)(A)(Y)=\Tr f'(A)Y,\mbox{ }\mbox{ }\ds\mbox{ }\mbox{ } Y\in{\mathcal B}({\mathcal H})_{\mathrm{sa}},
\end{align}
where $f'$ is the derivative of $f$ as a real-valued function.
\medskip
An operator $A\in{\mathcal B}({\mathcal H}^{\otimes n})$ is \ki{symmetric}, if $U_{\pi}AU_{\pi}^*=A$ for all permutations
$\pi\in S_n$, where $U_{\pi}$ is defined by
$U_{\pi}x_1\otimes\ldots\otimes x_n=x_{\pi^{-1}(1)}\otimes\ldots\otimes x_{\pi^{-1}(n)}$, $x_i\in{\mathcal H}$, $i\in[n]$. As it was shown in \cite{universalcq}, for every finite-dimensional Hilbert space ${\mathcal H}$ and every $n\in\mathbb{N}$, there exists a
\ki{universal symmetric state} $\sigma_{u,n}\in{\mathcal S}({\mathcal H}^{\otimes n})$ such that it is symmetric, it commutes with every symmetric state, and for every symmetric state $\omega\in{\mathcal S}({\mathcal H}^{\otimes n})$,
\begin{align*}
\omega\le v_{n,d}\sigma_{u,n},
\end{align*}
where $v_{n,d}$ only depends on $d=\dim{\mathcal H}$ and $n$, and it is polynomial in $n$.
\medskip
By a \ki{generalized classical-quantum (gcq) channel} we mean a map $W:\,{\mathcal X}\to{\mathcal B}({\mathcal H})_+$, where ${\mathcal X}$ is a non-empty set, and ${\mathcal H}$ is a finite-dimensional Hilbert space. It is a \ki{classical-quantum (cq) channel} if
$\ran W\subseteq{\mathcal S}({\mathcal H})$, i.e., each output of the channel is a normalized quantum state.
A (generalized) classical-quantum channel is \ki{classical}, if $W(x)W(y)=W(y)W(x)$ for all $x,y\in {\mathcal X}$.
We remark that we do not require any further structure of ${\mathcal X}$ or the map $W$, and in particular, ${\mathcal X}$ need not be finite.
Given a finite number of gcq channels $W_i:\,{\mathcal X}_i\to{\mathcal B}({\mathcal H}_i)_+$, their \ki{product} is the gcq channel
\begin{align*}
W_1\otimes\ldots\otimes W_n:\,&{\mathcal X}_1\times\ldots\times {\mathcal X}_n\to{\mathcal B}({\mathcal H}_1\otimes\ldots\otimes{\mathcal H}_n)_+\\
&(x_1,\ldots,x_n)\mapsto W_1(x_1)\otimes\ldots\otimes W_n(x_n).
\end{align*}
In particular, if all $W_i$ are the same channel $W$ then we use the notaion $W^{\otimes n}=W\otimes\ldots\otimes W$.
\medskip
We say that a function $P:\,{\mathcal X}\to[0,1]$ is a \ki{probability density function} on a set ${\mathcal X}$ if
$\sum_{x\in{\mathcal X}}P(x)=1$. The \ki{support} of $P$ is
$\supp P:=\{x\in{\mathcal X}:\,P(x)>0\}$. We say that $P$ is finitely supported if $\supp$ is a finite set, and we denote by ${\mathcal P}_f({\mathcal X})$ the set of all finitely supported probability distributions.
The \ki{Shannon entropy} of a $P\in{\mathcal P}_f({\mathcal X})$ is defined as
\begin{align*}
H(P):=-\sum_{x\in{\mathcal X}}P(x)\log P(x).
\end{align*}
For a sequence
$\vecc{x}\in{\mathcal X}^n$, the \ki{type} $\tp{\vecc{x}}\in{\mathcal P}_f({\mathcal X})$ of $\vecc{x}$ is
the empirical distribution of $\vecc{x}$, defined as
\begin{align*}
\tp{\vecc{x}}:=\frac{1}{n}\sum_{i=1}^n\delta_{x_i}:\mbox{ }\mbox{ } y\mapsto
\frac{1}{n}\abs{\{k:\,x_k=y\}},\mbox{ }\mbox{ }\ds\mbox{ }\mbox{ } y\in{\mathcal X},
\end{align*}
where $\delta_x$ is the Dirac measure concentrated at $x$.
We say that a probability distribution $P$ on ${\mathcal X}$ is an \ki{$n$-type} if there exists an $\vecc{x}\in{\mathcal X}^n$
such that $P=\tp{\vecc{x}}$. We denote the set of $n$-types by ${\mathcal P}_n({\mathcal X})$. For an $n$-type $P$, let
${\mathcal X}^n_{P}:=\{\vecc{x}\in{\mathcal X}^n:\,\tp{\vecc{x}}=P\}$ be the set of sequences with the same type $P$.
A key property of types is that $\vecc{x},\vecc{y}\in{\mathcal X}^n$ have the same type if and only if they are permutations of each other, and for any $\vecc{x},\vecc{y}$ with $\tp{\vecc{x}}=\tp{\vecc{y}}$, we have
\begin{align}\label{type prob}
\tp{\vecc{x}}^{\otimes n}(\vecc{y})=e^{-nH(\tp{\vecc{x}})}.
\end{align}
By Lemma 2.3 in \cite{CsiszarKorner2}, for any $P\in{\mathcal P}_n({\mathcal X})$,
\begin{align}\label{type card}
(n+1)^{-|\supp P|}e^{nH(P)}\le |{\mathcal X}^n_{P}|\le e^{nH(P)}.
\end{align}
\medskip
The following lemma is an extension
of the minimax theorems due to Kneser \cite{Kneser} and Fan \cite{Fan}
to the case where $f$ can take the value $+\infty$. For a proof, see
\cite[Theorem 5.2]{FarkasRevesz2006}.
\begin{lemma}\label{lemma:KF+ minimax}
Let $X$ be a compact convex set in a topological vector space $V$ and $Y$ be a convex
subset of a vector space $W$. Let $f:\,X\times Y\to\mathbb{R}\cup\{+\infty\}$ be such that
\smallskip
\mbox{ }(i) $f(x,.)$ is concave on $Y$ for each $x\in X$, and
\smallskip
(ii) $f(.,y)$ is convex and lower semi-continuous on $X$ for each $y\in Y$.
\smallskip
\noindent Then
\begin{align}\label{minimax statement}
\inf_{x\in X}\sup_{y\in Y}f(x,y)=
\sup_{y\in Y}\inf_{x\in X}f(x,y),
\end{align}
and the infima in \eqref{minimax statement} can be replaced by minima.
\end{lemma}
\section{Divergence radii}
\subsection{General divergences}
\label{sec:gendivrad}
By a \ki{divergence} $\divv$ we mean a family of maps
$\divv_{{\mathcal H}}:\,{\mathcal B}({\mathcal H})_+\times{\mathcal B}({\mathcal H})_+\to[-\infty,+\infty]$, defined for every finite-dimensional Hilbert
space ${\mathcal H}$. We will normally not indicate the dependence on the Hilbert space, and simply use the notation
$\divv$ instead of $\divv_{{\mathcal H}}$. We will
only consider divergences that are invariant under isometries, i.e.,
for any $\varrho,\sigma\in{\mathcal B}({\mathcal H})_+$ and $V:\,{\mathcal H}\to{\mathcal K}$ isometry,
$\divv\left( V\varrho V^*\|V\sigma V^*\right)=\divv(\varrho\|\sigma)$. Note that this implies that $\divv$ is invariant under extensions with pure states, i.e., $\divv(\varrho\otimes\pr{\psi}\|\sigma\otimes\pr{\psi})=\divv(\varrho\|\sigma)$, where $\psi$ is an arbitrary unit vector in some Hilbert space.
Further properties will often be important. In particular, we say that
a divergence $\divv$ is
\begin{itemize}
\item
\ki{positive} if $\divv(\varrho\|\sigma)\ge 0$ for all density operators $\varrho,\sigma$, and it is
\ki{strictly positive} if $\divv(\varrho\|\sigma)=0\Longleftrightarrow\varrho=\sigma$, again for density operators;
\item
\ki{monotone under CPTP maps} if for any $\varrho,\sigma\in{\mathcal B}({\mathcal H})_+$ and
any CPTP (completely positive and trace-preserving) map $\Phi:\,{\mathcal B}({\mathcal H})\to{\mathcal B}({\mathcal K})$,
\begin{align*}
\divv\left(\Phi(\varrho)\|\Phi(\sigma)\right)
\le
\divv\left(\varrho\|\sigma\right);
\end{align*}
\item
\ki{jointly convex} if for all $\varrho_i,\sigma_i\in{\mathcal B}({\mathcal H})$, $i\in[r]$, and probability distribution $(p_i)_{i=1}^r$,
\begin{align*}
\divv\left(\sum_{i=1}^r p_i\varrho_i\Bigg\|\sum_{i=1}^r p_i\sigma_i\right)
\le
\sum_{i=1}^r p_i\divv\left(\varrho_i\|\sigma_i\right);
\end{align*}
\item
\ki{block additive} if for any $\varrho_1,\varrho_2$, $\sigma_1,\sigma_2$ such that $\varrho_1^0\vee\sigma_1^0\perp\varrho_2^0\vee\sigma_2^0$, we have
\begin{align*}
\divv(\varrho_1+\varrho_2\|\sigma_1+\sigma_2)=
\divv(\varrho_1\|\sigma_1)+
\divv(\varrho_2\|\sigma_2);
\end{align*}
\item
\ki{homogeneous} if
\begin{align*}
\divv(\lambda\varrho\|\lambda\sigma)=\lambda\divv(\varrho\|\sigma),\mbox{ }\mbox{ }\ds\mbox{ }\mbox{ }
\varrho,\sigma\in{\mathcal B}({\mathcal H})_+,\mbox{ }\mbox{ }\lambda\in(0,+\infty).
\end{align*}
\end{itemize}
Typical examples for divergences with some or all of the above properties are the relative entropy and some
R\'enyi divergences and related quantities;
see Section \ref{sec:Renyi divrad}.
\begin{rem}
It is well-known \cite{P86,Uhlmann1973} that a block additive and homogenous divergence is monotone under CPTP maps if and only if it is jointly convex. The ``only if'' direction follows by applying monotonicity to
$\widehat\varrho:=\sum_ip_i\pr{i}_E\otimes\varrho_i$ and
$\widehat\sigma:=\sum_ip_i\pr{i}_E\otimes\sigma_i$ under the partial trace over the $E$ system, where
$(\ket{i})_{i=1}^r$ is an ONS in ${\mathcal H}_E$. The ``if'' direction follows by using a Stinespring dilation
$\Phi(.)=\Tr_E V(.)V^*$ with an isometry $V:\,{\mathcal H}\to{\mathcal K}\otimes{\mathcal H}_E$,
and writing the partial trace as a convex combination of unitary conjugations (e.g., by the discrete Weyl unitaries).
\end{rem}
\medskip
Given a non-empty set of positive semi-definite operators $S\subseteq{\mathcal B}({\mathcal H})_+$, its \ki{$\divv$-radius} $R_{\divv}(S)$ is defined as
\begin{align}\label{div radius def}
R_{\divv}(S):=\inf_{\sigma\in{\mathcal S}({\mathcal H})}\sup_{\varrho\in{\mathcal S}}\divv(\varrho\|\sigma).
\end{align}
If the above infimum is attained at some $\sigma\in{\mathcal S}({\mathcal H})$ then $\sigma$ is called a \ki{$\divv$-center} of $S$.
A variant of this notion is when, instead of minimizing the maximal $\divv$-distance,
we minimize an averaged distance according to some
finitely supported probability distribution $P\in{\mathcal P}_f(S)$. This yields the notion of the
\ki{$P$-weighted $\divv$-radius:}
\begin{align}\label{P div rad}
R_{\divv,P}(S):=\inf_{\sigma\in{\mathcal S}({\mathcal H})}\sum_{\varrho\in S}P(\varrho)\divv(\varrho\|\sigma).
\end{align}
If the above infimum is attained at some $\sigma\in{\mathcal S}({\mathcal H})$ then $\sigma$ is called a \ki{$P$-weighted $\divv$-center} for $S$.
\begin{rem}
For applications in channel coding, $S$ will be the image of a classical-quantum channel, and hence a subset of the state space.
In this case minimizing over density operators $\sigma$ in \eqref{div radius def} and
\eqref{P div rad} seems natural, while it is less obviously so when the elements of $S$ are general positive semi-definite operators. We discuss this further in Appendix \ref{sec:divrad further}.
\end{rem}
\begin{rem}
Note that for any finitely supported probability distribution $P$ on ${\mathcal B}({\mathcal H})_+$, and any $\supp P\subseteq S\subseteq{\mathcal B}({\mathcal H})_+$, we have
\begin{align*}
R_{\divv,P}(S)=R_{\divv,P}(\supp P)=R_{\divv,P}({\mathcal B}({\mathcal H})_+).
\end{align*}
That is, $R_{\divv,P}(S)$ does not in fact depend on $S$, it is a function only of $P$.
Hence, if no confusion arises, we may simply denote it as $R_{\divv,P}$.
\end{rem}
\begin{rem}
The concepts of the divergence radius and $P$-weighted divergence radius can be unified (to some extent) by the notion of the \ki{$(P,\beta)$-weighted
$\divv$-radius}, which we explain in Section \ref{sec:general divrad}.
\end{rem}
We will mainly be interested in the above concepts when ${\mathcal S}$ is the image of a gcq
channel $W:\,{\mathcal X}\to{\mathcal B}({\mathcal H})_+$, in which case we will use the notation
\begin{align}
\chi_{\divv}(W,P)&:=R_{\divv,P\circ W^{-1}}(\ran W)=\inf_{\sigma\in{\mathcal S}({\mathcal H})}\sum_{x\in{\mathcal X}}P(x)\divv(W(x)\|\sigma),\label{Pdivrad}
\end{align}
where $(P\circ W^{-1})(\varrho):=\sum_{x\in{\mathcal X}:\,W(x)=\varrho}P(x)$. Note that,
as far as these quantities are concerned, the channel simply gives a parametrization of its image set,
and the previously considered case can be recovered by parametrizing the set by itself, i.e., by taking the gcq channel ${\mathcal X}:={\mathcal S}$ and $W:=\id_{{\mathcal S}}$.
We will call \eqref{Pdivrad} the \ki{$P$-weighted $\divv$-radius} of the channel $W$, and any state achieving the infimum in its definition a \ki{$P$-weighted $\divv$-center} for $W$.
We define the \ki{$\divv$-capacity} of the channel $W$ as
\begin{align*}
\chi_{\divv}(W)&:=\sup_{P\in{\mathcal P}_f({\mathcal X})}\chi_{\divv}(W,P)
\end{align*}
In the relevant cases for information theory, the $\divv$-capacity coincides with the $\divv$-radius of the image of the channel, i.e.,
\begin{align*}
\chi_{\divv}(W)=R_{\divv}(\ran W)=\inf_{\sigma\in{\mathcal S}({\mathcal H})}\sup_{x\in{\mathcal X}}\divv(W(x)\|\sigma);
\end{align*}
see Proposition \ref{prop:radius equality}.
We will mainly be interested in the above quantities when $\divv$ is a quantum R\'enyi divergence. For some further properties of these quantities for general divergences, see Appendix \ref{sec:general divrad}.
\subsection{Quantum R\'enyi divergences}
\label{sec:Renyi divrad}
In this section we specialize to
various notions of quantum R\'enyi divergences.
For every pair of positive definite operators $\varrho,\sigma\in{\mathcal B}({\mathcal H})_{++}$
and every $\alpha\in(0,+\infty)\setminus\{1\}$, $z\in(0,+\infty)$ let
\begin{align*}
Q_{\alpha,z}(\varrho\|\sigma):=\Tr\left(\varrho^{\frac{\alpha}{2z}}\sigma^{\frac{1-\alpha}{z}}\varrho^{\frac{\alpha}{2z}}\right)^z.
\end{align*}
These quantities were first introduced in \cite{JOPP} and further studied in \cite{AD}.
The cases
\begin{align}
Q_{\alpha}(\varrho\|\sigma)&:=Q_{\alpha,1}(\varrho\|\sigma)=
\Tr \varrho^{\alpha}\sigma^{1-\alpha},\label{quasi}\\
Q_{\alpha}^{*}(\varrho\|\sigma)&:=Q_{\alpha,\alpha}(\varrho\|\sigma)=
\Tr \left( \varrho^{\frac{1}{2}}\sigma^{\frac{1-\alpha}{\alpha}}\varrho^{\frac{1}{2}}\right)^{\alpha},\label{sand}
\end{align}
and
\begin{align}
Q_{\alpha}^{\flat}(\varrho\|\sigma)&:=Q_{\alpha,+\infty}(\varrho\|\sigma):=\lim_{z\to+\infty}Q_{\alpha,z}(\varrho\|\sigma)=
\Tr e^{\alpha\log\varrho+(1-\alpha)\log\sigma}\label{exp}
\end{align}
are of special significance. (The last identity in \eqref{exp} is due to the Lie-Trotter formula.)
Here and henceforth $(t)$ stands for one of the three possible values
$(t)=\{\s\},\,(t)=*$ or $(t)=\flat$, where $\{\s\}$ denotes the empty string, i.e.,
$Q_{\alpha}^{(t)}$ with $(t)=\{\s\}$ is simply $Q_{\alpha}$.
These quantities are extended to general, not necessarily invertible positive semi-definite operators $\varrho,\sigma\in{\mathcal B}({\mathcal H})_+$ as
\begin{align}
Q_{\alpha,z}(\varrho\|\sigma)
&:=
\lim_{\varepsilon\searrow 0}Q_{\alpha,z}(\varrho+\varepsilon I\|\sigma+\varepsilon I)\label{ext0}\\
&=
\lim_{\varepsilon\searrow 0}Q_{\alpha,z}(\varrho\|\sigma+\varepsilon I)
=
\lim_{\varepsilon\searrow 0}Q_{\alpha,z}(\varrho\|(1-\varepsilon)\sigma+\varepsilon I/d)
=
s(\alpha)\sup_{\varepsilon>0}\overline Q_{\alpha,z}(\varrho\|\sigma+\varepsilon I),\label{ext1}
\end{align}
for every $z\in(0,+\infty)$,
where $d:=\dim{\mathcal H}$,
\begin{align*}
s(\alpha):=\sgn(\alpha-1)=
\begin{cases}
-1,&\alpha<1,\\
1,&\alpha>1
\end{cases},
\mbox{ }\mbox{ }\ds\mbox{ }\mbox{ }\ds\mbox{ }\mbox{ }\ds\mbox{ }\mbox{ }\ds\mbox{ }\mbox{ }
\overline Q_{\alpha,z}:=s(\alpha)Q_{\alpha,z},
\end{align*}
and the identities are easy to verify.
For $z=+\infty$, the extension is defined by \eqref{ext0}; see \cite{HP_GT,MO-cqconv} for details.
Various further divergences can be defined from the above quantities.
The \ki{quantum $\alpha$-$z$ R\'enyi divergences} \cite{AD} are defined as
\begin{align}\label{Renyi div def}
D_{\alpha,z}(\varrho\|\sigma):=\frac{1}{\alpha-1}\log \frac{Q_{\alpha,z}(\varrho\|\sigma)}{\Tr\varrho}
\end{align}
for any $\alpha\in(0,+\infty)\setminus\{1\}$ and $z\in(0,+\infty]$.
It is easy to see that
\begin{align*}
\alpha>1,\mbox{ }\mbox{ }\ds\varrho^0\nleq\sigma^0\mbox{ }\mbox{ }\Longrightarrow\mbox{ }\mbox{ } Q_{\alpha,z}=D_{\alpha,z}=+\infty
\end{align*}
for any $z$.
Moreover, if $\alpha\mapsto z(\alpha)$ is continuously differentiable in a neighbourhood of $1$,
on which $z(\alpha)\ne 0$, or $z(\alpha)=+\infty$ for all $\alpha$, then, according to
\cite[Theorem 1]{LinTomamichel15} and \cite[Lemma 3.5]{MO-cqconv},
\begin{align*}
D_1(\varrho\|\sigma):=
\lim_{\alpha\to1}D_{\alpha,z(\alpha)}(\varrho\|\sigma)=\frac{1}{\Tr\varrho}D(\varrho\|\sigma)
=:D_{1,z}(\varrho\|\sigma),\mbox{ }\mbox{ }\ds\mbox{ }\mbox{ } z\in(0,+\infty],
\end{align*}
where $D(\varrho\|\sigma)$ is Umegaki's relative entropy \cite{Umegaki}, defined as
\begin{align*}
D(\varrho\|\sigma):=\Tr\varrho(\log\varrho-\log\sigma)
\end{align*}
for positive definite operators, and extended
as above for non-zero positive semidefinite operators.
Of the R\'enyi divergences corresponding to the special $Q_{\alpha}$ quantities discussed above,
$D_{\alpha}$ is usually called the \ki{Petz-type} R\'enyi divergence,
$D_{\alpha}^{*}$ the \ki{sandwiched} R\'enyi divergence \cite{Renyi_new,WWY},
and $D_{\alpha}^{\flat}$ the \ki{log-Euclidean} R\'enyi divergence. For more on the above definitions and a more detailed reference to their literature, see, e.g., \cite{MO-cqconv}.
To discuss some important properties of the above quantities, let us introduce the following regions of the $\alpha$-$z$ plane:
\begin{align*}
& K_0:\,0<\alpha<1,\,z<\min\{\alpha,1-\alpha\}; & & K_1:\,0<\alpha<1,\,\alpha\le z\le 1-\alpha;\\
&K_2:\,0<\alpha<1,\,\max\{\alpha,1-\alpha\}\le z\le 1; & & K_3:\,0<\alpha<1,\,1-\alpha\le z\le \alpha;\\
& K_4:\,0<\alpha<1,\,1\le z; & & K_5:\,1<\alpha,\,\alpha/2\le z\le 1; \\
&K_6:\,1<\alpha,\,\max\{\alpha-1,1\}\le z\le \alpha; & &
K_7:\,1<\alpha\le z;
\end{align*}
The $(\alpha,z)$ values for which $D_{\alpha,z}$ is monotone under CPTP maps have been completely characterized in
\cite{Hi3,An,Beigi,FL,CFL,Zhang2018} (cf.~also \cite[Theorem 1]{AD}). This can be summarized as follows.
\begin{lemma}\label{lemma:az monotonicity}
$D_{\alpha,z}$ is monotone under CPTP maps
$\Longleftrightarrow$ $\overline Q_{\alpha,z}$ is monotone under CPTP maps
$\Longleftrightarrow$ $\overline Q_{\alpha,z}$ is jointly convex
$\Longleftrightarrow$ $(\alpha,z)\in K_2\cup K_4\cup K_5\cup K_6$.
\end{lemma}
\begin{cor}\label{cor:az joint convexity}
$D_{\alpha,z}$ is jointly convex if $(\alpha,z)\in K_2\cup K_4$.
\end{cor}
\begin{proof}
Immediate from Lemma \ref{lemma:az monotonicity}, as the joint convexity of $\overline Q_{\alpha,z}$ implies the joint convexity of $D_{\alpha,z}=\frac{1}{\alpha-1}\log s(\alpha)\overline Q_{\alpha,z}$ whenever $\alpha\in(0,1)$.
\end{proof}
Recall that a function $f:\,C\to\mathbb{R}\cup\{+\infty\}$ on a convex set $C$ is \ki{quasi-convex} if
$f((1-t)x+ty)\le\max\{f(x),f(y)\}$ for all $x,y\in C$ and $t\in[0,1]$.
\begin{lemma}\label{lemma:2nd convexity}
On top of the cases discussed in Lemma \ref{lemma:az monotonicity} and Corollary \ref{cor:az joint convexity},
$D_{\alpha,z}$ is convex in its second argument if $(\alpha,z)\in K_3\cup K_6\cup K_7$, and
$\overline Q_{\alpha,z}$ is convex in its second argument if
$(\alpha,z)\in K_3\cup K_7$. Moreover, $D_{\alpha,z}$ is jointly quasi-convex if
$(\alpha,z)\in K_5$.
\end{lemma}
\begin{proof}
The assertion about the quasi-convexity of $D_{\alpha,z}$ is immediate from the joint convexity of
$\overline Q_{\alpha,z}$ when $(\alpha,z)\in K_5$.
Note that it is enough to prove convexity in the second argument for positive definite operators,
due to \eqref{ext1}.
Assume that $(\alpha,z)\in K_2\cup K_3$, i.e., $0<\alpha<1$, $1-\alpha\le z\le 1$. Then
$0<\frac{1-\alpha}{z}\le 1$, and hence $\sigma\mapsto \sigma^{\frac{1-\alpha}{z}}$ is concave. Since
${\mathcal B}({\mathcal H})_+\ni A\mapsto\Tr A^z$ is both monotone and concave (see Lemma \ref{lemma:trace function properties}), we get that
$\sigma\mapsto Q_{\alpha,z}(\varrho\|\sigma)$ is concave, from which the
convexity of both $\overline Q_{\alpha,z}$ and $D_{\alpha,z}$ in their second argument follows for
$(\alpha,z)\in K_3$ (and also for $K_2$, although that is already covered by joint convexity).
Assume next that $(\alpha,z)\in K_6\cup K_7$, i.e., $1<\alpha$, and $\max\{1,\alpha-1\}\le z$. Then
$-1\le\frac{1-\alpha}{z}<0$, and hence $f:\,t\mapsto t^{\frac{1-\alpha}{z}}$ is a non-negative operator monotone decreasing function on $(0,+\infty)$. Applying the duality of the Schatten $p$-norms to $p=z$, we have
\begin{align*}
D_{\alpha,z}(\varrho\|\sigma)=\sup_{\tau\in{\mathcal S}({\mathcal H})}\frac{z}{\alpha-1}\log
\Tr\varrho^{\frac{\alpha}{2z}}\sigma^{\frac{1-\alpha}{z}}\varrho^\frac{\alpha}{2z}\tau^{1-\frac{1}{z}}
=
\sup_{\tau\in{\mathcal S}({\mathcal H})}\frac{z}{\alpha-1}\log\omega_{\tau}\left( f(\sigma)\right),
\end{align*}
where $\omega_{\tau}(.):=\Tr\varrho^{\frac{\alpha}{2z}}(.)\varrho^\frac{\alpha}{2z}\tau^{1-\frac{1}{z}}$
is a positive functional.
By \cite[Proposition 1.1]{AH}, $D_{\alpha,z}(\varrho\|.)$ is the supremum of convex functions
on ${\mathcal B}({\mathcal H})_{++}$, and hence is itself convex.
This immediately implies that $\overline Q_{\alpha,z}$ is convex in its second argument when
$(\alpha,z)\in K_6\cup K_7$ (of which the case $K_6$ also follows from joint convexity).
\end{proof}
\begin{lemma}\label{lsc}
For any fixed $\varrho\in{\mathcal B}({\mathcal H})_+$, the maps
\begin{align*}
\sigma\mapsto \overline Q_{\alpha,z}(\varrho\|\sigma)\mbox{ }\mbox{ }\ds\text{and}\mbox{ }\mbox{ }\ds
\sigma\mapsto D_{\alpha,z}(\varrho\|\sigma)
\end{align*}
are lower semi-continuous on ${\mathcal B}({\mathcal H})_+$ for any $\alpha\in(0,+\infty)\setminus\{1\}$ and $z\in(0,+\infty)$, and for $z=+\infty$ and $\alpha>1$.
\end{lemma}
\begin{proof}
The cases $\alpha\in(0,+\infty)\setminus\{1\}$ and $z\in(0,+\infty)$ are obvious from the last expression in \eqref{ext1}, and the case $z=+\infty$ was discussed in \cite[Lemma 3.27]{MO-cqconv}.
\end{proof}
It is known that $D_{\alpha}$, $D_{\alpha}^*$ and $D_{\alpha}^{\flat}$ are non-negative on pairs of states
\cite{Renyi_new,P86,MO-cqconv}, but it seems that the non-negativity of general $\alpha$-$z$ R\'enyi divergences
has not been analyzed in the literature so far. We show in Appendix \ref{sec:positive Renyi} that they are indeed non-negative for any pair of parameters $(\alpha,z)$.
\subsection{The R\'enyi divergence center}
\label{sec:Renyi center}
Let $W:\,{\mathcal X}\to{\mathcal S}({\mathcal H})$ be a gcq channel. Specializing to $\divv=D_{\alpha,z}$ in \eqref{Pdivrad} yields the
\ki{$P$-weighted R\'enyi $(\alpha,z)$ radii} of the channel for a finitely supported input probability distribution $P\in{\mathcal P}_f({\mathcal X})$,
\begin{align}\label{weighted alpha divrad def}
\chi_{\alpha,z}(W,P)
:=
\chi_{D_{\alpha,z}}(W,P)
=
\inf_{\sigma\in{\mathcal S}({\mathcal H})}\sum_{x\in{\mathcal X}}P(x)D_{\alpha,z}(W(x)\|\sigma)
=
\min_{\sigma\in{\mathcal S}({\mathcal H})}\sum_{x\in{\mathcal X}}P(x)D_{\alpha,z}(W(x)\|\sigma).
\end{align}
The existence of the minimum is guaranteed by the lower semi-continuity stated in Lemma \ref{lsc}.
We will call any state $\sigma$ achieving the minimum in \eqref{weighted alpha divrad def} a
\ki{$P$-weighted $D_{\alpha,z}$ center} for $W$.
It is sometimes convenient that it is enough to consider the infimum above over invertible states, i.e., we have
\begin{align}\label{positive minimization}
\chi_{\alpha,z}(W,P)=\inf_{\sigma\in{\mathcal S}({\mathcal H})_{++}}\sum_{x\in{\mathcal X}}P(x)D_{\alpha,z}(W(x)\|\sigma),
\end{align}
which is obvious from the second expression in \eqref{ext1}.
Moreover, any minimizer of \eqref{weighted alpha divrad def} has the same support as the joint support of the
channel states $\{W_x\}_{x\in\supp P}$, with projection
\begin{align*}
\bigvee_{x\in\supp P}W(x)^0=W(P)^0,
\end{align*}
at least for a certain range of $(\alpha,z)$ values, as we show below.
\begin{lemma}\label{lemma:minimizer support}
Let $\sigma$ be a $P$-weighted $D_{\alpha,z}$ center for $W$.
If $(\alpha,z)$ is such that $D_{\alpha,z}$ is quasi-convex in its second argument
then $\sigma^0\le W(P)^0$.
\end{lemma}
\begin{proof}
Define ${\mathcal F}(X):=W(P)^0XW(P)^0+(I-W(P)^0)X(I-W(P)^0)$, $X\in{\mathcal B}({\mathcal H})$, and let
$\tilde\sigma:=W(P)^0\sigma W(P)^0/\Tr W(P)^0\sigma$.
We will show that $D_{\alpha,z}(W(x)\|\tilde\sigma)\le D_{\alpha,z}(W(x)\|\sigma)$ for all
$x\in\supp P$, which will yield the assertion. Note that we can assume without loss of generality that
$W(P)^0\sigma\ne 0$, since otherwise $D_{\alpha,z}(W(x)\|\sigma)=+\infty$ for all $x\in\supp P$, and hence
$\sigma$ clearly cannot be a minimizer for \eqref{weighted alpha divrad def}.
According to the decomposition ${\mathcal H}=\ran W(P)^0\oplus\ran(I-W(P)^0)$, define the block-diagonal unitary
$U:=\begin{bmatrix}I & 0\\ 0& -I\end{bmatrix}$, so that
${\mathcal F}(.)=\frac{1}{2}\left((.)+U(.)U^*\right)$. For every $x\in{\mathcal X}$,
\begin{align*}
D_{\alpha,z}(W(x)\|{\mathcal F}(\sigma))
&\le
\max\left\{D_{\alpha,z}(W(x)\|\sigma),D_{\alpha,z}(W(x)\|U\sigma U^*)\right\}\\
&=
\max\left\{D_{\alpha,z}(W(x)\|\sigma),D_{\alpha,z}(UW(x)U^*\|U\sigma U^*)\right\}
=
D_{\alpha,z}(W(x)\|\sigma),
\end{align*}
where the first inequality is due to quasi-convexity, and the first equality is due to the fact that
$UW(x)U^*=W(x)$. On the other hand,
\begin{align*}
D_{\alpha,z}(W(x)\|{\mathcal F}(\sigma))
&=
D_{\alpha,z}(W(x)\|(\Tr W(P)^0\sigma)\tilde\sigma)\\
&=
D_{\alpha,z}\left( W(x)\|\tilde\sigma\right)-\log\Tr W(P)^0\sigma
\ge
D_{\alpha,z}\left( W(x)\|\tilde\sigma\right),
\end{align*}
where the inequality is strict unless $\sigma^0\le W(P)^0$.
\end{proof}
\medskip
For fixed $W$ and $P$, we define
\begin{align*}
F(\sigma):=\sum_{x\in{\mathcal X}}P(x)D_{\alpha,z}(W(x)\|\sigma),\mbox{ }\mbox{ }\ds\mbox{ }\mbox{ } \sigma\in{\mathcal B}({\mathcal H})_+.
\end{align*}
In the following, we may naturally interpret $W(x)$ as an operator acting on
$\ran W(x)$ or on $\ran W(P)$.
\begin{lemma}\label{lemma:F derivative}
$F$ is Fr\'echet-differentiable at every $\sigma\in{\mathcal B}({\mathcal H})_{++}$, with
Fr\'echet-derivative $DF(\sigma)$ given by
\begin{align}
DF(\sigma):\,Y\mapsto&
\frac{z}{\alpha-1}\sum_{x\in{\mathcal X}}P(x)\frac{1}{Q_{\alpha,z}(W(x)\|\sigma)}\nonumber\\
&\cdot
\Tr\sum_{a,b}h_{\alpha,z}^{[1]}(a,b)P^{\sigma}_a W(x)^{\frac{\alpha}{2z}}\left( W(x)^{\frac{\alpha}{2z}}\sigma^{\frac{1-\alpha}{z}} W(x)^{\frac{\alpha}{2z}}\right)^{z-1}W(x)^{\frac{\alpha}{2z}}P^{\sigma}_b\,Y,
\label{F derivative}
\end{align}
where $h_{\alpha,z}^{[1]}$ is the first divided difference function of $h_{\alpha,z}(t):=t^{\frac{1-\alpha}{z}}$.
\end{lemma}
\begin{proof}
We have $F=\sum_{x\in{\mathcal X}}P(x) (g_x\circ\iota_x\circ H_{\alpha,z})$, where
$H_{\alpha,z}:\,{\mathcal B}({\mathcal H})_+\to{\mathcal B}({\mathcal H})$, $H_{\alpha,z}(\sigma):=\sigma^{\frac{1-\alpha}{z}}$ is
Fr\'echet differentiable at every $\sigma\in{\mathcal B}({\mathcal H})_{++}$ with
$DH_{\alpha,z}(\sigma):\,Y\mapsto \sum_{a,b}h_{\alpha,z}^{[1]}(a,b)P^{\sigma}_a YP^{\sigma}_b$, according to Lemma \ref{lemma:op function derivative}.
For a fixed $x$, $\iota_x:\,{\mathcal B}({\mathcal H})\to{\mathcal B}(\ran (W(x))$ is defined as
$A\mapsto W(x)^{\frac{\alpha}{2z}}AW(x)^{\frac{\alpha}{2z}}$, and, as a linear map, it is Fr\'echet differentiable at every $A\in{\mathcal B}({\mathcal H})$, with its derivative being equal to itself. Finally,
$g_x:\,{\mathcal B}(\ran W(x))\to\mathbb{R}$ is defined as $g_x(T):=\Tr T^z$, and it is Fr\'echet differentiable at every
$T\in{\mathcal B}(\ran W(x))_{++}$, with Fr\'echet derivative
$Dg_x(T):\,Y\mapsto z\Tr T^{z-1}Y$, according to \eqref{Tr derivative}.
If $\sigma\in{\mathcal B}(\ran W(P))_{++}$ then
$H_{\alpha,z}(\sigma)\in{\mathcal B}(\ran W(P))_{++}$, and
$\iota_x(H_{\alpha,z}(\sigma))\in{\mathcal B}(\ran W(x))_{++}$. Hence, we can apply the chain rule for derivatives, and obtain \eqref{F derivative}.
\end{proof}
\begin{lemma}\label{lemma:support2}
Let $\sigma$ be a $P$-weighted $D_{\alpha,z}$ center for $W$.
If
$\alpha\ge 1$ or $\alpha\in(0,1)$ and
$1-\alpha<z<+\infty$ then
$W(P)^0\le\sigma^0$.
\end{lemma}
\begin{proof}
When $\alpha>1$ and $W(P)^0\nleq\sigma^0$, there exists an $x\in\supp P$ with $W_x^0\nleq\sigma^0$ so that
$D_{\alpha,z}(W(x)\|\sigma)=+\infty$. Hence, $\sigma$ cannot be a minimizer for \eqref{weighted alpha divrad def}.
Assume for the rest that $\alpha\in(0,1)$, and $\sigma$ is such that $W(P)^0\nleq \sigma^0$;
this is equivalent to the existence of an $x_0\in\supp P$ such that
$W_{x_0}P^{\sigma}_0\ne 0$.
Let us define the state $\omega:=cP^{\sigma}_0$, with $c:=1/\Tr P^{\sigma}_0$.
For every $t\in[0,1]$, let
\begin{align*}
\sigma_t:=(1-t)\sigma+t\omega=
\sum_{\lambda\in\spec(\sigma)\setminus\{0\}}(1-t)\lambda P^{\sigma}_{\lambda}+tcP^{\sigma}_0,
\end{align*}
so that $\sigma_t\in{\mathcal B}({\mathcal H})_{++}$ for every $t\in(0,1]$.
Note that if $t<t_0:=\lambda_{\min}(\sigma)/(c+\lambda_{\min}(\sigma))$, where
$\lambda_{\min}(\sigma)$ is the smallest non-zero eigenvalue of $\sigma$, then
$P^{\sigma_t}_{ct}=P^{\sigma}_0$, and
$P^{\sigma_t}_{(1-t)\lambda}=P^{\sigma}_{\lambda}$, $\lambda\in\spec(\sigma)\setminus\{0\}$.
By Lemma \ref{lemma:F derivative}, the derivative of $f(t):=F(\sigma_t)$ at any $t\in (0,1)$ is given by
\begin{align*}
&f'(t)=DF(\sigma_t)(\omega-\sigma)\\
&=
\frac{z}{\alpha-1}\sum_{x\in{\mathcal X}}P(x)\frac{1}{Q_{\alpha,z}(W(x)\|\sigma_t)}
\Bigg[
h_{\alpha,z}'\left( ct\right) c\Tr A_{x,t}P^{\sigma}_0\\
&\hspace{5.5cm}
-
\sum_{\lambda\in\spec(\sigma)\setminus\{0\}}h_{\alpha,z}'\left((1-t)\lambda\right)\lambda
\Tr A_{x,t}P^{\sigma}_{\lambda}
\Bigg]\\
&=
\sum_{x\in{\mathcal X}}P(x)\frac{1}{Q_{\alpha,z}(W(x)\|\sigma_t)}
\Bigg[
(1-t)^{\frac{1-\alpha}{z}-1}\sum_{\lambda\in\spec(\sigma)\setminus\{0\}}\lambda^{\frac{1-\alpha}{z}}\Tr A_{x,t}P^{\sigma}_{\lambda}\\
&\hspace{4.5cm}
-
t^{\frac{1-\alpha}{z}-1}c^{\frac{1-\alpha}{z}}\Tr A_{x,t}P^{\sigma}_0
\Bigg],
\end{align*}
where
$A_{x,t}:=W(x)^{\frac{\alpha}{2z}}\left( W(x)^{\frac{\alpha}{2z}}\sigma_t^{\frac{1-\alpha}{z}} W(x)^{\frac{\alpha}{2z}}\right)^{z-1}W(x)^{\frac{\alpha}{2z}}$.
Our aim will be to show that $\lim_{t\searrow 0}f'(t)=-\infty$. This implies that
$f(t)<f(0)$ for small enough $t>0$, contradicting the assumption that $F$ has a global minimum at $\sigma$.
Note that $\lim_{t\searrow 0}Q_{\alpha,z}(W(x)\|\sigma_t)=Q_{\alpha,z}(W(x)\|\sigma)$, which is strictly positive for every $x\in\supp P$. Indeed, the contrary would mean that
$D_{\alpha,z}(W(x)\|\sigma)=+\infty$, contradicting again the assumption that $F$ has a global minimum at $\sigma$.
Hence, the proof will be complete if we show that
$t^{\frac{1-\alpha}{z}-1}\Tr A_{x_0,t}P^{\sigma}_0$ diverges to $+\infty$ while
$\Tr A_{x,t}P^{\sigma}_{\lambda}$ is bounded as $t\searrow 0$ for any $x\in\supp P$ and $\lambda\in\spec(\sigma)\setminus\{0\}$.
Note that for any $t\in(0,t_0)$ and $z\ge 1$,
\begin{align}
tc I\le\sigma_t\le I &\mbox{ }\mbox{ }\Longrightarrow\mbox{ }\mbox{ }
(tc)^{\frac{1-\alpha}{z}} I\le\sigma_t^{\frac{1-\alpha}{z}}\le I\nonumber\\
&\mbox{ }\mbox{ }\Longrightarrow\mbox{ }\mbox{ }
(tc)^{\frac{1-\alpha}{z}}W(x)^{\frac{\alpha}{z}} \le
W(x)^{\frac{\alpha}{2z}}\sigma_t^{\frac{1-\alpha}{z}}W(x)^{\frac{\alpha}{2z}}
\le W(x)^{\frac{\alpha}{z}}\nonumber\\
&\mbox{ }\mbox{ }\Longrightarrow\mbox{ }\mbox{ }
t^{\frac{1-\alpha}{z}}c_1W(x)^0\le W(x)^{\frac{\alpha}{2z}}\sigma_t^{\frac{1-\alpha}{z}}W(x)^{\frac{\alpha}{2z}}
\le c_3W(x)^0\nonumber\\
&\mbox{ }\mbox{ }\Longrightarrow\mbox{ }\mbox{ }
t^{\frac{1-\alpha}{z}(z-1)}c_2 W(x)^0\le
\left[W(x)^{\frac{\alpha}{2z}}\sigma_t^{\frac{1-\alpha}{z}}W(x)^{\frac{\alpha}{2z}}\right]^{z-1}
\le
c_4W(x)^0\label{operator bound1}\\
&\mbox{ }\mbox{ }\Longrightarrow\mbox{ }\mbox{ }
t^{\frac{1-\alpha}{z}(z-1)}c_2 W(x)^{\frac{\alpha}{z}}\le
A_{x,t}
\le
c_4W(x)^{\frac{\alpha}{z}},\label{operator bound2}
\end{align}
where $c_1:=c^{\frac{1-\alpha}{z}}\lambda_{\min}(W(x))^{\frac{\alpha}{z}}>0$,
$c_2:=c_1^{z-1}>0$,
$c_3:=\norm{W(x)}^{\frac{\alpha}{z}}>0$,
$c_4:=c_3^{z-1}>0$,
and the inequalities in \eqref{operator bound1}--\eqref{operator bound2} hold in the opposite direction when $z\in(0,1)$.
This immediately implies that
\begin{align*}
&t^{\frac{1-\alpha}{z}-1}\Tr A_{x_0,t}P^{\sigma}_0
\ge
t^{\frac{1-\alpha}{z}-1+\frac{1-\alpha}{z}(z-1)}c_2\Tr W(x_0)^{\frac{\alpha}{z}}P^{\sigma}_0
\xrightarrow[t\searrow 0]{}+\infty,&z\ge 1,\\
&t^{\frac{1-\alpha}{z}-1}\Tr A_{x_0,t}P^{\sigma}_0
\ge
t^{\frac{1-\alpha}{z}-1}c_4\Tr W(x_0)^{\frac{\alpha}{z}}P^{\sigma}_0
\xrightarrow[t\searrow 0]{}+\infty,&z\in(0,1),
\end{align*}
since
$\Tr W(x_0)^{\frac{\alpha}{z}}P^{\sigma}_0>0$ by assumption,
$\frac{1-\alpha}{z}-1+\frac{1-\alpha}{z}(z-1)=-\alpha<0$, and
$\frac{1-\alpha}{z}-1<0$ iff $1-\alpha< z$ when $z\in(0,1)$.
Next, observe that
\begin{align*}
(1-t)P^{\sigma}_{\lambda}\le\sigma_t &\mbox{ }\mbox{ }\Longrightarrow\mbox{ }\mbox{ }
(1-t)^{\frac{1-\alpha}{z}}P^{\sigma}_{\lambda}\le\sigma_t^{\frac{1-\alpha}{z}}
\end{align*}
where the inequality follows since, by assumption, $0<\frac{1-\alpha}{z}<1$, and
$x\mapsto x^{\gamma}$ is operator monotone on $(0,+\infty)$ for $\gamma\in(0,1)$.
Hence,
\begin{align*}
0\le\Tr A_{x,t}P^{\sigma}_{\lambda}
\le
(1-t)^{\frac{\alpha-1}{z}}
\Tr A_{x,t}\sigma_t^{\frac{1-\alpha}{z}}
=
(1-t)^{\frac{\alpha-1}{z}}Q_{\alpha,z}(W(x)\|\sigma_t)\xrightarrow[t\searrow 0]{}Q_{\alpha,z}(W(x)\|\sigma),
\end{align*}
which is finite. This finishes the proof.
\end{proof}
\begin{rem}
Note that the region of $(\alpha,z)$ values given in Lemma \ref{lemma:support2} covers $z=1$ for all
$\alpha\in(0,+\infty]$, i.e., all the Petz-type R\'enyi divergences,
and
$\{(\alpha,\alpha):\,\alpha\in(1/2,+\infty]\}$, i.e.,
the sandwiched R\'enyi divergences for every parameter $\alpha$ for which they are monotone under CPTP maps, except for $\alpha=1/2$.
It is an open question whether the condition $z> 1-\alpha$
in Lemma \ref{lemma:support2} can be improved, or maybe completely removed.
\end{rem}
\begin{rem}
Note that the case $\alpha>1$ in Lemma \ref{lemma:support2} is trivial, and this is the case that we actually need for the strong converse exponent of constant composition classical-quantum channel coding in Section \ref{sec:sc}; more precisely, we need the case $z=\alpha>1$.
\end{rem}
Let us define $\Gamma_D$ to be the set of $(\alpha,z)$ values such that
for any gcq channel $W$ and any input probability distribution $P$,
any $P$-weighted $D_{\alpha,z}$ center $\sigma$ for $W$ satisfies
$\sigma^0=W(P)^0$.
Then Corollary \ref{cor:az joint convexity} and Lemmas \ref{lemma:2nd convexity}, \ref{lemma:minimizer support}
and \ref{lemma:support2} yield
\begin{align*}
\Gamma_D\supseteq\left\{(\alpha,z):\,\alpha\in(0,1),\,1-\alpha< z+\infty\right\}
\cup
\left\{(\alpha,z):\,\alpha>1,\, z\ge\max\{\alpha/2,\alpha-1\}\right\}.
\end{align*}
\smallskip
The following characterization of the weighted $D_{\alpha,z}$ centers
will be crucial in proving the additivity of the weighted sandwiched R\'enyi divergence radius of a gcq channel.
\begin{theorem}\label{prop:fixed point characterization}
Assume that $(\alpha,z)\in\Gamma_D$ are such that
$D_{\alpha,z}$ is convex in its second variable.
Then $\sigma$
is a $P$-weighted $D_{\alpha,z}$ center for $W$
if and only if it is a fixed point of
the map
\begin{align}\label{D fixed point eq}
\Phi_{W,P,D_{\alpha,z}}(\sigma):=&
\sum_{x\in{\mathcal X}}P(x)\frac{1}{Q_{\alpha,z}(W(x)\|\sigma)}
\left(\sigma^{\frac{1-\alpha}{2z}}W(x)^{\frac{\alpha}{z}}\sigma^{\frac{1-\alpha}{2z}}\right)^{z}
\end{align}
defined on ${\mathcal S}_{W,P}({\mathcal H})_{++}:=\{\sigma\in{\mathcal S}({\mathcal H})_+:\,\sigma^0=W(P)^0\}$.
\end{theorem}
\begin{proof}
By the assumption that $(\alpha,z)\in\Gamma_D$,
we may restrict the Hilbert space to be $\ran W(P)^0$, and assume that $\sigma$ is invertible.
Let $F(A):=\sum_{x\in{\mathcal X}}P(x)D_{\alpha,z}(W(x)\|A)$, $A\in{\mathcal B}({\mathcal H})_{++}$.
Due to the assumption that $D_{\alpha,z}$ is convex in its second variable, $\sigma$ is a minimizer of $F$
if and only if $DF(\sigma)(Y)=0$ for all self-adjoint traceless $Y$.
By Lemma \ref{lemma:F derivative}, this condition is equivalent to
\begin{align*}
\lambda I=
\frac{z}{\alpha-1}\sum_{x\in{\mathcal X}}P(x)\frac{1}{Q_{\alpha,z}(W(x)\|\sigma)}
\sum_{a,b}h_{\alpha,z}^{[1]}(a,b)P^{\sigma}_a W(x)^{\frac{\alpha}{2z}}\left( W(x)^{\frac{\alpha}{2z}}\sigma^{\frac{1-\alpha}{z}} W(x)^{\frac{\alpha}{2z}}\right)^{\alpha-1}W(x)^{\frac{\alpha}{2z}}P^{\sigma}_b
\end{align*}
for some $\lambda\in\mathbb{R}$.
Multiplying both sides by $\sigma^{1/2}$ from the left and the right, and taking the trace, we get
$\lambda=-1$. Hence, the above is equivalent to (by multiplying both sides by $\sigma^{1/2}$ from the left and the right)
\begin{align}
\sigma&=
\frac{z}{1-\alpha}\sum_{x\in{\mathcal X}}P(x)\frac{1}{Q_{\alpha,z}(W(x)\|\sigma)}
\sum_{a,b}a^{1/2}b^{1/2}h_{\alpha,z}^{[1]}(a,b)P^{\sigma}_a W(x)^{\frac{\alpha}{2z}}\left( W(x)^{\frac{\alpha}{2z}}\sigma^{\frac{1-\alpha}{z}} W(x)^{\frac{\alpha}{2z}}\right)^{z-1}W(x)^{\frac{\alpha}{2z}}P^{\sigma}_b\nonumber\\
&=
\sum_{a,b}P^{\sigma}_a\left(\frac{z}{1-\alpha}a^{1/2}b^{1/2}h_{\alpha,z}^{[1]}(a,b)
\widehat\Phi_{W,P,\alpha,z}(\sigma)\right) P^{\sigma}_b,\label{fixed point1}
\end{align}
where
\begin{align*}
\widehat\Phi_{W,P,D_{\alpha,z}}(\sigma)&:=\sum_{x\in{\mathcal X}}P(x)\frac{1}{Q_{\alpha,z}^{*}(W(x)\|\sigma)}
W(x)^{\frac{\alpha}{2z}}\left( W(x)^{\frac{\alpha}{2z}}\sigma^{\frac{1-\alpha}{z}} W(x)^{\frac{\alpha}{2z}}\right)^{z-1}W(x)^{\frac{\alpha}{2z}}.
\end{align*}
Writing the operators in \eqref{fixed point1} in block form according to the spectral decomposition of $\sigma$,
we see that \eqref{fixed point1} is equivalent to
\begin{align*}
\forall a,b:\mbox{ }\mbox{ } \delta_{a,b}a^{\frac{\alpha-1}{z}+1}P^{\sigma}_a
=P^{\sigma}_a\widehat\Phi_{W,P,\alpha,z}(\sigma)P^{\sigma}_b
&\mbox{ }\mbox{ }\Longleftrightarrow\mbox{ }\mbox{ }
\sigma^{\frac{\alpha-1}{z}+1}=\widehat\Phi_{W,P,D_{\alpha,z}}(\sigma)\\
&\mbox{ }\mbox{ }\Longleftrightarrow\mbox{ }\mbox{ }\sigma=\sigma^{\frac{1-\alpha}{2z}}\widehat\Phi_{W,P,D_{\alpha,z}}(\sigma)\sigma^{\frac{1-\alpha}{2z}}.
\end{align*}
This can be rewritten as
\begin{align*}
\sigma&=
\sum_{x\in{\mathcal X}}P(x)\frac{1}{Q_{\alpha,z}(W(x)\|\sigma)}
\sigma^{\frac{1-\alpha}{2z}}W(x)^{\frac{\alpha}{2z}}
\left( W(x)^{\frac{\alpha}{2z}}\sigma^{\frac{1-\alpha}{z}}W(x)^{\frac{\alpha}{2z}}\right)^{z-1}
W(x)^{\frac{\alpha}{2z}}\sigma^{\frac{1-\alpha}{2z}}\\
&=
\sum_{x\in{\mathcal X}}P(x)\frac{1}{Q_{\alpha,z}(W(x)\|\sigma)}
\left(\sigma^{\frac{1-\alpha}{2z}}W(x)^{\frac{\alpha}{z}}\sigma^{\frac{1-\alpha}{2z}}\right)^{z},
\end{align*}
where the last identity follows from $Xf(X^*X)X^*=(\id_{\mathbb{R}}f)(XX^*)$.
\end{proof}
\begin{rem}
The special case $z=1$ yields the characterization of the $P$-weighted Petz-type R\'enyi divergence center as the fixed point of the map
\begin{align*}
\Phi_{W,P,D_{\alpha}}(\sigma):=\sum_{x\in{\mathcal X}}P(x)\frac{1}{Q_{\alpha}(W(x)\|\sigma)}\sigma^{\frac{1-\alpha}{2}}
W(x)^{\alpha}\sigma^{\frac{1-\alpha}{2}},\mbox{ }\mbox{ }\ds\mbox{ }\mbox{ }
\sigma\in{\mathcal S}_{W,P}({\mathcal H})_{++},
\end{align*}
for any $\alpha\in(0,+\infty)\setminus\{1\}$.
Note that in the classical case $D_{\alpha,z}$ is independent of $z$, i.e., $D_{\alpha,z}=D_{\alpha}$ for all
$z>0$, and the above characterization of the minimizer has been derived recently by
Nakibo\u glu in \cite[Lemma 13]{NakibogluAugustin}, using very different methods.
Following Nakibo\u glu's approach, Cheng, Li and Hsieh has derived the
above characterization for the Petz-type R\'enyi divergence center in
\cite[Proposition 4]{Cheng-Li-Hsieh2018}.
The advantage of Nakibo\u glu's approach is that it also provides quantitative bounds of the deviation of
$\sum_xP(x)D_{\alpha}(W(x)\|\sigma)$ from $\chi_{\alpha,z}(W,P)$ for an arbitrary state $\sigma$; however, it is
not clear whether this approach can be extended
to the case $z\ne 1$, in particular, for $z=\alpha$, which is the relevant case
for the strong converse exponent of constant composition classical-quantum channel coding, as we will see in Section \ref{sec:sc}.
\end{rem}
\begin{rem}
A similar approach as in the above proof of Theorem \ref{prop:fixed point characterization}
was used by Hayashi and Tomamichel in \cite[Appendix C]{HT14} to characterize the optimal state for the
sandwiched R\'enyi mutual information as the fixed point of a non-linear map on the state space.
We comment on this in more detail in Section \ref{sec:generalized mutual informations}.
\end{rem}
\begin{example}\label{ex:noiseless}
We say that a cq channel $W$ is \ki{noiseless} on $\supp P$ if
$W(x)W(y)=0$ for all $x,y\in\supp P$, $x\ne y$, i.e.,
the output states corresponding to inputs in $\supp P$ are perfectly distinguishable. A straightforward computation shows that if $W$ is noiseless on $\supp P$ then
$\sigma:=W(P)=\sum_x P(x)W(x)$ satisfies the fixed point equation
\eqref{D fixed point eq} for any pair $(\alpha,z)$.
Hence, if $(\alpha,z)$ satisfies the conditions of Proposition \ref{prop:fixed point characterization} then
$W(P)$ is a minimizer for \eqref{weighted alpha divrad def}, and we have
\begin{align*}
\chi_{\alpha,z}(W,P)=\sum_{x\in{\mathcal X}}P(x)D_{\alpha,z}(W(x)\|W(P))=H(P):=-\sum_{x\in{\mathcal X}}P(x)\log P(x).
\end{align*}
Thus, the R\'enyi $(\alpha,z)$ radius of $W$ is equal to the Shannon entropy of the input distribution, independently of the value of $(\alpha,z)$.
\end{example}
\begin{cor}
If $(\alpha,z)$ satisfies the conditions of Proposition \ref{prop:fixed point characterization}, and $D_{\alpha,z}$ is monotone under CPTP maps then
\begin{align}\label{entropy bound}
\chi_{\alpha,z}(W,P)\le H(P)
\end{align}
for any cq channel $W$ and input distribution $P$.
\end{cor}
\begin{proof}
We may assume without loss of generality that ${\mathcal X}=\supp P$.
Let $\tilde W(x):=\pr{e_x}$ for some orthonormal basis $(e_x)_{x\in\supp P}$ in a Hilbert space ${\mathcal K}$, and let
$\Phi(.):=\sum_{x\in\supp P}W(x)\bra{e_x}(.)\ket{e_x}$, which is a CPTP map from
${\mathcal B}({\mathcal K})$ to ${\mathcal B}({\mathcal H})$. We have $W=\Phi\circ \tilde W$, and the assertion follows from Example \ref{ex:noiseless}.
\end{proof}
\begin{rem}
Our approach to prove \eqref{entropy bound} follows that of Csisz\'ar \cite{Csiszar}.
A (much) simpler approach to prove the inequality \eqref{entropy bound} was given by Nakibo\u glu
\cite[Lemma 13]{NakibogluAugustin}
(see also \cite[Proposition 4]{Cheng-Li-Hsieh2018} for an adaptation to various quantum R\'enyi divergences).
Obviously,
\begin{align}\label{upper bound}
\chi_{\alpha,z}(W,P)\le\sum_{x\in{\mathcal X}}P(x)D_{\alpha,z}(W(x)\|W(P)).
\end{align}
Assume now that $D_{\alpha,z}$ satisfies the monotonicity property
${\mathcal B}({\mathcal H})_+\ni\sigma_1\le\sigma_2$ $\Longrightarrow$ $D_{\alpha,z}(\varrho\|\sigma_1)\ge D_{\alpha,z}(\varrho\|\sigma_2)$ for any
$\varrho\in{\mathcal B}({\mathcal H})_+$. It is easy to see that this holds for every $(\alpha,z)$ with $z\ge |\alpha-1|$. In this case, we can lower bound
$W(P)$ by $P(x)W(x)$, and hence
$D_{\alpha,z}(W(x)\|W(P))\le D_{\alpha,z}(W(x)\|P(x)W(x))=-\log P(x)$, whence the RHS of \eqref{upper bound} can be upper bounded by $H(P)$.
\end{rem}
\subsection{Additivity of the weighted R\'enyi radius}
\label{sec:additivity}
Let $W^{(i)}:\,{\mathcal X}^{(i)}\to{\mathcal B}({\mathcal H}^{(i)})_+$, $i=1,2$, be gcq channels, and $P^{(i)}\in{\mathcal P}_f({\mathcal X}^{(i)})$ be input probability distributions.
For any $\alpha\in(0,+\infty)$ and $z\in(0,+\infty]$,
\begin{align}
&\chi_{\alpha,z}\left( W^{(1)}\otimes W^{(2)},P^{(1)}\otimes P^{(2)}\right)\\
&\mbox{ }\mbox{ }\le
\inf_{\sigma_i\in{\mathcal S}({\mathcal H}_i)}\sum_{x_1\in{\mathcal X}^{(1)},\,x_2\in{\mathcal X}^{(2)}}
P^{(1)}(x_1)P^{(2)}(x_2)D_{\alpha,z}\left( W^{(1)}(x_1)\otimes W^{(1)}(x_1)\|\sigma_1\otimes\sigma_2\right)
\label{subadditivity}\\
&\mbox{ }\mbox{ }=
\chi_{\alpha,z}\left( W^{(1)},P^{(1)}\right)
+
\chi_{\alpha,z}\left( W^{(2)},P^{(2)}\right),
\end{align}
by definition, i.e., $\chi_{\alpha,z}$ is subadditive. In particular, for fixed $W:\,{\mathcal X}\to{\mathcal S}({\mathcal H})$ and $P\in{\mathcal P}_f({\mathcal X})$,
the sequence $m\mapsto \chi_{\alpha,z}(W^{\otimes m},P^{\otimes m})$ is subadditive, and hence
\begin{align*}
\lim_{m\to+\infty}\frac{1}{m}\chi_{\alpha,z}(W^{\otimes m},P^{\otimes m})
=
\inf_{m\in\mathbb{N}}\frac{1}{m}\chi_{\alpha,z}(W^{\otimes m},P^{\otimes m})
\le
\chi_{\alpha,z}(W,P).
\end{align*}
In fact,
\begin{align}\label{weak subadditivity}
\frac{1}{m}\chi_{\alpha,z}(W^{\otimes m},P^{\otimes m})
\le
\chi_{\alpha,z}(W,P)
\end{align}
for all $m\in\mathbb{N}$.
As it turns out, we also have the stronger property of additivity, at least for $(\alpha,z)$ pairs for which the optimal $\sigma$ can be characterized by the fixed point equation \eqref{D fixed point eq}.
\begin{theorem}\label{thm:additivity}
\ki{(Additivity of the weighted R\'enyi radius)}
Let $W^{(1)}:\,{\mathcal X}^{(1)}\to{\mathcal S}({\mathcal H}^{(1)})$ and
$W^{(2)}:\,{\mathcal X}^{(2)}\to{\mathcal S}({\mathcal H}^{(2)})$ be gcq channels,
and $P^{(i)}\in{\mathcal P}_f({\mathcal X}^{(i)})$, $i=1,2$, be input distributions.
Assume, moreover, that $\alpha$ and $z$ satisfy the conditions of
Theorem \ref{prop:fixed point characterization}.
Then
\begin{align}
\chi_{\alpha,z}\left( W^{(1)}\otimes W^{(2)},P^{(1)}\otimes P^{(2)}\right)=
\chi_{\alpha,z}\left( W^{(1)},P^{(1)}\right)
+
\chi_{\alpha,z}\left( W^{(2)},P^{(2)}\right).
\end{align}
\end{theorem}
\begin{proof}
Let $\sigma_i$ be a minimizer of \eqref{weighted alpha divrad def} for $(W^{(i)},P^{(i)})$. By Theorem \ref{prop:fixed point characterization}, this means that
$\Phi_{W^{(i)},P^{(i)},D_{\alpha,z}}(\sigma_i)=\sigma_i$.
It is easy to see that
\begin{align*}
\Phi_{W^{(1)}\otimes W^{(2)},P^{(1)}\otimes P^{(2)},D_{\alpha,z}}(\sigma_1\otimes\sigma_2)
=
\Phi_{W^{(1)},P^{(1)},D_{\alpha,z}}(\sigma_1)\otimes
\Phi_{W^{(2)},P^{(2)},D_{\alpha,z}}(\sigma_2)
=
\sigma_1\otimes\sigma_2.
\end{align*}
Hence, again by Proposition \ref{prop:fixed point characterization},
$\sigma_1\otimes\sigma_2$ is a minimizer of \eqref{weighted alpha divrad def} for
$(W^{(1)}\otimes W^{(2)},P^{(1)}\otimes P^{(2)})$.
This proves the assertion.
\end{proof}
\begin{cor}\label{cor:additivity0}
For any gcq channel $W:\,{\mathcal X}\to{\mathcal B}({\mathcal H})_+$, any $P\in{\mathcal P}_f({\mathcal X})$, and any
pair $(\alpha,z)$ satisfying the conditions in Theorem \ref{prop:fixed point characterization}, we have
\begin{align*}
\chi_{\alpha,z}(W^{\otimes m},P^{\otimes m})
=
m\chi_{\alpha,z}(W,P),\mbox{ }\mbox{ }\ds\mbox{ }\mbox{ } m\in\mathbb{N}.
\end{align*}
\end{cor}
We will need the following special case for the application to classical-quantum channel coding in the next section:
\begin{cor}\label{cor:additivity}
For any gcq channel $W:\,{\mathcal X}\to{\mathcal B}({\mathcal H})_+$, any $P\in{\mathcal P}_f({\mathcal X})$, and any $\alpha\in(1/2,+\infty]$,
\begin{align*}
\chi_{\alpha}^{*}(W^{\otimes m},P^{\otimes m})
=
m\chi_{\alpha}^{*}(W,P),\mbox{ }\mbox{ }\ds\mbox{ }\mbox{ } m\in\mathbb{N}.
\end{align*}
\end{cor}
\begin{rem}
As far as we are aware, the idea of proving the additivity of an information quantity by characterizing some optimizer state as the fixed point of a non-linear operator on the state space appeared first in \cite{HT14}.
We comment on this in more detail in Appendix \ref{sec:generalized mutual informations}.
\end{rem}
\section{Strong converse exponent with constant composition}
\label{sec:sc}
Let $W:\,{\mathcal X}\to{\mathcal S}({\mathcal H})$ be a classical-quantum channel.
A \ki{code} ${\mathcal C}_n$ for $n$ uses of the channel is a pair ${\mathcal C}_n=(\mathrm{E}_n,{\mathcal D}_n)$, where
$\mathrm{E}_n:\,[M_n]\to{\mathcal X}^n$, ${\mathcal D}_n:\,[M_n]\to{\mathcal B}({\mathcal H}^{\otimes n})_+$, where
$|{\mathcal C}_n|:=M_n\in\mathbb{N}$ is the size of the code, and
${\mathcal D}_n$ is a POVM, i.e., $\sum_{i=1}^{M_n}{\mathcal D}_n(i)=I$.
The average success probability of a code ${\mathcal C}_n$ is
\begin{align*}
P_s(W^{\otimes n},{\mathcal C}_n):=\frac{1}{|{\mathcal C}_n|}\sum_{m=1}^{|{\mathcal C}_n|}\Tr W^{\otimes n}(\mathrm{E}_n(m)){\mathcal D}_n(m).
\end{align*}
A sequence of codes ${\mathcal C}_n=(\mathrm{E}_n,{\mathcal D}_n)$, $n\in\mathbb{N}$, is called a sequence of \ki{constant composition codes with asymptotic composition $P\in{\mathcal P}_f({\mathcal X})$} if there exists a sequence of types
$P_n\in{\mathcal P}_n({\mathcal X})$, $n\in\mathbb{N}$, such that $\lim_{n\to+\infty}\norm{P_n- P}_1=0$, and $\mathrm{E}_n(k)\in {\mathcal X}^n_{P_n}$ for all $k\in\{1,\ldots,|{\mathcal C}_n|\}$, $n\in\mathbb{N}$.
(See Section \ref{sec:Preliminaries} for the notation and basic facts concerning types.)
For any rate $R\ge 0$,
the strong converse exponents of $W$ with composition constraint $P$ are defined as
\begin{align}
\underline{\mathrm{sc}}(W,R,P):=\inf\left\{\liminf_{n\to+\infty}-\frac{1}{n}\log P_s(W^{\otimes n},{\mathcal C}_n):\,\liminf_{n\to+\infty}\frac{1}{n}\log|{\mathcal C}_n|\ge R\right\},\label{sci}\\
\overline{\mathrm{sc}}(W,R,P):=\inf\left\{\limsup_{n\to+\infty}-\frac{1}{n}\log P_s(W^{\otimes n},{\mathcal C}_n):\,\liminf_{n\to+\infty}\frac{1}{n}\log|{\mathcal C}_n|\ge R\right\},\label{scs}
\end{align}
where the infima are taken over code sequences of constant composition $P$.
Our main result is the following:
\begin{theorem}\label{thm:main result}
For any classical-quantum channel $W$, and finitely supported probability distribution $P$ on the input of $W$, and any rate $R$,
\begin{align}\label{main result}
\underline{\mathrm{sc}}(W,R,P)=\overline{\mathrm{sc}}(W,R,P)=\sup_{\alpha>1}\frac{\alpha-1}{\alpha}\left[R-\chi_{\alpha}^{*}(W,P)\right].
\end{align}
\end{theorem}
We will prove the equality in \eqref{main result} as two separate inequalities in
Propositions \ref{prop:sc lower} and \ref{prop:upper reg}. Before starting with that, we point out the following complementary result by Dalai and Winter \cite{DW2015}:
\begin{rem}
Similarly to the strong converse exponents, one can define the direct exponents as
\begin{align*}
\underline{d}(W,R,P):=\sup\left\{\liminf_{n\to+\infty}-\frac{1}{n}\log (1-P_s(W^{\otimes n},{\mathcal C}_n)):\,\liminf_{n\to+\infty}\frac{1}{n}\log|{\mathcal C}_n|\ge R\right\},\\
\overline{d}(W,R,P):=\sup\left\{\limsup_{n\to+\infty}-\frac{1}{n}\log (1-P_s(W^{\otimes n},{\mathcal C}_n)):\,\liminf_{n\to+\infty}\frac{1}{n}\log|{\mathcal C}_n|\ge R\right\},
\end{align*}
where the suprema are taken over code sequences of constant composition $P$.
The following, so-called sphere packing bound has been shown in \cite{DW2015}:
\begin{align}\label{sp bound}
\overline{d}(W,R,P)\le\sup_{0<\alpha<1}\frac{\alpha-1}{\alpha}\left[R-\chi_{\alpha}(W,P)\right].
\end{align}
Note that the right-hand sides of \eqref{main result} and \eqref{sp bound} are very similar to each other, except that the range of optimization is $\alpha>1$ in the former and $\alpha\in (0,1)$ in the latter,
and the weighted R\'enyi radii corresponding to the sandwiched R\'enyi divergences appear in the former, and
to the Petz-type R\'enyi divergences in the latter.
Also, while
\eqref{main result} holds for any $R>0$ (and is non-trivial for $R>\chi_1(W,P)$),
it is known that \eqref{sp bound} holds as an equality only for high enough rates (and is non-trivial for
$R<\chi_1(W,P)$) for classical channels, and it is a long-standing open problem if the same equality is true for classical-quantum channels.
\end{rem}
The following lower bound follows by a standard argument, due to Nagaoka \cite{N}, as was also observed, e.g., in
\cite{ChengHansonDattaHsieh2018}.
For readers' convenience, we write out the details in Appendix \ref{sec:lower bound}.
\begin{prop}\label{prop:sc lower}
For any $R>0$,
\begin{align*}
\sup_{\alpha>1}\frac{\alpha-1}{\alpha}\left[R-\chi_{\alpha}^{*}(W,P)\right]\le \underline{\mathrm{sc}}(W,R,P).
\end{align*}
\end{prop}
Our aim in the rest is to show that the second term is upper bounded by the rightmost term in \eqref{main result}.
We will follow the approach of \cite{MO-cqconv}, which in turn was inspired by \cite{DK}.
We start with the following:
\begin{prop}\label{prop:sc upper}
For any $R>0$,
\begin{align}\label{sc upper}
\overline{\mathrm{sc}}(W,R,P)\le\sup_{\alpha>1}\frac{\alpha-1}{\alpha}\left[R-\chi_{\alpha}^{\flat}(W,P)\right].
\end{align}
\end{prop}
\begin{proof}
We will show that
\begin{align}\label{sc upper2}
\overline{\mathrm{sc}}(W,R,P)\le\min\{F_1(W,R,P),F_2(W,R,P)\},
\end{align}
where
\begin{align*}
F_1(W,R,P)&:=\inf_{V:\,\chi(V,P)>R}D(V\|W|P),\\
F_2(W,R,P)&:=\inf_{V:\,\chi(V,P)\le R}\left[D(V\|W|P)+R-\chi(V,P)\right].
\end{align*}
Here, the infima are over channels $V:\,{\mathcal X}\to{\mathcal S}({\mathcal H})$ satisfying the indicated properties.
It was shown in \cite[Theorem 5.12]{MO-cqconv} that the RHS of \eqref{sc upper2} is the same as the RHS of \eqref{sc upper}.
We first show that $\overline{\mathrm{sc}}(W,R,P)\le F_1(W,R,P)$. To this end, let $r>F_1(W,R,P)$; then, by definition, there exists a channel $V:\,{\mathcal X}\to{\mathcal S}({\mathcal H})$ such that
\begin{align*}
D(V\|W|P)<r\mbox{ }\mbox{ }\ds\mbox{ }\mbox{ }\text{and}\mbox{ }\mbox{ }\ds\mbox{ }\mbox{ }
\chi(V,P)>R.
\end{align*}
Due to $\chi(V,P)>R$, Corollary \ref{cor:random coding exponent} yields the existence of a sequence
of constant composition codes ${\mathcal C}_n$ with composition $P_n$, $n\in\mathbb{N}$, such that
$\supp P_n\subseteq\supp P$ for all $n$, $\lim_{n\to+\infty}\norm{P_n-P}_1=0$,
the rate is lower bounded as $\frac{1}{n}\log|{\mathcal C}_n|\ge R$, $n\in\mathbb{N}$,
and
$\lim_{n\to+\infty}P_s(V^{\otimes n},{\mathcal C}_n)=1$.
Note that for any message $k$,
\begin{align*}
\Tr\left( V^{\otimes n}(\mathrm{E}_n(k))-e^{nr}W^{\otimes n}(\mathrm{E}_n(k))\right)_+
\ge
\Tr\left( V^{\otimes n}(\mathrm{E}_n(k))-e^{nr}W^{\otimes n}(\mathrm{E}_n(k))\right){\mathcal D}_n(k),
\end{align*}
and hence
\begin{align*}
\Tr W^{\otimes n}(\mathrm{E}_n(k)){\mathcal D}_n(k)
&\ge
e^{-nr}\left[\Tr V^{\otimes n}(\mathrm{E}_n(k)){\mathcal D}_n(k)
-
\Tr\left( V^{\otimes n}(\mathrm{E}_n(k))-e^{nr}W^{\otimes n}(\mathrm{E}_n(k))\right)_+
\right],
\end{align*}
This in turn yields, by averaging over $k$, that
\begin{align*}
P_s(W^{\otimes n},{\mathcal C}_n)&\ge
e^{-nr}\left[P_s(V^{\otimes n},{\mathcal C}_n)
-
\frac{1}{|{\mathcal C}_n|}\sum_{k=1}^{|{\mathcal C}_n|}\Tr\left( V^{\otimes n}(\mathrm{E}_n(k))-e^{nr}W^{\otimes n}(\mathrm{E}_n(k))\right)_+
\right]\\
&=
e^{-nr}\left[P_s(V^{\otimes n},{\mathcal C}_n)
-
\Tr\left( V^{\otimes n}(\vecc{x}^{(n)})-e^{nr}W^{\otimes n}(\vecc{x}^{(n)})\right)_+
\right],
\end{align*}
where $\vecc{x}^{(n)}$ is any sequence in ${\mathcal X}^n$ with type $P_n$. Since
$D(V\|W|P)<r$, Corollary \ref{cor:infospec limit} yields that
$\lim_{n\to+\infty}\Tr\left( V^{\otimes n}(\vecc{x}^{(n)})-e^{nr}W^{\otimes n}(\vecc{x}^{(n)})\right)_+=0$, and so finally
\begin{align*}
\liminf_{n\to+\infty}\frac{1}{n}\log P_s(W^{\otimes n},{\mathcal C}_n)&\ge
-r,\mbox{ }\mbox{ }\ds\text{whence}\mbox{ }\mbox{ }\ds
\overline{\mathrm{sc}}(W,R,P)\le r.
\end{align*}
Since this holds for any $r>F_1(W,R,P)$, we get $\overline{\mathrm{sc}}(W,R,P)\le F_1(W,R,P)$.
From this, one can prove that also $\overline{\mathrm{sc}}(W,R,P)\le F_2(W,R,P)$, the same way as it was done in
\cite[Lemma 5.11]{MO-cqconv} (which in turn followed the proof in \cite[Lemma 2]{DK}); one only has to make sure that the extension of the code can be done in a way that it remains constant composition with composition $P$,
but that is easy to verify.
\end{proof}
\medskip
From the above result, we can obtain the desired upper bound.
\begin{prop}\label{prop:upper reg}
For any $R>0$,
\begin{align}\label{sc upper reg}
\overline{\mathrm{sc}}(W,R,P)\le
\sup_{\alpha>1}\frac{\alpha-1}{\alpha}\left[R-\chi_{\alpha}^{*}(W,P)\right].
\end{align}
\end{prop}
\begin{proof}
We employ the asymptotic pinching technique from \cite{MO-cqconv}. Let
$W_m:\,{\mathcal X}^m\to{\mathcal S}({\mathcal H}^{\otimes m})$ be defined as
\begin{align*}
W_m(\vecc{x}):=\pin_m W^{\otimes m}(\vecc{x}),
\end{align*}
where $\pin_m$ is the pinching by the universal symmetric state $\sigma_{u,m}$, introduced in Section \ref{sec:Preliminaries}. Employing Proposition
\ref{prop:sc upper} with $W\mapsto W_m$, $R\mapsto Rm$ and $P\mapsto P^{\otimes m}$, we get that for any $R>0$, there exists a sequence of codes ${\mathcal C}^{(m)}_k=(\mathrm{E}^{(m)}_k,{\mathcal D}^{(m)}_k)$ with constant composition $P^{(m)}_k\in{\mathcal P}_k({\mathcal X}^n)$, $k\in\mathbb{N}$, such that
$\frac{1}{k}\log|{\mathcal C}^{(m)}_k|\ge mR$ for all $k$,
$\lim_{k\to+\infty}\norm{P^{(m)}_k-P^{\otimes m}}_1=0$,
and
\begin{align*}
\limsup_{k\to+\infty}-\frac{1}{k}\log P_s(W_m^{\otimes k},{\mathcal C}^{(m)}_k)
\le
\sup_{\alpha>1}\frac{\alpha-1}{\alpha}\left[mR-\chi_{\alpha}^{\flat}(W_m,P^{\otimes m})\right].
\end{align*}
For every $k\in\mathbb{N}$, define ${\mathcal C}_{km}:=(\mathrm{E}^{(m)}_k,\pin_m{\mathcal D}^{(m)}_k)$, which can be considered a code
for $W^{\otimes km}$, with the natural identifications
$({\mathcal X}^m)^k\equiv{\mathcal X}^{km}$ and $({\mathcal H}^{\otimes m})^{\otimes k}\equiv{\mathcal H}^{\otimes km}$. For a general $n\in\mathbb{N}$, choose $k\in\mathbb{N}$ such that $km\le n< (k+1)m$, and for every
$i=1,\ldots,|{\mathcal C}^{(m)}_k|$, define $\mathrm{E}_n(i)$ to be $\mathrm{E}_{km}(i)$ concatenated with $n-km$ copies of some fixed
$x_0\in\supp P$, independent of $i$ and $n$, and let ${\mathcal D}_n(i):={\mathcal D}_{km}(i)\otimes I_{{\mathcal H}}^{\otimes (n-km)}$.
Then it is easy to see that
\begin{align*}
\liminf_{n\to+\infty}\frac{1}{n}\log|{\mathcal C}_n|\ge R,
\end{align*}
and
\begin{align}\label{sc upper4}
\limsup_{n\to+\infty}-\frac{1}{n}\log P_s(W^{\otimes n},{\mathcal C}_n)
\le
\sup_{\alpha>1}\frac{\alpha-1}{\alpha}\left[R-\frac{1}{m}\chi_{\alpha}^{\flat}(W_m,P^{\otimes m})\right].
\end{align}
We need to show that the above sequence of codes is of constant composition $P$.
Let ${\bf x}=(\vecc{x}_1,\ldots,\vecc{x}_k)\in({\mathcal X}^m)^k\equiv{\mathcal X}^{km}$ be a codeword for ${\mathcal C}^{(m)}_k$, and let $P^{(m)}_{\bf{x}}$
and $P_{\bf{x}}$ denote the corresponding types when $\bf{x}$ is considered as an element of
$({\mathcal X}^m)^k$ and of ${\mathcal X}^{km}$, respectively. For any $a\in{\mathcal X}$,
\begin{align*}
P_{{\bf x}}(a)&=\frac{1}{km}\sum_{\vecc{x}\in{\mathcal X}^m}\#\{i:\,{\bf x}_i=\vecc{x}\}\cdot\#\{j:\,x_j=a\}
=
\sum_{\vecc{x}\in{\mathcal X}^m}P^{(m)}_{{\bf x}}(\vecc{x})P_{\vecc{x}}(a)
=
\sum_{\vecc{x}\in{\mathcal X}^m}P^{(m)}_k(\vecc{x})P_{\vecc{x}}(a)
\end{align*}
only depends on ${\bf x}$ through its type $P^{(m)}_{{\bf x}}=P^{(m)}_k$, which is independent of ${\bf x}$. Thus, the type of $\mathrm{E}_{km}(i)$ is independent of $i$, i.e., ${\mathcal C}_{km}$ is a constant composition code for every
$k\in\mathbb{N}$. For a general $n\in\mathbb{N}$ with $km\le n<(k+1)m$, we have
\begin{align*}
P_{\mathrm{E}_n(i)}(a)=\frac{km}{n}P_{\mathrm{E}_{km}(i)}(a)+\delta_{a,x_0}\frac{n-km}{n},\mbox{ }\mbox{ }\ds\mbox{ }\mbox{ }
i\in\{1,\ldots,|{\mathcal C}_n|=|{\mathcal C}_{km}|\},
\end{align*}
and hence ${\mathcal C}_n$ is also of constant composition.
Next, we show that $\lim_{n\to+\infty}\norm{P_n-P}_1=0$, where $P_n$ is the type of ${\mathcal C}_n$. For $km\le n<(k+1)m$, we have
\begin{align*}
\sum_{a\in{\mathcal X}}|P_n(a)-P(a)|
&\le
\sum_{a\in{\mathcal X}}|P_n(a)-P_{km}(a)|+\sum_{a\in{\mathcal X}}|P_{km}(a)-P(a)|,
\end{align*}
and
\begin{align*}
\sum_{a\in{\mathcal X}}|P_n(a)-P_{km}(a)|
&=
\sum_{a\in{\mathcal X}}\left( 1-\frac{km}{n}\right) P_{km}(a)+\left( 1-\frac{km}{n}\right)=
2\left( 1-\frac{km}{n}\right)\to 0
\end{align*}
as $k\to+\infty$. For the second term, we get
\begin{align*}
\sum_{a\in{\mathcal X}}|P_{km}(a)-P(a)|
&=
\sum_{a\in{\mathcal X}}
\abs{
\sum_{\vecc{x}\in{\mathcal X}^m}P^{(m)}_k(\vecc{x})P_{\vecc{x}}(a)
-
\sum_{\vecc{x}\in{\mathcal X}^m}P^{\otimes m}(\vecc{x})P_{\vecc{x}}(a)
}\\
&\le
\sum_{\vecc{x}\in{\mathcal X}^m}\abs{P^{(m)}_k(\vecc{x})-P^{\otimes m}(\vecc{x})}
\sum_{a\in{\mathcal X}}P_{\vecc{x}}(a)
=\norm{P^{(m)}_k-P^{\otimes m}}_1,
\end{align*}
where in the first identity we used Lemma \ref{lemma:type lemma},
and the last expression goes to $0$ as $k\to+\infty$ by assumption.
Since we have established that the codes used in \eqref{sc upper4} are of constant composition $P$, we get that
for any $m\in\mathbb{N}$,
\begin{align}
\overline{\mathrm{sc}}(W,R,P)\le\sup_{\alpha>1}\frac{\alpha-1}{\alpha}\left[R-\frac{1}{m}\chi_{\alpha}^{\flat}(W_m,P^{\otimes m})\right].
\end{align}
According to \cite[Lemma 4.10]{MO-cqconv},
$\chi_{\alpha}^{\flat}(W_m,P^{\otimes m})\ge\chi_{\alpha}^{*}(W^{\otimes m},P^{\otimes m})-3\log v_{m,d}$, and hence
\begin{align}
\overline{\mathrm{sc}}(W,R,P)
&\le\sup_{\alpha>1}\frac{\alpha-1}{\alpha}\left[R-\frac{1}{m}\chi_{\alpha}^{*}(W^{\otimes m},P^{\otimes m})\right]+3\frac{\log v_{m,d}}{m}\\
&\le\sup_{\alpha>1}\frac{\alpha-1}{\alpha}\left[R-\inf_{m\in\mathbb{N}}\frac{1}{m}\chi_{\alpha}^{*}(W^{\otimes m},P^{\otimes m})\right]+3\frac{\log v_{m,d}}{m}
\end{align}
for every $m\in\mathbb{N}$, from which
\begin{align}
\overline{\mathrm{sc}}(W,R,P)
&\le\sup_{\alpha>1}\frac{\alpha-1}{\alpha}\left[R-\inf_{m\in\mathbb{N}}\frac{1}{m}\chi_{\alpha}^{*}(W^{\otimes m},P^{\otimes m})\right].
\end{align}
Finally, Corollary \ref{cor:additivity} yields the desired bound \eqref{sc upper reg}.
\end{proof}
\medskip
As it has been shown in \cite{MO-cqconv}, the strong converse exponent of a cq channel $W$ with unconstrained
coding is given by
\begin{align*}
\mathrm{sc}(W,R):=\underline{\mathrm{sc}}(W,R)=\overline{\mathrm{sc}}(W,R)=\sup_{\alpha>1}\frac{\alpha-1}{\alpha}\left[R-\chi_{\alpha}^{*}(W)\right]
\end{align*}
for any $R>0$, where $\chi_{\alpha}^{*}(W)=\sup_{P\in{\mathcal P}_f({\mathcal X})}\chi_{\alpha}^{*}(W,P)$, and
$\underline{\mathrm{sc}}(W,R)$ and $\overline{\mathrm{sc}}(W,R)$ are defined analogously to \eqref{sci}--\eqref{scs} by dropping the constant composition constraint.
It is natural to ask whether this optimal value can be achieved, or at least arbitrarily well approximated, by constant composition codes, i.e., whether we have
\begin{align}\label{constant approximation}
\inf_{P\in{\mathcal P}_f({\mathcal X})}sc(W,R,P)=sc(W,R).
\end{align}
In view of Theorem \ref{thm:main result}, this is equivalent to whether
\begin{align}\label{minimax1}
\inf_{P\in{\mathcal P}_f({\mathcal X})}\sup_{\alpha>1}\frac{\alpha-1}{\alpha}\left[R-\chi_{\alpha}^{*}(W,P)\right]
=
\sup_{\alpha>1}\inf_{P\in{\mathcal P}_f({\mathcal X})}\frac{\alpha-1}{\alpha}\left[R-\chi_{\alpha}^{*}(W,P)\right].
\end{align}
By introducing the new variable $u:=\frac{\alpha-1}{\alpha}$, and $f(P,u):=u\chi_{\frac{1}{1-u}}^{*}(W,P)$,
\eqref{minimax1} can be rewritten as
\begin{align}\label{minimax2}
\inf_{P\in{\mathcal P}_f({\mathcal X})}\sup_{0<u<1}\{uR-f(P,u)\}=
\sup_{0<u<1}\inf_{P\in{\mathcal P}_f({\mathcal X})}\{uR-f(P,u)\}.
\end{align}
One can extend $f$ to $[0,1]$ by $f(0):=0$ and $f(1):=\lim_{u\nearrow 1}f(u)=\chi_{+\infty}^{*}(W,P)$, where the
latter is the $P$-weighted max-relative entropy \cite{RennerPhD,Datta} radius of $P$. It is not difficult to see
(similarly to \cite[Lemma 5.13]{MO-cqconv}) that \eqref{minimax2} is equivalent to
\begin{align}\label{minimax3}
\inf_{P\in{\mathcal P}_f({\mathcal X})}\sup_{0\le u\le 1}\{uR-f(P,u)\}=
\sup_{0\le u\le 1}\inf_{P\in{\mathcal P}_f({\mathcal X})}\{uR-f(P,u)\}.
\end{align}
It is clear that $f$ is a concave function of $P$, hence the above minimiax equality follows from
Lemma \ref{lemma:KF+ minimax} if $f(P,.)$ is convex on $[0,1]$ for every $P\in{\mathcal P}_f({\mathcal X})$.
This turns out to be highly non-trivial, and has been proved very recently in
\cite{Cheng-Li-Hsieh2018} using interpolation techniques.
The identity \eqref{minimax2} has also been stated in \cite{Cheng-Li-Hsieh2018}, although only as a formal
identity, as the operational interpretation of
$\sup_{\alpha>1}\frac{\alpha-1}{\alpha}\left[R-\chi_{\alpha}^{*}(W,P)\right]$, (i.e.,
Theorem \ref{thm:main result}), and hence the equivalence of
\eqref{minimax2} and \eqref{constant approximation},
had not yet been known then.
\medskip
The convexity of $f(P,.)$ also plays an important role in establishing the weighted sandwiched
R\'enyi divergence radii as generalized cutoff rates in the sense of Csisz\'ar \cite{Csiszar}.
Following \cite{Csiszar}, for a fixed $\kappa>0$,
we define the
generalized $\kappa$-cutoff rate $C_{\kappa}(W,P)$ for a cq channel $W$ and input distribution $P$ as
the smallest number $R_0$ satisfying
\begin{align}\label{cutoff def}
\underline{\mathrm{sc}}(W,R,P)\ge \kappa(R-R_0),\mbox{ }\mbox{ }\ds\mbox{ }\mbox{ } R>0.
\end{align}
The following extends the analogous result for classical channels in \cite{Csiszar} to classical-quantum channels,
and gives a direct operational interpretation of the weighted sandwiched R\'enyi divergence radius of a cq channel as a generalized cutoff rate.
\begin{prop}\label{prop:cutoff}
For any $\kappa\in(0,1)$,
\begin{align*}
C_{\kappa}(W,P)=\chi^{*}_{\frac{1}{1-\kappa}}(W,P),
\end{align*}
or equivalently, for any $\alpha>1$,
\begin{align*}
\chi^{*}_{\alpha}(W,P)=C_{\frac{\alpha-1}{\alpha}}(W,P).
\end{align*}
\end{prop}
\begin{proof}
By Theorem \ref{thm:main result}, we have
\begin{align*}
\underline{\mathrm{sc}}(R,W,P)=\sup_{0<u<1}\{uR-f(P,u)\}\ge \kappa R-f(P,\kappa)=\kappa\left( R-\frac{1}{\kappa} f(P,\kappa)\right),\mbox{ }\mbox{ }\ds\mbox{ }\mbox{ }
\kappa\in(0,1),
\end{align*}
where the inequality is trivial. Since $f(P,.)$ is convex according to \cite{Cheng-Li-Hsieh2018}, its left and right derivatives at $\kappa$, $\derleft{f(P,.)}(\kappa)$ and $\derright{f(P,.)}(\kappa)$, exist. Obviously,
for any $\derleft{f(P,.)}(\kappa)\le R\le \derright{f(P,.)}(\kappa)$,
\begin{align*}
\sup_{0<u<1}\{uR-f(P,u)\}= \kappa R-f(P,\kappa)=\kappa\left( R-\frac{1}{\kappa} f(P,\kappa)\right),
\end{align*}
showing that $\frac{1}{\kappa} f(P,\kappa)=\chi^{*}_{\frac{1}{1-\kappa}}(W,P)$ is the minimal $R_0$ for which \eqref{cutoff def} holds for all $R>0$.
\end{proof}
|
1,108,101,564,563 | arxiv | \section*{Introduction}
The beautiful theory of cluster tilting in triangulated categories has been
developed by Iyama and Yoshino; as an important outcome of this the authors gave in \cite[Theorem 1.2 and Theorem
1.3]{iyama-yoshino} the classification of rigid indecomposable MCM modules over two
Veronese embeddings in $\mathbb P^9$ given, respectively, by plane cubics and space
quadrics.
Another proof, that makes use of Orlov's singularity
category, appears in \cite{keller-murfet-van_den_bergh}, where the link
between power series Veronese rings and the graded rings of the
corresponding varieties is also explained.
Also, \cite{keller-reiten:acyclic} contains yet another argument.
The goal of this note is to present a simple proof of Iyama-Yoshino's
classification of rigid MCM modules over the aforementioned Veronese
rings, making use of
vector bundles and Beilinson's theorem.
This proof works over a field ${\boldsymbol{k}}$ which is algebraically
closed or finite.
Consider the embedding of the projective space $\mathbb P^n$ given by
homogeneous forms of degree $d$, i.e. the {\it $d$-fold
Veronese variety}.
A coherent sheaf $E$ on $\mathbb P^n$ is arithmetically Cohen-Macaulay (ACM)
with respect to this embedding if and only
if $E$ is locally free and has {\it no intermediate cohomology}:
\begin{equation}
\label{sonmalato}
\HH^i(\mathbb P^n,E(d t))=0, \qquad \mbox{for all $t \in \mathbb Z$ and all $0<i<n$}.
\end{equation}
This is equivalent to ask that the module of global sections
associated with $E$ is MCM over the corresponding Veronese ring.
For $d$-fold Veronese embeddings of $\mathbb P^n$ in $\mathbb P^9$ (i.e. $\{n,d\}=\{2,3\}$), we are going to classify ACM
bundles $E$ which are {\it rigid}, i.e. $\Ext^1_{\mathbb P^n}(E,E)=0$.
We set $\ell = {n+1 \choose 2}$.
To state the classification,
we define
the {\it Fibonacci numbers} $a_{\ell,k}$ by the relations:
$a_{\ell,0}=0$, $a_{\ell,1}=1$ and $a_{\ell,k+1}=\ell a_{\ell,k}-a_{\ell,k-1}$.
For instance $(a_{3,k})$ is given by the odd values of the
usual Fibonacci sequence:
\[
a_{3,k}=0,1,3,8,21,55,144,\ldots \qquad \mbox{for $k=0,1,2,3,4,5,6,\ldots$}
\]
\begin{thm}[\{n,d\}=\{2,3\}] \label{class2}
Let $E$ be an indecomposable bundle on $\mathbb P^n$ satisfying
\eqref{sonmalato}.
\begin{enumerate}[i)]
\item \label{perforzasteiner} If $E$ has no endomorphism factoring
through $\mathcal{O}_{\mathbb P^n}(t)$, then there are
$a,b \ge 0$ such that, up to a twist by
$\mathcal{O}_{\mathbb P^n}(s)$, $E$ or $E^*$ is the cokernel of an injective map:
\[
\Omega_{\mathbb P^n}^2(1)^b \to \mathcal{O}_{\mathbb P^n}(-1)^a
\]
\item \label{quandoerigido} If $E$ is indecomposable and rigid, then there is $k\ge 1$
such that, up to tensoring with $\mathcal{O}_{\mathbb P^n}(s)$, $E$ or $E^*$ is the cokernel of an injective map:
\[
\Omega_{\mathbb P^n}^2(1)^{a_{\ell,k-1}} \to \mathcal{O}_{\mathbb P^n}(-1)^{a_{\ell,k}}
\]
\item \label{nienteH1} Conversely for any $k \ge 1$, there is a unique indecomposable
bundle $E_k$ having a resolution of the form:
\[
0 \to \Omega_{\mathbb P^n}^2(1)^{a_{\ell,k-1}} \to \mathcal{O}_{\mathbb P^n}(-1)^{a_{\ell,k}} \to E_k \to 0,
\]
and both $E_k$ and $E^*_k$ are ACM and exceptional.
\end{enumerate}
\end{thm}
In the previous statement, it is understood that a bundle $E$ is
{\it exceptional} if it is rigid, {\it simple}
(i.e. $\Hom_{\mathbb P^n}(E,E) \simeq {\boldsymbol{k}}$) and $\Ext^i_{\mathbb P^n}(E,E)=0$ for $i \ge 2$.
Next, we write $\Omega_{\mathbb P^n}^p=\wedge^p \Omega_{\mathbb P^n}^1$ for the bundle
of differential $p$-forms on $\mathbb P^n$.
\begin{rmk} Part \eqref{perforzasteiner} of Theorem \ref{class2}
is a version of Iyama-Yoshino's general results on Veronese
rings \cite[Theorem 9.1 and 9.3]{iyama-yoshino}, to the
effect that for $\{n,d\}=\{2,3\}$ the stable category of MCM modules is equivalent to
the category of representations of a certain Kronecker quiver.
However our result is algorithmic, for it provides
the representation associated with an MCM
module via Beilinson spectral sequence applied to the corresponding ACM bundle.
\end{rmk}
\begin{rmk}
The rank of the bundle $E_k$ is given by the Fibonacci number between $a_{3,k-1}$ and
$a_{3,k}$ in case $(n,d)=(2,3)$.
In this case $E_{2k}$ (respectively, $E_{2k+1}$) is the $k$-th
sheafified syzygy occurring in the
resolution of $\mathcal{O}_{\mathbb P^2}(1)$ (respectively, of $\mathcal{O}_{\mathbb P^2}(2)$)
over the Veronese ring, twisted by
$\mathcal{O}_{\mathbb P^2}(3(k-1))$. A similar result holds for $(n,d)=(3,2)$.
\end{rmk}
As for notation, we write small letters for the dimension
of a space in capital letter, for instance $\hh^i(\mathbb P^n,E)=\dim_{\boldsymbol{k}} \HH^i(\mathbb P^n,E)$.
We also write $\chi(E,F)=\sum(-1)^i \ext^i_{\mathbb P^n}(E,F)$ and
$\chi(E)=\chi(\mathcal{O}_{\mathbb P^n},E)$. $\delta_{i,j}$ is Kronecker's delta.
\section{Fibonacci bundles}
\subsection{}
Let us write
$\Upsilon_\ell$ for the $\ell$-th Kronecker quiver, namely the oriented
graph with two vertices $\mathbf{e_0}$ and $\mathbf{e_1}$, and $\ell$
arrows from $\mathbf{e_0}$ to $\mathbf{e_1}$.
A representation $R$ of $\Upsilon_\ell$, with
dimension vector $(a,b)$ is the choice of $\ell$ matrices of size $a
\times b$.
\[\begin{tikzpicture}[scale=1]
\draw (-2.5,0) node [] {$\Upsilon_3$:};
\draw (-1,0) node [above] {$ \mathbf{e_1}$};
\draw (1,0) node [above] {$\mathbf{e_2}$};
\node (1) at (-1,0) {$\bullet$};
\node (1) at (-1,0) {$\bullet$};
\node (2) at (1,0) {$\bullet$};
\draw[->,>=latex] (1) to (2);
\draw[->,>=latex] (1) to[bend left] (2);
\draw[->,>=latex] (1) to[bend right] (2);
\end{tikzpicture}
\]
We identify a basis of $\HH^0(\mathbb P^n,\Omega_{\mathbb P^n}(2))$ with the set
of $\ell={n+1 \choose 2}$ arrows of $\Upsilon_\ell$.
Then the derived category of finite-dimensional representations of
$\Upsilon_\ell$
embeds into the derived category of $\mathcal{O}_{\mathbb P^n}$-modules by sending $R$ to
the cone $\Phi(R)$ of the morphism $e_R$
associated with $R$ according to this identification:
\[
\Phi(R)[-1] \to \mathcal{O}_{\mathbb P^n}(-1)^{a} \xrightarrow{e_R} \Omega_{\mathbb P^n}(1)^{b},
\]
where we denote by $[-1]$ the shift to the right of complexes. It is clear that:
\[
\Ext_{\mathbb P^n}^i(\Phi(R),\Phi(R)) \simeq \Ext^i_{\Upsilon_\ell}(R,R), \qquad \mbox{for all $i$}.
\]
\subsection{} \label{KAC}
We will use Kac's classification of rigid $\Upsilon_\ell$-modules as
Schur roots (hence the restriction on ${\boldsymbol{k}}$), which is also one of the main ingredients in
Iyama-Yoshino's proof.
By \cite[Theorem 4]{kac}, any non-zero rigid
$\Upsilon_\ell$-module is a direct sum of rigid simple
representations of the form $R_k$, for some
$k \in \mathbb Z$, where $R_k$ is defined as the unique indecomposable
representation of $\Upsilon_\ell$ with dimension vector
$(a_{\ell,k-1},a_{\ell,k})$ for $k \ge 1$, or $(a_{\ell,-k},a_{\ell,1-k})$
for $k \le 0$.
Set $F_k=\Phi(R_k)$ for $k \ge 1$, and $F_k=\Phi(R_k)[-1]$ for $k \le 0$.
It turns out that $F_k$ is an exceptional locally free sheaf, called a
{\it Fibonacci bundle}, cf. \cite{brambilla:fibonacci}.
We rewrite the defining exact sequences of $F_k$:
\begin{align}
\label{defining+}
&0 \to \mathcal{O}_{\mathbb P^n}(-1)^{a_{\ell,k-1}} \to \Omega_{\mathbb P^n}(1)^{a_{\ell,k}} \to F_k \to 0, && \mbox {for $k \ge 1$},\\
\nonumber &0 \to F_k \to \mathcal{O}_{\mathbb P^n}(-1)^{a_{\ell,-k}} \to \Omega_{\mathbb P^n}(1)^{a_{\ell,1-k}} \to 0, && \mbox {for $k \le 0$}.
\end{align}
\subsection{} Here is a lemma on the cohomology of Fibonacci bundles.
\begin{lem} \label{cohom}
For $k \ge 1$, the only non-vanishing intermediate cohomology of $F_k$
is:
\[
\hh^1(\mathbb P^n,F_k(-1))=a_{\ell,k-1} \qquad \hh^{n-1}(\mathbb P^n,F_k(-n))=a_{\ell,k}.
\]
\end{lem}
\begin{proof}
We consider the left and right mutation
endofunctors of the derived category of coherent sheaves on $\mathbb P^n$,
that associate with a pair $(E,F)$ of complexes, two complexes denoted respectively by $\rR_FE$ and $\rL_E F$. These are the cones of the natural
evaluation maps $f_{E,F}$ and $g_{E,F}$:
\[
E \xrightarrow{f_{E,F}} \RHom_{\mathbb P^n}(E,F)^* \ts F \to \rR_FE, \qquad
\rL_E F \to \RHom_{\mathbb P^n}(E,F) \ts E \xrightarrow{g_{E,F}} F.
\]
It is well-known (cf. \cite{brambilla:fibonacci}) that the Fibonacci bundles $F_k$ can be defined recursively
from $F_0=\mathcal{O}_{\mathbb P^n}(-1)$ and $F_1=\Omega_{\mathbb P^n}(1)$ by setting:
\begin{align}
\label{+} &F_{k+1}=\rR_{F_k} F_{k-1}, && \mbox{for $k \ge 1$}, \\
\label{-} &F_{k-1}=\rL_{F_k} F_{k+1}, && \mbox{for $k \le 0$}.
\end{align}
This way, for any $k\in \mathbb Z$ we get a natural exact sequence:
\begin{equation}
\label{recurrence}
0 \to F_{k-1} \to (F_k)^\ell \to F_{k+1} \to 0.
\end{equation}
Over $\mathbb P^n$, we consider the full exceptional
sequence:
\[
(\mathcal{O}_{\mathbb P^n}(-1),\Omega_{\mathbb P^n}(1),\mathcal{O}_{\mathbb P^n},\mathcal{O}_{\mathbb P^n}(2),\mathcal{O}_{\mathbb P^n}(3),\ldots,\mathcal{O}_{\mathbb P^n}(n-1)),
\]
obtained from the standard collection $(\mathcal{O}_{\mathbb P^n}(-1),\ldots,\mathcal{O}_{\mathbb P^n}(n-1))$,
by the mutation $\Omega_{\mathbb P^n}(-1) \simeq
\rL_{\mathcal{O}_{\mathbb P^n}}\mathcal{O}_{\mathbb P^n}(1)$ (all the terminology and results we
need on exceptional collections are contained in
\cite{bondal:representations-coherent}).
By \eqref{+}, we can replace the previous
exceptional sequence with:
\[
(F_{k-1},F_{k},\mathcal{O}_{\mathbb P^n},\mathcal{O}_{\mathbb P^n}(2),\ldots,\mathcal{O}_{\mathbb P^n}(n-1)),
\]
Right-mutating $F_{k-1}$ through the full collection, we must get back
$F_{k-1} \ts \omega_{\mathbb P^n}^* \simeq F_{k-1}(n+1)$.
So, using \eqref{recurrence}, we get a long exact sequence:
\begin{equation}
\label{lunga}
0 \to F_{k+1} \to \mathcal{O}_{\mathbb P^n}^{u_{1}} \to
\mathcal{O}_{\mathbb P^n}(2)^{u_{2}} \to \cdots \to \mathcal{O}_{\mathbb P^n}(n-1)^{u_{n-1}}
\to F_{k-1}(n+1) \to 0,
\end{equation}
for some integers $u_{i}$.
Now by \eqref{defining+} we get:
\[
\HH^i(\mathbb P^n,F_k(t))=0 \quad \mbox {for:} \quad
\left\{
\begin{array}[h]{ll}
2 \le i \le n-2, & \forall t, \\
i = 0, & t \le 0, \\
i = 1, & t \ne -1, \\
i = n-1, & t \ge 1-n.
\end{array}
\right.
\]
The required non-vanishing cohomology of $F_k$ appears again from
\eqref{defining+}.
So it only remains to check that $\HH^{n-1}(\mathbb P^n,F_k(t))=0$ for $t
\le -n-1$. But this is clear by induction once we twist \eqref{lunga} by
$\mathcal{O}_{\mathbb P^n}(t)$, and take cohomology.
\end{proof}
\subsection{} We compute the Ext groups
between pairs of Fibonacci bundles.
\begin{lem} \label{reciproci}
For any pair of integers $j \ge k+1$ we have:
\[
\ext^i_{\mathbb P^n}(F_j,F_k) = \delta_{1,i} a_{\ell,j-k-1}, \qquad \ext^i_{\mathbb P^n}(F_k,F_j) = \delta_{0,i} a_{\ell,j-k+1}.
\]
\end{lem}
\begin{proof}
The formulas hold for $k=j$ since $F_k$ is exceptional, and
we easily compute $\chi(F_j,F_{k})=-a_{\ell,j-k-1}$ and
$\chi(F_k,F_{j})=a_{\ell,j-k+1}$ (for instance by computing $\chi$ of
$\Upsilon_\ell$-modules via the Cartan form and using faithfullness of
$\Phi$).
The second formula is proved once we show
$\Ext^i_{\mathbb P^n}(F_k,F_j) = 0$ for $i \ge 1$.
In fact, since the category of
$\Upsilon_\ell$-representations is hereditary, the second formula
holds if $k \le 0$ for in this case $F_k \simeq \Phi(R_k)[-1]$.
By the same reason, we only have to check it for $i=1$.
Using \eqref{recurrence}, this vanishing holds for $j$ if it does
for $j-1$ and $j-2$. Since the statement is clear when extended to $j=k$, it
suffices to check $\Ext^1_{\mathbb P^n}(F_k,F_{k-1}) = 0$.
Since $\chi(F_k,F_{k-1})=0$, $\Hom_{\mathbb P^n}(F_k,F_{k-1}) = 0$ will do
the job. However, any nonzero map $F_k \to F_{k-1}$ would give,
again by \eqref{recurrence}, a non-scalar endomorphism of $F_k$,
which cannot exist since $F_k$ is simple. The second formula is now proved.
As for the first formula, again we see that
it holds if $k \le 0$ and $j \ge 1$ once we check it for
$i=0$. However using repeatedly \eqref{recurrence} we see that a
non-zero map $F_j \to F_k$ leads to an endomorphism of
$F_j$ which factors through $F_k$ this is absurd for $F_j$ is simple.
When $j,k$ have the same sign, the first formula has to be checked for
$i=2$ only.
Moreover, we have just proved the statement for $j=k+1$, and using
\eqref{recurrence} and exceptionality of $F_k$ we get it for
$j=k+2$. By iterating this argument we get the statement for any $j
\ge k+1$.
\end{proof}
\begin{rmk}
This lemma holds more generally (with the same proof) for any exceptional pair
$(F_0,F_1)$ of objects on a projective ${\boldsymbol{k}}$-variety $X$, with
$\hom_X(F_0,F_1)=\ell$, by defining recursively $F_k$ for all $k \in
\mathbb Z$ by \eqref{+} and \eqref{-}.
\end{rmk}
\section{Rigid ACM bundles on the third Veronese surface}
We prove here Theorem \ref{class2} in case $(n,d)=(2,3)$.
\subsection{}
Let us first prove \eqref{perforzasteiner}. So let $E$ be an
indecomposable vector
bundle on $\mathbb P^2$ satisfying \eqref{sonmalato}.
Without loss of generality, we may replace $E$ by $E(s)$, where $s$ is the smallest integer such that
$\hh^0(\mathbb P^2,E(s)) \ne 0$.
Set $\alpha_{i,j}=\hh^i(\mathbb P^2,E(-j))$. Of course, $\alpha_{0,j}=0$ if and only if $j \ge 0$.
The Beilinson complex $F$ associated with $E$ (see for instance \cite[Chapter 8]{huybrechts:fourier-mukai}) reads:
\[
0
\to
\mathcal{O}_{\mathbb P^2}(-1)^{\alpha_{1,2}}
\xrightarrow{d_0}
\begin{array}{c}
\mathcal{O}_{\mathbb P^2}(-1)^{\alpha_{2,2}} \\
\oplus \\
\Omega_{\mathbb P^2}(1)^{\alpha_{1,1}} \\
\oplus \\
\mathcal{O}_{\mathbb P^2}^{\alpha_{0,0}}
\end{array}
\xrightarrow{d_1}
\begin{array}{c}
\Omega_{\mathbb P^2}(1)^{\alpha_{2,1}}\\
\oplus \\
\mathcal{O}_{\mathbb P^2}^{\alpha_{1,0}}
\end{array}
\xrightarrow{d_2}
\mathcal{O}_{\mathbb P^2}^{\alpha_{2,0}}
\to 0.
\]
The term consisting of three summands in the above complex sits in
degree $0$ (we call it {\it middle term}), and the
cohomology of this complex is $E$.
By condition \eqref{sonmalato}, at least one of the $\alpha_{1,j}$ is
zero, for $j=0,1,2$.
If $\alpha_{1,2}=0$, then $d_0=0$. By minimality of the Beilinson complex the
restriction of $d_1$ to the summand $\mathcal{O}_{\mathbb P^2}^{\alpha_{0,0}}$ of the
middle term is also zero.
Therefore $\mathcal{O}_{\mathbb P^2}^{\alpha_{0,0}}$ is a direct summand of $E$, so $E
\simeq \mathcal{O}_{\mathbb P^2}$ by indecomposability of $E$.
If $\alpha_{1,1}=0$, then the non-zero component of $d_0$ is just a map
$\mathcal{O}_{\mathbb P^2}(-1)^{\alpha_{1,2}} \to \mathcal{O}_{\mathbb P^2}^{\alpha_{0,0}}$, and a direct
summand of $E$ is the cokernel of this map.
By indecomposability of $E$, in this case $E(-1)$ has a resolution
of the desired form with $a=\alpha_{0,0}$ and $b=\alpha_{1,2}$.
\subsection{}
It remains to look at the case $\alpha_{1,0}=0$.
Note that
the restriction of $d_1$ to
$\Omega_{\mathbb P^2}(1)^{\alpha_{1,1}} \oplus \mathcal{O}_{\mathbb P^2}^{\alpha_{0,0}}$ is zero,
which implies that a direct summand of $E$ (hence all of $E$ by
indecomposability) has the resolution:
\begin{equation}
\label{generale}
0 \to \mathcal{O}_{\mathbb P^2}(-1)^{\alpha_{1,2}} \xrightarrow{d_0} \Omega_{\mathbb P^2}(1)^{\alpha_{1,1}}
\oplus \mathcal{O}_{\mathbb P^2}^{\alpha_{0,0}} \to E \to 0
\end{equation}
and $\alpha_{2,j}=0$ for $j=0,1,2$.
We compute $\chi(E(-3))=3\alpha_{1,2} - 3\alpha_{1,1}+\alpha_{0,0}$, so:
\[
\hh^0(\mathbb P^2,E^*)= \hh^2(\mathbb P^2,E(-3))= 3\alpha_{1,2} - 3\alpha_{1,1}+\alpha_{0,0}.
\]
If this value is positive, then there is a non-trivial morphism
$g : E \to \mathcal{O}_{\mathbb P^2}$, and since $\alpha_{0,0} \ne 0$ there also
exists
$0 \ne f : \mathcal{O}_{\mathbb P^2} \to E$. So $E$ has an endomorphism
factoring through $\mathcal{O}_{\mathbb P^2}$, a contradiction.
Hence we may assume $3\alpha_{1,2} - 3\alpha_{1,1}+\alpha_{0,0}$, in
other words $\alpha_{0,3}=0$.
Therefore, the Beilinson complex associated with $E(-1)$ gives a
resolution:
\[
0 \to E(-1) \to \Omega_{\mathbb P^2}(1)^{\alpha_{1,2}} \to \mathcal{O}_{\mathbb P^2}^{\alpha_{1,1}} \to 0.
\]
It it easy to convert this resolution into the form we want by
the diagram:
\[
\[email protected]{
& 0 \ar[d] & 0 \ar[d] \\
0 \ar[r] & E(-1) \ar[r] \ar[d]& \Omega_{\mathbb P^2}(1)^{\alpha_{1,2}} \ar[r] \ar[d]& \mathcal{O}_{\mathbb P^2}^{\alpha_{1,1}} \ar[r] \ar@{=}[d]& 0\\
0 \ar[r] &\mathcal{O}_{\mathbb P^2}^{3\alpha_{1,2}-\alpha_{1,1}} \ar[r]\ar[d] & \mathcal{O}_{\mathbb P^2}^{3\alpha_{1,2}} \ar[r]\ar[d] & \mathcal{O}_{\mathbb P^2}^{\alpha_{1,1}} \ar[r] & 0\\
&\mathcal{O}_{\mathbb P^2}(1)^{\alpha_{1,2}} \ar@{=}[r] \ar[d]&\mathcal{O}_{\mathbb P^2}(1)^{\alpha_{1,2}}\ar[d]\\
& 0 & 0
}\]
From the leftmost column, it follows that $E^*$ has a resolution
of the desired form, with $a=3\alpha_{1,2}-\alpha_{1,1}$ and $b=\alpha_{1,2}$.
Claim \eqref{perforzasteiner} is thus proved.
\subsection{} \label{case}
Let us now prove \eqref{quandoerigido}. We may assume that
$\rk(E)>1$. We check that $E$
being indecomposable and rigid forces $E$ to be simple, so that it
has no endomorphism factoring through $\mathcal{O}_{\mathbb P^2}(t)$ and thus
\eqref{perforzasteiner} applies.
In the previous proof, we used this condition only for
$\alpha_{1,0} = 0$, and \eqref{perforzasteiner} will apply if $\alpha_{0,3}=0$.
\subsection{}
We work with $\alpha_{1,0} = 0$.
Let $e$ be the restricted map $e : \mathcal{O}_{\mathbb P^2}(-1)^{\alpha_{1,2}} \to
\Omega_{\mathbb P^2}(1)^{\alpha_{1,1}}$ extracted from $d_0$ and let $F$
be its cone, shifted by $1$:
\begin{equation}
\label{Fagain}
F \to \mathcal{O}_{\mathbb P^2}(-1)^{\alpha_{1,2}} \xrightarrow{e} \Omega_{\mathbb P^2}(1)^{\alpha_{1,1}}.
\end{equation}
This is a complex with two terms, and its cohomology is
concentrated in degrees zero and one, namely
$\mathcal{H}^0 F \simeq \ker(e)$ and $\mathcal{H}^1 F \simeq \coker(e)$.
From \eqref{generale} we easily see that $F$ fits into a
distinguished triangle:
\begin{equation}
\label{F}
F \to \mathcal{O}_{\mathbb P^2}^{\alpha_{0,0}} \to E.
\end{equation}
Applying $\Hom_{\mathbb P^2}(\mathcal{O}_{\mathbb P^2},-)$ to \eqref{Fagain}, we get
$\Ext_{\mathbb P^2}^i(\mathcal{O}_{\mathbb P^2},F)=0$ for all $i$, so:
\[
\Ext_{\mathbb P^2}^i(F,F) \simeq \Ext^i_{\mathbb P^2}(E,F[1]), \qquad \mbox{for all $i$}.
\]
Also, we know that $\HH^2(\mathbb P^2,E^*)=0$, so applying $\Hom_{\mathbb P^2}(E,-)$ to \eqref{F} we
get:
\[
\Ext^1_{\mathbb P^2}(E,E) \to \Ext^1_{\mathbb P^2}(E,F[1]) \to 0.
\]
Putting this together, we obtain a surjection:
\[
\Ext^1_{\mathbb P^2}(E,E) \twoheadrightarrow \Ext^1_{\mathbb P^2}(F,F) \simeq \Ext^1_{\Upsilon_3}(R,R),
\]
with $F \simeq \Phi(R)$.
We understand now that, if $E$ is rigid, then also $R$ is.
\subsection{}
If $R$ is rigid, then by \S \ref{KAC}, $R$ is a direct sum of
rigid simple representations of the form $R_k$.
Therefore, cohomology of \eqref{F} gives an exact
sequence:
\[
0 \to \oplus_{i\le 0} F^{r_i}_{i} \to \mathcal{O}_{\mathbb P^2}^{\alpha_{0,0}} \to E \to
\oplus_{i \ge 1} F^{r_i}_i \to 0,
\]
for some integers $r_i$.
If only $R_i$ with $i\le 0$ appear, then we are done by \S \ref{case}.
Indeed, in that case
$E$ is globally generated, so $\HH^0(\mathbb P^2,E) \ne 0$
implies $\HH^0(\mathbb P^2,E^*)=0$ for otherwise $\mathcal{O}_{\mathbb P^2}$ would be a
direct summand of $E$.
If some $R_i$ appears with $i \ge 1$, we call $I$ the (non-zero) image
of the middle map in the previous exact sequence, and we check
$\Ext^1_{\mathbb P^2}(F_j,I)=0$ for all $j \ge 1$, which contradicts $E$ being
indecomposable.
To check this, note that $\Hom_{\mathbb P^2}(F_j,-)$ gives an exact
sequence:
\[
\Ext^1_{\mathbb P^2}(F_j,\mathcal{O}_{\mathbb P^2})^{\alpha_{0,0}} \to \Ext^1_{\mathbb P^2}(F_j,I) \to \oplus_{i\le 0}\Ext^2_{\mathbb P^2}(F_j,F_i)^{r_i}.
\]
The leftmost term vanishes by Serre duality and Lemma \ref{cohom}.
The rightmost term is zero by Lemma \ref{reciproci}.
Part \eqref{quandoerigido} is now proved.
\subsection{} The statement \eqref{nienteH1}
is clear by Lemma \ref{cohom} and by exceptionality of Fibonacci bundles.
The fact that $E^*$ is also ACM is obvious by Serre duality.
\begin{rmk}
If ${\boldsymbol{k}}$ is algebraically closed of characteristic zero, we may
apply \cite[Corollaire 7]{drezet:beilinson}, to the effect that a
rigid bundle
is a direct sum of exceptional bundles.
So, at the price of relying on this result,
from \eqref{perforzasteiner} we may deduce directly
\eqref{quandoerigido} via Kac's theorem.
\end{rmk}
\section{ACM bundles on the second Veronese threefold}
The techniques we have just seen apply to the embedding of $\mathbb P^3$ in
$\mathbb P^9$ by quadratic forms.
Again we replace $E$ with the $E(s)$, where $s$ is the smallest integer such
that $E$ has non-zero global sections, and set
$\alpha_{i,j}=\hh^i(\mathbb P^3,E(-j))$.
If \eqref{sonmalato} gives $\alpha_{1,1}=\alpha_{2,1}=0$, then
$E(-1)$ has the desired resolution.
On the other hand, if \eqref{sonmalato} tells
$\alpha_{1,0}=\alpha_{2,0}=\alpha_{1,2}=\alpha_{2,2}=0$, then we are
left with a resolution of the form:
\[
0 \to \mathcal{O}_{\mathbb P^3}(-1)^{\alpha_{1,3}} \xrightarrow{d_0} \Omega_{\mathbb P^3}(1)^{\alpha_{1,1}}
\oplus \mathcal{O}_{\mathbb P^3}^{\alpha_{0,0}} \to E \to 0.
\]
This time we also have $\alpha_{0,4}=0$, and $\alpha_{1,4}=\alpha_{2,4}=0$ again by
\eqref{sonmalato}, and simplicity of $E$ gives
$\alpha_{3,4}=\hh^0(\mathbb P^3,E^*)=0$.
So $E(-1)$ has a resolution like:
\[
0 \to E(-1) \to \Omega_{\mathbb P^3}^2(2)^{\alpha_{1,3}} \to \mathcal{O}_{\mathbb P^3}^{\alpha_{1,1}} \to 0.
\]
Then, using the same trick as in the proof of the previous theorem,
we see that $E^*$ has the desired resolution, with
$a=6\alpha_{1,3}-\alpha_{1,1}$ and $b=\alpha_{1,3}$.
This proves the first statement. The rest follows by the same path.
Drezet's theorem as shortcut for $\eqref{perforzasteiner}
\Rightarrow \eqref{quandoerigido}$ may be replaced by
\cite{happel-zacharia:self-extensions}.
\begin{rmk}
It should be noted that, in \cite[Theorem 1.2 and Theorem
1.3]{iyama-yoshino}, the ACM bundle $E$ on the given Veronese variety is assumed to
have a rigid module of global sections. This implies, respectively,
$\Ext^1_{\mathbb P^2}(E,E(3t))=0$, or $\Ext^1_{\mathbb P^3}(E,E(2t))=0$, for all
$t \in \mathbb Z$.
A priori, this is a stronger requirement than just
$\Ext^1_{\mathbb P^n}(E,E)=0$.
However, our proof shows that the two conditions are equivalent for
ACM bundles.
\end{rmk}
\section{Rigid ACM bundles higher Veronese surfaces}
Assume ${\boldsymbol{k}}$ algebraically closed.
The next result shows that, for $d \ge 4$, the class of rigid ACM bundles
on $d$-fold Veronese surfaces contains the class
of exceptional bundles on $\mathbb P^2$, which is indeed quite
complicated, cf. \cite{drezet-lepotier:stables}.
At least if $\mathrm{char}({\boldsymbol{k}})=0$, the two classes coincide by
\cite[Corollaire 7]{drezet:beilinson}.
\begin{thm}
Let $F$ be an exceptional bundle on $\mathbb P^2$ and fix $d \ge 4$.
Then there is an integer $t$ such that $E=F(t)$ satisfies \eqref{sonmalato}.
\end{thm}
\begin{proof}
It is known that $F$ is actually stable by \cite{drezet-lepotier:stables}.
This implies that $F$ has natural cohomology by
\cite{hirschowitz-laszlo:generiques}, i.e. for all $t \in \mathbb Z$ there
is at most one $i$ such that $\HH^i(\mathbb P^2,F(t)) \ne 0$.
Then, $\HH^1(\mathbb P^2,F(t)) \ne 0$ if and only if $\chi(F(t))<0$.
Let now that $r$, $c_1$ and $c_2$ be the rank and the Chern classes
of $F$. Computing $\chi$ by additivity, we see that $\chi(F(t))$ is a polynomial of
degree $2$ in $t$, of dominant term $r/2$, whose discriminant is:
\[
\Delta = {c}_{1}^{2}(1-r)+r (2{c}_{2}+r/4 ) = -\chi(F,F)+5r^2/4.
\]
So, using $\chi(F,F)=1$, we get $\Delta = -1+5r^2/4$.
Therefore, the roots of $\chi(F(t))$ differ at most by:
\[
\mbox{$\lceil \frac{2\sqrt{\Delta}}{r} \rceil = \lceil\frac{\sqrt{5r^2-4}}{r} \rceil \le 3$}.
\]
Then, there are at most three consecutive integers $t_0$, $t_0+1$, $t_0+2$ such that
$\HH^1(\mathbb P^2,F(t_0+j))\ne 0$ for $j=0,1,2$. This means that
$E=F(t_0-1)$ satisfies \eqref{sonmalato} for any choice of $d \ge 4$.
\end{proof}
\noindent {\bf Acknowledgements}. I would like to thank
F.-O. Schreyer, J. Pons Llopis and M. C. Brambilla for useful comments
and discussions.
\bibliographystyle{alpha-my}
|
1,108,101,564,564 | arxiv | \section{INTRODUCTION}
Clusters of galaxies are the most massive gravitationally bound objects in
the Universe. They include hundreds of galaxies within a radius of 1--2 Mpc
(e.g., Abell 1958). Dispersions of the member galaxy redshifts indicate
that the gravitational potential of the cluster is much deeper than can be
created by the total mass of its galaxies, revealing the presence of
smoothly distributed dark matter (Zwicky 1937). Its nature is still
unknown, except that it is probably cold and collisionless. At present, its
distribution can be directly mapped using the gravitational lensing
distortion that it introduces to the images of distant background galaxies
(e.g., Bartelmann \& Schneider 2001). In Fig.\ \ref{1e}, panels ({\em a})
and ({\em b}), we show an optical image of the field containing a relatively
distant cluster \mbox{1E\,0657--56}, and a map of its total projected mass derived from
lensing. Within the radius covered by the image, this cluster has a mass of
about $10^{15}$ $M_{\odot}$\ ($2\times 10^{48}$ g), of which only 1--3\% is
stellar mass in the member galaxies.
\begin{figure}[p]
\centering{
\noindent
\includegraphics[width=0.75\textwidth, bb=61 389 403 706]%
{1e_opt.ps}
\vspace{4mm}\hspace{0.1mm}
\includegraphics[width=0.75\textwidth, bb=61 389 403 706]%
{1e_opt_lens1.ps}
}
\caption{({\em a}) Optical image of a field containing the \mbox{1E\,0657--56}\ cluster
({\em Magellan}\/ 6.5m telescope; Clowe et al.\ 2006). Image size is
about 10$^\prime$\ and the ruler shows linear scale for the redshift of the
cluster, $z=0.3$. The cluster consists of two concentrations of faint red
galaxies in the middle. ({\em b}) Contours show a map of projected total
mass derived from gravitational lensing (Clowe et al.\ 2006; the lowest
contour is noisy). The mass is dominated by dark matter. Peaks of the two
mass concentrations are coincident with the galaxy concentrations.}
\end{figure}
\addtocounter{figure}{-1}
\begin{figure}[p]
\centering{
\noindent
\includegraphics[width=0.75\textwidth, bb=61 389 403 706]%
{1e_xray.ps}
\vspace{4mm}\hspace{0.1mm}
\includegraphics[width=0.75\textwidth]%
{1e_xray_rhalo.ps}
}
\caption{---continued. ({\em c}) \emph{Chandra}\ X-ray image of the same region
of the sky containing \mbox{1E\,0657--56}\ (M06). Diffuse X-ray emission traces the hot
gas. Compact sources are mostly unrelated projected AGNs, left in the
image to illustrate the 1$^{\prime\prime}$\ angular resolution. ({\em d}) Contours show
the surface brightness of the diffuse radio emission (ATCA telescope, 1.3
GHz; Liang et al.\ 2000). The resolution is 24$^{\prime\prime}$; compact sources are
removed. The radio emission is synchrotron from the ultrarelativistic
electrons in the cluster magnetic field, coexisting with the hot gas.}
\label{1e}
\end{figure}
\begin{figure}[t]
\centering
\noindent
\includegraphics[width=\textwidth,bb=58 587 480 726]%
{merg3.ps}
\caption{Clusters form and grow via mergers. Panels from left to
right show X-ray images of a pair of clusters about to merge (A399--A401),
a system undergoing a merger (A754), and a relaxed, more massive cluster
(A2029) that emerges in a few Gyr as a result.}
\label{merg3}
\end{figure}
X-ray observations showed that intergalactic space in clusters is filled
with hot plasma (Kellogg et al.\ 1972; Forman et al.\ 1972; Mitchell et al.\
1976; Serlemitsos et al.\ 1977). It is the second most massive cluster
component, and their dominant baryonic component. A theoretical and
observational review of this plasma and other cluster topics can be found in
Sarazin (1988). Here we summarize the basics which will be needed in the
sections below.
The intracluster medium (ICM) has temperatures $T_e\sim 10^7-10^8$ K ($1-10$
keV) and particle number densities steeply declining from $n\sim 10^{-2}$
cm$^{-3}$\ near the centers to $10^{-4}$ cm$^{-3}$\ in the outskirts. It
consists of fully ionized hydrogen and helium plus traces of highly ionized
heavier elements at about a third of their solar abundances, increasing to
around solar at the centers. It emits X-rays mostly via thermal
bremsstrahlung. At densities and temperatures typical for the ICM, the
ionization equilibrium timescale is very short. The electron-ion
equilibration timescale via Coulomb collisions is generally shorter than the
age of the cluster, so $T_e=T_i$ can be assumed in most cluster regions,
except perhaps in the low-density outskirts and at shock fronts (and
probably even there, as will be seen in \S\ref{sec:tei}). The spectral
density of the X-ray continuum emission at energy $E$\/ from such a plasma
is
\begin{equation}
\epsilon_X \;\propto\; \overline{g}\; n^2\; T_e^{-1/2}\, e^{-E/kT_e}
\label{eq:brems}
\end{equation}
where $\overline{g}$ is the effective Gaunt factor, which includes all
continuum mechanisms and depends weakly on $E$, $T_e$\/ and ion abundances
(e.g., Gronenschild \& Mewe 1978; Rybicki \& Lightman 1979). On top of this
continuum, there is line emission, discussed, e.g., by Mewe \& Gronenschild
(1981). The timescale of radiative cooling of the ICM is generally very
long, longer than the cluster age, with the exception of small, dense
central regions. Thus, nonradiative approximation is applicable to all the
phenomena discussed in this review.
The ICM is optically thin for X-rays for all densities encountered in
clusters (except for the possible resonant scattering at energies of strong
emission lines in the dense central regions; Gilfanov, Sunyaev, \& Churazov
1987). An X-ray telescope can thus map the ICM density and electron
temperature in projection. The current X-ray imaging instruments are
sensitive mostly to X-rays with $E\approx 0.3-2$ keV. From eq.\
(\ref{eq:brems}), an X-ray image of a hot cluster at $E\lesssim kT_e$\/ is
essentially a map of the projected $n^2$. In Fig.\ \ref{1e}{\em c}, we show
an X-ray image of \mbox{1E\,0657--56}\ as an example. This highly disturbed cluster has one
of the hottest and most X-ray luminous plasma halos (with
$\overline{T_e}\simeq 14$ keV and a bolometric luminosity of $10^{46}$
erg$\;$s$^{-1}$), and will feature in several sections below.
The electron temperature, $T_e$, averaged along the line of sight, can be
determined from the shape of the continuum component, and sometimes from the
relative intensities of emission lines, using an X-ray spectrum collected
from a spatial region of interest. The ion temperature, $T_i$, cannot be
directly measured at present. In principle, it can be determined from
thermal broadening of the emission lines, but this requires an energy
resolution of a calorimeter. Because of the strong dependence of
$\epsilon_X$ on $n$, it is often possible to ``deproject'' the ICM
temperature and density in three dimensions under reasonable assumptions
about the symmetry of the whole cluster or within a certain region of the
cluster. In such a way, the mass of the hot plasma can be determined for
many clusters whose gas atmospheres are spherically symmetric. It is found
to comprise 5--15\% of the total mass, several times more than the stellar
mass in galaxies (e.g., Allen et al.\ 2002; Vikhlinin et al.\ 2006).
If a cluster is undisturbed by collisions with other clusters for a
sufficient time, its dark matter distribution should acquire a centrally
peaked, slightly ellipsoidal, symmetric shape. After several sound crossing
times (of order $10^9$ yr), the ICM comes to hydrostatic equilibrium in the
cluster gravitational potential $\Phi$, so that the pressure of the ICM
$p$\/ and its mass density $\rho_{\rm gas}$ satisfy the equation $\nabla
p=-\rho_{\rm gas}\nabla \Phi$. For a spherically symmetric cluster, and
assuming that the intracluster plasma can be described as ideal gas, it can
be written as
\begin{equation}
M(r)=-\frac{kT(r)\,r}{\mu m_p G}
\left(\frac{d\ln\rho_{\rm gas}}{d\ln r} +
\frac{d\ln T}{d\ln r}\right),
\label{eq:mass}
\end{equation}
where $M(r)$ is the total mass of the cluster enclosed within the radius
$r$, $T(r)$ is the gas temperature at that radius, and $\mu=0.6$ is the mean
atomic weight of the plasma particles. That is, by measuring radial
distributions of the gas temperature and density, one can derive the cluster
{\em total}\/ mass (Bahcall \& Sarazin 1977; Sarazin 1988). This method of
measuring the cluster masses is independent of, and complementary to, those
using galaxy velocity dispersions and gravitational lensing. Unlike the
lensing mass measurement, it works only for clusters in equilibrium;
however, it is less affected by the line-of-sight projections. For those
clusters where a comparison is possible, different total mass measurement
methods usually agree to within a factor of 2.
Cluster masses are interesting because the ratio of the baryonic mass (ICM
plus stars) to dark matter mass for a cluster should be close to the average
for the Universe as a whole, which enables some powerful cosmological tests
(e.g., White et al.\ 1993; Allen et al.\ 2004). Furthermore, the number
density of clusters as a function of mass and its evolution with redshift
depend sensitively on cosmological parameters, which is the basis for
another class of tests (e.g., Sunyaev 1971; Press \& Schechter 1974; Eke,
Cole, \& Frenk 1996; Henry 1997; Vikhlinin et al.\ 2003). Hot electrons in
the ICM also introduce a distortion in the spectrum of the Cosmic Microwave
Background (CMB), which at $\lambda>1$ mm turns clusters into negative radio
``sources'' (Sunyaev \& Zeldovich 1972). By comparing the Sunyaev-Zeldovich
decrement and the X-ray brightness and temperature, one can derive absolute
distances to the clusters and, again, use them for a cosmological test (Silk
\& White 1978). The best estimates of the cluster baryonic and total masses
currently come from the X-ray data. To rely on them for cosmological
studies, we have to understand in detail the physical processes in the ICM,
how well the quantities required for those tests can be determined from the
X-ray images and spectra, and how valid are the underlying assumptions about
the ICM. This has been the main motivation for the studies discussed in
this review.
Clusters form via gravitational infall and mergers of smaller mass
concentrations, as illustrated by a time sequence in Fig.\ \ref{merg3}. Such
mergers are the most energetic events in the Universe since the Big Bang,
with the total kinetic energy of the colliding subclusters reaching
$10^{65}$ ergs (Markevitch, Sarazin, \& Vikhlinin 1999a). In the course of a
merger, a significant portion of this energy, that carried by the gas, is
dissipated (on a Gyr timescale) by shocks and turbulence. Eventually, the
gas heats to a temperature that approximately corresponds to the depth of
the newly formed gravitational potential well.
A fraction of the merger energy may be channeled into the acceleration of
ultrarelativistic particles and amplification of magnetic fields. These
nonthermal components manifest themselves most clearly in the radio band.
Polarized radio sources located inside and behind clusters are known to
exhibit Faraday rotation, which is caused by magnetic fields in the ICM. In
the radial range $r\sim 0.1-1$ Mpc (outside the dense central regions often
affected by the central AGN), the field strengths are in the range $B\sim
0.1-3\;\mu$G (with different measurement methods giving somewhat diverging
values; for a review see, e.g., Carilli \& Taylor 2002). For such fields
and the typical ICM temperatures, gyroradii for thermal electrons and
protons are of order $10^{8}$ cm and $10^{10}$ cm, respectively, many orders
of magnitude smaller than the particle collisional mean free paths
($10^{21}-10^{23}$ cm). The plasma electric conductivity is very high and
the magnetic field is frozen in. For typical ICM densities and a $1\;\mu$G
field, the Alfv\'en velocity is $v_A \sim 50$ km$\;$s$^{-1}$, much lower than the
typical thermal sound speeds $c_s \sim 1000$ km$\;$s$^{-1}$. Thus, the plasma is
``hot'' in the sense that the ratio of thermal pressure to magnetic pressure
$\beta\equiv p/p_B \approx c_s^2/v_A^2 \gg 1$, except in special places
which will be discussed in \S\ref{sec:drap}. This means, among other things,
that the magnetic pressure contribution is negligible for the hydrostatic
mass determination using eq.\ (\ref{eq:mass}).
The magnetic field is tangled, with coherence scales of order 10 kpc
(Carilli \& Taylor 2002). This suppresses collisional thermal conduction on
scales greater than this linear scale, by a large factor that depends on
the exact structure of the field (Chandran \& Cowley 1998; Narayan \&
Medvedev 2001). As a result, plasma with temperature gradients (for example,
created by a merger) comes to pressure equilibrium much faster than those
gradients dissipate (e.g., Markevitch et al.\ 2003a). Indeed, we are yet to
find a cluster without spatial temperature variations.
Merging clusters often exhibit faint radio halos, such as that shown in
Fig.\ \ref{1e}{\em d} (for a review see, e.g., Feretti 2002; radio halos are
not to be confused with the more localized radio ``relics'' of different
origin). The radio emission at $\nu\sim 1$ GHz is produced by synchrotron
radiation of ultrarelativistic electrons with Lorentz factor $\gamma\sim
10^4$ in a microgauss magnetic field. These relativistic electrons coexist
with thermal ICM, bound to it by the magnetic field.
Their exact origin is uncertain; one possibility is acceleration by merger
turbulence (for a review see, e.g., Brunetti 2003; we will touch on this in
\S\ref{sec:halos}). Relativistic electrons also produce X-ray emission by
inverse Compton (IC) scattering of the CMB photons. A detection of such
nonthermal emission at $E>10$ keV (where thermal bremsstrahlung falls off
exponentially with energy) was reported for some clusters (e.g.,
Fusco-Femiano et al.\ 2005 and references therein). The energy density in
the relativistic electrons should be of the order of the magnetic pressure
and thus negligible compared to thermal pressure of the ICM. However, it
was suggested that the currently unobservable relativistic protons that may
accompany them can have a significant energy density (V\"olk et al.\ 1996
and later works).
A part of our review will deal with hydrodynamic phenomena near the cluster
centers, and a brief description of these rather special regions will be
helpful. One of the models for the radial dark matter density distribution,
widely used until recently, is the King (1966) profile, $\rho(r) \propto
(1+r^2/r_c^2)^{-3/2}$. It has a flat core in the center with typical sizes
$r_c \sim 200$ kpc (see, e.g., Sarazin 1988 for a motivation for this
model). An isothermal gas in hydrostatic equilibrium within such a
potential also has a flat density core (Cavaliere \& Fusco-Femiano 1976).
This is an adequate description of the observed gas density profiles for
about 1/3 of the clusters. However, most clusters exhibit sharp central gas
density peaks (e.g., Jones \& Forman 1984; Peres et al.\ 1998).
Coincidentally, Navarro, Frenk \& White (1997, hereafter NFW) found that
density profiles of equilibrium clusters in their cosmological Cold Dark
Matter simulations can be approximated by a functional form $\rho(r) \propto
(r/r_s)^{-1}(1+r/r_s)^{-2}$. Its $r^{-1}$ dark matter density cusp in the
center corresponds to a finite, but sharp density peak of the gas in
equilibrium. The NFW model is a good description for the total mass
profiles derived from the X-ray data for such centrally peaked clusters
(e.g., Markevitch et al.\ 1999b; Nevalainen et al.\ 2001; Allen, Schmidt, \&
Fabian 2001; Pointecouteau, Arnaud, \& Pratt 2005; Vikhlinin et al.\ 2006).
These clusters usually have relatively undisturbed ICM (see, e.g., the last
panel in Fig.\ \ref{merg3}) and a giant elliptical galaxy in the center (a
cD galaxy) which marks the dark matter density peak. Within $r\sim 100$ kpc
of this peak, the ICM temperature declines sharply toward the center (e.g.,
Fukazawa et al.\ 1994; Kaastra et al.\ 2004; Sanderson, Ponman, \&
O'Sullivan 2006), while the gas density increases, along with the relative
abundance of heavy elements in the gas. This creates a rather distinct
central region of low-entropy gas. Outside this region, the radial
temperature gradient reverses and $T_e$\/ declines outward, but the entropy
continues to increase, so on the whole, the clusters are convectively
stable. The high central gas densities correspond to X-ray radiative
cooling times shorter than the cluster ages (a few Gyr). This gave rise to
a ``cooling flow'' scenario, in which central regions of such peaked
clusters are thermally unstable. Recent data indicate that there has to be a
process that partially compensates for the radiative cooling; for a recent
review see, e.g., Peterson \& Fabian (2006). We will use the term ``cooling
flow'' to signify this region that encloses the observed gas density and
temperature peaks (positive and negative, respectively), without any
particular physical model in mind.
Gas density and temperature distributions in clusters have been studied
extensively by all imaging X-ray observatories (e.g., by \emph{Einstein}, Forman
\& Jones 1982; Jones \& Forman 1999; \emph{ROSAT}, Briel, Henry, \& Boehringer
1992; Henry \& Briel 1995; Peres et al.\ 1998; Vikhlinin, Forman, \& Jones
1997, 1999; \emph{ASCA}, Fukazawa et al.\ 1994; Honda et al.\ 1996; Markevitch et
al. 1996a, 1998; \emph{SAX}, Nevalainen et al.\ 2001; De Grandi \& Molendi 2002;
\emph{XMM}, Arnaud et al.\ 2001; Briel, Finoguenov, \& Henry 2004; Piffaretti et
al.\ 2005). Temperature maps proved to be more difficult to obtain than
maps of the gas density, but the currently operating \emph{XMM}\ and \emph{Chandra}\
observatories have the right combination of spectroscopic and imaging
capabilities to derive them with a good linear resolution for a large number
of clusters at a range of redshifts. At $z<0.05$, \emph{Chandra}'s 1$^{\prime\prime}$\ angular
resolution, the best among the X-ray observatories, corresponds to linear
scales $<1$ kpc. This is less than the typical collisional mean free path
in the ICM or a typical galaxy size, and provides an exquisitely detailed
view of the physical processes in the cluster Megaparsec-sized gas halos.
With \emph{Chandra}, we are able to see the classic bow shocks driven by infalling
subclusters, as well as ``cold fronts'' --- unexpected sharp features of a
different nature. While \emph{XMM}\ can measure temperatures with a higher
statistical accuracy, its angular resolution is not sufficient to see these
sharp features in full detail, so our review of these two phenomena will be
based almost exclusively on the \emph{Chandra}\ results.
\emph{Chandra}'s main detector, ACIS, is sensitive in the 0.5--8 keV energy band,
with a peak sensitivity between 1--2 keV, and has a FWHM energy resolution
of $50-150$ eV for extended sources, sufficient to disentangle emission
lines in the uncrowded cluster X-ray spectra, but far from that needed to
resolve the line Doppler widths. A \emph{Chandra}\ overview can be found, e.g.,
in Weisskopf et al.\ (2002). Uncertainties of the cluster gas temperatures
from typical \emph{Chandra}\ exposures are limited by the photon counting
statistics, while uncertainties of the gas densities are usually limited by
the assumed three-dimensional geometry of a cluster. Physical quantities
below are given for a spatially flat cosmology with $\Omega_0=0.3$,
$\Omega_\Lambda=0.7$, and $H_0=70$~km$\;$s$^{-1}\,$Mpc$^{-1}$\ (unless the dependence on $H_0$ is
given explicitly via a factor $h\equiv H_0/100$ km$\;$s$^{-1}\,$Mpc$^{-1}$).
\section{COLD FRONTS}
Those interested primarily in physics that can be learned from this recently
found phenomenon may read \S\ref{sec:merg}, which gives a general
description of cold fronts, then skip the rest of this chapter (which
discusses the various kinds of cold fronts and their origin and evolution
based on hydrodynamic simulations), and go directly to \S\ref{sec:tools}.
\begin{figure}[p]
\centering{
\noindent
\includegraphics[width=0.7\textwidth,bb=1 14 490 512,clip]%
{a2142_img.ps}
\vspace{4mm}\hspace{0.1mm}
\includegraphics[width=0.7\textwidth,bb=1 14 490 512,clip]%
{a3667_07-4_4.ps}
}
\caption{\emph{Chandra}\ X-ray images of clusters with the first discovered cold
fronts, A2142 and A3667. In A2142, at least two sharp brightness edges are
seen, between blue and black in the NW and between purple and blue south
of center. In A3667, there is a prominent edge SE of center. Unrelated
compact X-ray sources are not removed. (Images are created from the recent
long \emph{Chandra}\ exposures.)}
\label{2142_3667}
\end{figure}
\begin{figure}[t]
\includegraphics[width=\textwidth]{a2142_edges.ps}
\caption{Cold fronts in A2142 (reproduced from M00). ({\em a}) X-ray image
with red overlays showing regions used for derivation of temperature
profiles (panel {\em b}). In panels ({\em b-e}), the southern edge is
shown in the left plot and the northwestern edge is in the right plot.
Panel ({\em c}) shows X-ray brightness profiles across the edges in the
same sectors. The red histogram is the brightness model that corresponds
to the best-fit gas density model shown in panel ({\em d}). Panel ({\em
e}) shows pressure profiles obtained from the temperature and density
profiles. Error bars are 90\%; vertical dashed lines show the positions
of the density jumps.}
\label{a2142_profs}
\end{figure}
\subsection{Cold fronts in mergers}
\label{sec:merg}
Among the first \emph{Chandra}\ cluster results was a discovery of ``cold fronts''
in merging clusters A2142 and A3667 (Markevitch et al.\ 2000, hereafter M00;
Vikhlinin, Markevitch, \& Murray 2001b, hereafter V01). Figure
\ref{2142_3667} shows ACIS images of the central regions of A2142 and A3667,
which show prominent sharp X-ray brightness edges. The edge in A3667 was
previously seen in a lower-resolution \emph{ROSAT}\ image (Markevitch et al.\
1999a), and at the time, we interpreted it as a shock front, even though the
crude \emph{ASCA}\ temperature map did not entirely support this explanation.
If these features were shocks, the gas on the denser, downstream side of the
density jump would have to be hotter than that on the upstream side. With
\emph{ROSAT}\ and \emph{ASCA}, we could not derive sufficiently accurate gas temperature
profiles across such edges. \emph{Chandra}\ provided this capability for the first
time, so now we can easily test this hypothesis. The \emph{Chandra}\ radial X-ray
brightness and temperature profiles across the two edges in A2142 are shown
in Fig.\ \ref{a2142_profs} (from M00). They were extracted in sectors shown
in panel ({\em a}). Both brightness profiles have a characteristic shape
corresponding to a projection of an abrupt, spherical (within a certain
sector) jump of the gas density. Best-fit radial density models of such a
shape are shown in panel ({\em d}), and their projections are overlaid on
the data as histograms in panel ({\em c}) --- they provide a very good fit.
Since there is no way of knowing the exact three-dimensional geometry of the
edge, for such fits we have to assume that the curvature of the
discontinuity surface along the line of sight is the same as in the sky
plane. To ensure the consistency with this assumption, it is important that
the radial profiles and the three-dimensional model for the gas inside the
discontinuity are centered at the center of curvature of the front, which is
often offset from the cluster center. At the same time, the model of the
outer, ``undisturbed'' gas may need to be centered elsewhere (e.g., the
cluster centroid).
Panel ({\em b}) in Fig.\ \ref{a2142_profs} shows the gas temperature
profiles across the edges. For a {\em shock discontinuity}, the
Rankine--Hugoniot jump conditions directly relate the gas density jump,
$r\equiv \rho_1/\rho_0$, and the temperature jump, $t\equiv T_1/T_0$, where
indices 0 and 1 denote quantities before and after the shock (e.g., Landau
\& Lifshitz 1959, \S89):
\begin{equation}
t=\frac{\zeta-r^{-1}}{\zeta-r}
\label{eq:t}
\end{equation}
or, conversely,
\begin{equation}
r^{-1}=\left[\frac{1}{4}\, \zeta^2\, (t-1)^2 +t\right]^{1/2}
-\frac{1}{2}\, \zeta\, (t-1),
\label{eq:r}
\end{equation}
where we denoted $\zeta \equiv (\gamma+1)/(\gamma-1)$; here $\gamma=5/3$ is
the adiabatic index for monoatomic gas.
For the observed density jump $r\sim 2$ and a presumably post-shock
temperature $T_1\sim 7.5$ keV observed inside the NW edge in A2142, one
would expect to find a $T_0\simeq 4$ keV gas in front of the shock. This
sign of the temperature change is opposite to that observed across the edge
--- the temperature in the less dense gas outside the edge is in fact higher
than that inside (Fig.\ \ref{a2142_profs}{\em b}). The same is true for the
smaller edge in A2142, as well as the one in A3667 (V01; see also Briel,
Finoguenov, \& Henry 2004), ruling out the shock interpretation.
What are these sharp edges then? One hint is given by the gas pressure
profiles across the edges (simply the product of the best-fit density models
and the measured temperatures; Fig.\ \ref{a2142_profs}{\em e}), which show
that there is approximate pressure equilibrium across the density
discontinuity (as opposed to a large pressure jump expected in a shock
front). One also notes a smooth, comet-like shape of the NW edge in A2142,
which looks as if the ambient gas flows around it. Given this evidence, we
proposed (M00) that these features are contact discontinuities at the
boundaries of the gas clouds moving sub- or transonically through a hotter
and less dense surrounding gas --- or ``cold fronts'', as V01 have termed a
similar feature in A3667
\footnote{We were certainly influenced by the Burns (1998) review, which
compared the gasdynamic phenomena in cluster mergers with ``stormy
weather''. The term ``cold front'' has since been commonly adopted to
denote either the discontinuity itself, or the discontinuity and the gas
cloud behind it; which of these two is usually clear from the context.}
Strictly speaking, a contact discontinuity implies continuous pressure and
velocity between the gas phases, but a cold front often has a discontinuous
tangential velocity, when the dense gas cloud is moving.
\begin{figure}[t]
\centering
\includegraphics[width=0.5\textwidth, bb=95 145 502 718, clip]%
{a2142_scheme.ps}
\caption{A model for the origin of cold fronts in A2142 proposed in M00 is
shown schematically in lower panel. The preceding stage of the merger is
shown in upper panel. In upper panel, shaded circles depict dense cores of
the two colliding subclusters (of course, in reality, there is a
continuous density gradient). Shock fronts 1 and 2 in the central region
of top panel have propagated to the cluster outskirts in lower panel,
failing to penetrate the dense cores that continue to move through the
shocked gas. The cores may develop additional shock fronts ahead of them,
shown by dashed lines. See Fig.\ \ref{mathis} for a simulation
illustrating these stages. (Reproduced from M00.)}
\label{a2142_scheme}
\end{figure}
In the particular scenario that was envisioned for A2142 in M00, these dense
gas clouds are remnants of the cool cores of the two merging subclusters
that have survived shocks and mixing of a merger (which would have to have a
nonzero impact parameter to avoid complete destruction of the less dense NW
core). They are observed after the passage of the point of minimum
separation and presently moving apart. The hotter, rarefied gas beyond the
NW edge can be the result of shock heating of the outer atmospheres of the
two colliding subclusters, as schematically shown in Fig.\
\ref{a2142_scheme}. In this scenario, the less dense outer subcluster gas
has been stopped by the collision shock, while the dense cores (or, more
precisely, regions of the subclusters where the pressure exceeded that of
the shocked gas in front of them, which prevented the shock from penetrating
them) continued to move ahead through the shocked gas, pulled along by their
host dark matter clumps.
With the benefit of a more recent, longer \emph{Chandra}\ observation of A2142,
and having seen images of numerous other clusters as well as hydrodynamic
simulations, we now think that the M00 scenario is not correct. Instead, it
seems more likely that A2142 is a cluster with a sloshing cool core (as
first pointed out by Tittley \& Henriksen 2005), a phenomenon that was
discovered later and which will be discussed in \S\ref{sec:slosh}. However,
the physical interpretation of the X-ray edges as contact discontinuities
between moving gases of different entropies still holds --- the only
difference is the origin of the two gas phases in contact (either from
different subclusters or from different radii in the same cluster). At the
same time, the scenario proposed in M00 is realized in a number of other
merging clusters. Two particularly striking examples are the textbook merger
\mbox{1E\,0657--56}\ and an elliptical galaxy NGC\,1401 in the Fornax cluster, although each
of these systems exhibits only one cold front. In both objects, there is an
independent (i.e., non X-ray) evidence of a distinct infalling subcluster
that hosts the gas cloud with a cold front. In Fornax, a cold front forms at
the interface between the atmosphere of the infalling galaxy NGC\,1404 seen
in the optical image, and the hotter cluster gas (Fig.\ \ref{n1404};
Machacek et al.\ 2005). In \mbox{1E\,0657--56}, a mass map derived from the gravitational
lensing data reveals a dark matter subcluster, which is also seen as a
concentration of galaxies in the optical image (Fig.\ \ref{1e}{\em ab};
Clowe, Gonzalez, \& Markevitch 2004; Clowe et al.\ 2006). Its X-ray image
(Fig.\ \ref{1e}{\em c}) shows a bright ``bullet'' of gas moving westward,
apparently pulled along by the smaller dark matter subcluster. Because of
ram pressure, the bullet lags behind the collisionless dark matter clump
(Fig.\ \ref{1e_lens}, which will be discussed in \S\ref{sec:dm}).
\begin{figure}[t]
\centering
\includegraphics[height=\textwidth, angle=90, bb=0 0 473 512, clip]%
{n1404_1.ps}
\caption{The NGC\,1404 elliptical galaxy falling into the Fornax
cluster. ({\em a}) \emph{Chandra}\ X-ray image, showing a sharp brightness edge
on the NW side of NGC\,1404 (the southern of the two halos). The 5$^\prime$\ bar
corresponds to 29 kpc. ({\em b}) Optical image (same scale as {\em a})
showing NGC\,1404 and the central galaxy of the cluster. ({\em c})
\emph{Chandra}\ temperature map of the NGC\,1404 region. The gas in the galaxy
is cool. ({\em d}) X-ray radial brightness profile in a sector across the
edge, showing the characteristic projected spherical discontinuity shape.
(Reproduced from Machacek et al.\ 2005.)}
\label{n1404}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[width=0.67\textwidth, bb=101 265 416 632]%
{1e_densprof_entro.ps}
\caption{Model radial profiles for the gas density $n$, pressure $P$\/ and
specific ``entropy'' $S\equiv T n^{-2/3}$ for \mbox{1E\,0657--56}\ in a sector crossing
the bullet boundary (a cold front at $r\approx 13''$) and the shock front
(at $r\approx 50''$), centered on the bullet. The pressure and
``entropy'' profiles are simply the combinations of the best-fit density
model and the average temperature values for each of the three regions.
Error bars on the pressure plot correspond to temperature uncertainties.
We omit the temperature variations inside the post-shock and bullet
regions, so the plot can serve only as a qualitative illustration of the
changes at shocks and cold fronts. (Reproduced from M02, with updated
temperatures and added entropy profile.)}
\label{1e_densprof}
\end{figure}
The gas bullet in \mbox{1E\,0657--56}\ has developed a sharp edge at its western side, which
is a cold front. Ahead of it is a genuine bow shock (a faint blue-black edge
in Fig.\ \ref{1e}{\em c}), confirmed by the temperature profile (Markevitch
et al.\ 2002, hereafter M02; \S\ref{sec:1e_M} below). It is instructive to
look at the radial profiles of the gas density, thermal pressure and
specific entropy derived in a narrow sector crossing both these edge
features (Fig.\ \ref{1e_densprof}). The two discontinuities have density
jumps of similar amplitudes (a factor of 3). As expected, the pressure has
a big jump at the shock, but is nearly continuous in comparison at the cold
front. In broad terms, thermal pressure in the cool gas behind a moving
cold front should be in balance with the thermal plus ram pressure of the
gas flowing around it. For a substantially supersonic motion (this shock
has $M=3$, see \S\ref{sec:1e_M}), the gas flow between the bow shock and the
driving body is very subsonic. So the ram pressure component is small
compared to thermal pressure, hence the near-continuity of thermal pressure
(a more detailed picture will be presented in \S\ref{sec:vel}). The
entropy, on the other hand, shows only a small increase at the shock (as
expected for this relatively weak shock), but a big drop at the cold front.
This is because in the past, the bullet apparently used to be a cooling flow
(M02). The merger brought this region of low-entropy
gas in direct contact with the high-entropy gas from the cluster outskirts.
These are the characteristic features of all cold fronts, regardless of the
exact origin of the two gas phases in contact.
In addition to the examples mentioned above, prominent if less clear-cut
cold fronts have been observed in a large number of other clusters (e.g.,
RXJ\,1720+26, Mazzotta et al.\ 2001; A2256, Sun et al.\ 2002; A85, Kempner,
Sarazin, \& Ricker 2002; A2034, Kempner, Sarazin, \& Markevitch 2003; A496,
Dupke \& White 2003; A754, Markevitch et al.\ 2003a; A2319, O'Hara, Mohr, \&
Guerrero 2004, Govoni et al.\ 2004, hereafter G04; A168, Hallman \&
Markevitch 2004; A2204, Sanders, Fabian, \& Taylor 2005). Many of such
features have been observed at smaller linear scales in the cool dense gas
near the cluster centers, which we will separate (somewhat artificially)
into a class of their own (\S\ref{sec:slosh}).
\begin{figure}[t]
\centering
\includegraphics[width=0.95\textwidth,clip]%
{a754_roettiger98.ps}
\caption{A snapshot from a simulated off-axis merger of two
subclusters. Colors show ({\em a}) projected gas temperature (increasing
from black to red), ({\em b}) projected dark matter density, ({\em d})
fraction of gas that initially belonged to each subcluster (red belonged
to the smaller subcluster that is now on the right side and moving away
from the collision site; black belonged to the bigger subcluster).
Contours in all panels show X-ray brightness. Arrows in ({\em c}) show gas
velocities. The interface between the two gases near the brightness peak
is a cold front. (Reproduced from Roettiger et al.\ 1998.)}
\label{roettiger}
\end{figure}
\begin{figure}[p]
\centering
~~\includegraphics[width=0.95\textwidth]%
{mathis_dm_1.ps}
\vspace{5mm}
\includegraphics[width=\textwidth, bb=132 288 478 489, clip]%
{mathis_tmaps.ps}
\caption{Simulated merger of two approximately equal clusters (selected from
a large cosmological simulation), resulting in the emergence of cold
fronts. The clusters collide along the NW-SE direction (but not exactly
head-on: their centers pass each other about 600 kpc apart). Upper panels
are 14 Mpc in size and show the density of dark matter in a 500 kpc thick
slice in the merger plane. Lower panels show the gas temperature in a
slice; their size is 6 Mpc (at centers of the corresponding upper panels).
Labels in each panel give the redshift of the snapshot. $z=0.22$ is the
moment right before core passage; there is a hot shock-heated gas strip (a
pancake in projection) between the cores. $z=0.20$ is right after the
core passage; from this moment on one can see two cold fronts moving in
the opposite directions. These two snapshots are similar to the two
stages shown schematically in Fig.\ \ref{a2142_scheme}. (Reproduced from
Mathis et al.\ 2005.)}
\label{mathis}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[width=0.45\textwidth, bb=0 0 550 550]%
{yago_f15d_ed.ps}
\includegraphics[width=0.45\textwidth, bb=0 0 550 550]%
{yago_f14g_ed.ps}
\caption{Simulated off-axis mergers with different subcluster masses and
trajectories. Contours show dark matter density, colors show gas
temperature (the scale bar gives its values in keV), and arrows show the
gas velocity field relative to the center of the bigger cluster (in the
center of each panel). The subcluster enters from upper-right. In the left
panel, the subcluster has retained some of its gas and developed a cold
front, preceded by a bow shock. However, in the right panel, a less
massive subcluster flying through denser regions has been completely
stripped of its gas by ram pressure. (Reproduced from A06.)}
\label{yago_strip}
\end{figure}
\subsection{Origin and evolution of merger cold fronts}
\label{sec:origin}
Since their discovery, cold fronts in merging clusters have been looked for,
and found, in hydrodynamic simulations with cosmological initial conditions
(e.g., Nagai \& Kravtsov 2003; Onuora, Kay \& Thomas 2003; Bialek, Evrard,
\& Mohr 2002; Mathis et al.\ 2005). Several other recent works used
idealized 2D or 3D merger simulations to model the effects of ram-pressure
stripping of a substructure moving through the ICM (e.g., Heinz et al.\
2003; Acreman et al.\ 2003; Takizawa 2005; Ascasibar \& Markevitch 2006,
hereafter A06). In fact, these features could already be seen in earlier
simulations of idealized mergers, such as those of Roettiger, Loken, \&
Burns (1997). For example, an obvious cold front is seen in a merger
simulated by Roettiger, Stone, \& Mushotzky (1998), although they have not
yet heard of this term then and therefore concentrated on other aspects of
their result. We present their simulated cluster in Fig.\ \ref{roettiger},
which shows maps of the gas temperature and velocity, dark matter density
and X-ray brightness for two subclusters right after a core passage. Panel
({\em d}) shows the fraction of gas that initially belonged to each
subcluster; a cold front is the boundary between the two gases that did not
mix.
The linear resolution of this and other contemporary simulations (as well as
most of the present-day ones) was limited to $\gg 1$ kpc, which is why they
could not predict that these interfaces would be so strikingly sharp in real
clusters when looked at with \emph{Chandra}. Nevertheless, keeping this
limitation in mind, we can use these and newer simulations to clarify the
origin and evolution of cold fronts.
Indeed, simulations show that when two subclusters collide, the outer
regions of their gas halos are shocked and stopped, while the lower-entropy
gas cores are often dense enough to resist the penetration of shocks and
stay attached to their host dark matter subclusters. This can be seen in a
time sequence for an interesting region selected from a cosmological
simulation (Fig.\ \ref{mathis}, from Mathis et al.\ 2005). The upper panels
show two similar dark matter subclusters colliding and passing nearly
through each other (the pericenter passage occurs between $z=0.22$ and
$z=0.20$). In lower panels, we see how the gas between the clusters is
first heated via compression and then by shocks, which propagate outwards
after the pericentric passage. At $z=0.20$ and later, we see the emergence
of two cold fronts, which are the boundaries of the unstripped remnants of
the two former subcluster gas cores. This is pretty much the picture
proposed in M00 (Fig.\ \ref{a2142_scheme}) and seen in other simulations
(e.g., Nagai \& Kravtsov 2003).
\begin{figure}[t]
\centering
\includegraphics[width=\textwidth,bb=75 278 553 490,clip]%
{a168_maps.ps}
\caption{({\em a}\/) Contours of the \emph{Chandra}\ X-ray brightness map
(smoothed) overlaid on the Palomar Digitized Sky Survey optical image of
the merging cluster A168. ({\em b}\/) Projected temperature map (colors)
overlaid with the image contours. The tip of the tongue in the north is a
cold front. The cD galaxy is the likely gravitational potential minimum.
The cold front apparently moved ahead of its host dark matter halo in a
ram pressure slingshot. (Reproduced from Hallman \& Markevitch 2004.)}
\label{a168}
\end{figure}
\begin{figure}[p]
\centering
\includegraphics[width=0.45\textwidth, bb=0 0 550 550]%
{yago_core_015_ed.ps}
\includegraphics[width=0.45\textwidth, bb=31 25 580 573]%
{yago_core_020.eps}\\
\includegraphics[width=0.45\textwidth, bb=82 76 529 523]%
{yago_Xcore_015.eps}
\includegraphics[width=0.45\textwidth, bb=82 76 529 523]%
{yago_Xcore_020.eps}
\caption{Ram-pressure slingshot illustrated by a simulated off-axis merger
of two subclusters. In top panels, contours show the dark matter density,
while colors show the gas temperature, in a slice along the merger plane.
Arrows show local gas velocities. In lower panels, black and white shows
gas particles that initially belonged to each subcluster, for the
corresponding upper panels. The subcluster (that entered from the
upper-right corner) has a cool core, while the main cluster (in the
center) was initially isothermal and did not have the usual sharp density
peak at the center. Such initial profiles are chosen here for
illustration, to increase the amplitude of the motions of the main central
gas. The main gas core is pushed back from its gravitational
potential, but rebounds and overshoots the dark matter peak as soon as the
ram pressure drops. (Reproduced from A06.)}
\label{yago_sling}
\end{figure}
\begin{figure}[p]
\centering
\includegraphics[width=1\textwidth,bb=65 480 573 580,clip]%
{heinz.ps}
\caption{A time sequence from a simulation of the passage of a planar shock
wave through a cluster-like isothermal gas halo (reproduced from Heinz et
al.\ 2003). The quantity shown is the gas specific entropy; arrows show
gas velocities. The shock propagates from the left. The gravitational
potential is fixed; its minimum is shown as a circle in each panel.}
\label{heinz}
\end{figure}
\paragraph*{Ram pressure stripping}
Ram pressure of the ambient gas first pushes these gas remnants out of the
gravitational potential wells of their respective subclusters. Depending on
the depth of the well, the density of the ambient gas and the merger
velocity, the ram pressure may or may not succeed in stripping the
subcluster gas completely, as shown in Fig.\ \ref{yago_strip}. As long as it
does not succeed, the cool dense gas core is dragged along by the gravity of
the subcluster, initially slightly lagging behind its dark matter peak. The
ambient shocked gas flows around it, separated by a sharp contact
discontinuity. This is the stage at which we observe the bullet subcluster
in \mbox{1E\,0657--56}\ (Fig.\ \ref{1e_lens}).
\paragraph*{Ram pressure slingshot}
For a subcluster that has managed to retain its cool core through the
pericenter passage where the ram pressure was the highest, an interesting
thing happens at a later stage. As the subcluster moves away from the
pericenter and slows down, it also enters the region with a lower density of
the ambient gas, and the ram pressure on the cool cores drops very rapidly
($p_{\rm ram}=\rho v^2$). As a result, the cool gas rebounds and overtakes
the dark matter core as if in a slingshot. The forward region of the cool
core moves away from the gravitational potential minimum which kept it at
high pressure, expands adiabatically and cools, further enhancing the
temperature contrast at the cold front (as noted by Bialek et al.\ 2002).
This is what we observe in A168 (Fig.\ \ref{a168}, from Hallman \&
Markevitch 2004) --- instead of lagging behind, a cold front in that cluster
is located ``ahead'' of the most likely center of the northern subcluster (a
giant galaxy seen in the optical image). This process is also seen at late
stages of the Mathis et al.\ simulations (note the crescent-shaped cool
regions appearing in the last panel in Fig.\ \ref{mathis}). This ``ram
pressure slingshot'' is further illustrated in Fig.\ \ref{yago_sling} (taken
from A06), which shows a small subcluster passing near the center of a
larger cluster. The corresponding black and white panels show the gas that
initially belonged to each of the subclusters. At first, ram pressure
exerted by the dense subcluster gas pushes the main cluster core far away
from the dark matter peak. However, as soon as the subcluster passes, that
ram pressure drops, and the main cluster gas (black) rebounds under the
effect of gravity and unbalanced thermal pressure behind the front,
overshooting the center. Note that in both panels, the boundary of the main
cluster core is a cold front, but at the latter moment, the temperature
contrast at the front is enhanced by adiabatic expansion. Interestingly, a
gas temperature map for A3667 obtained with \emph{XMM}\ (Heinz et al.\ 2003; Briel
et al.\ 2004), which has sufficient statistical accuracy to show the
small-scale detail, shows that the coolest gas is located right along the
cold front, suggesting that the front in A3667 is at this late,
``slingshot'' stage of its evolution. (Another possibility is that the cool
spot in A3667 is a remnant of a cooling flow-like initial temperature
distribution.)
There may be an additional effect that helps to enhance the temperature
contrast at the cold fronts. Heinz et al.\ (2003) used idealized
two-dimensional hydrodynamic simulation to model the evolution of a contact
discontinuity between a uniform wind flowing around a cool, initially
isothermal (and therefore, with the specific entropy declining toward the
center), cluster-like gas cloud in a stationary gravitational potential of
the underlying dark matter halo. The initially planar discontinuity has
developed into a spheroidal cold front with a flow of ambient (post-shock)
gas around it (Fig.\ \ref{heinz}). The gas halo in this simulation is first
displaced from the potential minimum along the direction of the wind (panel
2), but then the central, lowest-entropy gas rebounds, overshoots the
potential minimum (as in the ram-pressure slingshot described above) and
starts flowing toward the cold front (panel 3). At the same time, the
ambient flow around the contact discontinuity has generated a shear layer in
which the Kelvin-Helmholtz (KH) instabilities drag the cool gas located just
under the surface of the front to the sides and away from the tip (panel 3
and later). This is in addition to the usual flattening and sideways
expansion of a dense gas sphere subjected to a wind. If occurs in real
clusters, this ``circulation'' may help the low-entropy gas located deeper
under the surface to reach the tip.
\begin{figure}[t]
\centering
\includegraphics[width=0.48\textwidth]%
{RXJ1720.eps}~~
\includegraphics[width=0.48\textwidth]%
{A2204.eps}
\vspace{5mm}
\includegraphics[width=0.48\textwidth]%
{A2029.eps}~~
\includegraphics[width=0.48\textwidth]%
{Ophiuchus.eps}
\caption{X-ray images of several non-merging clusters exhibiting cold fronts
inside their cool cores. Horizontal bars are 100 kpc. A2029 (Clarke et
al.\ 2004) exhibits 2 edges in a spiral pattern at $r\sim 7$ kpc and 20
kpc (between white and pink and between light and dark pink).
RXJ\,$1720+26$ (Mazzotta et al.\ 2001) exhibits a large, $r\approx 250$
kpc edge (pink--blue), while A2204 (Sanders et al.\ 2005) shows a spiral
pattern consisting of at least two edges at 20 kpc and 70 kpc (white--pink,
pink--blue). Ophiuchus has some evidence in the outskirts of a recent
merger; note three edges on scales around $r\sim 3$ kpc, 8 kpc and 40 kpc
(white--pink, white--darker pink, blue--darker blue). (Reproduced from
A06.)}
\label{slosh_examples}
\end{figure}
\subsection{Cold fronts in cluster cool cores}
\label{sec:slosh}
When the subclusters merge, one does expect to see vigorous gas flows,
including moving remnants of the subcluster cores which give rise to cold
fronts. Surprisingly, though, cold fronts are also observed near the centers
of most ``cooling flow'' clusters, many of which are relaxed and show little
or no signs of recent merging (e.g., Mazzotta et al.\ 2001; Markevitch,
Vikhlinin, \& Mazzotta 2001, hereafter M01; Mazzotta, Edge, \& Markevitch
2003; Churazov et al.\ 2003; Dupke \& White 2003; Sanders et al.\ 2005).
These fronts are typically more subtle in terms of the density jump than
those in mergers, and occur on smaller linear scales close to the center
($r\lesssim 100$ kpc), with their arcs usually curved around the central gas
density peak. There are often several such arcs at different radii around
the density peak. Cooling flow clusters by definition have a sharp
temperature decline and an accompanying density increase toward the center
(that is, a sharp decline of specific entropy). The edges are seen inside
or on the boundaries of this cool central region. This is a very common
variety of the cold fronts; we found them in more than a half of the cooling
flow clusters (Markevitch, Vikhlinin, \& Forman 2003b). Given the
projection, this means that most, if not all, cooling flow clusters may have
one or several such fronts. Some of the clusters with such fronts are shown
in Fig.\ \ref{slosh_examples}. One of them is A2029, which on scales
$r>100-200$ kpc is the most undisturbed cluster known (e.g., Buote \& Tsai
1996). As in mergers, cold fronts in these clusters must indicate gas
motion; however, the moving gas clearly does not belong to any infalling
subcluster.
\begin{figure}[t]
\centering
\includegraphics[width=1\textwidth,bb=59 223 553 612,clip]%
{a1795_profs.ps}
\caption{Profiles from a sector in the A1795 cluster, centered on the
cluster peak and containing a brightness edge (which is concentric with
the cluster peak). Vertical dotted lines show the edge position. ({\em
a}) X-ray surface brightness. The red line is a projection of the
best-fit density model shown in panel ({\em c}). ({\em b}) Gas temperature
profile, corrected for projection. ({\em d}) Pressure profile. ({\em e})
Total mass within a given radius derived using the hydrostatic equilibrium
assumption (with an error band shown in gray). For comparison, a fit in
the sector opposite to the edge, where the gas distribution is continuous,
is shown by dashed line. If the gas were indeed in hydrostatic
equilibrium, they would show the same mass. (Reproduced from M01.)}
\label{a1795_profs}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[width=0.5\textwidth]%
{wine_glass.eps}
\caption{The origin of cold fronts in the dense cluster cores.}
\label{glass}
\end{figure}
We studied such a feature in A1795 --- another one of the most-relaxed
nearby clusters (Buote \& Tsai 1996) --- and showed that the gas forming a
cold front is not in hydrostatic equilibrium in the cluster gravitational
potential (M01). Figure \ref{a1795_profs} (reproduced from M01) shows radial
profiles of X-ray brightness and gas temperature in the sector of A1795 that
contains the cold front, along with the best-fit gas density model and the
resulting pressure profile. The brightness edge is barely noticeable (and
would certainly go undetected without the \emph{Chandra}'s arcsecond resolution)
and corresponds to a density jump by only a factor of 1.3 (for comparison,
in the more prominent merger cold fronts discussed in \S\ref{sec:merg}, the
density jumps by factors 2--5). The pressure profile turns out to be almost
exactly continuous, leaving little or no room for a relative gas motion,
since the ram pressure from such a motion would cause the inner thermal
pressure to be higher compared to that on the outside (beyond a small
stagnation region; see \S\ref{sec:vel} below). Thus the inner and outer
gases appear very nearly at rest and in pressure equilibrium. Therefore,
one might expect them to be in hydrostatic equilibrium in the cluster
gravitational potential.
For a spherically symmetric cluster in hydrostatic equilibrium, one can
derive the cluster total (mostly dark matter) mass from the above radial
profiles of the gas density and temperature using eq.\ (\ref{eq:mass}). The
resulting mass profile in the immediate vicinity of the edge in A1795 is
shown in Fig.\ \ref{a1795_profs}{\em e}. The profile reveals an unphysical
discontinuity by a factor of 2 at the front. For comparison, a similar
analysis was performed using a sector on the opposite side from the center,
which has smooth distributions of gas density and temperature. The total
mass profile derived using that sector is overlaid as a dashed line. If the
gas around the cluster center were in hydrostatic equilibrium, both sectors
would measure the same enclosed cluster mass. Indeed, outside the edge
radius, the masses derived from the two opposite sectors agree, strongly
suggesting that the gas immediately outside the edge is indeed near
hydrostatic equilibrium. But the gas inside the edge is not --- even though
there is pressure equilibrium between the two sides of the edge.
Such an unphysical mass discontinuity at the cold front was first reported
by Mazzotta et al.\ (2001) for the cluster RXJ\,1720+26, which is similarly
relaxed on large scales. Although the statistical accuracy of the available
temperature profile was not sufficient to exclude a significant bulk
velocity of the cool gas, the situation appears similar to A1795.
\subsubsection{Gas sloshing}
\label{sec:a1795}
Given the above evidence, we proposed (M01) that the low-entropy gas in the
A1795 core is ``sloshing'' in the central potential well (Fig.\
\ref{glass}). The observed edge delineates a parcel of cool gas that has
moved from the cluster center and is currently near the point of maximum
displacement, where it has zero velocity but nonzero centripetal
acceleration. In agreement with this scheme, there is a $30-40\;h^{-1}$ kpc
cool gas filament extending from the cD galaxy in the center of A1795 toward
this cold front (Fabian et al.\ 2001), suggesting that the bulk of the
central gas has indeed been flowing around the cD galaxy (which most
probably sits in the gravitational potential minimum). Such an oscillating
gas parcel would not be in hydrostatic equilibrium with the potential ---
instead, the gas distribution would reflect the reduced gravity force in the
accelerating reference frame, resulting in the above unphysical mass
underestimate. M01 made an estimate of this acceleration from the apparent
mass jump $\Delta M$, assuming that the gas outside the edge is hydrostatic:
$a\sim G \Delta M r^{-2} \approx 3\times 10^{-8}h$ cm~s$^{-2}$ or
$800h\;{\rm km}\;{\rm s}^{-1}\;(10^8\;{\rm yr})^{-1}$, where $r$ is the
radius of the edge, which is a sensible number for an oscillation on this
linear scale. More recent detailed simulations (A06, see below) have shown
that this picture is somewhat oversimplified, but the physics in it is
correct.
In M01 we suggested that this subsonic sloshing of the cluster's own cool,
dense central gas in the gravitational potential well may be the result of a
disturbance of the central potential by past subcluster infall. There are
striking examples suggesting that this is the case at least in some clusters
(Fig.\ \ref{a1644}). Alternatively, one can imagine some off-center
disturbance in the gas from the activity of the central AGN; AGNs blowing
bubbles in the intracluster gas are observed in many cooling flow clusters
(e.g., Fabian et al.\ 2000; Nulsen et al.\ 2005). However, the absence of
any visible merger or AGN disturbance in the X-ray images of two of the most
undisturbed clusters, A2029 and A1795, presented an apparent difficulty,
which has motivated some of the numerical studies reviewed below.
\begin{figure}[t]
\centering
\includegraphics[width=0.85\textwidth,bb=1 11 394 363,clip]%
{a1644.ps}
\caption{\emph{Chandra}\ X-ray image of the merging system A1644. An infalling
subcluster (the northeastern clump) apparently has passed near the center
of the main cluster, disturbed its mass distribution and set off sloshing
in its core.}
\label{a1644}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[width=0.32\textwidth]%
{yago_f3a1.eps}
\includegraphics[width=0.32\textwidth]%
{yago_f3d1.eps}
\includegraphics[width=0.32\textwidth]%
{yago_f3b1.eps}\\[1mm]
\includegraphics[width=0.32\textwidth]%
{yago_f3c1.eps}
\includegraphics[width=0.32\textwidth]%
{yago_f3e1.eps}
\includegraphics[width=0.32\textwidth]%
{yago_f3f1.eps}\\[1mm]
\includegraphics[width=0.32\textwidth]%
{yago_f3g1.eps}
\includegraphics[width=0.32\textwidth]%
{yago_f3h1.eps}
\includegraphics[width=0.32\textwidth]%
{yago_f3i1.eps}
\caption{
Time sequence from a simulated infall of a small, purely dark matter
satellite into a relaxed, cooling flow cluster. Color shows the gas
temperature (in keV) in the orbital plane. Arrows represent the gas
velocity field w.r.t.\ the main dark matter density peak. Contours show
the dark matter density. The white cross shows the center of mass of the
main cluster's particles. The panel size is 1 Mpc. Such a merger induces
gas sloshing in the center, but not much disturbance elsewhere, as will be
seen in the next figure. (Reproduced from A06.)}
\label{yago_dm}
\end{figure}
\begin{figure}[p]
\centering
\includegraphics[width=0.31\textwidth]%
{yago_f21a.eps}
\includegraphics[width=0.31\textwidth]%
{yago_f21b.eps}
\includegraphics[width=0.31\textwidth]%
{yago_f21c.eps}\\
\includegraphics[width=0.31\textwidth]%
{yago_f21d.eps}
\includegraphics[width=0.31\textwidth]%
{yago_f21e.eps}
\includegraphics[width=0.31\textwidth]%
{yago_f21f.eps}
\caption{
Projected X-ray brightness for the merger shown in Fig.\ \ref{yago_dm}.
The panel size is 1 Mpc. With the possible exception of short moments when
the subcluster flyby generates a conical wake (at 1.5 Gyr and 4.2 Gyr),
the cluster stays very symmetric on large scales; the only visible
disturbance is cold fronts in the center. (Reproduced from A06.)}
\label{yago_lx_dm}
\end{figure}
\begin{figure}[p]
\centering
\includegraphics[width=0.31\textwidth]%
{yago_f22a.eps}
\includegraphics[width=0.31\textwidth]%
{yago_f22b.eps}
\includegraphics[width=0.31\textwidth]%
{yago_f22c.eps}\\
\includegraphics[width=0.31\textwidth]%
{yago_f22d.eps}
\includegraphics[width=0.31\textwidth]%
{yago_f22e.eps}
\includegraphics[width=0.31\textwidth]%
{yago_f22f.eps}
\caption{
Projected X-ray brightness for a merger similar to that in Figs.\
\ref{yago_dm} and \ref{yago_lx_dm}, but now the subcluster has gas. Cold
fronts still form in the center of the main cluster, but the system is
very disturbed on all scales. It needs 3--4 Gyr after the first core
passage to regain a symmetric appearance on large scales (last panel).
(Reproduced from A06.)}
\label{yago_lx_gas}
\end{figure}
\subsubsection{Simulations of gas sloshing}
\label{sec:simslosh}
Several simulation works addressed the possibility that mergers can create
cold fronts in the cluster centers. Churazov et al.\ (2003) and Fujita,
Matsumoto \& Wada (2004) used 2D simulations to show that a weak shock or
acoustic wave propagating toward the center of a cooling flow cluster can
displace the cool gas from the gravitational potential well and cause gas
sloshing, giving rise to cold fronts. On the other hand, Tittley \&
Henriksen (2005), using mergers extracted from a cosmological simulation,
suggested that cold fronts in the cores arise when the gas peak is dragged
along as the dark matter peak oscillates around the cluster centroid because
of a gravitational disturbance from a merging subcluster.
The most detailed simulation that addressed the specific question of
whether a merger can generate cold fronts in the cores, but no visible
disturbance elsewhere, was presented by A06. They found that sloshing is
indeed easily set off by any minor merger and can persist for gigayears,
producing concentric cold fronts just as those observed. The only necessary
(and obvious) condition for their formation is a steep radial entropy drop
in the gas peak, such as that present in all cooling flows. Most
interestingly, fronts form even if the infalling subcluster has not had any
gas during core passage. This may occur if the gas was stripped early in the
merger, leaving a clump of only the dark matter and galaxies. It is mergers
with such gasless clumps that can set the central cool dense gas in motion,
while leaving no other long-lived visible traces in X-rays.
Figure \ref{yago_dm} (from A06) shows a time sequence from a simulated
merger with the dark matter-only subcluster whose mass is 1/5 of that of the
main cluster and which falls in with a nonzero impact parameter (or angular
momentum). The figure shows how the subcluster makes two passes, around 1.4
Gyr and 4.2 Gyr from the start of the simulation run. During the first
pass, the gravitational disturbance created by the subhalo causes the
density peak of the main cluster (DM and gas together) to swing along a
spiral trajectory relative to the center of mass of the main cluster (white
cross in Fig.\ \ref{yago_dm}). The gas and DM peaks feel the same gravity
force and initially start moving together toward the subcluster. However,
during the core passage, the direction of this motion quickly changes,
leading to a rapid change of the ram-pressure force exerted on the gas peak.
Mainly as a result of this change, the cool gas core shoots away from the
potential minimum in a ram-pressure slingshot similar to the one described
above in \S\ref{sec:origin}. Subsequently, the densest gas turns around and
starts falling back toward the minimum of the gravitational potential (a
cuspy NWF mass profile has a well-defined sharp potential minimum). It then
starts sloshing around the DM peak and generating long-lived cold fronts, as
will be discussed below.
Mock X-ray images of this simulated merger (Fig.\ \ref{yago_lx_dm}) show
these cold fronts quite clearly, at the same time exhibiting very little
disturbance on the cluster-wide scale. The exception is several brief
periods when the DM satellite crosses the cluster and generates a subtle
conical wake (first and last panels in Fig.\ \ref{yago_lx_dm}), but the
subcluster spends most of the orbiting time in the outskirts. This
simulation looks very much like the most relaxed cooling-flow clusters in
the real world, such as A2029 and A1795.
For comparison, Fig.\ \ref{yago_lx_gas} shows mock X-ray images from a
simulation of a similar merger, but with the gas subcluster. Hydrodynamic
effects of the collision of two gas clouds are now overwhelming and the gas
is disturbed on all scales for a long time. Sloshing and central cold
fronts are generated, too --- in fact, sloshing occurs on a wider scale,
because the initial displacement caused by the subcluster shock and stripped
gas is much stronger than that caused by a swinging motion of the DM peak.
There are many real clusters that look similar to this simulation.
\begin{figure}[p]
\centering
\includegraphics[width=0.45\textwidth]%
{yago_f7a.eps}
\includegraphics[width=0.45\textwidth]%
{yago_f7b.eps}\\[1mm]
\includegraphics[width=0.45\textwidth]%
{yago_f7c.eps}
\includegraphics[width=0.45\textwidth]%
{yago_f7d.eps}\\[1mm]
\includegraphics[width=0.45\textwidth]%
{yago_f7e.eps}
\includegraphics[width=0.45\textwidth]%
{yago_f7f.eps}
\caption{
A zoom-in view of the center of the main cluster in the merger simulation
shown in Fig.\ \ref{yago_dm} (where the infalling subcluster did not have
any gas). The panel size is 250 kpc. Color shows gas temperature (between
2--10 keV) in the merger plane; contours show dark matter density; arrows
show gas velocities relative to the dark matter peak. One can see the
onset of sloshing and the emergence of cold fronts. (Reproduced from
A06.)}
\label{yago_dm_zoom}
\end{figure}
\subsubsection*{The emergence of multiple cold fronts}
Let us now examine in details how the sloshing central gas gives rise to
cold fronts. Figure \ref{yago_dm_zoom} presents a zoomed-in view of the gas
temperature and velocity field from the A06 simulation of a merger with the
gasless subcluster. The dark matter peak has a cuspy NFW density profile,
and the initial gas density and temperature profiles are similar to those
observed in cooling flow clusters, so this is what should happen in clusters
such as A2029. The figure shows several snapshots following the initial
displacement of the gas density peak from the central potential dip. The
displaced cool gas expands adiabatically as it is carried further out by the
flow of the surrounding gas (the orange plume above the center in the
1.6--1.7 Gyr panels of Fig.\ \ref{yago_dm_zoom}). However, in a process
similar to the onset of a Rayleigh-Taylor (RT) instability, the densest,
coolest gas turns around and starts sinking towards the minimum of the
gravitational potential, as seen most clearly in the 1.6 Gyr and again in
the 1.8 Gyr snapshots. The coolest gas overshoots the center at 1.7 Gyr
and, subjected to ram pressure from the gas on the opposite side still
moving in the opposite direction, eventually spreads into a characteristic
mushroom head with velocity eddies (von Karman vortices) at the sides. This
is a classic structure seen in numeric and real-life experiments involving a
gas jet flowing through a less dense gas (for cluster-related works see,
e.g., Heinz et al.\ 2003, our Fig.\ \ref{heinz}; Takizawa 2005). The
mushroom head forms where the dense gas is slowed by the ambient ram
pressure and spreads sideways into the regions of lower pressure created by
the flow of the ambient gas around a blunt obstacle (Bernoulli's law). The
front edges of these mushroom heads are sharp discontinuities, as confirmed
by a detailed look at these structures (A06). We will discuss how exactly
they arise in \S\ref{sec:discont} below.
At 1.8 Gyr, one can see how the inner side of the first mushroom head
sprouts a new RT tongue --- the densest, lowest-entropy gas separates and
again starts sinking toward the potential minimum (compare this, e.g., with
a structure in the center of the Ophiuchus cluster in Fig.\
\ref{slosh_examples}). Meanwhile, the rest of the gas in the first mushroom
head continues to move outwards, expanding adiabatically as it moves into
the lower-pressure regions of the cluster, and roughly delineating the
equipotential surfaces. The process repeats itself, and these mushroom heads
are generated on progressively smaller linear scales. Sloshing of the
densest gas that is closest to the center occurs with a smaller period and
amplitude than that of the gas initially at greater radii (the reason is
explained in Churazov et al.\ 2003). It is this period difference that
eventually brings into contact the gas phases that initially were at
different off-center distances and had different entropies (recall that a
cooling flow gas profile has a sharp radial entropy gradient).
\begin{figure}[t]
\centering
\includegraphics[width=1\textwidth]%
{ricker01.ps}
\caption{Sloshing of the low-entropy central gas set off by an off-axis
merger in an initially isothermal cluster with a cuspy dark matter
profile. Color shows the gas specific entropy in the merger plane
(increasing from black to white), contours show the gravitational
potential. Panel size is 1.5 Mpc. One can see the ram pressure slingshot
(the low-entropy gas is pushed to the upper-right side and then shoots
back), followed by the development of a convective plume flowing toward
the center. The resulting edges in the entropy map are analogs of the cold
fronts seen in Fig.\ \ref{yago_dm_zoom}. (Reproduced from Ricker \&
Sarazin 2001.)}
\label{ricker_slosh}
\end{figure}
The picture is qualitatively similar even if there is no cooling flow-like
temperature drop --- as long as there is entropy gradient. Ricker \&
Sarazin (2001) simulated a merger in an isothermal cluster with a cuspy
potential, which shows the formation of similar multiple mushroom heads
(Fig.\ \ref{ricker_slosh}). The specific entropy declines toward the center,
but less steeply than in a cooling flow cluster, hence the linear scale of
the resulting sloshing is bigger. (Another reason for that is this merger
involves a gas-containing subcluster, so the initial disturbance was
greater.)
Note that the oscillation of the DM peak caused by the subcluster flyby has
a much longer period, of order 1 Gyr (Fig.\ \ref{yago_dm}), than the $\sim
0.1$ Gyr timescale of gas sloshing. Indeed, as seen in Fig.\
\ref{yago_dm_zoom}, the DM distribution in the core stays largely centrally
symmetric, while the gas sloshes back and forth in its potential well. The
former timescale is determined by the subcluster masses and impact
parameter, while the latter is determined by the gas and DM profiles at the
main peak (Churazov et al.\ 2003). Thus, sloshing is mostly a hydrodynamic
effect in a quasi-static central gravitational potential (cf.\ Tittley \&
Henriksen 2005), although in the long term, the DM peak oscillation can feed
additional kinetic energy to the sloshing gas. For this reason, it is
unlikely that looking at the pattern of cold fronts in the center of the
cluster, one will be able to determine, for example, the mass and impact
parameter of the subcluster. It may be possible to get an upper limit on the
time since the disturbance if the velocity of the sloshing gas can be
determined (\S\ref{sec:vel}).
\paragraph*{Long-term evolution of a cold front}
The A06 simulations further showed that, although the lowest-entropy gas
indeed oscillates back and forth in the potential minimum, a cold front,
once formed, always propagates outward from the center, and does not ``turn
around'' with the gas or ``straighten out'' (Figs.\ \ref{yago_dm} and
\ref{yago_dm_zoom}). This is somewhat counterintuitive, because it has to
be difficult to move the low-entropy central gas out to large radii against
convective stability in the radially increasing entropy profile. But in
fact, the central gas does not move out to large radii. In later panels of
Fig.\ \ref{yago_dm_zoom}, one can discern a flow pattern inside the cold
front in which the lowest-entropy gas initially forming the front, turns
around and sinks back towards the center. It is replaced at the front by
higher-entropy gas that arrives later and whose origin traces back to larger
radii. In other words, the cold front as a geometric feature moves out, but
the low-entropy gas stays close to the center of the potential.
\paragraph*{Spiral pattern}
Finally, Fig.\ \ref{yago_dm} reveals a curious spiral pattern that the
central cold fronts develop with time. A similar spiral structure (if not so
well-developed) is seen in the X-ray image of A2029 (Fig.\
\ref{slosh_examples}; Clarke et al.\ 2004) and in the temperature map of
Perseus (Fabian et al.\ 2006, discussed in A06). The simulated merger in
A06 (along with most real-life mergers) has a nonzero impact parameter. So
when the cool gas is displaced from the center for the first time, it
acquires angular momentum from the gas in the wake and does not fall back
radially. As a result, the subsequent cold fronts of different radii are
not exactly concentric, but combine into a spiral pattern (Fig.\
\ref{yago_dm_zoom}). Initially, it does not represent any coherent
spiraling motion --- each edge is an independent structure. However, as the
time goes by and the linear scale of the structure grows, circular motions
that are against the average angular momentum subside, and the ``mushrooms''
become more and more lopsided. On large scales, the spiral does indeed
become a largely coherent spiraling-in of cool gas --- the mushroom stems,
through which the low-entropy gas flows from one mushroom cap toward the
smaller-scale mushroom cap, shift more and more to the edge of the cap.
As a side note, the spiraling-in central gas should have the same direction
of the angular momentum as the infalling subcluster. Thus, looking at the
brightness peak in A1644 (Fig.\ \ref{a1644}), we can immediately say that
the subcluster must have passed it on the eastern side. Indeed, Reiprich et
al.\ (2004) conclude the same from their analysis of the temperature and
abundance distributions obtained with \emph{XMM}.
\begin{figure}[b]
\centering
\includegraphics[width=0.85\textwidth, bb=20 280 590 576, clip]%
{slosh_scheme.ps}
\caption{The emergence of a density discontinuity from an initially
continuous density distribution. The area-proportional drag force causes
different deceleration of gases with different densities, which eventually
brings the densest gas forward.}
\label{slosh_scheme}
\end{figure}
\subsubsection{Origin of density discontinuity}
\label{sec:discont}
While cold fronts may be caused by different events in the cluster, the
density discontinuities in them form for the same basic reason, which is
worth a clarifying aside. Simulations show that whenever a gas density peak
encounters a flow of ambient gas, a contact discontinuity quickly forms.
This occurs even when the initial gas distribution was perfectly smooth (no
shocks, etc.), as in a merger with a pure dark matter subcluster considered
above. Stripping by a shear flow is usually quoted as the cause of the
discontinuity (e.g., M00; V01). Indeed, the gas pressure immediately inside
the cold front in A3667 was found to be equal to the pressure of the outer
gas everywhere along the front, if one models the velocity field around the
spherical front and uses the Bernoulli equation (Vikhlinin \& Markevitch
2002, hereafter V02; see \S\ref{sec:RT} below). This suggests that the
outer, less dense layers of the subcluster's gas are quickly removed until
the radius is reached where the pressure in the cold gas equals the pressure
outside. It is easy to imagine how a shear flow would strip the
subcluster's gas at the sides of the front. However, at the forward tip of
the front (the stagnation point), there is no shear flow for symmetry
reasons, but the fronts are just as sharp.
A simple reason for the emergence of a discontinuity at the stagnation point
is illustrated in Fig.\ \ref{slosh_scheme}. When an initially smooth
spherical density peak starts moving w.r.t.\ the surrounding gas, it starts
experiencing ram pressure, which creates roughly the same
(area-proportional) net force for each cubic centimeter of the gas in the
core (near the symmetry axis and assuming subsonic motions). Denser gas
experiences smaller resulting acceleration. This produces a velocity
gradient inside the core along the direction of the force. The
lower-density, outer layer of the core gas is then squeezed to the sides,
and the ambient gas eventually meets the dense gas for which the
forward-pulling, density-proportional gravity force (as in the bullet
cluster) or inertial force (as in the sloshing central gas) prevails over
the area-proportional ram pressure force. The initial density peak has to be
sufficiently sharp to ensure that the compressed intermediate gas does not
decelerate the denser gas behind it before being squeezed to the sides, a
condition which appears to be easily satisfied in real clusters. Thus, a
contact discontinuity at the stagnation point forms by ``squeezing out'' the
gas layers not in pressure equilibrium with the flow. Of course, stripping
by the shear flow does occur away from the axis of symmetry of the cold
front. For an illustration based on simulations, see Fig.\ 23 in A06.
\subsubsection{Effect of sloshing on cluster mass estimates}
Any motion of gas in the cluster core obviously represents a deviation from
hydrostatic equilibrium and thus poses a problem for the derivation of the
total masses based on this assumption (eq.\ \ref{eq:mass}). As we saw above
(A06), in clusters that may be perfectly relaxed outside their cool cores,
the central low-entropy region is easily disturbed and may not subsequently
come to equilibrium for a long time. Unfortunately, ``relaxed'' clusters
almost always have those easily disturbed cooling flow regions.
How this may affect the hydrostatic mass estimates is illustrated by the
example of A1795 summarized above (Fig.\ \ref{a1795_profs}). Using the gas
profiles from the sector containing the front, the total mass within the
edge radius was underestimated by a factor of 2. If one uses the radial
profiles averaged over the full 360$^{\circ}$\ azimuth, the effect is diluted; on
the other hand, the edge in A1795 is relatively small. Pending a more
quantitative analysis of this issue (e.g., emulating the hydrostatic mass
estimates for the simulated clusters with sloshing), we can estimate roughly
that masses within the cooling flow regions can be underestimated by up to a
factor of 2. (The average result should always be an underestimate, since
the cool gas is gravitationally bound but has a mechanical component to its
total pressure, which we omit by measuring only the thermal component.)
Recall that even if a cooling flow cluster does not exhibit cold fronts,
statistically, it is likely to have one (or more) hidden by projection. To
keep this in proper perspective, the radii of the cooling flow regions,
$r\lesssim 100$ kpc, contain only a few percent of the cluster total mass, so
this systematic mass error is relevant only for a narrow range of studies,
such as the exact shapes of the central dark matter cusps in clusters, or
comparison of X-ray derived masses with those from strong gravitational
lensing.
\subsubsection{Effects of sloshing on cooling flows}
\emph{XMM}\ and \emph{Chandra}\ observations have not found the amounts of cool gas in
the centers of cooling flow clusters predicted by simple models based on the
cluster X-ray brightness profiles (see Peterson \& Fabian 2006 for a
review). This means that that there has to be a steady energy supply to
compensate for the (directly observed) radiative cooling. Several
mechanisms have been proposed; the currently favored view is that AGNs,
found in most cD galaxies in the centers of cooling flows, provide the
heating via the interaction of AGN jets with the ICM (e.g., Voit \& Donahue
2005; Fabian et al.\ 2005). A difficulty of this mechanism is that heating
has to be steady and finely tuned (to avoid blowing up the entire cluster
core), whereas AGNs have different powers in different clusters, and some
clusters do not even have a presently active central AGN. In the latter
clusters, other heating mechanisms may be needed for the cooling flow
suppression. One of the possible alternatives is sloshing, which may have
two effects. First one is obvious --- M01 estimated that the mechanical
energy in the sloshing gas in A1795 is around half of its thermal energy (an
estimate for an analogous edge in Perseus is 10--20\%, Churazov et al.\
2003). As the gas sloshes, this energy is converted into heat at a steady
rate.
Another effect is more interesting and possibly more significant. As seen in
Fig.\ \ref{yago_dm_zoom}, sloshing brings hot gas from outside the cool core
into the cluster center, where it comes in close contact with the cool gas
that oscillates with a different period, as discussed in
\S\ref{sec:simslosh}. Provided that the two phases can mix, this should
result in heat inflow from the large reservoir of thermal energy in the gas
outside of the cool core. A classic electron heat conduction was proposed
to tap that reservoir, but was shown to be insufficient to balance the
cooling (Voigt \& Fabian 2004 and references therein), mainly because of the
strong temperature dependence of this process. However, a ``heat
conduction'' caused by the above mixing may be an attractive mechanism
(Markevitch \& Ascasibar, in preparation).
\subsubsection{Effect of sloshing on central abundance gradients}
Heavy elements in cooling flow clusters are concentrated toward the center
(e.g., Fukazawa et al.\ 1994; Tamura et al.\ 2004; for ideas why see, e.g.,
B{\"o}hringer et al.\ 2004). Their relative abundance starts to increase
just at the radii where the temperature starts to decrease (e.g., Vikhlinin
et al.\ 2005). Cold fronts found around these gas density peaks form as a
result of displacement of the central, higher-abundance gas outwards. Thus,
the abundance should be discontinuous across these fronts, as long as
sloshing occurs within the region with the strong gradient. Such abundance
discontinuities were indeed observed, e.g., in A2204 (Sanders et al.\ 2005)
and Perseus (Fabian et al.\ 2006), although Dupke \& White (2003) did not
see them in A496 (but their measurement uncertainties were relatively
large). In general, sloshing should spread the heavy elements from the
center outwards --- but not too far, because, as we have seen above
(\S\ref{sec:simslosh}), the low-entropy, high-abundance gas eventually flows
back into the center even as a cold front continues to propagate outwards.
\subsection{Zoology of cold fronts}
\label{sec:zoo}
In the sections above, we have discussed cold fronts in merging clusters and
in cooling flows. Since we now know more than two clusters with cold fronts,
we ought to propose a classification scheme, which will also help to
summarize the above observations and simulations. First, in mergers with
cold fronts that are a boundary between gases from two distinct subclusters,
the front can be at the ``stripping'' and the ``slingshot'' stages. At the
most intuitive ``stripping'' stage, ram pressure of the ambient gas pushes
the cool subcluster gas back from its dark matter host; the examples are
\mbox{1E\,0657--56}\ (Figs.\ \ref{1e} and \ref{1e_lens}) and NGC\,1404 (Fig.\ \ref{n1404}).
This is likely to occur on the inbound part of the subcluster trajectory or
around the time of core passage, when the ram pressure increases and reaches
its maximum. A less massive subcluster may be completely stripped of gas at
this stage (e.g., right panel in Fig.\ \ref{yago_strip}). If it does retain
gas, on the outbound leg of the trajectory, the ram pressure drops rapidly
(because both the ambient gas density and the velocity decrease), and the
displaced gas rebounds as in a ``slingshot'', overtaking the subcluster's
mass peak. An example is A168 (Fig.\ \ref{a168}). In Fig.\
\ref{yago_sling}, the subcluster in the left panel and the main cluster in
the right panel exhibit cold fronts at the ``slingshot'' stage, while the
main cluster in the left panel is at the ``stripping'' stage.
A third variety is the ``sloshing'' cold fronts observed in the centers of
clusters that exhibit sharp radial entropy gradients (i.e., cooling flows).
Here, multiple near-concentric discontinuities divide gas parcels from
different radii of the same cluster that came into contact due to sloshing
(\S\ref{sec:simslosh}). Simulations show (A06) that it is set off easily by
any minor merger and may last for gigayears. This is why this cold front
species is very common, often with multiple fronts in the same cluster. For
comparison, the core passage stage of a merger is very short (of order
$10^8$ yr), and we should also be lucky enough to observe it from the right
angle, which makes ``stripping'' cold fronts the rarest.
\subsection{Non-merger cold fronts and other density edges}
Because the central dense gas in cooling flow clusters is so easily
disturbed, in principle, sloshing can also be induced by bubbles blown by
the central AGNs. This possibility has not yet been addressed with detailed
simulations, although some works suggest that such disturbance is possible
(Quilis, Bower, \& Balogh 2001). A rising bubble can also push the
low-entropy gas in front of it (provided the ensuing instabilities can be
suppressed), which would develop a cold front when it moves into a
lower-density, lower-pressure outer region.
Edge-like features in the X-ray images of clusters and groups may have an
altogether different nature. The obvious bow shocks caused by mergers will
be discussed later. Edges in the cores of clusters harboring powerful AGNs
may be weak shocks propagating in front of large AGN-blown bubbles, as
observed, e.g., in the Hydra-A cluster (Nulsen et al.\ 2005). Such edges
look somewhat different from the ``sloshing'' edges considered above,
spanning a larger sector --- in Hydra-A, it can be traced almost all the way
around the cluster core. In addition, very subtle brightness edges or
``ripples'' observed in the core of the Perseus cluster have been attributed
to sound waves from the central AGN explosions (Fabian et al.\ 2006).
Because such features always have very low brightness contrast and therefore
are strongly affected by line-of-sight projection, and because weak shocks
have inherently low temperature contrast, it is difficult to distinguish
such features from cold fronts by simply looking at their temperature
profiles.
Finally, we mention a more exotic possibility of an ``iron front'', as
reported for the NGC\,507 group (Kraft et al.\ 2004). An X-ray image of this
cool group exhibits an edge, and the spectral analysis shows that most of
the brightness difference is due to a higher abundance of heavy elements on
one side of the edge (which strongly increases the emissivity for a plasma
at $T\lesssim 1$ keV). Physically, this is still a contact discontinuity similar
to a cold front.
\section{COLD FRONTS AS EXPERIMENTAL TOOL}
\label{sec:tools}
\begin{figure}[t]
\centering
\includegraphics[width=0.6\textwidth]%
{a3667_p_M.ps}
\caption{({\em a}) Geometry of a flow past a spheroidal cold front. Zones
0, 1, and 2 are those near the stagnation point, in the undisturbed free
stream, and past the possible bow shock, respectively. Zone $0^\prime$ is
within the body. ({\em b}) Ratio of pressures at the stagnation point 0
before the tip of the cold front (which is equal to that just inside the
front in zone $0^\prime$) and in the free stream 1, as a function of the
Mach number in the free stream. The solid and dashed line corresponds to
the sub- and supersonic regimes, respectively (eqs.\
\ref{eq:p:ratio:subsonic} and \ref{eq:p:ratio:supersonic}). The dotted
lines show the confidence interval for the pressure ratio in A3667.
(Reproduced from V01.)}
\label{p_M}
\end{figure}
The origin of cold fronts is certainly interesting, but the most useful
thing about this phenomenon is that it provides a unique tool to study the
cluster physics, including determining the gas bulk velocity (and sometimes
acceleration, as we already saw above), the growth of hydrodynamic
instabilities (or lack thereof), strength and structure of the intracluster
magnetic fields, thermal conductivity, and perhaps viscosity of the ICM. We
will discuss some of these possibilities below. From a technical viewpoint,
what makes these studies possible is the high contrast and symmetric shape
of cold fronts in the X-ray images, which enables accurate deprojection of
various three-dimensional quantities near the front.
\subsection{Velocities of gas flows}
\label{sec:vel}
As we mentioned above, in broad terms, thermal pressure of the gas inside
the cold front balances the sum of thermal and ram pressures of the gas
outside. Both components of thermal pressure, the gas density and
temperature, can be measured directly from the X-ray data, but not the
velocity in the plane of the sky. The difference of thermal pressures across
the front gives the ram pressure and thus the velocity of the gas cloud.
This method was first applied by V01 to the cold front in A3667 (Fig.\
\ref{2142_3667}).
For a quantitative estimate, we must consider a more exact physical picture,
schematically shown in Fig.\ \ref{p_M} (reproduced from V01). Panel {\em
a}\/ shows a uniform flow around a stationary blunt body of dense gas. The
flow forms a stagnation region at the tip of the body (zone 0), where the
velocity component along the axis of symmetry goes to zero. Note that
thermal pressure increases in the stagnation region as one moves closer to
the front, and is continuous across the front (unlike that across a shock).
The gas is compressed adiabatically, i.e., there will be a density and
temperature increase in the stagnation region compared to the values in the
flow (absent complications such as those discussed in \S\ref{sec:depl}
below). The ``outer gas pressure'' in the argument above is the pressure in
the free-stream region of the flow (zone 1), at a sufficient distance from
the front beyond the stagnation region ($\sim 0.5$ of the front's radius of
curvature for a transonic flow), or ahead of the shock front if $M>1$. In
practice, the stagnation region is small and difficult to detect because of
the line-of-sight projection, so a typical observed pressure profile derived
in wide radial bins across a moving cold front would exhibit a jump.
The ratio of thermal pressures at the stagnation point, $p_0$, and in the
free stream, $p_1$, is a function of the cloud speed (Landau \& Lifshitz
1959, \S114):
\begin{equation}
\label{eq:p:ratio:subsonic}
\frac{p_0}{p_1}=\left(1+\frac{\gamma-1}{2}\;M_1^2\right)^{
\frac{\scriptstyle\gamma}{\scriptstyle\gamma-1}},\quad M_1\le1
\end{equation}
\begin{equation}\label{eq:p:ratio:supersonic}
\frac{p_0}{p_1}=
\left(\frac{\gamma+1}{2}\right)^{
\frac{\scriptstyle\gamma+1}{\scriptstyle\gamma-1}}
M_1^2
\left(\gamma-\frac{\gamma-1}{2M_1^2}\right)^{
-\frac{\scriptstyle 1}{\scriptstyle\gamma-1}},
\quad M_1>1,
\end{equation}
where $M_1$ is the Mach number of the cloud relative to the sound speed in
the free stream region and $\gamma=5/3$ is the adiabatic index of the gas.
The subsonic equation~(\ref{eq:p:ratio:subsonic}) follows from Bernoulli's
equation. The supersonic equation~(\ref{eq:p:ratio:supersonic}) accounts
for the gas entropy jump at the bow shock. Figure \ref{p_M}{\em b}\/ shows
these ratios $p_0/p_1$ as a function of $M_1$.
The gas parameters at the stagnation point usually cannot be measured
directly, because the stagnation region is physically small and its X-ray
emission is strongly affected by projection. However, as we mentioned,
thermal pressure at the stagnation point equals thermal pressure within the
cloud, which is easily determined.
Because the cluster has a gradient of the gravitational potential, the gas
pressure increases toward the center of a cluster in hydrostatic
equilibrium, which is of course not included in eqs.\
(\ref{eq:p:ratio:subsonic}-\ref{eq:p:ratio:supersonic}). This change may
not be negligible on a distance between zones 1 and 0. Because most clusters
are reasonably centrally symmetric on large scales, one can usually correct
the free-stream pressure for this effect with sufficient accuracy by fitting
a centrally symmetric pressure model in a representative image area that
excludes the front and its disturbed vicinity, and evaluating it at the
radius of the front.
For the cold front in A3667, V01 obtained the ratio of the pressures
$p_0/p_1=2.1\pm0.5$ (horizontal dashed lines in Fig.\ \ref{p_M}), which
corresponds to $M_1=1.0\pm0.2$, i.e.\ the gas cloud moves at the sound speed
of the hotter gas. Evaluating the sound speed from the X-ray temperature,
the cloud velocity is $1400\pm300$ km$\;$s$^{-1}$. In another example, Machacek et
al.\ (2005) performed a similar analysis of the cold front at the boundary
of the galaxy NGC\,1404 and derived $M=0.8-1.0$, which corresponds to the
galaxy's velocity of $530-660$ km$\;$s$^{-1}$\ relative to the ambient gas in the
Fornax cluster. Mazzotta et al.\ (2003) obtained $M\approx 0.75\pm 0.2$ for
a prominent cold front in 2A\,0335+096, and O'Hara et al.\ (2004) obtained
$M\approx 1$ for a front in A2319, which are examples of clusters with
sloshing cool cores. In A1795, the pressures at two sides of the cold front
are equal, which corresponds to zero velocity (M01 and \S\ref{sec:slosh}
above). There are only a few observed mergers with $M>1$.
\subsection{Thermal conduction and diffusion across cold fronts}
\label{sec:cond}
Cold fronts are remarkably sharp, both in terms of the density and the
temperature jumps. Ettori \& Fabian (2000) first pointed out that the
observed temperature jumps in A2142 require thermal conduction across cold
fronts to be suppressed by a factor of order 100 compared to the collisional
Spitzer or saturated values. Furthermore, V01 have found that for A3667, the
gas density discontinuity at the cold front is several times narrower than
the electron mean free path with respect to Coulomb collisions. Figure
\ref{a3667_lambda} (an update of a similar plot in V01) shows a detailed
X-ray surface brightness profile across the tip of the front. The X-ray
brightness increases sharply within 2--3 kpc from the front position. We can
compare this width with the Coulomb mean free path of electrons (and
protons, $\lambda_e=\lambda_p$) in the plasma on both sides of the front.
The Coulomb scattering of particles traveling across the front can be
characterized by four different mean free paths: that of thermal particles
in the gas on each side of the front, $\lambda_{\rm in}$ and $\lambda_{\rm
out}$, and that of particles from one side of the front crossing into the
gas on the other side, $\lambda_{\rm in\rightarrow out}$ and $\lambda_{\rm
out\rightarrow in}$. From Spitzer (1962), we have for $\lambda_{\rm in}$
or $\lambda_{\rm out}$:
\begin{equation}
\label{eq:lambda:thermal}
\lambda = 15~{\rm kpc}\left(\frac{T}{7~{\rm keV}}\right)^2
\left(\frac{n_e}{10^{-3}~{\rm cm}^{-3}}\right)^{-1},
\end{equation}
and for $\lambda_{\rm in\rightarrow out}$ and $\lambda_{\rm out\rightarrow
in}$:
\begin{equation}
\label{eq:lambda:in:out}
\lambda_{\rm in\rightarrow out} = \lambda_{\rm out}\,
\frac{T_{\rm in}}{T_{\rm out}}\,
\frac{G(1)}{G\left(\sqrt{T_{\rm in}/T_{\rm out}}\right)}
\end{equation}
\begin{equation}
\label{eq:lambda:out:in}
\lambda_{\rm out\rightarrow in} = \lambda_{\rm in}\,
\frac{T_{\rm out}}{T_{\rm in}}\,
\frac{G(1)}{G\left(\sqrt{T_{\rm out}/T_{\rm in}}\right)},
\end{equation}
where $G(x)=[\Phi(x)-x\Phi'(x)]/2x^2$ and $\Phi(x)$ is the error function.
For the front in A3667, $\lambda_{\rm out}\approx 20-40$~kpc, $\lambda_{\rm
in}\approx 2$~kpc, $\lambda_{\rm in\rightarrow out}\approx 10-13$~kpc,
$\lambda_{\rm out\rightarrow in}\approx 4$~kpc. The upper bounds in the
above intervals correspond to the expected temperature increase in the
stagnation region (which is difficult to measure due to strong projection
effects), and the lower bounds correspond to no increase from the observed
outer temperature.
\begin{figure}[t]
\centering
\includegraphics[width=0.65\textwidth, bb=29 183 534 640]%
{a3667_briprof3a_lambda.ps}
\caption{X-ray surface brightness profile in a narrow sector near the tip of
the cold front in A3667 (whose image is shown in Fig.\ \ref{2142_3667}).
Blue line shows the best-fit projected spherical density discontinuity
with an infinitely small width, which describes the data well. Red line
shows a discontinuity smeared with a Gaussian with $\sigma=13$ kpc, which
corresponds to the collisional m.f.p.\ $\lambda_{\rm in\rightarrow out}$
(eq.\ \ref{eq:lambda:in:out}). It is ruled out by the data. The $r$\/
coordinate is from the center of curvature of the front. (This figure is
an update of a similar plot in V01, using a much longer \emph{Chandra}\
observation and a narrower sector.)}
\label{a3667_lambda}
\end{figure}
The hotter gas in the stagnation region has a low velocity relative to the
cold front. Therefore, diffusion, undisturbed by the gas motions, should
smear any density discontinuity by at least several mean free paths on a
very short time scale. Diffusion in our case is mostly from the inside of
the front to the outside, because the particle flux through the unit area is
proportional to $nT^{1/2}$. Thus, if Coulomb diffusion is not suppressed,
the front width should be at least several times $\lambda_{\rm in\rightarrow
out}$. Indeed, the time for $T=4$ keV protons to travel 10 kpc is $10^7$
yr, compared to the age of the structure of at least $R/v \approx 2\times
10^8$ yr, where $R$\/ and $v$\/ are the front radius and velocity. Such a
smearing is ruled out by the sharp rise in the X-ray brightness at the front
(Fig.\ \ref{a3667_lambda}). Fitting the observed surface brightness profile
by a projected abrupt density discontinuity smeared with a Gaussian, we
obtain a formal upper limit on the Gaussian $\sigma$ of 4 kpc. One should
remember that because the front is seen along the surface in projection, any
deviations from the ideal spherical shape would smear the edge, and yet the
observed front is sharper than the Coulomb m.f.p. This can be explained only
if the diffusion coefficient is suppressed by at least a factor of 3 with
respect to the Spitzer value. (This is a very conservative upper limit,
simply equal to the ratio of the Spitzer m.f.p.\ and our upper limit on the
front width; in fact, the front should spread by much more than one m.f.p.
during its presumed lifetime.)
The suppression of transport processes in the intergalactic medium is most
naturally explained by the presence of a magnetic field perpendicular to the
density or temperature gradient. Even a very small field is sufficient for
the electron and proton gyro radii to be many orders of magnitude smaller
than the Coulomb m.f.p. in the ICM, so electrons and ions would move mostly
along the field lines. The observation of a sharp density discontinuity in
A3667 is a first direct indication that such a suppression is possible in
the ICM (although this has, of course, been expected, since radio
observations have provided evidence for microgauss-level magnetic fields in
the ICM, see, e.g., Carilli \& Taylor 2002). In many other clusters, e.g.,
A2142, the front width is also unresolved by \emph{Chandra}, but the data quality
does not allow such accurate constraints.
The suppression of diffusion and collisional thermal conduction is most
effective if the magnetic field lines do not cross the front surface, that
is, the two sides of the front are magnetically isolated. Below we will see
that the cold front in A3667 provides another, indirect indication of just
such a field configuration, with field lines mostly parallel to the front
surface. We will also see why such a configuration should arise naturally in
a cluster merger.
\subsection{Stability of cold fronts}
Cold fronts are remarkably smooth in shape, considering that they form in a
violent merger environment. This property contains information on their
underlying dark matter distribution, as well as conditions in the ICM,
possibly including its viscosity, the prevalence of turbulence, and strength
and structure of the magnetic fields. These ICM properties can
significantly impact such diverse problems as the energy balance in the
cluster cooling flows and estimates of the cluster total masses.
\begin{figure}[t]
\centering
\includegraphics[width=0.5\textwidth]%
{jones96_fig7d.ps}
\caption{A simulation of the gas flow around an initially round dense gas
bullet without gravity or significant magnetic field support. Density
increases from dark to light gray; time in the two snapshots increases
from top to bottom. The bullet is a white blob in the top panel, preceded
by a shock front. The RT instabilities quickly develop and destroy the
bullet. (Reproduced from Jones, Ryu, \& Tregillis 1996.)}
\label{jones}
\end{figure}
\subsubsection{Rayleigh-Taylor instability and underlying mass}
\label{sec:RT}
When a dense gas cloud moves through a more rarefied medium, ram pressure of
the ambient flow slows it down. In the decelerating reference frame of the
cloud, there is an inertial force directed from the dense phase to the
less-dense phase, which makes the front interface of the cloud
Rayleigh-Taylor unstable. As a result, the cloud quickly disintegrates, as
observed in laboratory and numerical experiments, see Fig.\ \ref{jones}. If
a gas cloud is bound by gravity, it can prevent the onset of the RT
instability. It is interesting to see if, for example, the cold front in
A3667 is stable in this respect (V02). The drag force on the cloud is
$F_{\rm d}=C \rho_{\rm out} v^2 A/2$, where $\rho_{\rm out}$ and $v$\/ are
the gas density and velocity of the ambient flow, $A$\/ is the cloud
cross-section area, and $C$\/ is the drag coefficient. For the particular
geometry of the front in A3667 (a cylinder with a rounded head), $C\approx
0.4$. From the measured density inside the cloud, $\rho_{\rm in}$, we can
roughly estimate the mass of the cloud as a sphere with radius $r$,
$M\approx (4/3)\pi r^3 \rho_{\rm in}$. Then the drag acceleration is:
\begin{equation}
g_{\rm d}=\frac{F_{\rm d}}{M}
\approx 0.15\,\frac{\rho_{\rm out}}{\rho_{\rm in}}\,\frac{v^2}{r}.
\end{equation}
For the measured quantities in A3667, $g_{\rm d}\approx 8\times 10^{-10}$
cm~s$^{-2}$. One can also estimate the gravitational acceleration at the
front surface created by the gas mass inside the cloud, which points in the
opposite direction. It turns out to be $\sim 2$ times smaller, which means
that gravity of the gas itself is insufficient to suppress the RT
instability. Because the front is apparently stable, this means that there
has to be a massive underlying dark matter subcluster, centered inside the
front, that holds the gas cloud together. This is, of course, what we expect
in a merger. If the total mass of the underlying dark matter halo is the
usual factor of $\sim 10$ higher than the gas mass, its gravity at the front
surface would be more than sufficient to compensate for the drag force,
thereby removing the RT instability condition.
\begin{figure}[t]
\centering
\includegraphics[width=1\textwidth]%
{a3667_p_azim.ps}
\caption{({\em a}) Gas pressure under the surface of the cold front in A3667
as a function of angle from the symmetry axis. The line is a prediction
for a $M=1.05$ flow. ({\em b}) Schematic depiction of the front surface
and the gravitational potential $\Phi$\/ of the underlying dark matter
subcluster. The presence of an underlying massive subcluster is necessary
to prevent the development of Rayleigh-Taylor instability (Fig.\
\ref{jones}). The mass center should be offset from the center of
curvature of the front in order for the front to be in pressure
equilibrium with the ambient flow. (Reproduced from V02.)}
\label{a3667_p_azim}
\end{figure}
\paragraph*{Pressure along the front and the underlying dark matter halo}
The A3667 cold front also provides another indication of the existence of a
massive dark matter halo binding the gas. Following the Bernoulli law, the
pressure of the ambient flow at the surface of the front should have a
maximum at the stagnation point and decline as one moves along the front
surface away from the symmetry axis, as the shear velocity increases.
Figure \ref{a3667_p_azim}{\em a}\/ (from V02) shows the measured thermal
pressure just inside the surface of the front as a function of the angle
from the axis of symmetry.%
\footnote{The data used by V02 were consistent with an isothermal gas
inside the front, so the pressure in Fig.\ \ref{a3667_p_azim}{\em a}\/ was
derived assuming a constant temperature (i.e., what changes in the plot is
the gas density). A more recent \emph{XMM}\ observation uncovered spatial
temperature variations there (Briel et al.\ 2004), in particular, a cooler
spot at the tip of the front. This would reduce the peak pressure
somewhat; however, qualitatively and methodologically, the V02 result
still holds.}
It behaves as expected if the front surface was in pressure equilibrium with
the ambient flow. The expected ambient pressure for a $M\approx 1$ flow
(from a simple simulation) is also shown for comparison; it describes the
measured profile very well. This indicates that the front is stationary
(i.e., it is not an expanding shell, for example).
Furthermore, because the density is not constant inside the cool gas along
the front, there has to be an underlying mass concentration that supports
the resulting pressure gradient. Indeed, under the simplifying assumption
that the gas temperature is constant, the hydrostatic equilibrium equation
(\ref{eq:mass}) can be written as $\rho=\rho_0 \exp (-\mu m_p \Phi /kT)$,
where $\rho$\/ is the gas density and $\Phi$\/ is the gravitational
potential. The declining gas pressure along the front requires a
corresponding rise of the potential --- Fig.\ \ref{a3667_p_azim}{\em b}\/
schematically shows the required configuration.
In V02, we presented an estimate of what kind of mass concentration is
needed for the gas inside the cold front to be in hydrostatic equilibrium
and exhibit such a pressure profile at the front. For a spherical halo with
a King radial mass profile, the center of the halo should be located $\sim
70$ kpc under the front surface (compare this to the front radius of
curvature of 250 kpc), and a total mass within $r=70$ kpc should be
approximately $3\times 10^{12}$$M_{\odot}$. This is about 20 times higher that the
gas mass in the corresponding region. Thus, the dark halo and the gas inside
the cold front together look like a typical merging subcluster. The overall
picture of A3667 is quite similar to the Roettiger et al.\ (1998) simulated
merger shown in Fig.\ \ref{roettiger}, and to that in the cosmological
simulations by Nagai \& Kravtsov (2003).
Let us now take another look at the X-ray and lensing maps for \mbox{1E\,0657--56}\ (Figs.\
\ref{1e}{\em bc}\/ or \ref{1e_lens}). The ram pressure has just pushed the
gas bullet out of its host dark matter halo. The halo's gravity can no
longer prevent the RT instability, and the gas bullet is expected to fall
apart very quickly, although it has not happened yet (Fig.\
\ref{1e_a520}{\em a}). Indeed, another cluster, A520, exhibits a cool
subcluster at a slightly later stage, see Fig.\ \ref{1e_a520}{\em b}.
Similarly to \mbox{1E\,0657--56}, A520 is going through a supersonic merger in the plane of
the sky. Its mass map also reveals a small dark matter halo located ahead of
what remains of the gas subcluster (Okabe \& Umetsu 2007; Clowe et al.\ in
preparation). Apparently, A520 has already proceeded to a stage when the
instabilities have broken up the cool subcluster into several pieces
(Markevitch et al.\ in preparation).
\begin{figure}[t]
\centering
\includegraphics[width=1\textwidth, bb=50 488 489 703]%
{1e_a520.ps}
\caption{X-ray images of cool subclusters in \mbox{1E\,0657--56}\ and A520 (their
larger-scale images can be found in Figs.\ \ref{1e}{\em c}\/ and
\ref{a520_profs}{\em a}, respectively). The A520 image is smoothed;
small-scale fluctuations are photon noise. Both are experiencing mergers
in the plane of the sky, and both have cool gas subclusters stripped by
ram pressure from their associated mass halos, making them Rayleigh-Taylor
unstable. In \mbox{1E\,0657--56}, the gas bullet has a sharp cold front not yet disrupted
by instabilities, while in A520, the cool subcluster has already been
broken up.}
\label{1e_a520}
\end{figure}
\subsubsection{Kelvin-Helmholtz instability and magnetic field}
\label{sec:KH}
Vikhlinin, Markevitch, \& Murray (2001a) pointed out that
the shape of the cold front in A3667 provides an independent constraint of
the magnetic field at the location of the front. Their argument is that the
cold front in A3667 (Fig.\ \ref{2142_3667}) is sharp and has a smooth shape
within a certain sector ($\pm 30^\circ$) around the symmetry axis. At the
same time, given the measured subcluster velocity, the front should be
quickly disturbed by the Kelvin-Helmholtz (KH) instability. It would be
observed in projection as smearing of the sharp edge on a width scale
comparable to the wavelength of the mode that has reached the nonlinear
growth stage. Thus, within that sector, the KH instability appears to be
suppressed, at least for perturbations with $\lambda$ greater than the
observed upper limit on the width of the front there.
The gravity of the subcluster can in principle suppress the KH instability,
just as it does the RT instability (\S\ref{sec:RT}). However, an estimate
shows that it is far too small (V02). Assuming the density discontinuity is
sharp (to which we will return below), the next most natural stabilizing
mechanism is the formation of a layer of magnetic field parallel to the
front. Such a layer would provide surface tension and make it difficult for
any deformations of the surface to grow (Fig.\ \ref{a3667_magn}). As we will
discuss below, a layer of increased magnetic field parallel to the surface
is indeed expected to emerge as a result of ``magnetic draping'', i.e.,
stretching of the field in the ambient ICM as it flows around the cold front
(e.g., Asai et al.\ 2005; Lyutikov 2006).
Because the shear velocity of the flow increases as one moves along the
front away from the stagnation point, the surface tension of a magnetic
layer at a certain angle ($\varphi_{\rm cr}$ in Fig.\ \ref{a3667_magn}) may
become insufficient, and the KH instability starts to grow. Thus, the
extent of the undisturbed sector of a cold front $\varphi_{\rm cr}$ may be
used to derive a lower limit on the strength of the stabilizing magnetic
field. Detailed calculations for a front in A3667 can be found in Vikhlinin
et al.\ (2001a).%
\footnote{A minor algebraic error in that paper was pointed out by P.
Mazzotta and corrected in V02, which did not change the result.}
Denoting the magnetic field strengths on the hot and cold sides of the front
as $B_h$ and $B_c$, the respective gas temperatures as $T_h$ and $T_c$, the
gas pressure at the front as $p_{\rm gas}$, and the Mach number of the local
shear flow as $M$, the KH instability is suppressed when
\begin{equation}
\label{eq:kh:mag:P:condition}
\frac{B_h^2}{8\pi}+\frac{B_c^2}{8\pi}>\frac{1}{2}\,\frac{\gamma
M^2}{1+T_c/T_h}\, p_{\text{gas}}.
\end{equation}
For the observed temperatures and flow velocities, the observed stability of
the front within the sector $\varphi<30^\circ$ (where $M\le0.55$), and
taking into account the uncertainties, this gives a lower limit on the sum
of magnetic pressures in the two gas phases:
\begin{equation}
\label{eq:pmag/pgas}
\frac{B_h^2}{8\pi}+\frac{B_c^2}{8\pi} > (0.1-0.2) p_{\text{gas}}.
\end{equation}
This gives $B> (7-16)\,\mu G$ for the maximum of the two quantities $B_h$
and $B_c$. If the apparent smearing of the front beyond the stable sector is
interpreted as the onset of the KH instability, from a lower limit this
becomes an estimate of the magnetic field. However, there may be other
mechanisms disturbing the front at large $\varphi$, so it is best to
consider it a lower limit.
\begin{figure}[t]
\centering
\includegraphics[width=0.5\textwidth, bb=411 421 653 663,clip]%
{a3667_magn.ps}
\caption{A schematic illustration of the suppression of Kelvin-Helmholtz
instability at the surface of the A3667 cold front. The magnetic layer
(shown by parallel curves along the front) can provide surface tension
that suppresses the growth of perturbations in the region where the
tangential velocity is smaller than some critical value $V_{\rm cr}$. The
velocity field (arrows) corresponds to the flow of incompressible
fluid around a sphere. (Reproduced from Vikhlinin et al.\ 2001a.)}
\label{a3667_magn}
\end{figure}
Looking at eq.\ (\ref{eq:pmag/pgas}), it is interesting to realize that even
though a magnetic field is far from being ``dynamically important'' in the
usual sense --- it does not dominate the total pressure in the ICM (and thus
will not significantly affect the cluster total mass estimates, for
example), it can still be strong enough to qualitatively alter the evolution
of cold fronts, mixing of gases, and perhaps the development of turbulence.
A more rigorous analysis of the growth of KH instability in the absence of a
magnetic layer was presented by Churazov \& Inogamov (2004). Their growth
factor estimate for a sharp discontinuity differed from a simplified
estimate in Vikhlinin et al.\ (2001a), but it did not change the conclusion
that such a front would be unstable. More interestingly, they pointed out
that the KH instability will not develop if the density discontinuity had a
finite intrinsic width, for example, because of diffusion. This would
completely suppress the growth of perturbations with wavelengths shorter
than a certain fraction of the front radius, dependent on the smearing
width. An upper limit on the width of the front from the current data
(\S\ref{sec:cond}) does not rule out the Churazov \& Inogamov scenario
(although it comes close). We note, however, that the observed limit on the
front width is already smaller by a factor of several than the collisional
m.f.p., so the natural candidate to create such a stabilizing layer,
collisional diffusion, does not work. A different physical process would
have to widen the front within a factor of 2--3 of the current upper limit,
which would require a coincidence.
While the above analyses assume a spherical shape for the front, a recent,
much deeper \emph{Chandra}\ exposure of A3667 (Fig.\ \ref{2142_3667}) shows that
it is not exactly spherical. This would modify the flow pattern, and so the
above instability suppression calculations are, of course, only qualitative.
\begin{figure}[t]
\centering
\includegraphics[width=\textwidth]%
{asai_1.ps}
\caption{Simulation of the magnetic field draping around a cold front in the
course of a gas cloud's motion through an ICM with a uniform magnetic
field. ({\em a}) The subcluster gas density (color) and the magnetic
field lines. ({\em b}) A horizontal cross-section through the subcluster:
color shows magnetic field strength, arrows show gas velocities.
Compression and shear of the field in the incoming flow creates a narrow
layer around the front, in which the magnetic field is strongly amplified.
(Reproduced from Asai et al.\ 2005.)}
\label{asai}
\end{figure}
\subsubsection{Origin of magnetic layer}
\label{sec:drap}
Assuming the KH instability is indeed suppressed by the surface tension of a
magnetic layer, the lower limit on the field in this layer obtained above is
significantly higher than the $B\sim 1\,\mu G$ estimates typically given for
the cluster regions outside cooling flows (e.g., Carilli \& Taylor 2002).
This is not unexpected, because a moving cold front is a special place,
where the field should align with the surface and significantly strengthen
via ``draping'' around an obstacle, a phenomenon originally proposed by
Alfv\'en (1957) to explain the comet tails. It is illustrated in Fig.\
\ref{asai} (from Asai et al.\ 2005). Panel ({\em a}) shows a simulation of
the magnetic field lines draping around a gas cloud as the cloud moves
through an ICM with a frozen-in magnetic field. These simulations used an
ordered field; however, even if the field is tangled on scales less than the
size of the cold front (as expected in clusters), the loops will stretch and
the field at the front will consist of large two-dimensional patches
(perhaps disconnected), which will have the same qualitative effect on the
instabilities. Panel ({\em b}) shows the increase of the field strength at
the front as a result of this shear and compression. As pointed out by
Lyutikov (2006), the field can reach equipartition with the gas thermal
pressure, i.e., the ratio of thermal to magnetic pressures $\beta\approx 1$,
in a narrow layer along the surface of the front (recall that in most of the
ICM, $\beta\sim 10^2-10^3$). This effect has long been known to space
physicists --- it is observed by space probes at locations where the solar
wind flows around the Earth magnetosphere (at magnetopause) and around the
atmospheres of Mars and Venus. Similarly to the ICM, outside of such
special locations, solar wind has $\beta>1$, so the analogy is physically
meaningful. Such a magnetic layer would be more than sufficient to suppress
KH instabilities and completely suppress thermal conduction and diffusion
across any moving cold front.
\begin{figure}[t]
\centering
\includegraphics[width=0.8\textwidth]%
{oieroset.ps}
\caption{Magnetic draping observed in the solar wind as it flows around the
Earth's magnetosphere. Profiles of the field strength and plasma density
between the Earth's bow shock (vertical dashed line marked BS) and
magnetopause (MP; an analog of a cluster cold front) as measured by a
satellite. Draping results in an increase of the field strength and
formation of a ``plasma depletion layer'' (PDL) near the magnetopause.
(Reproduced from {\O}ieroset et al.\ 2004.)}
\label{oieroset}
\end{figure}
\subsection{Possible future measurements using cold fronts}
\label{sec:futurecf}
\subsubsection{Plasma depletion layer and magnetic field}
\label{sec:depl}
There are several interesting tests that are yet to be done using cold
fronts. First, apart from stabilizing the fronts, magnetic draping
mentioned in the previous section should have another directly observable
effect. When magnetic pressure near the front reaches equipartition, the
total (thermal plus magnetic) pressure should still satisfy the constraints
imposed by the hydrodynamics of the flow. The result is that some of the
plasma is squeezed out of the narrow layer near the front. The width of such
a layer, $\Delta r$, is estimated to be
\begin{equation}
\frac{\Delta r}{R}\approx \frac{1}{M} \frac{p_{\text{gas}}}{p_B},
\label{eq:drap}
\end{equation}
where $R$\/ is the radius of the front, $M$\/ is the Mach number of the
cloud, and $p_{\text{gas}}$ and $p_B$ are thermal and magnetic pressures in
the undisturbed gas ahead of the front (Lyutikov 2006). For cluster-sized
transonic cold fronts and $1\;\mu G$ fields, $\Delta r\sim 1$ kpc. Such
``plasma depletion layers'' have indeed been observed from the satellites in
the corresponding locations of the solar wind (e.g., Zwan \& Wolf 1976;
{\O}ieroset et al.\ 2004; see Fig.\ \ref{oieroset} reproduced from the
latter work). If such a dip is found in the X-ray brightness profile of a
cluster cold front, its width could be used for an independent estimate of
the magnetic field strength in the {\em undisturbed}\/ cluster gas.
We note that this effect will compete with the density increase in the
stagnation region of the flow (\S\ref{sec:vel}). There is also a
possibility of a trivial geometric ``depletion'' such as that seen, e.g., in
the X-ray brightness profile of NGC\,1404. The gas temperature inside that
cold front drops well below 1 keV (Fig.\ \ref{n1404}; Machacek et al.\
2005). Even though the dense gas is present, it does not emit much at $E>1$
keV. Combined with projection, this creates a gap along the front in the
\emph{Chandra}\ wide-band X-ray image (curious readers can extract it from the
\emph{Chandra}\ data archive). Thus, detecting a plasma depletion layer will
require accurate modeling.
\subsubsection{Effective viscosity of intracluster plasma}
Another interesting physical quantity that may be constrained using cluster
cold fronts is the effective viscosity of the ICM. For example, this unknown
property of the ICM has recently attracted the attention of those studying
the cluster cooling flows as a way to transfer the mechanical energy of AGN
explosions into heat (e.g., Fabian et al.\ 2005). As we have already
mentioned, the observed cold fronts appear very smooth and undisturbed by
turbulence. A particularly interesting example is the bullet subcluster in
\mbox{1E\,0657--56}\ (Fig.\ \ref{1e_a520}), which exhibits long and straight ``wisps'' at
its sides, where one would expect strong turbulence. There are several
mechanisms that would prevent turbulence from developing; collisional
viscosity is one. Even if it is suppressed (e.g., for similar reasons why
thermal conduction is suppressed), there may be other kinds of viscosity
specific to a magnetized plasma. Another plausible mechanism is a
stabilizing magnetic layer at the cold front interface, such as the one
discussed above. Such magnetic structures should not be rare; in fact, the
plasma may be filled with them, because any flow in the ICM would create
velocity shear and stretch the magnetic field lines around each moving gas
parcel. Their net damping effect on gas mixing and turbulence may be
similar to an effective viscosity. Hydrodynamic simulations that explicitly
include viscosity (e.g., Sijacki \& Springel 2006) would be required to
derive any quantitative constraints.
For an order of magnitude estimate of what one can expect, we can look at
\mbox{1E\,0657--56}. To prevent the development of turbulence around the gas bullet, the
Reynolds number of the gas flow should be of order 10 or lower (e.g., Landau
\& Lifshitz 1959, \S26). One can use a gasdynamic expression $\text{Re}\sim
ML/\lambda$, where $M$\/ is the Mach number of the flow of gas around the
bullet, $L$\/ is the size the bullet, and $\lambda$ is the m.f.p.\ of the
gas particles that determines viscosity. The condition $\text{Re}\lesssim 10$
gives $\lambda$ in the several kpc range. Interestingly, this is comparable
to the Spitzer m.f.p.\ in the plasma around the bullet. Thus, the effective
viscosity, whatever its physical nature, may be of the order of the
collisional Spitzer viscosity. Note that it may also be possible to derive
upper limits on viscosity, for example, from the observed RT instability in
A520 (\S\ref{sec:RT}), and perpahs from the presence of turbulence necessary
to explain the cluster radio halos (\S\ref{sec:halos}).
\section{SHOCK FRONTS AS EXPERIMENTAL TOOL}
We now turn to the ICM density discontinuities of another physical nature
which we always expected to find in clusters --- shock fronts. There are
three types of phenomena that create shocks in the ICM. As we already
mentioned, in the cluster central regions, powerful AGNs often blow bubbles
in the ICM, which may generate shocks within the central few hundred kpc
regions (e.g., Jones et al.\ 2002; McNamara et al.\ 2005; Nulsen et al.\
2005; Fabian et al.\ 2006; Forman et al.\ 2006). These shocks have $M\sim
1$. It is difficult to derive accurate temperature and density profiles for
such low-contrast shocks because of large corrections for the projected
emission, so they are poorly suited for our purpose of using shocks as a
diagnostic tool.
At very large off-center distances (several Mpc), cosmological simulations
predict that intergalactic medium (IGM) should continue to accrete onto the
clusters though a system of shocks that separate the IGM from the hot,
mostly virialized inner regions. The IGM is much cooler than the ICM, so
these accretion (or infall) shocks should be strong, with $M\sim 10-100$
(e.g., Miniati et al.\ 2000; Ryu et al.\ 2003). As such, they are likely to
be the sites of effective cosmic ray acceleration, with consequences for the
cluster energy budget and the cosmic $\gamma$-ray background (see, e.g.,
Blasi, Gabici, \& Brunetti 2007 for a review). However, these shocks have
never been observed in X-rays or at any other wavelengths, and may not be in
the foreseeable future, because they are located in regions with very low
X-ray surface brightness.
If an infalling subcluster has a deep enough gravitational potential to
retain at least some of its gas when it enters the dense, X-ray bright
region of the cluster into which it is falling, we may observe a spectacular
merger shock, such as the one in the X-ray image of \mbox{1E\,0657--56}\ (Fig.\ \ref{1e}{\em
c}). Such shocks can be used as tools for a number of interesting studies,
which will be the subject of the rest of this review.
\subsection{Cluster merger shocks}
\label{sec:mergshocks}
Merger shocks have been predicted in the earliest hydrodynamic simulations
of cluster mergers (e.g., Schindler \& M\"uller 1993; Roettiger, Burns, \&
Loken 1993; Burns 1998). They have relatively low Mach numbers, $M\lesssim 3$
(e.g., Gabici \& Blasi 2003; Ryu et al.\ 2003), simply because the sound
speed in the gas of the main (bigger) cluster and the velocity of the
infalling subcluster both reflect the same gravitational potential of the
main cluster. Indeed, the central depth of the King potential is
$\Phi_0=-9\sigma_r^2$, where $\sigma_r$ is the radial velocity dispersion of
galaxies in this potential, which in turn is close to the average velocity
of particles in the intracluster gas in hydrostatic equilibrium (e.g.,
Sarazin 1988). So an infalling test particle, or a small subcluster, should
acquire $M\sim 3$ at the center. In practice, for a merger of clusters with
comparable masses (what is usually called a ``major merger''), it is
unlikely to observe a shock with $M$\/ much greater than 1. Such a merger
would generate multiple successive shocks, each one significantly preheating
the gas of the whole combined system, thereby reducing the Mach numbed for
any subsequent shock.
Simulations also indicate that it is unlikely to find a shock front during
the inbound leg of the infalling subcluster's trajectory, because the front
must climb up the steep gas density gradient of the main cluster. At the
merger stage shown in top panel of Fig.\ \ref{a2142_scheme}, shock fronts
may slow down and disappear while ascending the density peaks, if the peaks
are high enough. At the same time, regions of the front away from the
symmetry axis will continue to propagate past the density peak, and may
eventually form a continuous surface behind it.
Regions of high-entropy gas in clusters have been observed with
lower-resolution X-ray telescopes such as \emph{ROSAT}, \emph{ASCA}\ and \emph{XMM}\ and
interpreted as the result of shock heating (e.g., Henry \& Briel 1995, 1996;
Markevitch, Sarazin, \& Irwin 1996b; Markevitch et al.\ 1999a; Belsole et
al.\ 2003, 2004). Shock-heated regions are also routinely found by
\emph{Chandra}\ (e.g., Markevitch \& Vikhlinin 2001; Markevitch et al.\ 2003a;
Kempner \& David 2004; G04). However, as of this writing, only two shock
{\em fronts}, exhibiting both a sharp gas density edge and an unambiguous
temperature jump, were found by \emph{Chandra}, those in \mbox{1E\,0657--56}\ (Fig.\
\ref{1e_profs}; M02; Markevitch 2006, hereafter M06) and A520 (Fig.\
\ref{a520_profs}; Markevitch et al.\ 2005, hereafter M05). Such finds are
so rare because one has to catch a merger when the shock has not yet moved
to the low-brightness outskirts, and is propagating nearly in the plane of
the sky, to give us a clear view of the shock discontinuity. In addition,
the shocks in A520 and \mbox{1E\,0657--56}\ have $M=2-3$, which provides a big enough gas
density jump to enable accurate deprojection. Merger shock fronts may have
been found in a couple of other clusters (e.g., in A3667, V01; A754,
Krivonos et al.\ 2003; Henry, Finoguenov, \& Briel 2004), but temperature
data for them either do not exist or are uncertain. Below, we will discuss
what can be learned from the fronts in \mbox{1E\,0657--56}\ and A520.
\begin{figure}
\centering
\includegraphics[width=0.63\textwidth]%
{1e_profs.ps}
\caption{Radial profiles of the \mbox{1E\,0657--56}\ X-ray brightness ({\em a}) and projected
temperature ({\em b}) in a narrow sector crossing the tip of the bullet
(the first big drop of the brightness) and the shock front (the second big
drop). The $r$\/ coordinate in panel {\em a}\/ is measured from the
shock's center of curvature; the $x$\/ coordinate in panel {\em b} is
measured from the shock surface. Red line in panel {\em a}\/ shows the
best-fit model for the shock jump (a projected sharp spherical density
discontinuity by a factor of 3). Vertical dotted lines in panel {\em b}\/
show the boundaries of the cool bullet and the shock; dashed line shows
the average pre-shock temperature. There is a subtle additional edge
between the bullet and the shock; the gas temperature inside it is lower
(the lower two crosses between the vertical lines). That region is not
used for any shock models. Error bars are 68\%. (Reproduced from M06.)}
\label{1e_profs}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[width=\linewidth,bb=70 274 507 733]%
{a520_profs.ps}
\caption{Shock front in A520. ({\em a}) A \emph{Chandra}\ image
(slightly smoothed) with point sources removed. The bow shock is a faint
blue edge southwest of the bright irregular remnant of a dense core. White
dashed lines mark a sector used for radial X-ray brightness and projected
temperature profiles across the shock (panels {\em b,c}). The profiles are
extracted excluding the core remnant. ({\em d}) A three-dimensional model
fit to the brightness profile; its projection is shown as a red line in
panel ({\em c}). (Reproduced from M05.)}
\label{a520_profs}
\end{figure}
\subsection{Mach number determination}
\label{sec:1e_M}
Let us now look at the profiles of the X-ray brightness and projected gas
temperature derived in a narrow sector crossing the bullet nose and the
shock front in \mbox{1E\,0657--56}. They are shown in Fig.\ \ref{1e_profs} (M06). From these
data, we can derive a Mach number and velocity of the shock, which should
also give the approximate velocity of the subcluster. In Fig.\
\ref{1e_densprof}, we showed an approximate gas density model with two
abrupt jumps, which in projection describes this brightness profile. (This
fit corresponds to an early subset of the data shown in Fig.\
\ref{1e_profs}. Note also the different reference points for the $r$\/
coordinate used in Figs.\ \ref{1e_densprof} and \ref{1e_profs}.) The inner
brightness edge in Fig.\ \ref{1e_profs}{\em a}\/ is the bullet; its boundary
at $r=65''$ is a cold front, as seen from the temperature jump in panel
({\em b}) at $x=-30''$. The edge at $r=90''$ is the shock front, as
confirmed by the temperature jump of the right sign. There is another
subtle brightness edge between these main features (Fig.\ \ref{1e_profs}).
It is unrelated to the shock, so we will exclude it when deprojecting the
shock temperatures and densities.
We can determine the density discontinuity across the shock just as we did
for the cold fronts, by fitting the brightness profile with a model with an
abrupt spherical density jump (\S\ref{sec:merg}). The best-fit model (red
line in Fig.\ \ref{1e_profs}{\em a}) has a density jump by a factor of 3,
which includes a small correction for the observed gas temperature change
across the front.
The Rankine-Hugoniot jump conditions (e.g., Landau \& Lifshitz 1959, \S89)
relate the density jump at the shock, $r$, and the Mach number of the shock,
$M\equiv v/c_s$, where $c_s$ is the velocity of sound in the pre-shock gas
and $v$\/ is the velocity of that gas w.r.t.\ the shock surface:
\begin{equation}
M=\left[\frac{2r}{\gamma+1-r(\gamma-1)}\right]^{1/2}.
\label{eq:M}
\end{equation}
Using this equation, the above density jump gives a Mach number $M=3.0\pm
0.4$.
Note that $M$\/ can be derived {\em independently}\/ using the temperature
jump (using eq.\ \ref{eq:r} and the above equation), provided that the
effective adiabatic index $\gamma$ is known (i.e., no dominant relativistic
particle component, significant energy leakage into particle acceleration,
etc.) and that there is temperature equilibrium between electrons and ions
(the latter assumption is needed only for the estimate based on the
temperature jump). We will see in the next section that the observed
temperature jump in \mbox{1E\,0657--56}\ is in good agreement with the value of $M$\/ from
the density jump. A similar analysis can be performed for the shock in A520
(Fig.\ \ref{a520_profs}), where a density jump by a factor of 2.3 gives
$M=2.1^{+0.4}_{-0.3}$, again consistent with the temperature jump (M05). For
relatively weak shocks such as these, the accuracy with which we can
determine $M$\/ from the X-ray derived density discontinuity is better than
what we can do from the temperatures. However, as $M$\/ increases, the
density jump asymptotically goes to a finite value ($r=4$ for $\gamma=5/3$),
while the temperature jump continues to grow, so the situation reverses.
Sound speed in the pre-shock gas can be determined from its observed
temperature. The above Mach numbers then give the shock velocities of 4700
km$\;$s$^{-1}$\ and 2300 km$\;$s$^{-1}$\ for \mbox{1E\,0657--56}\ and A520, respectively. From the continuity
condition, the respective velocities of the post-shock gas w.r.t.\ the shock
surface are $r$\/ times lower, 1600 km$\;$s$^{-1}$\ and 1000 km$\;$s$^{-1}$. For \mbox{1E\,0657--56}, the
distance between the centers of merging subclusters is 720 kpc (see the weak
lensing map in Fig.\ \ref{1e_lens}). Assuming that the bullet subcluster's
velocity is close to the shock velocity (although it can be lower, Springel
\& Farrar 2007; Milosavljevic et al.\ 2007), the two subclusters have passed
through each other just 0.15 Gyr ago.
\subsection{Front width}
\label{sec:shockwidth}
The X-ray brightness profile in Fig.\ \ref{1e_profs} does not exclude a
shock front of a finite width; in fact, it marginally prefers a density jump
widened by a Gaussian with $\sigma\approx 8''$ (35 kpc). Curiously, this is
of the order of the collisional m.f.p.\ in the gas around the shock --- but
it may simply be a coincidence, since, e.g., any deformations of the front
shape seen in projection would smear the edge. The best-fit amplitude of
the jump that was used above to derive the Mach numbers does not depend
noticeably on this width.
In this regard, it is interesting to look at simulations by Heinz \&
Churazov (2005), who studied propagation of a shock in an ICM filled with
bubbles of relativistic plasma. Such a mixture in pressure equilibrium may
form as a result of AGN activity over the cluster lifetime; cluster radio
relics may be examples of such regions containing fossil relativistic plasma
(e.g., En{\ss}lin \& Gopal-Krishna 2001). Because the sound speed inside
such a bubble is very high, a shock front surface will be deformed on a
linear scale of these bubbles. Heinz \& Churazov proposed such smearing as
an explanation for the lack of visible strong AGN jet-driven shocks in the
cluster centers. The fact that the fronts in \mbox{1E\,0657--56}\ and A520 can be seen at
all in the X-ray images indicates that, at least in these clusters, the bulk
of the ICM is not filled with fossil bubbles greater than several tens of
kpc in size. On the other hand, there is an example of A665, in which the
overall X-ray morphology and the temperature map suggest that there should
be a shock front, and there is a gas density excess, but no density
discontinuity is seen (Markevitch \& Vikhlinin 2001; G04). It might be that
an observable sharp discontinuity did not form there because the shock has
to propagate in such a mixture.
\subsection{Mach cone and reverse shock?}
Before proceeding to some of the interesting shock-based measurements, we
would like to address two questions that people familiar with shocks in
other astrophysical contexts frequently have: the Mach cone%
\footnote{The Mach cone discussion here differs from the version published
in {\em Physics Reports}\/ --- it is updated with results from simulations
by Springel \& Farrar (2007).}
and the reverse shock.
A small body moving supersonically creates a Mach cone with an opening angle
$\sin\varphi = M^{-1}$. For $M=3$, the shock in \mbox{1E\,0657--56}\ should have an
asymptotic angle of 20$^{\circ}$\ from the symmetry axis. However, the image
(Fig.\ \ref{1e}{\em c}) shows a more widely open ``Mach cone''. A possible
reason is easy to understand if we consider that the bullet is not a solid
body, but a gas cloud whose outer, less dense gas is being continuously
stripped by the flow of the shocked gas. The tip of the bullet is becoming
smaller with time, as shown schematically in Fig.\ \ref{slosh_scheme} (see
\S\ref{sec:discont}). The off-axis parts of the shock front that we see at
present are not driven by the bullet that we see at present, but by a bigger
bullet that has existed just a short while ago. Thus, the shape of the front
(at least of its bright region which we can follow in the image) does not
correspond to a Mach cone, but rather reflects the change of the bullet size
and velocity with time. Another significant effect is the inflow of the
pre-shock gas caused by the gravity of the subcluster (Springel \& Farrar
2007). This inflow is faster at the nose of the shock (which currently
coincides with the subcluster's mass peak, see Figs.\ \ref{1e}{\em bc}),
which flattens the front shape.
We can test this by deriving an X-ray brightness profile across an off-axis
stretch of the shock front, e.g., in a narrow sector pointing 30$^{\circ}$\
clockwise from the bullet velocity direction. For a stationary oblique shock
with such an angle to the uniform upstream flow, the density jump should be
reduced from 3.0 at the nose to 2.8 (e.g., Landau \& Lifshitz 1959, \S92).
However, the observed jump is a factor of 2.1, which corresponds to a
smaller $M$. This is consistent with a higher inflow velocity of the
pre-shock gas at the nose of the shock.
In addition, the main cluster's radially declining density profile, in which
the shock propagates, and deceleration of the bullet by gravity and ram
pressure should also affect the shape of the front (M02).
Those working with supernova remnants (SNR) may also ask where is the
reverse shock in \mbox{1E\,0657--56}\ --- a front that propagates inwards from the outer
surface of an SNR. If we look at the scheme in Fig.\ \ref{a2142_scheme}, it
is clear that the reverse shock for shock 1 is shock 2, and vice versa. The
current stage of the merger in \mbox{1E\,0657--56}\ corresponds to the bottom panel in Fig.\
\ref{a2142_scheme}. By this time, the reverse shock has attempted to climb
the sharp density peak of the bullet subcluster, but failed to penetrate
inside the radius where we now have a cold front. As mentioned in
\S\ref{sec:mergshocks}, the outer regions of that shock must have moved past
the bullet and away from the picture; from the symmetry, it should be
somewhere on the opposite side from the cluster center.
\subsection{Test of electron-ion equilibrium}
\label{sec:tei}
The post-shock temperature that enters the Rankine-Hugoniot jump conditions
is the temperature that all plasma particle species acquire when they reach
equilibrium after the shock passage. In a collisional plasma, protons, whose
thermal velocity is lower than the shock velocity, are heated dissipatively
at the shock layer that has a width of order the collisional m.f.p. The
faster-moving electrons do not feel the shock (for $M\ll
(m_p/m_e)^{1/2}=43$) and are compressed adiabatically. Subsequently,
electrons and protons equilibrate via Coulomb collisions on a timescale
(e.g., Zeldovich \& Raizer 1967)
\begin{equation}
\tau_{\rm c}= 2\times 10^8\;{\rm yr}
\left(\frac{n_e}{10^{-3}\;{\rm cm}^{-3}}\right)^{-1}
\left(\frac{T_e}{10^8\,{\rm K}}\right)^{3/2}.
\label{eq:tei}
\end{equation}
For a shock in a magnetized plasma such as the ICM, the final post-shock
temperature should be the same (provided that the kinetic energy does not
leak into cosmic rays, etc.), but the shock structure can be very different.
Indeed, studies of solar wind shocks in-situ showed that the electron and
proton temperature jump occurs on a linear scale of order several proton
gyroradii, many orders of magnitude smaller than their collisional m.f.p.
(Montgomery, Asbridge, \& Bame 1970 and later works), which is why these
shocks are called collisionless. Therefore, it would not be surprising to
find a different rate of electron heating at the shock, and a shorter
electron-proton equilibration timescale. (To find a longer timescale would
be surprising, because Coulomb collisions do occur even in the
``collisionless'' plasma.)
The bow shock in \mbox{1E\,0657--56}\ offers a unique experimental setup to determine how
long it takes for post-shock electrons to come to thermal equilibrium with
protons in a magnetized plasma (M06). Given the likely turbulent widening of
the heavy ion emission lines (and, of course, for lack of the required
energy resolution at present), we cannot directly measure $T_i$ in X-rays,
only $T_e$. However, we can use the accurately measured gas density jump at
the front (\S\ref{sec:1e_M}) and the pre-shock electron temperature to
predict the post-shock adiabatic and instant-equilibration electron
temperatures, using the adiabatic and the Rankine-Hugoniot jump conditions,
respectively, and compare them with the data. Furthermore, we also know the
downstream velocity of the shocked gas flowing away from the shock (1600
km$\;$s$^{-1}$, \S\ref{sec:1e_M}). This flow effectively spreads out the time
dependence of the electron temperature along the spatial coordinate in the
plane of the sky. The Mach number of the \mbox{1E\,0657--56}\ shock is conveniently high,
such that the adiabatic and shock electron temperatures are sufficiently
different (for $M\lesssim 2$, they become close and difficult to distinguish,
given the temperature uncertainties). It is also not a strong shock, for
which the density jump would just be a factor of 4 and would not let us
directly determine $M$. Furthermore, the distance traveled by the
post-shock gas during the time given by eq.\ (\ref{eq:tei}), $\Delta x\simeq
230$ kpc $=50''$, is well-resolved by \emph{Chandra}.
\begin{figure}[t]
\centering
\includegraphics[width=0.65\linewidth,bb=42 186 548 685,clip]%
{1e_tei_bw.ps}
\caption{Electron-ion equilibrium at shock in \mbox{1E\,0657--56}.
Deprojected electron temperatures for the two outer post-shock bins of the
temperature profile from Fig.\ \ref{1e_profs}, overlaid on the model
predictions (with error bands) for instant equilibration (labeled
``shock'', light gray) and adiabatic compression followed by collisional
equilibration (dark gray). The velocity shown is for the post-shock gas
relative to the shock. Error bars are 68\%. (Reproduced from M06.)}
\label{1e_tei}
\end{figure}
As we already mentioned, there is a subtle secondary brightness edge between
the bullet and the shock, behind which the temperature decreases (Fig.\
\ref{1e_profs}). It is most likely caused by residual cool gas from the
subcluster in one form or another, and is unrelated to the shock.
Therefore, we can use only the temperature profile between the shock and this
edge, where we have two bins. Using the gas density profile in front of the
shock, we can subtract the contributions of the cooler pre-shock gas
projected onto these post-shock bins, assuming spherical symmetry in this
small segment of the cluster. The shock brightness contrast is high, which
makes this subtraction sufficiently accurate. In Fig.\ \ref{1e_tei}, the
deprojected temperatures are overlaid on the two models: instant
equilibration and adiabatic compression with subsequent equilibration on a
timescale given by eq.\ (\ref{eq:tei}). (The plot assumes a constant
post-shock gas velocity, which of course is not correct, but we are only
interested in the immediate shock vicinity.) The deprojected post-shock gas
temperatures are so high compared with the \emph{Chandra}\ energy band that only
their lower limits are meaningful. The temperatures are consistent with
instant heating; equilibration on the collisional timescale is excluded,
although with a relatively low 95\% confidence. The equilibration timescale
should be at least 5 times shorter than $\tau_{\rm c}$.
A few sanity checks have been performed in M06. In particular, the
possibility of a non-thermal contamination of the spectra was considered.
\mbox{1E\,0657--56}\ has a radio halo (Fig.\ \ref{1e}{\em d}; Liang et al.\ 2000), which has
an edge right at the shock front (M02; see below). Therefore, there may be
an IC contribution from relativistic electrons accelerated at the shock.
However, in the \emph{Chandra}\ energy band, the power-law spectrum of such
emission for any $M$\/ would be {\em softer}\/ than thermal (which for these
temperatures has a flat effective photon index of $-1.4$), so it cannot bias
our temperature measurements high. It is unfortunate that a cluster with
such a perfect geometric setup and $M$\/ is so hot that \emph{Chandra}\ can barely
measure the post-shock temperatures; however, there are not many of them to
choose from. The available A520 data do not have the accuracy for such a
measurement.
\begin{figure}[t]
\centering
\includegraphics[width=0.8\textwidth,bb=48 157 559 456]%
{hull.ps}
\caption{Structure of the Earth's bow shock observed in-situ from a
satellite. Top two panels show proton (solid lines) and electron (dashed
lines) density (in units of cm$^{-3}$) and temperature ($10^4$ K)
profiles. Lower panel shows temperature components parallel (solid) and
perpendicular (dashed) to the magnetic field. The linear scale unit
CIIL$_2$ (the downstream proton cyclotron frequency times the flow
velocity) corresponds to about 35 km. This shock propagates
quasi-perpendicularly to the upstream magnetic field and has $M\approx 3$,
$M_A\approx 4.7$ and electron $\beta=1.5$. Electrons are not heated at
the shock as much as protons, which is typical for solar wind shocks.
(Reproduced from Hull et al.\ 2001.)}
\label{hull}
\end{figure}
\subsubsection{Comparison with other astrophysical plasmas}
To our knowledge, the above result is the first direct indication for any
astrophysical plasma that after a shock, electrons reach the equilibrium
temperature on a timescale shorter than collisional. However, it cannot
distinguish between two interesting possibilities: electrons being heated to
the ``correct'' temperature right at the shock, or electrons and protons
equilibrating soon after via a mechanism unrelated to the shock. This is
where the solar wind and supernova remnants (SNR) provide complementary
evidence.
A large amount of data has been gathered by space probes for the
heliospheric shocks, especially for the Earth's bow shock (for recent
reviews see, e.g., Russell 2005 and references therein). Many of these
shocks have high Alfv\'en Mach numbers $M_A$, moderate sonic Mach numbers,
and a ratio of pre-shock thermal to magnetic pressure $\beta>1$, which
should make them qualitatively similar to merger shocks in clusters
(although $M_A$ and $\beta$ in clusters are typically much higher; e.g.,
upstream of the \mbox{1E\,0657--56}\ shock, $M_A=70\; (B/1\,\mu{\rm G})^{-1}$ and
$\beta=650\; (B/1\,\mu{\rm G})^{-2}$, so the similarity is not guaranteed).
In most observed heliospheric shocks, and in all the stronger ones,
electrons were heated much less than protons, barely above the adiabatic
compression temperature (e.g., Schwartz et al.\ 1988). Figure \ref{hull}
gives a typical example of a shock crossing by the {\em Wind}\/ satellite
(Hull et al.\ 2001). Both $T_p$ and $T_e$ quickly reach their respective
asymptotic values, which are $T_e < T_p$. These in-situ measurements cannot
tell us what happens after that on timescales comparable to $\tau_{\rm c}$
(which corresponds to a linear scale of several A.U.), but they do show that
equilibration at least does not occur on scales $\sim 10^{-7} \tau_{\rm c}$.
Another site of actively studied shocks is SNRs. The typical $\beta$ in SNR
plasmas is of the same order of that in clusters, much higher than that in
the solar wind. Outer shocks in SNRs are typically very strong and may be
modified by cosmic ray acceleration (see, e.g., Vink 2004 for a review).
Nevertheless, these very different shocks also exhibit $T_e/T_i<1$ in most
cases (e.g., Rakowski 2005; Raymond \& Korreck 2005), although with
considerable uncertainty due to the difficulty of estimating the velocity
and $T_i$ for a strong shock. (One cannot use the density jump as we did,
as it is at its asymptotic value; on the other hand, in some Galactic SNRs,
the shock velocity can be directly measured using high-resolution X-ray
images taken several years apart, which, of course, is impossible for
clusters.) The typical timescale $\tau_{\rm c}$ for an SNR plasma is of
order $10^3$ yr, which in many cases is comparable to the SNR's age.
Furthermore, the region for which the temperatures are derived is usually
close to the shock, so the timescales sampled by the temperature
measurements in the SNR plasma are usually $\ll \tau_{\rm c}$. Thus,
similarly to the solar wind results, the SNR data indicate that
collisionless shocks with a wide range of parameters produce $T_e/T_i<1$,
but, to our knowledge, do not constrain the timescale of the subsequent
equilibration.
The solar wind and SNR results on one hand, and our \mbox{1E\,0657--56}\ measurement on the
other, appear to leave only one of the two possibilities mentioned above,
namely, that shocks produce unequal electron and ion temperatures, which
quickly equalize outside the shock layer. Proposing an equilibration
mechanism that works faster than Coulomb collisions in plasma is beyond the
scope of this review (and our expertise).
\subsection{Shocks and cluster cosmic ray population}
\label{sec:halos}
Cluster mergers convert kinetic energy of the gas in colliding subclusters
into thermal energy by driving shocks and turbulence. A fraction of this
energy may be diverted into nonthermal phenomena, such as magnetic field
amplification and the acceleration of ultrarelativistic particles. Such
particles manifest themselves via synchrotron radio halos (recently reviewed
by, e.g., Feretti 2002, 2004; Kempner et al.\ 2004) and IC hard X-ray
emission (e.g., Fusco-Femiano et al.\ 2005; Rephaeli \& Gruber 2002). This
energy fraction may be significant and depends on the exact mechanism of the
production of halo-emitting electrons, which is not yet understood. However,
a consensus emerges that the underlying cause and energy source should
indeed be the subcluster mergers (e.g., Feretti 2002; Buote 2001; G04).
The radio halo generating electrons are relatively short-lived ($10^7-10^8$
yr) due to IC and synchrotron energy losses (e.g., Sarazin 1999). This is
much shorter than the diffusion time across a cluster-sized halo, so a
mechanism is required by which these electrons are locally and
simultaneously (re)accelerated over the halo volume. Several possibilities
were proposed, including radio galaxies, so-called ``secondary electron''
production by interactions of the long-lived cosmic ray protons with the ICM
protons, and ``primary electron'' mechanisms, in which electrons are
accelerated directly by merger-driven turbulence or shocks (see, e.g.,
Brunetti 2003 for a review).
Shock fronts provide a unique experimental tool in this area. They are those
rare locations in clusters where we can determine gas velocities in the sky
plane and the gas compression ratio (\S\ref{sec:1e_M}). They also create
high-contrast features in the cluster X-ray images and, as we will see, in
the radio images. Simply by comparing the X-ray and radio images of clusters
with shock fronts, one may be able to determine the contribution of shocks
to the halo production.
\begin{figure}[t]
\centering
\includegraphics[width=0.7\textwidth]%
{a520_halo.ps}
\caption{Radio halo brightness (contours; Govoni et al.\ 2001) overlaid on
the \emph{Chandra}\ X-ray image of A520 (shown without this overlay in Fig.\
\ref{a520_profs}{\em a}). This 1.4 GHz map has an angular resolution of
15$^{\prime\prime}$; unrelated compact radio sources are left in the image. The halo is
a low-brightness mushroom-like structure with a ``cap'' that coincides
with the X-ray bow shock southwest of center and a wide ``stem'' extending
along the NE-SW axis of the cluster. See Fig.\ \ref{1e}{\em d}\/ for a
similar overlay for \mbox{1E\,0657--56}.}
\label{a520_halo}
\end{figure}
Both clusters with known shock fronts, \mbox{1E\,0657--56}\ and A520, have prominent radio
halos (Liang et al.\ 2000; Govoni et al.\ 2001). Their radio brightness
contours are overlaid on the X-ray images in Figs.\ \ref{1e}{\em d}\/ and
\ref{a520_halo}. One immediately notices a striking coincidence of the SW
edge of the radio halo in A520 with the shock front (M05). A similar
extension of the radio halo edge to the bow shock is seen in the less
well-resolved halo in \mbox{1E\,0657--56}\ (M02; G04). In another merging cluster, A665, a
``leading'' edge of the radio halo also corresponds to a region of hot gas
that's probably behind a bow shock (Markevitch \& Vikhlinin 2001; G04),
although the X-ray image of A665 does not show a gas density edge at the
putative shock (\S\ref{sec:shockwidth}). The cold front cluster A3667
exhibits two radio relics at the opposite sides of the cluster (R\"ottgering
et al.\ 1997), along the merger axis suggested by the cold front. They are
often mentioned as evidence of bow shocks at those locations; however, they
are located too far in the cluster periphery, where X-ray imaging is not
currently feasible.
The overall structure of the radio halo in A520 suggests two distinct
components, a mushroom with a stem and a cap, where the main stem component
goes across the cluster and the cap ends at the bow shock. The main halo
component is located in the region of the cluster where one expects
relatively strong turbulence (G04), but about which we cannot say much
observationally at present. However, the part of the halo that coincides
with the shock front can already be used for interesting tests. Below we
discuss two main possibilities for its origin, how to distinguish between
them, and propose some future tests that can be done using this shock (M05).
\subsubsection{Shock acceleration}
\label{sec:accel}
One explanation for the radio edge is acceleration of electrons to
ultrarelativistic energies by the shock. The shock with a density jump
$r$\/ should generate electrons with a Lorentz factor $\gamma$ and energy
spectrum $dN/d\gamma=N_0 \gamma^{-p}$ with
\begin{equation}
p=\frac{r+2}{r-1}
\label{eq:p}
\end{equation}
via first-order Fermi acceleration (see, e.g., Blandford \& Eichler 1987 for
a review). For the A520 shock with $r\simeq 2.3$, $p\simeq 3.3$. The
synchrotron emission should have a spectrum $I_\nu \propto \nu^{-\alpha}$
with $\alpha=(p-1)/2 \simeq 1.2$ right behind the shock front. However,
these electrons are short-lived because of IC and synchrotron energy losses
(e.g., Rybicki \& Lightman 1979), and their spectrum will quickly steepen.
The respective electron lifetimes, $t_{\rm\,IC}$ and $t_{\rm syn}$, are
\begin{equation}
t_{\rm\,IC} = 2.3\times 10^{8}\; \left(\frac{\gamma}{10^4}\right)^{-1}
(1+z)^{-4}\; {\rm yr}
\label{eq:tic}
\end{equation}
and
\begin{equation}
t_{\rm syn} = 2.4\times 10^{9}\; \left(\frac{\gamma}{10^4}\right)^{-1}
\left(\frac{B}{1\,\mu G}\right)^{-2}\; {\rm yr}
\label{eq:tsyn}
\end{equation}
(e.g., Sarazin 1999). IC losses dominate for $B<3\,\mu G$; other losses are
negligible for our range of energies and fields. For a power-law electron
spectrum with $p=2-4$ (expected for shocks with $M>1.7$), the contribution
of different $\gamma$ at a given synchrotron frequency has a peak at
\begin{equation}
\gamma_{\rm peak} \approx 10^4\, \left(\frac{\nu}{1\,{\rm GHz}}\right)^{1/2}
\left(\frac{B}{1\,\mu G}\right)^{-1/2}.
\label{eq:gmax}
\end{equation}
Assuming $B\sim 1\,\mu G$, the lifetime for electrons with $\gamma\sim
1.2\times 10^4$ that emit at our radio image frequency of 1.4 GHz in A520 is
$\sim 10^8$ yr. Thus, given the 1000 km$\;$s$^{-1}$\ velocity of the downstream flow
(\S\ref{sec:1e_M}) that carries these electrons away from the shock, the
width of the synchrotron-emitting region should only be about 100 kpc,
beyond which the electrons cool out of the 1.4 GHz band. This scale is an
order of magnitude smaller than the size of the halo, so the whole halo
cannot be produced by particles accelerated at this shock. A similar
conclusion was reached for the front in \mbox{1E\,0657--56}\ by Siemieniec-Ozieb{\l}o
(2004), and this is expected for merger shocks in general (e.g., Brunetti
2003). However, the cap-like part of the radio halo appears to have just
the right width, $\Delta R\lesssim 100$ kpc (considering the finite angular
resolution). Thus, with the available data, this region is not inconsistent
with shock acceleration. While the relativistic electrons in this structure
cool down soon after the shock passage, some may later be picked up and
re-accelerated as they reach the turbulent region behind the subcluster
core, where the stem-like halo component forms.
Because the bow shock is spatially separated from the turbulent area further
downstream (except for the region around the small dense core fragments) and
there is no reason to expect significant turbulence and additional
acceleration in that intermediate region, the cap-like structure is likely
to exhibit a measurable spectral difference from the main halo. Within the
100 kpc-wide strip along the shock, the spectrum should quickly steepen
starting from $\alpha=1.2$. If the region is unresolved, the resulting
mixture would have a volume-averaged slope $\bar{\alpha}\approx \alpha+1/2$
(Ginzburg \& Syrovatskii 1964) which is significantly steeper than
$\alpha\simeq 1-1.2$ observed on average in most halos (e.g., Feretti 2004),
the bulk of which is probably continuously powered by turbulence.
Interestingly, Feretti et al.\ (2004) found that the presumed post-shock
region in A665 indeed exhibits the steepest radio spectrum in the spectral
index map of the cluster, which is consistent with the above two-component
cap + stem picture.
\subsubsection{Compression of fossil electrons}
\label{sec:compr}
The efficiency with which collisionless shocks can accelerate relativistic
particles is unknown, and may be insufficient to generate the observed radio
brightness. The radio edges in A520 and \mbox{1E\,0657--56}\ offer an interesting prospect
for constraining it. If the acceleration efficiency is low, the observed
radio edge may alternatively be explained by an increase in the magnetic
field strength and the energy density of the {\em pre-existing}\/
relativistic electrons, simply due to the gas compression at the shock. Such
pre-existing electrons with energies below those required to emit at our
radio frequencies (eq.\ \ref{eq:gmax}) may accumulate from past mergers
(e.g., Sarazin 1999). In this model, the pre-existing electrons must
produce low-brightness diffuse radio emission {\em in front of}\/ the bow
shock, whose intensity and spectrum may be predicted from the shock
compression factor and the post-shock radio spectrum.
M05 made such a prediction under the assumption that, as the plasma crosses
the shock surface, each volume element, with its frozen-in tangled magnetic
field, is compressed isotropically. Indeed, observations at the Earth's bow
shock show that a shock passage strengthens the field whether it is parallel
or perpendicular to the shock (e.g., Wilkinson 2003), so this assumption is
adequate. The average field strength $B$\/ then increases by a factor
\begin{equation}
B\propto r^{\,2/3}.
\label{eq:b}
\end{equation}
This increase would cause the particles to spin up as \mbox{$\gamma \propto
B^{1/2}$} (the adiabatic invariant). In addition, the number density of
relativistic electrons increases by another factor of $r$\/ due to the
compression. As a result, for a power-law fossil electron spectrum of the
form $dN/d\gamma=N_0\, \gamma^{-\delta}$, the synchrotron brightness at a
given radio frequency should exhibit a jump at the shock
\begin{equation}
I_\nu \propto r\rlap{\phantom{I}}^{%
\frac{\scriptstyle 2\delta}{\scriptstyle 3}+1}.
\label{eq:icompr}
\end{equation}
The power-law slope of the spectrum is preserved. Thus, for the A520 shock
with $r=2.3$, if the radio edge is due to the compression only, there must
be a pre-shock radio emission with the same spectrum as post-shock, but
fainter by a factor of $7-20$ for $\delta=2-4$, respectively. (In practice,
$\delta$\/ can be determined from the post-shock radio spectrum; this
measurement is currently feasible but has not yet been made.) Of course,
projection of a spheroidal shape of the shock should be taken into account,
as is done for deriving the amplitude of the X-ray brightness jump.
Although the low surface brightness within the radio halos is already near
the limit of detectability, an improvement of sensitivity by an order of
magnitude required to look for emission from pre-shock fossil electrons is
not completely out of reach. If future sensitive measurements do not detect
such a pre-shock emission at the level predicted for simple compression, it
would mean that the shock generates relativistic electrons and/or a magnetic
field, as opposed to simply compressing them. Observations of solar wind
shocks of a similar strength and $\beta>1$ as in cluster plasma, as well as
lower-$\beta$ shocks, seem to be consistent with a simple compression of the
field (e.g., Russell \& Greenstadt 1979; Hull et al.\ 2001), so significant
magnetic field generation is unlikely. Thus, such a non-detection would
provide a lower limit on the shock's particle acceleration efficiency, a
quantity that is interesting for a wide range of astrophysical problems.
\begin{figure}[t]
\centering
\includegraphics[width=0.65\linewidth]%
{a520_gamma_losses.ps}
\caption{Lifetime of the relativistic electrons that contribute the most to
the synchrotron emission at two frequencies, as a function of the magnetic
field strength (for a $p=3-4$ electron energy spectrum). Two frequencies
are shown as solid and dashed lines. Black and red lines correspond to
two different redshifts as labeled ($z=0.2$ for A520 and $z=0.3$ for \mbox{1E\,0657--56}).
This timescale can be determined from the width of the radio-bright strip
at the shock, and gives an estimate of $B$. (Reproduced from M05.)}
\label{a520_B}
\end{figure}
\subsubsection{Yet another method to measure intracluster magnetic field}
\label{sec:b}
The strength of the magnetic field in the ICM is an important quantity, but
its measurements are still very limited in accuracy and often restricted to
certain non-representative locations in clusters (e.g., Carilli \& Taylor
2002). The shock-related edge of the radio halo in A520, and perhaps a
similar region in \mbox{1E\,0657--56}, may allow an independent estimate of the magnetic
field strengths behind the shock, regardless of the exact origin of the
relativistic electrons. As discussed above, IC and synchrotron cooling cause
the electron spectrum to steepen and the electrons to drop out of the radio
image on timescales of order $10^8$ yr. This timescale depends on the field
strength as shown in Fig.\ \ref{a520_B}, which combines eqs.\
(\ref{eq:tic}--\ref{eq:gmax}). It gives the lifetime of the electrons that
contribute the most emission at a given frequency, for an electron energy
spectrum with the slope $p=3-4$ and an interesting range of $B$. Assuming
that the bow shock causes a momentary increase in electron energy and $B$,
and that diffusion of the relativistic particles is negligible (see M05 for
a discussion), the flow of the post-shock gas (that carries the relativistic
electrons with it) will spread the time evolution of the electron spectrum
along the spatial coordinate --- something we have already used for another
test in \S\ref{sec:tei}. Thus, the width of the cap-like region of the halo
can give us the magnetic field strength. Of course, the magnetic field
behind the shock will be amplified from its pre-shock value as discussed in
\S\ref{sec:compr}. This method can distinguish among the values in the
currently controversial range of $B\sim 0.1-3\,\mu G$, although, as seen
from Fig.\ \ref{a520_B}, it cannot give a unique value of $B$, because
$t_{\rm loss}$ is not a monotonic function of $B$. In practice, such a
measurement will need to be done at more than one frequency, in order to
determine the spectrum of the electrons and to verify that they cool as
predicted at different frequencies, that is, no acceleration occurs after
the shock has passed. The available single-frequency radio data (Fig.\
\ref{a520_halo}; Govoni et al.\ 2001) do not have the needed signal to noise
ratio or angular resolution, but are not inconsistent with $B\sim 1\,\mu$G.
\subsection{Constraints on the nature of dark matter}
\label{sec:dm}
We finally mention two interesting results that rely on the determination of
the velocity and the direction of motion of the subcluster in \mbox{1E\,0657--56}\ based on
its shock front. The fact that we observe a cold front or a shock front
tells that the subcluster responsible for the feature moves very nearly in
the plane of the sky, because otherwise, the sharp gas density jump would be
smeared by projection. From the shapes of either feature, usually the
direction of motion in the sky plane is also clear, as seen in the images of
A3667 (Fig.\ \ref{2142_3667}), NGC\,1404 (Fig.\ \ref{n1404}), and \mbox{1E\,0657--56}\
(Fig.\ \ref{1e}{\em c}). For the latter cluster, the shock and the cold
front (the bullet) in the X-ray image strongly suggest that the X-ray gas
distribution has axial symmetry, at least in its western half containing the
shock and the bullet. This allows a reasonably accurate estimate of the gas
mass in that region. As expected from observations of other clusters,
invariably showing that gas is the dominant baryonic mass component, the gas
mass of the bullet is several times higher than the stellar mass in galaxies
in a comparable aperture centered on the brightest galaxy of that
subcluster. At the same time, comparison of the gravitational lensing mass
map and the X-ray image (Fig.\ \ref{1e_lens}; Clowe et al.\ 2006; see also
Brada\v{c} et al.\ 2006) shows that the subcluster's total mass peak is
offset from the baryonic mass peak (the X-ray bullet). The same is true for
the bigger eastern subcluster. Clowe et al.\ (2004, 2006) interpret this as
the first direct evidence for the existence of dark matter, as opposed to
alternative theories that avoid the notion of dark matter by, e.g.,
modifying the gravitational force law on cluster scales (e.g., MOND: Milgrom
1983; Brownstein \& Moffat 2006). If cluster had contained only visible
matter (i.e., mostly the X-ray emitting gas), regardless of the form of the
gravity law, the gravitational lensing would have to show a mass peak
coincident with the gas bullet (see Clowe et al.\ 2006 for a more detailed
discussion, and Angus et al.\ 2007 for a MOND response admitting the need
for dark matter).
\begin{figure}
\centering
\includegraphics[width=0.7\textwidth,angle=-90]%
{1e0657_press.ps}
\caption{\emph{Chandra}\ X-ray image of \mbox{1E\,0657--56}\ (pink, same as in Fig.\ \ref{1e}{\em
c}) and the projected total mass map from weak lensing (blue; Clowe et
al.\ 2006; same as in Fig.\ \ref{1e}{\em b}) overlaid on the optical
image. The gas bullet lags behind its host dark matter subcluster, while
the two mass peaks coincide with centroids of the respective galaxy
concentrations. This overlay proves the existence of dark matter, as
opposed to alternative gravity theories (Clowe et al.\ 2004, 2006). (Image
created by the \emph{Chandra}\ press group.)}
\label{1e_lens}
\end{figure}
Furthermore, the current orientation of the shock and the bullet let us
reconstruct the trajectory of the merging subclusters back in time at least
to the core passage. It is clear that the small halo has passed near the
center of the bigger halo. The shock front also gives an estimate of the
current velocity of the subcluster (\S\ref{sec:1e_M}). Combined with the
mass measurements from lensing and the distribution of the member galaxies
from the optical images, this can be used to derive a limit on the
self-interaction cross-section of the dark matter particles. Although the
common wisdom is that dark matter is collisionless, a nonzero cross-section
was proposed as a solution for several observational problems (Spergel \&
Steinhardt 2000). \mbox{1E\,0657--56}\ offers an interesting opportunity to test this
hypothesis in a relatively robust and model-independent way (Markevitch et
al.\ 2004). If the dark matter particles experienced scattering by elastic
collisions, it would have several observable consequences. At large
cross-sections, the dark matter would behave as a fluid --- just like the
intracluster gas --- and there would be no offset between the gas peak and
the dark matter peak. This extreme is excluded simply by comparing the
X-ray and mass images (Fig.\ \ref{1e_lens}). Smaller values would lead to
subtler effects, such as an anomalously low mass-to-light ratio for the
subcluster caused by loss of dark matter particles by the subcluster, and a
lag of the subcluster mass peak from the centroid of the collisionless
galaxy distribution. The absence of these effects gives a conservative upper
limit on the cross-section of $\sigma/m<0.7$ cm$^{2}\,$g$^{-1}$, where $m$
is the (unknown) mass of the dark matter particle (Randall et al.\ 2007).
This excludes almost the entire range of values (0.5--5 cm$^{2}\,$g$^{-1}$)
required to solve the problems for which the self-interacting dark matter
has been invoked.
\section{SUMMARY}
Recent observations of sharp gas density edges in galaxy clusters provided
us with novel ways to study the cluster mergers, and even to perform remote
sensing of some otherwise inaccessible physical properties of the
intracluster plasma. Classic bow shocks have been observed in two clusters
undergoing violent mergers at the right stage and with the right orientation
for us to see the front discontinuities unobstructed. They have been used
to determine the velocities and trajectories of the infalling subclusters.
The velocity and the unique geometry of the subcluster in \mbox{1E\,0657--56}, whose gas is
clearly stripped from its dark matter by ram pressure, were combined with
the gravitational lensing and optical data to place an upper limit on the
dark matter's self-interaction cross-section, and to directly demonstrate
the dark matter existence.
A jump of the electron temperature at the shock in \mbox{1E\,0657--56}\ provides evidence
that electron-ion equilibration in the ICM occurs on a timescale shorter
than the collisional timescale. To our knowledge, this is the first such
test in any astrophysical plasma. Combined with finding based on the shocks
in solar wind and SNRs that electrons are not heated to equilibrium
temperature immediately in the shock, the \mbox{1E\,0657--56}\ result indicates the presence
of a fast electron-ion equilibration mechanism unrelated to shocks (although
the uncertainty of the \mbox{1E\,0657--56}\ result is quite large).
The shock front in A520 coincides with a distinct edge-like feature in the
radio halo observed in this cluster. This indicates that ultrarelativistic
electrons responsible for this part of the radio halo emission are either
accelerated by this weak shock, or that there is a pre-existing relativistic
population that is being compressed by the shock. More sensitive radio
observations will be able to distinguish among these interesting
possibilities and constrain the efficiency of electron acceleration by
shocks. The A520 shock also provides a setup for a measurement of the
strength of the magnetic field in this cluster by determining the cooling
time of the relativistic electrons.
In addition to observing shock fronts, \emph{Chandra}\ has discovered a new kind
of transient features in the intracluster gas, named ``cold fronts.'' These
sharp density edges are contact discontinuities between gas phases with
different specific entropies in approximate pressure equilibrium. They form
as a result of bulk motion of a dense gas cloud through the hotter ICM. In
merging clusters, these clouds are the surviving remnants of the cores of
the infalling subclusters. In cooling flow clusters, such edges are produced
by displacement and subsequent sloshing of the low-entropy central gas.
Cold fronts are much more common than shock fronts. Just as shock fronts,
they can be used to determine the velocity and the direction of motion of
the gas flow.
Cold fronts are remarkably sharp and stable. The observed temperature jumps
across cold fronts require thermal conduction across the front to be
severely suppressed. Furthermore, the density jump in the best-observed
front in A3667 is narrower than the collisional mean free path in the
plasma. These observations demonstrate that transport processes in the ICM
can be easily suppressed. The KH stability of the front in A3667 also
suggests the presence of a layer of the compressed magnetic field oriented
along the front surface, such as the one expected to form as a result of
magnetic draping. Such a layer is exactly what is needed to completely
suppress thermal conduction and diffusion across the front surface. In
addition, Rayleigh-Taylor stability of the front in A3667 reveals the
presence of an underlying massive dark matter subcluster.
This area of cluster research is relatively new (although much of the
underlying hydrodynamics is textbook). It is not yet completely clear what
these findings mean for the more general questions of how the energy is
dissipated in a cluster merger, how much of it goes into relativistic
particles and magnetic fields, how prevalent and long-lived is turbulence,
how effective is mixing of the various gas phases, what is the effect of the
central gas sloshing and turbulence on cooling flows and on the hydrostatic
mass estimates, how representative of the hydrostatic temperatures are the
average temperatures derived from the X-ray data, etc. Our goal for this
mostly observational review was to encourage detailed theoretical and
numerical studies of these interesting phenomena that can be used as novel
diagnostic tools in clusters.
\ack{The phenomena discussed in this review owe their discovery to the
sharpness of the \emph{Chandra}\ X-ray mirror, whose development was led by the
late Leon VanSpeybroeck. These subtle X-ray features in clusters may have
gone unnoticed without the convenience of the software tool SAOimage
written by M. VanHilst, and its later versions. We thank Y.~Ascasibar,
P.~Blasi, G.~Brunetti, E.~Churazov, W.~Forman, L.~Hernquist, N.~Inogamov,
C.~Jones, A.~Loeb, M.~Lyutikov, P.~Mazzotta, P.~Nulsen, A.~Petrukovich,
C.~Sarazin, P.~Slane, and A.~Schekochikhin for stimulating discussions,
and the referee, A. Kravtsov, for helpful comments. This work was
supported by NASA contract NAS8-39073 and grants NAG5-9217, GO4-5152X and
GO5-6121A.}
|
1,108,101,564,565 | arxiv | \section{Introduction}\label{sec:intro}
The white-light continuum of a solar flare (WLF) was the first manifestation of a flare ever detected \citep{1859MNRAs..20...13C,1859MNRAs..20...16H}.
Nevertheless its origin has remained enigmatic over the intervening centuries.
This continuum contains a large fraction of the total luminous energy of a flare \citep[e.g.,][]{1989SoPh..121..261N}, and so its identification has always posed an important problem for solar and stellar physics.
Although the most obvious flare effects appear in the chromosphere and corona, the physics of the lower solar atmosphere has great significance for the reasons described by Neidig.
The association of white-light continuum with the impulsive phase of a solar flare has long been known, and cited as an indication that high-energy nonthermal particles have penetrated deep into the lower solar atmosphere \citep{1970SoPh...15..176N,1970SoPh...13..471S}.
Such an association would link the acceleration of ``solar cosmic rays,'' as they were then known, with the intense energy release of the impulsive phase, and this association has proven to be crucial to our understanding of flare physics \citep{1976SoPh...50..153L,2003ApJ...595L..69L,2005JGRA..11011103E,2007ApJ...656.1187F}.
Because the continuum appears in the emission spectrum, and because this nominally originates in an optically-thick source, this energy release may drastically distort the structure of the lower solar atmosphere during a flare.
Accordingly the usual methods for modeling the atmosphere \citep[e.g.,][]{1981ApJS...45..635V}, which assume hydrostatic equilibrium, may be too simple.
\begin{figure*}[htpb]
\includegraphics[width=0.95\textwidth]{fig1.eps}
\caption{
HMI intensity continuum difference image (prime, 07:31:13.40~UT; reference, images from 07:25-07:28~UT) combined with white-light difference and RHESSI CLEAN contour plots (red and blue, respectively). The Hard X-ray images (30-80 keV) made with the CLEAN technique for the interval 07:30:50.9 -- 07:31:35.9~UT, exactly that of the HMI cadence. These images were made with \textit{RHESSI} subcollimators 1-4, with uniform weighting, giving an angular resolution (FWHM) of 3.1$''$. The orange contours in panel a) show the 6-8 keV Soft X-ray source at the same time, defining a loop structure connecting the footpoints. The dotted line shows the locus of the STEREO/EUVI source positions, which in this projection show the projected angular location of the photosphere (see Section~\ref{sec:absolute}).}
\label{fig:footpoints}
\end{figure*}
The association with hard X-ray bremsstrahlung has always suggested $>$10~keV electrons in particular \citep{1972SoPh...24..414H,1975SoPh...40..141R,1992PASJ...44L..77H,1993SoPh..143..201N}.
According to the standard thick-target model the primary particle acceleration occurs in the corona above the flare sources.
The presence of fast electrons automatically implicates the chromosphere as well, rather than the photosphere, because of the relatively short collisional ranges of such particles.
Nevertheless the indirect nature of the bremsstrahlung emission mechanism \citep[e.g.,][]{1971SoPh...18..489B} has made it difficult to rule out other processes, such as energy transport by protons \citep{1969sfsr.conf..356E,1970SoPh...15..176N,1970SoPh...13..471S}.
The bremsstrahlung signature may also result from acceleration directly in the lower atmosphere, rather than in the corona \citep{2008ApJ...675.1645F}.
Finally radiative backwarming could also provide a mechanism for explaining for the observed tight correlation of hard X-rays and white-light continuum \citep{1989SoPh..124..303M}.
This mechanism involves irradiation and heating of the photosphere, rather than the chromosphere, as would correspond to the shorter stopping distance of energetic electrons in a thick-target model \citep{1972SoPh...24..414H}.
In the modern era we are seeing a rapid increase in our understanding of these processes, thanks to the excellent new data from various spacecraft and ground-based observatories.
This paper reports on a first good example of a flare near the limb, observed at high resolution both in hard X-rays by \textit{RHESSI}, and at 6173~\AA~in the visible continuum by the \textit{Solar Dynamics Observatory (SDO)} spacecraft via its Helioseismic Magnetic Imager (HMI) instrument \citep{2012SoPh..275..229S}.
This event, SOL2011-02-24T07:35 (M3.5), occurred just inside the limb (NOAA coordinates N14E87), so that source heights could be compared by simple projection.
We analyze data from \textit{RHESSI}, HMI, and the Extreme Ultraviolet Imager (EUVI) on \textit{STEREO-B)}.
\cite{2011A&A...533L...2B} have already studied this flare, using the same data but without reference to the \textit{STEREO-B} observations.
\section{The observations}\label{sec:data}
The flare we study here (Figure~\ref{fig:footpoints}) occurred close to the geocentric east limb.
Both hard X-rays and white-light emission come clearly from the visible hemisphere, as we establish below via the use of
STEREO-B/EUVI images. Figure~~\ref{fig:footpoints} shows the HMI intensity continuum difference image at 07:31:13.40~UT, with the reference image constructed as the average of HMI data from 07:25-07:28~UT.
The white-light contrast and area for this M3.5 flare are consistent with the trends found in surveys \citep{2003A&A...409.1107M,2006SoPh..234...79H,2009RAA.....9..127W} based on \textit{Yohkoh}, \textit{TRACE}, and \textit{Hinode} observations, respectively.
The emission time series (Figure~\ref{fig:ts}) shows the timing behavior of hard X-ray and soft X-ray emissions typical of white-light flares \citep[e.g.][]{1975SoPh...40..141R,2010ApJ...715..651W}, and (again typically) we find a close match between the hard X-ray and white-light variations \citep[e.g.,][]{2011ApJ...739...96K}.
The time series show white-light differences relative to a pre-flare reference interval (07:25-07:28~UT).
The gradually increasing disagreement between the two footpoint light curves is the behavior expected for base-difference images in the presence of normal photospheric variability, which has time scales of minutes due to p-modes and granulation.
\begin{figure}[htpb]
\includegraphics[width=0.95\columnwidth, trim = 0 0 0 -20]{fig2.eps}
\caption{Time series for SOL2011-02-24, showing the \textit{GOES} 1-8~\AA~soft X-rays in gray and the \textit{RHESSI} 30-80~keV flux in black.
The blue and red histograms show the mean white-light contrasts relative to the reference image (average between 07:25\,--\,07:28~UT), integrated over the footpoint areas shown in Figures~\ref{fig:footpoints}b and~\ref{fig:footpoints}c, respectively.
The gradual mismatch of the two white-light footpoints results at least partly from the base-difference technique, and does not accurately represent flare emission. The lightcurves were scaled to fit on the plot.
}\label{fig:ts}
\end{figure}
We determined the locations of the hard X-ray (HXR) sources by using the RHESSI visibilities-based forward fitting procedure, using data for the same 45-s time interval as observed by HMI.
We used a single 30-80~keV energy interval for simplicity and assumed a circular Gaussian model for each of the two bright sources.
The application of least-squares fitting then yielded the best-fit centroid coordinates of the two footpoints with statistical errors.
These positions have statistical errors $\approx$~0.16$''$.
Systematic errors in these positions come only from the RHESSI telescope metrology, which we believe to be substantially better than the statistical errors \citep{2003SPIE.4853...41Z}.
Since the uncertainty of the centroid position of the forward-fit model is smaller than its fitted size, there is the possibility
of systematic error if the true source shape is not well-approximated by a circular Gaussian.
Thus, in addition to the forward-fit image locations, we have separately characterized the source positions via the distribution of CLEAN component sources, which does not require the assumption of a specific model.
This check shows excellent consistency for the source centroids.
For the position of the white-light continuum sources, we have used the full-disk HMI image as a reference, fitting its limb via a standard inflection-point method.
Since the flare occurred near the limb, this greatly reduces the error in radial position due to any uncertainty in the HMI image properties, such as platescale.
We adopt the limb correction described by \cite{1998ApJ...500L.195B} to relate this measurement to the height of the photosphere (R$_\odot$); the mean of their two methods gives 498~km for the altitude separation, and we adopt 18~km to represent the uncertainty of this number.
At the time of the observation described here, SDO was near its antisolar point, and we have made the correction
to geocentric perspective.
\section{Relative source heights}\label{sec:relative}
The images in Figure~\ref{fig:footpoints} show good agreement in the radial coordinates of the source centroids in hard X-rays and white light.
We have made a small \textit{ad hoc} roll adjustment for the HMI observations, which has no effect on the source heights
because of the foreshortening.
Table~\ref{tab:big} summarizes the data from the three spacecraft and our estimates of uncertainties.
The STEREO entries characterize the heliographic (Stonyhurst) coordinates of the two footpoint sources, and serve to locate the projected position of the sub-flare photosphere on the HMI and RHESSI images.
The RHESSI positions include an estimate of the apparent source displacement due to scattering albedo,
based on the Monte Carlo simulations of \citet{2010A&A...513L...2K}.
In our observations, the primary and albedo sources would merge together, with a centroid slightly lower (closer to disk center) than the position of the primary source \citep[see Figure~4 in][for an example]{2011ApJ...739...96K}.
The anisotropy of the primary emission is likely to be small; in principle large anisotropies could lead to
larger albedo corrections \citep{2010A&A...513L...2K}.
At the extreme limb these simulations are not complete, since they do not incorporate photospheric rough structure \citep[e.g.,][]{1969SoPh....9..317S,1976ApJ...203..753L,1994IAUS..154..139C}.
This co-alignment analysis differs from that in Fig. 2 of \cite{2011A&A...533L...2B}, as the result of a different interpretation of the image metadata, and the overlay shown here is correct (M. Battaglia \& E. Kontar, personal communication 2011).
\begin{table}
\caption{Source Positions}
{\footnotesize
\bigskip
\begin{tabular}{| l | l | l | r | r | r | r |}
\hline
Quantity & Type & Units & Value & Err & Value & Err \\
\hline
\multicolumn{3}{| c |}{STEREO$^a$} & \multicolumn{2}{|c|}{X/Lon/EW} & \multicolumn{2} {|c|} {Y/Lat/NS} \\
\hline
N source$^b$ & Meas & px & 1118.8 & 1.4 & 1206.5 & 1.0 \\
S source$^c$ & Meas & px & 1122.7 & 1.1 & 1198.5 & 1.0 \\
N Helio & Calc & deg & 276.33 & 0.14 & 16.10 & 0.02\\
S Helio & Calc & deg & 276.56 & 0.11 & 15.25 & 0.02\\
N Geo & Calc & arc s & $-926.03$ & 0.15 & 279.53 & 0.05\\
S Geo & Calc & arc s & $-929.43$ & 0.14 & 266.41& 0.04\\
\hline
\multicolumn{3}{| c |}{HMI} & \multicolumn{2}{|c|}{X} & \multicolumn{2} {|c|} {Y} \\
\hline
N Geo & Meas & arc s & $-926.76$ & 0.10 & 278.22 & 0.10 \\
S Geo & Meas & arc s & $-931.06$ & 0.10 & 261.47 & 0.10 \\
\hline
\multicolumn{3}{| c |}{RHESSI} & \multicolumn{2}{|c|}{X} & \multicolumn{2} {|c|} {Y} \\
\hline
N Geo &Meas & arc s & $-926.93$ & 0.16 & 278.06& 0.16 \\
S Geo & Meas & arc s & $-931.19$ & 0.16 & 260.76 & 0.16 \\
\hline
\end{tabular}\label{tab:big}
}
\smallskip
$^a$ Sun center [1035.77, 1051.13] px (measured)
$^b$ Y pixel range 1205-1208
$^c$ Y pixel range 1196-1201
\end{table}
\section{Absolute source heights}\label{sec:absolute}
The proximity of the flare to the limb means that we can compare its central distance (the root-sum-square of its angular position coordinates) to the projected position of the photosphere, at the flare's heliographic location, to estimate the absolute source height.
We were fortunate in that this flare was observed near disk center by \textit{STEREO-B}, then at a spacecraft heliolongitude of $-$94.5$^\circ$.
The heliographic coordinates of the flare footpoints cannot be well determined from a geocentric perspective (\textit{RHESSI, SDO}) because of the extreme foreshortening, but this is minimal for the \textit{STEREO-B} view.
One EUV (195~\AA) image (Figure~\ref{fig:stereo_b}) was taken during the integration time of the WL image shown in Figure~\ref{fig:footpoints}.
This 8-s EUVI image was saturated in about 16 of the columns of the CCD containing the flare itself.
The bright hard X-ray sources correspond to the southern two of the saturation regions, and a fainter HXR source, not detected in white light, lies to the north of these footpoint sources.
\begin{figure}[htbp]
\includegraphics[width=0.95\columnwidth, trim = 30 40 90 70]{fig3.eps}
\caption{The STEREO-B/EUVI image in the 195~\AA~band taken at 07:31:01.095~UT.
The flare occurs near disk center and results in image saturation as shown.
The line shows the geocentric limb, demonstrating that the flare was not occulted.
}\label{fig:stereo_b}
\end{figure}
On each column of the CCD that is saturated, the excess charge should spread equally in both directions (M. Waltham, personal communication 2011).
Thus, to a first approximation, the mean row on each column (horizontal in the raw image) give the centroid of the image brightness on that column.
For the set of columns associated with each of the footpoints, we therefore estimate the heliographic coordinates and their uncertainties (Table~\ref{tab:big}) from the scatter of the data.
These coordinates agree, to within a few arcsec, with the positions of the WL and HXR sources (see Figure~\ref{fig:footpoints}).
The component of the uncertainty in the radial direction is small, as described below in the height measurement, because of the foreshortening.
The geometrical assumption here is that the heliographic coordinates of the EUV and hard X-ray sources coincide,
as they do in the impulsive phase of a flare \citep[e.g.,][]{2001SoPh..204...69F,2011ApJ...739...96K}.
We also implicitly assume that the EUV source is at zero height; we checked the uncertainty resulting from this systematic term and incorporate it in the height measurements. The result of these calculations give us a height for the Northern and southern HXR sources of $4.2 \times 10^2$ and $2.1 \times 10^2$~km respectively with an uncertainty of $2.4 \times 10^2$~km in both cases. The height of the intensity continuum sources was of $2.3 \times 10^2$ and $1.6 \times 10^2$~km for the Northern and southern sources, with uncertainties of $1.0 \times 10^2$~km for both measurements. These results are shown schematically in Figure~\ref{fig:arrows}.
We note several other unknowns present in this procedure: the EUV and HXR sources may not actually have coincided in heliographic position; the images are not exactly cotemporal, since the EUV image has only an 8-s exposure time, so source variability could contribute to misalignment; finally, although impulsive-phase EUV and HXR sources can coincide precisely, some do not \citep[see references in][]{2011SSRv..159...19F}.
We mention these items for completeness but note that the extreme foreshortening minimizes their significance in this analysis.
\begin{figure}[htdp]
\includegraphics[width=0.47\textwidth]{fig4.eps}
\caption{Schematic view of the source heights determined for the centroids of the two footpoint sources of SOL2011-02-24.
Red points with errors show the hard X-ray source centroids ($>30$~keV), and black points the white-light sources.
The horizontal solid line with error bar is the zero point of the height measurement, set at the projected position of the corresponding STEREO sources.
The two dashed lines show the $\tau = 1$ points for Compton scattering opacity (red) and a scaled optical opacity (black).
The heights and their uncertainties for the (N, S) footpoints are $4.2 (2.1) \pm 2.4 \times 10^2$~km for HXR, and
$2.3 (1.6) \pm 1.0 \times 10^2$~km for WL.
}
\label{fig:arrows}
\end{figure}
Figure~\ref{fig:arrows} shows the computed height of unit optical depth to Compton scattering at about 350~km.
We estimated this from the 1D semi-empirical models of \cite{2009ApJ...707..482F}, taking the Compton cross-section
at 40~keV, a slightly higher energy than the 30~keV threshold for our HXR imaging.
In such a simple model atmosphere, we would not expect appreciable HXR emission to be detectable.
Note that this consideration would not affect the albedo source expected from a higher-altitude source, such as that implied by the thick-target model.
Figure~\ref{fig:arrows} also shows an estimate of the height of unit optical depth for the white-light continuum, derived here just
as a scaling from the quantities at the limb.
\section{Conclusions}\label{sec:concl}
In this study we have compared hard X-ray and white-light observations of a limb flare, SOL201-02-24T07:35:00.
The relative positions of the sources agree well, for each of the double-footpoint sources; since the local vertical maps almost exactly onto the solar radial coordinate, this means that the source heights match well.
Our uncertainties on this centroid matching are of order 0.2$''$.
This result strongly associates the white-light continuum with the collisional losses of the non-thermal electrons observed via bremsstrahlung hard X-rays in the impulsive phase of the flare.
We have also used the EUVI data in the 195~\AA~band from \textit{STEREO-B} to determine the heliographic coordinates of the flare footpoints, from a near-vertical vantage point.
To our knowledge, this enables the first direct determination of the absolute height of a white-light flare and its associated hard X-ray sources. Surprisingly, our estimates lie close to (if not below) the projected heights of optical depth unity for both 40~keV hard X-rays and for optical continuum at 6173~\AA.
They also lie well below the expected penetration depth of the $\sim$50-keV electrons needed to produce the hard X-rays (about 800~km for the Fontenla et al. quiet-Sun atmosphere, vs. 200-400~km as observed).
At present we have no explanation for this striking result and do not speculate about it, because it depends upon only a
single flare event.
We are sure that there are other comparable events in the existing data and hope to see a generalization of these results based on similar analyses.
\bigskip\noindent
{\bf Acknowledgements:} This work was supported by NASA under Contract NAS5-98033 for \textit{RHESSI} for authors Hudson, Hurford, Krucker, Lin, and Mart{\' i}nez Oliveros. R. Lin was also supported in part by the WCU grant (R31-10016) funded by the Korean Ministry of Education, Science, and Technology. Jesper Schou and Sebastien Couvidat are supported by NASA contract NAS5-02139 to Stanford University. The HMI data used are courtesy of NASA/SDO and the HMI science team.
We thank Martin Fivian for helpful discussions of the RHESSI aspect system.
Alex Zehnder's precise metrology of RHESSI has made this analysis possible in the first place.
We further thank M. Waltham for comments on the saturation properties of the SECCHI CCD detectors.
\nocite{2011A&A...533L...2B}
\nocite{1971SoPh...18..489B}
\nocite{1859MNRAs..20...13C}
\nocite{1969sfsr.conf..356E}
\nocite{2005JGRA..11011103E}
\nocite{2001SoPh..204...69F}
\nocite{1859MNRAs..20...16H}
\nocite{1972SoPh...24..414H}
\nocite{1992PASJ...44L..77H}
\nocite{1976SoPh...50..153L}
\nocite{2003ApJ...595L..69L}
\nocite{1976ApJ...203..753L}
\nocite{1989SoPh..124..303M}
\nocite{1970SoPh...15..176N}
\nocite{1989SoPh..121..261N}
\nocite{1993SoPh..143..201N}
\nocite{1975SoPh...40..141R}
\nocite{1970SoPh...13..471S}
\nocite{1981ApJS...45..635V}
\nocite{2009RAA.....9..127W}
\nocite{2010ApJ...715..651W}
\nocite{1998ApJ...500L.195B}
\bibliographystyle{apj}
|
1,108,101,564,566 | arxiv | \section{Introduction}
Supersymmetry~(SUSY)~\cite{haberkane,Nilles:1983ge} is one of the best
motivated candidates for physics beyond the Standard Model~(SM).
Besides providing a unification of the strong and electroweak gauge
couplings and a suitable cold dark matter candidate, the lightest
stable SUSY particle, SUSY could offer new sources of CP
violation~\cite{Haber:1997if,Ibrahim:2002ry,Ibrahim:2007fb,Ellis:2007kb}.
In the Minimal Supersymmetric Standard Model (MSSM), the complex
parameters are conventionally chosen to be the Higgsino mass parameter
$\mu$, the ${\rm U(1)}$ and ${\rm SU(3)}$ gaugino mass parameters
$M_1$ and $M_3$, respectively, and the trilinear scalar coupling
parameters $A_f$ of the third generation sfermions ($f=b,t,\tau$),
\begin{eqnarray}
\label{eq:phases}
\mu = |\mu| e^{i \phi_\mu}, \quad
M_1 = |M_1| e^{i \phi_{1}}, \quad
M_3 = |M_3| e^{i \phi_3}, \quad
A_f = |A_f| e^{i \phi_{A_f}}.
\end{eqnarray}
The sizes of these phases are constrained by experimental bounds from
the electric dipole moments (EDMs). Such experimental limits
generally restrict the CP phases to be small, in particular the phase
$\phi_\mu$~\cite{Barger:2001nu}. However, the extent to which the EDMs
can constrain the SUSY phases depends strongly on the considered model
and its parameters~\cite{Ellis:1982tk,Barger:2001nu,Bartl:2003ju,
Choi:2004rf,YaserAyazi:2006zw,Ellis:2008zy,Deppisch:2009nj}.
\medskip
Due to cancellations among different contributions to the EDMs, large
CP phases can give CP-violating signals at colliders, as shown for
example in Ref.~\cite{Deppisch:2009nj}. It is important to search for
these signals, since the cancellations could be a consequence of an
unknown underlying structure that correlates the phases. In addition,
the existing EDM bounds could also be fulfilled by including lepton
flavor violating couplings in the slepton
sector~\cite{Bartl:2003ju}.
\medskip
Thus, direct measurements of SUSY CP-sensitive observables are
necessary to determine or constrain the phases independently of EDM
measurements. The phases can change SUSY particle masses, their cross
sections, branching
ratios~\cite{Bartl:2003he,Bartl:2003pd,Bartl:2002uy,Bartl:2002bh,Rolbiecki:2009hk},
and longitudinal polarizations of final
fermions~\cite{Gajdosik:2004ed}. Although such CP-even observables can
be very sensitive to the CP phases, CP-odd (T-odd) observables have to
be measured for a direct evidence of CP violation.
\medskip
CP asymmetries in particle decay chains can be defined with triple
products of final particle
momenta~\cite{Valencia:1994zi,Branco:1999fs}. Due to spin
correlations, such asymmetries show unique hints for CP phases already
at tree level. Thus, triple product asymmetries have been proposed in
many theoretical papers. For the Large Hadron Collider (LHC), triple
product asymmetries have been studied for the decays of
neutralinos~\cite{Bartl:2003ck,Langacker:2007ur,MoortgatPick:2009jy},
stops~\cite{Bartl:2004jr,Ellis:2008hq,Deppisch:2009nj,MoortgatPick:2010wp},
sbottoms~\cite{Bartl:2006hh,Deppisch:2010nc}, and
staus~\cite{Dreiner:2010wj}. In a Monte Carlo~(MC) analysis for stop
decays\cite{Ellis:2008hq,MoortgatPick:2010wp}, it could be shown that
the decay chain can be reconstructed and asymmetries be measured at a
$3\sigma$ level for luminosities of the order of $300~{\rm fb}^{-1}$.
At the International Linear Collider
(ILC)~\cite{Behnke:2007gj,:2007sg,Djouadi:2007ik,TDR} a clearer
identification and a more precise measurement is expected to be
achievable. However, in this context only theoretically-based papers
exist: for instance, neutralino production with
two-~\cite{Bartl:2003tr,Bartl:2003ck,Bartl:2003gr,Choi:2003pq,
Bartl:2004ut,AguilarSaavedra:2004dz,Choi:2003fs,Bartl:2009pg,Kittel:2004rp}
and three-body
decays~\cite{Kizukuri:1990iy,Choi:1999cc,Bartl:2004jj,AguilarSaavedra:2004hu,
Choi:2005gt}, charginos with
two-~\cite{Choi:2000ta,Bartl:2004vi,Kittel:2004kd,
AguilarSaavedra:2004ru,Bartl:2008fu,Dreiner:2010ib} and three-body
decays~\cite{Kizukuri:1993vh,Bartl:2006yv}, also with transversely
polarized beams~\cite{MoortgatPick:2005cw,Bartl:2004xy,Bartl:2005uh,
Choi:2006vh,Bartl:2006bn,Bartl:2007qy}, have been studied.
\medskip
Therefore, we present in this paper the first experimentally-oriented
analysis with regard to the observation of CP asymmetries on the basis
of a full detector simulation. To show the feasibility of a
measurement of triple product asymmetries, we focus on neutralino
production~\cite{Bartl:2003tr,AguilarSaavedra:2004dz}
\begin{eqnarray}
e^+ + e^-&\to& \tilde\chi^0_i+\tilde\chi^0_1
\label{production}
\end{eqnarray}
with longitudinally polarized beams and the subsequent leptonic
two-body decay of one of the neutralinos into the near lepton
\begin{eqnarray}
\tilde\chi^0_i&\to& \tilde\ell_R + \ell_N,
\label{decay_1}
\end{eqnarray}
and that of the slepton into a far lepton
\begin{eqnarray}
\tilde\ell_R&\to&\tilde\chi^0_1+\ell_F; \qquad \ell= e,\mu.
\label{decay_2}
\end{eqnarray}
Fig.~\ref{shematic picture} shows a schematic picture of neutralino
production and its decay chain.
The CP-sensitive spin correlations of the neutralino in its production
process allow us to probe the phase of the Higgsino mass parameter
$\mu$, and the gaugino parameter~$M_1$.
\begin{figure}[t]
\centering
\includegraphics[scale=1]{schematic.eps}
\caption{\label{shematic picture}
Schematic picture of neutralino production and decay.}
\end{figure}
\medskip
In order to effectively disentangle different signal samples and
reduce SM and SUSY backgrounds we apply a method of kinematic
reconstruction. A similar approach has been studied successfully for
the LHC~\cite{MoortgatPick:2010wp}. The kinematic reconstruction was
also considered to study selectron and neutralino properties at the
ILC~\cite{AguilarSaavedra:2003hw,AguilarSaavedra:2003ur,Berggren:2005gp}.
Here, we demonstrate that it can be used as an effective signal
selection method, greatly improving the sensitivity to the effects of
CP violation. In particular, compared to the previous studies of
process~\eqref{production}, Ref.~\cite{AguilarSaavedra:2004dz}, we are
able to suppress slepton and $WW$ contamination to $\mathcal{O}(10\%)$
level.
\medskip
To investigate in detail the prospect of measuring CP-sensitive
observables at the ILC we perform a full detector simulation of the
International Large Detector (ILD) concept. We include all relevant SM
and SUSY background processes in our study, simulated with a realistic
beam energy spectrum (beamstrahlung and initial state radiation (ISR)), beam backgrounds and a beam polarization of
$(P_{e^-}, P_{e^+}) = (0.8,-0.6)$, which enhances the cross section of
our signal and the size of the asymmetry. We apply the method of
kinematic reconstruction to a preselected sample of signal event
candidates in order to efficiently reject any background and to
disentangle the decays of $\tilde{\chi}^0_2$ from $\tilde{\chi}^0_3$.
We determine the CP asymmetries with the selected signal events and
study the sensitivity to determine the values of the CP phases via a
fit.
\medskip
The paper is organized as follows. In Sec.~\ref{theo frame} we
introduce the theoretical framework for the used CP-sensitive
observable and we apply it for the studied process in the chosen
benchmark scenario. Section~\ref{sec:kinrecoTheo} discusses the
kinematic aspects of signal versus background selection.
Section~\ref{Numerical results} treats the full detector simulation.
Finally in Sec.~\ref{fitting-proc} the SUSY parameters including the
CP phases are derived via a fit of the CP-odd asymmetries together
with masses and cross sections. Appendix~\ref{sec:app3} provides
details for the reconstruction of $W$ and $\tilde{\ell}$ production
and App.~\ref{sec:NeutralinoMixing} recapitulates the neutralino
mixing and its parameters.
\section{Theoretical framework\label{theo frame}}
\subsection{CP-odd observables\label{CP-odd observables}}
CP-violating observables in collider-based experiments are based on
the invariance under CPT$_{\rm N}$, where C is charge conjugation, P
stands for parity transformation and T for time reversal. The index
N denotes 'naive' time reversal, i.e. time reversal but without
interchanging initial with final states and therefore can be tested in
collider-based experiments. At tree level of perturbation theory,
observables odd under T$_{\rm N}$ transformations are also odd under the
'true' time reversal T.
Therefore, it is useful to categorize CP-violating observables in two
classes~\cite{Atwood:2000tu}: those that are even under T$_{\rm N}$
and those that are odd under T$_{\rm N}$ operation. Under the absence
of final state interactions (FSI), CPT$_{\rm N}$-even operators relate
T$_{\rm N}$-odd symmetries uniquely with CP-odd
transformations\cite{Valencia:1994zi}. Contrary, CPT$_{\rm N}$-odd
operators (i.e.\ CP-odd but T$_{\rm N}$-even) can have nonzero
expectation values only if FSI are present that give a non-trivial
phase (absorptive phase) to the amplitude. Such a phase can arise for
instance in loop diagrams.
Examples for T$_{\rm N}$-odd observables are triple products that
arise from the terms $\epsilon[p_1,p_2,p_3,p_4]$, where $p_i$ are
4-vectors representing spins or momenta and $\epsilon$ is the
antisymmetric Levi-Civita tensor. Consequently, such T$_{\rm N}$-odd
signals can only be observed in processes where at least four
independent momenta (or their spin orientations) are involved. The
$\epsilon$-tensor can then be expanded in a series of four triple
products $\epsilon[p_1,p_2,p_3,p_4]=E_1\, \mathbf{p}_2 \cdot
(\mathbf{p}_3\times \mathbf{p}_4)\pm \ldots$ that can be evaluated in
a suitable and specific kinematical system. The T$_{\rm N}$-odd
asymmetries are then composed by the corresponding triple products.
\subsection{Neutralino production and decay processes \label{CP-neutralino}}
Neutralinos are mixed states of the supersymmetric partners of the
neutral gauge and Higgs bosons and depend on the phases $\phi_1$ and
$\phi_{\mu}$, see App.~\ref{sec:NeutralinoMixing}. CP-violating
effects in the neutralino production and decay arise at tree level and
can lead to CP-sensitive asymmetries due to neutralino spin
correlations.
In neutralino production effects from CP-violating phases can only
occur if two different neutralinos are produced, $e^+ e^- \to
\tilde\chi^0_i\tilde\chi^0_j$, $i\neq j$. Each of the produced
neutralinos has a polarization with a component normal to the
production plane~\cite{Choi:1999cc,Choi:2001ww,MoortgatPick:1999di}.
This polarization leads to asymmetries in the angular distributions of
the decay products.
In our process the only T$_{\rm N}$-odd contribution originates from
the production process. It is proportional to $\epsilon[p_{e^+}
p_{e^-} s_{\tilde{\chi}^0_i} p_{\tilde{\chi}^0_i}]$ leading to
$\epsilon[p_{e^+} p_{e^-} p_{{\ell}_N} p_{\tilde{\ell}_R}]$ due to
spin correlations caused by the mentioned neutralino polarization
normal to the production plane. Applying momentum conservation
$p_{\tilde{\ell}_R}=p_{\tilde{\chi}^0_1}+p_{\ell_F}$ allows one to
extract the T$_{\rm N}$-odd triple product of the beam and the final lepton
momenta~\cite{Bartl:2003tr},
\begin{eqnarray}
{\mathcal T} &=&
({\mathbf p}_{e^-} \times {\mathbf p}_{\ell_N^+}) \cdot {\mathbf p}_{\ell_F^-}
\; \equiv \;
({\mathbf p}_{e^-},{\mathbf p}_{\ell_N^+}, {\mathbf p}_{\ell_F^-}),
\label{AT}
\end{eqnarray}
which projects out the CP-sensitive parts.
The corresponding T$_{\rm N}$-odd asymmetry is then
\begin{eqnarray}
{\mathcal A}({\mathcal T} ) &=&
\frac{\sigma({\mathcal T}>0) - \sigma({\mathcal T}<0)}
{\sigma({\mathcal T}>0) + \sigma({\mathcal T}<0)},
\label{eq:asyth}
\end{eqnarray}
where $\sigma$ is the cross section for neutralino production and
decay, Eqs.~(\ref{production})-(\ref{decay_2}).
Note: in the case of the 3-body neutralino decay, $\tilde{\chi}^0_i\to
\ell^+\ell^-\tilde{\chi}^0_1$, one also obtains T$_{\rm N}$-odd
contributions originating only from the decay
process~\cite{Choi:2005gt,Ellis:2008hq}. Therefore further T$_{\rm
N}$-odd asymmetries can be composed that contribute also in case of
same-neutralino pair production and can offer tools for disentangling
the different phases.
The CP-sensitive asymmetries, Eq.~\eqref{eq:asyth}, depend on the
charge of the leptons~\cite{Deppisch:2009nj} and the following
relations are given:
\begin{eqnarray} \label{eq:leptonexch}
{\mathcal A}({\mathbf p}_{e^-}, {\mathbf p}_{\ell_N^+}, {\mathbf p}_{\ell_F^-})
&=& -{\mathcal A}({\mathbf p}_{e^-}, {\mathbf p}_{\ell_N^-}, {\mathbf p}_{\ell_F^+})
\nonumber \\
&=& -{\mathcal A}({\mathbf p}_{e^-}, {\mathbf p}_{\ell_F^-}, {\mathbf p}_{\ell_N^+})
\nonumber \\
& =& +{\mathcal A}({\mathbf p}_{e^-}, {\mathbf p}_{\ell_F^+}, {\mathbf p}_{\ell_N^-}),
\end{eqnarray}
neglecting FSI contributions.
Note that a true CP-odd asymmetry, where also an absorptive phase from FSI or
finite-width effects is automatically eliminated,
can be defined as
\begin{eqnarray}\label{eq:genuinecp}
{\mathcal A}^{\rm CP} &=& \frac{1}{2}({\mathcal A} -\bar{\mathcal A} ),
\end{eqnarray}
where $\bar{\mathcal A}$ denotes the T$_{\rm N}$-odd asymmetry for the
CP-conjugated process~\cite{Jarlskog:1989bm}. In our case this leads
to a separate measurement of asymmetries for a positive ($\ell^+_N$)
and negative ($\ell^-_N$) near lepton. If ${\mathcal A}^{\rm CP} \neq
0$ holds than we observe a genuine CP-violating effect. Therefore, it
is important to tag the charge of the near and far leptons in order to
establish CP violation in the process
\eqref{production}-\eqref{decay_2}.
\medskip
\begin{figure}[t]
\centering
\subfigure[]{ \hspace{-1.3cm} \includegraphics[width=0.44\textwidth]{asym_phi1_pol08-06.eps}
}
\hspace{1.5cm}
\subfigure[]{ \hspace{-1.3cm} \includegraphics[width=0.44\textwidth]{asym_phimu_pol08-06.eps}}
\caption{
Dependence of the asymmetry ${\mathcal A}$, Eq.~\eqref{eq:asyth},
(a) on the phase $\phi_1$ (with $\phi_\mu=0$),
(b) the phase $\phi_\mu$ (with $\phi_1=0$),
for neutralino production $e^+e^- \to \tilde\chi^0_1\tilde\chi^0_{2}$
(solid), and
$e^+e^- \to \tilde\chi^0_1\tilde\chi^0_{3}$ (dashed),
and subsequent decay
$\tilde\chi_i^0\to\tilde\ell_{R} \ell_N$,
$\tilde\ell_{R} \to \tilde\chi_1^0 \ell_F$,
(for $\ell=e$ or $\mu$), at $\sqrt s =500$~GeV and
polarized beams $(P_{e^-}, P_{e^+}) = (0.8,-0.6)$.
The other MSSM parameters are given in Tab.~\ref{tab:scenario}.
In the left panel, along the flat line of the asymmetry (solid)
the decay $\tilde\chi^0_{2}\to \tilde\ell_{R} \ell$ is closed.
}
\label{fig:Asym}
\end{figure}
\subsection{Benchmark scenario}\label{sec:benchmark}
\begin{table}[t]
\renewcommand{\arraystretch}{1.7}
\vspace{1cm}
\begin{center}
\begin{tabular}{cccccccc} \toprul
\multicolumn{1}{c}{$M_2$}
& \multicolumn{1}{c}{$|M_1|$}
& \multicolumn{1}{c}{$|\mu|$}
& \multicolumn{1}{c}{$\phi_{1}$}
& \multicolumn{1}{c}{$\phi_\mu$}
& \multicolumn{1}{c}{$\tan{\beta} $}
& \multicolumn{1}{c}{$M_{\tilde{E}}$}
& \multicolumn{1}{c}{$M_{\tilde{L}}$}
\\\hline
\multicolumn{1}{c}{$300~{\rm GeV}$}
& \multicolumn{1}{c}{$150~{\rm GeV}$}
& \multicolumn{1}{c}{$165~{\rm GeV}$}
& \multicolumn{1}{c}{$0.2\pi$}
& \multicolumn{1}{c}{$0$}
& \multicolumn{1}{c}{$10$}
& \multicolumn{1}{c}{$166~{\rm GeV}$}
& \multicolumn{1}{c}{$280~{\rm GeV}$}
\\\bottomrul
\end{tabular}
\end{center}
\renewcommand{\arraystretch}{1.0}
\caption{
MSSM parameters of the benchmark scenario at the electroweak scale, see Sec.~\ref{sec:benchmark}.
\label{tab:scenario}}
\end{table}
For our full simulation study, we have chosen a benchmark scenario
with the relevant MSSM parameters given in Tab.~\ref{tab:scenario}.
Since the phase of the Higgsino mass parameter is strongly constrained
by EDM bounds, we have set it to zero. The value of the gaugino phase
$\phi_1=0.2\pi$ approximately corresponds to the maximum of the CP
asymmetries, see Fig.~\ref{fig:Asym}.
The scenario was chosen to have an enhanced neutralino mixing close to
a level-crossing of the neutralino states $\tilde\chi_2^0$ and
$\tilde\chi_3^0$ for $\phi_1 = 0$, and of $\tilde\chi_1^0$ and
$\tilde\chi_2^0$ for $\phi_1 = \pi$, which leads to large
CP asymmetries.
\medskip Further, we have assumed beam polarizations of
$(P_{e^-},P_{e^+})=(0.8,-0.6)$ enhancing slightly the SUSY cross
section and the asymmetries. At the same time, this choice suppresses
the background from $WW$-pair production, $\sigma(e^+e^-\to WW) =
0.7$~pb (compared with 7~pb for unpolarized beams), and also chargino
pair production $\sigma(e^+e^-\to\tilde\chi_i^\pm\tilde\chi_j^\mp) =
110$~fb (410~fb, respectively).
Thus, the scenario is optimized to yield large asymmetries, large
cross sections and sizable neutralino branching ratios into electrons
and muons, as listed in Tab.~\ref{tab:masses}. We have set
$A_{\tau}=-250$ GeV in the stau sector, which has low impact on the
neutralino branching ratios. Also we have chosen the slepton masses
such that $\tilde\chi_2^0$ is close in mass with the slepton
$\tilde\ell_{R}$. This leads to soft leptons from the decay
$\tilde\chi_2^0 \to \tilde\ell_{R} \ell$, as can be seen in
Fig.~\ref{fig:mu160_edist_plot}, which has to be taken care of in the
lepton identification described in Sec.~\ref{sec:preselection}.
\begin{table}[t]
\renewcommand{\arraystretch}{1.7}
\begin{center}
\begin{tabular}{lcc}
\toprule
masses & $ m_{\tilde{\chi}^0_1} = 117~{\rm GeV}$ & $ m_{\tilde \ell_R} = 166~{\rm GeV} $ \\
& $ m_{\tilde{\chi}^0_2} = 169~{\rm GeV}$ & $ m_{\tilde \ell_L} = 280~{\rm GeV}$ \\
& $ m_{\tilde{\chi}^0_3} = 181~{\rm GeV}$ & $m_{\tilde\tau_1} = 165~{\rm GeV}$ \\
& $ m_{\tilde{\chi}^0_4} = 330~{\rm GeV}$ & $m_{\tilde\tau_2} = 280~{\rm GeV}$ \\
& $ m_{\tilde{\chi}^\pm_1} = 146~{\rm GeV} $ & $ m_{\tilde\nu} = 268~{\rm GeV} $ \\
& $m_{\tilde{\chi}^\pm_2} = 330~{\rm GeV}$ & $m_{\tilde\nu_\tau} = 268~{\rm GeV}$ \\ \hline
cross sections & $\sigma(e^+e^-\to\tilde\chi_1^0\tilde\chi_2^0) = 244~{\rm fb} $ & $\sigma(e^+e^-\to\tilde{e}^+_R\tilde{e}^-_R) = 304~{\rm fb}$ \\
& $\sigma(e^+e^-\to\tilde\chi_1^0\tilde\chi_3^0) = 243~{\rm fb} $ & $\sigma(e^+e^-\to\tilde{\mu}^+_R\tilde{\mu}^-_R) = 97~{\rm fb}$ \\ \hline
branching ratios & ${\rm BR}(\tilde\chi_2^0\to \tilde \ell_R \ell) = 55 \%$ & ${\rm BR}(\tilde\chi_2^0\to \tilde \tau_1 \tau) = 45 \%$ \\
& ${\rm BR}(\tilde\chi_3^0\to \tilde \ell_R \ell) = 64 \%$ & ${\rm BR}(\tilde\chi_3^0\to \tilde \tau_1 \tau) = 36 \%$ \\ \hline
asymmetries & ${\mathcal A}({\mathcal T} )_{\tilde\chi_1^0\tilde\chi_2^0} = -9.2\% $ & ${\mathcal A}({\mathcal T} )_{\tilde\chi_1^0\tilde\chi_3^0} = 7.7\% $ \\ \bottomrule
\end{tabular}
\end{center}
\renewcommand{\arraystretch}{1.0}
\caption{Masses, production cross sections, neutralino branching ratios and asymmetries, Eq.~\eqref{eq:asyth},
for the benchmark scenario, see Tab.~\ref{tab:scenario}, calculated using the formulas presented in~\cite{Kittel:2004rp}.
The ILC cross sections are for $\sqrt s =500$~GeV and
polarized beams $(P_{e^-}, P_{e^+}) = (0.8,-0.6)$.
The branching ratios are summed over
$\ell=e,\mu$ and both slepton charges.
\label{tab:masses}}
\end{table}
\begin{figure}[t]
\vspace{1cm}
\centering
\subfigure[]{\label{fig:neu2edist
\includegraphics[scale=1.1]{energydist_N2.eps}
} \hspace{0.8cm}
\subfigure[]{\label{fig:neu3edist
\includegraphics[scale=1.1]{energydist_N3.eps}
}
\caption{ Energy distributions (each normalized to 1) of the near
lepton $\ell_N$ (solid), and the far lepton $\ell_F$ (dashed),
from neutralino production $e^+e^- \to
\tilde\chi^0_1\tilde\chi^0_i$ for (a)~$i=2$, and (b)~$i=3$, with
subsequent decay $\tilde\chi_i^0\to \tilde \ell_R \ell_N$,
$\tilde\ell_R\to \tilde\chi_1^0\ell_F$, for the benchmark scenario
as given in Tabs.~\ref{tab:scenario} and \ref{tab:masses}.
\label{fig:mu160_edist_plot}
}
\end{figure}
In general, our analysis is relevant for scenarios
with strong gaugino-higgsino mixing in the neutralino sector, usually leading to sizable asymmetries.
In particular, for $|\mu|\lsim |M_2| \lsim 300$~GeV the asymmetries can reach several
percent, the neutralino pair-production cross sections reach more than
$50$~fb, and the neutralino branching ratios into electrons and muons are
of the order of several $10$\%, see also Ref.~\cite{Bartl:2003tr} for more details. In any case, the selectron and smuon masses should fulfil $m_{\tilde{\chi}_1^0} < m_{\tilde{\ell}_R} < m_{\tilde{\chi}_2^0}$, so that at least one relevant decay channel remains open. Decreasing the selectron mass will result in larger asymmetries and production cross sections.
\section{Kinematic selection of signal and background\label{sec:kinrecoTheo}}
In order to measure the CP asymmetries, Eq.~\eqref{eq:asyth}, we have
to separate the signal lepton pairs originating from
$\tilde\chi_1^0\tilde\chi_2^0$ and $\tilde\chi_1^0\tilde\chi_3^0$
production, respectively. This is essential, since in our scenario
the corresponding CP asymmetries, $ {\mathcal
A}(\tilde\chi_1^0\tilde\chi_2^0) = -9.2\% $ and ${\mathcal
A}(\tilde\chi_1^0\tilde\chi_3^0) = 7.7\% $, have opposite sign.
Large CP asymmetries naturally occur when the neutralinos are mixed
states of gauginos and Higgsinos, which often implies that they are
close in mass. In addition we need an efficient method for background
separation. The CP-even backgrounds will reduce the asymmetries, since
they contribute to the denominator, but cancel out in the numerator of the
asymmetries, see Eq.~\eqref{eq:asyth}. Among the most severe SM and
SUSY background processes are $W$ pair production and slepton pair
production.
\medskip
Owing to a known center-of-mass energy and a well-defined initial
state one may attempt to perform a full kinematic reconstruction of
the events at the ILC. Unlike in the case of the
LHC~\cite{MoortgatPick:2009jy,MoortgatPick:2010wp}, this is already
possible with very short decay chains. Assuming that the masses of
intermediate and invisible particles are known from other
measurements, the full reconstruction can be performed even when only
two particles are visible in the final state.
\medskip
In the following, we extend the method of Ref.~\cite{Bartl:2005uh} to
reconstruct the pair of signal leptons from the neutralino decay,
Eq.~\eqref{production}. We show that even for two neutralino states
that are close in mass, here $\neu{2}$ and $\neu{3}$, the final
leptons can be correctly assigned to their mother particle. A similar
procedure to identify and suppress background from $W$ and slepton
pair production is described in App.~\ref{sec:app3}, see also
Ref.~\cite{Buckley:2007th}. Finally, we discuss how well the kinematic
selection and reconstruction works at the MC level.
\subsection{Kinematic constraints from neutralino production}\label{sec:neurec}
In the center-of-mass system of neutralino pair production
the momenta and energies are fixed~\cite{MoortgatPick:1999di} :
\begin{eqnarray}
E_{\neu{i}} =\frac{s+m_{\neu{i}}^2-m_{\neu{j}}^2}{2 \sqrt{s}},\quad
E_{\neu{j}} =\frac{s+m_{\neu{j}}^2-m_{\neu{i}}^2}{2 \sqrt{s}},\quad
|{\mathbf p}_{\neu{i,j}}| =\frac{\lambda^{\frac{1}{2}}
(s,m_{\neu{i}}^2,m_{\neu{j}}^2)}{2 \sqrt{s}},
\end{eqnarray}
with the beam energy $E=\sqrt{s}/2$, the neutralino masses
$\mneu{i}, \mneu{j}$, and $\lambda(x,y,z) = x^2+y^2+z^2-2(xy+xz+yz)$.
The neutralino production is followed by the two-body decay chain of
one of the neutralinos $\neu{i}$ via a slepton,
\begin{eqnarray}
\tilde{\chi}^0_i&\to& \tilde{\ell}_R + \ell_N \to \tilde{\chi}^0_1 + \ell_F + \ell_N,
\quad \ell= e,\mu.
\end{eqnarray}
In our signal process, Eqs.~\eqref{production}-\eqref{decay_2}, we
have $\neu{j} = \neu{1}$ and it escapes undetected.
In the following, we assume that the near and far leptons can be
distinguished via their different energy distributions.\footnote{This
is not needed for the determination of the asymmetry, see
Eq.~\eqref{eq:leptonexch}, but will be exploited for the event
selection.} For our scenario, the leptons from $\neu{1}\neu{2}$
production and decay have distinct energy ranges, see
Fig.~\ref{fig:mu160_edist_plot}(a). The leptons from $\neu{1}\neu{3}$
production and decay only have a small overlap in the energy window
$E_\ell\in [18,38] \gev$, see Fig.~\ref{fig:mu160_edist_plot}(b).
Events are discarded if both leptons happen to fall into this energy
range. We now choose a coordinate system such that the measured
momenta are
\begin{eqnarray}
{\mathbf p}_{\ell_N} &=& |{\mathbf p}_{\ell_N}|~ (0,0,1), \label{defl1}\\
{\mathbf p}_{\ell_F} &=& |{\mathbf p}_{\ell_F}| ~
(\sin \theta_{NF},\;0, \;\cos \theta_{NF}),
\quad \theta_{NF} \in[0,\pi],
\label{def2}\label{defl2}
\end{eqnarray}
where $\theta_{NF}$ is the angle between the near and the far leptons.
In order to fully reconstruct the event, the decay angles of the sleptons need to be resolved
\begin{eqnarray}
{\mathbf p}_{\tilde\ell} &=& |{\mathbf p}_{\tilde\ell}|~
(\sin b \cos B,\; \sin b \sin B, \;\cos b),
\quad b\in[0,\pi],\quad B\in[0,2\pi].
\label{defmomenta}
\end{eqnarray}
The slepton momentum, ${\mathbf p}_{\tilde\ell}^2=
E_{\tilde\ell}^2-m_{\tilde\ell}^2$, is already fixed due to energy
conservation, $E_{\tilde\ell}=E_{\neu{i}}-E_{\ell_N}$. Using also
momentum conservation, $ {\mathbf p}_{\neu{i}}^2 = ({\mathbf
p}_{\ell_N}+{\mathbf p}_{\tilde\ell})^2, $ the polar angle, $b =
\varangle \, ({\mathbf p}_{\ell_N},{\mathbf p}_{\tilde\ell})$, can be
determined
\begin{eqnarray}
\cos b &=& \frac{{\mathbf p}_{\neu{i}}^2- {\mathbf p}_{\ell_N}^2-
{\mathbf p}_{\tilde\ell}^2}{2 |{\mathbf p}_{\ell_N}||{\mathbf
p}_{\tilde\ell}|}, \qquad
\sin b = +\sqrt{1-\cos^2 b}.
\label{eq:cosb}
\end{eqnarray}
Using also momentum conservation in the slepton decay, $ {\mathbf
p}_{\tilde\ell} = {\mathbf p}_{\ell_F}+{\mathbf p}_{\neu{1}}, $ a
similar relation can be obtained for the azimuthal angle
\begin{eqnarray}
\cos B &=&
\frac{1}{ \sin b \sin \theta_{NF}}
\left(
\frac{{\mathbf p}_{\ell_F}^2+{\mathbf p}_{\tilde\ell}^2
-{\mathbf p}_{\neu{1}}^2
}{2 |{\mathbf p}_{\ell_F}||{\mathbf p}_{\tilde\ell}|}
- \cos b \cos \theta_{NF}
\right),
\label{eq:angleb}
\end{eqnarray}
and ${\mathbf p}_{\neu{1}}^2= E_{\neu{1}}^2-m_{\neu{1}}^2$ is obtained
from energy conservation
$E_{\neu{1}}=E_{\neu{i}}-E_{\ell_N}-E_{\ell_F}$. The kinematic
variables $\cos b$ and $\cos B$ solely depend on the center-of-mass
energy $s$, the lepton energies, $ E_{\ell_N}$ and $E_{\ell_F}$, the
angle between the leptons, ${\mathbf p}_{\ell_N} \cdot {\mathbf
p}_{\ell_F}$, and finally on the contributing particle masses,
$m_{\tilde\chi_i^0}$, $m_{\tilde\chi_1^0}$, and $m_{\tilde\ell_R}$.
Thus, there only remains an ambiguity for $\sin B$, since for
$B\in[0,2\pi]$ we have $\sin B = \pm\sqrt{1-\cos^2 B}$. This ambiguity
in the azimuthal angle is irrelevant for the efficiency of the event
selection\footnote{ The ambiguity is related to the neutralino
momentum, for which we have two possible solutions
\begin{eqnarray}
{\mathbf p}_{\neu{i}}={\mathbf p}_{\ell_N}
+ {\mathbf p}_{\tilde\ell}=
\left(\begin{array}{c}
\phantom{\pm}
|{\mathbf p}_{\tilde\ell}|\sin b \cos B\\
\pm|{\mathbf p}_{\tilde\ell}|\sin b \sin B\\
|{\mathbf p}_{\ell_N}|+|{\mathbf p}_{\tilde\ell}|\cos b
\label{chipm}
\end{array}\right).
\end{eqnarray}
For this reason, the neutralino production plane cannot be resolved
in $e^+e^-\to\tilde\chi_i^0\tilde\chi_j^0$ processes for $j=1$.
For $j\ge2$, the decay of $\tilde\chi_j^0$ could be included
in order to reconstruct the production plane~\cite{Bartl:2005uh}.
In that case larger triple product asymmetries can be studied,
which include the neutralino momentum itself~\cite{Bartl:2003tr}.
}.
\subsection{Method of kinematic event selection}\label{sec:BGSupp}
For a given lepton pair, we apply the following kinematic selection
method. The aim is to assign the correct origin of the lepton pair,
which can be signal, $e^+e^-\to\tilde\chi_i^0\tilde\chi_1^0$, or
background $e^+e^-\to W^+W^-$, $\tilde{\ell}^+_R\tilde{\ell}^-_R$.
Thus, we have four systems of equations, one for each possible
production process. For each candidate event we employ the following
kinematic selection:
\begin{itemize}
\item We apply the reconstruction procedure from
Sec.~\ref{sec:neurec}, assuming $\neu{1}\neu{2}$ and
$\neu{1}\neu{3}$ production. Thus, we calculate $\cos b$ and $\cos
B$, Eqs.~\eqref{eq:cosb} and \eqref{eq:angleb}, with $m_{\neu{i}} =
m_{\neu{2}}$ ($m_{\neu{i}} = m_{\neu{3}}$) for $\neu{1}\neu{2}$
($\neu{1}\neu{3}$) production.
\item We apply the reconstruction procedure from App.~\ref{sec:app3},
assuming $WW$ and slepton pair production. Thus, we calculate two
values of $y^2$, Eq.~\eqref{eq:defy}.
\item The event solves the system of equations if
\begin{equation}
|\cos b| < 1 \qquad \mathrm{and} \qquad |\cos B| < 1 \;, \label{eq:reco_neu}
\end{equation}
when neutralino production has been assumed, and
\begin{equation}
y^2 > 0 \;, \label{eq:reco_slWW}
\end{equation}
when $W$/slepton production has been assumed.
\item The event is accepted and labeled as coming from a given process
only if it solves {\it exactly one} out of the four above mentioned
systems of equations, i.e. it fulfills condition~\eqref{eq:reco_neu}
for $\neu{1}\neu{2}$ or $\neu{1}\neu{3}$ production, or
condition~\eqref{eq:reco_slWW} for $W$ or slepton production.
\end{itemize}
In order to demonstrate the efficiency of this procedure we perform a
Monte Carlo simulation of $\neu{1}\neu{2}$, $\neu{1}\neu{3}$,
$W^+W^-$, and $\tilde{\ell}_R^+\tilde{\ell}_R^-$ production and their
leptonic decays, using \texttt{Whizard 1.96}~\cite{Kilian:2007gr}. We
use the MSSM parameters, Tab.~\ref{tab:scenario}, with an integrated
luminosity of ${\mathcal{L}} = 500 \fb^{-1}$, and a beam polarization
of $(P_{e^-}, P_{e^+}) = (0.8,-0.6)$ with realistic beam
spectra\footnote{We include ISR and beamstrahlung, which slightly degrade the number of
reconstructed events even if the correct process is assumed for a
given event. Additionally, these effects will increase the number of
false solutions, leading to wrong assignments.}. In
Tab.~\ref{tab:reconstruction} the results of the event selection are
summarized. Without any additional cuts, the selection method gives an
excellent separation between the different samples at the MC level.
This method is still performing well after a full detector simulation,
as demonstrated in the following section.
\begin{table}[!t] \renewcommand{\arraystretch}{1.3}
\begin{center}
\vspace{0.3cm}
\begin{tabular}{ccccccc} \cmidrule[\heavyrulewidth]{4-7} & & &
\multicolumn{4}{c}{system solved {\emph{only}}} \\ \cmidrule{4-7} &
& & $\neu{1}\neu{2}$& $\neu{1}\neu{3}$ & $\tilde{\ell}_R^+
\tilde{\ell}_R^-$ & $W^+W^-$ \\ \cmidrule[\heavyrulewidth]{1-7}
\multirow{4}*{\begin{sideways}true
process\phantom{bb}\end{sideways}} & $64$ k & $\neu{1}\neu{2}$ &
41566 & 788 & 64 & 856 \\ \cmidrule{2-7} & $74$ k & $\neu{1}\neu{3}$
& 100 & 25513 & 369 & 873 \\ \cmidrule{2-7} & $200$ k &
$\tilde{\ell}_R^+ \tilde{\ell}_R^-$ & 181 & 1801 & 43919 & 3400 \\
\cmidrule{2-7} & $8.8$ k & $W^+W^-$ & 0 & 13 & 37 & 6802 \\
\bottomrule && purity & 99\% & 91\% & 99\% & 57\% \\ \cmidrule{3-7}
&&efficiency& 65\% & 34\% & 22\% & 77\% \\ \bottomrule
\end{tabular}
\end{center}
\caption{The numbers of leptonic events from the
pair production of neutralinos, sleptons and $W$ bosons,
with their identification according to the kinematic selection
procedure at the generator level, see Sec.~\ref{sec:BGSupp}. The events are simulated
for our benchmark scenario, Tab.~\ref{tab:scenario},
with an integrated luminosity of ${\mathcal{L}} = 500 \fb^{-1}$
and beam polarization $(P_{e^-}, P_{e^+}) = (0.8,-0.6)$ for $\sqrt{s}= 500$~GeV.
\label{tab:reconstruction} }
\end{table}
The method can be successfully applied also for different particle mass spectra. In our benchmark scenario, due to the small difference between $\tilde{\ell}_R$ and $\tilde{\chi}_2^0$ masses, a separation of the near and far leptons, and of the $\tilde{\chi}^0_2$ and $\tilde{\chi}^0_3$ signals was rather straightforward, see Fig.~\ref{fig:mu160_edist_plot}. However, scenarios with different $m_{\tilde{\chi}_2^0} - m_{\tilde{\ell}_R}$ can turn out to be more demanding. In order to test the applicability of the kinematic reconstruction, we consider a scenario with $m_{\tilde{\ell}_R} = 146$~GeV and the other parameters kept as in the benchmark point. It can be regarded as the worst case scenario since the energies of leptons from neutralino $\tilde{\chi}^0_2$, $\tilde{\chi}^0_3$, and slepton decays are all in the 10 to 80~GeV range. Therefore, in the kinematic reconstruction, one has to take into account two possible assignments of the near and far leptons. This, in principle, could result in a reduction of the efficiency. Nevertheless, in the case of $\tilde{\chi}^0_1 \tilde{\chi}^0_3$ production the efficiency is about 40\%. It drops to 8\% for $\tilde{\chi}^0_1 \tilde{\chi}^0_2$ pairs, since they likely also solve the kinematic on-shell conditions for the $\tilde{\chi}^0_1 \tilde{\chi}^0_3$ production process. The purity, however, remains at about 90\%. In half of the cases, one can also correctly and unambiguously assign near and far leptons.
\section{Full detector simulation study}
\label{Numerical results}
The next step of our analysis is passing the generated signal
events and all relevant SM and SUSY background events through
a full ILD simulation and event reconstruction. After
discussing the preselection cuts for the leptonic event
candidates, we apply the kinematic selection as described in
the previous section.
\subsection{Detector simulation and event reconstruction}
For the present study we have performed a full simulation of the ILD
detector designed for the ILC. A detailed description of the detector
concept can be found in Ref.~\cite{Group:2010eu}.
The ILD is a concept under study for a multipurpose particle detector
with a forward-backward symmetric cylindrical geometry. It is designed
for an excellent precision in momentum and energy measurements over a
large solid angle. The tracking system consists of a multi-layer
pixel-vertex detector, surrounded by a system of strip and pixel
detectors and a large volume time projection chamber. The track
finding efficiency is 99.5\% for momenta above 1~GeV and angles down
to $\theta=7^\circ$, while the transverse momentum resolution is
$\delta\left(1/{p}_{\rm T}\right)\sim 2\cdot
10^{-5}$~GeV$^{-1}$. The SiW electromagnetic calorimeter~(ECAL) is
highly segmented with a transverse cell size of 5~mm $\times$ 5~mm and
20 layers. It provides an energy resolution of $(16.6\pm
0.1)/\sqrt{E(\text{GeV})}\oplus (1.1\pm 0.1)\%$ for the measurement of
electrons and photons, and also the steel-scintillator hadronic
calorimeter is highly granular and optimized for Particle Flow
reconstruction. The calorimeters are surrounded by a large
superconducting coil, creating an axial magnetic field of 3.5 Tesla.
For the simulation of the ILD, we use the {\it ILD\_00} detector
model, as implemented in the \texttt{Geant4}-based
\texttt{Mokka}~\cite{Agostinelli:2002hh,Musat:2004sp,MoradeFreitas:2004sq}
package. We have taken into account all active elements, and also
cables, cooling systems, support structures and dead regions. We have
used the radiation hard beam calorimeter~(BCAL) to reject forward
$\gamma\gamma$ events at low angles. In particular the modeling of
the response of the BCAL is relevant for the estimation of the
background from events with activity in the very forward regions.
This background was estimated by tracking electrons to the BCAL and
determining the probability of detection from a map of the expected
energy density from beamstrahlung pairs~\cite{Bechtle:2009em}.
All relevant SM backgrounds\footnote{We consider the final states
listed in Tab.~\ref{tab:cutflow}.} and SUSY signal and background
events are generated using \texttt{Whiz\-ard}~\cite{Kilian:2007gr},
for $\mathcal L = 500~\rm{fb}^{-1}$ and
$(P_{e^-},P_{e^+})=(0.8,-0.6)$. The \texttt{Whiz\-ard} generator provides an ISR structure function that resums leading soft and collinear logarithms, and hard-collinear terms up to the third order~\cite{Skrzypek:1990qs}. The beamstrahlung is simulated using the \texttt{Circe} package~\cite{Ohl:1996fi}. After the detector simulation the
events are reconstructed with \texttt{MarlinReco}~\cite{Wendt:2007iw}.
We have used the Particle Flow concept, as it is implemented in
\texttt{Pandora}~\cite{Thomson:2007zza}.
\subsection{Backgrounds and event preselection}\label{sec:preselection}
In order to clearly measure the CP-violating effects in the
production of neutralinos, we need to have
a clean sample of signal events.
Otherwise the CP asymmetry would be reduced by the CP-even
backgrounds, which enter in the denominator, see Eq.~\eqref{eq:asyth}.
We therefore apply a number of preselection cuts listed
in Tab.~\ref{tab:preselectionCuts},
to reject as much background as possible before applying
the final selection.
\begin{table
\begin{center}
\vspace{0.5cm}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{ll} \toprule
initial selection & no significant activity in BCAL \\
& number of all tracks $N_{\rm tracks} \leq 7$ \\ \hline
lepton selection & $\ell^+\ell^-$ pair with $\ell = e,\mu$ \\
& $|\cos\theta|<0.99$, min. energy $E>3$~GeV \\
& lower energetic $\ell$ with $E<18$~GeV, or \\
& \phantom{x} higher energetic $\ell$ with $E>38$~GeV \\
& higher energetic $\ell$ with $E\in[15,150]$~GeV \\
& $\theta_{\rm acop}>0.2\pi$, $\theta_{\rm acol}>0.2\pi$ \\ \hline
final preselection & ${p}_{\rm T}^{\rm miss}> 20$~GeV \\
& $E_{\rm vis}< 150$~GeV \\
& $m_{\ell\ell}<55$~GeV \\ \bottomrule
\end{tabular}
\end{center}
\renewcommand{\arraystretch}{1.0}
\caption{Preselection cuts, see Sec.~\ref{sec:preselection} for details.\label{tab:preselectionCuts}}
\end{table}
\subsubsection{Initial selection}
For efficient electron and muon identification, we apply the following
initial selection on the tracks and clusters reconstructed by the
\texttt{Pandora} Particle Flow algorithm:
\begin{itemize}
\item $\displaystyle{\frac{E_{\rm ECAL}}{E_{\rm tot}}} > 0.6 $ and
$\displaystyle{\frac{E_{\rm tot}}{p_{\rm track}}}> 0.9$ for electrons;
\item $\displaystyle{\frac{E_{\rm ECAL}}{E_{\rm tot}}} < 0.5$ and
$\displaystyle{\frac{E_{\rm tot}}{p_{\rm track}}} < 0.3$ (0.8)
for muons, with energy $E >(<)\,12$~GeV,\footnote{The cut on the
ratio of the total calorimeter energy and the track momentum
is relaxed for low-energetic muons, which deposit more energy
in the calorimeters. This ensures a reasonably high muon
identification efficiency even for low-energetic muons.}
\end{itemize}
where $E_{\rm ECAL}$ is the energy measured in the electromagnetic
calorimeter, $E_{\rm tot}$ is the total measured energy in the
calorimeters, and $p_{\rm track}$ is the measured track momentum in
the tracking detectors. $E$ is the energy of the Particle Flow object
assigned by \texttt{Pandora}, which is derived from the track momentum
in case a track is present or from the deposited energy in the
calorimeters. We require no significant activity in the BCAL to reject
$\gamma\gamma$ events. We select those events with less than eight
tracks\footnote{Although we expect to have only two isolated leptons
in our signal events, we do not tighten this cut to avoid removing
signal events due to overlaid $\gamma\gamma$ events.}, $N_{\rm
tracks} \leq 7$, which efficiently removes all sorts of hadronic
background.
\subsubsection{Lepton selection}\label{sec:lepselection}
We require that two of the tracks form a pair of opposite-sign
same-flavor leptons $\ell^+ \ell^-$, with $\ell =e$ or $\mu$. Only
electrons or muons are selected with a polar angle $|\cos\theta|<0.99$
and a minimum energy $E>3$~GeV.
There is a large contribution from beam induced
$e^+e^-\rightarrow\gamma\gamma e^+e^-\rightarrow \ell\ell e^+e^-$
($\ell=e,\mu,\tau$) background events\footnote{In this study we only
consider the class of $\gamma\gamma$ background events where both
photons from the $e^+$ and $e^-$ beam interact via a virtual
fermion. Interactions where the photon fluctuates into a
vector-meson or where the photon is highly virtual and the
interaction is best described as deep inelastic electron scattering
on a vector-meson are not considered. Since their transverse
momentum distributions are narrower, it is less likely that they
contribute to the overall
background~\cite{Bambade:2004tq,Bechtle:2009em}.}~\cite{Group:2010eu,Bechtle:2009em}.
The two outgoing beam electrons are high-energetic with a small
scattering angle, while the rest of the event forms a system of low
energy and mass. If the beam remnants escape close to the beam pipe,
and cannot be rejected by a low angle veto, the missing transverse
momentum of the event is limited, such that the remaining leptons are
almost back-to-back in the transverse projection~(T). Thus, we apply a
cut on their acoplanarity angle
\begin{equation}
\theta_{\rm acop}>0.2\pi
\qquad {\rm with} \qquad
\theta_{\rm acop}=\pi - \arccos\left(
\frac{{\mathbf p}_{\ell^+}^{\rm T} \cdot {\mathbf p}_{\ell^-}^{\rm T}}
{|{\mathbf p}_{\ell^+}^{\rm T}| \; |{\mathbf p}_{\ell^-}^{\rm T}|}
\right),
\end{equation}
where $\theta_{\rm acop} = 0$ for back-to-back ($180^\circ$) events.
Electrons or muons from $\gamma\gamma$ induced $\tau\tau$ events
usually have energies below 10 GeV and can therefore be suppressed by
exploiting that the far leptons from SUSY signal decays usually have
higher energies. To do this and to select the signal lepton pairs
$\ell^+ \ell^-$ from the neutralino $\tilde\chi^0_{2,3}$ decays, we
first use their energy distributions, see
Fig.~\ref{fig:mu160_edist_plot}. We keep events where either the
lower energetic lepton has $E<18$~GeV, or the higher energetic lepton
has $E>38$~GeV. In addition the higher energetic lepton is required to
have an energy $E\in[15,150]$~GeV.
Since the signal lepton pairs $\ell^+\ell^-$ originate from the same
parent neutralino, they follow its direction in first approximation.
The lepton pairs from SM decay processes, and also from slepton pair
decays, tend to be more back-to-back, since the leptons originate from
different mother particles. We therefore apply a cut on the
acollinearity angle between the leptons
\begin{equation}
\theta_{\rm acol} >0.2\pi
\qquad {\rm with} \qquad
\theta_{\rm acol}=\pi - \arccos\left(
\frac{{\mathbf p}_{\ell^+}\cdot{\mathbf p}_{\ell^-}}
{|{\mathbf p}_{\ell^+}|\, |{\mathbf p}_{\ell^-}|}\right).
\end{equation}
\begin{figure}[t]
\centering
\subfigure[]{\label{fig:p_T_miss_distribution}
\includegraphics[scale=0.35,angle=0]{p_T_miss_plot.eps}
}
\subfigure[]{\label{fig:InvMass_distribution}
\includegraphics[scale=0.35,angle=0]{InvMass_plot.eps}
}
\caption{ (a) Missing transverse momentum ${p}_{\rm T}^{\rm
miss}$ distribution of SM background, SUSY background and SUSY
signal after the lepton selection, see
Sec.~\ref{sec:lepselection}. (b) Invariant mass $m_{\ell\ell}$
distribution of the lepton pair after all preselection cuts except
the cut on $m_{\ell\ell}$. The events are simulated for
${\mathcal{L}} = 500 \fb^{-1}$, beam polarization $(P_{e^-},
P_{e^+}) = (0.8,-0.6)$ at $\sqrt{s}= 500$~GeV, and MSSM parameters
for our benchmark scenario, Tab.~\ref{tab:scenario}. }
\label{fig:distributions}
\end{figure}
\subsubsection{Final preselection}
In Fig.~\ref{fig:p_T_miss_distribution}, we show the missing
transverse momentum, ${p}_{\rm T}^{\rm miss}$, distribution of
the SM background, the SUSY background, and the SUSY signal after the
lepton selection. The ${\mathbf p}_{\rm T}^{\rm miss}$ is calculated
to balance out the sum of all reconstructed transverse particle
momenta in an event.
Our signal neutralinos $\tilde\chi^0_{2,3}$, but also the background
sleptons, decay into the lightest neutralino, which escapes detection,
thus giving signatures with high ${p}_{\rm T}^{\rm miss}$.
However, most background lepton pairs from beam induced $\gamma\gamma$
events have a transverse momentum typically below $10$~GeV, and are
removed by the cut ${p}_{\rm T}^{\rm miss}>
20$~GeV.\footnote{The spike in the 3rd bin of the ${p}_{\rm
T}^{\rm miss}$ distribution in
Fig.~\ref{fig:p_T_miss_distribution} is due to 4
$\gamma\gamma\rightarrow\ell\ell$ events that have a high event
weight. Due to limited CPU time and the large cross section of these
events, it is not possible to simulate an event sample corresponding
to ${\mathcal{L}} = 500 \fb^{-1}$. The final preselection cuts are
chosen such that this remaining high cross section background is
safely removed.} Due to the escaping neutralinos, we also expect a
limited total visible energy $E_{\rm vis}$ in the signal events, and
we apply the cut $E_{\rm vis}< 150$~GeV. The visible energy is
calculated as the sum of all reconstructed particle energies.
Finally we apply a cut $m_{\ell\ell}<55$~GeV on the invariant mass of
the lepton pair, see the distribution in
Fig.~\ref{fig:InvMass_distribution}, after all preselection cuts,
except the cut on $m_{\ell\ell}$. The signal lepton pair from
$\tilde{\chi}^0_3$ ($\tilde{\chi}^0_2$) decays has a sharp endpoint at
$51$~GeV ($22$~GeV), which is also exploited for mass
measurements~\cite{ball,D'Ascenzo:2009zz,Desch:2003vw}. The invariant
mass cut also removes SM backgrounds from $ZZ$ and $WW$ production. In
Fig.~\ref{fig:InvMass_distribution}, we can see the invariant mass
peak of one of the $Z$ bosons decaying into two electrons or muons,
while the other decays into a neutrino pair. The $WW$ events
contribute to the background if they either both decay directly into
same-flavor leptons, or if one of them decays into a $\tau$, which in
turn can complete the same-flavor lepton pair in the event by its
subsequent decay.
The number of remaining events after the lepton selection and the
entire event preselection is listed in Tab.~\ref{tab:cutflow}. The
most severe remaining SM background stems from $WW$ and $ZZ$
production, while the slepton pair production is the dominant SUSY
background. The difference in the numbers of selected
$\tilde{\chi}^0_2$ and $\tilde{\chi}^0_3$ decays is due to different
cross sections times branching ratios, see Tab.~\ref{tab:masses}, and
due to a reduced muon identification efficiency at low muon momenta,
which reduces the efficiency for the selection of $\tilde{\chi}^0_2$
decays.
\begin{table
\begin{center}
\vspace{0.5cm}
\renewcommand{\arraystretch}{1.2}
\begin{tabular}{llcc} \toprule class & final state & after lepton
selection & after preselection \\ \hline signal &
$\tilde\chi^0_1\tilde\chi^0_2\rightarrow\tilde\chi^0_1\tilde\chi^0_1\ell\ell$
($\ell\neq\tau$) & 31543 & 28039 \\
&
$\tilde\chi^0_1\tilde\chi^0_3\rightarrow\tilde\chi^0_1\tilde\chi^0_1\ell\ell$
($\ell\neq\tau$) & 49084 & 45966 \\ \hline SUSY &
$\tilde\ell\tilde\ell\rightarrow\tilde\chi^0_1\tilde\chi^0_1\ell\ell$
($\ell\neq\tau$)
& 108302 & 34223 \\
& $\tilde\chi^0_1\tilde\chi^0_1\tau\tau$ & 5147 & 4076 \\
& $\tilde\chi^0_1\tilde\chi^0_1\ell\ell\nu\nu$ & 681 & 528 \\ \hline
SM & $\ell\ell\nu\nu$ & 8241 & 1196 \\
& $\tau\tau$ & 13017 & 360 \\
& $\ell\ell$ ($\ell\neq\tau$) & 24113 & 0 \\
& $qq$ & 1380 & 0 \\
& $\gamma\gamma$ & 917355 & 272 \\ \bottomrule
\end{tabular}
\end{center}
\renewcommand{\arraystretch}{1.0}
\caption{Number of selected events after lepton selection and final preselection,
for ${\mathcal{L}} = 500 \fb^{-1}$, $(P_{e^-}, P_{e^+}) = (0.8,-0.6)$
at $\sqrt{s}= 500$~GeV. The MSSM parameters are given in Tab.~\ref{tab:scenario}.
\label{tab:cutflow}}
\end{table}
\subsection{Signal identification with kinematic event selection
\label{sec:kinreco}}
In order to measure our CP
asymmetry
from the preselected events, we now apply the kinematic
selection procedure, which we have described for the signal
in Sec.~\ref{sec:kinrecoTheo}, and for the $WW$ and
$\tilde\ell\tilde\ell$ backgrounds in App.~\ref{sec:app3}.
The kinematic selection allows us not only to reduce the
remaining SUSY background from slepton pair production, but
also to distinguish the lepton pairs which stem from
$\tilde\chi^0_1\tilde\chi^0_2$ or
$\tilde\chi^0_1\tilde\chi^0_3$ production and decay. This
will be essential, since in our benchmark scenario,
Tab.~\ref{tab:scenario}, the corresponding CP asymmetries
have roughly equal size, but opposite sign, see Fig.~\ref{fig:Asym}.
\medskip
For each preselected lepton pair, we require that it exclusively
solves only one of the systems of equations, as discussed in
Sec.~\ref{sec:BGSupp}. We reject all other events that solve more than
one system of equations. In Tab.~\ref{tab:reconstruction_results}, we
list the number of preselected events that fulfill this requirement.
For the lepton pairs coming from $\tilde\chi^0_1\tilde\chi^0_2$
decays, the final signal selection efficiency is 29\%, and the total
background contamination of the selected sample is about 8\%. Lepton
pairs from $\tilde\chi^0_1\tilde\chi^0_3$ decays reach a signal
selection efficiency of 27\%, while the total background contamination
of the selected sample is about 16\%.
\begin{table
\begin{center}
\label{tab:reconstruction_results}
\vspace{0.5cm}
\renewcommand{\arraystretch}{1.2}
\begin{tabular}{lcccc} \toprule
class & only $\tilde\chi^0_1\tilde\chi^0_2$ & only $\tilde\chi^0_1\tilde\chi^0_3$ & only $\tilde\ell^+_R\tilde\ell^-_R$ & only $W^+W^-$ \\ \hline
$\tilde\chi^0_1\tilde\chi^0_2\rightarrow\tilde\chi^0_1\tilde\chi^0_1\ell\ell$ ($\ell\neq\tau$) & 18343 & 615 & 51 & 855 \\
$\tilde\chi^0_1\tilde\chi^0_3\rightarrow\tilde\chi^0_1\tilde\chi^0_1\ell\ell$ ($\ell\neq\tau$) & 290 & 20132 & 372 & 635 \\
all SUSY background & 1153 & 3055 & 5626 & 951 \\
all SM background & 87 & 256 & 44 & 81 \\ \bottomrule
\hfill purity & 92\% & 84\% & -- & -- \\ \hline
\hfill efficiency & 29\% & 27\% & -- & -- \\ \bottomrule
\end{tabular}
\end{center}
\renewcommand{\arraystretch}{1.0}
\caption{Number of preselected events from Tab.~\ref{tab:cutflow},
that fulfill the requirements of the kinematic selection procedure,
discussed in Sec.~\ref{sec:kinreco}.}
\end{table}
\subsection{Measurement of the CP asymmetries}\label{sec:asymmetryexp}
The CP asymmetries, Eq.~\eqref{eq:asyth}, can now be calculated as the
difference between the number of events $N_+$ and $N_-$, with the
triple product $\mathcal{T}>0$ or $\mathcal{T}<0$, respectively,
\begin{eqnarray}
{\mathcal A}(\mathcal{T}) &=& \frac{N_+ - N_-}{N_+ + N_-}.
\label{eq:asyex}
\end{eqnarray}
We obtain
\begin{eqnarray}
{\mathcal A}({\mathbf p}_{e^-},{\mathbf p}_{\ell_N^+},{\mathbf p}_{\ell_F^-})_{\tilde\chi^0_1\tilde\chi^0_2} &=& -10.2\pm 1.0\%, \\
{\mathcal A}({\mathbf p}_{e^-},{\mathbf p}_{\ell_N^-},{\mathbf p}_{\ell_F^+})_{\tilde\chi^0_1\tilde\chi^0_2} &=& +10.7\pm 1.0\%, \\
{\mathcal A}({\mathbf p}_{e^-},{\mathbf p}_{\ell_N^+},{\mathbf p}_{\ell_F^-})_{\tilde\chi^0_1\tilde\chi^0_3} &=& +9.3\pm 1.0\%, \\
{\mathcal A}({\mathbf p}_{e^-},{\mathbf p}_{\ell_N^-},{\mathbf p}_{\ell_F^+})_{\tilde\chi^0_1\tilde\chi^0_3} &=& -8.8\pm 1.0\%,
\end{eqnarray}
with the statistical uncertainty~\cite{Desch:2006xp}
\begin{eqnarray}
\delta(\mathcal{A})_{\mathrm{stat}} &=& \sqrt{\frac{1-\mathcal{A}^2}{N}},
\end{eqnarray}
and the total number of events $N = N_+ + N_-$. Exchanging the near
and far leptons gives, within the uncertainties, the same size of the
asymmetry but with opposite sign, see Eq.~\eqref{eq:leptonexch}.
However, the values of the asymmetries are different from the
theoretical values, see Tab.~\ref{tab:masses}, which is mainly due to:
\begin{enumerate}
\item CP-even background events cancel in the numerator,
but contribute to the denominator in Eq.~\eqref{eq:asyex}.
\item Events are removed by the experimental selection cuts and
by the kinematic selection procedure, which can bias the
measured asymmetry.
\end{enumerate}
CP-even backgrounds shift the asymmetry to slightly lower values,
whereas the selection cuts have the opposite effect. If we assume that
the background contributions are known, we obtain
\begin{eqnarray}
{\mathcal A}({\mathbf p}_{e^-},{\mathbf p}_{\ell_N^+},{\mathbf p}_{\ell_F^-})_{\tilde\chi^0_1\tilde\chi^0_2} &=& -11.0 \pm 1.0\%,\label{asy:1}\\
{\mathcal A}({\mathbf p}_{e^-},{\mathbf p}_{\ell_N^-},{\mathbf p}_{\ell_F^+})_{\tilde\chi^0_1\tilde\chi^0_2} &=& +11.6 \pm 1.0\%,\label{asy:2}\\
{\mathcal A}({\mathbf p}_{e^-},{\mathbf p}_{\ell_N^+},{\mathbf p}_{\ell_F^-})_{\tilde\chi^0_1\tilde\chi^0_3} &=& +11.1 \pm 1.0\%,\label{asy:3}\\
{\mathcal A}({\mathbf p}_{e^-},{\mathbf p}_{\ell_N^-},{\mathbf p}_{\ell_F^+})_{\tilde\chi^0_1\tilde\chi^0_3} &=& -10.6 \pm 1.0\%.\label{asy:4}
\end{eqnarray}
Additionally, we have studied the bias due to the selection procedure
by calculating the asymmetry after each cut and comparing the results
obtained from \texttt{Whizard} with the results obtained from
\texttt{Herwig++}~\cite{Bahr:2008pv,Gieseke:2011na,Gigg:2007cr}. Both
programs show consistently a total shift of about 2\% towards larger
values after the application of the complete selection procedure; see
also Ref.~\cite{MoortgatPick:2010wp}. The cuts remove events with a
small value of the asymmetry inducing an upward shift. As an example,
Fig.~\ref{fig:tripcuts} shows the dependence of the asymmetry on
${\mathbf p}_{\rm T}^{\rm miss}$ and $\theta_{\rm acol}$, indicating
also our cut value. The shift will be taken into account in the
parameter fit described in the next section.
\begin{figure}
\centering
\subfigure[]{\label{fig:p_T_miss_trip}
\includegraphics[scale=0.35,angle=0]{Asymmetry_pTmiss.eps}
}
\subfigure[]{\label{fig:acol_trip}
\includegraphics[scale=0.35,angle=0]{Asymmetry_Acol.eps}
}
\caption{ The (a) ${\mathbf p}_{\rm T}^{\rm miss}$ and (b)
$\theta_{\rm acol}$ dependence of the asymmetries ${\mathcal
A}({\mathbf p}_{e^-},{\mathbf p}_{\ell_N^-},{\mathbf
p}_{\ell_F^+})_{\tilde\chi^0_1\tilde\chi^0_2}$ and ${\mathcal
A}({\mathbf p}_{e^-},{\mathbf p}_{\ell_N^+},{\mathbf
p}_{\ell_F^-})_{\tilde\chi^0_1\tilde\chi^0_3}$, solid and dashed
lines, respectively. The cut value used in our analysis is
indicated by the dashed line. In each case $10^7$ events were
generated and no detector effects are included. Statistical
uncertainties are shown. }
\label{fig:tripcuts}
\end{figure}
\section{Fit of the parameters in the neutralino sector\label{fitting-proc}}
In the final step of our analysis, we estimate the accuracy to
determine the parameters in the neutralino sector of the MSSM. These
are the six free parameters of the neutralino mass matrix, see
App.~\ref{sec:NeutralinoMixing},
\begin{eqnarray} \label{eq:parameters}
|M_1|, \quad M_2, \quad |\mu|, \quad \tan\beta, \quad \phi_1, \quad\phi_\mu.
\end{eqnarray}
We have a number of observables at hand that can be used in the fit.
These are cross sections, masses, and asymmetries. Masses will be
measured with high precision using different methods~\cite{TDR}. For
the neutralino masses we assume the uncertainties as in
Ref.~\cite{Desch:2006xp}, since no detailed analysis has been done for
our parameter point. However, these uncertainties are rather
conservative, since we expect that in our scenario a similar precision
can be achieved as in~\cite{Martyn:2003av}. In case of the cross
sections, the uncertainty is dominated by the statistical uncertainty,
\begin{equation}
\frac{\Delta \sigma}{\sigma} = \frac{\sqrt{S+B}}{S},
\end{equation}
where $S$ and $B$ are the signal and background contributions,
respectively; see Tab.~\ref{tab:cutflow}. Since experimentally the
number of events is recorded, not the cross section itself, we have to
take into account branching ratios for the relevant decays. These will
depend on the masses and, in case of staus, on the stau mixing angle,
$\cos\theta_{\tilde{\tau}}$. The stau mixing angle can be obtained
from $\tau$ polarization measurements in stau pair
production~\cite{Bechtle:2009em}, with an accuracy of
5\%~\cite{Boos:2002wu,Boos:2003vf,Nojiri:1996fp}.
\medskip
After our procedure of the kinematic event selection, see
Sec.~\ref{sec:kinreco}, to disentangle contributions from
$\neu{1}\neu{2}$ and $\neu{1}\neu{3}$ production and decay, the
background contributions are below $15$\%. The small uncertainties in
the beam polarizations of $0.5$\%~\cite{MoortgatPick:2005cw}, in the
luminosity, and in the SUSY masses are negligible, see also
Ref.~\cite{ball,Desch:2003vw,D'Ascenzo:2009zz}. For the CP
asymmetries, we have estimated relative uncertainties of the order of
$10\%$ in Sec.~\ref{sec:asymmetryexp}. For the fit we take into
account a bias due to cuts on the asymmetry derived from the MC
simulation, as described in Sec.~\ref{sec:asymmetryexp}. Thus, the
analytical value of the asymmetry, given in Tab.~\ref{tab:masses}, is
shifted accordingly. Furthermore, we use Eq.~\eqref{eq:genuinecp} to
calculate the measured value of the asymmetry, which is free of FSI
effects. In summary, we have the following set of input observables
and uncertainties:
\begin{eqnarray*}
&& \mneu{1} = 117.3 \pm 0.2 \gev, \\
&& \mneu{2} = 168.5 \pm 0.5 \gev, \\
&& \mneu{3} = 180.8 \pm 0.5 \gev, \\
&& \sigma(\neu{1}\neu{2}) \times {\rm BR}(\neu{2}\to\tilde{\ell}_R\ell) = 130.9 \pm 1.4 \fb ,\\
&& \sigma(\neu{1}\neu{3}) \times {\rm BR}(\neu{3}\to\tilde{\ell}_R\ell) = 155.7 \pm 1.6 \fb , \\
&& \sigma(\neu{2}\neu{2}) \times {\rm BR}(\neu{2}\to\tilde{\ell}_R\ell)^2 = 4.8 \pm 0.3 \fb , \\
&& \sigma(\neu{3}\neu{3}) \times {\rm BR}(\neu{3}\to\tilde{\ell}_R\ell)^2 = 26.3 \pm 0.7 \fb , \\
&& \sigma(\neu{2}\neu{3}) \times {\rm BR}(\neu{2}\to\tilde{\ell}_R\ell) \times {\rm BR}(\neu{3}\to\tilde{\ell}_R\ell) = 28.9 \pm 0.7\fb , \\
&& {\mathcal{A}}^{\rm CP}({\mathbf p}_{e^-},{\mathbf p}_{\ell_N},{\mathbf p}_{\ell_F})_{\tilde\chi^0_1\tilde\chi^0_2} = +11.3\% \pm 0.7\%, \\
&& {\mathcal{A}}^{\rm CP}({\mathbf p}_{e^-},{\mathbf p}_{\ell_N},{\mathbf p}_{\ell_F})_{\tilde\chi^0_1\tilde\chi^0_3} = -10.9\% \pm 0.7\%.
\end{eqnarray*}
The uncertainties for the cross sections correspond to an integrated
luminosity of ${\mathcal{L}} = 500 \fb^{-1}$. We perform a six
dimensional $\chi^2$ fit using
\texttt{Minuit}~\cite{James:1975dr,minuit}
\begin{equation}
\chi^2 = \sum_i \left| \frac{{\mathcal{O}}_i - \bar{{\mathcal{O}}}_i }
{ \delta {\mathcal{O}}_i } \right|^2 ,
\end{equation}
where the sum runs over the input observables ${\mathcal{O}}_i$
mentioned above, with their corresponding experimental uncertainties
$\delta {\mathcal{O}}_i$. The theoretical values calculated using the
fitted MSSM parameters, Eq.~\eqref{eq:parameters}, are denoted by
$\bar{{\mathcal{O}}}_i$. The parameter dependence of branching ratios
(e.g. the stau mixing angle) is also included in the fit, but has
negligible impact. We then obtain the following fitted values for the
MSSM parameters:
\begin{eqnarray*}
|M_1| &=& 150.0 \pm 0.7 \gev ,\\
M_2 &=& 300 \pm 5 \gev ,\\
|\mu| &=& 165.0 \pm 0.3 \gev ,\\
\tan\beta &=& 10.0 \pm 1.6 , \\
\phi_1 &=& 0.63 \pm 0.05 ,\\
\phi_\mu &=& 0.0 \pm 0.2 .
\end{eqnarray*}
The best estimates are obtained for the $|M_1|$ and $|\mu|$ mass
parameters, since the neutralino states $\neu{1}$, $\neu{2}$, and
$\neu{3}$ are mostly composed of bino and Higgsino. The fourth
neutralino is heavy and cannot be measured, so the limit on the wino
mass $M_2$ is not as good. Also a rather large uncertainty is obtained
for $\tan\beta$. However, if additional measurements from other
sectors will be added, it should be improved significantly. We note
that the precision obtained in this study is similar to the results of
Ref.~\cite{Desch:2003vw}, which uses a similar set of
observables.\footnote {The high precision achieved in the fit calls
for the inclusion of higher order corrections which can be in the
$\mathcal{O}(20\%)$ regime in the neutralino system, see
e.g.~\cite{Oller:2004br,Fritzsche:2004nf}. These corrections will in
turn depend on the full parameter set of the MSSM. Therefore, the
proper treatment would require the inclusion of observables from
other sectors, in particular from the third generation of squarks,
cf.~Ref.~\cite{Bechtle:2005vt,Lafaye:2007vs}. This issue is beyond
the scope of this paper, however, it should stimulate further
studies.}
\medskip
It is remarkable that the moduli of the phases $\phi_1$ and $\phi_\mu$
can also be determined with high precision, using the CP-even
observables alone. However, only an inclusion of CP-odd asymmetries in
the fit allows us to resolve the sign ambiguities of the phases.
Without the CP-odd asymmetries in the fit we would have a twofold
ambiguity, $\phi_1 = \pm 0.6$, and even fourfold if $\phi_\mu \neq 0$.
Thus, the triple product asymmetries are not only a direct test of CP
violation, but are also essential to determine the correct values of
the phases.
\section{Summary and conclusions
\label{Summary and conclusion}}
We have presented the first full detector simulation study to
measure SUSY CP phases at the ILC. We have considered
CP-sensitive triple-product asymmetries in neutralino production
$e^+e^- \to\tilde\chi^0_i \tilde\chi^0_1$ and the subsequent
leptonic two-body decay chain $\tilde\chi^0_i \to \tilde\ell_R
\ell$, $ \tilde\ell_R \to \tilde\chi^0_1 \ell$, for $ \ell=
e,\mu$. Large asymmetries typically arise due to strong
neutralino mixing. This causes on the one side that asymmetries
for $\tilde\chi^0_1\tilde\chi^0_2$ and
$\tilde\chi^0_1\tilde\chi^0_3$ production have about the same
size but opposite sign. On the other side the strong mixing
implies two close-in-mass neutralino states, that can have a
mass separation of the order of $10$~GeV. This quasi-degeneracy
would potentially pose a problem for the separation of both
signal components.
\medskip
Therefore we have developed a kinematic selection method, to identify
the lepton pairs from the signal events. At the Monte Carlo
level, we have shown that this method allows one to separate the
leptons from the two signal processes $\tilde\chi^0_1\tilde\chi^0_2$
and $\tilde\chi^0_1\tilde\chi^0_3$, and also to reduce the major SM
and SUSY backgrounds, in particular from $W$-pair and slepton-pair
production.
\medskip
Then we have performed a detailed case study, which includes a full
ILD detector simulation and event reconstruction. A detailed cut flow
analysis has been done to preselect leptonic event candidates, which
then have been passed to our method of kinematic selection. Even after
the detector simulation, our method has worked efficiently to reduce
background and separate the signal.
After the full simulation with kinematic selection, the efficiencies
of signal event selection is of the order of $27\%$ with a purity of
about $90$\% of the event samples. That allows one to measure the
asymmetry with a relative precision of about $10\%$. {Our method of kinematic event reconstruction also works well in
scenarios with different mass splittings of the neutralinos and the
selectron. In the worst case scenario we found that the efficiency
will go down to some $10\%$, but still with a high purity of the
correctly identified signal sample of the order of $90\%$.}
\medskip
We have performed a global fit of the neutralino masses, cross
sections, and CP asymmetries to reconstruct the MSSM parameters of the
neutralino sector, including the CP phases. The relative uncertainties
of the parameters $|M_1|$ and $|\mu|$ are below $1\%$, those for $M_2$
about $1\%$, and for $\tan\beta$ and the CP phases $\phi_1$,
$\phi_\mu$ about $10\%$. Although the moduli of the phases $\phi_1$,
$\phi_\mu$ can also be determined by using the CP-even observables
alone, we have shown that only an inclusion of CP-odd asymmetries in
the fit allows us to resolve the sign ambiguities of the phases.
\medskip
To summarize, we have shown that a measurement of the neutralino
sector seems to be feasible, including CP phases. In particular the
triple product asymmetries are not only a direct test of CP violation,
but are also essential to determine the correct values of the phases
in the neutralino sector.
\section*{Acknowledgments}
We would like to thank Steve Aplin, Mikael Berggren, Jan Engels, Frank
Gaede, Nina Herder, Jenny List, and Mark Thomson for useful discussions
and help with the detector simulations.
This work was supported by MICINN project FPA.2006-05294 and CPAN. We
acknowledge the support of the DFG through the SFB (grant SFB 676/1-2006).
\begin{appendix}
\renewcommand{\thesubsection}{\Alph{section}.\arabic{subsection}}
\renewcommand{\theequation}{\Alph{section}.\arabic{equation}}
\setcounter{equation}{0}
\section{Reconstruction of $W$ and $\tilde\ell$ pair production}\label{sec:app3}
We consider a template process
\begin{equation}
e^+ + e^- \;\to\; A + \bar A \;\to\; \ell + \bar\ell + B + B,
\end{equation}
where $(A,B) = (\tilde\ell,\neu{1})$ or $(W,\nu)$. In both cases
$B=\neu{1}$ or $\nu$ escapes detection. Since the system of the lepton
pair has to obey different kinematic constraints, we consider the
question, whether the final lepton pair can be assigned to its mother
production process, if the lepton momenta are measured, and the
slepton and LSP masses are known. We follow closely
Ref.~\cite{Buckley:2007th}, and define the notation
\begin{eqnarray}
c_1 &\equiv& {\mathbf p}_A \cdot {\mathbf p}_\ell \;= \;\phantom{-}
\frac{1}{2} ( m^2_B -m_A^2 +
E_\ell \sqrt{s}),\label{eq:c1}\\
c_2 &\equiv& {\mathbf p}_A \cdot {\mathbf p}_{\bar\ell} \;=\;
-\frac{1}{2} ( m^2_B - m^2_A + E_{\bar\ell} \sqrt{s}),
\label{eq:c2}\\
b_2 &\equiv& {\mathbf p}_A \cdot {\mathbf p}_A \;=\; \frac{s}{4} - m^2_A,
\label{eq:b2} \\[2mm]
a_{11} &\equiv& {\mathbf p}_\ell \cdot {\mathbf p}_\ell, \quad
a_{12} \;\equiv\; {\mathbf p}_\ell \cdot {\mathbf p}_{\bar\ell}, \quad
a_{22} \;\equiv\; {\mathbf p}_{\bar\ell} \cdot {\mathbf p}_{\bar\ell}.
\end{eqnarray}
The momentum ${\mathbf p}_A$ can be decomposed into the final lepton momenta
\begin{eqnarray}
{\mathbf p}_A &= & t_1 \, {\mathbf p}_\ell + t_2\, {\mathbf p}_{\bar\ell} +
y \, {\mathbf p}_\bot,
\end{eqnarray}
where ${\mathbf p}_\bot = {\mathbf p}_\ell \times {\mathbf p}_{\bar\ell}$.
The expansion coefficients follow from Eqs.~\eqref{eq:c1} and \eqref{eq:c2}
\begin{eqnarray}
\left|
\begin{array}{ccc}
c_1 &=& t_1 a_{11} + t_2 a_{12} \\[3mm]
c_2 &=& t_1 a_{12} + t_2 a_{22}
\end{array}
\right|
&\Rightarrow&
\left|
\begin{array}{ccc}
t_1 &=& \displaystyle{\frac{a_{22} c_1 - a_{12} c_2}{a_{11}a_{22} - a_{12}^2}} \\[5mm]
t_1 &=& \displaystyle{\frac{a_{11} c_2 - a_{12} c_1}{a_{11}a_{22} - a_{12}^2}}
\end{array}
\right|.
\label{eq:ct12}
\end{eqnarray}
We finally obtain, from Eqs.~\eqref{eq:b2} and \eqref{eq:ct12},
\begin{eqnarray}
b_2 &=& (t_1^2 a_{11} + 2 t_1 t_2 a_{12} + t_2^2 a_{22}) + y^2
|{\mathbf p}_\bot|^2,
\qquad\\[4mm]
\Rightarrow
y^2
&=& \frac{b_2 - (t_1^2 a_{11} + 2 t_1 t_2 a_{12} + t_2^2
a_{22})}{|{\mathbf p}_\bot|^2}.\label{eq:defy}
\end{eqnarray}
The equation for $y$ constitutes a condition for existence of physical
solutions of the system, i.e.\ $y^2 \geq 0$, where $y^2$ is computed
from the kinematic variables $s$, $m_A$, $m_B$, $E_\ell$,
$E_{\bar\ell}$, and ${\mathbf p}_\ell \cdot {\mathbf p}_{\bar\ell}$.
Similar to neutralino pair production, Sec.~\ref{sec:neurec},
Eq.~\eqref{chipm}, there remains a twofold ambiguity in solving the
$W$ or $\tilde{\ell}$ system, $y = \pm \sqrt{y^2}$.
\setcounter{equation}{0}
\section{Neutralino mixing}
\label{sec:NeutralinoMixing}
The complex symmetric mass matrix of the neutralinos in the photino,
zino, Higgsino basis ($\tilde{\gamma},\tilde{Z}, \tilde{H}^0_a, \tilde{H}^0_b$),
is given by~\cite{Choi:2001ww}
\begin{equation}
{\mathcal M}_{\chi^0} =
\left(\begin{array}{cccc}
M_2 \, s^2_W + M_1 \, c^2_W &
(M_2-M_1) \, s_W c_W & 0 & 0 \\
(M_2-M_1) \, s_W c_W &
M_2 \, c^2_W + M_1 \, s^2_W & m_Z & 0 \\
0 & m_Z & \mu \, s_{2\beta} & -\mu \, c_{2\beta} \\
0 & 0 & -\mu \, c_{2\beta} & -\mu \, s_{2\beta}
\end{array}\right),
\label{eq:neutmass}
\end{equation}
with the short hand notation for the angles $s_W = \sin\theta_W$, $c_W
= \cos\theta_W$, and $s_{2\beta} = \sin(2\beta)$, $c_{2\beta} =
\cos(2\beta)$, and the $SU(2)$ gaugino mass parameter $M_2$. The
phases of the complex parameters $M_1=|M_1|e^{i\phi_{1} }$ and
$\mu=|\mu|e^{i\phi_\mu } $ can lead to CP-violating effects in the
neutralino system. The phase of $M_2$ can be rotated away by a
suitable redefinition of the fields. We diagonalize the neutralino
mass matrix with a complex, unitary $4\times 4$ matrix $N$,
\begin{equation}
N^* \cdot {\mathcal M}_{\chi^0} \cdot N^{\dagger} =
{\rm diag}(m_{\chi^0_1},\dots,m_{\chi^0_4}),
\label{eq:neutn}
\end{equation}
with the real neutralino masses $ 0 < m_{\chi^0_1} < m_{\chi^0_2} <
m_{\chi^0_3} < m_{\chi^0_4}$.
\end{appendix}
\bibliographystyle{utphys}
|
1,108,101,564,567 | arxiv | \section{Introduction}\label{introduction}
Recently, the interest in studying new alternative theories of gravity has been increasing see, for instance, \cite{alternative}. In spite of its efficiency, the unmodified general theory of relativity is not able to answer some fundamental questions. Some of the most important problems are the construction of non-contradictory quantum gravity, the singularity problem, the problems of dark matter and dark energy. The problem of the construction of non-contradictory quantum gravity is connected with the non-renormalizability of General Relativity. This can usually be solved by adding the higher order terms in curvature to the theory \cite{tHooft}. In this paper, we will consider two different approaches: the first approach is related to adding of the Gauss Bonnet term coupled to a dilaton \cite{Blazquez-Salcedo:2016enn,Pani:2009wy,Nampalliwar:2018iru}, while the second theory consists of the Weyl term \cite{Einstein-Weyl:2018pfe} added to the Einstein action. Both theories are inspired by the low energy limit of string theory \cite{low-energy}, which contain quadratic corrections in curvature, but the Gauss-Bonnet term alone leads to the full divergence and does not contribute to the equations of motions, so that there remaining only two options for adding higher curvature corrections: either coupling of the Gauss-Bonnet term to other fields or choosing essentially non-Gauss-Bonnet quadratic corrections. Thus, here we will consider example of the both options.
The Lagrangian of the Einstein-dilaton-Gauss-Bonnet gravity is:
\begin{eqnarray} \label{lagranzianEdGB}
{\cal L}_{EdGB}&=&\frac{1}{2}R - \frac{1}{4} \partial_\mu \phi \partial^\mu \phi \\\nonumber&&+ \frac{\alpha '}{8g^2} e^{\phi }\left(R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma} - 4 R_{\mu\nu}R^{\mu\nu} + R^2\right),
\end{eqnarray}
where $\alpha '$ is the Regge slope, $g$ is the gauge coupling constant and $\phi$ is the dilaton field function. Black holes in the Einstein-dilaton-Gauss-Bonnet gravity has been recently investigated in number of papers \cite{Nampalliwar:2018iru}, \cite{Ayzenberg:2014aka,Maselli:2014fca,Maselli:2015tta,Cunha:2016wzk,Konoplya:2016jvv,Zhang:2017unx,Prabhu:2018aun,Nair:2019iur,Konoplya:2019hml}.
For the Einstein-Weyl gravity the Lagrangian can be written as follows:
\begin{eqnarray} \label{lagranzianEW}
{\cal L}_{EW}&=&\sqrt{-g} (\gamma R - \alpha C_{\mu\nu \rho\sigma} C^{\mu\nu \rho\sigma} + \beta R^2),
\end{eqnarray}
where $\alpha$, $\beta$ and $\gamma$ are coupling constants, $C_{\mu\nu\rho\sigma}$ is the Weyl tensor. For spherically symmetric and asymptotically flat solutions we can choose $\gamma =1$ and $\beta =0$ \cite{EW}, so that the only new coupling constant is $\alpha$. The condition $R=0$ is evidently satisfied in this case, so that the Schwarzschild solution is also the solution of the above theory. The static spherically symmetric and asymptotically flat black holes in the Einstein-Weyl theory represent the generic class of black hole solutions in the quadratic theories of gravity if no other matter fields are added. They have been recently studied in \cite{Lin:2016kip,Zinhailo:2018ska,Konoplya:2019ppy,Wu:2019uvq}.
Recently black holes in the both theories have been extensively studied. In particular, quasinormal modes were found for test scalar and electromagnetic fields \cite{Zinhailo:2018ska,Konoplya:2019hml}. Although quasinormal modes of a Dirac field around black holes in the Einstein gravity were studied in detail in a number of papers (see \cite{Cho:2003qe,Jing:2003wq,Giammatteo:2004wp,Jing:2005dt,Blazquez-Salcedo:2018bqy} and reference therein), to the best of our knowledge there are no works devoted to Dirac quasinormal modes in theories with higher curvature corrections. When considering the neutrino field, the special attention must be paid to the presence of a negative region of a potential curve with negative chirality. The positive definite effective potential guarantees dynamical stability of perturbations, that is, absence of unboundedly growing modes. For the one of the chiralities of Dirac field in the Schwarzschild background the negative gap does not lead to the instability because the other chirality provides positive definite potential and the both chiralities are proved to be iso-spectral. However, the iso-spectrality has never been proved for the considered non-Einsteinian theories, so that the instability cannot be excluded a priori. Because of this, it would be interesting to study the quasinormal spectrum of the Dirac field in the above non-Einsteinian theories of gravity and see whether there is an instability. After all, the test of stability is extremely important for higher curvature corrected theories because of the so called eikonal instability which occurs in a abroad class of theories with various higher curvature corrections, spacetime dimensions and asymptotics \cite{Dotti:2005sq,Gleiser:2005ra,Takahashi:2010gz,Grozdanov:2016fkt,Konoplya:2017zwo,Cuyubamba:2018jdl}, and not only for gravitational, but also for test fields \cite{Gonzalez:2017gwa}.
We will analyze values of modes at the low angular parameter $\ell$ and in the eikonal regime. We will find dependencies of the complex frequency on the new dimensionless parameter $p$ (related to the coupling constant in each theory). In addition, we will compare quasinormal modes of both theories between each other and with modes of other fields in each theory. In addition here we will test the quasinormal modes of Dirac field in the above two theories as to the Hod's conjecture \cite{Hod:2006jw} who claims that there must always be a minimal mode whose damping rate is limited by the Hawking temperature multiplied by some factor.
This work is organized as follows. In Sec.~\ref{sec:metricsection} we introduce a metric and a general wave equation and consider the effective potential for the Dirac field for a spherically symmetric black hole. For this case we prove that the Dirac perturbations are linearly stable in both theories. In Sec.~\ref{sec:massless}, the basic principles of the WKB method are briefly considered, an analytical approximation in the eikonal regime is analysed Sec.~\ref{sec:eikonal}, the quasinormal modes for test massless Dirac field are found, a comparative analysis is made for the our result with the results for other fields in these theories of gravity Sec.~\ref{sec:lsmall}. In Sec.~\ref{sec:hods} we will check the Hod's conjecture for the Dirac field in Einstein-dilaton-Gauss-Bonnet and Einstein-Weyl gravities.
\section{Black hole metric and analytics for the wave equation}\label{sec:metricsection}
In the general case the metric for a spherically symmetric black hole can be written in the form:
\begin{eqnarray}\label{metric}
ds^2 &=& -e^{\mu(r)}dt^2+e^{\nu(r)}{dr^2}+r^2 (\sin^2 \theta d\phi^2+d\theta^2),
\end{eqnarray}
where $e^{\mu(r)}$ and $e^{\nu(r)}$ are the metric coefficients. The explicit expression for the metric coefficients were obtained numerically in \cite{Kanti:1995vq} for Einstein-dilaton-Gauss-Bonnet gravity and in \cite{EW} for Einstein-Weyl gravity. The approximate analytical expressions (which will be used here) were obtained in \cite{Kokkotas:2017ymc} for the Einstein-dilaton-Gauss-Bonnet metric, in \cite{Kokkotas:2017zwt} for the Einstein-Weyl metric. They are also written down in Appendixs \ref{Appendix1}, \ref{Appendix2}.
We parameterize the both black-hole solutions in theories (\ref{lagranzianEdGB}, \ref{lagranzianEW}) via the following dimensionless parameter $p$ up to the rescaling:
\begin{subequations}\label{parameter}
\begin{eqnarray}\label{parameterpedgb}
p_{EdGB}\equiv6e^{2\phi_0}=\frac{6\alpha'^2}{g^4r_0^4}e^{2(\phi_0-\phi_{\infty})} \qquad{(Einstein-dilaton-Gauss-Bonnet)}\,
\end{eqnarray}
\begin{eqnarray}\label{parameterpew}
p_{EW}=\frac{r_0}{\sqrt{2\alpha}} \qquad{(Einstein-Weyl)}.
\end{eqnarray}
\end{subequations}
For convenience we fix radius of the black-hole event horizon to be $r_0=1$. For all $p$ the Schwarzschild metric is the exact solution of the Einstein-Weyl equations as well, but only at some minimal nonzero $p_{min}$, in addition to the Schwarzschild solution, there appears the non-Schwarzschild branch which describes the asymptotically flat black hole, whose mass is decreasing, when $p$ grows. The approximate maximal and minimal values of $p$ are:
\begin{subequations}\label{paraneter}
\begin{eqnarray}\label{parameteredgb}
p_{min,EdGB} \geq 0, \quad p_{max,EdGB}\leq0.97,
\end{eqnarray}
\begin{eqnarray}\label{parameterew}
p_{min,EW} \approx 1054/1203 \approx 0.876, \quad p_{max,EW} \approx 1.14.
\end{eqnarray}
\end{subequations}
The general covariant Dirac equation has the form \cite{Brill:1957fx}:
\begin{equation}\label{covdirac}
\gamma^{\alpha} \left( \frac{\partial}{\partial x^{\alpha}} - \Gamma_{\alpha} \right) \Psi=0,
\end{equation}
where $\gamma^{\alpha}$ are noncommutative gamma matrices and $\Gamma_{\alpha}$ are spin connections in the tetrad formalism. We separate of angular variables in equation (\ref{covdirac}) and rewrite the wave equation in the following general master form in terms of the ``tortoise coordinate'' $r_*$ \cite{Brill:1957fx}:
\begin{equation} \label{klein-Gordon}
\dfrac{d^2 \Psi}{dr_*^2}+(\omega^2-V(r))\Psi=0, \quad dr_*=\sqrt{e^{\nu(r)-\mu(r)}}dr.
\end{equation}
The effective potentials of test Dirac ($s=\pm 1/2$) field in the general background (\ref{metric}) can be written as follows:
\begin{equation}
V_{\pm}(r) = \frac{k}{r}\left(\frac{e^{\mu(r)} k}{r}\mp\frac{e^{\mu(r)}\sqrt{e^{\nu(r)}}}{r}\pm\sqrt{e^{\mu(r)-\nu(r)}}(\sqrt{e^{\mu(r)}})'\right),
\end{equation}
where the prime designates the differentiation with respect to the ``tortoise coordinate'' $r_{*}$.
\vspace*{0.5em plus .6em minus .5em}
\begin{figure}[ht]
\vspace{-4ex} \centering \subfigure[]{
\includegraphics[width=0.45\linewidth]{EdGBminusl1.eps} \label{fig:EdGB1_a} }
\hspace{4ex}
\subfigure[]{
\includegraphics[width=0.45\linewidth]{EdGBplusl1.eps} \label{fig:EdGB2_b} }
\caption{The effective potential $V(r)$ for the EdGB gravity for $\ell=1$; the blue line
corresponds to $p=0$, the red line corresponds to $p=0.5$ and the green line is $p=0.97$:
\subref{fig:EdGB1_a} $V_{-}(r)$;
\subref{fig:EdGB2_b} $V_{+}(r)$.} \label{fig:vedgb}
\label{ris:one1}
\end{figure}
\begin{figure}[ht]
\vspace{-4ex} \centering \subfigure[]{
\includegraphics[width=0.45\linewidth]{EWVminusl1.eps} \label{fig:EW1_a} }
\hspace{4ex}
\subfigure[]{
\includegraphics[width=0.45\linewidth]{EWVplusl1.eps} \label{fig:EW2_b} }
\caption{The effective potential $V(r)$ for the EW gravity for $\ell=1$; the blue line
corresponds to $p=0.876$, the red line corresponds to $p=0.9816$ and the green line is $p=1.14$:
\subref{fig:EdGB1_a} $V_{-}(r)$;
\subref{fig:EdGB2_b} $V_{+}(r)$.} \label{fig:vew}
\label{ris:one2}
\end{figure}
In the both cases for the ``plus'' (``minus'') potential of the Dirac field $k =\ell+1$ ($k =\ell$). As can be seen from figs. (\ref{ris:one1}(a), \ref{ris:one2}(a)) the potential $V_{-}(r)$ has a negative gap near the event horizon. The same behavior is appropriate to the potential $V_{-}(r)$ in the Schwarzschild case. However, as it was shown earlier for black holes for which both metric coefficients are equal (like for the Schwarzschild case $e^{\mu(r)}=e^{-\nu(r)}$ \cite{potentials}), the potentials of opposite chiralities can be transformed into each another with help of the Darboux transformation. This means that from both potentials we get the same quasinormal spectrum. It allows us to ignore the negative gap of the ``minus'' potential an and talk about overall stability for the Schwarzschild case. When both metric coefficients are not the same anymore, to the best of our knowledge the iso-spectrality of both chiralities was not shown. Here we can see that the following replacements:
\begin{equation} \label{psi}
\Psi_{+}=q (W+\dfrac{d}{dr_*}) \Psi_{-}, \quad W=\sqrt{e^{\mu(r)}}, \quad q=const;
\end{equation}
provides the Darboux transformation of equations (\ref{klein-Gordon}, \ref{psi}) for transition between ``minus'' (given by $V_{-}(r)$ and ``plus'' ($V_{+}(r)$) perturbations. As the potential for one of the chiralities is positive definite, this immediately guarantees the stability of the Dirac field for the other chirality in both considered theories.
Therefore, we can use only stable potential. Later in the work, we will use the potential $V_{+}(r)$.
\section{Quasinormal modes of massless Dirac field for Einstein-dilaton-Gauss-Bonnet and Einstein-Weyl gravities}\label{sec:massless}
For finding quasinormal modes, it is necessary to solve the spectral problem with the appropriate boundary conditions: for a functions $\Psi$ there are only incoming waves at the horizon ($r_*\rightarrow-\infty$) and only the outgoing waves at the infinity ($r_*\rightarrow+\infty$). Quite effectively this problem can be solved using the WKB-method \cite{WKB,Matyjasek:2017psv,Konoplya:2003ii,Konoplya:2019hlu}. The advantages of this method over numerical methods is the ability to obtain low-lying quasinormal modes with sufficient accuracy automatically for a broad class of effective potentials, and, thereby, not to tailor the method for each case. The method gives good accuracy when $n \leq \ell$, where $n=0,1,2,..$ is a overtone number. The general formula for the m-order of the WKB approach can be written in form:
\begin{equation}\label{wkb}
\dfrac{i(\omega^2-V_0)}{\sqrt{-2 V_0''}}-\sum\limits_{i = 2}^{m}\Lambda_i=n+\dfrac{1}{2}.
\end{equation}
Here, the $\Lambda_i$ are the correction term of the i-th order and $\Lambda_i$ depend on the value of the potentials $V(r)$ and its derivative at the maximum, ${V_0}$ is a value of $V(r)$ in $r_{max}$ and $V_0''$ is a second derivative in $r_{max}$. But the WKB series converges only asymptotically, there is no strict criterium for evaluation of an error. The higher accuracy of the WKB approach can be achieved the averaging of the Padé approximation \cite{Matyjasek:2017psv}. We will use the fourth-order of the WKB approximation and apply further Padé expansion of the order which provides the best accuracy in the Schwarzschild limit \cite{Matyjasek:2017psv,Konoplya:2019hlu}.
\subsection{An analytical approximation in the eikonal regime}\label{sec:eikonal}
In the regime of high multipole numbers $\ell$ (eikonal regime) it is sufficient to use the first order WKB formula:
\begin{equation}\label{wkbone}
\omega=\sqrt{{V_0}-i \left(n+\frac{1}{2}\right) \sqrt{-2 {V_0''}}}.
\end{equation}
When the multipole numbers $\ell$ is high the behavior of test fields of different spin obey the same law in the dominant order and the expression for $\omega$ for the Dirac field will be identical to the formulas for other spin. For Einstein-dilaton-Gauss-Bonnet case it was found for electromagnetic field \cite{Konoplya:2019hml} for small $1/\ell$:
\begin{subequations}
\begin{eqnarray}\label{wedgb}
\omega_{EdGB}=\frac{2}{3 \sqrt{3}r_0} \left(\left(\ell+\frac{1}{2}\right) \left(1-0.065 p\right)-i \left(n+\frac{1}{2}\right) \left(1-0.094 p\right)\right)+\mathcal{O}(p^2,\ell^{-1}),
\end{eqnarray}
\begin{eqnarray}
r_{max}= \frac{3 r_0}{2} + 0,055 r_{0} p+\mathcal{O}(p^2,\ell^{-1}),
\end{eqnarray}
\end{subequations}
where $r_{max}$ is the position of peak of the effective potential.
For the Einstein-Weyl gravity the values of $\omega$ was found in \cite{Kokkotas:2017zwt} for small $1/\ell$, where $t=1054-1203 p$ is a deviations from the Schwarzschild branch:
\begin{subequations}
\begin{eqnarray}\label{wew}
\omega_{EW}=\frac{2}{3 \sqrt{3}r_0} \left(\left(\ell+\frac{1}{2}\right)\left(1-0.001308 t\right)-i \left(n+\frac{1}{2}\right) \left(1-0.002743 t\right)\right)+\mathcal{O}(t^2,\ell^{-1}),
\end{eqnarray}
\begin{eqnarray}
r_{max}= \frac{3 r_0}{2} (1+0.000393 t)+\mathcal{O}(t^2,\ell^{-2}).
\end{eqnarray}
\end{subequations}
When $p = 0$ in the formula (\ref{wedgb}) and $t = 0$ in (\ref{wew}) these formulas go over into the well-known eikonal formula for the Schwarzschild black hole. A general approach to finding eikonal quasinormal modes for static asymptotically flat and spherically symmetric black holes has been recently suggested in \cite{Churilova:2019jqx}.
It is worthwhile mentioning that the real and imaginary parts of the above eikonal formulas for test fields will coincide with the oscillation frequency and the Lyapunov exponents of the null geodesics in the background of the Einstein-dilaton-Gauss-Bonnet and Einstein-Weyl black holes \cite{Cardoso:2008bp}. However, this is not expected for the gravitational or other non-test (non-minimally coupled) fields \cite{Konoplya:2017wot,Breton:2017hwe,Toshmatov:2018ell}.
\subsection{Quasinormal modes for low $\ell$}\label{sec:lsmall}
For obtaining accurate values of quasinormal modes at low numbers $\ell$ we will use the fourth-order of the WKB approximation (\ref{wkb}) and apply further Padé expansion of the order. In the figs. (\ref{ris:one}, \ref{ris:two}) we construct the real (oscillation frequency) and imaginary (damping rate of oscillation) parts of the frequency $\omega$ on the values of the parameter $p$ for various multipole numbers $\ell$. As can be seen, the function $\omega(p)$ tends to be linear for all cases. This behavior is also characteristic of other test fields that were previously considered \cite{Konoplya:2019hml}, \cite{Zinhailo:2018ska}. Comparing figs. (\ref{ris:one}, \ref{ris:two}), we can see that the deviations from Schwarzschild branch by the Weyl correction are much larger than the Einstein-dilaton-Gauss-Bonnet gravity. For the Einstein-dilaton-Gauss-Bonnet case values of the modes are decreasing when increasing the dimensionless parameter $p$. On the contrary, for Weyl case we see, that the oscillation frequency and the damping rate of oscillations are increasing with increasing $p$. This also follows from the form of curves for potentials figs. (\ref{ris:one1}, \ref{ris:one2}). With an increase the dimensionless parameter $p$ for the Einstein-dilaton-Gauss-Bonnet gravity, the height of the potential barrier decreases, which is on the favor of lower bound states. For the Einstein-Weyl gravity, the maximum of potential $V(r)$ increases with increasing $p$. It means, that with increasing $p$ the corresponding frequencies are higher.
\vspace*{0.5em plus .6em minus .5em}
\begin{figure}[ht]
\vspace{-4ex} \centering \subfigure[]{
\includegraphics[width=0.29\linewidth]{EdGBl1w.eps} \label{fig:EdGBl1_a} }
\hspace{4ex}
\subfigure[]{
\includegraphics[width=0.29\linewidth]{EdGBl2w.eps} \label{fig:EdGBl2_b} }
\hspace{4ex}
\subfigure[]{ \includegraphics[width=0.29\linewidth]{EdGBl3w.eps} \label{fig:EdGBl3_c} }
\caption{The fundamental quasinormal mode of EdGB ($n=0$) for the Dirac field ($s=1/2$), blue line is real part of frequency, red line is imaginary part; positive values for the real part and negative values for the imaginary part:
\subref{fig:EdGBl1_a} $\ell = 1$;
\subref{fig:EdGBl2_b} $\ell = 2$;
\subref{fig:EdGBl3_c} $\ell = 3$.} \label{fig:qnmedgb}
\label{ris:one}
\end{figure}
\begin{figure}[ht]
\vspace{-4ex} \centering \subfigure[]{
\includegraphics[width=0.29\linewidth]{EWl1w.eps} \label{fig:EWl1_a} }
\hspace{4ex}
\subfigure[]{
\includegraphics[width=0.29\linewidth]{EWl2w.eps} \label{fig:EWl2_b} }
\hspace{4ex}
\subfigure[]{ \includegraphics[width=0.29\linewidth]{EWl3w.eps} \label{fig:EWl3_c} }
\caption{The fundamental quasinormal mode of EW ($n=0$) for the Dirac field ($s=1/2$), blue line is real part of frequency, red line is imaginary part; positive values for the real part and negative values for the imaginary part:
\subref{fig:EWl1_a} $\ell = 1$;
\subref{fig:EWl2_b} $\ell = 2$;
\subref{fig:EWl3_c} $\ell = 3$.} \label{fig:qnmew}
\label{ris:two}
\end{figure}
Approximate calculation formulas for the complex frequency were found from the obtained data for different values $\ell$. For Einstein-dilaton-Gauss-Bonnet it is (\ref{EdGB}):
\begin{subequations}\label{EdGB}
\begin{eqnarray}
Re(\omega_{s=0.5,\ell=1})\approx0.444-0.089 p,
\nonumber\\
Im(\omega_{s=0.5,\ell=1})\approx-0.284+0.100 p;\\
Re(\omega_{s=0.5,\ell=2})\approx0.927-0.189 p,
\nonumber\\
Im(\omega_{s=0.5,\ell=2})\approx-0.269+0.086 p;\\
Re(\omega_{s=0.5,\ell=3})\approx1.399-0.285 p,
\nonumber\\
Im(\omega_{s=0.5,\ell=3})\approx:-0.266+0.083 p.
\end{eqnarray}
\end{subequations}
Accordingly, for Einstein-Weyl $\ell = 1$, $\ell = 2$, $\ell = 3$, we have (\ref{EW}):
\begin{subequations}\label{EW}
\begin{eqnarray}
Re(\omega_{s=0.5,\ell=1})\approx0.011+0.407 p,
\nonumber\\
Im(\omega_{s=0.5,\ell=1})\approx0.515-0.803 p;\\
Re(\omega_{s=0.5,\ell=2})\approx-0.227+1.128 p,
\nonumber\\
Im(\omega_{s=0.5,\ell=2})\approx0.499-0.783 p;\\
Re(\omega_{s=0.5,\ell=3})\approx-0.407+1.776 p,
\nonumber\\
Im(\omega_{s=0.5,\ell=3})\approx0.496-0.780 p.
\end{eqnarray}
\end{subequations}
Approximate dependencies for parts of the complex frequency in $p$ were obtained in formulas (\ref{EdGB}) and (\ref{EW}) for low $\ell$. For all options, we have a reasonable linear approximation, which is clearly visible in figs. (\ref{ris:one}) and (\ref{ris:two}).
\subsection{The checking of Hod's conjecture}\label{sec:hods}
In work \cite{Hod:2006jw}, Hod put forward the statement for damping rate of the fundamental oscillation. In other words in the spectrum of quasinormal modes there always must exist a frequency whose absolute value of the imaginary part is smaller than $\pi$ times Hawking temperature of the black hole. According to this statement, for asymptotically flat black holes as well as for nonasymptotically flat ones, the following inequality holds:
\begin{equation}\label{hod}
|Im(\omega)|\leq\pi T_H,
\end{equation}
where $T_H$ is the Hawking temperature. The Hawking temperature $T_H$ for Einstein-dilaton-Gauss-Bonnet and Einstein-Weyl gravities can be written in the next form:
\begin{equation}\label{hod}
T_{H}=\frac{1}{4 \pi} \sqrt{\frac{de^{\mu(r)}}{dr} \frac{de^{-\nu(r)}}{dr}}\bigg|_{r=r_{0}}.
\end{equation}
\begin{figure}[ht]
\vspace{-4ex} \centering \subfigure[]{
\includegraphics[width=0.45\linewidth]{THpEdGB.eps} \label{fig:EdGBT1_a} }
\hspace{4ex}
\subfigure[]{
\includegraphics[width=0.45\linewidth]{THpEW.eps} \label{fig:EWT2_b} }
\caption{The dependencies of Hod's conjecture on $p$:
\subref{fig:EdGBT1_a} $EdGB$;
\subref{fig:EWT2_b} $EW$.} \label{fig:th}
\label{ris:th}
\end{figure}
From the fig. (\ref{ris:th}) it can be seen that for the whole interval of values of parameter $p$ for Einstein-dilaton-Gauss-Bonnet and Einstein-Weyl metrics $\frac{|Im(\omega)|}{\pi T_H}\leq1$. It means, that for the Hod's conjecture also holds for the cases considered.
\section{Conclusions}\label{sec:conclusions}
In this work we considered test massless Dirac field in Einstein-dilaton-Gauss-Bonnet and Einstein-Weyl gravities. It was shown that although the potential $V_{-}(r)$ of the Dirac field has a negative gap near the event horizon, we have proved that the Dirac field is stable in both considered theories. This is possible because of the stability of the second potential $V_{+}(r)$ and the iso-spectrality of both potentials. Quasinormal modes were obtained for both metrics for different values of the angular parameter $\ell$. The dependence of the complex frequency on the new parameter $p$ was constructed. The Einstein-Weyl gravity allows for much stronger deviations form the Schwarzschild geometry. Therefore, quasinormal modes of Einstein-Weyl black hole are more different from the Schwazrschild case than those of Einstein-dilaton-Gauss-Bonnet black hole, achieving the effect of tens of percents. In the last part of this work we shown, that the Hod's conjecture holds for Einstein-dilaton-Gauss-Bonnet and Einstein-Weyl gravities for the Dirac field.
In the future, it would be interesting to investigate the Dirac field including the massive term and check the possibility of the existence of the arbitrarily long-lived quasinormal modes, called quasiresonances \cite{Ohashi:2004wr}, for this case. In \cite{Konoplya:2017tvu} it has recently been shown that the quasiresonances exist for the massive Dirac field in the Einstein theory, but no such study was performed in the higher curvature corrected theories. Our approach could also be extended to the case of Einstein-Gauss-Bonnet black holes with other types of coupling of the scalar field \cite{Konoplya:2019fpy} as well as to scalarized black holes for whose metrics analytical approximations are known \cite{Konoplya:2019goy}.
\acknowledgments{
The author acknowledges the support of the grant 19-03950S of Czech Science Foundation ($GA\check{C}R$) and acknowledges the SU grant SGS/12/2019}. The author would like to acknowledge Roman Konoplya for useful discussions.
\newpage
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1,108,101,564,568 | arxiv | \section{\@startsection {section}{1}{\z@}%
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\begin{document}
\title{\sc\bf\large\MakeUppercase{Model identification for ARMA time series through convolutional neural networks}}
\author{\sc Wai Hoh Tang and Adrian R\"ollin}
\date{\it National University of Singapore}
\maketitle
\begin{abstract}
In this paper, we use convolutional neural networks to address the problem of model identification for autoregressive moving average time series models.
We compare the performance of several neural network architectures, trained on simulated time series, with likelihood based methods, in particular the Akaike and Bayesian information criteria. We find that our neural networks can significantly outperform these likelihood based methods in terms of accuracy and, by orders of magnitude, in terms of speed.
\end{abstract}
\noindent\textbf{Keywords: autoregressive moving average time series; ARMA; model identification; model selection; ; convolutional neural networks; residual neural networks (ResNet); Akaike information criterion; Bayesian information criterion.}
\bigskip
\section{Introduction}
The autoregressive moving average (ARMA) time series model is a classical stochastic model that appears in diverse fields from foreign exchange to biomedical science to rainfall prediction. Using ARMA model for time series analysis typically involves three parts: identification of model orders, estimation of model coefficients and forecasting. Identification of ARMA orders is crucial as this has an impact on the subsequent two parts. While the equation describing an ARMA model is simple and easy to interpret, the task of correctly identifying the orders of the AR and MA components to fit a time series is not straightforward. Various methods have been proposed, such as graphical approaches based on autocorrelation function and partial autocorrelation function by \cite{Box1976} and, more commonly, likelihood based methods, such as Akaike information criterion (AIC); see \cite{Akaike1969} and \cite{Durbin2001}, Bayesian information criterion (BIC); see \cite{Akaike1977}, \cite{Rissanen1978} and \cite{Schwarz1978}, Hannan-Quinn information criterion (HQC); see \cite{HannanQuinn1979}, and minimum eigenvalue criterion (MEV); see \cite{Liang1993}.
Artificial intelligence methods such as genetic algorithms; see \cite{Ong2005}, \cite{Palaniappan2006}, \cite{Abo-Hammour2011} and \cite{Calster2017}, and artificial neural networks; see \cite{Lee1991}, \cite{Jhee1992}, \cite{LeeJhee1994}, \cite{LeeOh1996}, \cite{Chenoweth2000} and \cite{AlQawasmi2010}, are other paradigms in model identification. There are common features amongst the methodologies of these artificial neural networks studies, namely limiting the range of ARMA orders, limiting the length of time series and the need for pre-processing of time series. Firstly, the range of ARMA orders in these neural networks papers is typically small, for example, \cite{Chenoweth2000} evaluated orders up to 2 and \cite{LeeOh1996} evaluated orders up to 5. \cite{LeeOh1996} explained that most time series in the real world falls within ARMA(5,5) model. Secondly, the length of time series can vary substantially. \cite{Chenoweth2000} used time series of length 100 and 3,000 while \cite{AlQawasmi2010} used length 1,500. \cite{Chenoweth2000} reasoned that the longer length was chosen to examine an upper limit in accuracy of identification because estimation errors were expected to be minimal for time series of such long length while the shorter length was more representative of real economic data. Thirdly, it is notable that raw time series data are not used directly as inputs in their unprocessed form. Instead, statistical properties or features of time series are used as inputs. \cite{AlQawasmi2010} used special covariance matrix of MEV criterion as input and \cite{Jhee1992}, \cite{LeeJhee1994} and \cite{Chenoweth2000} used extended sample autocorrelation function (ESACF) as inputs. Most papers reckoned that there are promises in using neural networks, and identifications are reasonably accurate but additional work is required for improvement.
In recent years, neural networks have received an increased amount of attention, in particular after \cite{Krizhevsky2012} introduced a convolutional neural network (CNN) architecture, called \emph{AlexNet}, which had won the \emph{ImageNet Large Scale Visual Recognition Challenge} in 2012 and dramatically improved the state-of-the-art of visual recognition and object detection; see the survey by \cite{LeCun2015}. Although the key ideas of the neural network has already been introduced by \cite{LeCun1990}, only advancements in computing power and availability of large data sets have enabled such deep CNN architectures to be built and trained efficiently. Many embellishments of the original architecture have been proposed since; particularly important has been the introduction of so-called \emph{skip connections} by \cite{He2015}, which allowed their architecture \emph{ResNet} to win the ImageNet competition in 2015.
In this paper, we attempt to harness the strength of CNNs --- which are ultimately just hierarchical non-linear filters --- for the purpose of ARMA time series model selection. At the most fundamental level, likelihood-based methods and, as matter of fact, all other methods, are simply non-linear functions from the data space into some decision space. In our setting, the data space consists of the raw time series, and the decision space the order of autoregressive (AR) and moving average (MA) components predicted by the non-linear function. Our aim will thus be to find a suitable neural network (serving as the non-linear function) that maximises some objective, which, in our case, is the probability of getting the correct order of the AR or MA component. In contrast to earlier artificial neural networks studies, we seek to evaluate a larger range of ARMA orders with length of time series at 1,000, which is shorter but in the same order as studies by \cite{Chenoweth2000} and \cite{AlQawasmi2010}. In order to keep the computational load manageable, we have restricted ourselves to a maximal order of the each of the AR and MA components of 9, amounting to a total of 100 different combinations of ARMA($p,q$) models. This is not a conceptual restriction --- higher orders can easily be achieved by increasing architecture sizes at the cost of also increasing computational time, in particular during training. An important difference in our approach is that time series are used directly as inputs to CNNs without prior needs of computing or using any statistical properties of the time series, apart from centering and scaling.
The article consists of mainly two parts: First, finding a suitable CNN architecture and training strategy to solve the task of model selection, and second, to compare the best-performing neural network against two classical likelihood-based methods, namely the Akaike and Bayesian information criteria. The two likelihood-based methods each come in two flavours: step-wise model selection and full search.
\section{Methods}
\subsection{Simulation of ARMA($p,q$) time series}\label{SSec:arma}
An ARMA time series model consists of an autoregressive part and a moving average part. We follow the usual convention and denote $p$ for AR order and $q$ for MA order. A time series $X_n$ following an ARMA($p,q$) model can then be expressed recursively as
\ben{
X_n = \varepsilon_n + \sum_{i=1}^{p} \phi_i X_{n-i} + \sum_{j=1}^{q} \theta_j \varepsilon_{n-j},
\label{eq:ARMA}}
where $\phi_1,\dots,\phi_p$ and $\theta_1,\dots,\theta_q$ are fixed real-valued coefficients (typically unknown), and where noise $\varepsilon_n$ are independent and identically distributed random values.
For the distribution of $\varepsilon_n$, we consider two cases in this paper: a standard normal distribution and a $t$-distribution with 2 degrees of freedom, the latter representing a very heavy-tailed distribution. These two types of time series are denoted as \acrshort{acr:tsnorm} and \acrshort{acr:tstdist} respectively.
Model identification by AIC and BIC selection criteria in \cite{HyndmanKhandakar2007} requires time series to be stationary and invertible. Time series used for training of CNN are therefore stationary and invertible as well. Coefficients $\phi_i$ and $\theta_j$ are generated such that all roots of AR polynomial and MA polynomial are larger than 1 (i.e. outside of unit circle) in order to guarantee that the time series defined by \eqref{eq:ARMA} is stationary and invertible. When we generate a time series for a given order $p$, we start by assigning all coefficients independent standard normal random values. If the stationarity condition is not met, we pick one of the autoregressive coefficients $\phi_1,\dots,\phi_p$ uniformly at random and halve its value. If the condition is met, we stop, otherwise we again pick one of the coefficients $\phi_1,\dots,\phi_p$ uniformly at random, halve it, and repeat this procedure until the condition for stationarity is met.
Likewise, we apply the same approach when generating the $\theta$ coefficients that fulfill the invertibility condition for a given order $q$. Invertibility condition is the counterpart to stationarity for the moving average coefficients. It implies that noise can be expressed as weighted sum of current and past observations; that is, we can write
\ben{
\varepsilon_n = \sum_{i=0}^{\infty} \pi_i X_{n-i}
}
for some real numbers $\pi_0, \pi_1, \dots$. In other words, information of noise (which is typically not observable) is equivalent to the information of current and past observable data.
While, for example, \cite{Minerva2001}, \cite{Cigizoglu2003} and \cite{Zhang2005} use simulated time series to augment a given dataset in training, we only use simulated data in this paper. Both training and testing data are generated based on the model~\eqref{eq:ARMA}. We use a standardized length of 1,000 time steps for each time series, excluding the respective burn-in time\footnote{Burn-in period in R arima.sim function is computed as $p + q + \ceil{6/\log(minroots)}$ when $p > 0$, where $minroots$ is the smallest absolute value of complex roots of AR polynomial. We modify the last term of burn-in calculation in our code to $\min(50,000, \ceil{10/\log(minroots)})$. Although this may result in a slight computational overhead, we want to be sure that stationarity has been reached. When $p=0$, the burn-in period is set to $q$.}. An effective training of CNN architectures requires a large number of input data and moreover, in order to avoid over-fitting in computer vision tasks, synthetic training data is commonly generated through manipulation of images by cropping, reflecting, etc. Since all our data is simulated and no-time series is seen twice by any CNN, over-fitting is not expected to be an issue --- for the same reason we do also not make use dropout like \cite{Ioffe2015} and \cite{He2015}.
\subsection{CNN architectures}\label{SSec:CNNArchitectures}
We consider four different architectures of convolutional neural networks in this article: one `vanilla' CNN without skip connections and three `residual' CNNs with skip connections, but with different arrangements of the convolutional layers, rectifier linear unit layers, batch normalisation layers and skip connections. The three residual architectures are adapted from \cite{He2016}, and the detailed arrangement of the various layers is illustrated in Figure~\ref{fig:Diagram}. Benefit and importance of skip connections are discussed by \cite{He2016}. Whereas \cite{He2015,He2016} introduce skip connections from the second layer onwards, we introduce skip connections right from the first layer onwards by `fanning out' the input to multiple features from the first layer.
\begin{figure}
\centering
\includegraphics[width=\textwidth]{Diagram.png}
\caption{Different architectures tested. The ResNet blocks are based on \cite{He2016} paper.}
\label{fig:Diagram}
\end{figure}
The size of a CNN architecture is determined by the total number of tunable parameters in the architecture. Architecture size can be varied by changing the hyper-parameters depth (counted as the number of convolutional layers), \acrfull{acr:filterwidth} and number of features. \cite{He2015} introduced very deep ResNet architectures such as ResNet-50, ResNet-101 and most famously ResNet-152, which won the 2015 ImageNet competition, where the digits represent the number of convolutional layers in the respective networks. \cite{Zagorukyo2016} highlighted the large sizes of these ResNet architectures: 25.6, 44.5 and 60.2 million parameters in these three networks respectively. Furthermore, \cite{Zagorukyo2016} showed that a wide but shallower network can achieve the same accuracy of a very deep but thin network of comparable size and it is several times faster to train the former network. Widening of a network is done by increasing the number of features.
In the same vein, we experiment with a range of hyper-parameters and at the same time evaluate whether a small network can perform ARMA model identification adequately. Moreover, a smaller network can be trained faster and thus enabling more experiments to be conducted. In this paper, depth ranges from 8 to 24 in steps of 4, \acrshort{acr:filterwidth} takes values 7, 11 and 15, and the number of features ranges from 8 to 68 in steps of 12. Resulting architecture sizes cover a wide range from 3,502 to 1,541,862 parameters. This allows us to investigate how the performance of model identification may vary when architecture size increases. The overall layout of our CNNs are illustrated in Figure~\ref{fig:CNNLayout}. Each CNN architecture is trained for estimating either AR orders or MA orders but not both orders concurrently in this paper. For ease of reference, the architectures are denoted by \acrshort{acr:cnnAR} and \acrshort{acr:cnnMA} respectively. Each output channel in a CNN represents one order. A mean operation, called average pooling, maps each feature in the last convolutional layer to a corresponding output channel, which can then be interpreted as probabilities using a softmax function. When a time series is fed through the networks, the predicted order is read off directly from the output channel with the highest value, i.e. the highest probability.
\begin{figure}
\centering
\includegraphics[width=\textwidth]{CNNLayout.png}
\caption{A general layout of convolutional neural network architecture to illustrate filtering of information from time series input and in between convolutional layers. The rectangle blocks represent stacks of convolutional, batch normalization and/or rectifier linear unit layers as illustrated in Figure~\ref{fig:Diagram}. Intermediate blocks are represented in dotted lines. Mean operator, i.e. average pooling, is applied to each of last 10 blocks to generate an output list. AR or MA order is identified from the index of largest value in this output list.}
\label{fig:CNNLayout}
\end{figure}
\subsection{Training of a CNN architecture}\label{SSec:CNNtraining}
Training data are generated on the fly during training. Each training batch comprises 100 time series of length 1,000: one time series for each of the 100 possible combinations of $p$ and $q$ orders from 0 to 9. For each time series, the coefficients are chosen at random as described in Section~\ref{SSec:arma}, independently of all other time series, and independently of other batches. The noise component, $\varepsilon$, is drawn from a standard normal distribution. Due to memory constrains, these 100 time series are further divided into two mini-batches randomly, and each mini-batch is used for one iteration of a forward and back-propagation step. Since all data is simulated and no two time series ever used twice on a particular neural network, it is not meaningful to use the concept of `epochs' in our setting.
As loss function, we use the cross-entropy criterion, and as gradient descent optimizers, we use \emph{Adam} and \acrfull{acr:nag}. The Adam optimizer is a popular method in computer vision deep learning. The first moment coefficient $\beta_1$ and second moment coefficient $\beta_2$ of the Adam optimizer are set at recommended default values of 0.9 and 0.999 respectively; see \cite{KingmaBa2015}. Momentum parameter $\mu$ of \acrshort{acr:nag} is tested at two different values, 0.95 (see \cite{Sutsveker2013}) and 0.75. We keep track of mean error, which is the average of loss function during training. Learning rate starts from 0.1 and is halved if there is no reduction in mean error after 6 consecutive 100 batches. Training stops when learning rate is lower than 0.0001.
We make use of GPUs for faster training of the neural networks. Forward and backward propagations can be done with matrix multiplications, which are computed efficiently by GPU. We use NVIDIA(R) GeForce(R) GTX 1080 GPUs for training in this paper.
\subsection{Selection of a ResNet architecture}\label{SSec:ResNetSelection}
When we repeat the training of an architecture for a fixed set of hyper-parameters, the outcome will naturally vary because the weights and biases of the architecture are initialized randomly and the simulated time series are completely different. In order to ascertain it is not by random chance that one ResNet architecture performs better than the other two, these architectures are `pitched' against each other. To do so, we firstly identify the best performing neural network of each ResNet architecture in Section~\ref{SSec:CNNtraining} and obtain its corresponding set of hyper-parameters. With these three sets of hyper-parameters, we then consider the combinations of all ResNet architectures and these hyper-parameters. We conduct 100 repeated training for all the nine resultant CNNs. In other words, each architecture will be trained with 3 different sets of hyper-parameters, 100 repetitions for each set of hyper-parameters. Through the box plots of this experiment, we choose the ResNet architecture in Figure~\ref{fig:Diagram} that works best.
\subsection{Progressive retraining of CNN architectures}\label{SSec:Retraining}
After determining the best architecture and corresponding hyper-parameters, the performance of this selected CNN can be improved by a series of progressive retraining. To illustrate this method, assume we are retraining a \acrshort{acr:cnnAR}. We perform a series of training where we vary the values of MA orders by increasing the orders gradually, thus the term progressive. The $q$ values are fixed throughout in one training and increase in a subsequent training. For instance, we start by restricting $q=0$, retrain the network, then repeat the retraining at $q=1$ and so on until we reach $q=9$. Subsequently, we use a range of $q= \{0,1,2,\dots,k \}$, where $k$ is increased from 1 to 9 progressively\footnote{Before we change $q$ from one value to another, we impose a stopping condition and perform recursive rounds of training so that we are indeed improving the performance of the network. Every time a \acrshort{acr:cnnAR} is fully trained, the resulting mean error is compared with the mean error of previous round of training. If the new mean error is lower, we earmark the newly trained CNN as the selected CNN, otherwise it is discarded. We then reset learning rate to a higher value of 0.5 in hope of the optimizer finding a lower minima and perform another round of training. These steps are repeated over and over until there is no further reduction in mean error for 6 consecutive rounds. We then proceed with a different $q$ value.}. In a way, the network adapts to the increasing variation in the time series due to increasing MA orders. Vice versa, we will control the AR orders in the same fashion when we retrain \acrshort{acr:cnnMA}. This retraining approach is analogous to the idea of transfer learning in computer vision. We are able to move from one local minima to a lower minima with this process.
\subsection{Validation of test suites}
We generate two test suites for each type of time series where the white noise comes from different underlying distributions, as mentioned in Section~\ref{SSec:arma}. The test suites are validated by AIC and BIC selection criteria and our trained CNN architectures. In our training process, a CNN architecture is trained with only one type of time series but it will be tested against both types of time series during validation. This will examine the robustness of CNN architectures against variation in underlying white noise of time series. Each test suite contains 10,000 time series (i.e. 100 batches of 100 time series, one time series for each of the 100 possible combinations of $p$ and $q$ orders from 0 to 9 in each batch).
Accuracy is the first metric for performance of identifications. We examine the percentage of correct identifications and \acrfull{acr:mse}, which is the average of square of difference between classified orders and actual orders. A lower \acrshort{acr:mse} means that classified orders are more concentrated around actual values. These two measurements are usually positively correlated. Furthermore, confidence intervals of the percentage of correct identifications are estimated by assuming a binomial distribution. Computational time is the second metric.
Since there is a separate CNN architecture for identification of AR and MA orders respectively, the architectures are eventually assembled together to evaluate each test suite. There are two configurations of such assembly. The first configuration is done by feeding the test suite to \acrshort{acr:cnnAR} and \acrshort{acr:cnnMA} independently. The layout is in Figure~\ref{fig:ArchitectureSeparate} and this configuration is denoted as CNN (Separate). The second configuration makes use of an ensemble of intermediate \acrshort{acr:cnnMA} that are produced during the progressive retraining in Section~\ref{SSec:Retraining}. The layout in Figure~\ref{fig:ArchitectureJoint} illustrates the mechanism where an appropriate \acrshort{acr:cnnMA} is chosen accordingly, conditional on the output of \acrshort{acr:cnnAR}. This configuration is denoted as CNN (Joint).
\begin{figure}
\centering
\includegraphics[width=0.75\textwidth]{ArchitectureSeparate.png}
\caption{CNN (Separate). \acrshort{acr:cnnAR} and \acrshort{acr:cnnMA} evaluate time series independently.}
\label{fig:ArchitectureSeparate}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=\textwidth]{ArchitectureJoint.png}
\caption{CNN (Joint). A corresponding \acrshort{acr:cnnMA} is chosen accordingly based on identification of AR order by preceding \acrshort{acr:cnnAR}.}
\label{fig:ArchitectureJoint}
\end{figure}
For the completeness of our validation, we train another two separate \acrshort{acr:cnnAR} and \acrshort{acr:cnnMA} with \acrshort{acr:tstdist}, i.e. drawing the noise, $\varepsilon$, in Section~\ref{SSec:CNNtraining} from a heavy-tailed distribution now. These two CNNs will also be tested against the two test suites in the same CNN (Separate) and CNN (Joint) configurations. Instead of repeating the process of selecting ResNet architecture and set of hyper-parameters, we utilize the same ResNet architecture and corresponding hyper-parameters that we have obtained in the steps earlier when we are working with \acrshort{acr:tsnorm}.
Validation by AIC and BIC selection criteria is done by \emph{auto.arima} function in R. It is done in both stepwise and non-stepwise settings. We shall call the non-stepwise setting as full evaluation for ease of comparison. Stepwise algorithm is faster because fewer combinations of $p$ and $q$ are considered. A full evaluation produces better accuracy naturally because all combinations of $p$ and $q$ are considered, but this approach takes much longer. Parameters of auto.arima function are set such that unnecessary computations are avoided\footnote{Setting of some parameters of auto.arima function. Order of first-differencing is set to 0, seasonal is set to FALSE, stationary is set to TRUE and allowmean is set to FALSE since the inputs are stationary ARMA($p,q$) time series.}, which will maximize processing time without affecting identification accuracy.
We perform the validations on different machines concurrently because of the long processing time in R. AIC and BIC tests are performed on Intel(R) Xeon(R) CPU E7-4850 v2 @ 2.30GHz and Intel(R) Xeon(R) CPU E7- 4870 @ 2.40GHz computers while CNN architectures are tested on a Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz computer. In order to compare the processing time of these different methods, an exact computational task\footnote{A code to compute the first 800 digits of $\pi$ for 200,000 iterations. It is based on the code from https://crypto.stanford.edu/pbc/notes/pi/code.html.} was timed on all three machines. The performance on Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz computer is used as the denominator to standardize all processing time.
\section{Results}
\subsection{Training: Selection of CNN architectures and hyper-parameters}
We observe from Figure~\ref{fig:ARidentificationsADAM}~to~\ref{fig:MAidentificationsNAG095} that ResNet architectures outperform `vanilla' CNN architecture. \emph{ReLU before activation} ResNet architecture produces the best \acrshort{acr:cnnAR} and \acrshort{acr:cnnMA} when training are conducted with \acrshort{acr:nag} optimizer, momentum of 0.75. Furthermore, progressive retraining methodology also improves accuracy of identifications. Mean error of \acrshort{acr:cnnAR} decreases from 1.7104 in Figure~\ref{fig:ARidentificationsNAG075} to 1.5490. Mean error of \acrshort{acr:cnnMA} decreases from 1.8486 in Figure~\ref{fig:MAidentificationsNAG075} to 1.6203.
\begin{figure}
\centering
\includegraphics[width=0.75\textwidth]{ARidentificationsADAM.png}
\caption{Mean errors of \acrshort{acr:cnnAR} architectures trained with Adam optimizer.}
\label{fig:ARidentificationsADAM}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.75\textwidth]{ARidentificationsNAG075.png}
\caption{Mean errors of \acrshort{acr:cnnAR} architectures trained with \acrshort{acr:nag} optimizer; momentum of 0.75.}
\label{fig:ARidentificationsNAG075}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.75\textwidth]{ARidentificationsNAG095.png}
\caption{Mean errors of \acrshort{acr:cnnAR} architectures trained with \acrshort{acr:nag} optimizer; momentum of 0.95.}
\label{fig:ARidentificationsNAG095}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.75\textwidth]{MAidentificationsADAM.png}
\caption{Mean errors of \acrshort{acr:cnnMA} architectures trained with Adam optimizer.}
\label{fig:MAidentificationsADAM}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.75\textwidth]{MAidentificationsNAG075.png}
\caption{Mean errors of \acrshort{acr:cnnMA} architectures trained with \acrshort{acr:nag} optimizer; momentum of 0.75.}
\label{fig:MAidentificationsNAG075}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.75\textwidth]{MAidentificationsNAG095.png}
\caption{Mean errors of \acrshort{acr:cnnMA} architectures trained with \acrshort{acr:nag} optimizer; momentum of 0.95.}
\label{fig:MAidentificationsNAG095}
\end{figure}
Difference in arrangement of layers in ResNet architectures does have a significant effect. We shortlist the hyper-parameters of the best performing \emph{ReLU before activation}, \emph{Original} and \emph{Full pre-activation} architectures from Figure~\ref{fig:ARidentificationsNAG075}~and~\ref{fig:MAidentificationsNAG075}. These hyper-parameters are tabulated in Table~\ref{tbl:ARnag075ResNetArchitectures} and \ref{tbl:MAnag075ResNetArchitectures} respectively. The box plots of 100 repetitive training of each combination of ResNet architecture and these hyper-parameters in Figure~\ref{fig:ARboxplot}~and~\ref{fig:MAboxplot} show clearly that `ReLU before activation' architecture outperforms the other two ResNet architectures regardless of hyper-parameters for both identification of AR and MA orders. Moreover, we observe from Figure~\ref{fig:ARMAboxplotNewVsOriginal} that introduction of skip connections from first convolutional layer instead of second convolutional layer improves accuracy of identifications.
\begin{table}\centering
\ra{1.3}
\resizebox{\columnwidth}{!}{
\begin{tabular}{@{}rcccccc@{}}
\toprule
\tbf{ResNet~architecture} & \tbf{Depth} & \mline{\tbf{\acrshort{acr:filterwidth}}} & \mline{\tbf{Number~of}\\\tbf{Features}} & \tbf{Size} & \mline{\tbf{Mean}\\\tbf{Error}} & \mline{\tbf{Training}\\\tbf{time~(hours)}}\\
\midrule
\tbf{ReLU before activation} & 16 & 15 & 68 & 985,350 & 1.710 & 1.606 \\
\tbf{Original} & 8 & 15 & 68 & 428,838 & 1.751 & 1.079 \\
\tbf{Full pre-activation} & 20 & 15 & 44 & 532,518 & 1.750 & 1.675 \\
\bottomrule
\end{tabular}
}
\caption{Mean errors and hyper-parameters of respective \acrshort{acr:cnnAR} trained with \acrshort{acr:tsnorm} and \acrshort{acr:nag} optimizer; momentum of 0.75.}
\label{tbl:ARnag075ResNetArchitectures}
\end{table}
\begin{table}\centering
\ra{1.3}
\resizebox{\columnwidth}{!}{
\begin{tabular}{@{}rcccccc@{}}
\toprule
\tbf{ResNet~architecture} & \tbf{Depth} & \mline{\tbf{\acrshort{acr:filterwidth}}} & \mline{\tbf{Number~of}\\\tbf{Features}} & \tbf{Size} & \mline{\tbf{Mean}\\\tbf{Error}} & \mline{\tbf{Training}\\\tbf{time~(hours)}}\\
\midrule
\tbf{ReLU before activation} & 24 & 15 & 44 & 649,206 & 1.849 & 3.065 \\
\tbf{Original} & 12 & 15 & 56 & 481,518 & 1.967 & 2.056 \\
\tbf{Full pre-activation} & 20 & 15 & 68 & 1,263,606 & 1.901 & 3.266 \\
\bottomrule
\end{tabular}
}
\caption{Mean errors and hyper-parameters of respective \acrshort{acr:cnnMA} trained with \acrshort{acr:tsnorm} and \acrshort{acr:nag} optimizer; momentum of 0.75.}
\label{tbl:MAnag075ResNetArchitectures}
\end{table}
\begin{figure}
\centering
\includegraphics[width=\textwidth]{ARboxplot.png}
\caption{Average mean error of training of 100 \acrshort{acr:cnnAR} for each combination of architecture and corresponding hyper-parameters. See Section~\ref{SSec:ResNetSelection}.}
\label{fig:ARboxplot}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=\textwidth]{MAboxplot.png}
\caption{Average mean error of training of 100 \acrshort{acr:cnnMA} for each combination of architecture and corresponding hyper-parameters. See Section~\ref{SSec:ResNetSelection}.}
\label{fig:MAboxplot}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.75\textwidth]{ARMAboxplotNewVsOriginal.png}
\caption{An experiment to show that introducing skip connections right from the first layer improves the performance of our CNNs. The word `(Original)' in the diagram denotes the architecture, which has skip connections from second layer onwards, just like those in \cite{He2015} and \cite{He2016}. The box plots show the average mean errors of training of 100 \acrshort{acr:cnnAR} and \acrshort{acr:cnnMA} respectively for each architecture. The hyper-parameter values are arbitrary chosen at depth of 20, \acrshort{acr:filterwidth} of 11 and number of features of 56 for \acrshort{acr:cnnAR} and depth of 24, \acrshort{acr:filterwidth} of 15 and number of features of 68 for \acrshort{acr:cnnMA}.}
\label{fig:ARMAboxplotNewVsOriginal}
\end{figure}
In general, it is seen in our linear regression analysis that mean error decreases as architecture size increases or in other words, accuracy of identifications improves when architecture size increases. This is observed in Figure~\ref{fig:ARregressplot}~and~\ref{fig:MAregressplot}, where training of `ReLU before activation' for the entire range of architecture size from 3,502 to 1,541,862 is repeated 5 times. Table~\ref{tbl:ARregresstable}~and~\ref{tbl:MAregresstable} summarize the negative correlation between mean error and log of architecture size. The results tell us that architecture size is an important component. Moreover, for a given architecture size, it seems depth and \acrshort{acr:filterwidth} hyper-parameters are more significant than number of features. As the hyper-parameters are intertwined given an architecture size, increasing one hyper parameter will require reduction in one or both of the other two hyper-parameters. Therefore when architecture size is a limitation, the results suggest that focus should be placed on depth and \acrshort{acr:filterwidth}.
\begin{figure}
\centering
\includegraphics[width=0.75\textwidth]{ARregressplot.png}
\caption{Training of \acrshort{acr:cnnAR} architectures is repeated 5 times for all combinations of hyper-parameters. Result of regression analysis is in Table~\ref{tbl:ARregresstable}.}
\label{fig:ARregressplot}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.75\textwidth]{MAregressplot.png}
\caption{Training of \acrshort{acr:cnnMA} architectures is repeated 5 times for all combinations of hyper-parameters. Result of regression analysis is in Table~\ref{tbl:MAregresstable}.}
\label{fig:MAregressplot}
\end{figure}
\begin{table}\centering
\begin{tabular}{l}
\toprule
\begin{tabular}{lccccc}
\tbf{Coefficients} & \tbf{Estimate} & \tbf{Std.~Error} & \tbf{t-value} & \tbf{p-value} \\
\midrule
\tbf{Intercept} & 2.037 & 0.060 & 34.059 & $\ll$0.001 \\
\tbf{log(ParamSize)} & -0.022 & 0.005 & -4.310 & $\ll$0.001 \\
\tbf{Depth} & -0.081 & 0.049 & -1.659 & 0.098 \\
\tbf{Depth\textsuperscript{2}} & 0.0379 & 0.016 & 2.299 & 0.022 \\
\tbf{kW} & -0.085 & 0.036 & -2.374 & 0.018 \\
\tbf{kW\textsuperscript{2}} & 0.017 & 0.015 & 1.135 & 0.257 \\
\tbf{Features} & 0.139 & 0.144 & 0.963 & 0.336 \\
\tbf{Features\textsuperscript{2}} & 0.053 & 0.043 & 1.216 & 0.225 \\
\end{tabular}
\\
\midrule
\begin{tabular}{ll}
\\
Residual~standard~error & 0.014 \\
$R^2$ & 0.826 \\
Adjusted~$R^2$ & 0.823 \\
N & 450 \\
\end{tabular}
\\
\bottomrule
\end{tabular}
\caption{Regression analysis of \acrshort{acr:cnnAR} architectures. Mean error is regressed against log of architecture size, depth, filter size (kW) and number of features. Architecture depth, filter size (kW) and number of features are set as polynomial of degree 2.}
\label{tbl:ARregresstable}
\end{table}
\begin{table}\centering
\begin{tabular}{l}
\toprule
\begin{tabular}{lccccc}
\tbf{Coefficients} & \tbf{Estimate} & \tbf{Std.~Error} & \tbf{t-value} & \tbf{p-value} \\
\midrule
\tbf{Intercept} & 2.526 & 0.210 & 12.018 & $\ll$0.001 \\
\tbf{log(ParamSize)} & -0.036 & 0.018 & -2.000 & 0.046 \\
\tbf{Depth} & -0.531 & 0.171 & -3.110 & 0.002 \\
\tbf{Depth\textsuperscript{2}} & 0.205 & 0.058 & 3.542 & $\ll$0.001 \\
\tbf{kW} & -1.031 & 0.126 & -8.173 & $\ll$0.001 \\
\tbf{kW\textsuperscript{2}} & 0.249 & 0.051 & 4.840 & $\ll$0.001 \\
\tbf{Features} & -0.160 & 0.508 & -0.315 & 0.753 \\
\tbf{Features\textsuperscript{2}} & 0.091 & 0.152 & 0.594 & 0.553 \\
\end{tabular}
\\
\midrule
\begin{tabular}{ll}
\\
Residual~standard~error & 0.050 \\
$R^2$ & 0.784 \\
Adjusted~$R^2$ & 0.781 \\
N & 450 \\
\end{tabular}
\\
\bottomrule
\end{tabular}
\caption{Regression analysis of \acrshort{acr:cnnMA} architectures. Mean error is regressed against log of architecture size, depth, filter size (kW) and number of features. Architecture depth, filter size (kW) and number of features are set as polynomial of degree 2.}
\label{tbl:MAregresstable}
\end{table}
The role of \acrshort{acr:nag} optimizer in the training of \acrshort{acr:cnnMA} is an interesting finding from our training. While Adam is a popular optimizer in the field of computer vision, it does not train \acrshort{acr:cnnMA} properly however. We find that the predicted MA orders tend to degenerate to a few orders regardless of CNN architectures trained with Adam optimizer, thus explaining why the mean errors of these architectures remain relatively unchanged between 2.20 to 2.35 in Figure~\ref{fig:MAidentificationsADAM} even when architecture size increases. On the other hand, we observe a decreasing mean error trend for architectures trained with \acrshort{acr:nag} optimizer in Figure~\ref{fig:MAidentificationsNAG075}~and~\ref{fig:MAidentificationsNAG095}. CNN architectures perform comparably well in identification of AR orders when they are trained with either Adam or \acrshort{acr:nag} optimizers.
\subsection{Validation: Comparison of CNN architectures against AIC and BIC selection criteria}
The results of first performance benchmarking with a set of 10,000 \acrshort{acr:tsnorm} are in Table~\ref{tbl:CompareNormalWN}. As expected, full evaluations by AIC and BIC selection criteria perform better than stepwise evaluations in terms of accuracy of identifications of AR and MA orders. The improvement in accuracy when performing full evaluations comes at a significant increase in computational costs however. Processing time increases from 3 hours in a stepwise BIC evaluation to 79 hours in a full BIC evaluation and from 8 hours in a stepwise AIC evaluation to 165 hours in full AIC evaluation. While the improvement from a stepwise to full BIC evaluation is significant, it is only marginal in the case of AIC selection criteria. Accuracies of classification of AR and MA orders by a full BIC evaluation are at 35.23\% and 35.58\% respectively vis-\'a-vis corresponding 23.84\% and 26.71\% by full AIC evaluation. Our observation of BIC selection criteria outperforming AIC selection criteria is consistent with \cite{Hannan1980} where BIC is also found to be a better criterion than AIC.
When we contrast accuracies of CNN architectures with full BIC evaluation in Table~\ref{tbl:CompareNormalWN}, we see that CNN architectures yield better results. The best performing CNN setup, \acrshort{acr:cnnnorm} (Joint), records 41.01\% and 40.00\% accuracies for identifications of AR and MA orders respectively. Moreover, the time taken is less than an hour. While \acrshort{acr:cnnnorm} (Joint) performs well in identifying individual AR and MA orders, its accuracy in identifying both orders concurrently is 20.47\%, which is marginally lower than 20.98\% by full BIC evaluation. A CNN architecture that is trained with \acrshort{acr:tstdist}, i.e. a different type of time series, fairs poorly however when evaluating \acrshort{acr:tsnorm} test suite, especially in terms of identification of MA orders. The output of \acrshort{acr:cnntdist} (Joint) is 30.91\% and 12.25\% for corresponding identifications of AR and MA orders.
\begin{table}\centering
\ra{1.3}
\resizebox{\columnwidth}{!}{
\begin{tabular}{@{}rcccccccc@{}}
\toprule
& \mline{\tbf{AIC}\\\tbf{stepwise}} & \mline{\tbf{AIC}\\\tbf{full}} & \mline{\tbf{BIC}\\\tbf{stepwise}} & \mline{\tbf{BIC}\\\tbf{full}} & \mline{\tbf{\acrshort{acr:cnnnorm}}\\\tbf{(Separate)}} & \mline{\tbf{\acrshort{acr:cnnnorm}}\\\tbf{(Joint)}} & \mline{\tbf{\acrshort{acr:cnntdist}}\\\tbf{(Separate)}} & \mline{\tbf{\acrshort{acr:cnntdist}}\\\tbf{(Joint)}} \\
\midrule
\tbf{AR(\%)} & 21.87 & 23.84 & 22.79 & 35.23 & 41.01 & 41.01 & 30.91 & 30.91 \\
\tbf{AR 95\%~C.I.} & [21.05, 22.69] & [23.00, 24.70] & [21.96, 23.63] & [34.26, 36.21] & [40.00, 42.02] & [40.00, 42.02] & [29.98, 31.85] & [29.98, 31.85] \\
\tbf{AR~\acrshort{acr:mse}} & 3.037 & 2.993 & 3.192 & 2.538 & 2.133 & 2.133 & 2.826 & 2.826 \\
\tbf{MA(\%)} & 23.94 & 26.71 & 23.71 & 35.58 & 39.10 & 40.00 & 12.03 & 12.25 \\
\tbf{MA 95\%~C.I.} & [23.09, 24.79] & [25.83, 27.61] & [22.87, 24.57] & [34.61, 36.57] & [38.10, 40.10] & [39.00, 41.01] & [11.41, 12.67] & [11.62, 12.89] \\
\tbf{MA~\acrshort{acr:mse}} & 3.115 & 2.854 & 3.429 & 2.626 & 2.240 & 2.230 & 4.427 & 4.347 \\
\tbf{Both~correct(\%)} & 10.55 & 11.66 & 11.88 & 20.98 & 17.82 & 20.47 & 3.73 & 3.63 \\
\tbf{Both~correct 95\%~C.I.} & [9.96, 11.14] & [11.05, 12.29] & [11.26, 12.51] & [20.18, 21.79] & [17.07, 18.57] & [19.68, 21.28] & [3.38, 4.09] & [3.28, 3.98] \\
\tbf{Time~taken~(hours)} & 7.967 & 164.919 & 3.005 & 78.964 & 0.439 & 0.454 & 0.438 & 0.445 \\
\bottomrule
\end{tabular}
}
\caption{Classifications of ARMA($p,q$) orders for 10,000 \acrshort{acr:tsnorm}. Time taken has been standardized.}.
\label{tbl:CompareNormalWN}
\end{table}
Table~\ref{tbl:CompareTDistWN} shows the second performance benchmarking with a set of 10,000 \acrshort{acr:tstdist}. When we examine \acrshort{acr:cnntdist}, i.e. the CNN architecture trained with time series of the same type of white noise, we see that CNN (Joint) setup perform better than CNN (Separate) setup again. Other observations are similar to Table~\ref{tbl:CompareNormalWN} with an exception that identification of MA orders by \acrshort{acr:cnntdist} (Joint) at 35.53\% falls short of full BIC evaluation at 38.20\%. One notable observation however is the performance of \acrshort{acr:cnnnorm}. Unlike the significant drop in accuracies when there is a mismatch of trained CNN architectures and types of time series in Table~\ref{tbl:CompareNormalWN}, \acrshort{acr:cnnnorm} (Joint) is able to evaluate \acrshort{acr:tstdist} at a comparable accuracies of 36.21\% and 34.89\% for respective identifications of AR and MA orders.
\begin{table}\centering
\ra{1.3}
\resizebox{\columnwidth}{!}{
\begin{tabular}{@{}rcccccccc@{}}
\toprule
& \mline{\tbf{AIC}\\\tbf{stepwise}} & \mline{\tbf{AIC}\\\tbf{full}} & \mline{\tbf{BIC}\\\tbf{stepwise}} & \mline{\tbf{BIC}\\\tbf{full}} & \mline{\tbf{\acrshort{acr:cnnnorm}}\\\tbf{(Separate)}} & \mline{\tbf{\acrshort{acr:cnnnorm}}\\\tbf{(Joint)}} & \mline{\tbf{\acrshort{acr:cnntdist}}\\\tbf{(Separate)}} & \mline{\tbf{\acrshort{acr:cnntdist}}\\\tbf{(Joint)}} \\
\midrule
\tbf{AR(\%)} & 23.43 & 30.48 & 22.44 & 34.71 & 36.21 & 36.21 & 46.81 & 46.81 \\
\tbf{AR 95\%~C.I.} & [22.59, 24.28] & [29.55, 31.41] & [21.62, 23.28] & [33.74, 35.68] & [35.23, 37.19] & [35.23, 37.19] & [45.78, 47.84] & [45.78, 47.84] \\
\tbf{AR~\acrshort{acr:mse}} & 2.958 & 2.689 & 3.173 & 2.471 & 2.095 & 2.095 & 1.699 & 1.699 \\
\tbf{MA(\%)} & 25.47 & 34.97 & 22.67 & 38.20 & 33.38 & 34.89 & 31.47 & 35.53 \\
\tbf{MA 95\%~C.I.} & [24.60, 26.34] & [34.00, 35.95] & [21.84, 23.50] & [37.21, 39.20] & [32.43, 34.35] & [33.92, 35.86] & [30.53, 32.41] & [34.56, 36.51] \\
\tbf{MA~\acrshort{acr:mse}} & 3.000 & 2.570 & 3.386 & 2.489 & 2.512 & 2.470 & 2.740 & 2.554 \\
\tbf{Both~correct~(\%)} & 13.01 & 18.60 & 12.38 & 22.20 & 13.48 & 15.91 & 16.40 & 20.09 \\
\tbf{Both~correct 95\%~C.I.} & [12.36, 13.66] & [17.84, 19.37] & [11.75, 13.02] & [21.38, 23.03] & [12.82, 14.14] & [15.20, 16.63] & [15.68, 17.13] & [19.30, 20.88] \\
\tbf{Time~taken~(hours)} & 6.253 & 77.714 & 2.957 & 54.380 & 0.437 & 0.451 & 0.439 & 0.438 \\
\bottomrule
\end{tabular}
}
\caption{Classifications of ARMA($p,q$) orders for 10,000 \acrshort{acr:tstdist}. Time taken has been standardized.}.
\label{tbl:CompareTDistWN}
\end{table}
Table~\ref{tbl:ARnorm}~to~\ref{tbl:MAtdist} show detailed breakdown of identification of respective AR and MA orders by full BIC evaluation and CNN (Joint). The highest values of each row in these tables are generally located along the diagonals for both approaches, which show that both methods identify well across the whole range of $p$ and $q$. We notice a pattern where accuracy is higher at lower $p$ or $q$ orders but it drops as the orders increase. Classifications of $q$ of order 7 and 8 by \acrshort{acr:cnnMA} in Table~\ref{tbl:MAtdist} are not as good as full BIC evaluation, which results in an overall lower accuracy in terms of identification of MA orders in Table~\ref{tbl:CompareTDistWN}.
\begin{table}\centering
\resizebox{\columnwidth}{!}{
\begin{tabular}{c c}
\toprule
\begin{tabular}{c c}
& BIC \\
& Classified AR order \\
\rotatebox[origin=c]{90}{Actual AR order} &
\begin{tabular}{rccccccccccc}
&& \tbf{0} & \tbf{1} & \tbf{2} & \tbf{3} & \tbf{4} & \tbf{5} & \tbf{6} & \tbf{7} & \tbf{8} & \tbf{9} \\
\cmidrule{3-12}
\tbf{0} && \tbf{\color{red} 65.40} & 13.50 & 6.00 & 3.10 & 1.30 & 1.20 & 1.10 & 1.70 & 2.00 & 4.70 \\
\tbf{1} && 27.80 & \tbf{\color{red} 51.20} & 7.10 & 3.00 & 2.70 & 0.60 & 0.80 & 1.20 & 2.40 & 3.20 \\
\tbf{2} && 13.30 & 19.20 & \tbf{\color{red} 45.50} & 7.90 & 3.80 & 2.00 & 1.10 & 0.60 & 2.60 & 4.00 \\
\tbf{3} && 6.20 & 11.00 & 19.40 & \tbf{\color{red} 40.20} & 8.60 & 4.90 & 2.80 & 1.90 & 1.90 & 3.10 \\
\tbf{4} && 4.50 & 7.10 & 13.90 & 22.80 & \tbf{\color{red} 34.50} & 6.40 & 3.30 & 1.90 & 1.80 & 3.80 \\
\tbf{5} && 1.70 & 4.50 & 7.40 & 14.30 & 20.90 & \tbf{\color{red} 31.90} & 7.70 & 4.90 & 2.80 & 3.90 \\
\tbf{6} && 1.60 & 3.80 & 7.30 & 9.80 & 15.20 & 22.30 & \tbf{\color{red} 25.40} & 6.20 & 3.90 & 4.50 \\
\tbf{7} && 1.90 & 2.80 & 7.10 & 10.00 & 12.40 & 16.00 & 17.00 & \tbf{\color{red} 20.90} & 6.20 & 5.70 \\
\tbf{8} && 2.10 & 2.80 & 6.10 & 8.00 & 10.10 & 13.00 & 16.60 & \tbf{\color{red} 16.70} & 15.80 & 8.80 \\
\tbf{9} && 2.10 & 2.00 & 5.00 & 6.70 & 8.60 & 11.20 & 11.50 & 15.60 & 15.80 & \tbf{\color{red} 21.50} \\
\end{tabular}
\end{tabular}
&
\begin{tabular}{c c}
& CNN \\
& Classified AR order \\
&
\begin{tabular}{cccccccccc}
\tbf{0} & \tbf{1} & \tbf{2} & \tbf{3} & \tbf{4} & \tbf{5} & \tbf{6} & \tbf{7} & \tbf{8} & \tbf{9} \\
\cmidrule{1-10}
\tbf{\color{red} 83.40} & 10.60 & 2.60 & 1.10 & 0.90 & 0.30 & 0.10 & 0.00 & 0.40 & 0.60 \\
33.30 & \tbf{\color{red} 55.20} & 6.10 & 2.20 & 0.70 & 0.60 & 0.40 & 0.50 & 0.40 & 0.60 \\
18.40 & 26.40 & \tbf{\color{red} 40.70} & 8.60 & 2.10 & 1.50 & 0.50 & 0.70 & 0.40 & 0.70 \\
6.60 & 12.00 & 18.20 & \tbf{\color{red} 44.20} & 9.70 & 4.20 & 1.60 & 1.20 & 0.60 & 1.70 \\
4.50 & 6.80 & 13.10 & 25.50 & \tbf{\color{red} 31.90} & 8.20 & 2.40 & 3.20 & 1.90 & 2.50 \\
4.30 & 3.80 & 6.50 & 15.10 & 21.90 & \tbf{\color{red} 33.90} & 5.30 & 3.80 & 3.00 & 2.40 \\
3.60 & 2.40 & 4.70 & 7.40 & 10.50 & 22.70 & \tbf{\color{red} 32.40} & 8.50 & 4.00 & 3.80 \\
4.20 & 2.50 & 4.80 & 6.90 & 13.90 & 14.60 & 16.30 & \tbf{\color{red} 24.10} & 6.70 & 6.00 \\
2.10 & 1.80 & 3.00 & 7.00 & 8.70 & 9.50 & 16.60 & 13.60 & \tbf{\color{red} 26.10} & 11.60 \\
2.20 & 1.70 & 2.80 & 3.80 & 8.20 & 6.30 & 8.90 & 15.50 & 12.40 & \tbf{\color{red} 38.20} \\
\end{tabular}
\end{tabular}
\\ \bottomrule
\end{tabular}
}
\caption{Classification of AR orders by full BIC evaluation and \acrshort{acr:cnnnorm} (Joint) in Table~\ref{tbl:CompareNormalWN}. Values are in percentages and normalized along the rows.}
\label{tbl:ARnorm}
\end{table}
\begin{table}\centering
\resizebox{\columnwidth}{!}{
\begin{tabular}{c c}
\toprule
\begin{tabular}{c c}
& BIC \\
& Classified MA order \\
\rotatebox[origin=c]{90}{Actual MA order} &
\begin{tabular}{rccccccccccc}
&& \tbf{0} & \tbf{1} & \tbf{2} & \tbf{3} & \tbf{4} & \tbf{5} & \tbf{6} & \tbf{7} & \tbf{8} & \tbf{9} \\
\cmidrule{3-12}
\tbf{0} && \tbf{\color{red} 78.40} & 10.30 & 4.50 & 1.90 & 1.10 & 0.50 & 0.40 & 0.70 & 0.40 & 1.80 \\
\tbf{1} && 31.20 & \tbf{\color{red} 55.30} & 5.90 & 2.90 & 1.20 & 0.90 & 0.50 & 0.40 & 0.50 & 1.20 \\
\tbf{2} && 13.90 & 23.70 & \tbf{\color{red} 46.10} & 7.00 & 3.80 & 1.20 & 1.00 & 0.80 & 0.50 & 2.00 \\
\tbf{3} && 8.50 & 14.50 & 24.10 & \tbf{\color{red} 39.00} & 6.30 & 2.90 & 0.90 & 0.80 & 0.70 & 2.30 \\
\tbf{4} && 7.40 & 9.50 & 16.30 & 20.70 & \tbf{\color{red} 33.80} & 4.70 & 3.90 & 1.00 & 0.90 & 1.80 \\
\tbf{5} && 7.60 & 6.70 & 11.60 & 14.30 & 22.50 & \tbf{\color{red} 26.80} & 4.70 & 2.80 & 1.80 & 1.20 \\
\tbf{6} && 5.50 & 8.50 & 9.40 & 11.40 & 15.30 & 19.70 & \tbf{\color{red} 20.70} & 3.90 & 2.40 & 3.20 \\
\tbf{7} && 5.60 & 5.50 & 7.80 & 8.20 & 11.70 & 13.10 & 19.30 & \tbf{\color{red} 21.10} & 4.80 & 2.90 \\
\tbf{8} && 4.60 & 6.00 & 7.30 & 7.60 & 10.00 & 12.00 & 13.50 & \tbf{\color{red} 17.00} & 15.90 & 6.10 \\
\tbf{9} && 4.00 & 4.50 & 6.30 & 7.40 & 10.10 & 8.70 & 12.40 & 14.00 & 13.90 & \tbf{\color{red} 18.70} \\
\end{tabular}
\end{tabular}
&
\begin{tabular}{c c}
& CNN \\
& Classified MA order \\
&
\begin{tabular}{cccccccccc}
\tbf{0} & \tbf{1} & \tbf{2} & \tbf{3} & \tbf{4} & \tbf{5} & \tbf{6} & \tbf{7} & \tbf{8} & \tbf{9} \\
\cmidrule{1-10}
\tbf{\color{red} 83.70} & 8.90 & 2.90 & 1.10 & 0.60 & 0.80 & 0.30 & 0.80 & 0.30 & 0.60 \\
30.70 & \tbf{\color{red} 53.80} & 7.70 & 3.10 & 1.10 & 0.70 & 0.70 & 0.60 & 0.60 & 1.00 \\
12.50 & 22.40 & \tbf{\color{red} 44.10} & 9.90 & 3.40 & 2.40 & 1.10 & 1.30 & 1.40 & 1.50 \\
6.60 & 11.60 & 23.10 & \tbf{\color{red} 41.20} & 7.50 & 4.80 & 1.50 & 1.90 & 0.80 & 1.00 \\
5.70 & 6.10 & 11.50 & 22.70 & \tbf{\color{red} 33.30} & 8.40 & 4.10 & 3.10 & 2.10 & 3.00 \\
3.90 & 4.90 & 6.80 & 12.40 & 23.20 & \tbf{\color{red} 30.00} & 6.80 & 4.50 & 3.40 & 4.10 \\
2.90 & 2.60 & 5.50 & 10.70 & 15.40 & 15.40 & \tbf{\color{red} 30.40} & 9.20 & 3.60 & 4.30 \\
2.90 & 3.50 & 3.10 & 5.90 & 8.40 & 14.00 & 19.20 & \tbf{\color{red} 28.00} & 7.20 & 7.80 \\
3.10 & 3.20 & 2.80 & 4.50 & 6.90 & 12.50 & 13.40 & 17.60 & \tbf{\color{red} 26.90} & 9.10 \\
4.00 & 2.30 & 3.10 & 4.00 & 6.00 & 10.10 & 10.10 & 13.80 & 18.00 & \tbf{\color{red} 28.60} \\
\end{tabular}
\end{tabular}
\\ \bottomrule
\end{tabular}
}
\caption{Classification of MA orders by full BIC evaluation and \acrshort{acr:cnnnorm} (Joint) in Table~\ref{tbl:CompareNormalWN}. Values are in percentages and normalized along the rows.}
\label{tbl:MAnorm}
\end{table}
\begin{table}\centering
\resizebox{\columnwidth}{!}{
\begin{tabular}{c c}
\toprule
\begin{tabular}{c c}
& BIC \\
& Classified AR order \\
\rotatebox[origin=c]{90}{Actual AR order} &
\begin{tabular}{rccccccccccc}
&& \tbf{0} & \tbf{1} & \tbf{2} & \tbf{3} & \tbf{4} & \tbf{5} & \tbf{6} & \tbf{7} & \tbf{8} & \tbf{9} \\
\cmidrule{3-12}
\tbf{0} && \tbf{\color{red} 75.10} & 7.90 & 5.10 & 2.10 & 1.90 & 1.20 & 1.30 & 1.00 & 1.90 & 2.50 \\
\tbf{1} && 30.50 & \tbf{\color{red} 46.80} & 7.70 & 3.10 & 2.60 & 1.40 & 1.90 & 1.40 & 1.30 & 3.30 \\
\tbf{2} && 13.60 & 27.50 & \tbf{\color{red} 39.00} & 5.70 & 3.10 & 2.20 & 1.90 & 1.80 & 1.80 & 3.40 \\
\tbf{3} && 6.00 & 12.10 & 22.20 & \tbf{\color{red} 40.20} & 7.40 & 3.70 & 2.10 & 2.20 & 1.60 & 2.50 \\
\tbf{4} && 5.80 & 11.40 & 16.20 & 18.80 & \tbf{\color{red} 30.40} & 6.30 & 2.70 & 2.70 & 2.60 & 3.10 \\
\tbf{5} && 1.90 & 4.50 & 12.60 & 17.70 & 21.10 & \tbf{\color{red} 27.90} & 4.50 & 3.20 & 3.10 & 3.50 \\
\tbf{6} && 2.30 & 4.00 & 8.60 & 13.30 & 19.00 & 18.50 & \tbf{\color{red} 22.00} & 5.40 & 4.20 & 2.70 \\
\tbf{7} && 2.40 & 3.40 & 6.10 & 8.50 & 11.60 & 18.10 & 18.70 & \tbf{\color{red} 19.30} & 7.00 & 4.90 \\
\tbf{8} && 1.20 & 2.80 & 3.70 & 4.90 & 10.00 & 12.30 & 14.50 & 20.30 & \tbf{\color{red} 23.50} & 6.80 \\
\tbf{9} && 1.70 & 2.60 & 5.20 & 7.00 & 6.70 & 10.20 & 11.70 & 16.50 & 15.50 & \tbf{\color{red} 22.90} \\
\end{tabular}
\end{tabular}
&
\begin{tabular}{c c}
& CNN \\
& Classified AR order \\
&
\begin{tabular}{cccccccccc}
\tbf{0} & \tbf{1} & \tbf{2} & \tbf{3} & \tbf{4} & \tbf{5} & \tbf{6} & \tbf{7} & \tbf{8} & \tbf{9} \\
\midrule
\tbf{\color{red} 91.40} & 6.50 & 2.10 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 \\
28.70 & \tbf{\color{red} 58.60} & 10.20 & 1.90 & 0.40 & 0.10 & 0.00 & 0.00 & 0.00 & 0.10 \\
9.00 & 23.90 & \tbf{\color{red} 52.90} & 11.40 & 1.90 & 0.60 & 0.00 & 0.00 & 0.00 & 0.30 \\
2.40 & 6.70 & 20.90 & \tbf{\color{red} 54.80} & 9.90 & 2.30 & 1.40 & 0.80 & 0.20 & 0.60 \\
3.30 & 5.00 & 10.40 & 23.60 & \tbf{\color{red} 45.10} & 7.00 & 3.30 & 1.00 & 0.50 & 0.80 \\
1.50 & 1.40 & 6.60 & 14.20 & 25.00 & \tbf{\color{red} 36.50} & 8.90 & 2.70 & 1.60 & 1.60 \\
1.40 & 2.40 & 2.30 & 9.10 & 19.10 & 23.00 & \tbf{\color{red} 25.70} & 10.80 & 4.70 & 1.50 \\
1.00 & 1.50 & 1.30 & 4.80 & 8.30 & 12.80 & 19.70 & \tbf{\color{red} 35.10} & 8.50 & 7.00 \\
0.50 & 0.40 & 1.70 & 2.10 & 8.10 & 11.20 & 9.50 & 19.50 & \tbf{\color{red} 34.00} & 13.00 \\
1.00 & 1.10 & 2.80 & 3.80 & 4.00 & 9.50 & 11.30 & 16.80 & 15.70 & \tbf{\color{red} 34.00} \\
\end{tabular}
\end{tabular}
\\ \bottomrule
\end{tabular}
}
\caption{Classification of AR orders by full BIC evaluation and \acrshort{acr:cnntdist} (Joint) in Table~\ref{tbl:CompareTDistWN}. Values are in percentages and normalized along the rows.}
\label{tbl:ARtdist}
\end{table}
\begin{table}\centering
\resizebox{\columnwidth}{!}{
\begin{tabular}{c c}
\toprule
\begin{tabular}{c c}
& BIC \\
& Classified MA order \\
\rotatebox[origin=c]{90}{Actual MA order} &
\begin{tabular}{rccccccccccc}
&& \tbf{0} & \tbf{1} & \tbf{2} & \tbf{3} & \tbf{4} & \tbf{5} & \tbf{6} & \tbf{7} & \tbf{8} & \tbf{9} \\
\cmidrule{3-12}
\tbf{0} && \tbf{\color{red} 79.20} & 8.70 & 4.40 & 2.10 & 1.10 & 0.70 & 0.80 & 0.60 & 0.40 & 2.00 \\
\tbf{1} && 27.60 & \tbf{\color{red} 55.50} & 7.00 & 3.30 & 1.90 & 1.10 & 0.90 & 0.50 & 0.90 & 1.30 \\
\tbf{2} && 14.00 & 22.60 & \tbf{\color{red} 51.30} & 5.40 & 1.90 & 1.00 & 1.30 & 0.30 & 0.70 & 1.50 \\
\tbf{3} && 10.10 & 13.20 & 23.40 & \tbf{\color{red} 40.10} & 5.20 & 2.50 & 1.10 & 0.90 & 1.30 & 2.20 \\
\tbf{4} && 9.50 & 7.60 & 14.50 & 22.70 & \tbf{\color{red} 32.30} & 5.80 & 3.00 & 1.50 & 1.20 & 1.90 \\
\tbf{5} && 6.80 & 5.80 & 10.10 & 12.20 & 21.10 & \tbf{\color{red} 31.90} & 5.80 & 2.20 & 1.50 & 2.60 \\
\tbf{6} && 5.70 & 4.80 & 6.80 & 9.30 & 14.40 & 21.00 & \tbf{\color{red} 29.20} & 4.40 & 2.50 & 1.90 \\
\tbf{7} && 4.80 & 4.90 & 7.00 & 7.60 & 11.60 & 16.00 & 19.40 & \tbf{\color{red} 21.70} & 4.50 & 2.50 \\
\tbf{8} && 4.40 & 3.50 & 4.10 & 6.10 & 10.70 & 11.60 & 14.70 & 19.00 & \tbf{\color{red} 20.60} & 5.30 \\
\tbf{9} && 4.90 & 3.60 & 3.90 & 6.80 & 6.20 & 8.50 & 9.90 & 16.20 & 19.80 & \tbf{\color{red} 20.20} \\
\end{tabular}
\end{tabular}
&
\begin{tabular}{c c}
& CNN \\
& Classified MA order \\
&
\begin{tabular}{cccccccccc}
\tbf{0} & \tbf{1} & \tbf{2} & \tbf{3} & \tbf{4} & \tbf{5} & \tbf{6} & \tbf{7} & \tbf{8} & \tbf{9} \\
\cmidrule{1-10}
\tbf{\color{red} 79.80} & 11.50 & 2.70 & 1.60 & 1.60 & 0.70 & 0.60 & 0.30 & 0.60 & 0.60 \\
24.30 & \tbf{\color{red} 55.60} & 7.80 & 3.90 & 2.70 & 2.10 & 1.20 & 0.60 & 0.80 & 1.00 \\
10.70 & 24.20 & \tbf{\color{red} 44.50} & 8.20 & 5.10 & 2.60 & 1.50 & 1.50 & 0.70 & 1.00 \\
10.40 & 11.10 & 19.60 & \tbf{\color{red} 38.80} & 8.20 & 5.60 & 2.30 & 1.70 & 0.70 & 1.60 \\
8.90 & 8.00 & 12.20 & 22.60 & \tbf{\color{red} 30.50} & 7.60 & 3.90 & 2.20 & 1.80 & 2.30 \\
5.50 & 6.90 & 9.40 & 12.20 & 19.90 & \tbf{\color{red} 28.60} & 8.50 & 3.30 & 2.40 & 3.30 \\
5.70 & 6.50 & 6.90 & 11.90 & 12.60 & 20.00 & \tbf{\color{red} 25.00} & 4.50 & 3.90 & 3.00 \\
4.80 & 5.30 & 6.80 & 9.60 & 11.00 & 15.80 & \tbf{\color{red} 20.00} & 17.70 & 5.20 & 3.80 \\
4.70 & 4.80 & 6.60 & 6.80 & 11.30 & 14.10 & \tbf{\color{red} 16.00} & 13.70 & 13.60 & 8.40 \\
3.20 & 4.50 & 5.40 & 6.00 & 8.50 & 9.60 & 14.20 & 13.10 & 14.30 & \tbf{\color{red} 21.20} \\
\end{tabular}
\end{tabular}
\\ \bottomrule
\end{tabular}
}
\caption{Classification of MA orders by full BIC evaluation and \acrshort{acr:cnntdist} (Joint) in Table~\ref{tbl:CompareTDistWN}. Values are in percentages and normalized along the rows.}
\label{tbl:MAtdist}
\end{table}
\subsection{Validation: Comparison against results by \cite{Chenoweth2000}}
\cite{Chenoweth2000} evaluated a smaller range of ARMA models namely up to order 2 with the exclusion of ARMA(0,0), in other words, a total of 8 ARMA models. 800 test time series corresponding to these 8 models were used in their evaluations, which equates to 100 time series for each ARMA model. They obtained a mean correct percentage of 49.38\% (with 95\% confidence interval [42.34\%,~56.36\%]) accurate identifications for time series of length 3,000 and 20.38\% (with 95\% confidence interval [16.44\%,~24.31\%]) for time series of length 100. We examine a subset of our results for the same ARMA models to get a ballpark comparison to the results by \cite{Chenoweth2000}. The number of time series in the subset of our test suites for this range of ARMA models happen to be 800 time series as well. Our results show that a full BIC evaluation correctly identifies 63.0\% (with 95\% confidence interval [59.62\%,~66.52\%]) of both orders while \acrshort{acr:cnnnorm} (Joint) gets both orders correctly at 50.1\% (with 95\% confidence interval [46.62\%,~53.84\%]).
\section{Discussion}
In this paper, CNN architectures can identify ARMA orders as well as, if not better than, likelihood based methods. Furthermore, CNN architectures perform the task at a fraction of the time taken by AIC and BIC methods. Although the time taken by AIC and BIC methods can be shortened by choosing a stepwise approach, it adversely affects accuracy and CNNs are still much faster. It is a common knowledge that a neural network typically does not perform well on a task or input that it has not been trained on or seen before. We test the robustness our CNNs by evaluating test time series that are statistically different from training data. We generate two types of time series of different white noise; one which is coming from a standard normal distribution and another from a $t$-distribution with 2 degrees of freedom, which represents a very heavy-tailed distribution. While performance of CNN architectures does depend on training data used, we notice that \acrshort{acr:cnnnorm} is more robust than \acrshort{acr:cnntdist}. Put it differently, \acrshort{acr:cnnnorm} is still able to evaluate \acrshort{acr:tstdist} relatively well.
Out of the three ResNet architectures evaluated, we find `ReLU before activation' to be the best candidate for ARMA model identification. This finding is contrary to the result in \cite{He2016} where it is the worst performing ResNet architecture for the task of computer vision. Moreover, contrary to common usage of Adam optimizer in computer vision, we find that \acrshort{acr:nag} optimizer with momentum of 0.75 is key in the training of \acrshort{acr:cnnMA} and produces better outcome too for both \acrshort{acr:cnnAR} and \acrshort{acr:cnnMA}. We observe that accuracy of identifications improves as architecture size increases. Our linear regression analysis seem to suggest a deeper architecture with wider filter width is needed to capture the correlation information of the time series.
Apart from choosing a good architecture, we introduce a few improvements that produce better trained CNNs. Training outcomes of two CNNs, despite having the same architecture and same set of hyper-parameters, can be very different. The outcomes vary because firstly, the weights and biases of each CNN are initialized randomly and secondly, training time series, which are generated on the fly, are different. We describe a progressive retraining strategy that sharpens a trained CNN. Introducing skip connections right from the first convolutional layer is another subtle tweak that improves the performance. When we assemble individual \acrshort{acr:cnnAR} and \acrshort{acr:cnnMA}, which are trained separately, we find the arrangement in a CNN(Joint) setup works better.
\printacronyms[title=Abbreviations, nonumberlist]
\setlength{\bibsep}{0.5ex}
\def\small{\small}
\section*{Acknowledgements}
The authors thanks Chen Ying and Alexandre Thiery for helpful discussions.
|
1,108,101,564,569 | arxiv | \section{Motivations}
Without doubts the construction of a quantum theory of gravity is the single most pressing challenge the contemporary high energy physics community is facing. We still do not know exactly what this theory is; however we can investigate some particular limits of it and build toy models, whose predictions might be, as I am going to argue here, even tested with current and near future observational technologies (see \cite{Addazi:2021xuf} for a recent comprehensive review.) Let us investigate how this limit, called the \textit{relative locality regime} comes about.
Quantum gravity is assumed to unify the two most successful theories of twentieth century physics, General Relativity and Quantum Mechanics. These theories are characterized by dimensionful constants, $G_N$ and $\hbar$, respectively, and describe systems and processes for which these constants are large, in an appropriate sense. For example, General Relativity describes systems for which the size $L$ and mass $M$ satisfy $R\sim G_N M$, and Quantum Mechanics is relevant when the action is of order $\hbar$. Similarly, Quantum Gravity describes systems/effect for which $G_N$ and $\hbar$ are large at the same time, meaning, essentially, that the Compton wavelength is of order of Schwarzschild radius. For the system of energy $E$ in the rest frame this means that $\hbar/E \sim G_N E$.
It is therefore commonly believed that a quantum theory of gravity, in which both GR and QM will play a fundamental role, becomes relevant when the scales of distance and energy are at the same time of the order of the Planck length and the Planck energy respectively
\begin{align}
l_P = \sqrt{\hbar G_N} \approx 10^{-35} m
\qquad
E_P = \sqrt{\frac{\hbar}{G_N}} \approx 10^{19} GeV
\end{align}
In what follows, we will call $\kappa$ the energy scale of the order of $E_P$ at which quantum gravitational effects begin to become relevant.
For example, a scattering with impact parameter $l_P$ between two particles of energy $\kappa$ is expected to be best described by a quantum theory of gravity. Indeed one can see that when the impact parameter of the Planck energy particles scattering approaches $l_P$ then new phenomena begin to appear that requires full quantum gravity to understand (see e.g. \cite{Giddings:2011xs} and references therein.) Unfortunately, a description of such phenomena is still beyond reach.
One can however study the `relative locality regime' \cite{Amelino-Camelia:2011lvm}, characterized by the condition that the size of the Planck energy system is much larger than the Planck length
\begin{align}
l \gg l_P
\qquad
\text{\textit{and}}
\qquad
E \approx \kappa
\end{align}
This regime can be obtained as a limit of quantum gravity by sending both $\hbar$ and $G_N$ to zero in such a way that their ratio remains constant. In physical terms, it is expected to describe phenomena at energies comparable to Planck energies, with the characteristic size much larger than the Planck length, so that the spacetime foam effects could be safely ignored.
In the relative locality regime, by the principle of correspondence, we expect to have to do with a somehow modified but still relativistic theory. Indeed both in the low energy limit and in the ultraviolet one we have to do with the theories possessing relativistic symmetries (quantum field theory at the one end and quantum gravity at the another). This regime can be modelled by a theory that possesses two observer independent scales: one of velocity $c$\footnote{In what follows we will set $c=1$.}, since the theory is to be relativistic, and one of mass/energy/momentum $\kappa$, which reflects the presence of the energy scale, a remnant of quantum gravity. Such theory was proposed two decades ago under the name of `Doubly (or Deformed) Special Relativity' \cite{Amelino-Camelia:2000cpa}, \cite{Amelino-Camelia:2000stu}, \cite{Kowalski-Glikman:2001vvk}. The introduction of an energy scale to relativistic theories is a nontrivial endeavour because the standard Lorentz transformations change energy. Therefore, in order to keep all the relativistic symmetries while at the same time incorporating an invariant energy scale, one needs to appropriately modify the Lorentz transformations as well as composition laws of various physical quantities.
This line of reasoning is analogous to the historical development of relativity in physics. The principle of relativity was first formulated by Galileo, who stated that the laws of physics have the same form in any inertial reference frame. Since in Galilean relativity there is no observer-independent velocity scale, the only possible velocity composition law is linear (just for dimensional reasons, since only in the case of linear composition $V = v+u$ one does not need to introduce a velocity scale to make the dimensions of terms match). At the same time, the transformation laws between different inertial observer are also linear in the velocity. The principle of relativity was later extended by Einstein, who incorporated a fundamental scale of velocity $c$. Because of this, now a new composition of 3-velocities is possible, and in the case of Special Relativity the formula is highly non-linear, non-symmetric and non-associative
\begin{align}
\mathbf{v}\oplus \mathbf{u}
=
\frac{1}{1 + \frac{\mathbf{u}\mathbf{v}}{c^2}}
\left(
\mathbf{v}
+
\frac{\mathbf{u}}{\gamma_{\mathbf{v}}}
+
\frac{1}{c^2}
\frac{\gamma_{\mathbf{v}}}{1 + \gamma_{\mathbf{v}}}
(\mathbf{v}\mathbf{u})\mathbf{v}
\right)
\qquad
\qquad
\gamma_{\mathbf{v}}
=
\sqrt{1 - \frac{\mathbf{v}^2}{c^2}}
\end{align}
At the same time, in order to preserve the relativity principle, Galilean transformations were extended into Lorentz transformations. The addition of a new invariant mass scale $\kappa$ gives rise to a yet new framework called Doubly Special Relativity (DSR in short), and once again this gives rise, this time, to modified energy and momentum composition laws as well as deformed Poincar\'e symmetry, which will be described in these lecture notes. As we will also see, the presence of the invariant scale $\kappa$ enforces the relaxation of the absolute locality postulate of Special Relativity, and locality becomes relative to the observer.
The aim is to study possible quantum-gravitational effects which become important at the relative locality regime. We will see that such deformed model predict results which slightly deviate from the ones obtained from Special Relativity. One such prediction is the different time of arrival of photons of different energy which have been simultaneously emitted by a distant source. Such a phenomenon can be matched against measurements in order to look for deviations from Special Relativity.
The reader interested in a more detailed description of the topics briefly discussed in this notes may consult a recent monograph \cite{Arzano:2021scz}.
\section{Relativistic point particle and LIV models}
We start from the standard action for a point-like relativistic particle, written in the first order formalism (linear in velocities)
\begin{align}\label{action}
S = \int d\tau \dot{x}^\mu p_\mu - N(p^2 + m^2)
\end{align}
where we are using the signature $-+++$. The equations of motion (EoM) resulting from variations over $x$, $p$, and $N$ are given by
\begin{align}\label{undefaction}
\dot{p}_\mu = 0
\qquad
\dot{x}^\mu = 2Np^\mu
\qquad
p^2 = -E^2 + \mathbf{p}^2 = -m^2.
\end{align}
The variable $N$ is a kind of gauge variable and can be fixed a posteriori. As an example, for a massive particle it can be fixed to $N=\frac{1}{2m}$ so that the EoM for $x^\mu$ become
$\dot{x}^\mu = \frac{p^\mu}{m}$. One can easily compute the speed of light in this model because we now have $\dot{x}^\mu \propto p^\mu$, and for a massless particle $p^2 = 0$, hence
\begin{align}
v^2 =
\left(
\frac{d \mathbf{x}}{d x^0}
\right)^2
=
\left(
\frac{\dot{\mathbf{x}}}{\dot{x}^0}
\right)^2
=
\left(
\frac{\mathbf{p}^2}{p_0^2}
\right)_{\mathbf{p}^2 = p_0^2}
=
1
\end{align}
The symmetries that leave the the action \eqref{action} invariant and the EoM \eqref{undefaction} covariant are\footnote{We encourage the reader to check it explicitly.} translations
\begin{align}
\delta x^\mu = \epsilon^\mu\,,
\end{align}
rotations
\begin{align}
\delta x^i = \rho^k \tensor{\epsilon}{^i_{jk}} x^j
\qquad
\delta x^0 = 0
\qquad
\delta p_i = \rho^k \tensor{\epsilon}{_i^j_k} p_j
\qquad
\delta p_0 = 0\,,
\end{align}
and boosts
\begin{align}
\delta x^i = -\lambda^i x^0
\qquad
\delta x^0 = -\lambda_i x^i
\qquad
\delta p_i = \lambda_i p_0
\qquad
\delta p_0 = \lambda^i p_i\,.
\end{align}
We have now 10 independent infinitesimal transformations associate with 10 independent parameters $\{\epsilon^\mu, \rho^k, \lambda^i\}$, and therefore a 10-dimensional algebra of the symmetry group. We can rewrite the above transformations in a more abstract way, separating the generators (which contain the actual physical information about the transformations) from the parameters in the following way
\begin{align}\label{abstractvariations}
\delta_T(\bullet)
&=
\epsilon^\mu P_\mu \triangleright \bullet
\qquad \qquad \qquad \qquad
P_\mu \triangleright x^\nu = \delta_\mu^\nu
\\
\delta_R(\bullet)
&=
\rho^i R_i \triangleright \bullet
\qquad \text{acting as} \qquad \, \, \,
R_i \triangleright x^j = \tensor{\epsilon}{^j_{ik}} x^k\\
\delta_N(\bullet)
&=
\lambda^i N_i \triangleright \bullet
\qquad \qquad \qquad \qquad \, \,
N_i \triangleright x^j = \delta_i^j x^0
\end{align}
where the generators $\{P_\mu, R_i, N_i\}$ satisfy the Poincar\'e algebra
\begin{align}
[P_\mu, P_\nu] = 0
\qquad
[R_i, P_j] = i \tensor{\epsilon}{_{ij}^k}P_k
\qquad
[R_i, P_0] = 0
\qquad
[N_i, P_j] = -i \eta_{ij} P_0 \nonumber
\end{align}
\begin{align}
[N_i, P_0] = -i P_i
\qquad
[R_i, R_j] = i \tensor{\epsilon}{_{ij}^k} R_k
\qquad
[R_i, N_j] = i \tensor{\epsilon}{_{ij}^k} N_k
\qquad
[N_i, N_j] = -i \tensor{\epsilon}{_{ij}^k} R_k \nonumber
\end{align}
So far we discussed the relativistic particle theory that preserved all the relativistic symmetries of Poincar\'e algebra/group. The simplest way to obtain a Lorentz-invariance-violating (LIV) theory is to keep the kinetic term in the action \eqref{undefaction} as it is, changing the dispersion relation to the modified form
\begin{align*}
p^2 + m^2 \, \rightarrow \, \mathcal{C}_\kappa(p) + m^2
\end{align*}
where $\mathcal{C}_\kappa(p)$ must satisfy the following conditions
\begin{itemize}
\item It cannot depend only on $p^2$ (otherwise the relativistic symmetries will be preserved, and, in particular the speed of massless particles will still be equal 1 -- the reader is encouraged to check this statement);
\item $ \lim_{\kappa \rightarrow \infty} \mathcal{C}_\kappa (p) = p^2$, which is just the natural complementarity condition.
\end{itemize}
The modified mass-shell condition $\mathcal{C}_\kappa (p)=-m^2$ is therefore assumed to be not Lorentz invariant anymore. Using the momentum representation of the boost, we can in general write this as a requirement
\begin{align}
p_0 \frac{\partial \mathcal{C}_\kappa(p)}{\partial \mathbf{p}^i} + \mathbf{p}_i \frac{\partial \mathcal{C}_\kappa(p)}{\partial p_0} \neq 0
\end{align}
How could this kind of model arise? An idea is that the quantum gravity vacuum (for example string theory vacuum), describing the spacetime in which we live in, is assumed to violate Lorentz symmetry \cite{Kostelecky:1988zi}, \cite{Colladay:1998fq}, \cite{Mattingly:2005re}. This Lorentz invariance violation must disappear at low energies, which is the reason for the conditions that in the limit $\kappa\rightarrow \infty$ the mass shell condition returns to its special-relativistic form.
The equations of motion for photons in such a model are given by
\begin{align*}
\dot{p}_\mu = 0
\qquad
\dot{x}^\mu = N \frac{\partial \mathcal{C}_\kappa}{\partial p_\mu}
\qquad
\mathcal{C}_\kappa (p) = 0
\end{align*}
which imply that in general
\begin{align}
v^2 =
\left(
\frac{\dot{\mathbf{x}}}{\dot{x}^0}
\right)^2
=
\left(
\frac{\partial \mathcal{C}_\kappa(p)}{\partial \mathbf{p}_i}
\Big/
\frac{\partial \mathcal{C}_\kappa(p)}{\partial p_0}
\right)_{\mathcal{C}_\kappa = 0}
\neq 1
\end{align}
and $v$ is momentum dependent. In physical terms this means that photons of different energies move with different velocities. This effect is very small, dumped by some power of the ratio $E/\kappa$, but in principle measurable if the source is at cosmological distances \cite{Amelino-Camelia:1997ieq}.
\section{From deformed dispersion relations to curved momentum space}
After the previous discussion about LIV models it is natural to ask ourselves the following question: is it possible to choose $\mathcal{C}_\kappa(p)$ in such a way that the Lorentz transformations are \textit{not} violated anymore (in the appropriate sense)?
The answer is surprisingly affirmative for a very large class of possible $\mathcal{C}_\kappa(p)$. The reasoning goes as follows. Let us assume for simplicity that $\mathcal{C}_\kappa(p)$ is invariant under rotations\footnote{This is just a simplifying assumption and not a fundamental requirement for the validity of the following reasoning.}, so that is is function of $p_0, \mathbf{p}^2$. Then, consider the generalized boost
\begin{align}\label{defboost}
N_i^\kappa
=
A(p_0, \mathbf{p}^2) \frac{\partial}{\partial \mathbf{p}_i}
+
B_i(p_0, \mathbf{p}^2) \frac{\partial}{\partial p_0}
\end{align}
and impose that it satisfy the requirements
\begin{align}\label{requirements}
[N_i^\kappa, N_j^\kappa] \propto \tensor{\epsilon}{_{ij}^k}R_k
\qquad
[R_i^\kappa, N_j^\kappa] \propto \tensor{\epsilon}{_{ij}^k}N_k
\qquad
N_i^\kappa \triangleright \mathcal{C}_\kappa(p_0, \mathbf{p}^2)=0
\end{align}
It turns out that the requirements in \eqref{requirements} can be easily satisfied \cite{Kowalski-Glikman:2002eyl} so that the Lorentz symmetry algebra is \textit{not} modified and the dispersion relation is still relativistic invariant. It is important to notice, however that the kinetic term $\dot x^\mu p_\mu$ is \textit{not} invariant in general under such deformed transformations, and it should be properly modified to render the whole action invariant.
This surprising fact can be understood if one realizes that the deformation of the boost and of the dispersion relation can be associated with coordinate changes in the momentum space, which, in the presence of the scale $\kappa$, are not necessarily confined to the linear ones. A historically relevant example (which we will also use later) of such a coordinate transformation is give by $p \mapsto p(k)$
\begin{align}\label{changeofcoord}
p_0(k_0, \mathbf{k}) &= \kappa \sinh \frac{k_0}{\kappa}
+
\frac{\mathbf{k}^2}{2\kappa} e^{\frac{k_0}{\kappa}} \\
\mathbf{p}(k_0, \mathbf{k})
&=
\mathbf{k} e^{\frac{k_0}{\kappa}}
\end{align}
where the deformed $\mathcal{C}_\kappa(p)$ is given by
\begin{align}\label{defdisprel}
\mathcal{C}_\kappa(p) = -2\kappa^2
\sqrt{1+\frac{-p_0^2 + \mathbf{p}^2 }{\kappa^2} } +2 \kappa^2\simeq p_0^2 + O\left(\frac1{\kappa^2}\right)
\end{align}
which can be written in terms of the new coordinates $(k_0, \mathbf{k})$ as
\begin{align}
\mathcal{C}_\kappa(k) = -4 \kappa^2 \sinh^2
\left(
\frac{k_0}{2\kappa}
\right)
+
\mathbf{k}^2 e^{\frac{k_0}{\kappa}}.
\end{align}
$\mathcal{C}_\kappa(k)$ is invariant under deformed boosts of the form of \eqref{defboost}, because $\mathcal{C}_\kappa(p)$ is invariant under standard boosts.
At this point, two important observations need to be made.
The first is that changes of coordinates cannot be physical. The classical argument involves a mad saboteur which is able to hack their way into the CERN systems with the aim of replacing each and every measured momentum with a function of the momenta themselves. Of course, the physics measured by the LHC experiment remains unchanged, although it is expressed in a different way.
Therefore deformation alone cannot bear any physical meaning, unless it is achieved in a way that it is not cancellable through a coordinate transformations. Furthermore, physical observables have to be momentum space coordinate independent objects.
The second observation is that, in the same way as one goes from special to general relativity by introducing a velocity scale and a nontrivial geometry of the (three) velocities manifold, one can very well conceive to go from a flat to a curved momentum space, as was first considered by Max Born around 1938. It is natural to look for deformations of the dispersion relation and of the Lorentz transformations coming from the curvature of momentum space. In this way one could get a deformation inspired by the one described in \eqref{defboost}, \eqref{changeofcoord}, \eqref{defdisprel}, which cannot be removed by a coordinate transformation (since curvature is not coordinate dependent). The curvature of momentum space can naturally be associated to the scale $\kappa$ at hand. The idea to associate a fundamental scale to nontrivial geometry is also not new, and it was expressed by Carl Friedrich Gauss at the very dawn of differential geometry:
\begin{quote}
The assumption that the sum of the three angles [of a triangle] is smaller than 180◦ leads to a geometry which is quite different from our (Euclidean) geometry, but which is in itself completely consistent. I have satisfactorily constructed this geometry for myself [\ldots ], except for the determination of one constant, which cannot be ascertained a priori. [\ldots] Hence I have sometimes in jest expressed the wish that Euclidean geometry is not true. For then we would have an absolute a priori unit of measurement\footnote{As cited in J. Milnor “Hyperbolic geometry: the first 150 years,”Bull. Am. Math. Soc.69 (1982).}.
\end{quote}
This idea can be summarized by saying that one can expect a non-trivial geometry if there is a scale which allows it. In other words, everything is curved unless it cannot be.
Motivated by these considerations, we start now the more detailed studies of curved momentum space. The conditions that such a space must satisfy are simple:
\begin{itemize}
\item[1)] The (deformed) Lorentz group must act on it;
\item[2)] The scale $\kappa$ must be related to to its geometry, for example being the curvature of momentum space;
\item[3)] In the limit $\kappa \rightarrow \infty$ one should get back the canonical, flat momentum space.
\end{itemize}
The simplest case that we can consider is the one of constant positive curvature $\frac{1}{\kappa^2}$, i.e. (some submanifold of) the de Sitter space\footnote{One could very well consider the interesting case of a momentum space whose curvature depends on the momentum, i.e. with non-trivial local curvature. However, in this case the scale $\kappa$ would not have an immediate fundamental meaning, and the treatment of such a case would be more difficult than the simpler case of constant curvature. }. But how to build such a momentum space?
The idea is simple, and requires us to go back to eq. \eqref{changeofcoord}. We can use the coordinates $(p_0, \mathbf{p})$ which form a Lorentz vector, and add a fifth component $p_4$ which is a Lorentz scalar. In this five dimensional space, we then find out what is the manifold individuated by the coordinates $(k_0, \mathbf{k})$ defined in \eqref{changeofcoord}. Of course, also $p_4$ needs to be adequately expressed as a function of $k_0, \mathbf{k}$ in order to obtain the correct submanifold. More precisely, the map $(k_0, \mathbf{k})\mapsto (p_0(k_0, \mathbf{k}), \mathbf{p}(k_0, \mathbf{k}), p_4(k_0, \mathbf{k}))$ defines a surface in the 5-dimensional space described by the coordinates $(p_0, \mathbf{p}, p_4)$.
In this way we obtain a 4-dimensional submanifold of the 5-dimensional space described by $(p_0,\mathbf{p}, p_4)$. In particular, the following coordinates
\begin{align}\label{5dcoord}
p_0(k_0, \mathbf{k}) &= \kappa \sinh \frac{k_0}{\kappa}
+
\frac{\mathbf{k}^2}{2\kappa} e^{\frac{k_0}{\kappa}} \nonumber \\
\mathbf{p}_i(k_0, \mathbf{k})
&=
\mathbf{k}_i e^{\frac{k_0}{\kappa}} \\
p_4(k_0, \mathbf{k})
&=
\kappa \cosh \frac{k_0}{\kappa}
-
\frac{\mathbf{k}^2}{2\kappa} e^{\frac{k_0}{\kappa}} \nonumber
\end{align}
satisfy
\begin{align}\label{deSitterrel}
-p_0^2 + \mathbf{p}^2 + p_4^2 = \kappa^2
\end{align}
so that they describe a 4-dimensional submanifold of the de Sitter space \eqref{deSitterrel} defined by the condition
\begin{align}\label{deSitterre2}
p_0+p_4 = \kappa e^{\frac{k_0}{\kappa}} >0.
\end{align}
Of course, $(k_0, \mathbf{k})$ are coordinates on this submanifold. In other words, the coordinates $(k_0, \mathbf{k})$ cover half of de Sitter space. Incidentally, these coordinates are in the direct analogy with to the Friedman-Robertson-Walker-de Sitter universe. The reader is encouraged to check this explicitly by computing the induced metric on the submanifold \eqref{deSitterrel}, \eqref{deSitterre2} using \eqref{5dcoord} (or look at \eqref{momspacemetric} for the answer).
\section{Deformed action and its properties}
We now have a curved momentum space, and we want to write an action for a particle with momenta in this space.
To illustrate the process, we first generalize the action in eq. \eqref{action} to the case of curved spacetime. After this we will use a similar procedure to instead generalize \eqref{action} to the case of a curved momentum space.
The generalization of \eqref{action} to a curved momentum space is straightforward, and makes use of the vierbein (also called tetrad). In this section, we will use Latin letters $a, b, \dots$ to indicate flat spacetime (momentum space) indices, and Greek letters $\mu, \nu, \dots$ for curved spacetime (momentum space)\footnote{Using this convention, all the indices in eq. \eqref{action} have to be Latin indices.}. The vierbein is defined through the relation
\begin{align}
g_{\mu\nu}(x) = \eta_{ab}\tensor{e}{_\mu^a}(x)\tensor{e}{_\nu^b}(x)
\end{align}
and using it we can write the action as follows
\begin{align}\label{actioncurvedst}
S = \int d\tau \,\, \dot{x}^\mu \tensor{e}{_\mu^a}(x) p_a - N(\eta^{ab} p_a p_b + m^2)
\end{align}
which is the correct action for a pointlike particle in curved spacetime (but flat momentum space, as one can see from the dispersion relation). The equations of motion of this action give the standard geodesic equations in the metric $g_{\mu\nu}(x)$. This is easy to see by noticing that $p_a = \tensor{e}{^\mu_a}(x) p_\mu$ so that the action in \eqref{actioncurvedst} reduces to the more familiar
\begin{align}
S = \int d\tau \,\, \dot{x}^\mu p_\mu - N(g^{\mu\nu}(x) p_\mu p_\nu + m^2)
\end{align}
whose equations of motion are now (ignoring for the moment the on-shell relation $g^{\mu\nu}(x) p_\mu p_\nu + m^2 = 0$)
\begin{align}
\dot{x}^\mu =& 2N g^{\mu\nu}(x)p_\nu \\
\dot{p}_\alpha + N (&\partial_\alpha g^{\mu\nu}) p_\mu p_\nu = 0.
\end{align}
Substituting the first one into the second we get
\begin{align}
\frac{d}{d\tau}
\left(
\frac{1}{2N}
g_{\alpha\mu} \dot{x}^\mu
\right)
+
N
\partial_\alpha g^{\mu\nu}
\left(
\frac{1}{2N}
g_{\mu\beta} \dot{x}^\beta
\right)
\left(
\frac{1}{2N}
g_{\nu\gamma} \dot{x}^\gamma
\right) = 0
\end{align}
which reduces to
\begin{align}
\frac{d}{d\tau}
\left(
g_{\alpha\mu} \dot{x}^\mu
\right)
+
\frac{1}{2}
(\partial_\alpha g^{\mu\nu})
g_{\mu\beta} \dot{x}^\beta
g_{\nu\gamma} \dot{x}^\gamma
= 0
\end{align}
and hence
\begin{align}
g_{\alpha\mu} \ddot{x}^\mu
+
\frac{1}{2}
(\partial_\sigma g_{\alpha\mu} +
\partial_\mu g_{\alpha\sigma} - \partial_\alpha g_{\mu\sigma})\dot{x}^\mu \dot{x}^\sigma
=0
\end{align}
which are indeed the canonical geodesic equations. Notice that we used the fact that
\begin{align}
(\partial_\alpha g^{\mu\nu})
g_{\mu\beta}g_{\nu\gamma}
=
-\partial_\alpha g_{\beta\gamma}.
\end{align}
Coming back to our discussion, we now have an action in \eqref{actioncurvedst} that correctly represents the motion of a particle in curved spacetime and flat momentum space. It is now immediate to see how the same construct can be applied to the case of a flat spacetime but curved momentum space. We just need to introduce a non-trivial vierbein which allow us to proceed in the same manner. Therefore, we write the action
\begin{align}\label{actioncurvedms}
S_\kappa =
-\int d\tau \,\, x^a \tensor{E}{_a^\mu}(k, \kappa) \dot{k}_\mu - N(\mathcal{C}_\kappa(k) + m^2).
\end{align}
Notice that now the time derivative, denoted by dot acts on the momentum and not the coordinate. In the flat space the placement of the dot is irrelevant if one ignores boundary contributions (since the choices differ just by a total derivative), but it is essential for what follows and relative locality to be discussed below that it acts on the momentum. Notice that this action is written in terms of the coordinates $(k_0, \mathbf{k})$ which cover our curved momentum space. In order to obtain the expression for $\tensor{E}{_a^\mu}(k, \kappa)$ we need the form of the metric $G$ in momentum space, and this is easily obtained from \eqref{5dcoord} and \eqref{deSitterrel}. The five-dimensional line element is given by $ds_5^2 = -dp_0^2 + d\mathbf{p}^2 + dp_4^2$, and from this we can get the line element in the submanifold of the de Sitter surface simply by substituting \eqref{5dcoord}. We have
\begin{align}
dp_0(k_0, \mathbf{k}) &= d k_0 \cosh \frac{k_0}{\kappa}
+
d\mathbf{k}\frac{\mathbf{k}}{\kappa}
e^{\frac{k_0}{\kappa}}
+
\frac{\mathbf{k}^2}{2\kappa^2} e^{\frac{k_0}{\kappa}} dk_0
\nonumber \\
d\mathbf{p}(k_0, \mathbf{k})
&=
d\mathbf{k} e^{\frac{k_0}{\kappa}}
+
\frac{\mathbf{k}}{\kappa} e^{\frac{k_0}{\kappa}} dk_0
\\
dp_4(k_0, \mathbf{k})
&=
d k_0 \sinh \frac{k_0}{\kappa}
-
d\mathbf{k}\frac{\mathbf{k}}{\kappa}
e^{\frac{k_0}{\kappa}}
-
\frac{\mathbf{k}^2}{2\kappa^2} e^{\frac{k_0}{\kappa}} dk_0 \nonumber
\end{align}
and therefore
\begin{align}\label{momspacemetric}
-dp_0^2 + d\mathbf{p}^2 + dp_4^2
\mapsto
dk_0^2
\left(
-\cosh^2 \frac{k_0}{\kappa}
+
\sinh^2 \frac{k_0}{\kappa}
\right)
+
d\mathbf{k}^2 e^{2\frac{k_0}{\kappa}}
=
-d^2 k_0
+
d\mathbf{k}^2 e^{2\frac{k_0}{\kappa}}
\end{align}
which means that $G = \text{diag} \left(-1,e^{2\frac{k_0}{\kappa}},e^{2\frac{k_0}{\kappa}},e^{2\frac{k_0}{\kappa}}\right)$ and, since the momentum space tetrad is defined in the usual way by\footnote{ Notice that since momenta are objects with lower indices, the metric in momentum space has upper indices.}
\begin{align}\label{momentumspacemetric}
G^{\mu\nu}(k, \kappa) = \eta^{ab} \tensor{E}{_a^\mu}(k, \kappa) \tensor{E}{_b^\nu}(k, \kappa)
\end{align}
we immediately obtain
\begin{align}\label{momentumspacetetrad}
\tensor{E}{_0^0}(k, \kappa) = 1
\qquad
\tensor{E}{_i^\mu}(k, \kappa) = e^{\frac{k_0}{\kappa}} \delta_i^\mu.
\end{align}
Using these explicit expressions we can rewrite the action \eqref{actioncurvedms} as
\begin{align}\label{explicitdefaction}
S_\kappa =
\int d\tau \,\, \dot{x}^0 k_0
-
e^{\frac{k_0}{\kappa}}
\mathbf{x} \cdot \dot{\mathbf{k}}
+
N(\mathcal{C}_\kappa(k) + m^2).
\end{align}
Notice that we integrated by parts ignoring boundary terms, and furthermore in the limit $\kappa \rightarrow \infty$ the above action reduces to the action in eq. \eqref{action}. Another way of rewriting this action that is found in the literature is obtained by using the rescaling $\mathbf{k}\mapsto e^{-\frac{k_0}{\kappa}} \mathbf{k}$ so that the action becomes
\begin{align}
S_\kappa =
\int d\tau \,\, \dot{x}^0 k_0
+
\dot{\mathbf{x}} \cdot \mathbf{k}
+
\mathbf{x} \mathbf{k}
\frac{\dot{k_0}}{\kappa}
+
N(\mathcal{C}_\kappa(k) + m^2).
\end{align}
where once again we integrated by parts ignoring the boundary terms. This reformulation does not affect the physical properties of the action.
We can now study the properties of the deformed action \eqref{explicitdefaction}. From the fact that $(p_0, \mathbf{p})$ is a Lorentz vector and $p_4$ is a Lorentz scalar, we can derive the deformed Lorentz transformations for the basis $(k_0, \mathbf{k})$. It turns out \cite{Arzano:2021scz} that their Lorentz transformations are
\begin{align}
\delta_\lambda k_0
=
\mathbf{\lambda} \cdot \mathbf{k}
\qquad
\delta_\lambda \mathbf{k}_i
=
\mathbf{\lambda}_i
\left(
\frac{\kappa}{2}
\left(
1 - e^{-2\frac{k_0}{\kappa}}
\right)
+ \frac{\mathbf{k}^2}{2\kappa}
\right)
-
\frac{1}{\kappa}
\mathbf{k}_i \mathbf{\lambda} \cdot \mathbf{k}
\end{align}
and this in turn allows us to compute the Lorentz transformations of the position coordinates that leave the deformed action invariant. After long computations, the final result is given by the following relations.
\begin{align}
\delta_\lambda x^0
=
-\mathbf{\lambda} \cdot \mathbf{x} e^{-\frac{k_0}{\kappa}}
\qquad
\delta_\lambda \mathbf{x}^i
=
-\mathbf{\lambda}^i x^0 e^{-\frac{k_0}{\kappa}}
-\frac{1}{\kappa}
(\mathbf{k}^i \mathbf{\lambda} \cdot \mathbf{x} - \mathbf{\lambda}^i \mathbf{x} \cdot \mathbf{k})
\end{align}
Furthermore, the action is also invariant under the translations\footnote{Recall that momenta don't change under spacetime translations.}
\begin{align}\label{deformedtranslations}
\delta_T x^0 = \epsilon^0
\qquad
\delta_T \mathbf{x}^i
=
\epsilon^i e^{-\frac{k_0}{\kappa}}
\end{align}
This is easily checked by noticing that, under these translations, the kinetic part of the action \eqref{explicitdefaction} acquires an extra term $\mathbf{\epsilon} \cdot \dot{\mathbf{k}}$ which is however a total derivative (and recall that for the moment we are ignoring boundary terms). This fact will be important for us when we will talk about relative locality.
The EoM of the action \eqref{explicitdefaction}, assuming that $\mathcal{C}_\kappa(p)$ only depends on the momenta through $p_4$ and ignoring the on-shell relation $\mathcal{C}_\kappa(p) = -m^2$, are given by\footnote{The assumption that $\mathcal{C}_\kappa(p)$ depends on $p_4$ is not a restriction since a general class of models fall under this category. In any case, recall that $\kappa^2 - p_4^2 = -p_0^2 + \mathbf{p}^2$. }
\begin{align}
\dot{k}^\mu &= 0 \\
\dot{x}^0
&=
N\frac{\partial \mathcal{C}_\kappa(p)}{\partial p_4}
\left(
\sinh \frac{k_0}{\kappa} - \frac{\mathbf{k}^2}{2\kappa^2} e^{\frac{k_0}{\kappa}}
\right) \\
\dot{\mathbf{x}}^i
&=
-N \frac{\partial \mathcal{C}_\kappa(p)}{\partial p_4}
\frac{\mathbf{k}^i}{\kappa}
\end{align}
and these allow us to verify that the speed of light in the model described by eq. \eqref{explicitdefaction} is actually equal to one. In fact on-shell massless particles satisfy $-p_0^2 + \mathbf{p}^2 = 0$, and since $-p_0^2 +\mathbf{p}^2 + p_4^2 = \kappa^2$ and $p_4>0$, they are described by the condition $p_4 = \kappa$, which written explicitly amounts to
\begin{align}
\kappa \cosh \frac{k_0}{\kappa}
-
\frac{\mathbf{k}^2}{2\kappa} e^{\frac{k_0}{\kappa}} = \kappa.
\end{align}
Then we can easily compute that
\begin{align}\label{c=1}
v^2 =
\left(
\frac{\dot{\mathbf{x}}}{\dot{x}^0}
\right)^2_{p_4=\kappa}
&=
\frac{\frac{\mathbf{k}^2}{\kappa^2}}{\left(
\sinh \frac{k_0}{\kappa} - \frac{\mathbf{k}^2}{2\kappa^2} e^{\frac{k_0}{\kappa}}
\right)^2} \Bigg|_{p_4=\kappa} \nonumber \\
&=
\frac{2\cosh \frac{k_0}{\kappa} e^{-\frac{k_0}{\kappa}} - 2e^{-\frac{k_0}{\kappa}}}{(\sinh\frac{k_0}{\kappa} +1-\cosh\frac{k_0}{\kappa})^2}
\Bigg|_{p_4=\kappa} \nonumber \\
&=
\frac{1 + e^{-2\frac{k_0}{\kappa}} - 2e^{-\frac{k_0}{\kappa}}}{\left(1 - e^{-\frac{k_0}{\kappa}}\right)^2}
\Bigg|_{p_4=\kappa} \nonumber \\
&= 1
\end{align}
and therefore the speed of light is energy-independent. This however does not automatically mean that photons of different energies emitted by a distant sources arrive at the same time to our detector. In fact, the phenomenon of relative locality (which we will discuss later) implies that indeed these two photons arrive at our detector at different times, but not because they have a different velocity.
Last but not least, from the action \eqref{explicitdefaction} we can also get the symplectic form, which in turn gives us information about the Poisson brackets in our model. The starting point is the kinetic part of the action
\begin{align}
K
=
\int d\tau \,\, \dot{x}^0 k_0
-
e^{\frac{k_0}{\kappa}}
\mathbf{x} \cdot \dot{\mathbf{k}}
\end{align}
from which we can obtain the pre-symplectic form
\begin{align}
\Theta_\kappa
=
\int d\tau \,\, \delta x^0 k_0
-
e^{\frac{k_0}{\kappa}}
\mathbf{x} \cdot \delta \mathbf{k}.
\end{align}
Notice that in the above notation the symbol $\delta$ denotes the exterior differential in phase space. The symplectic form is then obtained by taking the exterior derivative of $\Theta_\kappa$, obtaining
\begin{align}\label{symplecticform}
\Omega_\kappa
=
\delta\Theta_\kappa
=
\int d\tau \,\, \delta k_0 \wedge \delta x^0
+
e^{\frac{k_0}{\kappa}}
\delta \mathbf{k}_i \wedge \delta \mathbf{x}^i
-
\frac{\mathbf{x}^i}{\kappa}
e^{\frac{k_0}{\kappa}}
\delta k_0 \wedge \delta \mathbf{k}_i
.
\end{align}
Notice that indeed we have $\delta\Omega_\kappa = 0$. The inverse of the symplectic form then gives the Poisson brackets. To understand that better consider an example inspired by \cite{Crnkovic:1986ex}. Consider a system with two degrees of freedom parametrized by the coordinates $(x^1, x^2)$ with conjugate momenta $(p^1, p^2)$. We can consider them together as a single object $Q = (p^1, p^2, x^1, x^2)$. Assume that the symplectic form is given by
\begin{align}
\Omega = \frac{1}{2} \omega_{ab} dQ^a \wedge dQ^b = f \, dp^1 \wedge dx^1 + g \, dp^2 \wedge dx^2 + h \, dp^1 \wedge dp^2
\end{align}
where $f, g, h$ are chosen such that $d\Omega=0$. Therefore we have
\begin{align}
\omega
=
\begin{pmatrix}
0 & h & f & 0 \\
-h & 0 & 0 & g \\
-f & 0 & 0 & 0 \\
0 & -g & 0 & 0
\end{pmatrix}
\implies
\omega^{-1}
=
\begin{pmatrix}
0 & 0 & -1/f & 0 \\
0 & 0 & 0 & -1/g \\
1/f & 0 & 0 & h/fg \\
0 & 1/g & -h/fg & 0
\end{pmatrix}.
\end{align}
We can then define the Poisson brackets between the components of $Q$ using the relation \cite{Crnkovic:1986ex}
\begin{align}
\{A,B\}
=
(\omega^{-1})^{ab} \frac{\partial A}{\partial Q^a}
\frac{\partial B}{\partial Q^b}.
\end{align}
One can immediately see from the above relations that
\begin{align}
\{x^1, p^1\} = \frac{1}{f}
\qquad
\{x^2, p^2\} = \frac{1}{g}
\qquad
\{x^1, x^2\} = \frac{h}{fg}
\end{align}
The same reasoning (with the necessary modifications) can be applied to the symplectic form in \eqref{symplecticform}, and therefore we obtain the following Poisson brackets between the canonical variables.
\begin{align}\label{Poissonb}
\{x^0, p_0\} = 1
\qquad
\{\mathbf{x}^i, \mathbf{k}_j\} = e^{-\frac{k_0}{\kappa}}\delta^i_j
\qquad
\{x^0, \mathbf{x}^i\}
=
-\frac{1}{\kappa} \mathbf{x}^i
\end{align}
Notice that, upon quantization, the non-trivial Poisson brackets $\{x^0, \mathbf{x}^i\}$ give rise to a non-trivial commutator between spacetime coordinates, so that this model gives rise to a non-commutative spacetime. This fact can be understood as the dual counterpart to the curvature of momentum space \cite{Majid:1999tc}. This type of non-commutative spacetime is called $\kappa$-Minkowski spacetime.
\section{Group theoretical perspective} \label{GTP}
In the previous sections, we started from an adequately defined deformed action and we then derived several interesting results. In particular, we saw that the model described by the action \eqref{explicitdefaction} predicts (upon quantization) a non-commutative spacetime defined by the following Lie algebra\footnote{In the literature, one commonly uses the relation $X^0=-t$, which accounts for the difference in sign between the classical relation \eqref{Poissonb} and the algebra in eq. \eqref{kappalgebra}.}.
\begin{align}\label{kappalgebra}
[X^0, \mathbf{X}^i] = \frac{i}{\kappa} \mathbf{X}^i
\qquad
[\mathbf{X}^i, \mathbf{X}^j] = 0
\end{align}
This algebra is called the $AN(3)$ algebra, where $A$ stands for abelian (because $[\mathbf{X}^i, \mathbf{X}^j] = 0$) and $N$ for nilpotent (since $(X_i)^3 = 0$).
Notice that the objects $X^0$ and $\mathbf{X}^i$ are not the operators corresponding to spacetime positions.
It is also possible to build the theory starting from the Lie algebra \eqref{kappalgebra}. It turns out that the simplest matrix representation of this algebra is (maybe unsurprisingly, given the discussion in the previous section) 5-dimensional, and it is given by the following matrices
\begin{equation}\label{rep}
X^0 = -\frac{i}{\kappa} \,\left(\begin{array}{ccc}
0 & \mathbf{0} & 1 \\
\mathbf{0} & \mathbf{0} & \mathbf{0} \\
1 & \mathbf{0} & 0
\end{array}\right) \quad
\mathbf{X} = \frac{i}{\kappa} \,\left(\begin{array}{ccc}
0 & {\mathbf{\epsilon}\,{}^T} & 0\\
\mathbf{\epsilon} & \mathbf{0} & \mathbf{\epsilon} \\
0 & -\mathbf{\epsilon}\,{}^T & 0
\end{array}\right).
\end{equation}
Given a Lie algebra, we can define an element of the associated Lie group as follows\footnote{Notice that since $X^0$ and $\mathbf{X}^i$ do not comute, we have to choose an ordering for the product of the two objects $e^{i\mathbf{k}_i \mathbf{X}^i}$ and $e^{ik_0 X^0}$, and this is just a matter of convention. For example, one could have chosen to define the group element as $\tilde \Pi (k) = e^{i(\mathbf{k}_i \mathbf{X}^i - k_0 X^0)}$, and the final result would be the same as the one which we get with the convention \eqref{groupelement} but expressed in a different basis of momentum space (in this case, the basis is called `normal basis' which we will meet later).}
\begin{equation}\label{groupelement}
\Pi(k) =e^{i\mathbf{k}_i \mathbf{X}^i} e^{ik_0 X^0}\,.
\end{equation}
and using the explicit representation \eqref{rep} one can check that
\begin{align}
\exp(i k_0 X^0)= \left(\begin{array}{ccc}
\cosh\frac{k_0}\kappa & \mathbf{0} & \sinh\frac{k_0}\kappa \\&&\\
\mathbf{0} & \mathbf{1} & \mathbf{0} \\&&\\
\sinh\frac{k_0}\kappa\; & \mathbf{0}\; & \cosh\frac{k_0}\kappa
\end{array}\right)\, , \quad
\exp(i \mathbf{k}_i\mathbf{X}^i) = \left(\begin{array}{ccc}
1+\frac{\mathbf{k}^2}{2\kappa^2}\; &\; \frac{\mathbf k}\kappa \;& \; \frac{\mathbf{k}^2}{2\kappa^2}\\&&\\
\frac{\mathbf k}\kappa & \mathbf{1} & \frac{\mathbf k}\kappa \\&&\\
-\frac{\mathbf{k}^2}{2\kappa^2}\; & -\frac{\mathbf k}\kappa\; & 1-\frac{\mathbf{k}^2}{2\kappa^2}
\end{array}\right)
\end{align}
and
therefore
\begin{equation}\label{groupelement2}
\Pi(k) =\left(\begin{array}{ccc}
\frac{ \bar p_4}\kappa \;&\; \frac{\mathbf k}\kappa \; &\;
\frac{ p_0}\kappa\\&&\\
\frac{\mathbf p}\kappa & \mathbf{1} & \frac{\mathbf p}\kappa \\&&\\
\frac{\bar p_0}\kappa\; & -\frac{\mathbf k}\kappa\; &
\frac{ p_4}\kappa
\end{array}\right)\, .
\end{equation}
where $p_0, \mathbf{p}, p_4$ are defined in eq. \eqref{5dcoord} and
\begin{align}
\bar{p}_0(k_0, \mathbf{k}) &= \kappa \sinh \frac{k_0}{\kappa}
-
\frac{\mathbf{k}^2}{2\kappa} e^{\frac{k_0}{\kappa}} \nonumber \\
\bar{p}_4(k_0, \mathbf{k})
&=
\kappa \cosh \frac{k_0}{\kappa}
+
\frac{\mathbf{k}^2}{2\kappa} e^{\frac{k_0}{\kappa}} \nonumber
\end{align}
Notice that $\Pi(k)$ is completely determined once $p_0, \mathbf{p}, p_4$ are known\footnote{One can easily express also $\bar{p}_0, \bar{p}_4, \mathbf{k}$ as a function of $p_0, \mathbf{p}, p_4$. For example, one can check that $\bar{p}_0 = p_0 - \frac{\mathbf{p}^2}{p_0+p_4}$.}. Now that we have a generic group element, we can also get a description of the group manifold. In order to get it, we act with all possible group elements on a fixed vector $\mathcal{O}$ in the 5-dimensional vector space on which $\Pi(k)$ acts\footnote{More technically, one can say that the group acts transitively on $\mathcal{O}$, i.e. the orbit of any point in the group is the whole group.}. We choose $\mathcal{O} = (0,0,0,0,\kappa)^T$, which represents the momentum space origin and can physically be interpreted as a point of zero energy and zero momentum (notice that this point is Lorentz invariant). As a result, for any fixed $\Pi(k)$, the object $\Pi(k) \mathcal{O}$ is a point in the 5-dimensional momentum space with coordinates $(p_0, \mathbf{p}, p_4)$ which is in one-to-one correspondence with the group element $\Pi(k)$. However, we already saw that the coordinates $(p_0, \mathbf{p}, p_4)$ describe a 4-dimensional (submanifold of the) de Sitter space, which is therefore our group manifold.
\section{Relative locality and interactions}
We now discuss relative locality, an unexpected feature of theories with nontrivial momentum space geometry \cite{Amelino-Camelia:2011lvm}, \cite{Amelino-Camelia:2011hjg}. Let us use as a starting point a local interaction of point-like objects, like the one depicted in the Figure \ref{FIG1}.
\begin{figure}
\centering
\begin{tikzpicture}
\draw[color=black, thick] (0,0) -- (1.72,-1);
\draw[color=black, thick] (0,0) -- (-1.72,-1);
\draw[color=black, thick] (0,0) -- (0,2);
\draw[color=black, thick, -<] (0,0) -- (0.86,-0.5);
\draw[color=black, thick, -<] (0,0) -- (-0.86,-0.5);
\draw[color=black, thick, ->] (0,0) -- (0,1);
\filldraw[color=black] (0,0) circle (0.8mm);
\draw[color=blue, ->] (-2,0) node[label={[xshift=-0.3cm, yshift=-0.2cm]$A$}]{} -- (-1.5,0);
\draw[color=blue, ->] (-2,0) -- (-2,0.5);
\draw[color=blue, ->] (-2,0) -- (-2.3,-0.3);
\end{tikzpicture}
\caption{Local interaction with respect to the observer $A$ \\ local with the interaction}
\label{FIG1}
\end{figure}
This type of graph of an interaction has actually a wider scope than just a representation of point-like particles, since the same structure can be found for Feynman diagrams in QFT. Furthermore, the only relativistic invariant potential in SR is the Dirac delta potential, which once again describes a contact interaction. Because of this universality, an interaction like this is usually referred to as an event, and indeed in canonical GR one can uniquely define a spacetime point by assigning an event to it. In fact, an event is an absolute concept in GR abd QFT since, although different observers can have different measurement of the scattering properties (such as the momenta of the particles involved), they all agree on the fact that the scattering happened in the first place, and that it happened locally at the same spacetime point\footnote{Of course, general covariance in GR only describes our freedom to give this spacetime point any name we like, i.e. any numerical value in terms of our favourite coordinates, but the construction of a spacetime point in terms of event is not influenced by this.}. Incidentally, this is also one of the problematic points of a potential theory of quantum gravity, since such contact interaction between gravitons leads to renormalization issues \cite{Goroff:1985th}.
The universality of contact interactions make them the ideal starting point for our considerations on relative locality. As a matter of fact, we already encountered one of the key aspects of relative locality when we wrote down the transformations our coordinates under translations in eq. \eqref{deformedtranslations} which leave the deformed action invariant (which we reproduce here for simplicity).
\begin{align}
\delta_T x^0 = \epsilon^0
\qquad
\delta_T \mathbf{x}^i
=
\epsilon^i e^{-\frac{k_0}{\kappa}}
\end{align}
Contrary to SR, translations act in different way on particles of different energies, and therefore an observer $B$ translated by a distance $D$ with respect to an observer $A$ near the interaction will see a different interaction, represented in the Figure \ref{FIG2}.
\begin{figure}
\centering
\begin{tikzpicture}
\draw[color=black, thick] (0,0) -- (1.72,-1);
\draw[color=black, thick] (0,0) -- (-1.72,-1);
\draw[color=black, thick] (0,0) -- (0,2);
\draw[color=black, thick, -<] (0,0) -- (0.86,-0.5);
\draw[color=black, thick, -<] (0,0) -- (-0.86,-0.5);
\draw[color=black, thick, ->] (0,0) -- (0,1);
\filldraw[color=white] (0,0) circle (4mm);
\draw[color=gray, ->] (-2,0) node[label={[xshift=-0.3cm, yshift=-0.1cm]$A$}]{} -- (-1.5,0);
\draw[color=gray, ->] (-2,0) -- (-2,0.5);
\draw[color=gray, ->] (-2,0) -- (-2.3,-0.3);
\draw[color=blue, ->] (-4,0.5) node[label={[xshift=-0.3cm, yshift=-0.2cm]$B$}]{} -- (-3.5,0.5);
\draw[color=blue, ->] (-4,0.5) -- (-4,1);
\draw[color=blue, ->] (-4,0.5) -- (-4.3,0.2);
\draw[color=red, dashed, <->] (-2,0) -- (-4,0.5) node[label={[xshift=0.8cm, yshift=-0.9cm]$D$}]{};
\end{tikzpicture}
\caption{\footnotesize{Same interaction as before, but seen from \\ $\qquad$ an observer $B$ distant from $A$}}
\label{FIG2}
\end{figure}
Therefore, in this theory locality is not observer independent. An event that is local for one observer, Figure \ref{FIG1}, is not local to another, Figure \ref{FIG2}. There is now the issue of defining spacetime, because the canonical description of a spacetime point as event does not hold anymore in our context. Indeed in Special Relativity a spacetime point is defined as an event that happens at this point, ie., by some physical local process taking place there. The only elementary process one can think of is the interaction as depicted in Figure \ref{FIG1}. In Special Relativity such interaction defines an event and the spacetime point because all the observers agree that the interaction is local so the definition of the spacetime point as the point where the interaction takes place is observer-independent. In the case of relative locality this is not the case, and the points are sharply defined by interactions only to the observers who are local to the interaction point.
One of the important properties of the interaction point is that some quantities, for example momenta, are conserved in the course of the interaction. It is therefore reasonable to start our investigation by listing all the properties that an interaction should have.
First of all, in a trivial (2-valent) vertex with one particle coming in, nothing happening in between, and one particle going out, if the initial momentum is $p$ then also the final momentum should be $p$ (by conservation of energy and momentum). Mathematically we can write that
\begin{align}
0\oplus p = p \oplus 0 = p
\end{align}
where $\oplus$ is some abstract composition of momenta. Of course, since we have this new composition law, we also expect to be able to have particles with momenta $S(p)$ which is opposite to $p$, i.e. such that
\begin{align}
p \oplus S(p) = S(p) \oplus p = 0.
\end{align}
Notice that at this point we are still not assuming associativity, so we have the structure of a quasigroup. Consider now a 3-valent vertex like the one in the Figure \ref{FIG3}.
\begin{figure}
\centering
\begin{tikzpicture}
\draw[color=black, thick] (0,0) -- (1.72,-1);
\draw[color=black, thick] (0,0) -- (-1.72,-1);
\draw[color=black, thick] (0,0) -- (0,2);
\draw[color=black, thick, -<] (0,0) -- (0.86,-0.5) node[label={[xshift=0cm, yshift=0cm]$p$}]{};
\draw[color=black, thick, -<] (0,0) -- (-0.86,-0.5) node[label={[xshift=0cm, yshift=0cm]$q$}]{};
\draw[color=black, thick, ->] (0,0) -- (0,1) node[label={[xshift=0.3cm, yshift=0cm]$r$}]{};
\filldraw[color=black, fill=white, thick] (0,0) circle (4mm);
\end{tikzpicture}
\caption{3-valent vertex}
\label{FIG3}
\end{figure}
In this case we have two particles coming in and one particle coming out, and we expect that the following relations hold.
\begin{align}\label{defmomcons}
p\oplus q = r
\qquad
S(r) \oplus (p\oplus q) = (p\oplus q) \oplus S(r) = 0
\end{align}
Now impose the condition that the above relations are Poincar\'e covariant in an appropriate, deformed sense. Considering boosts (because spacetime rotations are assumed to act trivially) and using the more abstract notation used in eq. \eqref{abstractvariations} we impose the covariance of the relations \eqref{defmomcons}
\begin{align}
N_i \triangleright (p\oplus q)
\equiv
\sum_i
(\{\mathcal{P}_i^1 \triangleright p \}
+
\{\mathcal{P}_i^2 \triangleright q \})
=
N_i \triangleright r
\end{align}
where we made use of the so-called Sweedler notation, and where $\mathcal{P}_i^1, \mathcal{P}_i^2$ are appropriate generators of the Poincar\'e algebra. This consistency condition ensures that the Poincar\'e generators form a Hopf algebra (which is a generalization of Lie algebras), but we will not go into more details in this direction. In the undeformed case, after a boost one would have $p_\mu \mapsto N_i\triangleright p_\mu = p_\mu + \delta_i p_\mu$ and $q\mu \mapsto N_i\triangleright q_\mu = q_\mu + \delta_i q_\mu$, and therefore
\begin{align}
p_\mu + q_\mu
\mapsto
N_i\triangleright p_\mu
+
N_i\triangleright q_\mu
=
p_\mu + q_\mu
+
\delta_i p_\mu + \delta_i q_\mu
=
r_\mu+\delta_i r_\mu
=
N_i \triangleright r_\mu
\end{align}
which therefore implies
\begin{align}
\mathcal{P}_1^1 = \mathcal{P}_1^2 = 1
\qquad
\mathcal{P}_2^1 = \mathcal{P}_2^2 = \delta_i.
\end{align}
Notice that the action of $N_i$ is linear because in the undeformed case we don't have a scale, which would make it possible for nonlinearities to appear.
Now we have all the properties that we would like to have for the deformed composition of momenta, and we only need to specify explicitly what this composition rule actually is. In order to do it, we come back to the group theoretical perspective. Given two elements $\Pi(k), \Pi(l)$ of the group, we know that also their product $\Pi(k)\Pi(l)$ will be an element of the group, and we can use this group multiplication to define the addition of momenta.
\begin{align}
e^{i(\mathbf{k}\oplus \mathbf{l})_i \mathbf{X}^i} e^{i(k_0\oplus l_0) X^0}
=
\Pi(k \oplus l)
:=
\Pi(k)\Pi(l)
=
e^{i\mathbf{k}_i \mathbf{X}^i} e^{ik_0 X^0}
e^{i\mathbf{l}_i \mathbf{X}^i} e^{il_0 X^0}
\end{align}
The above product can be easily preformed using the Baker–Campbell–Hausdorff formula or the explicit expression in eq. \eqref{groupelement2}, obtaining
\begin{align}
e^{i(\mathbf{k}\oplus \mathbf{l})_i \mathbf{X}^i} e^{i(k_0\oplus l_0) X^0}
=
e^{i\left(\mathbf{k}_i + e^{-k_0/\kappa} \mathbf{l}_i\right) \mathbf{X}^i} e^{i(k_0 + l_0) X^0}
\end{align}
which means that
\begin{align}
(\mathbf{k}\oplus \mathbf{l})_i
&=
\mathbf{k}_i + e^{-k_0/\kappa} \mathbf{l}_i \label{comprule1}\\
(k_0 \oplus l_0)
&=
k_0 + l_0 \label{comprule2}
\end{align}
Notice that the energies sum like in the undeformed case, but the composition law for spatial momenta is not linear due to the factor $e^{-\frac{k_0}{\kappa}}$. Furthermore, this composition law is non-abelian and associative (due to the group properties).
More in general, one can view the deformed composition law for momenta more geometrically by relating it to the properties of momentum space. In fact, one can write
\begin{align}\label{connectiondef}
(p\oplus q)_\mu
=
p_\mu + q_\mu
-
\tensor{\Gamma}{_\mu^{\alpha\beta}} p_\alpha q_\beta
+
\dots
\qquad
\tensor{\Gamma}{_\mu^{\alpha\beta}}
=
-
\frac{\partial^2 (p\oplus q)_\mu}{\partial p_\alpha \partial q_\beta} \Bigg|_{p=q=0}.
\end{align}
Recall that $p,q$ from a geometrical point of view describe the coordinates of some points in the 4-dimensional de Sitter space, and $(p\oplus q)$ is a different point in momentum space, and to go from one point to another in a curved momentum space one really needs to define a parallel transport, and therefore a connection. Notice however that the connection in \eqref{connectiondef} is associated with the $\oplus$ operation which involves two points, and it is therefore not immediately related to the metric of momentum space, which only involves one (see for example \eqref{momentumspacemetric}). Indeed, having defined a connection in \eqref{connectiondef}, and since we already know the metric in momentum space, it is now possible to see that the following hold:
\begin{itemize}
\item The connection $\Gamma$ is torsion-free iff the composition rule $\oplus$ is symmetric;
\item The connection $\Gamma$ has curvature (i.e. the curvature associated with the connection $\Gamma$ is non-zero) iff the composition rule $\oplus$ is not associative;
\item The connection $\Gamma$ is not in general metric since $\nabla_\Gamma G^{\mu\nu} \neq 0$.
\end{itemize}
Referring back to the composition rules in \eqref{comprule1}, \eqref{comprule2}, our connection would therefore be with torsion, zero curvature, and non-metric.
We now want to describe the interaction at the level of the action, i.e. we want to generalize the single particle action in \eqref{actioncurvedms} to the case of many interacting particles (for simplicity, here we consider only scalar particles without spin). To start with we notice that an interacting particle's worldline is semi-infinite, and in particular for incoming particles $\tau \in (-\infty, 0)$ and for outgoing particles $\tau \in (0, +\infty)$ (here we are choosing a $\tau$ such that the interaction happens at $\tau = 0$). Therefore the single incoming particle action is first rewritten as
\begin{align}\label{actioncurvedmsincoming}
S_\kappa =
-\int_{-\infty}^0 d\tau \,\, x^a \tensor{E}{_a^\mu}(k, \kappa) \dot{k}_\mu - N(\mathcal{C}_\kappa(k) + m^2).
\end{align}
and similarly for outgoing particles, so that the total free Lagrangian of incoming and outgoing particles is now given by
\begin{align}\label{freeaction}
S_\kappa^{\text{free}} =
-\sum_j
\int_{\text{in}/\text{out}} d\tau \,\, x^a_j \tensor{E}{_a^\mu}(k^j, \kappa) \dot{k}_\mu^j - N^j(\mathcal{C}_\kappa^j(k^j) + m^2_j).
\end{align}
For scalar particles, the only quantity which is conserved in an interaction is momentum, so we can define the interaction term in the action as
\begin{align}\label{intaction}
S_\kappa^{\text{int}}
:=
z^\mu
(k^1 \oplus k^2 \oplus \dots )_\mu
\end{align}
where $z^\mu$ is just a Lagrange multiplier enforcing momentum conservation at the vertex. Notice that in this case the importance of having the time derivative on $k$ and not on $x$ in the free action \eqref{freeaction} is highlighted even more. In fact, while previously we didn't have a boundary so that we could have in principle integrated by parts, now we do have a boundary, and therefore the choice of whether to use $\dot{x}^\mu k_\mu$ or $x^\mu \dot{k}_\mu$ in the deformed action is crucial, since the two are not equivalent.
The equations of motion following from the action $S_\kappa^{\text{free}} + S_\kappa^{\text{int}}$ will now have both a bulk contribution and a surface one. The bulk equations of motion are given by
\begin{align}
&\dot{k}_\mu^j = 0 \nonumber \\
&\dot{x}^\mu_j = N^j \tensor{E}{_a^\mu}(k^j, \kappa) \frac{\partial \mathcal{C}_\kappa^j(k^j)}{\partial k_\mu^j} \\
&\mathcal{C}_\kappa^j(k^j) + m^2 = 0
\end{align}
and the equations of motion at the interaction point has the form
\begin{align}
x^a_j(0)
=
z^\nu
\tensor{E}{^a_\mu}
\frac{\partial}{\partial k^j_\mu}
(k^1 \oplus k^2 \oplus \dots )_\nu.
\end{align}
The boundary equations of motion highlight again the effect of relative locality already depicted in Figure \ref{FIG2}. Let us look at translations for simplicity (there is a similar effect for Lorentz transformations). We have
\begin{align}\label{rellocvertex}
\delta x^a_j(0)
=
\delta z^\nu
\tensor{E}{^a_\mu}
\frac{\partial}{\partial k^j_\mu}
(k^1 \oplus k^2 \oplus \dots )_\nu.
\end{align}
Notice that since $\delta x^a_j(0)\neq \delta x^a_i(0)$ for $i\neq j$, it follows from \eqref{rellocvertex} that translations are momentum dependent, which means that particles of different momenta will be translated differently, which is indeed the behaviour shown in Figure \ref{FIG2}. The only way to avoid translations being momentum dependent would be to have a flat momentum space, which translates to a trivial momentum space tetrad and a linear composition of momenta, in which case \eqref{rellocvertex} reduces to $\delta x^a_j(0) = \delta z^a$, so that the endpoints of all the worldlines translate by the same amount. Of course, in this case the interaction event behaves as in canonical SR, with the standard locality.
One could also think to extend this reasoning to many vertices and to loops, but the situation in this case is still not completely clear (especially when loops are involved).
Finally, as a final demonstration of phenomena related to relative locality, we show how two photons with different energies emitted simultaneously (with respect to an observer nearby the source) in a distant objects can be detected on Earth as having a time delay, despite the fact that both travel at the same speed $c=1$ (as shown in \eqref{c=1}). The reason why this is possible is of course related to the fact that something that can be local for a distant observer, because of relative locality it is not local anymore from the point of view of our detectors on Earth. Let us consider the situation (depicted below) of one high-energy photon with momentum $(k_0^2, 0,0,\mathbf{k}_z^2)$ and a low-energy one with momentum $(k_0^1, 0,0,\mathbf{k}_z^1)$ emitted at the same time locally to an observer $A$ distant $d$ from Earth.
\begin{center}
\begin{tikzpicture}
\draw[color=black] (10,-1) node[label={[xshift=0cm, yshift=0.6cm]$Earth$}]{} arc (270:90:1cm);
\draw[x=0.052cm,y=0.2cm, blue]
(3,0) sin (4,1) cos (5,0) sin (6,-1) cos (7,0)
sin (8,1) cos (9,0) sin (10,-1) cos (11,0) sin (12,1) cos (13,0) sin (14,-1) cos (15,0) sin (16,1) cos (17,0) sin (18,-1) cos (19,0) sin (20,1) cos (21,0) sin (22,-1) cos (23,0) sin (24,1) cos (25,0);
\draw[x=0.3cm,y=0.2cm, red]
(0.5,0) sin (1.5,1) cos (2.5,0) sin (3.5,-1) cos (4.5,0);
\begin{scope}[shift={(6,0)}]
\draw[x=0.052cm,y=0.2cm, blue]
(3,0) sin (4,1) cos (5,0) sin (6,-1) cos (7,0)
sin (8,1) cos (9,0) sin (10,-1) cos (11,0) sin (12,1) cos (13,0) sin (14,-1) cos (15,0) sin (16,1) cos (17,0) sin (18,-1) cos (19,0) sin (20,1) cos (21,0) sin (22,-1) cos (23,0) sin (24,1) cos (25,0);
\end{scope}
\begin{scope}[shift={(7.6,0)}]
\draw[x=0.3cm,y=0.2cm, red]
(0.5,0) sin (1.5,1) cos (2.5,0) sin (3.5,-1) cos (4.5,0);
\end{scope}
\draw[color=ForestGreen, ->] (0,-0.5) -- (9,-0.5) node[label={[xshift=-9cm, yshift=-1cm]$z=0$}]{} node[label={[xshift=0cm, yshift=-1cm]$z=d$}]{};
\draw[color=ForestGreen, dashed] (0,0) -- (0,-0.5);
\draw[color=ForestGreen, dashed] (9,0) -- (9,-0.5);
\filldraw[color=black] (0,0) node[label={[xshift=0cm, yshift=0cm]$A$}]{} circle (0.8mm);
\filldraw[color=black] (9,0) node[label={[xshift=-0.2cm, yshift=0cm]$B$}]{} circle (0.8mm);
\end{tikzpicture}
\end{center}
Since $c^2 = \left(\frac{\dot{\mathbf{X}}^z}{\dot{X}^0}\right)^2 = 1$, the equations of motion for the low-energy photon will be given by
\begin{align}
\dot{X}_1^0 = A(k^1) := A_1
\qquad
\dot{\mathbf{X}}_1^z = A(k^1) := A_1
\end{align}
which can immediately be integrated to give
\begin{align}\label{solAlow}
X^0_1 = A_1 \tau
\qquad
\mathbf{X}^z_1 = A_1 \tau
\end{align}
where there is no additional constant $C$ because we are at the moment in the reference frame $A$. In the same way, for the energetic photon we get
\begin{align}\label{solAhigh}
X^0_2 = A_2 \tau
\qquad
\mathbf{X}^z_2 = A_2 \tau
\end{align}
Now we have to translate the solutions \eqref{solAlow}, \eqref{solAhigh} to the observer $B$, so that we can predict what they will measure. To do so, assume that $k_0^1$ is small enough that $e^{-\frac{k_0}{\kappa}} \approx 1$, so that for the low-energy photon the translation acts in the standard way so that for $B$ we have.
\begin{align}\label{solBlow}
X^0_1 = A_1 \tau
\qquad
\mathbf{X}^z_1 = A_1 \tau - d
\end{align}
If instead $k_0^2$ is high enough, then we cannot neglect the factor $e^{-\frac{k_0^2}{\kappa}}$ coming from the shift of the second photon. Therefore from $B$ point of view we would have
\begin{align}\label{solBhigh}
X^0_2 = A_2 \tau
\qquad
\mathbf{X}^z_2 = A_2 \tau - e^{-\frac{k_0^2}{\kappa}} d.
\end{align}
Now notice that the low-energy photon reaches us when $\tau = \frac{d}{A_1}$, and at this instant the high energy photon is still distant
\begin{align}
\mathbf{X}^z_2(\tau = d/A_1)
=
d
\left(
\frac{A_2}{A_1}
-
e^{-\frac{k_0^2}{\kappa}}
\right)
\end{align}
from $B$. In general this distance is different from zero, which provides the anticipated difference in arrival times.
Notice one subtlety of this example. In the whole discussion about relative locality, interactions play a crucial role. Indeed, relative locality can be understood as the fact that an interaction which is local for an observer is not local for some other one. However, in the above example we have non-interacting photons, which are created in different processes and measured with different apparatuses, and yet we still have effects due to relative locality. The key point is that, although the photons are not interacting between themselves, they were (individually) created by some interaction between other particles\footnote{For example, the low energy photon could have been created due to a low energy scattering between charged particles, and the high energy photon by a positron-electron annihilation}. These interactions are both local for $A$ (since both photons are created locally to $A$), but are not local for $B$. However, the amount by which these interactions are non-local for $B$ depends on the energy of the photons. In particular, the event which generated the low-energy photon will seem approximately local also for $B$, while the one which generated the energetic photon will not be local for $B$. The situation can be schematically represented by the picture below.
\begin{center}
\begin{tikzpicture}
\begin{scope}[shift={(0.1,0)}]
\draw[x=0.052cm,y=0.2cm, blue]
(3,0) sin (4,1) cos (5,0) sin (6,-1) cos (7,0)
sin (8,1) cos (9,0) sin (10,-1) cos (11,0) sin (12,1) cos (13,0) sin (14,-1) cos (15,0) sin (16,1) cos (17,0) sin (18,-1) cos (19,0) sin (20,1) cos (21,0) sin (22,-1) cos (23,0) sin (24,1) cos (25,0);
\end{scope}
\begin{scope}[shift={(0,3)}]
\draw[x=0.3cm,y=0.2cm, red]
(0.5,0) sin (1.5,1) cos (2.5,0) sin (3.5,-1) cos (4.5,0);
\end{scope}
\begin{scope}[shift={(0.2,0)}]
\filldraw[color=black] (0,0) circle (0.9mm);
\draw[color=black] (-0.5,-0.86) -- (0,0) -- (-0.5, 0.86);
\end{scope}
\begin{scope}[shift={(0.2,3)}]
\filldraw[color=black] (0,0) circle (0.9mm);
\draw[color=black] (-0.5,-0.86) -- (0,0) -- (-0.5, 0.86);
\end{scope}
\begin{scope}[shift={(4.4,1)}]
\draw[color=blue, ->] (-4,0.5) node[label={[xshift=-0.3cm, yshift=-0.2cm]$A$}]{} -- (-3.5,0.5);
\draw[color=blue, ->] (-4,0.5) -- (-4,1);
\draw[color=blue, ->] (-4,0.5) -- (-4.3,0.2);
\end{scope}
\begin{scope}[shift={(6.6,0)}]
\draw[x=0.052cm,y=0.2cm, blue]
(3,0) sin (4,1) cos (5,0) sin (6,-1) cos (7,0)
sin (8,1) cos (9,0) sin (10,-1) cos (11,0) sin (12,1) cos (13,0) sin (14,-1) cos (15,0) sin (16,1) cos (17,0) sin (18,-1) cos (19,0) sin (20,1) cos (21,0) sin (22,-1) cos (23,0) sin (24,1) cos (25,0);
\end{scope}
\begin{scope}[shift={(6,3)}]
\draw[x=0.3cm,y=0.2cm, red]
(0.5,0) sin (1.5,1) cos (2.5,0) sin (3.5,-1) cos (4.5,0);
\end{scope}
\begin{scope}[shift={(6.2,0)}]
\draw[color=black] (-0.5,-0.86) -- (0,0) -- (-0.5, 0.86);
\filldraw[color=white] (0,0) circle (3mm);
\end{scope}
\begin{scope}[shift={(6.2,3)}]
\filldraw[color=black] (0,0) circle (0.9mm);
\draw[color=black] (-0.5,-0.86) -- (0,0) -- (-0.5, 0.86);
\end{scope}
\begin{scope}[shift={(10.4,1)}]
\draw[color=blue, ->] (-2,0) node[label={[xshift=-0.3cm, yshift=-0.1cm]$B$}]{} -- (-1.5,0);
\draw[color=blue, ->] (-2,0) -- (-2,0.5);
\draw[color=blue, ->] (-2,0) -- (-2.3,-0.3);
\draw[color=gray, ->] (-4,0.5) node[label={[xshift=-0.3cm, yshift=-0.2cm]$A$}]{} -- (-3.5,0.5);
\draw[color=gray, ->] (-4,0.5) -- (-4,1);
\draw[color=gray, ->] (-4,0.5) -- (-4.3,0.2);
\draw[color=red, dashed, <->] (-2,0) -- (-4,0.5) node[label={[xshift=0.8cm, yshift=-0.9cm]$d$}]{};
\end{scope}
\end{tikzpicture}
\end{center}
\section{The soccer ball problem}
The soccer ball problem for theories with deformed dispersion relation and/or modified composition laws has been formulated almost immediately after deformed theories were first formulated\footnote{According to J.\ Lukierski it was already in early 1990th when during one of his seminars I. Bia\l{}ynicki-Birula objected that if the deformed dispersion relation was universally valid for macroscopic bodies it would contradict every day observations. Unfortunately, part of the community still keeps thinking that the soccer ball problem is an unsolved paradox.}. The problem can be roughly summarized as follows. In the deformed context we deal with deformed dispersion relations like
\begin{align}\label{defdisp}
\mathcal{C}_\kappa(k)
=
-k_0^2 + \mathbf{k}^2
+
\frac{1}{\kappa}
\Delta^{\mu\nu\rho}
k_\mu k_\nu k_\rho
+
\dots
=
-m^2
\end{align}
and composition rules like the one in eq. \eqref{connectiondef}. All of them depend on the scale $\kappa$ which has been central in our discussion. Since the deformation in DSR models is assumed to be of quantum gravitational origin, one usually expects $\kappa$ to be of the order of the Planck mass, which is around $10^{19} \, GeV$. Furthermore, when one deals about composition of momenta and scattering of particles, one usually has fundamental particles in mind (like electron, photons and so on) for which one usually has $E/\kappa <<1$, where $E$ is the particle energy. One then can argue that since $\kappa$ is universal, the same composition rules and deformed dispersion relation should hold no matter how many particles are involved. In particular, we could also consider a soccer ball with all its particles, and for this object, the effects of the deformed composition rules are at the first sight very big. Indeed, although every single particle has the energy by many orders of magnitudes smaller than the Planck energy (i.e. $E/\kappa <<1$), there are so many particles that the energy of the system is much higher than $\kappa$, so that the deformations used in DSR should have a large macroscopic effect. The problem now arises because such large deformations are obviously not observed in macroscopic objects.
In other words, how can composition rules and dispersion relations be universally deformed, and yet a soccer ball, for which $p/\kappa \approx 10^8$ (and therefore the contributions due to deformations should be enormous) behaves classically?
To describe the solution to this problem it is convenient to use the so-called normal basis \cite{Amelino-Camelia:2011dwc} (since the choice of basis, being a choice of coordinates, cannot have any physical effect, it is just a technical simplification). In fact, mathematically, the soccer ball problem relies on the fact that, once we generalize \eqref{connectiondef} to the sum of $N$ momenta, the number of quadratic contributions grows like $N^2$. A solution of the soccer problem would therefore amount to some way of dealing with this growth. Notice that the qualitative argument on the growth of terms like $N^2$ relies on \eqref{connectiondef}, which however requires an explicit choice of basis to be made. To deal with the soccer ball problem, we pick the so called normal basis. We already encountered normal coordinates $\tilde{p}_\mu$ when talking about the convention used to write eq. \eqref{groupelement} and they are defined by
\begin{align}
e^{i\mathbf{k}_i \mathbf{X}^i} e^{ik_0 X^0}
=
e^{i \tilde{p}_\mu X^\mu}.
\end{align}
Because of their definition, since $[X^\mu, X^\mu]=0$, they satisfy the relation
\begin{align}
(\tilde{p}\oplus \tilde{p})_\mu = 2\tilde{p}_\mu
\end{align}
To give more explicit expressions, proceeding as in section \ref{GTP}, one can show that \cite{Kowalski-Glikman:2017ifs}
\begin{align}
\tilde{p}_0
=
k_0
\qquad
\tilde{\mathbf{p}}_i
=
\mathbf{k}_i
\frac{\frac{k_0}{\kappa}}{1- e^{-\frac{k_0}{\kappa}}}
\end{align}
with the composition rules
\begin{align}
(\tilde{p}\oplus \tilde{q})_0 = \tilde{p}_0 + \tilde{q}_0
\qquad
(\tilde{\mathbf{p}}\oplus \tilde{\mathbf{q}})_i
=
\left(
\tilde{\mathbf{p}}_i \frac{1}{f(\tilde{p}_0)}
+
\tilde{\mathbf{q}}_i \frac{e^{-\frac{\tilde{p_0}}{\kappa}}}{f(\tilde{q}_0)}
\right)
f(\tilde{p}_0 + \tilde{q}_0)
\end{align}
where
\begin{align}
f(\tilde{p}_0)
=
\frac{\tilde{p_0}}{\kappa}
\frac{1}{1-e^{-\frac{\tilde{p}_0}{\kappa}}}
\end{align}
Using these one can indeed verify that $(\tilde{p}\oplus \tilde{p})_0 = 2\tilde{p}_0$ and
\begin{align}
(\tilde{\mathbf{p}}\oplus \tilde{\mathbf{p}})_i
=
\frac{f(2\tilde{p}_0)}{f(\tilde{p}_0)}
\left(
1 + e^{-\frac{\tilde{p}_0}{\kappa}}
\right)
\tilde{\mathbf{p}}_i
=
2 \tilde{\mathbf{p}}_i.
\end{align}
In the normal basis, if we consider the ball to be formed by $N$ particles\footnote{This of course now bear the question of what are the fundamental constituents of the ball, because depending on what they are the number $N$ changes. However, even restricting our attention to atoms, then $N$ would be big enough to make our reasoning valid. Of course, if one chooses then to consider electrons, protons and so on, then one would get a bigger $N$.} each with momentum $\tilde{p}$, then the total momentum of the ball would just be $(\tilde{p}\oplus(\tilde{p}\oplus \dots)) = N \tilde{p} := \tilde{P}$, without any contribution coming from deformations. At the same time, the dispersion relation for the whole ball can be easily obtained from eq. \eqref{defdisp} by summing the individual dispersion relations of the individual particles, obtaining
\begin{align}
-P_0^2 + \mathbf{P}^2
+
\frac{1}{N\kappa}
\Delta^{\mu\nu\rho}
P_\mu P_\nu P_\rho
+
\dots
=
-N^2m^2
=
-M^2
\end{align}
where $M=Nm$ is the total mass of the ball. We see that the deviation from the undeformed dispersion relation is of the order $\frac{P}{N\kappa}$, i.e. it is negligibly small.
Notice that we chose the normal basis because the solution to the soccer ball problem is particularly transparent, but if one prefers one can chose any other coordinates and work out explicitly the whole argument with them. A similar procedure shows that for scattering of macroscopic bodies one can get
\begin{align}
P^{in}_1 + P^{in}_2 = P^{out}_1 + P^{out}_2 + O\left(\frac{1}{N\kappa}\right)
\end{align}
This shows that for macroscopic bodies the effective deformation parameter is not $\kappa$ but $N\kappa$, which is a number at least by the factor of order of $10^{23}$ times larger. The reader is encouraged to investigate how the argument would change if one replaces the normal coordinates in momentum space with different ones.
There is however yet another potential problem \cite{Hossenfelder:2012vk} that needs to be solved when one allows for fluctuations around an average value, i.e. $k = \bar{k} + \delta k$ with the average fluctuation equal zero $\langle\delta k \rangle = 0$. The issue is that, even though $\langle\delta k \rangle = 0$, the amplitude of the fluctuation must be small too. If this amplitude were to be relevant, the soccer ball would be macroscopically fluctuating around its classical trajectory, which is once again a behaviour which is not observed in reality. Not all deformed theories pass this test since some indeed predict macroscopic fluctuations of large bodies, and indeed this condition restrict the possible geometries of momentum space \cite{Amelino-Camelia:2013zja}.
\section{Relations between deformations and quantum gravity}
In this concluding section we will briefly discuss the relation between the deformation of relativistic symmetries discussed up to now, and the problem of quantum gravity. The starting point is given by general relativity in $2+1$ dimensions.
In 3-dimensional gravity there are no local degrees of freedom (i.e. no local gravitational degrees of freedom, no gravitational waves, no Newtonian interactions), only a finite number of topological ones. One important feature of 3-dimensional gravity comes from dimensional analysis (here we are considering the case $c=1$ for simplicity). In fact, it can be shown that the Newton constant $G_N$ has the dimension of inverse mass. This is easily seen by remembering that Newtonian gravity satisfies the local Gauss law
\begin{align}
\nabla \cdot \mathbf{g} = 4\pi G_N \rho
\end{align}
where $\mathbf{g}$ is the Newtonian acceleration. Assuming to have only two spatial dimension and considering for simplicity rotational symmetry, the above law implies
\begin{align}
|\mathbf{g}| = \frac{2 G_N m}{r}
\end{align}
which in turn means that
\begin{align}
\frac{L}{T^2}
=
[G_N] \frac{M}{L}
\qquad
\implies
\qquad
[G_N] = \frac{1}{M}
\end{align}
where recall that the assumption $c=1$ also implies that $[c] = \frac{L}{T} = 1$. Therefore, the Newton constant $G_N$ has the dimensions of inverse mass, which means that classical gravity in 3 dimension already has a length scale built-in from the outset. As a consequence, keeping in mind Gauss motto, we expect that some kind of deformation has to be present already at the classical level. Skipping the details (which can for example be found in \cite{Arzano:2021scz} and in several literature papers) the way one proceeds is along the following steps.
\begin{itemize}
\item[1)] The starting point is given by the action of a point particle coupled with gravity. As said above, the number of degrees of freedom of gravity in 3 dimension is finite, so we can actually get an explicit solution of the equations of motion for them\footnote{This is not possible in general in 4 dimensions, which is the reason why this procedure cannot be repeated in the more realistic setting of a universe with $3+1$ dimensions.};
\item[2)] Substitute these solutions back to the action of a point particle coupled with gravity, obtaining an effective action of a particle `dressed' in its own gravitational field;
\item[3)] This action can then be written in a way analogous to the action in \eqref{explicitdefaction}.
\end{itemize}
The way to relate the above discussion about 3-dimensional gravity and quantum gravity is given by the following argument \cite{Freidel:2003sp},\cite{Arzano:2021scz}. We will skip the technicalities, which can be found on the provided references, but the main idea of the argument can be summarized as follows.
Assume we have a spatially planar system in $3+1$ dimensions consisting of quantum particles coupled to the gravitational field. Then one can show that this system can be described as classical particles in $2+1$ dimensions. Hence they are described by 3-dimensional gravity, and (using the above procedure) the system will therefore be characterized by the deformation of spacetime symmetries. More precisely, the Hilbert space $\mathcal{H}^3$ of the system will have some symmetry group $\mathcal{P}^3$. At the same time, this original planar system is just a particular planar type of physical system in a $(3+1)$-dimensional spacetime. Therefore, the Hilbert space $\mathcal{H}^3$ of our subsystem is a subspace of $\mathcal{H}^4$, which is the vector field in the context of a full quantum theory of gravity. In particular, the symmetry group $\mathcal{P}^3$ must then be a subgroup of the full symmetry group $\mathcal{P}^4$ acting on $\mathcal{H}^4$. But now we know that the symmetries encoded in $\mathcal{P}^3$ are the deformed ones that we talked about, so that $\mathcal{P}^4$ cannot just be the Lorentz group (because in that case $\mathcal{P}^3$ couldn't be its subgroup). Therefore, we also expect that $\mathcal{P}^4$ (which we recall describes the symmetries of a full quantum gravity) must be some deformed symmetry group.
\section*{Acknowledgment}
This work was supported by funds provided by the Polish National Science Center, the project number 2019/33/B/ST2/00050, and for JKG also by the project number 2017/27/B/ST2/01902.
|
1,108,101,564,570 | arxiv | \section{Introduction} \label{sec:intro}
Let $\Omega = \{\omega_J : \, J \subseteq [n]\}$
be the sample space freely generated by $n$ random events $A_1, \dots, A_n$, so that $\{\omega_J\} = \bigcap_{j \in J} A_j \cap \bigcap_{j \notin J} \overline{A}_j =: \A^J$
for all subsets $J \subseteq [n]$.
Here, as usual, for any integer $n \ge 0$, let $[n] := \{1, \dots, n\}$, let $\subseteq$ (resp.\ $\subset$) denote the subset (resp.\ proper subset) relation, let $\overline{A}$ denote the complement of $A$ and let $|J|$ denote the cardinality of a set $J$.
With $\Sigma$ as the $\sigma$-algebra of all subsets of $\Omega$,
the unique probability measure $P$ on the measurable space $(\Omega, \Sigma)$
with respect to which $A_1, \dots, A_n$ are mutually independent
and whose marginal probabilities are $P(A_j) =: a_j$ for all $j \in [n]$
is given by:
\begin{equation} \label{eq: mutualIndependenceSolution}
P(\A^J) = \prod_{j \in J} a_j \times \prod_{j \notin J} (1 - a_j) =: \a^J,
\quad \text{for all } J \subseteq [n].
\end{equation}
We may relax the condition that the $n$ events are mutually independent
to require only that every $(n - 1)$ events among $A_1, \dots, A_n$
are mutually independent.
This weaker condition is sometimes known as \emph{$(n - 1)$-wise independence}. It is known that $(n - 1)$-wise independence does not imply that the $n$ events are mutually independent in general (see \cite{Wang,wangstoy} for counterexamples), although the converse is true. Comparisons of various notions of independence and dependence for random variables can be found in the literature \cite{Feller, Mukhopadhyay, Stoyanov, Wong}. The special case of Bernoulli random variables correspond to the setting of random events in this paper.
Bernstein \cite[p.\ 126]{Feller} constructed his classic example of $n = 3$ pairwise independent events which are not mutually independent. The example in \cite{Wang} generalizes his construction to $n \ge 3$ random events that are $(n - 1)$-wise independent but not mutually independent. We discuss this construction next.
\begin{example} [Construction of $(n-1)$-wise independent events] \label{ex:wang}
Let $a_j = 1/2$ for all $j \in [n]$ where $A_1,\ldots,A_{n-1}$ are mutually independent events and suppose that the event $A_n$ occurs given that an even number of events among $A_1,\ldots,A_{n-1}$ occur. It can be verified that these $n$ events are $(n - 1)$-wise independent but not mutually independent.
The induced probability measure is not the unique one with respect to which $A_1,\ldots,A_{n}$ are $(n - 1)$-wise independent but not mutually independent. For example, if we suppose instead that $A_n$ occurs given that an odd number of events among $A_1,\ldots,A_{n-1}$ occur, we obtain a distinct probability measure.
\end{example}
\subsection*{Overview}
In Section \ref{sec:characterization_(n-1)independence}, we characterize all probability measures with respect to which $n$ random events are $(n - 1)$-wise independent (Theorem \ref{thm: characterisationOfProbabilityMeasures}). Now we briefly outline the steps leading to Theorem \ref{thm: characterisationOfProbabilityMeasures}. First, in Proposition \ref{prop: formalEquivalence}, we identify a system of $2^n - 1$ equations linear in $P(\A^I)$ for all $I \subseteq [n]$ that is satisfied if and only if $A_1, \dots, A_n$ are $(n - 1)$-wise independent. We then relax the condition that $P(\A^I)$ are nonnegative and show in Lemma \ref{lem:1} and Corollary \ref{cor: characterisationOfUnsignedMeasures} that the solutions are parameterized by a single real parameter. Reimposing the condition that $P(\A^I)$ are nonnegative forces this parameter to lie within a compact interval which is identified in \eqref{eq: solveSystemForPositivityOfMeasure}.
Lemmas \ref{lem2}--\ref{lem: minimizeAtomicProbabilityForUpperBound} then explicitly identifies the endpoints of this interval in terms of the marginal probabilities. Theorem \ref{thm: characterisationOfProbabilityMeasures} then follows.
Section \ref{sec:atleastksharpbounds} applies Theorem \ref{thm: characterisationOfProbabilityMeasures} to identify sharp bounds on the probability that at least $k$ out of $n$ events which are $(n - 1)$-wise independent occurs (Theorem \ref{thm:atleastksharpbounds}) while Theorem \ref{thm:complexity} shows that these bounds are computable in polynomial time.
Examples \ref{ex:reductiontopairwise} and \ref{ex:reductiontobonferroni} in Section \ref{sec:examples} give cases when the newly derived bounds are instances of known universal bounds such as the classical Bonferroni bounds. Example \ref{ex:lovaszlocallemma} illustrates the connection of the results to the probabilistic method which provides conditions for the non-occurrence of ``bad" events when events are mostly independent. Example \ref{ex:comparisonmakarov} illustrates the usefulness of the bounds in providing robust estimates when mutual independence breaks down or when existing bounds are not sharp.
\section{Characterization of $(n-1)$-wise independence}\label{sec:characterization_(n-1)independence}
\begin{proposition} \label{prop: formalEquivalence}
A collection of $n$ random events $A_1, \dots, A_n$ is $(n - 1)$-wise independent with $P(A_j) = a_j$ for all $j \in [n]$
if and only if:
\begin{equation} \label{eq: n-1wiseIndependence}
\sum_{I \supseteq J} P(\A^I) = \prod_{j \in J} a_j,
\quad \text{for all } J \subset [n].
\end{equation}
\end{proposition}
\begin{proof}
Since the $\A^I$'s are mutually exclusive, we have $\sum_{I \supseteq J} P(\A^I) = P(\bigcap_{j \in J} A_j)
$ for any $J$.
Thus \eqref{eq: n-1wiseIndependence} is equivalent to
$$P(\bigcap_{j \in J} A_j) = \prod_{j \in J} a_j,
\quad \text{for all } J \subset [n],$$
which is in turn equivalent to the $(n - 1)$-wise independence of $A_1, \dots, A_n$.
\end{proof}
The system of linear equations in \eqref{eq: n-1wiseIndependence} is inhomogeneous with \eqref{eq: mutualIndependenceSolution}
as a particular solution.
\begin{lemma} \label{lem:1}
The general solution of the system of linear equations:
\begin{equation} \label{eq: associatedHomogeousSystem}
\sum_{I \supset J} P(\A^I) = 0,
\quad \text{for all } J \subset [n],
\end{equation}
in the $2^n$ variables $\{P(\A^J)\}_{J \subseteq [n]}$ (not necessarily nonnegative),
is given by $P(\A^J) = (-1)^{|J|} s$, where $s \in \mathbb{R}$ is a free parameter.
\end{lemma}
\begin{proof}
The following inhomogeneous linear system has a unique solution, namely the mutually independent probability measure in \eqref{eq: mutualIndependenceSolution}:
$$\sum_{I \supseteq J} P(\A^I) = \prod_{j \in J} a_j,
\quad \text{for all } J \subseteq [n].$$
Hence the square coefficient matrix of its associated homogeneous linear system is invertible.
Thus, the equations of the homogeneous subsystem \eqref{eq: associatedHomogeousSystem} are linearly independent.
Therefore its solution space has dimension $1$,
because there are $2^n - 1$ equations and $2^n$ variables. But, substituting $P(\A^I) = (-1)^{|I|} s$ into \eqref{eq: associatedHomogeousSystem}, where $s \in \mathbb{R}$ is a parameter, we have for any $J \subset [n]$:
\begin{align*}
\sum_{I \supset J} (-1)^{|I|} s &= s \sum_{q = |J|}^n \binom{n - |J|}{q - |J|}(-1)^q\\
&= s (-1)^{|J|} \sum_{q = |J|}^n \binom{n - |J|}{q - |J|}(-1)^{q - |J|} \notag \\
&= s(-1)^{|J|}(1 - 1)^{n - |J|} \\
&= 0,
\end{align*}
where we use the Binomial Theorem in the third equality and the fact that $n - |J| > 0$ for all $J \subset [n]$ in the last equality.
Thus $P(\A^J) = (-1)^{|J|} s$.
\end{proof}
\noindent Recall that a measure $P$ is \emph{unitary} if $P(\Omega) = 1$.
\begin{corollary} \label{cor: characterisationOfUnsignedMeasures}
Every unitary measure $P$ (not necessarily nonnegative) on $(\Omega, \Sigma)$
with $P(A_j) = a_j$ for all $j \in [n]$ and with respect to which $A_1, \dots, A_n$
are $(n - 1)$-wise independent has the form:
\begin{equation} \label{eq:characterisationOfN-1wiseUnsignedMeasures}
P(\A^J) = \a^J + (-1)^{|J|} s, \quad \text{for all } J \subseteq [n],
\end{equation}
for some scalar parameter $s \in \mathbb{R}$.
\end{corollary}
\begin{proof}
Add the particular solution $P(\A^J) = \a^J$
(when the collection of events $\{A_1, \dots, A_n\}$ is mutually independent)
with the general solution of the associated homogenous linear system
given in the previous lemma.
\end{proof}
\noindent To characterise when $P$ in \eqref{eq:characterisationOfN-1wiseUnsignedMeasures} is a valid probability measure, we have to ensure that the following $2^n$ nonnegativity conditions are satisfied:
\begin{equation} \label{eq: systemOfLinearInequalitiesToEnsurePositivity}
P(\A^J)=\a^J + (-1)^{|J|} s \ge 0, \quad \text{for all } J \subseteq [n].
\end{equation}
Simplifying, this system gives:
\begin{equation}
\begin{cases}
s \ge -\a^J, &\text{ for all }J \subseteq [n]:\, |J| \text{ is even}, \\
s \le \a^J, &\text{ for all } J \subseteq [n]:\, |J| \text{ is odd}.
\end{cases}
\end{equation}
Therefore it is a valid probability measure for all values of $s$ that satisfy:
\begin{equation} \label{eq: solveSystemForPositivityOfMeasure}
- \min_{J \subseteq [n]:\, |J| \text{ is even}} \a^J \le s \le \min_{J \subseteq [n]:\, |J| \text{ is odd}} \a^J.
\end{equation}
\noindent From this point onwards, we make the following assumption.
\begin{assumption} \label{ass:order}
The events are ordered by nondecreasing value of their marginal probabilities, i.e. $a_1 \le \cdots \le a_n$.
\end{assumption}
\noindent The next lemma provides a lower bound on $\a^J$ for any set $J \subseteq [n]$, which will be used to establish the precise interval for the parameter $s$ in \eqref{eq: solveSystemForPositivityOfMeasure} and thus to identify all probability measures with respect to which $A_1, \dots, A_n$ are $(n - 1)$-wise independent.
\begin{lemma} \label{lem2}
$\a^J \ge \a^{\initialSubset{|J|}}$ for all $J \subseteq [n]$.
\end{lemma}
\begin{proof}
Use the notation $J \succeq I$ to denote $\a^J \ge \a^I$.
Say $|J| = \ell$. We need to show that $J \succeq \initialSubset{\ell} = \{1,\ldots,\ell\}$. If $J=[\ell]$, we are done.
Otherwise, if $J \neq \initialSubset{\ell}$,
then there is a smallest index $r \le \ell$ that is not in $J$.
Indeed, if there did not exist such an $r$, then the smallest index not in $J$ is strictly greater than $\ell$, hence $J \supset [\ell]$, which contradicts $|J| = \ell$. Hence the smallest index not in $J$, which we denote by $k$, satisfies $k > \ell$.
Let $J^\prime := J \cup \{r\} \setminus \{k\}$
be the set obtained by replacing $k$ with $r$ in $J$;
hence $|J^\prime| = |J| = \ell$ have the same cardinality.
Now, a common factor of $\a^J$ and $\a^{J^\prime}$ is
$C := \prod_{j \in J \setminus \{k\}} a_j
\times \prod_{j \notin J \cup \{r\}} (1 - a_j)$,
so
\begin{equation*}
\a^{J} - \a^{J^\prime} = C \big((1 - a_{r}) a_k - a_{r}(1- a_k) \big)
= C (a_k - a_{r}) \ge 0,
\end{equation*}
since $a_k \ge a_{r}$ because $k > r$. Thus
\begin{equation*}
J \succeq J^\prime.
\end{equation*}
Now, if $J^\prime=[\ell]$, we are done, else, repeating this procedure, we get a finite sequence of subsets, each of cardinality $\ell$ which terminates at $J^{\prime \dots \prime}=[\ell]$ after $|J \setminus [\ell]|$ replacements (since each iteration replaces exactly one element of $J \setminus [\ell]$ with one element of $[\ell] \setminus J$).
\iffalse
Now $|J| = |J^\prime|$ is invariant.
However, since $r \le \ell$ is the smallest index not in $J$,
so $\initialSubset{r - 1} \subset J$ but $\initialSubset{r} \subsetneqq J$.
But $r \in J^\prime$ and $k > r$,
so $\initialSubset{r} \subset J^\prime$.
What this means is that the initial segment of indices
has lengthened as we replace $J$ with $J^\prime$.
Formally, $\max\{k = 1, \dots, n:\, \initialSubset{k} \subset J\}
< \max\{k = 1, \dots, n:\, \initialSubset{k} \subset J^\prime\}$.
Repeating this procedure, we get a sequence of subsets, all of cardinality $\ell$:
\begin{equation*}
J \succeq J^\prime \succeq \cdots \succeq \initialSubset{\ell}.
\end{equation*}
The reason why this sequence terminates is
because the length of the initial segment is strictly increasing,
but the total cardinality is (bounded by) $\ell$.
Thus the last subset is $\initialSubset{\ell}$, the unique subset of cardinality $\ell$,
whose initial segment is equal to itself.
\fi
\end{proof}
\noindent We next define two integer invariants $p$ and $m$ of the ordered sequence of marginal probabilities.
These invariants are used to formulate the sharp lower bound on $\a^J$ for odd $|J|$ and that for even $|J|$ respectively.
Lemma \ref{lem2} is used to obtain these bounds.
Associate to $a_1 \le \cdots \le a_n$ an integer $p \in \{0, 1, \dots, \lfloor (n - 1)/2 \rfloor\}$
defined as:
\begin{multline} \label{eq: defineInvariantForInf}
p \mbox{ is the largest integer such that } \\
a_2 + a_3 \le \cdots \le a_{2 p} + a_{2 p + 1} \le 1.
\end{multline}
\begin{lemma} \label{lem: minimizeAtomicProbabilityForLowerBound}
If $|J|$ is odd, then:
\begin{equation}
\a^J \ge \a^{\initialSubset{2p + 1}},
\end{equation}
where $p$ is defined in \eqref{eq: defineInvariantForInf}.
\end{lemma}
\begin{proof}
Say $|J| = 2q + 1$.
First use Lemma \ref{lem2} to get $\a^J \ge \a^{\initialSubset{2q + 1}}$.
If $q = p$, we are done. Otherwise, either $q < p$ or $q > p$.
\paragraph{Case 1: Suppose $q < p$}
Then, we compare $\a^{\initialSubset{2q + 1}}$ with $\a^{\initialSubset{2q + 3}}$,
which have $C := a_1 \cdots a_{2q + 1} (1 - a_{2q + 4}) \cdots (1 - a_n)$ as a common factor, hence
\begin{align*}
\a^{\initialSubset{2q + 1}} - \a^{\initialSubset{2q + 3}}
&= C\big(
(1 - a_{2q + 3})(1 - a_{2q + 2}) - a_{2q + 3} a_{2q + 2}
\big) \\
&= C\big(
1 - a_{2q + 2} - a_{2q + 3}
\big)\\
&\ge 0,
\end{align*}
since $a_{2q + 2} + a_{2q + 3} \le a_{2p } + a_{2p + 1} \le 1$.
Thus $\a^{\initialSubset{2q + 1}} \ge \a^{\initialSubset{2q + 3}}$. Repeating this process, we get
\begin{equation*}
\a^{\initialSubset{2q + 1}} \ge \a^{\initialSubset{2q + 3}} \ge \cdots \ge
\a^{\initialSubset{2p + 1}}.
\end{equation*}
\paragraph{Case 2: Suppose $q > p$} One can similarly show that $\a^{\initialSubset{2q + 1}} \ge a^{\initialSubset{2q - 1}}$ using $a_{2q} + a_{2q + 1} > 1$ because $p$ is the largest integer such that $a_{2p} + a_{2p + 1} \le 1$.
Repeat the process to get
$\a^{\initialSubset{2q + 1}} \ge \a^{\initialSubset{2q - 1}}
\ge \cdots \ge \a^{\initialSubset{2p + 1}}$.
\end{proof}
\noindent
Associate also to $a_1 \le \cdots \le a_n$ an integer $m \in \{0, 1, \dots, \lfloor n/2\rfloor\}$ defined as:
\begin{multline} \label{eq: defineInvariantForSup}
m \text{ is the largest integer such that } \\
a_1 + a_2 \le \cdots \le a_{2m - 1} + a_{2m} \le 1.
\end{multline}
The proof of the next lemma is similar to the previous lemma
and we omit it.
\begin{lemma} \label{lem: minimizeAtomicProbabilityForUpperBound}
If $|J|$ is even, then:
\begin{equation}
\a^J \ge \a^{\initialSubset{2m}},
\end{equation}
where $m$ is defined in \eqref{eq: defineInvariantForSup}.
\end{lemma}
\noindent This brings us to the following theorem.
\begin{theorem} \label{thm: characterisationOfProbabilityMeasures}
Let $p$ and $m$ be defined as in \eqref{eq: defineInvariantForInf} and \eqref{eq: defineInvariantForSup} respectively where Assumption \ref{ass:order} holds. Then every probability measure $P$ on $(\Omega, \Sigma)$
with $P(A_j) = a_j$ for all $j \in [n]$ and with respect to which $A_1, \dots, A_n$ are $(n - 1)$-wise independent has the form:
$$
P(\A^J) = \a^J + (-1)^{|J|} s,\quad \mbox{for all} \;J \subseteq [n],$$ where $s$ is a scalar parameter satisfying:
\begin{equation} \label{eq: rangeOfFreeParameterToGetPositivity}
-\prod_{i=1}^{2m}a_i\prod_{i=2m+1}^{n}(1-a_i) \leq s \leq
\prod_{i=1}^{2p+1}a_i\prod_{i=2p+2}^{n}(1-a_i).
\end{equation}
\end{theorem}
\begin{proof}
By Corollary \ref{cor: characterisationOfUnsignedMeasures},
the conditions that
$P(A_j) = a_j$ for all $j \in [n]$
and $A_1, \dots, A_n$ are $(n - 1)$-wise independent with respect to $P$ entails that
$
P(\A^J) = \a^J + (-1)^{|J|} s
$ for $J \subseteq [n]$ for some $s \in \mathbb{R}$. In order for $P$ to be a valid probability measure,
$s$ has to satisfy \eqref{eq: solveSystemForPositivityOfMeasure}. This gives:
\begin{align}
s &\in [-\min_{J \subseteq [n]: |J| \text{ is even}} \a^J , \min_{J \subseteq [n]: |J| \text{ is odd}} \a^J] \notag\\
&= [-\a^{\initialSubset{2m}}, \a^{\initialSubset{2p + 1}}],\notag
\end{align}
where the equality follows from Lemma \ref{lem: minimizeAtomicProbabilityForLowerBound} and Lemma \ref{lem: minimizeAtomicProbabilityForUpperBound}.
\end{proof}
\noindent
Previous results in the literature are limited to the construction of specific counterexamples showing that $(n - 1)$-wise independence does not imply the mutual independence of $n$ events (see \cite{Wang,wangstoy}). Theorem \ref{thm: characterisationOfProbabilityMeasures} comprehensively characterizes all such counterexamples.
Note that $s = 0$ corresponds to mutual independence.
\begin{remark}
Let us revisit the constructions given in Example \ref{ex:wang} in view of Theorem \ref{thm: characterisationOfProbabilityMeasures}. The probability measure where $A_n$ occurs given that an even number of events in $A_1, \dots, A_{n - 1}$ occur is given by
$$
P(\A^J) = \frac{1}{2^n} - \frac{(-1)^{|J|}}{2^n}, \quad \text{for all } J \subseteq [n].$$
Since $s = - 1/2^n \in [-1/2^n, 1/2^n]$, it follows from Theorem \ref{thm: characterisationOfProbabilityMeasures}
that the events are $(n - 1)$-wise independent. Note that the events are not mutually independent since $s \neq 0$.
The other construction where $A_n$ occurs given that an odd number of events in $A_1, \dots, A_{n - 1}$ occur is the case of $s = 1/2^n$.
\end{remark}
\begin{proposition} \label{prop1}
Either $m = p$ or $m = p + 1$.
\end{proposition}
\begin{proof}
Let $k$ be the largest integer such that $a_i + a_{i + 1} \le 1$ for all $i \in [k]$.
If $k$ is odd, then $k = 2m - 1$ and $k - 1 = 2 p$,
hence $m = (k + 1)/2 = p + 1$.
On the other hand, if $k$ is even, then $k = 2 p $ and $k - 1 = 2 m - 1$, hence $m = k/2 = p$.
\end{proof}
\section{Probability bounds on at least $k$ events occurring} \label{sec:atleastksharpbounds}
\noindent In this section, we derive sharp bounds on the probability that at least $k$ out of $n$ events that are $(n - 1)$-wise independent occur. Bounds of this type under differing assumptions on the dependence structure of the random events have been studied (see \cite{rusch,boros1989}). Here we provide results for $(n-1)$-wise independence. From Theorem \ref{thm: characterisationOfProbabilityMeasures}, we can represent all such probabilities by:
\begin{align*}
P_{s}(n,k,\a)
&:= \sum_{q \ge k} \sum_{J \subseteq [n]: \, |J|=q} P(\A^J)\\
& = \sum_{q \ge k}^n \sum_{J \subseteq [n]: \, |J|=q} \left(\a^J + (-1)^{|J|} s\right),
\end{align*}
where $s \in [- \a^{\initialSubset{2m}}, \a^{\initialSubset{2p + 1}}]$ .
Then $P_0(n,k,\a)$ is the probability of occurrence of at least $k$ out of $n$ mutually independent events
with the given marginal probabilities $\a$.
We show that $P_{s}(n,k,\a)$ is linear in $s$.
Recall the binomial coefficient given by $\binom{z}{m} = z(z - 1) \cdots (z - m + 1)/m!$ for integers $z \ge 0$ and $n$.
\begin{lemma}\label{lem:atleastkevents}
For any integer $k \ge 0$,
\begin{align}\label{eq:atleastkintermsofalln}
P_{s}(n,k,\a) = P_0(n,k,\a) +(-1)^k\dbinom{n-1}{k-1} s,
\end{align}
where $s \in [- \a^{\initialSubset{2m}}, \a^{\initialSubset{2p + 1}}]$.
\end{lemma}
\begin{proof}
We have:
\begin{align*}
P_{s}(n,k,\a)
&= P_0(n, k, \a) + s \sum_{q \ge k} (-1)^q \sum_{J \subseteq [n] : \, |J| = q} 1 \\
&= P_0(n, k, \a) + s \sum_{q \ge k} (-1)^q \binom{n}{q}
\end{align*}
The result then follows from the combinatorial identity $\sum_{q \ge k} (-1)^{q} \binom{n}{q} = (-1)^k\binom{n-1}{k-1} $.
\end{proof}
We next derive sharp upper and lower bounds on the probability that at least $k$ out of $n$ events occur under $(n-1)$-wise independence.
\begin{theorem}\label{thm:atleastksharpbounds}
Let $p$ and $m$ be defined as in \eqref{eq: defineInvariantForInf} and \eqref{eq: defineInvariantForSup} respectively where Assumption \ref{ass:order} holds. Then every probability measure $P$ on $(\Omega, \Sigma)$
with $P(A_j) = a_j$ for all $j \in [n]$ and with respect to which $A_1, \dots, A_n$ are $(n - 1)$-wise independent satisfies:
\begin{enumerate}[label=\roman*),wide=0pt]
\item For $k$ odd:
\begin{align}
P(n,k,\a) &\ge P_0(n,k,\a) - \binom{n-1}{k-1}\prod_{i=1}^{2p+1} a_i \prod_{i=2p+2}^{n} (1-a_i),
\label{eq: lowerBoundForatleastkProbability2} \\
P(n,k,\a) &\le P_0(n,k,\a) + \binom{n-1}{k-1}\prod_{i=1}^{2m} a_i \prod_{i=2m+1}^{n} (1-a_i), \label{eq: upperBoundForatleastkProbability2}
\end{align}
\item For $k$ even:
\begin{align}
P(n,k,\a) &\ge P_0(n,k,\a) - \dbinom{n-1}{k-1}\prod_{i=1}^{2m} a_i \prod_{i=2m+1}^{n} (1-a_i),
\label{eq: lowerBoundForatleastkProbability1} \\
P(n,k,\a) &\le P_0(n,k,\a) + \dbinom{n-1}{k-1}\prod_{i=1}^{2p+1} a_i \prod_{i=2p+2}^{n} (1-a_i). \label{eq: upperBoundForatleastkProbability1}
\end{align}
\end{enumerate}
Moreover, all the bounds are sharp.
The lower bound for odd $k$ in \eqref{eq: lowerBoundForatleastkProbability2} and the upper bound for even $k$ in \eqref{eq: upperBoundForatleastkProbability1} is uniquely achieved with $P = P_{\a^{\initialSubset{2p + 1}}}$.
The upper bound for odd $k$ is \eqref{eq: upperBoundForatleastkProbability2} and the lower bound for even $k$ in \eqref{eq: lowerBoundForatleastkProbability1} is uniquely achieved with $P = P_{-\a^{\initialSubset{2m}}}$.
\end{theorem}
\begin{proof}
The result is obtained from Lemma \ref{lem:atleastkevents} and optimally selecting $s$ in $[-\a^{\initialSubset{2m}},\a^{\initialSubset{2p + 1}}]$ from Theorem \ref{thm: characterisationOfProbabilityMeasures}.
\end{proof}
\noindent Probability bounds on the occurrence of at least $k$ out of $n$ events that are $\ell$-wise independent (i.e. every $\ell$ out of the $n$ events are mutually independent) have been studied for particular values of $\ell$. The case of $\ell = 1$ represents arbitrary dependence among the random events, for which the sharp upper bound is derived for $k = 1$ in \cite{boole} and for general $k$ in \cite{Ruger}.
At the other extreme is mutual independence ($\ell = n$), where the said probability is unique.
For $\ell = 2$, the sharp upper bound on the probability of the union ($k = 1$) of pairwise independent random events has been recently derived in \cite{ramanatarajan2021pairwise} and new bounds that are not necessarily sharp have been proposed for $k \ge 2$. Further, to the best of our knowledge, sharp bounds for other values of $\ell \in [3,n-1]$ have not been identified in the literature. Our results contribute to this line of work by finding sharp bounds for $l =n-1$.
We next demonstrate, as an immediate implication, the computability of the sharp bounds in Theorem \ref{thm:atleastksharpbounds}.
\begin{theorem} \label{thm:complexity}
The sharp upper and lower bounds in Theorem \ref{thm:atleastksharpbounds} are computable in polynomial time.
\end{theorem}
\begin{proof}
The value of $P_0(n,k,\a) $ is computable in polynomial time using dynamic programming. To see this, let $P_0(r,t,\a)$ denote the probability that at least $t$ events occur out of the first $r$ events where $r \geq t \geq 0$. Then the probabilities satisfy the recursive formula:
\begin{equation*}
P_0(r,t,\a) = P_0(r-1,t-1,\a)p_r + P_0(r-1,t,\a)(1-p_r),
\end{equation*}
where the boundary conditions are $P_0(r,0,\a) = 1$ for $r \geq 0$ and $P_0(r,t,\a) = 0$ for $t > r$ (see \cite{hong2013poissonbinomial}).
The probability $P_0(n,k,\a) $ is hence computable in $O(n^2)$ time. Since the additional term in the formulas \eqref{eq: lowerBoundForatleastkProbability2}-\eqref{eq: upperBoundForatleastkProbability1} is efficiently computable using sorting, evaluating binomial coefficients and multiplication, all the bounds are computable in polynomial time; specifically $O(n^2)$ time.
\end{proof}
\noindent The sharp bounds for $k = 1$ (union of events) and $k = n$ (intersection of events) are detailed next.
\begin{corollary} \label{unionintersect}
Let $p$ and $m$ be defined as in \eqref{eq: defineInvariantForInf} and \eqref{eq: defineInvariantForSup} respectively where Assumption \ref{ass:order} holds. Then every probability measure $P$ on $(\Omega, \Sigma)$
with $P(A_j) = a_j$ for all $j \in [n]$ and with respect to which $A_1, \dots, A_n$ are $(n - 1)$-wise independent satisfies:
\begin{enumerate}[label=\roman*),wide=0pt]
\item For the union of events:
\begin{align}
P(\bigcup_{j = 1}^n A_j) &\ge 1 - \left(\prod_{i=1}^{2p+1}(1-a_i)+\prod_{i=1}^{2p+1}a_i\right)\prod_{i=2p+2}^{n}(1 - a_i),
\label{eq: lowerBoundForUnionProbability} \\
P(\bigcup_{j = 1}^n A_j) &\le 1 - \left(\prod_{i=1}^{2m}(1-a_i)-\prod_{i=1}^{2m}a_i\right)\prod_{i=2m+1}^{n}(1 - a_i), \label{eq: upperBoundForUnionProbability}
\end{align}
\item For the intersection of an even number of events:
\begin{align}
P(\bigcap_{j = 1}^n A_j) &\ge \prod_{i=1}^{2m}a_i\left(\prod_{i=2m+1}^{n}a_i-\prod_{i=2m+1}^{n}(1-a_i)\right),
\label{eq: lowerBoundForIntersectProbabilityeven} \\
P(\bigcap_{j = 1}^n A_j) &\le\prod_{i=1}^{2p+1}a_i\left(\prod_{i=2p+2}^{n}a_i+\prod_{i=2p+2}^{n}(1-a_i)\right),
\label{eq: upperBoundForIntersectProbabilityeven}
\end{align}
\item For the intersection of an odd number of events:
\begin{align}
P(\bigcap_{j = 1}^n A_j) &\ge \prod_{i=1}^{2p+1}a_i\left(\prod_{i=2p+2}^{n}a_i-\prod_{i=2p+2}^{n}(1-a_i)\right),
\label{eq: lowerBoundForIntersectProbabilityodd} \\
P(\bigcap_{j = 1}^n A_j) &\le \prod_{i=1}^{2m}a_i\left(\prod_{i=2m+1}^{n}a_i+\prod_{i=2m+1}^{n}(1-a_i)\right)\label{eq: upperBoundForIntersectProbabilityodd}.
\end{align}
\end{enumerate}
Each of these bounds is sharp and is achieved by a unique probability measure $P(\A^J) = \a^J + (-1)^{|J|}s$, where either $s =-\a^{\initialSubset{2m}}$ or
$s =\a^{\initialSubset{2p + 1}}$.
\end{corollary}
\begin{proof}
With $k=1$, we have $P_0(n,1,\a)=1-\prod_{i=1}^n (1-a_i)$ and the result immediately follows from \eqref{eq: lowerBoundForatleastkProbability2} and \eqref{eq: upperBoundForatleastkProbability2} in Theorem \ref{thm:atleastksharpbounds}. With $k=n$, we have $P_0(n,n,\a)=\prod_{i=1}^n a_i$ and the result immediately follows from Theorem \ref{thm:atleastksharpbounds} depending on whether $k = n$ is even or odd.
\end{proof}
\section{Examples}\label{sec:examples}
\noindent In this section, we discuss several examples to illustrate the connection of the newly proposed bounds with existing bounds and provide numerical evidence of the quality of the bounds.
\begin{example} [Bounds for $n = 3$ pairwise independent events]\label{ex:reductiontopairwise}
For $n=3$ events, $(n-1)$-wise independence is pairwise independence.
In this case from \eqref{eq: defineInvariantForSup} and \eqref{eq: upperBoundForUnionProbability} where $a_1 \leq a_2 \leq a_3$, we obtain the sharp upper bound on the union as:
\begin{equation} \label{eq: sharpUpperBoundForThreeEvents}
P(\bigcup_{j = 1}^3 A_j)
\le \min\left(a_1+a_2+a_3-a_3(a_1+a_2),1\right)
\end{equation}
Another proof of the sharpness was given in \cite{ramanatarajan2021pairwise}.
Kounias \cite{kounias} showed that every probability measure
satisfies $P(\bigcup_{j = 1}^3 A_j) \le \min\left(a_1+a_2+a_3-a_{31}+a_{32}),1\right)$, where $a_{ij} := P(A_i \cap A_j)$ for $i, j \in [3]$ denotes the bivariate joint probability.
Therefore, \eqref{eq: sharpUpperBoundForThreeEvents} entails that the upper bound of Kounias is achieved by some probability measure with respect to which $A_1, A_2, A_3$ are pairwise independent.
For the sharp lower bound, from \eqref{eq: defineInvariantForInf} and \eqref{eq: lowerBoundForUnionProbability}, we get:
\begin{multline*}
P(\bigcup_{j = 1}^3 A_j)
\ge \max(a_2+a_3-a_2a_3, \\
a_1+a_2+a_3-a_1a_2-a_1a_3-a_2a_3),
\end{multline*}
Similarly, a corresponding universal lower bound of Kounias in terms of bivariate joint probabilities is therefore achievable under pairwise independence.
Likewise for the intersection of three pairwise independent events, we can verify that the sharp bounds are given as:
\begin{equation*}
P(\bigcap_{j = 1}^3 A_j) \le \min(a_1a_2,
(1 - a_1)(1 - a_2)(1 - a_3) + a_1 a_2 a_3),
\end{equation*}
and
\begin{equation*}
P(\bigcap_{j = 1}^3 A_j) \ge \max\left(a_1(a_2+a_3-1),0\right).
\end{equation*}
An alternative proof of the sharpness of the lower bound is given in \cite{ramanatarajan2021pairwise}.
\end{example}
\begin{example} [Bonferroni bounds]\label{ex:reductiontobonferroni}
Suppose the sum of the two largest marginal probabilities satisfies $a_{n-1} + a_{n} \leq 1$ and $n$ is even. Then $m=n/2$ in \eqref{eq: defineInvariantForSup}, hence from \eqref{eq: upperBoundForUnionProbability}, we get the sharp upper bound on the union:
\begin{align*}
P(\bigcup_{j = 1}^n A_j) &\le 1 - \prod_{i=1}^{n}(1 - a_i) + \prod_{i=1}^{n}a_i \\
&= \sum_{k = 0}^{n - 2} (-1)^k \sum_{1 \le i_0 < \cdots < i_k \le n} a_{i_0} \cdots a_{i_k}.
\end{align*}
Bonferroni \cite{Bonferroni} showed that every probability measure
satisfies
\begin{equation}
P(\bigcup_{j = 1}^n A_j) \le \sum_{k = 0}^{n - 2} (-1)^k \sum_{1 \le i_0 < \cdots < i_k \le n} a_{i_0 \cdots i_k},
\end{equation}
where $a_{i_0 \cdots i_k} := P(A_{i_0} \cap \cdots \cap A_{i_k})$ for $i_0, \dots, i_k \in [n]$ is the joint probability.
Therefore the Bonferroni upper bound is achieved by some probability measure with respect to which $A_1, \dots, A_n$ are $(n - 1)$-wise independent, in this case.
Similarly if $a_{n - 1} + a_n \le 1$ and $n$ is odd,
then $p = (n - 1)/2$ in \eqref{eq: defineInvariantForInf} and thus the sharp lower bound in \eqref{eq: lowerBoundForUnionProbability} becomes:
\begin{align*}
P(\bigcup_{j = 1}^n A_j) &\ge 1 - \prod_{i=1}^{n}(1 - a_i) - \prod_{i=1}^{n}a_i \\
&= \sum_{k = 0}^{n - 2} (-1)^k \sum_{1 \le i_0 < \cdots < i_k \le n} a_{i_0} \cdots a_{i_k}.
\end{align*}
Again, a corresponding lower bound of Bonferroni in terms of joint probabilities of up to $n - 1$ events is thus achievable under $(n - 1)$-wise independence.
\end{example}
The next example shows the connection of the bound to the probabilistic method which has proved to be very useful tool in combinatorics (see \cite{alon}).
\begin{example} [Probabilistic method]\label{ex:lovaszlocallemma} Suppose there are $n$ random ``bad'' events, each of which occurs with probability $a_j$ for $j \in [n]$. When the events are mutually independent, the probability of no bad event occurring is strictly positive when the probability of each bad event is strictly less than 1 (namely $\max_j a_j < 1$).
On the other hand, if the events can be arbitrarily dependent, from Boole's union bound \cite{boole}, the sum of the probabilities must be strictly less than 1 (namely $\sum_j a_j < 1$) to guarantee the same. The Lov\'{a}sz local lemma \cite{erdos} is a powerful tool that allows one to relax the assumption of mutual independence to weak dependence while allowing for the probability of each bad event to be fairly large and still guarantee that no bad event occurs with strictly positive probability. Specifically consider a graph $G$ on $n$ nodes where each node $i \in [n]$ is associated with an event $A_i$ and $A_i$ is independent of the collection of events $\{A_j: (i,j) \notin G\}$ for each $i \in [n]$. If $G$ has maximum degree $d$ and $\max_i a_i \leq 1/4d$, then the probability of no bad event occurring satisfies (see \cite{erdos,tetali}):
\begin{align}
P(\bigcap_{j = 1}^n \overline{A}_j)
\ge \prod_{i=1}^{n}(1-2a_i) > 0.\label{eq:lovaszlower}
\end{align}
Computing the tightest lower bound in terms of the dependency graph is known to be NP-complete \cite{shearer}. More generally, in \cite{shearer} it was shown that for $d \geq 2$, $\max_i a_i < (d-1)^{d-1}/d^d$ and for $d = 1$, $\max_i a_i < 1/2$ guarantees that there is a strictly positive probability that no bad event occurs. For the specific case of $d=1$, we can compare our results with the lower bound as shown next (although the Lov\'{a}sz local lemma holds more generally for lesser independence with $d \ge2 $). When the events are $(n-1)$-wise independent, using \eqref{eq: upperBoundForUnionProbability}, the probability that none of the events occur is strictly positive if $a_n< 1$ and $a_1+a_2 < 1$.
Indeed, then $(1 - a_1)(1 - a_2) > a_1 a_2$ and $(1 - a_{2k - 1})(1 - a_{2k}) \ge a_{2k - 1} a_{2k}$ for all $k \in \{2, \dots, m\}$. Hence $\prod_{i=1}^{2m}(1-a_i) = \prod_{k = 1}^{m}((1 - a_{2k - 1})(1 - a_{2k})) > \prod_{k = 1}^{m} (a_{2k - 1} a_{2k}) = \prod_{i=1}^{2m}a_i$ and from \eqref{eq: upperBoundForUnionProbability}:
$$
P(\bigcap_{j = 1}^n \overline{A}_j)
\ge \left(\prod_{i=1}^{2m}(1-a_i)-\prod_{i=1}^{2m}a_i\right)\prod_{i=2m+1}^{n}(1 - a_i)
> 0.
$$
\noindent When all the marginal probabilities $a_1=\ldots=a_n= a$, are identical, the condition $a_1 + a_2 < 1$ gives $a < 1/2$ which exactly corresponds to the condition identified in \cite{spencer,shearer} for $d = 1$. It is easy to verify that the lower bound on the probability of no bad event occurring in this case is given by $(1-a)^n-a^n$ for $n$ even and $(1-a)^n-a^{n-1}(1-a)$ for $n$ odd which is the sharp lower bound instance wise. In comparison, the lower bound identified above in \eqref{eq:lovaszlower} is $(1-2a)^n$. For example with $n = 6$ and $a = 0.1$, the sharp lower bound is $0.53144$ while the weaker lower bound is $0.262144$.
In fact for $a = 1/2$, the first construction in Example \ref{ex:wang} has a zero probability that no bad event occurs since $A_n$ must occur when none of the events in $\{A_1,A_2,\ldots A_{n-1}\}$ occur.
\end{example}
\noindent We next provide a numerical example to illustrate the performance of the bounds in Theorem \ref{thm:atleastksharpbounds} and compare it with an existing bound.
Specifically tail probability bounds on the sum of two random variables given their marginal distribution functions were derived by Makarov in \cite{makarov1982}. We adopt these closed-form bounds also known as ``standard" bounds in our context as follows.
Given that $n$ random events $A_1,\ldots A_n$ with respective marginal probabilities $a_1 \le \cdots \le a_n$ are $(n - 1)$-wise independent, define two random variables as follows: $Y_1=\sum_{i=1}^{n-1} \mathbb{1}_{A_i},\;Y_2=\mathbb{1}_{A_n}$
where $ \mathbb{1}_{A}$ is the indicator function of event $A$ occurring. Here $ Y_1 \sim \operatorname{PoissonBinomial}(n-1,a_1,a_2,\ldots a_{n-1})$ is an integer random variable taking values in $[0,n-1]$ while $Y_2 \sim \operatorname{Bernoulli}(a_n)$.
Let $F_1$ and $F_2$ be the resepective distribution functions of $Y_1$ and $Y_2$. Then the Makarov upper bound for the probability that the sum of $Y_1$ and $Y_2$ is at least an integer $k \in [n]$ is given from \cite{makarov1982,rusch} as follows:
\begin{equation}\label{eq: makarovupper}
\begin{array}{rll}
P(Y_1+Y_2\ge k)& \leq
& \min(2-(F_{1}\vee F_{2})^{-}(k),1),
\end{array}
\end{equation}
where $(F_{1}\vee F_{2})^{-}(k)=\underset{u \in \mathbb{R}}{\max}(F_{1}(k-u)^{-} +F_{2}(u))$ is the left continuous version of the supremum convolution $F_{1}\vee F_{2}$. Since $Y_2$ is a Bernoulli random variable, it is sufficient to maximize over $u \in \{0,1\}$ and thus we have :
\begin{align*}
& (F_{1}\vee F_{2})^{-}(k)
& =&\max(F_{1}(k-1) +a_n,\;F_{1}(k-2) +1).
\end{align*}
The Makarov lower bound can be similarly derived as
\begin{equation}\label{eq: makarovlower}
\begin{array}{rll}
P(Y_1+Y_2\ge k)
\ge\max(1-\min(F_{1}(k),F_{1}(k-1) +a_n),0).
\end{array}
\end{equation}
We next illustrate through a numerical example that the Makarov bound is not sharp in general under $(n-1)$-wise independence since we lose out on using additional independence information available in our context. For example, our bounds assume that any $n-2$ events from the first $n-1$ events $A_1,\ldots,A_{n-1}$ along with the last event $A_n$ are mutually independent while the Makarov bounds do not assume so.
\begin{example} [Numerical example] \label{ex:comparisonmakarov} Here we compute the exact probability for $n = 8$ with identical marginal probabilities $a_i = a \in \{0.1,0.2,0.3,0.4,0.5\}$ for different values of $k$ assuming mutual independence. In addition we compute the sharp lower and upper bounds with $7$-wise independence from Theorem \ref{thm:atleastksharpbounds} (here $p = 3$ and $m = 4$ for all considered values of $a$). We also provide the Makarov lower and upper bounds from (\ref{eq: makarovlower}) and (\ref{eq: makarovupper}) to highlight that if more information is known on the independence of the random variables, we can exploit it tightening the bounds.
\begin{table}[H]
\footnotesize
\caption{$k=1$ to $k=4$ - For each value of $a$, the first row provides the Makarov lower bound from \eqref{eq: makarovlower}, the second row provides the sharp lower bound with $7$-wise independence, the third row provides the exact value with $8$ mutually independent events, the fourth row provides the sharp upper bound with $7$-wise independence and the fifth row provides the Makarov upper bound from \eqref{eq: makarovupper}} \label{tab:heteroall3vstight1}
\begin{center}
\scriptsize{\begin{tabular}
{|l|c|c|l|l|l|l|l|l|l|}
\hline
\mbox{a} & $k = 1$ & $k = 2$ & $k = 3$ & $k = 4$ \\ \hline
0.1 & 4.6953e-01 & 8.6895e-02 & 5.0243e-03 & 4.3165e-04\\
0.1 & 5.6953e-01 & 1.8690e-01 & 3.8090e-02 & 5.0240e-03\\
0.1 & 5.6953e-01 & 1.8690e-01 & 3.8092e-02 & 5.0244e-03\\
0.1 & 5.6953e-01 & 1.8690e-01 & 3.8092e-02 & 5.0275e-03\\
0.1 & 1.0000e+00 & 5.6953e-01 & 1.8690e-01 & 3.8092e-02\\
\hline
0.2 & 6.3223e-01 & 2.9668e-01 & 5.6282e-02 & 1.0406e-02\\
0.2 & 8.3222e-01 & 4.9667e-01 & 2.0287e-01 & 5.6192e-02\\
0.2 & 8.3223e-01 & 4.9668e-01 & 2.0308e-01 & 5.6282e-02\\
0.2 & 8.3223e-01 & 4.9676e-01 & 2.0314e-01 & 5.6640e-02\\
0.2 & 1.0000e+00 & 8.3223e-01 & 4.9668e-01 & 2.0308e-01\\
\hline
0.3 & 7.4470e-01 & 4.4823e-01 & 1.9410e-01 & 5.7968e-02\\
0.3 & 9.4220e-01 & 7.4424e-01 & 4.4501e-01 & 1.9181e-01\\
0.3 & 9.4235e-01 & 7.4470e-01 & 4.4823e-01 & 1.9410e-01\\
0.3 & 9.4242e-01 & 7.4577e-01 & 4.4960e-01 & 1.9946e-01\\
0.3 & 1.0000e+00 & 9.4235e-01 & 7.4470e-01 & 4.4823e-01\\
\hline
0.4 & 8.9362e-01 & 6.8461e-01 & 4.0591e-01 & 1.7367e-01\\
0.4 & 9.8222e-01 & 8.8904e-01 & 6.6396e-01 & 3.8298e-01\\
0.4 & 9.8320e-01 & 8.9362e-01 & 6.8461e-01 & 4.0591e-01\\
0.4 & 9.8386e-01 & 9.0051e-01 & 6.9837e-01 & 4.4032e-01\\
0.4 & 1.0000e+00& 9.8320e-01 & 8.9362e-01 & 6.8461e-01\\
\hline
0.5 & 9.6484e-01 & 8.5547e-01 & 6.3672e-01 & 3.6328e-01\\
0.5 & 9.9219e-01 & 9.3750e-01 & 7.7344e-01 & 5.0000e-01\\
0.5 & 9.9609e-01 & 9.6484e-01 & 8.5547e-01 & 6.3672e-01\\
0.5 & 1.0000e+00 & 9.9219e-01 & 9.3750e-01 & 7.7344e-01\\
0.5 & 1.0000e+00 & 9.9610e-01 & 9.6484e-01 & 8.5547e-01\\
\hline
\end{tabular}}
\end{center}
\end{table}
\begin{table}[H]
\footnotesize
\caption{$k=5$ to $k=8$} \label{tab:heteroall3vstight2}
\begin{center}
\scriptsize{\begin{tabular}[t]
{|l|c|c|l|l|l|l|l|l|l|}
\hline
\mbox{a} & $k = 5$ & $k = 6$ & $k = 7$ & $k = 8$ \\ \hline
0.1 & 2.3410e-05 & 7.3000e-07 & 9.9999e-09 & 0.0000e+00\\
0.1 & 4.2850e-04 & 2.3200e-05 & 1.0000e-07 & 0.0000e+00\\
0.1 & 4.3165e-04 & 2.3410e-05 & 7.3000e-07 & 1.0000e-08\\
0.1 & 4.3200e-04 & 2.5300e-05 & 8.0000e-07 & 1.0000e-07\\
0.1 & 5.0244e-03 & 4.3165e-04 & 2.3410e-05 & 7.3000e-07\\
\hline
0.2 & 1.2314e-03 & 8.4480e-05 & 2.5600e-06 & 0.0000e+00\\
0.2 & 1.0048e-02 & 1.1776e-03 & 1.2800e-05 & 0.0000e+00\\
0.2 & 1.0406e-02 & 1.2314e-03 & 8.4480e-05 & 2.5600e-06\\
0.2 & 1.0496e-02 & 1.4464e-03 & 1.0240e-04 & 1.2800e-05\\
0.2 & 5.6282e-02 & 1.0406e-02 & 1.2314e-03 & 8.4480e-05\\
\hline
0.3 & 1.1292e-02 & 1.2903e-03 & 6.5610e-05 & 0.0000e+00\\
0.3 & 5.2610e-02 & 9.9144e-03 & 2.1870e-04 & 0.0000e+00\\
0.3 & 5.7968e-02 & 1.1292e-02 & 1.2903e-03 & 6.5610e-05\\
0.3 & 6.0264e-02 & 1.4507e-02 & 1.7496e-03 & 2.1870e-04\\
0.3 & 1.9410e-01 & 5.7968e-02 & 1.1292e-02 & 1.2903e-03\\
\hline
0.4 & 4.9807e-02 & 8.5200e-03 & 6.5536e-04 & 0.0000e+00\\
0.4 & 1.3926e-01 & 3.6045e-02 & 1.6384e-03 & 0.0000e+00\\
0.4 & 1.7367e-01 & 4.9807e-02 & 8.5197e-03 & 6.5536e-04\\
0.4 & 1.9661e-01 & 7.0451e-02 & 1.3107e-02 & 1.6384e-03\\
0.4 & 4.0591e-01 & 1.7367e-01 & 4.9807e-02 & 8.5197e-03\\
\hline
0.5 & 1.4453e-01 & 3.5156e-02 & 3.9063e-03 & 0.0000e+00\\
0.5 & 2.2656e-01 & 6.2500e-02 & 7.8125e-03 & 0.0000e+00\\
0.5 & 3.6328e-01 & 1.4453e-01 & 3.5156e-02 & 3.9063e-03\\
0.5 & 5.0000e-01 & 2.2656e-01 & 6.2500e-02 & 7.8125e-03\\
0.5 & 6.3672e-01 & 3.6328e-01 & 1.4453e-01 & 3.5156e-02\\
\hline
\end{tabular}}
\end{center}
\end{table}
\noindent As it can be observed, the sharp bounds with $(n-1)$-wise independence clearly improve upon the Makarov bounds, especially as $k$ increases (for the same $a$) and $a$ decreases (for the same $k$), where the bounds can be a couple or more magnitude of orders apart. In other words, the sharp bounds especially provide value in the regime where the right tail probabilities are more constrained i.e. large $k$ and small $a$. Such bounds are useful in providing robust estimates of the probabilities when the assumption of mutual independence breaks down.
\end{example}
\subsection*{Acknowledgements}
\noindent The research of the first and third authors was partly supported by MOE Academic Research Fund Tier 2 grant T2MOE1906, ``Enhancing Robustness
770 of Networks to Dependence via Optimization''. The authors would like to thank the Area Editor Henry Lam, the Associate Editor and the anonymous reviewer for valuable comments.
|
1,108,101,564,571 | arxiv | \section{Introduction}
\label{sec:intro}
\begin{figure*}
\begin{center}
\begin{tabular}{cc}
\includegraphics[clip,width=0.4\textwidth]{fig01a.pdf} \hspace{1.0cm} &
\includegraphics[clip,width=0.4\textwidth]{fig01b.pdf}\tabularnewline
\end{tabular}
\end{center}
\caption{Examples of the magnetospheric accretions in a stable (left panel) and an
unstable (right panel) regimes. The background colours show the volume rendering of the density (in
logarithmic scales and in arbitrary units). The sample magnetic
field lines are shown as red lines.}
\label{stable-unstable}
\end{figure*}
Classical T Tauri stars (CTTSs) show variability in their light curves
on time-scales from seconds up to decades (e.g.~ \citealt{herbst02},
\citealt{rucin08}). The origin of variability is likely different for
different time-scales. The long-term variability may be connected with
the viscous evolution of the disc (e.g.~\citealt{spruit93}). The
variability on the time-scales of a few stellar rotations may be
connected with the inflation and closing of the field lines of the
stellar magnetosphere (e.g.~\citealt{aly-kuijpers90};
\citealt*{goodson97}; \citealt{lovelace1995}), which is supported by
the observations of AA~Tau \citep{bouvier07}. The variability on the
time-scale of a stellar rotation are most likely connected with the
rotation of a star, or with the obscuration of a stellar surface by a
large-scale warped disc adjacent to the magnetosphere, which corotates
with the star (e.g.~\citealt{bouvier03,alencar10,roma13}). The
variability on even smaller time-scales may be connected with the
accretion of individual turbulent cells formed in an accretion disc
driven by magnetorotatinal instability (MRI)
(e.g.~\citealt{hawley00,stone2000,roma12}).
Another interesting possibility is that matter may accrete on to the
star as a result of the magnetic Rayleigh-Taylor (RT) instability
(e.g.~\citealt{arons76,spruit93,li:2004}). Global three-dimensional
(3D) magnetohydrodynamic (MHD) simulations (e.g.~\citealt*{roma08};
\citealt{kulk08,kulk09}) show that accretion proceeds through
several unstable `tongues', and the expected time-scale of
the variability induced by the instability is a few times smaller than
the rotation period of a star. The variability on the time-scale less
than a second is also expected if the interaction of the funnel stream
with the surface of the star leads to unstable radiative shocks which
oscillate in a very short time-scale
(e.g.~\citealt{kold08,sacco2010}). Finally, the variability can be
also caused by magnetic flaring activities, which may occur on
different time-scales (e.g.~\citealt{herbst94,wolk:2005,feigelson07}).
Here, we concentrate on the variability associated with the unstable accretion caused
by the RT instability, and study spectral properties of stars accreting in this
regime. The global 3D numerical simulations performed earlier
(e.g.~\citealt{roma08,kulk08}) show that randomly-forming tongues produce temporary
hot spots on the stellar surface with irregular shapes and positions, and that the
corresponding light curves are very irregular. Such irregular light-curves are
frequently observed in CTTSs (e.g.~\citealt{herbst94,rucin08,alencar10}). However, the
irregular light curve can be also produced, for example, by flaring activities. To
distinguish the different mechanisms of forming irregular variability, we perform
time-dependent modelling and analysis of emission line profiles of hydrogen.
To calculate the time-dependent line profiles from a CTTS accreting through the RT
instability, we first calculate the matter flows in a global 3D
MHD simulation with frequent writing of data. The fine time-slices of the MHD
simulation data are then used as the input of the separate 3D radiative code
\textsc{torus} (e.g.~\citealt{harries2000, harries2012, kurosawa04};
\citealt*{kurosawa05,kurosawa06}; \citealt*{symington2005}; \citealt*{kurosawa11};
\citealt{kurosawa12}). This allows us to follow the time evolution of the line
profiles that occurs as a consequence of the dynamical nature of the accretion flows
in the `unstable regime'. In earlier studies, we have used a similar procedure (of
3D+3D modelling) to calculate line spectra from a modelled star accreting in a `stable
regime' (using only a single time slice of MHD simulations) in which the gas accretes
in two ordered funnel streams \citep{kurosawa08}. More recently, we have performed a
similar modelling for a star with realistic stellar parameters (i.e.~V2129~Oph), and
have found a good agreement between the model and the time-series observed line
profiles \citep{alencar12}. This example has shown that the combination of the 3D MHD
and 3D radiative transfer (3D+3D) models is a useful diagnostic tool for studying
magnetospheric accretion processes. Examples of the magnetospheric accretion flows in
both stable and unstable regimes are shown in Fig.~\ref{stable-unstable}.
In this paper, we use a similar 3D+3D modelling method as in the
earlier studies, but will focus on the CTTSs accreting in the
`unstable regime'. A model accretion in a stable regime is also
presented as a comparison purpose. In the previous works, which
focused on the stable regime, we have fixed the magnetospheric
accretion at some moment of time (a single time slice of MHD
simulations) to calculate line profiles; however, in the current work,
we will use multiple moments of time from MHD simulations to
investigate variability of hydrogen lines caused by the dynamical
changes in the accretion flows. In particular, we focus on the
variability phenomenon in the time-scale comparable with or less than
a rotational period.
In Section~\ref{sec:mhd-rad}, we describe numerical simulations of
stable and unstable accretion flows along with our numerical methods
used in the MHD and radiative transfer models. The model results are
presented in Section~\ref{sec:results}. In Section
\ref{sec:observations}, the model results are compared with
observations. The discussion of the dependency of our models on
various model parameters is presented in Section~\ref{sec:modelpar}.
Our conclusions are summarized in
Section~\ref{sec:conclusions}. Finally, some additional plots are
given in Appendix~\ref{sec:appendix} and in Appendix~B in the online
supporting information.
\section{Model Description}
\label{sec:mhd-rad}
To investigate variable line spectra from CTTSs in stable and unstable
regimes of accretion, we perform numerical modelling in two
steps. Firstly, we calculate the magnetospheric accretion flows using
3D MHD simulations and store the data at different stellar rotational
phases. Secondly, we calculate hydrogen line profiles from the
accretion flows, using the 3D radiative transfer code. For all the MHD
and line profile models presented in this work, we adopt stellar
parameters of a typical CTTS, i.e.~its stellar radius
$R_{*}=2.0\,\mathrm{R_{\sun}}$ and its mass $M_{*}=0.8\,\mathrm{M_{\sun}}$.
Below, we briefly describe both models.
\subsection{MHD models}
\label{sec:mhd-rad:mhd}
We calculate matter flows around a rotating star with a dipole magnetic field with the
magnetic axis tilted from the rotational axis by an angle $\Theta$. The rotational
axis of the star coincides with that of the accretion disc. A star is surrounded by a dense cold
accretion disc and a hot low-density corona above and below the disc. Initially, a
pressure balance is present between the disc and corona, and the gravitational,
centrifugal and gas pressure forces are in balance in the whole simulation region
\citep{roma02}. The dipole field is strong enough to truncate the disc at a few
stellar radii. To calculate matter flow, we solve a full set of magnetohydrodynamic
equations (in 3D), using a Godunov-type numerical code (see~\citealt{kold02, roma03,
roma04}, hereafter ROM04). The equations are written in the coordinate system that is
corotating with the star. The magnetic field is split into the dipole component
($B_\mathrm{d}$) which is fixed and the component induced by currents in the
simulation region ($B'$) which is calculated in the model. A viscosity term has been
added to the code with a viscosity coefficient proportional to the $\alpha$ parameter
(e.g.~\citealt{shakura:1973}). The viscosity helps to form a quasi-stationary
accretion in the disc. Our numerical code uses the `cubed sphere' grid \citep{kold02},
and the Riemann solver used in the code is similar to that in \citet{powell99}. This
code has been used for modelling the magnetospheric flows in stable and unstable
regimes of accretion in the past (\citealt{roma03}; ROM04; \citealt{roma08,kulk08}).
It has been also used for modelling accretions to stars with a complex magnetic
field (e.g.~\citealt*{long07,long08}; \citealt{long2011,roma11}).
In this work, we study the simulations performed for the stable and unstable regimes
of accretion (Fig.~\ref{stable-unstable}).
Based on the previous numerical simulations of magnetospheric
accretions on to neutron stars by \citet{kulk08} (see also
\citealt{kulk05}), we select two sets of model parameters that are known
to produce a stable and a strongly unstable regimes. However, we adjust
some models parameters such that they become more suitable for a
typical CTTS, and perform new MHD simulations.
To keep a consistency, we adopt the same grid resolutions used in many of our earlier
simulations \citep{roma08, kulk08, kulk09}, i.e.~$N_r\times N^2=72\times 31^2$ grid
points in each of six blocks of the cubed sphere grid, which approximately corresponds
to $N_r\times N_\theta\times N_\phi=72\times 62 \times 124$ grid points in the
spherical coordinates.
The boundary conditions used here are similar to those in \citet{roma03} and ROM04. At
the stellar surface, `free' {[}$\partial(\cdots)/\partial
r=0${]} boundary conditions to the density and pressure are applied. A
star is treated as a perfect conductor so that the normal component of the magnetic
field does not vary in time. A `free' condition is applied to the azimuthal component
of the poloidal current: ${\partial(r B_\phi)}/{\partial r}=0$ such that the
magnetic field lines have a `freedom' to bend near the stellar surface. In the
reference frame that is corotating with the star, the flow velocity is adjusted to be
parallel to the magnetic field $\boldsymbol{B}$ on the stellar surface ($r=R_*$),
which corresponds to a frozen-in condition. The gas falls on to the surface of the
star supersonically, and most of its kinetic energy is expected to be radiated away in
the shock near the surface of the star (e.g.~\citealt{camenzind:1990};
\citealt{koenigl:1991}; \citealt{calvet98}). The evolution of the radiative shock
above the surface of CTTSs has been studied in detail by \citet{kold08}. In this
study, we assume all the kinetic energy of the flow is converted to thermal radiation
(see ROM04 and Section~\ref{sub:mhd-rad:rad}) at the stellar surface. At the outer
boundary ($r=R_{\rm max}$), free boundary conditions are applied for all variables.
The key model parameters in our MHD simulations are: the stellar mass
($M_{*}$), stellar radius ($R_{*}$), surface magnetic field on the
equator ($B_{\mathrm{eq}}$), corotation radius ($r_{\mathrm{cor}}$),
stellar rotation period ($P_{*}$), tilt angle of dipole magnetic field
with respect the disc axis ($\Theta$) and $\alpha$ (viscosity)
parameter. The parameter values adopted for the simulations in both stable
and unstable regimes are summarized in Table~\ref{tab:refval}.
For more details on the simulation method and on the difference between the stable and
unstable regimes, readers are referred to \citet{roma03}, ROM04,
\citet{roma08} and \citet{kulk08}.
Although we have adopted a relatively large viscosity
coefficient $\alpha=0.1$ for the model in the unstable regime
(Table~\ref{tab:refval}), to be
consistent with our earlier models (\citealt{roma08, kulk08, kulk09}),
our recent simulations have shown that the instability also develops
with a smaller viscosity (e.g.~$\alpha=0.02$--$0.04$). More detailed discussion
on the dependency of our model on different model parameters is
presented in Section~\ref{sec:modelpar}.
\begin{table}
\begin{tabular}{lccccccc}
\hline
& $M_{*}$ & $R_{*}$ & $B_{\mathrm{eq}}$ & $r_{\mathrm{cor}}$ & $\mathrm{P_{*}}$ & $\Theta$ & $\alpha$\tabularnewline
& $\left(\mathrm{M_{\sun}}\right)$ & $\left(\mathrm{R_{\sun}}\right)$ & $\left(\mathrm{G}\right)$ & $\left(R_{*}\right)$ & $\left(\mathrm{d}\right)$ & $\left(-\right)$ & $\left(-\right)$\tabularnewline
\hline
Stable & $0.8$ & $2$ & $10^{3}$ & $5.1$ & $4.3$ & $30^{\circ}$ & $0.02$\tabularnewline
Unstable & $0.8$ & $2$ & $10^{3}$ & $8.6$ & $9.2$ & $5^{\circ}$ & $0.1$\tabularnewline
\hline
\end{tabular}
\caption{Basic model parameters used for the stable and unstable regimes of accretions.}
\label{tab:refval}
\end{table}
\subsection{Radiative transfer models}
\label{sub:mhd-rad:rad}
For the calculations of hydrogen emission line profiles from the
matter flow in the MHD simulations (Section~\ref{sec:mhd-rad:mhd}), we
use the radiative transfer code \textsc{torus}
(e.g.~\citealt{harries2000, harries2012, kurosawa06, kurosawa11,
kurosawa12}). In particular, the numerical method used in the
current work is essentially identical to that in \citet{kurosawa11}; hence, for more
comprehensive descriptions of our method, readers are referred to the
earlier papers. In the following, we briefly summarize some
important aspects of our line profile models.
The basic steps for computing the line variability are as follows:
(1)~mapping the MHD simulation data on to the radiative transfer grid,
(2)~source function ($S_{\nu}$) calculations, and (3)~observed
line profile calculations as a function of rotational phase. In
step~(1), we use an adaptive mesh refinement (AMR) which allows for an
accurate mapping of the original MHD simulation data onto the
radiative transfer grid. The density and velocity values from the MHD
simulations are mapped here, but the gas temperatures are assigned
separately (see below). In step~(2), we use a method similar to that
of \citet{klein78} (see also \citealt{rybicki78}; \citealt*{hartmann94}) in which
the Sobolev approximation (e.g.~\citealt{sobolev1957,castor1970}) is
applied. The populations of the bound states of hydrogen are assumed
to be in statistical equilibrium, and the continuum sources are the
sum of radiations from the stellar photosphere and the hot spots
formed by the funnel accretion streams merging on the stellar surface.
For the photospheric contribution to the continuum flux, we adopt the
effective temperature of photosphere $T_{\mathrm{ph}}=4000\,\mathrm{K}$
and the surface gravity $\log g_{*}=3.5$ (cgs), and use the model
atmosphere of \citet{kurucz1979}. The sizes and shapes of the hot
spots are determined by the local energy flux on the stellar surface
(see Section~\ref{sec:results:continuum} for more detail).
Our hydrogen model atom consists of 20
bound and a continuum states. In step~(3), the line profiles are
computed using the source function computed in step~(2). The observed
flux at each frequency point in line profiles are computed using the
cylindrical coordinate system with its symmetry axis pointing toward an
observer. The viewing angles of the system (the central star and the
surrounding gas) are adjusted according to the rotational phase of the
star and the inclination angle of the system for each time-slice of
the MHD simulations. For both stable and unstable cases, the line
profiles are computed at 25 different phases per stellar rotation.
We follow the time evolution of the line profiles for about 3 stellar
rotations; therefore, in total we compute 75 profiles for a given line
transition.
The gas temperatures in the accretion funnels from the MHD simulations
are in general too high ($T > 10^4\,\mathrm{K}$) when directly applied to
the radiative transfer models, resulting in the lines being too
strong. This is perhaps due to the 3D MHD simulations being performed
for a pure adiabatic case (in which the equation for the entropy is
solved). Solving the full energy equation would not solve the problem
here because in both cases, the main issue is how the gas cools down
at the inner disc and in the magnetospheric accretion flows. The process of
cooling should involve cooling in a number of spectral lines, and is a
separate complex problem. Hence, we use the adiabatic approach for
solving 3D MHD equations in which we solve the equation for the
entropy. To control the gas temperature in the
radiative transfer calculations and to produce the line strengths comparable to
those seen in observations, we adopt the parameterized temperature
structure used by \citet{hartmann94} in which the gas temperatures are
determined by assuming a volumetric heating rate proportional to $r^{-3}$, and
by solving the energy balance of the radiative cooling rate
\citep*{hartmann1982} and the heating rate. The same technique was also
used by e.g.~\citet*{muzerolle:1998,muzerolle:2001} and \citet{lima:2010}.
In our earlier work (e.g.~\citealt{kurosawa08}), the high temperatures
from MHD simulations were `re-scaled' to lower values in order to
match the line strengths seen in observations. However, we found that
the line profiles computed with the re-scaled temperature were very
similar to those computed with the parameterized temperature structure
from \citet{hartmann94} (see Figure~7 in \citealt{kurosawa08}). Here,
we adopt the latter as it is more physically motivated, but we expect
to obtain similar line profiles even when we choose to use the
re-scaled MHD simulation temperatures.
\begin{figure}
\centering
\includegraphics[clip,width=0.45\textwidth]{fig02.pdf}
\caption{The mass accretion rates (in $10^{-8}\,\mathrm{M_{\sun}\,yr^{-1}}$) on to the
central star plotted as a function of rotational phase for the
stable (top) and unstable (bottom) cases.}
\label{mdot-2}
\end{figure}
\begin{figure*}
\centering
\includegraphics[clip,width=0.8\textwidth]{fig03.pdf}
\caption{Magnetospheric accretion in the stable regime and the
corresponding H$\delta$ model profiles are shown at different
rotational phases (indicated at the lower-left corner of each panel;
with 0.08 interval). For a demonstration purpose, only a subset
(only for about one rotation period of the star) of the MHD
simulations and model profiles is shown. The volume rendering of the
density shown in colour (in logarithmic scale) at each phase are
projected toward an observer viewing the system with its inclination
angle $i=60^{\circ}$. The model profiles are also computed at
$i=60^\circ$. Line profiles models extended to the rotational phase
$\phi=2.96$ are presented in Fig.~B1 (Appendix~B in the online
supporting information).
}
\label{stable-14}
\end{figure*}
\begin{figure*}
\centering
\includegraphics[clip,width=0.8\textwidth]{fig04.pdf}
\caption{Same as in Fig. \ref{stable-14} but for the unstable regime.
Line profiles models extended to the rotational
phase $\phi=2.96$ are presented in Fig.~B2 (Appendix~B in the
online supporting information).
}
\label{unstable-14}
\end{figure*}
\section{Results}
\label{sec:results}
We perform two separate numerical MHD simulations for the
magnetospheric accretion as described in Section~\ref{sec:mhd-rad:mhd}
with the parameters shown in Table~\ref{tab:refval}, which are known
to produce a stable and an unstable magnetospheric accretion
(e.g.~\citealt{kulk08}). Typical configurations of matter flows in the
stable and unstable regimes are shown in Fig.~\ref{stable-unstable}.
While matter accretes in two ordered funnel streams in a stable
regime, it accretes in several temporarily formed tongues, which
appear in random locations at the inner edge of the accretion disc, in
the unstable regime.
For the line and continuum variability calculations,
we select the moments of time in the MHD simulations when the
mass-accretion rates ($\dot{M}$) of the system become
quasi-stationary over a few rotation periods. Note that $\dot{M}$ for the
unstable regime can be still variable in a shorter time scale. The MHD simulation
outputs during 3 stellar rotation periods around these quasi-stationary
phases are used in the variability calculations. The
corresponding $\dot{M}$ as a function rotational phase ($\phi$) for the stable and
unstable cases are shown in Fig.~\ref{mdot-2}. According to the figure, for the stable
case, $\dot{M}$ rather steadily decreases by about 40~per~cent during the 3 rotation
periods. On the other hand, for the unstable case, $\dot{M}$ increases slightly by
about 20~per~cent in 3 rotation periods, but it changes rather stochastically with
smaller time-scales during those 3 rotation periods.
\subsection{Line variability: persistent redshifted absorptions in the unstable
regime}
\label{sec:results:spectrum}
The output from the MHD simulations are saved with the rotational
phase interval of $\Delta\phi=0.04$, for 3 rotation periods. This
corresponds to 75 total phase points at which line profiles are to
be computed. This frequency is sufficient for `catching' the main
variability features in both stable and unstable regimes. For these
moments of time, we calculate hydrogen lines profiles in
optical and near-infrared i.e.~H$\alpha$, H$\beta$, H$\gamma$,
H$\delta$, Pa$\beta$ and Br$\gamma$ as a function of rotational phase,
using the radiative transfer models as described in
Section~\ref{sub:mhd-rad:rad}. We set the gas temperatures in the accretion
funnels for both the stable and unstable cases to be between $\sim
6000\,\mathrm{K}$ and $\sim 7500\,\mathrm{K}$. In all the variability
calculations presented in this work, we adopt the intermediate
inclination angle $i=60^{\circ}$.
The subsets of the time-series line profile calculations for
H$\alpha$, H$\beta$, H$\gamma$, H$\delta$, Pa$\beta$ and Br$\gamma$
are summarised in Figs. \ref{app-spectra-1} and \ref{app-spectra-2}
(in Appendix~\ref{sec:appendix}), with $\Delta\phi=0.16$, for the first
rotation period. A quick inspection of the figures shows that with
these particular sets of model parameters (see above and
Table~\ref{tab:refval}), we find no clear redshifted absorption
component caused by the accretion funnel flows in H$\alpha$
profiles, and only a weak redshifted absorption component is seen in
H$\beta$ at some rotational phases. However, it is very notable in the
higher Balmer lines (H$\gamma$ and H$\delta$), and also in the
near-infrared lines (Pa$\beta$ and Br$\gamma$) partly because the peak
fluxes in the emission part of the line profiles are relatively
smaller than those of other lines (H$\alpha$ and H$\beta$). A similar
tendency is also seen in some observations
(e.g.~\citealt{edwards:1979}; \citealt*{appenzeller:1986};
\citealt*{Krautter:1990}; \citealt{edwards94}). As a demonstration
for differentiating the line variability seen in the stable and unstable
cases, we mainly use the evolution of the redshifted absorption
component in H$\delta$ profiles.
Fig.~\ref{stable-14}\footnote{The animated version of this figure,
extended to three rotation periods, can be downloaded from
\url{http://www.astro.cornell.edu/~kurosawa/research/ctts_instab.html}}
shows the 3D views of the matter flows (the volume rendering of the 3D
density distributions as seen by an observer with $i=60^{\circ}$) as a
function of rotational phases, for the star accreting in the stable
regime. The change in the mass-accretion rate during the first
rotation is fairly smooth according to Fig.~\ref{mdot-2}, and the
geometry of the accretion funnels is also very stable. The
corresponding model line profiles of H$\delta$ are also shown in the
same figure. It shows a subset of the time-series calculations,
i.e.~only the phases between $\phi=0$ and $\phi=1.04$ with the
interval $\Delta\phi=0.08$. The time-series of model H$\delta$
profiles during the whole 3 stellar rotations are given in Appendix~B
(Fig.~B1) in the online supporting information. Fig.~\ref{stable-14}
shows that the redshifted absorption component starts appearing around
$\phi \approx 0$, and its presence persists until $\phi \approx
0.5$. This phase span coincides with the time when the upper funnel
accretion stream is located in front of the star, i.e., in the line of
sight of the observer to the stellar surface. The redshifted
absorption occurs when the hotspot continuum is absorbed due to the
line opacity in the accretion flow which is, for the most part, cooler
than the hotspots and moving away from the observer. Between
$\phi\approx0.5$ and $0.96$, the redshifted absorption is not present
because the lower accretion stream cannot intersect the line of sight
of the observer to the hotspot. A similar behaviour was found in our
earlier calculations with a stable accretion flow \citep{kurosawa08}.
The appearance and disappearance of the redshifted absorption
component is fairly periodic with their periods corresponding to the
stellar rotation period (see also Fig.~B1 in the online supporting
information.).
A similar sequence of the 3D views of the matter flow in the unstable
regime and the corresponding model line profiles for H$\delta$ are
shown in Fig.~\ref{unstable-14}\footnote{The animated version of this
figure, extended to three rotation periods, can be downloaded from
\url{http://www.astro.cornell.edu/~kurosawa/research/ctts_instab.html}}.
Again, the figure only shows a subset of the whole calculations,
i.e.~only for the phases between $\phi=0$ and $\phi=1.04$ with the
interval $\Delta\phi=0.08$. As before, the time-series of H$\delta$
profiles computed for 3 stellar rotations are given in the
supplemental figure in Appendix~B (Fig.~B2) in the online supporting
information. The figure clearly shows that the geometry of the
accretion funnels and their evolution are very different from those of
the stable case. The accretion no longer occurs in two streams, but
rather occurs in the form of thin tongues of gas that penetrate the
magnetosphere from the inner edge of the accretion disc (see also
\citealt{roma08,kulk08}). The shape and the number (up to a several)
of the tongues change within one stellar rotation period. The
corresponding mass-accretion rate, during these rotational phases,
changes rather stochastically as seen in Fig.~\ref{mdot-2}. According
to Figs.~\ref{unstable-14} and B2 (Appendix~B in the online supporting
information), the peak strength of the line also changes
stochastically. Interestingly, in the unstable regime, the redshifted
absorption component is present at almost all rotational phases. This
is caused by the fact that there are a few to several accretion
streams/tongues in the system at all times, and at least one of the
accretion stream is almost always in the line of sight of the observer
to the stellar surface. This behaviour is clearly different from that
in the stable regime in which the redshifted absorption appears and
disappears periodically with a stellar rotation.
\begin{figure*}
\begin{center}
\begin{tabular}{cc}
\includegraphics[clip,width=0.45\textwidth]{fig05a.pdf} &
\includegraphics[clip,width=0.45\textwidth]{fig05b.pdf}\tabularnewline
\end{tabular}
\end{center}
\caption{The maps of hot spots, as seen by an observer at the
inclination angle $i=60^{\circ}$, at four different rotational phases
(top panels), the light-curves calculated at the wavelength of
H$\delta$ (middle panels; flux in arbitrary units) and the line
equivalent widths (EWs) of model
H$\delta$ profiles (lower panels, in \AA) are shown for the stable (left
panels) and unstable (right panels) regimes of accretions. The hot
spots are assumed to be radiating as a blackbody with $T=8000$~K,
and are shown for representative moments of time which are marked as
red dots in the light curves.}
\label{spots-stable-unstable}
\end{figure*}
\subsection{Photometric variability: stochastic variability in the unstable regime}
\label{sec:results:continuum}
In both stable and unstable regimes, matter accretes on to the star and forms hot spots
on its surface. In our earlier work (e.g.~ROM04), the total kinetic energy of
the gas accreting on to the stellar surface through the funnel streams
are assumed to be radiated away isotropically as a blackbody. The
effective temperature of a surface element in hot spots depends on the local energy
flux at the surface element. The simulations show the hot spots are inhomogeneous
with its highest energy flux and temperature in the innermost parts of
spots. ROM04 have shown that the light curves in a stable
regime are ordered/periodic, and have one or two maxima per period depending on the tilt angle
of the dipole and on the inclination angle of the system. On the other hand, in an unstable
regime, the light curves are usually stochastic because the equatorial
accretion streams/tongues form stochastically and deposit the
gas on to the stellar surface in a stochastic manner. This results in
the stochastic formation of hot spots which produce the stochastic
light curve \citep[see also][]{roma08,kulk08}. There is a wide range of
possibilities when accretion is only weakly unstable in which both
stable funnels and unstable tongues can coexist \citep{kulk09,bac10}. However, in this
paper, we focus on the case of a strongly unstable regime as a demonstration.
As mentioned earlier, our MHD simulations lack
an implementation of a proper cooling mechanism (e.g.~radiative
cooling), and consequently the adiabatically-heated gas in the funnel slows down
gravitational acceleration of gas near the stellar surface. This results in the
highest temperature of the hot spots to be only $T_\mathrm{hs}\approx 4500$K in the
centre of the spot. This is not sufficient in explaining the ultraviolet radiation
observed in many CTTSs (e.g. \citealt{calvet98,gullbring2000}). Here, we use a
slightly different approach for the light curve calculation from hot spots. To
overcome the low hot spot temperatures, we simply set the temperature to
$T_\mathrm{hs}=8000$K regardless of the local energy flux at the hot spots.
Consequently, the temperature is uniform over the hot spots. The shapes and
the locations of the hot spots are determined by the energy fluxes computed at the
inner boundary of the MHD simulation outputs. We apply a threshold energy-flux value
such that the total hot spot coverage is about 2~per~cent of the stellar surface
(cf.~\citealt*{valenti1993}; \citealt{calvet98,gullbring2000},
\citealt{valenti:2004}). The threshold energy-flux values are fixed for each sequence
of the line profile computations (in the stable and unstable regimes separately) so
that the hot spot sizes can evolve with time when the local energy flux on the stellar
surface changes. The continuum radiation from the hot spots is included
in the previous line variability calculation in
Section~\ref{sec:results:spectrum}. No separate continuum radiative transfer model
have been performed, as the continuum flux calculations are performed as a part of line
profile calculations. The resulting variability of the continuum flux (light curve)
near H$\delta$ are shown in Fig~\ref{spots-stable-unstable} along with the hot spot
distribution maps at four different rotational phases for both stable and unstable
cases. The line wavelength of H$\delta$ is $4101$\,\AA{} which falls in a $B$-band
filter; hence, the light curve shown here should be somewhat comparable to those from
$B$-band photometric observations. The same figure also shows the corresponding line
equivalent widths (EWs) for H$\delta$ as a function of rotational phase
(cf.~Figs.~B1 and B2 in Appendix~B in the online supporting
information). Similar variability curves for other hydrogen lines,
i.e.~H$\alpha$, H$\beta$, H$\gamma$, Pa$\beta$ and Br$\gamma$ are
summarized in Appendix~\ref{sec:appendix}
(Fig.~\ref{variability-all}).
The surface maps for the stable regime in
Fig.~\ref{spots-stable-unstable} show the presence of banana-shaped
hot spots on the stellar surface. For example, the hot spot on the
upper hemisphere, which is created by the upper accretion stream
(cf.~Figs.~\ref{stable-unstable} and \ref{stable-14}), is visible on
the surface maps at the rotational phases $\phi=1.00$ and $1.24$. The
spot on the lower hemisphere is visible, for example, at $\phi=1.60$
and $1.72$. The light curve is periodic with
its period corresponding to that of the stellar rotation. It is clearly seen that the
maxima (at $\phi \approx 0.25, 1.25, 2.25$) of the light curve occur when the upper
hot spot faces the direction of the observer. The smaller
secondary peaks at $\phi \approx 0.75, 1.75, 2.75$ in the light curve are produced by
the spot from the lower hemisphere. Interestingly, the line EW variability curves
almost mirrors the light curve, i.e.~it shows a similar variability pattern but the
maxima of the light curve corresponds to the minima of the EW values\footnote{In this
paper, the sign of line equivalent widths
(EWs) is opposite of that in a usual definition, i.e.~a EW value is
positive when the line is in emission. }. This is caused by the
combinations of the following: (1)~for a given amount of line
emission, the line EW decreases as the underlying continuum flux
increases and (2)~the amount of the redshifted absorption is largest
when the accretion funnel on the upper hemisphere is directly in
between the stellar surface and the observer. Geometrically, this
corresponds to a phase when the upper hot spot points toward the
observer. The shapes of the spots are fairly constant during the three
rotation periods for which the light curves are evaluated. However,
the peak of the light curves drop slightly in the later rotational
phases since the mass-accretion rate decreases slightly during this
time (see Fig.~\ref{mdot-2}) and consequently the size of the spot
decreases slightly.
In contrast to the smooth and periodic light curve seen in the stable
case, the light curve (at the H$\delta$ wavelength) from the unstable
case is highly irregular, and it resembles many of the observed light
curves from CTTSs (e.g., \citealt{herbst94}, see also
Section~\ref{sec:observations:continuum}). The figure shows that
there are typically several maxima per rotation period which
correspond to several accretion tongues rotating around the star
(Fig.~\ref{unstable-14}). The stellar surface maps show that the
number of hot spots visible to the observer changes in time. For
example, only one spot is visible at $\phi=0.52$, and multiple spots
are visible at $\phi=1.08$ (two spots), $2.12$ (two spots), $2.54$ (3
spots -- the third one is barely visible but present on the upper
right edge). The sizes of individual spots also change in time as this
should correlate with the mass-accretion rate seen in
Fig.~\ref{mdot-2}. The same figure also shows that the variability of
H$\delta$ EW is also stochastic, and no clear periodicity is found.
Unlike in the stable case, the line EW for the unstable case does not
clearly mirror the light curves, but they tend to correlate with each
other. This is likely caused by the fact the mass-accretion rate and
the density in the funnel flows are not constant in time for the
unstable case while they are almost constant in the stable case. An
increase in the mass-accretion rate would cause an increase not only
in the continuum flux but also in the line emission because the
density in the funnel flows should also increase for the higher
mass-accretion rate. This effect would make the light curve and the
line EW tend to correlate each other, and the mirroring effect seen in
the stable case becomes weaken or disappears. The light curves
computed at other line frequencies (at H$\alpha$, H$\beta$, H$\gamma$,
Pa$\beta$ and Br$\gamma$) have also shown similar qualitative
behaviours, i.e.~ ordered curves in the stable regime of accretion,
and stochastic curves in the unstable regime (see
Fig.~\ref{variability-all}). In addition to the variable but rather
persistent redshifted absorption component found in some lines
(Figs.~\ref{unstable-14}, \ref{app-spectra-2} and B2 -- Appendix~B in
the online supporting information), the stochastic light curves and
stochastic EW variability (Figs.~\ref{spots-stable-unstable} and
\ref{variability-all}) are also key signatures of the CTTSs accreting
in the unstable regime.
\section{Comparisons with Observations}
\label{sec:observations}
\subsection{Line Variability}
\label{sec:observations:spectrum}
Many spectral line variability observations of CTTSs, in different
time-scales, have been carried out in the past
(e.g.~\citealt{aiad:1984, edwards94, johns:1995a, johns:1995b,
gullbring:1996, petrov:1996, chelli:1997, smith:1999, petrov:1999,
oliveira:2000, petrov:2001, alencar:2002, stempels:2002,
alencar:2005, kurosawa05, bouvier07, donati:2007, donati:2008,
donati10, donati:2011a, donati:2011b, alencar12, costigan:2012,
faesi:2012}). The line variability is often used to probe the
geometry of magnetospheric accretion flows, and in many cases the
accretion flows are found to be non-axisymmetric, e.g.~SU~Aur
(e.g.~\citealt{johns:1995b, petrov:1996}) and RW~Aur~A
(e.g.~\citealt{petrov:2001}). One of the key signatures of magnetospheric
accretion is the presence of the redshifted absorption component in
line profiles. Such absorption components have been observed e.g.\,in
some Balmer lines
(e.g.~\citealt{aiad:1984}; \citealt*{appenzeller:1988}; \citealt{reipurth:1996}), in
near-infrared hydrogen lines (Pa$\beta$ and Br$\gamma$,
e.g.~\citealt{folha01}) and in some helium lines
(e.g.~\citealt*{beristain:2001}; \citealt{edwards:2006}).
In some objects, observations show that the redshifted absorption
component is rather
persistently present in higher Balmer lines, e.g.~in H$\gamma$ and
H$\delta$ in DR~Tau and YY~Ori (e.g.~\citealt{aiad:1984, appenzeller:1988,
edwards94}), in H$\delta$ in DF~Tau and RW~Aur~A
(e.g.~\citealt{edwards94}). This contradicts
with our stable accretion model (Fig.~\ref{stable-14}) in which
gas flows on to the star occurs in two ordered funnel
streams, and a redshifted absorption is present only during
a part of the whole rotational phase -- when the funnel stream on the
upper hemisphere is located in the line of sight of an observer to the
stellar surface. However, a persistent redshifted absorption could be
observed also in a stable accretion model when the system is viewed
pole-on because the hot spot and the accretion funnel stream on one
hemisphere is visible for an observer during a complete rotational
phase (e.g.~see Figure~5 in \citealt{kurosawa08}).
No clear periodicity in the line variability is found in
many CTTSs. For example, in the observation of TW~Hya, \citet{donati:2011b}
found that the variability of H$\alpha$ and H$\beta$ are not periodic (but see also
\citealt{alencar:2002}), and suggested the cause of the variability is intrinsic
(e.g.~changes in the mass-accretion rates) rather than the stellar rotation. TW~Hya
also shows the redshifted absorption component in near infrared hydrogen lines
(Pa$\beta$, Pa$\gamma$ and Br$\gamma$) in multiple occasions
(e.g.~\citealt{edwards:2006, vacca:2011}); however, the frequency of the occurrence is
unclear. In the studies of DR~Tau by \citet*{alencar:2001} and DF~Tau by
\citet{johns-krull:1997}, no clear periodicity was found in their line variability
observations. The non-periodic behaviour of the line variability is consistent with
our unstable accretion model (Figs.~\ref{unstable-14} and
\ref{spots-stable-unstable}). On the other hand, clear signs of periodicity are found
in the line variability in some of CTTSs, e.g.~BP~Tau
(e.g.~\citealt{gullbring:1996,donati:2008}) and V2129~Oph
(e.g.~\citealt{donati:2007,donati:2011a,alencar12}). In the line variability study of
V2129~Oph, \citet{alencar12} found that the redshifted absorption component in
H$\beta$ was visible only during certain rotational phases. This is consistent with
our stable accretion model (Fig.~\ref{stable-14}).
\subsection{Photometric Variability}
\label{sec:observations:continuum}
\begin{figure}
\begin{center}
\begin{tabular}{cc}
\includegraphics[clip,width=0.45\textwidth]{fig06a.pdf}
\vspace{0.25cm}
\tabularnewline
\includegraphics[clip,width=0.45\textwidth]{fig06b.pdf}
\end{tabular}
\end{center}
\caption{Top panel: the light-curve of TW Hya with the new broadband (located
between $V$ and $R$ band) filter on \textit{MOST}
satellite (filled circle) overplotted with the $V$-band observation (open circle) of All Sky
Automated Survey (from Figure~4 in \citealt{rucin08}). Bottom panel:
A typical stochastic light-curve (in `white light' broadband) obtained by \textit{CoRoT} satellite (from
panel~e of Fig.~1 in \citealt{alencar10}). Here, $F$ and $F0$ denote the flux and the
maximum flux in the light curve, respectively. Credit: (upper
panel) \citet{rucin08}, reproduced with permissions; (lower
panel) \citet{alencar10}, reproduced with permission \copyright\,ESO. }
\label{light-curve-obs}
\end{figure}
Observations show a variety of photometric variability in CTTSs
(e.g.~\citealt{bouvier:1993,gahm:1993,herbst94,bouvier:1995,bouvier03,stassun:1999};
\citealt*{herbst:2000}; \citealt{oudmaijer:2001,lamm:2004,alencar10}). In some stars,
clear periodic light-curves are observed, while in others, the light curves appear
stochastic (e.g., \citealt{herbst94,alencar10}). Although the variability in X-ray are
mostly stochastic, in some cases, signs of rotational modulation are found in soft
X-ray emission in CTTSs \citep*[e.g.][]{flaccomio:2005,flaccomio:2012}. The
well-ordered and periodic light curves are likely connected with rotation of hot spots
resulting from ordered magnetospheric accretion streams (e.g., \citealt{herbst94}), or
from cold spots which represent regions of strong magnetic field. The exact origin of
the stochastic variability is not well known, and there may be different causes.
It was earlier suggested that stars
with ordered light-curves are accreting in a stable regime while stars with stochastic
light curves are in the unstable regime (e.g., \citealt{roma08,kulk08,kulk09}).
Examples of periodic light curves can be found in
e.g. \citet{herbst94} and \citet{alencar10}. Using the COnvection ROtation and
planetary Transits (\textit{CoRoT}) satellite, \citet{alencar10} have shown that
about 34~per~cent of CTTSs in the young stellar cluster NGC~2264
display clearly periodic light curves with one or two maxima per
stellar rotation (see panels~a and b in their Fig.~1). Our model
light curve from the stable accretion regime
(Fig.~\ref{spots-stable-unstable}) is very similar to the periodic
light curves with two maxima per rotation. As in the observation, the
smaller and larger maxima alternately appear. The periodic light
curves with one maximum are also consistent with a stable accretion observed with a lower
inclination angle system (e.g.~Figure~4 in \citealt{kurosawa08}).
Examples of irregular light curves are shown in Fig.~\ref{light-curve-obs}. The figure
shows the light curve of TW Hya obtained with the Microvariability \& Oscillations of
STars (\textit{MOST}) satellite presented in \cite{rucin08} and the \textit{CoRoT}
light curve of a CTTS in NGC~2264 presented in \citet{alencar10}. Irregular light
curves are found in about 39~per~cent of the CTTSs sample in \citet{alencar10}.
\cite{rucin08} finds no clear periodicity in the light curve for TW~Hya in
Fig.~\ref{light-curve-obs}. However, they determined an approximate period of 3.7~d
based on the light curve in an earlier epoch which showed some periodic behaviour.
\footnote{This may be a quasi-period or drifting period (see also \citealt{siwak:2011}).}
Using the 3.7~d period, one can see that there are a few to several maxima per
rotation in the light curve for TW~Hya in Fig.~\ref{light-curve-obs}. Interestingly,
our model light curve for the unstable accretion regime
(Fig.~\ref{spots-stable-unstable}) also shows a few to several maxima per rotation.
Similarly no clear period is found in the light curve of \citet{alencar10} in
Fig.~\ref{light-curve-obs}; however, the pattern of the variability resembles that of
our model from the unstable accretion regime (Fig.~\ref{spots-stable-unstable}).
Further, the variability amplitudes of the irregular light curves in
Fig.~\ref{light-curve-obs} are also comparable to the model in the unstable regime
(Figs.~\ref{spots-stable-unstable} and \ref{variability-all}).
In summary, we confirm that the periodic light curves found in the observations are
consistent with a scenario in which the continuum variability are caused by the
rotation of two hot spots which are naturally formed in a stable accretion regime with
a tilted dipole magnetosphere. Although not included in our model,
the presence of stable cold spots on a stellar surface would also
produce a periodic light curve. On the other hand, the irregular light curves
found in observations are consistent with a scenario in which the continuum
variability are caused by the rotation of stochastically forming hot spots
which naturally occurs in the unstable regime. Yet the third
type of the light curve (AA Tau-like), in which the variability is partially caused by
the occultation of a stellar surface by an accretion `disc wall' or a
warp near the disc truncation radius
(e.g.~\citealt{bouvier:1999,bertout:2000,bouvier07,alencar10,roma13}), is not
considered in this paper.
\section{Dependency on model parameters}
\label{sec:modelpar}
In general, the light curves and line EW variability behaviours are
expected to be similar among the MHD models which clearly show stable
and ordered accretion flows. The stable accretion funnels will always
form smooth and periodic light curves and line variability. On the
other hand, light curves and line variability behaviour for the
unstable regimes will be slightly different, depending on the strength
of the instability. In earlier models, (e.g.~\citealt{kulk08};
\citealt{kulk09}), we have found that the number of unstable accretion
filaments or `tongues' present at a given moment of time decreases
from several to just one or two as the instability changes from a
strong regime to a weak regime. While the number of tongues
may change for different model parameters, the occurrences of the
unstable tongues are still non-periodic. Therefore, the corresponding
light curves and line variability will also remain non-periodic,
similar to those seen in the unstable model in this paper
(Fig.~\ref{spots-stable-unstable}). However, the frequency of the
peaks in the light curves and line EW curves per stellar rotation will
decrease when the instability becomes weaker. Such variations are
expected during the transition from a strongly unstable regime to a
stable regime. The strength of the instability depends on some MHD
model parameters such as $P_{*}$, $r_{\mathrm{cor}}$, $\Theta$ and
$\alpha$ in Table~\ref{tab:refval}. In the following we briefly
discuss the dependency of our model on these key model parameters that
can influence the stability of the accretion flows.
\textit{Dependency on $P_{*}$ and $r_{\mathrm{cor}}$}. These are
essentially the same model parameters since the corotation radius
($r_{\mathrm{cor}}$) is determined by the stellar rotation period
$P_{*}$ for a given stellar mass ($M_{*}$),
i.e.~$r_{\mathrm{cor}}=(GM_{*})^{1/3}(P_{*}/2\pi)^{2/3}$. The unstable
flows occur more easily when the effective gravity (including the
effect of the centrifugal force) is stronger at the inner rim of the
accretion disc (e.g. \citealt{spruit95}). A slower stellar rotation or
equivalently a larger $r_{\mathrm{cor}}$ will increase the effective
gravity; hence, it tends to enhance the instability. The effective
gravity becomes negative at the disc-magnetosphere boundary when
$r_{\mathrm{cor}}$ becomes larger than the truncation (magnetospheric)
radius $r_{\mathrm{m}}$ (where the matter stress in the disc becomes
comparable with the magnetic stress in the magnetosphere). Hence, in
this simplified approach, accretion flows are expected to become
unstable when $r_{\mathrm{m}} < r_{\mathrm{cor}}$
(e.g.~\citealt{spruit93}). However, the onset of instability depends
on a few other factors, such as the level of differential rotation in
the inner disc, and also on the gradient of the ratio between the disc
surface density and magnetic flux (e.g.~\citealt*{Kaisig:1992};
\citealt{Lubow:1995}; \citealt{spruit95}). Hence, the condition,
$r_{\mathrm{m}} < r_{\mathrm{cor}}$, is necessary for the onset of
instability, but it may not be sufficient. In general, we find the
instability is strong in our MHD simulations when stars are rotating
relatively slowly i.e. $r_{\mathrm{m}} <
r_{\mathrm{cor}}$. Consequently, we have chosen a larger $P_{*}$ for
the unstable case and a smaller $P_{*}$ for the stable case in this
paper.
\textit{Dependency on $\Theta$}. The instability can occur at
different misalignment angles of the dipole $\Theta$. However, at a
large $\Theta$ (when the dipole is strongly tilted), the accretion
flow becomes more stable because the magnetic poles are closer to the
inner rim of the disc. The potential barrier in the vertical direction
that the gas at the inner rim of the disc has to overcome to form the
stable funnel accretion flow is reduced at a high $\Theta$ angle
(\citealt{kulk09}). On the other hand, when $\Theta$ becomes small,
the potential barrier to overcome becomes higher, and the stable
accretion funnels cannot be formed so easily. In this case, the gas
accumulates at the inner rim of the disc, and the accretion on to the
star proceeds through instability `tongues' which penetrate through
the magnetosphere. In our MHD simulations, we find that a strong
instability occurs when $\Theta < 25^{\circ}$ (\citealt{kulk09}).
Therefore, in this paper, we have chosen the large tilt angle
$\Theta=30^{\circ}$ to demonstrate the accretion in the stable case,
and the small tilt angle $\Theta=5^{\circ}$ in the unstable case.
\textit{Dependency on $\alpha$}. We use the viscosity parameter
($\alpha$) to control the global mass-accretion rate. The initial
density conditions for the disc are common in all of our models,
including the ones presented in the past (e.g.~\citealt{kulk08}). The
higher the value of $\alpha$, the higher the mass-accretion rate
becomes. At a higher accretion rate, the inner edge of the disc moves
closer to the star. Consequently the ratio
$r_{\mathrm{m}}/r_{\mathrm{cor}}$ becomes smaller, and the condition
becomes more favourable for the instability. The compression of gas
near the magnetosphere (the enhanced surface density per unit magnetic
flux) may also play a role. The possible artifacts of the $\alpha$
viscosity on the formation of instability have been studied in a few
experiments. For example, we start from a stable case with a small
viscosity ($\alpha=0.02$), which is used for the stable case as in
Table~\ref{tab:refval}, and we increase the initial density in the
disc by a factor of 2. In this case, we have observed that the
accretion becomes unstable (\citealt{roma08}). In another experiment,
we start from a relatively small viscosity ($\alpha=0.04$), and
increase the period of the star $P_{*}$. In this case, we find the
accretion becomes strongly unstable (e.g.~see Fig.~10 in
\citealt{kulk09}). Moreover, in our recent simulations with the grid
resolutions twice of those in the simulations presented in this paper
(i.e.~$N_r\times N^2=160\times 61^2$) but with a smaller viscosity
($\alpha=0.02$), we find the accretions flow also become unstable, if
a star rotates relatively slowly (e.g.~see Fig.~17 in
\citealt{roma13}). Based on these experiments, we conclude that a
large $\alpha$ is not a necessary condition for the instability. For
the MHD simulations presented in this paper, we have used $\alpha=0.1$
in the unstable case and $\alpha=0.02$ in the stable case for the
consistency with earlier simulations (e.g.~\citealt{roma08};
\citealt{kulk09}).
\section{Conclusions}
\label{sec:conclusions}
We have investigated observational properties of CTTSs
accreting in the stable and unstable regimes using (a)~3D MHD
simulations to model matter flows, and (b)~3D radiative transfer
models to calculate time-series hydrogen line profiles and continuum
emissions. In modelling line profiles, we have introduced the
parameterized temperature structures of \citet{hartmann94} in the
accretion flow (Section~\ref{sub:mhd-rad:rad}) and a fixed hot spot
temperature (Section~\ref{sec:results:continuum}) in order to match
the predicted line strengths and the amplitudes of light curves more
closely to observations. Our approach is to introduce minimal
modifications to the MHD solutions to obtain reasonable line profiles
and light curves. Although this is not completely consistent with the
MHD solutions, we still retain the flow geometry, density and velocity
information from the MHD simulations.
To implement proper heating/cooling mechanisms in the flows and
more realistic hot spots, we need to develop proper physical models
for them, which we have not done yet.
In this work, we have focused on the variability in the time scale
comparable with or less than a rotational period. For the unstable
accretion model, we have considered the flows associated with the
Rayleigh-Taylor instability that occurs at the interfaces between the
accretion disc and stellar magnetosphere. In particular, we have
analyzed qualitative behaviours of the variability in the redshifted
absorption component in H$\delta$ and the continuum flux to find key
observational signatures that can distinguish CTTSs accreting in the
two different regimes (stable and unstable). Our main findings are
summarized in the following:
(1)~In the stable regime, the emission lines vary smoothly
(Fig.~\ref{stable-14}), and their line EW show one or two peaks per stellar rotation period
(Fig.~\ref{spots-stable-unstable}). In the
unstable regime, the EW of spectral lines vary irregularly and more
frequently --- showing several maxima per stellar rotation period
(Fig.~\ref{spots-stable-unstable}).
(2)~In the stable regime, the redshifted absorption is observed during
approximately one half of the rotation period (when one of two
accretion streams is located in front of an observer), and is absent
during another half of the period (Fig.~\ref{stable-14}). In the unstable regime, the
redshifted absorption is observed at most of the rotational phases due
to the presence of several unstable tongues/accretion streams which
frequently move into the line of sight of an observer to the stellar
surface (Fig.~\ref{unstable-14}).
(3)~In the stable regime, the continuum emission from hot spots (the
light-curve) varies smoothly, and it has two peaks per stellar
rotation period (Fig.~\ref{spots-stable-unstable}). In the unstable regime, the light-curve shows
irregular variability with several peaks per period corresponding to
several unstable tongues (Fig.~\ref{spots-stable-unstable}).
We find that the redshifted absorption components are in general more
visible in the higher Balmer series, e.g. in H$\delta$ and H$\gamma$
(see Figs.~\ref{app-spectra-1} and \ref{app-spectra-2}) and some
near-infrared hydrogen lines such as Pa$\beta$ and Br$\gamma$ than in
the lower Balmer lines (H$\alpha$ and H$\beta$). Although the
line variability analysis presented here is mainly based on H$\delta$,
similar conclusions can be reached if we use the other
redshifted absorption sensitive lines mentioned above. In addition to
these hydrogen lines, \ion{He}{i}~$\lambda$10830 may be also useful
for probing the accretion stream geometry of CTTSs, as demonstrated by
\citet{fischer:2008}.
The work presented here may help to differentiate possible mechanisms
of variability. Since the irregular light curves found in many CTTSs
can be connected with different physical processes (e.g.~magnetic
flares, an accretion from a turbulent disc, an accretion through
an instability), the additional information from a line profile
variability analysis may help to disentangle the different processes. For
example, magnetic flares are not relevant to an accretion process;
hence, no correlation is expected between a light curve and a
redshifted absorption (which characterizes the magnetospheric flows).
If a light curve is irregular and the redshifted absorption is
persistent and non-periodic, the system may be accreting through
an instability, which can occur either from a laminar, or from a
turbulent disc. On the other hand, if a light curve is irregular and
the redshifted absorption is periodic (seen only during a part of a
rotational phase), the system may be accreting through well-defined
accretion funnels originating from a turbulent disc, which provides
turbulent cells to the accretion funnels and consequently produces
an irregular light curve.
We have investigated only the case of a strong instability in which
most of the matter accretes in the unstable equatorial
tongues. However, in reality, many CTTSs may be in a moderately
unstable accretion regime (e.g.~\citealt{kulk09}) in which both
ordered funnel flows and chaotic tongues are present. In these cases,
both a persistent redshifted absorption component associated with the
unstable accretion tongues, and ordered variability pattern associated
with the stable accretion streams can be present
simultaneously. However, if the two ordered accretion streams from the
stable flow components dominate, one can expect only one redshifted
absorption per stellar rotation period as seen in the current work
(e.g.~Fig.~\ref{stable-14}). Interestingly, a recent study by
\citet{roma09a} has shown that the unstable flows can be more ordered
and corotate with the inner disc when the size of magnetosphere is
small and comparable with the radius of the star. They have also shown
that the formation of unstable tongues can be driven by the density
waves formed in the inner disc region and by the interaction of the
inner disc with the tilted magnetic dipole field of the star
\citep{roma12}.
Although we have concentrated on the line variability associated with the
magnetospheric accretion, some line variability could be also attributed to a variable
nature of the wind. The winds from the disc-magnetosphere boundary
(e.g.~\citealt{shu:1994}) are usually associated with episodic inflations and openings
of the field lines connecting the magnetosphere and the disc on a time-scale of a few
inner disc rotations \citep{aly-kuijpers90,goodson97,roma09b,zanni:2012}. If the line
emission from a wind (either stellar or disc) is not significant, a variable wind
would cause a variability mainly in the blueshifted absorption component. However,
the emission from the wind could contribute non-negligibly to a total line emission
(e.g.~\citealt{kurosawa06,lima:2010,kwan:2011,kurosawa12}). If this is so, it makes
the interpretation of a line variability a harder problem. On the other hand, some
near-infrared hydrogen lines e.g.~Pa$\beta$ and Br$\gamma$, which show relatively
strong redshifted absorption in our models (Fig~\ref{app-spectra-2}), are usually not
affected by the winds (e.g.~\citealt{folha01}); therefore, the diagnostic tools
presented in this paper are still valid. In addition, the main time-scale of
variability associated with winds is expected to be longer than that
of the magnetospheric accretion considered in this paper.
\section*{Acknowledgments}
We thank an anonymous referee who provided us valuable comments and
suggestions which helped improving the manuscript.
We thank S.~H.~P.~Alencar and S.~M.~Rucinski for allowing us to use their figures in
this work. RK thanks Vladimir Grinin for a helpful discussion and P.~P. Petrov for an
example of a variable CTTS. RK also thanks Tim Harries, the original
code author of \textsc{torus}, for his support. Resources supporting
this work were provided by the NASA High-End Computing (HEC) Program
through the NASA Advanced Supercomputing (NAS) Division at Ames
Research Center and the NASA Center for Computational Sciences (NCCS)
at Goddard Space Flight Center. The research was supported by NASA
grants NNX10AF63G, NNX11AF33G and NSF grant AST-1008636.
|
1,108,101,564,572 | arxiv | \section{Introduction}
Prototype learning is a type of supervised learning, where class labels are represented by `prototypes', that is, by points in a suitably chosen output space. Instead of minimizing a general loss function, the learning model is trained by minimizing the distance of its output to the prototype of the true class label. In particular for multi-class categorization, prototype learning is a viable alternative to more common representations of class labels, such as `one-hot encoding' or word2vec, see \cite{mettes2019hyperspherical}.\\
The crucial ingredients of a prototype learner are the choice of its output space (including its metric structure) and a method to embed prototypes into the output space. In \cite{mettes2019hyperspherical} \textit{hyperspherical prototype learning} is proposed, in which the output space is given by the $d$-dimensional sphere $\mathbb{S}^d$ and improvements over one-hot-encoding and word2vec are shown. Here, we formulate a theory of hyperbolic learning, in which the output space is given by $d$-dimensional \textit{hyperbolic space} $\mathbb{H}^d$ and prototypes are represented by \textit{ideal points} (points at infinity) of $\mathbb{H}^d$. Instead of hyperbolic distance, we propose to use the (penalized) \textit{Busemann function} which can be interpreted as a `distance to infinity' and can be meaningfully applied to ideal points. We show that in the one-dimensional case, hyperbolic prototype learning with penalized Busemann loss and a linear base learner is equivalent to logistic regression, or -- coupled with a general neural network -- equivalent to cross-entropy-loss combined with a logistic output function.
\section{Hyperbolic Geometry}
\subsection{The Poincar\'e ball model of hyperbolic space}
In the Poincar\'e ball model, $d$-dimensional hyperbolic space is represented by the open unit ball
\[\mathbb{H}^d = \set{z \in \mathbb{R}^d: z_1^2 + \dotsm + z_d^2 < 1}.\]
We parameterize $\mathbb{H}^d$ by hyperbolic polar (HP) coordinates $(r,u)_\mathrm{HP}$, consisting of a unit vector $u \in \mathbb{S}^d$ and the hyperbolic radius $r \in [0,\infty)$, such that
\[z = \tanh(r/2)\,u,\]
i.e., $|z| = \tanh(r/2)$ is the Euclidean norm of $z$. Equipped with the \textit{hyperbolic distance}
\[\mathrm{d}_H((r_1,u_1)_\mathrm{HP},(r_2,u_2)_\mathrm{HP}) = \arcosh\Big(\cosh(r_1)\cosh(r_2) - \sinh(r_1)\sinh(r_2)\, u_1 \cdot u_2\Big),\]
$\mathbb{H}^d$ becomes a metric space, cf. \cite{ratcliffe2006foundations, cannon1997hyperbolic}. In two dimensions, $\mathbb{H}^2$ is called the Poincar\'e disc and is convenient to visualize properties of hyperbolic geometry\footnote{An artistic rendition of the Poincar\'e disc is given by the woodcuts `Circle Limit I-IV' by the Dutch artist M.C. Escher; see \url{https://www.wikiart.org/en/m-c-escher}.}, see Figure~\ref{fig:schematic}. For $d=2$, the direction vector $u$ can be replaced by its angle $\theta \in [0,2\pi)$ such that the hyperbolic distance becomes
\[\mathrm{d}_H((r_1,\theta_1)_\mathrm{HP},(r_2,\theta_2)_\mathrm{HP}) = \arcosh\Big(\cosh(r_1)\cosh(r_2) - \sinh(r_1)\sinh(r_2) \cos(\theta_1 - \theta_2)\Big).\]
Endowed with the metric tensor
\[ds^2 = \frac{4|dz|^2}{(1 - |z|^2)^2}\]
$\mathbb{H}^d$ becomes a Riemannian manifold and $\mathrm{d}_H$ is precisely the corresponding Riemannian distance (see \cite{ratcliffe2006foundations}). Hence, all concepts from differential geometry, such as tangent space, geodesics, (sectional) curvature, and exponential maps have direct interpretations in the context of hyperbolic space. Here, we will only make use of the exponential map\footnote{Roughly speaking, the exponential map $\exp_p(y)$ returns the result of following a geodesic from $p$ with speed $|y|$ and in direction $y / |y|$.} at the origin of $\mathbb{H}^d$, which is given (in Euclidean and in hyperbolic polar coordinates) by
\[\exp_0: \quad \mathbb{R}^d \to \mathbb{H}^d, \quad y \mapsto \tanh(|y|/2) \frac{y}{|y|} = \left(|y|, \frac{y}{|y|}\right)_\mathrm{HP}.\]
\begin{wrapfigure}{r}{0.4\textwidth}
\begin{center}
\includegraphics[width=0.35\textwidth]{schematic_crop.png}
\caption{The Poincar\'e disc with six ideal points $p_1, \dotsc, p_6$ and an `ordinary' point $z$. Three geodesics are shown in red. The point $z$ is at infinite hyperbolic distance, but at finite `Busemann distance', from all ideal points $p_i$.\label{fig:schematic}}
\end{center}
\end{wrapfigure}
\subsection{Ideal points and the Busemann function}
Ideal points in hyperbolic geometry represent points at infinity. In the Poincar\'e model, the ideal points form the boundary of the unit ball, i.e., they are given by
\[\mathbb{I}^d = \set{z \in \mathbb{R}^d: z_1^2 + \dotsc z_d^2 = 1}\]
and each ideal point is naturally associated with a unit vector $p \in \mathbb{S}^d$. In hyperbolic coordinates this corresponds to a point with infinite radial coordinate, i.e. we can write $p = (\infty, p)_\mathrm{HP}$.\\
As outlined above, our goal is to represent class prototypes by ideal points. This raises the problem that ideal points are at infinite distance from all other points in $\mathbb{H}^d$ and hence that the hyperbolic distance cannot be used as a loss function for prototype learning. This problem can be avoided by replacing hyperbolic distance by the \textit{Busemann function}. The Busemann function, originally introduced in \cite{busemann1955geometry} (see also Def.~II.8.17 in \cite{bridson2013metric}) can be considered a `distance to infinity' and is defined (in any metric space) as follows: Let $p$ be an ideal point and $\gamma_p$ a geodesic ray (parameterized by arc length) tending to $p$. Then the Busemann function with respect to $p$ is defined for $z \in \mathbb{H}^d$ as
\[b_p(z) = \lim_{t \to \infty} (d_H(\gamma_p(t),z) - t).\]
In the Poincar\'e model this limit can be explicitly calculated and the Busemann function is given by
\[b_p((r,u)_\mathrm{HP}) = \log \Big(\cosh(r) - \sinh(r)\,u \cdot p\Big),\]
or, alternatively, in Euclidean coordinates by
\[b_p(z) = \log\left(\frac{|p - z|^2}{1 -|z|^2}\right).\]
In the two-dimensional case, replacing $u$ and $p$ by the angles $\theta$ and $\xi$, we obtain
\[b_\xi((r,\theta)_\mathrm{HP}) = \log \Big(\cosh(r) - \sinh(r) \cos(\xi - \theta)\Big).\]
\section{Hyperbolic Prototype Learning}
\subsection{Hyperbolic prototypes and the penalized Busemann loss}
A Hyperbolic Prototype Learner consists of
\begin{itemize}
\item A base learner $\mathcal{B}(x,w)$ which, given trainable weights $w$, maps an input $x \in \mathbb{R}^n$ to an output $y \in \mathbb{R}^d$. Examples are linear regression ($\mathcal{B}(x,w) = w^\top x + w_0$) or multi-layer feed-forward networks.
\item A representation of class labels $1, \dotsc, K$ as prototypes $P = \set{p_1, \dotsc p_K}$ in the set of ideal points $\mathbb{I}^d$.
\end{itemize}
For training, the base learner $\mathcal{B}$ is concatenated with the exponential map\footnote{See also \cite{chami2019hyperbolic}, where the exponential map of $\mathbb{H}^d$ is used as a transfer function between the layers of a graph convolutional network.} $\exp_0$, such that each input $x$ is mapped to a point
\[z = \exp_0(\mathcal{B}(x,w))\]
in $\mathbb{H}^d$. To evaluate the output $z \in \mathbb{H}^d$ against the prototype $p_j$ of the true class label $j \in [K]$, we propose to use the \textit{penalized Busemann (peBu) loss}
\[l(z;p) = b_{p}(z) - \log(1 - z^2) = 2 \log \left(\frac{|p-z|}{1 - |z|^2}\right),\]
or, alternatively in hyperbolic polar coordinates
\[l((r,u)_\mathrm{HP}; p) = \log \Big(\cosh(r) - \sinh(r)\,u \cdot p\Big) + \log\Big(\cosh(r) + 1\Big).\]
The role of $b_p(z)$ is to steer $z$ towards the prototype $p$, while the penalty term penalizes `overconfidence', i.e., values of $z$ close to the ideal boundary of $\mathbb{H}^d$. The exact form of the penalty can be motivated from the fact that the peBu-loss becomes identical (up to scaling) to cross-entropy-loss in dimension $d=1$; see below.
Given a training sample $(x_i,p_i)_{i=1}^N$ of inputs $x_i$ with class labels represented by prototypes $p_i$, the Hyperbolic Prototype Learner is trained by minimizing the sample peBu-loss of the embedded base learner output, i.e. by minimizing
\[\mathcal{L}(w) = \frac{1}{N} \sum_{i=1}^N l\Big(\exp_0(\mathcal{B}(x_i,w));p_i\Big).\]
\subsection{Prediction from a Hyperbolic Prototype Learner}
For prediction, we propose the same procedure as in \cite{mettes2019hyperspherical}: For a given input $x$, the class label of the closest prototype to $z = \exp_0(\mathcal{B}(x,w))$ is returned, i.e. the predicted prototype is
\[p_* = \argmin_{p \in P} l(z;p).\]
As the penalty term $-\log(1 - |z|^2)$ is the same for all prototypes, we can equivalently minimize the Busemann function $b_p(z)$ directly over prototypes. Moreover, as $b_p(z)$ is (for given $z$) a decreasing function of the cosine similarity $\tfrac{z}{|z|} \cdot p$, prediction is equivalent to maximizing the cosine similarity between $\tfrac{z}{|z|}$ and $p$, that is
\[p_* = \argmax_{p \in P} \frac{z}{|z|} \cdot p.\]
Note that this is exactly the same prediction procedure as in hyperspherical embedding (see eq.~(3) in \cite{mettes2019hyperspherical}). This also implies that each output $z = (r,u)_\mathrm{HP}$ can be separated into the directional coordinate $u$, which measures the \textit{similarity} to a given prototype $p$, and the radial coordinate $r$, which represents the \textit{confidence} of this assessment and which can be interpreted analogous to the `log-odds' in logistic regression.\footnote{Interestingly, a similar decomposition of hyperbolic coordinates into a `similarity' and a `popularity' component is at the heart of the influential network growth model (`PSO-model') of \cite{papadopoulos2012popularity}.}
\subsection{Gradients of the penalized Busemann loss}
The gradients of the penalized Busemann loss with respect to $(r,u)_\mathrm{HP}$ can be easily calculated and we obtain
\begin{align*}
\partial_r\,l((r,u)_\mathrm{HP};p) &= - \frac{p \cdot u - \tanh(r)}{1 - \tanh(r) \, p \cdot u} + \tanh(r/2)\\
\nabla_u\,l((r,u)_\mathrm{HP};p) &= - p\,\frac{\tanh(r)}{1 - \tanh(r) \, p \cdot u}.
\end{align*}
Also the gradient of $l(\exp_0(y);p)$ with respect to $y$, which is need for backpropagation to the base learner $\mathcal{B}$, can be calculated with some effort and is given by
\begin{multline*}
\nabla_y\,l(\exp_0(y);p) = (y - p) \frac{\tanh(|y|}{|y| - \tanh(|y|) \, p \cdot y} + \bm{1}\, p \cdot y \frac{\tanh(|y|)/|y| - 1}{|y| - \tanh(|y|) \, p \cdot y} + \tanh(|y|/2),
\end{multline*}
where $\bm{1}$ is a vector of $d$ ones.\\
\begin{figure}[!h]
\begin{center}
\includegraphics[width=0.4\textwidth]{hyperbolic02.png}
\hspace{2em}
\includegraphics[width=0.4\textwidth]{hyperbolic01.png}
\caption{Penalized Busemann loss (A) and its radial gradient (B) for a prototype located at the ideal point $(1,0)$. \label{fig:loss}}
\end{center}
\end{figure}
\subsection{Equivalence to logistic regression for $d=1$}
We proceed to show that in the one-dimensional case ($d=1$) hyperbolic prototype learning with linear base learner and penalized Busemann loss is equivalent to logistic regression. More generally, the case $d=1$ is equivalent to a concatenation of the base learner $\mathcal{B}$ with a logistic output function and cross-entropy-loss.\\
In one dimension, hyperbolic space $\mathbb{H}^1$ becomes the interval $(-1,1)$. The set of ideal points $\mathbb{I}^1$ consists of the two points $\pm 1$. Thus, only two prototypes $p_+, p_-$ can be embedded at the points $z = \pm 1$, which corresponds to a binary classification task. In binary classification, class labels are more commonly identified with $0/1$, and hence we introduce another simple (linear) change of coordinates to
\[z' = \frac{z+1}{2}.\]
Under this change of coordinates $\mathbb{H}^1$ becomes the unit interval $(0,1)$ and the prototypes $p'$ (the ideal points) become the endpoints $0$ and $1$. The exponential map, i.e., the embedding of the base learner output $y = w^\top x + w_0$ into $\mathbb{H}^1$, maps $y \in \mathbb{R}$ to
\[z' = \exp_0(y) = \frac{\tanh(y/2) + 1}{2} = \frac{1}{1 + e^{-y}},\]
which is the logistic function. In $z'$-coordinates, the penalized Busemann loss becomes
\[l(z';p') = 2 \log\Big(\frac{|p' - z'|}{2z'(1 - z')}\Big) = - 2p' \log \Big(2(1-z')) - 2(1 - p') \log(2z')\Big).\]
Using the cross entropy
\[h(z';p') = - p' \log(z') - (1 - p') \log(1-z'), \]
we can write the peBu-loss as
\[l(z';p') = 2 h(z';p') - 2 \log(2),\]
a scaled and shifted transformation of cross-entropy. We conclude that minimizing the peBu-loss is equivalent to minimizing cross-entropy in the one-dimensional case.
\subsection{Embedding of prototypes}
The set of ideal points of $\mathbb{H}^d$ is homeomorphic to the hypersphere $\mathbb{S}^d$. Thus, any of the methods proposed for prototype embedding into $\mathbb{S}^d$ in \cite{mettes2019hyperspherical} can also be used to embed prototypes into $\mathbb{I}^d$. In particular
\begin{itemize}
\item in dimension $d=2$ prototypes can be placed uniformly onto the unit sphere $\mathbb{S}^2$
\item in dimension $d > 2$ prototypes can be placed by minimizing their mutual cosine similarities, as proposed in \cite{mettes2019hyperspherical}. Since the peBu-loss is a decreasing function of the cosine similarity between directional coordinate and prototype, the arguments outlined in \cite{mettes2019hyperspherical} for hyperspherical prototypes equally apply to hyperbolic prototypes.
\item A hyperbolic embedding method, such as \texttt{hydra} (see \cite{keller2020hydra}), can be used to embed prototypes into $\mathbb{H}^d$. This results in coordinates $\tilde p_j = (r_j,u_j)_\mathrm{HP}$ for each class label $j \in [K]$, which can be projected to the ideal boundary by setting $p_j = (\infty,u_j)_\mathrm{HP}$.
\end{itemize}
\begin{wrapfigure}[14]{r}{0.4\textwidth}
\vspace{-3.5em}
\begin{center}
\includegraphics[width=0.35\textwidth]{shortcut_crop.png}
\caption{The `hyperbolic shortcut' \label{fig:shortcut}}
\end{center}
\end{wrapfigure}
\subsection{The `hyperbolic shortcut'}
Finally, we present a heuristic argument, why we expect hyperbolic prototype learning to be more efficient than hyperspherical prototype learning. Suppose that an untrained learning model gives an output which is `as wrong as possible', i.e., opposite from the correct prototype (see Figure~\ref{fig:shortcut}). To update its output toward the true prototype, the hyperspherical learner has to `walk along the sphere', passing through several other incorrect prototypes to reach the correct prototype.\footnote{This problem is somewhat alleviated in higher dimension.} The hyperbolic learner, on the other hand, can take a shortcut and `cross through the disc' without passing through other incorrect prototypes.
\section{Summary} We have proposed Hyperbolic Prototype Learning, a type of supervised learning, which uses hyperbolic space as an output space. Learning is achieved by minimizing the penalized Busemann loss function, which is given by the Busemann function of hyperbolic geometry, enhanced by a penalty term. As an interesting property, we have shown that Hyperbolic Prototype Learning becomes equivalent to logistic regression in dimension one, when a linear base learner is used. It remains to be seen whether the performance of a practical implementation of Hyperbolic Prototype Learning can live up to its attractive theoretical features.
\small
\bibliographystyle{plain}
|
1,108,101,564,573 | arxiv | \section{\textcolor{black}{Introduction\medskip{}
}}
\end{singlespace}
\begin{singlespace}
\textcolor{black}{}
\begin{figure}
\begin{centering}
\textcolor{black}{\includegraphics[scale=0.85]{NI_Figure1}}
\par\end{centering}
\textcolor{black}{\medskip{}
}
\begin{spacing}{0.5}
\raggedright{}\textbf{\textcolor{black}{\footnotesize Figure 1. }}\textcolor{black}{\footnotesize Mid-sagittal
and coronal plots of example coefficients from dense, sparse, and
structured sparse coefficients (in Talairach coordinates). Warm colored
coefficients indicate a positive relationship with the target variable
(here predicting the decision to buy a product), cool colors a negative
relationship. Sparse methods set many coefficients to zero, while
in dense methods almost all coefficients are nonzero. Structured sparse
methods use a penalty on differences between selected voxels to impose
a structure on the fit so that it yields coefficients that are both
sparse and structured (e.g., smooth). Log-histograms of the estimated
voxel-wise coefficients show that the sparse method coefficients have
a near-Laplacian (double-exponential) distribution, while the dense
coefficients have a near-Gaussian distribution. The structured sparse
coefficients are a product of these distributions (see also Figure
2). Coefficient penalties that yield each result and examples of related
methods are given below each column. }\end{spacing}
\end{figure}
\textcolor{black}{Accurately predicting subject behavior from functional
brain data is a central goal of neuroimaging research. In functional
magnetic resonance imaging (fMRI) studies, investigators measure the
blood oxygen-level dependent (BOLD) signal---a proxy for neural activity---and
relate this signal to psychophysical or psychological variables of
interest. Historically, modeling is performed one voxel at a time
to yield a map of univariate statistics that are then thresholded
according to some heuristic to yield a {}``brain map'' suitable
for visual inspection. Over the past decade, however, a growing number
of neuroimaging studies have applied machine learning analyses to
fMRI data to model effects across multiple voxels. Commonly referred
to as {}``multivariate pattern analysis'' \citep{Hanke:2009p615}
or {}``decoding'' (to distinguish them from more commonly-used {}``mass-univariate''
methods \citep{Friston:1995p2108}), these approaches have allowed
investigators to use activity patterns across multiple voxels to classify
image categories during visual presentation \citep{Peelen:2009p2210,Shinkareva:2008p2395},
image categories during memory retrieval \citep{Polyn:2005p2405},
intentions to move \citep{Haynes:2007p2478}, and even intentions
to purchase \citep{Grosenick:2008p2789} (to name just a few applications---see
also \citet{Norman:2006p824,Haynes:2006p862,Pereira:2009p606,OToole:2007p909,Bray2009},
and examples in }\textit{\textcolor{black}{NeuroImage}}\textcolor{black}{{}
Volume 56 Issue 2). In multiple cases, these statistical learning
algorithms have shown better predictive performance than standard
mass-univariate analyses \citep{Haynes:2006p862,Pereira:2009p606}.}
\textcolor{black}{Despite these advances, analysis of neuroimaging
data with statistical learning algorithms is still young. Most of
the research that has applied statistical learning algorithms to fMRI
data has been conducted by a few laboratories \citep{Norman:2006p824},
and most analyses have been conducted with off-the-shelf classifiers
(\citet{Norman:2006p824,Pereira:2009p606}, but cf. \citealt{Grosenick:2008p2789,Hutchinson:2009p2394,Chappell2009,Brodersen2011,Michel2011,NgVaroquaux2012}).
These classifiers are often applied to volume of interest (VOI) data
within subjects rather than whole-brain data across subjects (\citet{Etzel:2009p3221,Pereira:2009p606},
but cf. \citealt{Mitchell:2004p970,MouraoMiranda:2007p2565,HBM2009,HBM2010,Ryali2010,VanGervenHeskes2012,Michel2011,NgVaroquaux2012}).
While these classifiers have a venerable history in the machine learning
literature, they were not originally developed for application to
whole-brain neuroimaging data, and so suffer from inefficiencies in
this context. Specifically, the large number of features (usually
voxel data) and spatiotemporal correlations characteristic of fMRI
data present unique challenges for off-the-shelf classifiers.}
\textcolor{black}{Indeed, the purpose of off-the-shelf classifiers
in the machine learning literature (e.g., discriminant analysis (DA),
naive Bayes (NB), k-nearest neighbors (kNN), random forests (RF),
and support vector machines (SVM)) has been to quickly and easily
yield good classification accuracy---for example in example speech
recognition or hand-written digit identification \citep{Hastie:2009p2681}.
Beyond accuracy, however, neuroscientists often aim to understand
which neural features are related to particular stimuli or behaviors
at specific points in time. This distinct aim of interpretability
requires classification or regression methods that can yield clearly
interpretable sets of model coefficients. For this reason, the recent
literature on classification of fMRI data has recommended using linear
classifiers (e.g., logistic regression (LR), linear discriminant analysis
(LDA), Gaussian Naive Bayes (GNB), or linear SVM) rather than nonlinear
classifiers \citep{Haynes:2006p862,Pereira:2009p606}. }
\textcolor{black}{Linearity alone, however, does not guarantee that
a method will yield a stable and interpretable solution. For instance,
in the case of multiple correlated input variables LR, LDA, and GNB
yield unstable coefficients and degenerate covariance estimates, particularly
when applied to smoothed data \citep{Hastie:1995p2589,Hastie:2009p2681}.
In the context of classification, penalized least squares may over
smooth coefficients, complicating interpretation \citep{Friedman1997}.
Additionally, most linear classifiers return dense sets of coefficients
(as in Figure 1, left panels) that require subsequent thresholding
or feature selection to yield parsimonious solutions. Although heuristic
methods exist for coefficient selection, these are generally greedy
(e.g., forward/backward stage-wise procedures like Recursive Feature
Elimination \citep{Guyon2002,DeMartino2007,Bray2009}), yielding unstable
solutions when data are resampled (since these algorithms tend to
converge to local minima) \citep{Hastie:2009p2681}. Although principled
methods exist for applying thresholds to dense mass-univariate coefficient
maps (e.g. Random Field Theory \citep{AdlerTaylor2000,Worsley2004}),
these approaches do not currently extend to dense multivariate regression
or classification methods. }
\textcolor{black}{Recently, sparse regression methods have been applied
to neuroimaging data to yield reduced coefficient sets that are automatically
selected during model fitting. The first examples in the fMRI literature
include \citet{Yamashitaa2008}, who applied sparse logistic regression
\citep{Tibs1996} to classification of visual stimuli, and \citet{Grosenick:2008p2789}
who first developed sparse penalized discriminant analysis by converting
an {}``Elastic Net'' regression \citep{ZouHastie} into a classifier,
and then applied it to choice prediction. Subsequently, sparse methods
for regression \citep{Carroll:2009p2920,Hanke:2009p615} and classification
\citep{Hanke:2009p615} have been applied to fMRI data to yield reduced
sets of coefficients from volumes of interest, whole-brain volumes
\citep{Ryali2010,vanGerven2010}, and whole-brain volumes over multiple
time points \citep{HBM2009,HBM2010}. These methods typically impose
an $\ell_{1}$-penalty (sum of absolute values) on the model coefficients,
which sets many of the estimated coefficients to zero (see Figure
1, leftmost panels, and Figure 2b). When applied to correlated fMRI
data, however, $\ell_{1}$-penalized methods can select an overly
sparse solution--resulting in omission of relevant features as well
as unstable coefficient estimates during cross-validation \citep{ZouHastie,Grosenick:2008p2789}.
To allow relevant but correlated coefficients to coexist in a sparse
model fit, recent approaches to fMRI regression \citep{Carroll:2009p2920,Li:2009p2898}
and classification \citep{Grosenick:2008p2789,HBM2009,Ryali2010}
impose a hybrid of both $\ell_{1}$- and $\ell_{2}$-norm penalties
(the {}``Elastic Net'' penalty of \citet{ZouHastie}) on the coefficients.
These hybrid approaches allow the inclusion of correlated variables
in sparse model fits. }
\textcolor{black}{This paper explores modified methods that combine
the Elastic Net penalty with a general user-specified sparse graph
penalty. This sparse graph penalty allows the user to efficiently
incorporate physiological constraints and prior information (such
as smoothness in space or time or anatomical details such as topology
or connectivity) in the model. The resulting graph-constrained elastic
net (or {}``GraphNet'') regression \citep{HBM2009,HBM2010} has
the capacity to find {}``structured sparsity'' in correlated data
with many features (Figure 1, right panels), consistent with results
in the manifold learning \citep{Belkin2006} and gene microarray literatures
\citep{LiLi2008}. In the statistics literature, related {}``sparse
structured'' methods have been shown to have desirable convergence
and variable selection properties for large correlated data sets \citep{SlawskiTutz2010,Jenatton2011}.
These sparse, structured models can also be implemented within a Bayesian
framework \citep{vanGerven2010}. Here, we extend the performance
of GraphNet regression and classification methods to whole-brain fMRI
data by: (1) generalizing them to be robust to outliers in fMRI data
(for both regression and classification), (2) adding {}``adaptive''
penalization to reduce fit bias and improve variable selection, and
(3) developing a novel support vector GraphNet (SVGN) classifier.
Additionally, to efficiently fit GraphNet methods to whole-brain fMRI
data over multiple time-points, we adapt algorithms from the applied
statistics literature \citep{Friedman2010}. }
\textcolor{black}{After developing robust and adaptive GraphNet regression
and classification methods, we demonstrate the enhanced performance
of GraphNet classifiers on previously published data \citep{Knutson2007,Karmarkar:2012}.
Specifically, we use GraphNet methods to predict subjects' trial-to-trial
purchasing behavior with whole-brain data over several time points,
and then infer which brain regions best predict upcoming choices to
purchase or not purchase a product. Fitting these methods to 25 subjects'
whole-brain data over 7 time points (2s TRs) yielded classification
rates which exceeded those found previously in a volume of interest
(VOI) based classification analysis \citep{Grosenick:2008p2789},
as well as those obtained with a linear support vector machine (SVM)
classifier fit to the whole brain data. While the GraphNet results
on whole-brain data confirm the relevance of previously chosen volumes
of interest (i.e., bilateral nucleus accumbens (NAcc), medial prefrontal
cortex (MPFC), and anterior insula), they also implicate previously
unchosen areas (i.e., ventral tegmental area (VTA) and posterior cingulate).
We conclude with a discussion of the interpretation of GraphNet model
coefficients, as well as future improvements, applications, and extensions
of this family of GraphNet methods to neuroimaging data. Open source
code for solving the GraphNet problems in this paper is freely available
at \url{https://github.com/logang/neuroparser}.}
\end{singlespace}
\textcolor{black}{\medskip{}
}
\begin{singlespace}
\section{\textcolor{black}{Methods}}
\end{singlespace}
\subsection{Background\textcolor{black}{\medskip{}
}}
\subsubsection{\textcolor{black}{Penalized least squares }}
\textcolor{black}{\medskip{}
}
\begin{singlespace}
\textcolor{black}{Many classification and regression problems can
be formulated as modeling a response vector $y\in\mathbb{R}^{n}$
as a function of data matrix $X\in\mathbb{R}^{n\times p}$, which
consists of $n$ observations each of length $p$ (with $n\geq p)$.
In particular, a large number of models treat $y$ as a linear combination
of the predictors in the presence of noise $\epsilon\in\mathbb{R}^{n}$,
such that}
\textcolor{black}{
\begin{equation}
y=X\beta+\epsilon,\label{eq:1}
\end{equation}
where $\epsilon$ is a noise vector typically assumed to be normally
distributed $\epsilon\sim\mathcal{N}(0,I\sigma^{2})$ with vector
mean $0$ and diagonal variance-covariance matrix $I\sigma^{2}$,
and $\beta\in\mathbb{R}^{p}$ a vector of linear model coefficients.
In this case using squared error loss leads to the well-known ordinary
least squares (OLS) solution}
\textcolor{black}{
\begin{equation}
\widehat{\beta}=\underset{\beta}{\text{argmin}}\ \|y-X\beta\|_{2}^{2}=(X^{T}X)^{-1}X^{T}y,\label{eq:2}
\end{equation}
which yields the best linear unbiased estimator (BLUE) if the columns
of $X$ are uncorrelated \citep{LehmannCasella}. }
\textcolor{black}{However, this estimator is inefficient in general
for}\textbf{\textcolor{black}{{} }}\textcolor{black}{$p>2$---it is
dominated by biased estimators \citep{Stein1955}---and if the columns
of $X$ are correlated (i.e. are {}``multicollinear'') then the
estimated coefficient values can vary erratically with small changes
in the data, so the OLS fit can be quite poor. A common solution to
this problem is penalized (or {}``regularized'') least squares regression
\citep{Tikhonov1943}, in which the magnitude of the model coefficients
are penalized to stabilize them. This is accomplished by adding a
penalty term $\mathcal{P}(\beta)$ on the coefficient vector $\beta$,
yielding}
\textcolor{black}{
\begin{equation}
\widehat{\beta}=\underset{\beta}{\text{argmin}}\ \|y-X\beta\|_{2}^{2}+\lambda\mathcal{P}(\beta),\ \lambda\in\mathbb{R}_{+},\label{eq:3}
\end{equation}
where $\lambda$ is a parameter that trades off least squares goodness-of-fit
with the penalty on the model coefficients (or equivalently, trades
off fit variance for fit bias) and $\mathbb{R}_{+}$ is the set of
nonnegative scalars. These estimates are equivalent to maximum a posteriori
(MAP) estimates from a Bayesian perspective (with a Gaussian prior
on the coefficients if $\mathcal{P}(\beta)=\|\beta\|_{2}^{2}$ \citep{Hastie:2009p2681}),
or to the Lagrangian relaxation of a constrained bi-criterion optimization
problem \citep{BoydVandenberghe2004}. Such equivalencies motivate
various interpretations of the model coefficients and parameter $\lambda$
(see section 2.3). \medskip{}
}
\end{singlespace}
\begin{singlespace}
\subsubsection{\textcolor{black}{Sparse regression and automatic variable selection }}
\end{singlespace}
\textcolor{black}{\medskip{}
}
\begin{singlespace}
\textcolor{black}{}
\begin{figure}[t]
\begin{centering}
\textcolor{black}{\includegraphics[scale=0.65]{NI_Figure2}}
\par\end{centering}
\textcolor{black}{\medskip{}
}
\begin{spacing}{0.5}
\raggedright{}\textbf{\textcolor{black}{\footnotesize Figure 2. }}\textcolor{black}{\footnotesize (a)
Diagrammatic representation of squared-error (red), Huber (black),
and Huberized Hinge (green) loss functions. Dotted lines denote where
the Huber loss changes from penalizing residuals quadratically (where
$|y-\widehat{y}|\leq\delta$) to penalizing them linearly (where $|y-\widehat{y}|>\delta$).
The linear penalty on large residuals makes the Huber loss robust.
(b) Diagrammatic representation of convex penalty functions used in
this article (along one coordinate $\beta$). The red curve is a quadratic
penalty $\mathcal{P}(\beta)=\beta^{2}$ on coefficient magnitude,
often called the Tikhonov or {}``ridge'' penalty in regression.
The blue curve is the Lasso penalty on coefficient magnitude $\mathcal{P}(\beta)=|\beta|$.
The purple curve is a convex combination of the red and blue curves:
$\mathcal{P}(\beta)=\alpha\beta^{2}+(1-\alpha)|\beta|$ (where here
$\alpha=0.5)$, called the {}``Elastic Net'' penalty. The inset
shows the shape of the prior distribution on the coefficient estimates
that each of these penalties corresponds to: Gaussian (red), Laplacian
(blue), and mixed Gaussian and Laplacian (purple) (units arbitrary).
The priors become increasingly peaked around zero as the Elastic Net
penalty approaches the Lasso penalty, corresponding to a prior belief
that many coefficients will be exactly zero.}\end{spacing}
\end{figure}
\textcolor{black}{{} There are a few standard choices for the penalty
$\mathcal{P}(\beta)$. Letting $\mathcal{P}(\beta)=\|\beta\|_{2}^{2}=\sum_{j=1}^{p}\beta_{j}^{2}$
(the $\ell_{2}$ norm) gives the classical Tikhonov or {}``ridge''
regression estimates originally proposed for such problems \citep{Tikhonov1943,HoerlKennard1970}.
More recently, the choice $\mathcal{P}(\beta)=\|\beta\|_{1}=\sum_{j=1}^{p}|\beta_{j}|$
(the $\ell_{1}$ norm)---called the Least Absolute Shrinkage and Selection
Operator (or {}``Lasso'') penalty in the regression context \citep{Tibs1996}---has
become widely popular in statistics, engineering, and computer science,
leading some to call such $\ell_{1}$-regression the {}``modern least
squares'' \citep{CandesBoyd2008}. In addition to shrinking the coefficient
estimates, the Lasso performs variable selection by producing sparse
coefficient estimates (i.e., many are exactly equal to zero, see Figure
1 left panels). In many applications, having a sparse vector $\widehat{\beta}$
is highly desirable, since a fit with fewer non-zero coefficients
is simpler, and can help select predictors that have an important
relationship with the response variable $y$. }
\textcolor{black}{The $\ell_{1}$-norm used in the Lasso is the closest
convex relaxation of the $\ell_{0}$ pseudo-norm $\|\beta\|_{0}=\sum_{j=1}^{p}1_{\{\beta_{j}\ne0\}}$,
where $1_{\{\beta_{j}\ne0\}}$ is an indicator function that is $1$
if the $j$th coefficient $\beta_{j}$ is nonzero and $0$ otherwise.
This represents a penalty on the number of nonzero coefficients (their
cardinality). However, finding a minimal cardinality solution generally
involves a combinatorial search through possible sets of nonzero coefficients
(a form of {}``all subsets regression'' \citep{Hastie:2009p2681})
and so is computationally infeasible for even a modest number of input
features. An $\ell_{1}$-norm penalty can be used as a heuristic that
results in coefficient sparsity (which corresponds to the maximum
a posteriori (MAP) estimates under a Laplacian (double-exponential)
prior; for a fully Bayesian approach see \citet{vanGerven2010}).
Such $\ell_{1}$-penalized regression methods set many variables equal
to zero and automatically select only a small subset of relevant variables
to assign nonzero coefficients. While these methods yield the sparsest
possible fit in many cases \citep{DonohoElad2003,Donoho2006}, they
do not always do so, and reweighted methods (e.g., Automatic Relevance
Determination (ARD) \citep{Wipf2008} and iterative reweighting of
the $\ell_{1}$ penalty \citep{CandesBoyd2008}) exist for finding
sparser solutions. It is worth noting that while Bayesian methods
for variable selection (such as Relevance Vector Machines) have existed
in the literature for some time, these methods typically require using
EM-like or MCMC approaches that do not guarantee convergence to a
global minimum and that are relatively computationally inefficient
(}though see \citet{MohamedHeller2011} for an interesting counter-point\textcolor{black}{).
As an interesting exception, recent work on ARD and sparse Bayesian
learning \citep{Wipf2008} has provided an attractive alternative,
showing that the sparse Bayesian learning problem can be solved as
a sequence of reweighted Lasso problems, similar to the adaptive methods
discussed below. This approach no longer provides a full posterior,
but does provide an interesting and computationally tractable link
to the Bayesian formulation. In the future we expect that such links
will lead to better approaches for model selection in these methods
than the {}``brute force'' grid search employed here. \medskip{}
}
\end{singlespace}
\subsubsection{Elastic Net regression}
\begin{singlespace}
\textcolor{black}{\medskip{}
}
\textcolor{black}{Despite offering a sparse solution and automatic
variable selection, there are several disadvantages to using $\ell_{1}-$penalized
methods like the Lasso in practice. For example, from a group of highly
correlated predictors, the Lasso will typically select a subset of
{}``representative'' predictors to include in the model fit \citep{ZouHastie}.
This can make it difficult to interpret coefficients because those
that are set to $0$ may still be useful for modeling $y$ (i.e.,
false negatives are likely). Worse, entirely different subsets of
coefficients may be selected when the data are resampled (e.g., during
cross-validation). Moreover, the Lasso can select at most $n$ non-zero
coefficients \citep{ZouHastie}, which may prove undesirable when
the number of input features ($p$) exceeds the number of observations
($n$) (i.e., {}``$p\gg n$'' problems). Finally, as a global shrinkage
method, the Lasso biases model coefficients towards zero \citep{Tibs1996,Hastie:2009p2681},
making interpretation with respect to original data units difficult.
Other methods that use only an $\ell_{1}$ penalty (e.g., sparse logistic/multinomial
regression and sparse SVM \citep{Hastie:2009p2681}) are subject to
the same deficiencies. }
\textcolor{black}{In response to several of these concerns \citet{ZouHastie}
proposed the Elastic Net, which uses a mixture of $\ell_{1}$- and
$\ell_{2}$-norm regularization, and may be written }
\textcolor{black}{
\begin{equation}
\widehat{\beta}=\kappa\ \underset{\beta}{\text{argmin}}\ \|y-X\beta\|_{2}^{2}+\lambda_{1}\|\beta\|_{1}+\lambda_{2}\|\beta\|_{2}^{2},\label{eq:4}
\end{equation}
where the factor $\kappa=1+\lambda_{2}$ in \eqref{eq:4} and subsequent
equations is a rescaling factor discussed in further detail below.
This Elastic Net estimator overcomes several (though not all) of the
disadvantages discussed above, while maintaining many advantages of
Tikhonov ({}``Ridge'') regression and the Lasso. In particular,
the Elastic Net accommodates groups of correlated variables and can
select up to $p$ variables with non-zero coefficients. The amount
of sparsity in the solution vector can be tuned by adjusting the penalty
coefficients $\lambda_{1}$ and $\lambda_{2}$. In this case, the
$\ell_{1}$ penalty can be thought of as a heuristic for enforcing
sparsity, while the $\ell_{2}$ penalty allows correlated variables
to enter the model and stabilizes the sample covariance estimate.
This Elastic Net approach performs well on fMRI data in both regression
and classification settings \citep{Grosenick:2008p2789,Carroll:2009p2920,Ryali2010}.\medskip{}
}
\end{singlespace}
\begin{singlespace}
\subsubsection{\textcolor{black}{Graph-constrained Elastic Net (GraphNet) regression }}
\end{singlespace}
\textcolor{black}{\medskip{}
}
\begin{singlespace}
\textcolor{black}{So far we have seen that sparse regression methods
like the Elastic Net, which use a hybrid $\ell_{1}$- and $\ell_{2}$-norm
penalty, can be used to yield sparse model fits that do not exclude
correlated variables \citep{ZouHastie}, and that we can turn these
regression methods into classifiers that perform well when fit to
VOI data \citep{Grosenick:2008p2789}. However, the Elastic Net penalty
merely makes the model fitting procedure {}``blind'' to correlations
between input features (by shrinking the sample estimate of the covariance
matrix towards the identity matrix). Indeed, if $\lambda_{2}$ in
equation \eqref{eq:5} grows large, this method is equivalent to applying
a threshold to mass-univariate OLS regression coefficients (i.e.,
the estimate of the covariance matrix becomes a scaled identity matrix)
\citep{ZouHastie}.}
\textcolor{black}{In this section, we describe a modification of the
Elastic Net that explicitly imposes structure on the model coefficients.
This allows the analyst to pre-specify constraints on the model coefficients
(e.g., based on prior information like local smoothness or connectivity,
or other desirable fit properties), and then to tune how strongly
the fit adheres to these constraints. Since the user-specified constraints
take the general form of an undirected graph, we call this regression
method the graph-constrained Elastic Net (or {}``GraphNet'') \citep{HBM2009,HBM2010}. }
\textcolor{black}{The GraphNet model closely resembles the Elastic
Net model, but with a modification to the $\ell_{2}$-norm penalty
term: }
\textcolor{black}{
\begin{eqnarray}
\widehat{\beta} & = & \kappa\ \underset{\beta}{\text{argmin}}\ \|y-X\beta\|_{2}^{2}+\lambda_{1}\|\beta\|_{1}+\lambda_{G}\|\beta\|_{G}^{2}\label{eq:8}\\
\text{} & & \|\beta\|_{G}^{2}=\beta^{T}G\beta=\sum_{j=1}^{p}\sum_{k=1}^{p}\beta_{j}G_{jk}\beta_{k},\nonumber
\end{eqnarray}
where $G$ is a sparse graph. Note that in the case where $G=I$,
where $I$ denotes the identity matrix, the GraphNet reduces back
to the Elastic Net. Thus the Elastic Net is a special case of GraphNet
and we can replicate the effects of increasing an Elastic Net penalty
by adding a scaled version of the identity matrix $(\lambda_{2}/\lambda_{G})I$
to $G$ (for $\lambda_{G}>0$). }
\textcolor{black}{The example we will use for the matrix $G$ in the
remainder of this paper is the graph Laplacian, which formalized our
intuition that voxels that are neighbors in time and space should
typically have similar values. If we take the coefficients $\beta$
to be functions over the brain volume $V\in\mathbb{R}{}^{3}$ over
time points $T\in\mathbb{R}$ such that $\beta(x,y,z,t)$, then we
would like a penalty that penalizes roughness in the coefficients
as measured by their derivatives over space and time, such as
\begin{equation}
\mathcal{P}(\beta)=\int_{V,T}\left(\frac{\partial^{2}\beta}{\partial x^{2}}+\frac{\partial^{2}\beta}{\partial y^{2}}+\frac{\partial^{2}\beta}{\partial z^{2}}+\frac{\partial^{2}\beta}{\partial t^{2}}\right)dx\ dy\ dz\ dt=\int_{V,T}\Delta\beta\ dx\ dy\ dz\ dt,\label{eq:9}
\end{equation}
where $\Delta$ is the Laplacian operator, which here is a 4D isotropic
measure of the second spatio-temporal derivative of the volumetric
time-series. Since we are sampling discretely, we use the discrete
approximation to the Laplacian operator $\Delta$: the matrix Laplacian
$L=D-A$ (the difference between the degree matrix $D$ and the adjacency
matrix $A$, see e.g., \citep{Hastie:1995p2589}). This formulation
generalizes well to arbitrary graph connectivity and is widely used
in spectral clustering techniques and spectral graph theory \citep{BelkinNiyogi2008}. }
\textcolor{black}{In the case where $G=L$, the graph penalty, $\|\beta\|_{G}^{2}$,
has the appealingly simple representation
\[
\|\beta\|_{G}^{2}=\sum_{(i,j)\in\mathcal{E}_{G}}(\beta_{i}-\beta_{j})^{2},
\]
where $\mathcal{E}_{G}$ is the set of index pairs for voxels that
share an edge in graph $G$ (i.e. have a nonzero entry in the adjacency
matrix $A$). Written this way, the graph penalty induces smoothness
by penalizing the size of the pairwise differences between coefficients
that are adjacent in the graph. In the one dimensional case, if the
quadratic terms $(\beta_{i}-\beta_{j})^{2}$ were replaced by absolute
deviations $|\beta_{i}-\beta_{j}|$ then this would instead be an
instance of the \textquotedbl{}fused Lasso\textquotedbl{} \citep{Tibshirani2005}
or Generalized Lasso \citep{TibshiraniTaylor2012}. There are two
main reasons for preferring a quadratic penalty in the present application:}
\end{singlespace}
\begin{enumerate}
\begin{singlespace}
\item \textcolor{black}{The fused Lasso is closely related to Total Variation
(TV) denoising \citep{Rudin1992} and tends to set many of the pairwise
differences $\beta_{i}-\beta_{j}$ to zero, creating a sharp piecewise
constant set of coefficients that lacks the spatial smoothness often
expected in fMRI data. Extending this formulation to processes with
more that one spatial or temporal dimension is nontrivial \citep{Michel2011}.}\end{singlespace}
\item \textcolor{black}{Significant algorithmic complications can be avoided
by choosing a differentiable penalty on the pairwise differences \citep{Tseng2001,Friedman:2007p36},
speeding up model fitting and reducing model complexity considerably---especially
in the case of spatial data, where the Total Variation penalty must
be formulated as a more complicated sum of non-smooth norms on each
of the first-order forward finite difference matrices \citep{Wang2008_TV,Michel2011}.}
\end{enumerate}
Thus GraphNet methods provide a sparse and structured solution similar
to the Fused Lasso, Generalized Lasso, and Total Variation. However,
unlike these approaches, GraphNet methods allow for smooth rather
than piecewise constant structure in the non-sparse parts of the reconstructed
volume. This is of interest in cases where we might expect the magnitude
of nonzero coefficients to be different within a volume of interest.
Due to the smoothness of the graph penalty GraphNet methods are also
easier from an optimization perspective. Of course, there are certainly
situations in which the piecewise smoothness of Total Variation could
be a better prior (this depends on the data and problem formulation).\medskip{}
\begin{singlespace}
\subsubsection{\textcolor{black}{Adaptive GraphNet regression}}
\end{singlespace}
\textcolor{black}{\medskip{}
}
\begin{singlespace}
\textcolor{black}{The methods described above automatically select
variables by shrinking coefficient estimates towards zero, resulting
in downwardly biased coefficient magnitudes. This shrinkage makes
it difficult to interpret coefficient magnitude in terms of original
data units, and severely restricts the conditions under which the
Lasso can perform consistent variable selection \citep{Zou2006}.
Ideally, given infinite data, the method would select the correct
parsimonious set of features (i.e., the {}``true model'', were it
known), but avoid shrinking nonzero coefficients that remain in the
model (unbiased estimation). Together, these desiderata are known
as the {}``oracle'' property \citep{FanLi}. Note that in the neuroimaging
context, the first (consistent variable selection) corresponds to
correct localization of signal, while the second (consistent coefficient
estimation) relates to improving estimates of coefficient magnitude.}
\textcolor{black}{Several estimators possessing the oracle property
(given certain conditions on the data) have been reported in the literature,
including the Adaptive Lasso \citep{Zou2006,Zhou:2009p3000} and the
Adaptive Elastic Net \citep{Zou:2009p2991}. These estimators are
straightforward modifications of penalized linear models. They work
by starting with some initial estimates of the coefficients obtained
by fitting the non-adaptive model \citep{Zou:2009p2991}, and use
these to adaptively reweight the penalty on each individual coefficient
$\beta_{j},\ j=1,\ldots,p$. Recently \citep{SlawskiTutz2010} extended
the adaptive approach to a sparse, structure method equivalent to
GraphNet regression, and proved that the oracle properties previously
shown for the adaptive Lasso and Adaptive Elastic Net extend to the
sparse, structured case provided the true coefficients are in the
null space of $G$ (i.e. the nonzero entries of $\beta$ specify a
connected component in $G$). We refer the reader to \citep{SlawskiTutz2010}
for further details.}
\end{singlespace}
\textcolor{black}{As in \citep{SlawskiTutz2010}, we may rewrite the
GraphNet to have an adaptive penalty (the adaptive GraphNet) as follows:}
\begin{singlespace}
\textcolor{black}{
\begin{eqnarray}
\widehat{\beta} & = & \underset{\beta}{\text{argmin}}\ \|y-X\beta\|_{2}^{2}+\lambda_{1}^{*}\sum_{j=1}^{p}\widehat{w}_{j}|\beta_{j}|+\lambda_{G}||\beta||_{G}^{2}\label{eq:12}\\
\widehat{w}_{j} & = & \left|\tilde{\beta}_{j}\right|^{-\gamma}.\label{eq:reweights}
\end{eqnarray}
The idea here is that important coefficients will have large starting
estimates $\tilde{\beta}_{j}$ (where $\tilde{\beta_{j}}$ is a suitable
estimator of $\beta_{j}$) and so will be shrunk at a rate inversely
proportional to their starting estimates, leaving them asymptotically
unbiased. On the other hand, coefficients with small starting estimates
$\tilde{\beta}_{j}$ will experience additional shrinkage, making
them more likely to be excluded. We let $\gamma=1$ as in the finite
sample case \citep{Zou2006,Zou:2009p2991}, and by analogy to the
Adaptive Elastic Net \citep{Zou:2009p2991} set $\tilde{\beta}$ to
the standard GraphNet coefficient estimates for a fixed value of $\lambda_{G}$
(chosen based on the GraphNet performance at that value). We use $\lambda_{1}^{*}$
to differentiate the adaptive fit sparsity parameter from the parameter
associated with the GraphNet fit used to initialize the weights $\widehat{w}_{j}.$}
\end{singlespace}
\textcolor{black}{It is important to note that the oracle properties
that hold in the asymptotic case may not apply to the finite sample,
$p\gg n$ situation. Nevertheless, we include these methods for comparison
since oracle properties are desirable and since evidence suggests
that the adaptive elastic-net has improved fi{}nite sample performance
because it deals well with collinearity \citep{Zou:2009p2991}.}
\textcolor{black}{\medskip{}
}
\begin{singlespace}
\subsubsection{\textcolor{black}{Turning sparse regression methods into classifiers:
Optimal Scoring (OS) and Sparse Penalized Discriminant Analysis (SPDA)}}
\end{singlespace}
\textcolor{black}{\medskip{}
}
\begin{singlespace}
\textcolor{black}{Sparse regression methods like the Lasso or Elastic
Net can be turned into sparse classifiers \citep{Leng2008,Grosenick:2008p2789,Clemmensen2011}.
Naively, we might imagine performing a two-class classification simply
by running a regression with Lasso or the Elastic Net on a target
vector containing $1$'s and 0's depending on the class of each observation
$y_{i}\in\{0,1\}$. We would then take the predicted values from the
regression $\widehat{y}$ and classify to $0$ if the $i$th estimate
$\widehat{y}_{i}<0.5$ and to $1$ if the estimate $\widehat{y}_{i}>0.5$
(for example). In the multi-class case (i.e. $J$ classes with $J>2$),
multi-response linear regression could be used as a classifier in
a similar way. This would be done by constructing an indicator response
matrix $Y$, with $n$ rows and $J$ columns (where again $n$ is
the number of observations and $J$ is the number of classes). Then
the $i$th row of $Y$ has a $1$ in the $j$th column if the observation
is in the $j$th class and a $0$ otherwise. If we run a multiple
linear regression of $Y$ on predictors $X$, we can classify by assigning
the $i$th observation to the class having the largest fitted value
$\widehat{Y}_{i1},\widehat{Y}_{i2},...,\widehat{Y}_{iJ}$. With the
exception of binary classification on balanced data, this classifier
has several disadvantages. For instance, the estimates $\widehat{Y}_{ij}$
are not probabilities, and in the multi-class case certain classes
can be {}``masked'' by others, resulting in decreased classification
accuracy \citep{Hastie:2009p2681}. However, applying LDA to the fitted
values of such a multiple linear regression classifier is mathematically
equivalent to fitting the full LDA model \citep{BriemanIhaka1984},
yielding posterior probabilities for the classes and dramatically
improving classifier performance over the original multivariate regression
in some cases \citep{Hastie:1994p2648,Hastie:1995p2589,Hastie:2009p2681}. }
\textcolor{black}{\citet{Hastie:1994p2648} and \citet{Hastie:1995p2589}
exploit equivalences between multiple regression and LDA and between
LDA and canonical correlation analysis to develop a procedure they
call Optimal Scoring (OS). OS allows us to build a classifier by first
fitting a multiple regression to $Y$ using an arbitrary regression
method, and then linearly transforming the fitted results of this
regression using the OS procedure (see }\citet{Hastie:1994p2648}\textcolor{black}{{}
for further algorithmic and statistical details). This procedure yields
both class probability estimates and discriminant coordinates, and
allows us to use any number of regression methods as discriminant
classifiers. This approach is discussed in detail for nonlinear regression
methods applied to a few input features in \citet{Hastie:1994p2648},
and for regularized regression methods applied to numerous (i.e.,
hundreds) of correlated input features in \citet{Hastie:1995p2589}.
Here we extend the results of the latter work to include sparse structured
regression methods that can be fit efficiently to hundreds of thousands
of input features. }
\textcolor{black}{More formally, OS finds an optimal scoring function
$\theta:g\rightarrow\mathbb{R}$ that maps classes $g\in\{1,...J\}$
into the real numbers. In the case of a multi-class classification
using the Elastic Net, we can apply OS to yield estimates}
\textcolor{black}{
\begin{eqnarray}
(\widehat{\Theta},\widehat{\beta}) & = & \kappa\ \underset{\Theta,\beta}{\text{argmin}}\ \|Y\Theta-X\beta\|_{2}^{2}+\lambda_{1}\|\beta\|_{1}+\lambda_{2}\|\beta\|_{2}^{2}\label{eq:5}\\
& & \text{subject to }n^{-1}\|Y\Theta\|_{2}^{2}=1,
\end{eqnarray}
where $\Theta$ is a matrix that yields the optimal scores when applied
to indicator matrix $Y$, and where we add the constraint \eqref{eq:5}
to avoid degenerate solutions \citep{Grosenick:2008p2789}. Given
that this is just a sparse version of PDA \citep{Hastie:1995p2589},
we have called this combination Sparse Penalized Discriminant Analysis
(SPDA). It has also recently been called Sparse Discriminant Analysis
(SDA) \citet{Clemmensen2011} (and for an interesting alternative
approach for constructing sparse linear discriminant classifiers,
see \citet{Witten2011}).}
\textcolor{black}{For simplicity, we consider only a local spatiotemporal
smoothing penalty in the current study, although using more elaborate
spatial/temporal coordinates would follow similar logic. The SPDA-GraphNet
is defined as
\begin{eqnarray}
(\widehat{\Theta},\widehat{\beta}) & = & \kappa\ \underset{\Theta,\beta}{\text{argmin}}\ \|Y\Theta-X\beta\|_{2}^{2}+\lambda_{1}\|\beta\|_{1}+\lambda_{G}\|\beta\|_{G}^{2}\label{eq:10}\\
& & \text{subject to \ensuremath{n^{-1}\|}Y\ensuremath{\Theta\|_{2}^{2}}=1.}
\end{eqnarray}
It is important to note that the direct equivalence between penalized
OS and penalized LDA has only recently been proven in the binary classification
case, and does not hold for multi-class classification problems \citep{Merchante2012ICML}.
However, both approximate methods that iteratively minimize over $\Theta$
and $\beta$ \citep{Clemmensen2011} and equivalent methods based
on the Group Lasso \citep{Merchante2012ICML} could be used with GraphNet
regression methods to build multi-class GraphNet classifiers. We note
that in the binary classification case there are at least two options
to turn regression methods into classifiers: Optimal Scoring and logistic
regression (see e.g. \citealt{Friedman2010}). In the case of multiple
classes, the approaches of \citep{Clemmensen2011,Merchante2012ICML}
provide LDA or LDA-like classifiers. Sparse multinomial regression
could also be used in the multi-class case. Any of these approaches
may be used to turn GraphNet regression methods into GraphNet classifiers.
Because Optimal Scoring converts regression methods into equivalent
linear discriminant classifiers, it allows us to combine notions from
regression such as degrees of freedom with notions from discriminant
analysis such as class visualization in the discriminant space using
discriminant coordinates and trial-by-trial posterior probabilities
for individual observations \citep{Hastie:1995p2589}. This, and its
greater computational simplicity over logistic and multinomial regressions,
make OS an appealing approach.}
\end{singlespace}
\textcolor{black}{\medskip{}
}
\subsubsection{\textcolor{black}{Turning regression methods into classifiers: relating
Support Vector Machines (SVM) to penalized regression}}
\textcolor{black}{\medskip{}
}
In addition to the LDA and logistic/multinomial approaches to classification,
maximum margin classifiers like SVM have been very successful. As
we will also be developing a Support Vector GraphNet (SVGN) variant
below, we briefly discuss how support vector machines can be related
to regression methods like those described above. If the data is centered
such that an intercept term can be ignored, the SVM solution can be
written
\[
\widehat{\beta}=\underset{\beta}{\text{argmin}}\ \sum_{i=1}^{n}(1-y_{i}x_{i}^{T}\beta)_{+}+(\lambda/2)\|\beta\|_{2}^{2},
\]
where $(\cdot)_{+}$ indicates taking the positive part of the quantity
in parentheses. In this function estimation formulation of the SVM
problem, we see the similarity to the penalized regression methods
above: the only difference is that the usual squared error loss $L(y_{i},x_{i},\beta)=(y_{i}-x_{i}^{T}\beta)^{2}$
has been replaced by the {}``hinge loss'' function $L_{H}(y_{i},x_{i},\beta)=(1-y_{i}x_{i}^{T}\beta)_{+}$
. This function is non-differentiable, and more recent work \citep{WangZou2008}
uses a differentiable {}``Huberized hinge loss'' (Figure 2a), which
we will discuss in greater detail below. The important point here
is that formulating the SVM problem as a loss term and a penalty term
reveals how we might build an SVM with more general penalization,
such as that used in GraphNet regression methods above.\textcolor{black}{\medskip{}
}
\begin{singlespace}
\subsection{Novel extensions of GraphNet methods\textcolor{black}{\medskip{}
}}
\end{singlespace}
\subsubsection{\textcolor{black}{Robust GraphNet and Adaptive Robust GraphNet\medskip{}
}}
\begin{singlespace}
\textcolor{black}{More generally, we can formulate the penalized regression
problem of interest as minimizing the penalized empirical risk }\textbf{\textcolor{black}{$\mathcal{R}_{p}(\beta)$}}\textcolor{black}{{}
as a function of the coefficients, so that
\begin{equation}
\widehat{\beta}=\underset{\beta}{\text{argmin}\ }\mathcal{R}_{p}(\beta)=\underset{\beta}{\text{argmin}}\ \mathcal{R}(y,\widehat{y})+\lambda\mathcal{P}(\beta),\label{eq:6}
\end{equation}
where $\widehat{y}$ is the estimate of response variable $y$ (note
$\widehat{y}=X\widehat{\beta}$ in the linear models we consider)
and $\mathcal{R}(y,\widehat{y})=n^{-1}\sum_{i=1}^{n}L(y_{i},\widehat{y}_{i})$
is the average of the loss function over the training data (the {}``empirical
risk'') of the loss function $L(y_{i},\widehat{y}_{i})$ that penalizes
differences between the estimated and true values of $y$ at the $i$th
observation. For example, in \eqref{eq:3}--\eqref{eq:5} we used
$\mathcal{R}(y,\widehat{y})=\|y-\widehat{y}\|_{2}^{2}=\sum_{i=1}^{n}(y_{i}-\widehat{y}_{i})^{2}$
({}``squared error loss''). While squared error loss enjoys many
desirable properties under the assumption of Gaussian noise, it is
sensitive to the presence of outliers.}
\textcolor{black}{Outlying data points are an important consideration
when modeling fMRI data, in which a variety of factors ranging from
residual motion artifacts to field inhomogeneities can cause some
observations to fall far from the sample mean. In the case of standard
squared-error loss (as in equations \eqref{eq:2}--\eqref{eq:5}),
these outliers can have undue influence on the model fit due to the
quadratically increasing penalty on the residuals (see Figure 2a).
A standard solution in such cases is to use a robust loss function,
such as the Huber loss function \citep{Huber2009},
\begin{eqnarray}
\mathcal{R}_{H}(y,\widehat{y};\delta) & =n^{-1} & \sum_{i=1}^{n}L_{\delta}(y_{i}-\widehat{y}_{i})\label{eq:7}\\
\text{where } & & L_{\delta}(y_{i}-\widehat{y}_{i})=\begin{cases}
(y_{i}-\widehat{y}_{i})^{2}/2 & \text{for }|y_{i}-\widehat{y}_{i}|\leq\delta\\
\delta|y_{i}-\widehat{y}_{i}|-\delta^{2}/2 & \text{for }|y_{i}-\widehat{y}_{i}|>\delta
\end{cases}.\nonumber
\end{eqnarray}
This function penalizes residuals quadratically when they are less
than or equal to parameter $\delta$, and linearly when they are larger
than $\delta$ (Figure 2a). A well specified $\delta$ can thus significantly
reduce the effects of large residuals (outliers) on the model fit,
as they no longer have the leverage resulting from a quadratic penalty.
As $\delta\rightarrow\infty$ (or practically, when it becomes larger
than the most outlying residual) we recover the standard squared-error
loss. }
\textcolor{black}{Since GraphNet uses squared-error loss, it can now
be modified to include a robust penalty like the Huber loss defined
above. Replacing the squared error loss function with the loss function
\eqref{eq:7} yields }
\textcolor{black}{
\begin{equation}
\widehat{\beta}=\kappa\ \underset{\beta}{\text{argmin}\ }\mathcal{R}_{H}(y,X\beta;\delta)+\lambda_{1}\|\beta\|_{1}+\lambda_{G}\|\beta\|_{G}^{2}.\label{eq:11}
\end{equation}
}The Adaptive Robust GraphNet is then a straightforward generalization
(following section 2.1.5; see also the next section)
\end{singlespace}
\textcolor{black}{The SPDA-RGN classifier can be defined like the
standard GraphNet classifier \eqref{eq:10}. However, the SPDA-RGN
classifier now has an additional hyperparameter to be estimated (or
assumed). Specifically, the value of $\delta$ determines where the
loss function switches from quadratic to linear (Figure 2a). Further,
the loss function on the residuals is no longer quadratic and therefore
could slow down optimization convergence . We discuss a solution to
this issue next.\medskip{}
}
\begin{singlespace}
\subsubsection{\textcolor{black}{Infimal convolution for non-quadratic loss functions}}
\end{singlespace}
\textcolor{black}{\medskip{}
}
In order to solve both the Robust GraphNet, Adaptive Robust GraphNet,
and Support Vector GraphNet problems efficiently, we introduce a general
method for solving coordinate-wise descent problems with smooth, non-quadratic
convex loss functions as penalized least squares problems in an augmented
set of variables.
\textcolor{black}{Convergence speed of subgradient methods such as
coordinate-wise descent can be substantially improved when the loss
function takes a quadratic form, while non-quadratic loss functions
can take numerous iterations to converge for each coefficient, significantly
increasing computation time. However, we can circumvent these problems
and extend the applicability of coordinate-wise descent methods using
a trick from convex analysis to rewrite these loss functions as quadratic
forms in an augmented set of variables. This method is called infimal
convolution \citep{Rockafellar1970}, and is defined as}
\begin{flushleft}
\textcolor{black}{
\begin{equation}
(f\star_{\text{inf}}g)(x):=\inf_{y}\{f(x-y)+g(y)|y\in\mathbb{R}^{n}\},\label{eq:infimal convolution}
\end{equation}
where $f$ and $g$ are two functions of $x\in\mathbb{R}^{p}$. In
this way it is possible to rewrite the $i$th term in the the Huber
loss function \eqref{eq:7} as the infimal convolution of the squared
and absolute-value functions applied to the $i$th residual $r_{i}$:
\begin{equation}
\rho_{\delta}(r_{i})=((1/2)(\cdot)^{2}\star_{\text{inf}}|\cdot|)(r_{i})=\inf_{a_{i}+b_{i}=r_{i}}a_{i}^{2}/2+\delta|b_{i}|,\label{eq:infimal conv for huber loss}
\end{equation}
where $r_{i}=y_{i}-(X\widehat{\beta})_{i}$ (note that a dot $(\cdot)$
is used to indicate the functional nature of the expression without
having to add additional dummy variables). This yields the augmented
estimation problem
\begin{equation}
(\widehat{\alpha},\widehat{\beta})=\underset{\alpha,\beta}{\text{argmin}}\ (1/2)\|y-X\beta-\alpha\|_{2}^{2}+\lambda_{G}\beta^{T}G\beta+\delta\|\alpha\|_{1}+\lambda_{1}\|\beta\|_{1},\label{eq:infimal huberized graphnet}
\end{equation}
where we have introduced the auxiliary variables $\alpha\in\mathbb{R}^{n}$.
Considering the residuals $r_{i}$, the first term in the objective
of \eqref{eq:infimal huberized graphnet} can be written $(1/2)\|y-X\beta-\alpha\|_{2}^{2}=(1/2)\sum_{i}(r_{i}-\alpha_{i})^{2},$
and thus each $\alpha_{i}$ can directly reduce the residual sum of
squares corresponding to a single observation by taking a value close
to $r_{i}$. Since for some $\delta$ the penalty $\delta\|\alpha\|_{1}$
requires the $\alpha$ vector to be $k$-sparse, this formulation
intuitively allows a linear rather that quadratic penalty to be placed
on $k$ of the residuals (with $k$ tuned by choice of $\delta,$
as expressed in the Huber loss formulation). These will correspond
to those observations with the most leverage (the most {}``outlying''
points). We can then rewrite \eqref{eq:infimal huberized graphnet}
as
\begin{eqnarray}
\widehat{\gamma} & = & \underset{\gamma}{\text{argmin}}\ (1/2)\|y-Z\gamma\|_{2}^{2}+\lambda_{G}\gamma^{T}G'\gamma+\sum_{j=1}^{p+n}w_{j}|\gamma_{j}|\label{eq:infimal huberized graphnet rewritten}\\
\text{} & & Z=[X\ \ I_{n\times n}],\ \ \gamma=[\beta\ \ \alpha],\ \ w_{j}=\begin{cases}
\lambda_{1} & j=1,..,p\\
\delta & j=p+1,\ldots,p+n,
\end{cases}\nonumber \\
& & G'=\left[\begin{array}{ll}
G & 0_{1\times n}\\
0_{n\times1} & 0_{n\times n}
\end{array}\right]\in S_{+}^{(p+n)\times(p+n)},\nonumber
\end{eqnarray}
where $S_{+}^{m\times m}$ is the set of positive semidefinite $m\times m$
matrices. This is just a GraphNet problem in an augmented set of $p+n$
variables, and so can be solved using the fast coordinate-wise descent
methods discussed in section 2.4 below. After solving for augmented
coefficients $\widehat{\gamma}$ we can simply discard the last $n$
coefficients to yield $\widehat{\beta}$. A similar approach can be
taken with the hinge-loss of a support vector machine classifier (as
we show next), or more generally with any loss function decomposable
into an infimal convolution of convex functions (see Appendix). The
Adaptive Robust GraphNet is easily obtained by letting
\[
w_{j}=\begin{cases}
\lambda_{1}^{*}\widehat{w}_{j} & j=1,..,p\\
\delta & j=p+1,\ldots,p+n
\end{cases}
\]
in \eqref{eq:infimal huberized graphnet rewritten} (see section 2.1.5
for more details on adaptive estimation).\medskip{}
}
\par\end{flushleft}
\begin{singlespace}
\subsubsection{\textcolor{black}{Huberized Support Vector Machine (SVM) GraphNet
for classifications}}
\end{singlespace}
\textcolor{black}{\medskip{}
}
\textcolor{black}{In the $p\gg n$ classification problem, maximum-margin
classifiers such as the support vector machine (SVM) often perform
exceedingly well in terms of classification accuracy, but do not yield
readily interpretable coefficients. For this reason we also develop
a sparse SVM with graph constraints, the Support Vector GraphNet (SVGN),
related to the {}``Hybrid Huberized SVM'' of \citet{WangZou2008}
as an alternative to the SPDA method. Using a {}``Huberized-hinge''
loss function $\mathcal{R}_{HH}$ (see below) on the fit residuals,
we have }
\begin{singlespace}
\textcolor{black}{
\begin{equation}
\widehat{\beta}=\kappa\ \underset{\beta}{\text{argmin}\ }\mathcal{R}_{HH}(y^{T}X\beta;\delta)+\lambda_{1}\|\beta\|_{1}+\lambda_{G}\|\beta\|_{G}^{2},\label{eq:12-1}
\end{equation}
where $y\in\{-1,1\}$, and letting $\widehat{y}=X\widehat{\beta}$
be the estimates of the target variable,}
\end{singlespace}
\textcolor{black}{
\begin{eqnarray}
\mathcal{R}_{HH}\left(y,\widehat{y};\delta\right) & = & n^{-1}\sum_{i=1}^{n}L_{\delta}(y_{i},\widehat{y}_{i})\label{eq:huber loss}\\
\text{where} & & L_{\delta}(y_{i},\widehat{y}_{i})=\begin{cases}
\left(1-y_{i}\widehat{y}_{i}\right)^{2}/2\delta & \text{for }1-\delta<y_{i}\widehat{y}_{i}\leq1\\
1-y_{i}\widehat{y}_{i}-\delta/2 & \text{for }y_{i}\widehat{y}_{i}\leq1-\delta\\
0 & \text{for }y_{i}\widehat{y}_{i}>1,
\end{cases}\nonumber
\end{eqnarray}
which is the Huberized-hinge loss of \citet{WangZou2008}. As with
the Huber loss, there is an additional hyperparameter $\delta$ to
be estimated or assumed. In this case, $\delta$ determines where
the hinge-loss function switches from the quadratic to the linear
regime (see Figure 2a). This problem's loss function can also be written
using infimal convolution to yield a more convenient quadratic objective
term (see Appendix). Finally, we discuss a heuristic alternative to
adaptive methods for adjusting nonzero coefficient magnitudes to match
the scale of the original data. This approach can be used with any
of the above methods. \medskip{}
}
\begin{singlespace}
\subsubsection{\textcolor{black}{Effective degrees of freedom for GraphNet estimators}}
\end{singlespace}
\textcolor{black}{\medskip{}
}
Following results for the Lasso \citet{ZouHastieDFs} and the Elastic
Net \citet{VanDerKooij2007}, the effective degrees of freedom $\widehat{df}$
for the GraphNet regression are given by the trace of the {}``hat
matrix'' $H_{\lambda_{G}}(\mathcal{A})$ for the GraphNet estimator:
\[
\widehat{df}=\text{tr}(H_{\lambda_{G}}(\mathcal{A}))=\text{tr}\left(X_{\mathcal{A}}\left(X_{\mathcal{A}}^{T}X_{\mathcal{A}}+\lambda_{G}G\right)^{-1}X_{\mathcal{A}}^{T}\right),
\]
where $X_{\mathcal{A}}$ denotes the columns of $X$ containing just
the {}``active set'' (those variables with nonzero coefficients
corresponding to a particular choice of $\lambda_{1}$). This quantity
is very useful in calculating standard model selection criteria such
as the Akaike Information Criterion (AIC), Bayesian Information Criterion
(BIC), Mallow's $C_{p}$, Generalized Cross Validation (GCV), and
others. Importantly, it can also be used for the various GraphNet
methods, as each of these is solved as an equivalent GraphNet problem
(for example, equation \ref{eq:infimal huberized graphnet rewritten})
for the Adaptive Robust GraphNet. \textcolor{black}{\medskip{}
}
\begin{singlespace}
\subsubsection{\textcolor{black}{Rescaling coefficients to account for {}``double
shrinking''}}
\end{singlespace}
\textcolor{black}{\medskip{}
}
\begin{singlespace}
\textcolor{black}{The Elastic Net was originally formulated by \citet{ZouHastie}
in both {}``naive'' and rescaled forms. The authors noted that a
combination of $\ell_{1}$ and $\ell_{2}$ penalties can {}``double
shrink'' the coefficients. To correct this they proposed rescaling
the {}``naive'' solution by a factor of $\kappa=1+\lambda_{2}$
\citep{ZouHastie}. Heuristically, the aim is to retain the desirable
variable selection properties of the Elastic Net while rescaling the
coefficients to be closer to the original scale. However, as this
result is derived for an orthogonal design, it is not clear that $\kappa=1+\lambda_{2}$
is the correct multiplicative factor if the data are collinear, and
this can complicate the problem of choosing a final set of coefficients.
Following the arguments of \citet{ZouHastie}, for GraphNet regression
we might rescale each coefficient by $\kappa_{j}=k(\widehat{\Sigma}_{jj}+\lambda_{G}G_{jj})$
where $k$ is the number of iterations used in the coordinate-wise
descent optimization (and thus the number of times shrinkage related
to $G$ is applied, see equation \ref{eq:15-2} and derivations in
Appendix) and where $\widehat{\Sigma}=X^{T}X$. In the case of an
orthogonal design and $G=I$ we would have $\widehat{\Sigma}=1$ and
thus $\kappa_{j}=1+\lambda_{G}$---reducing to the Elastic Net rescaling
employed in \citet{ZouHastie}.}
\textcolor{black}{A simpler alternative is to fit the Elastic Net,
generating a fitted response $\hat{y}$, and then to regress $y$
on $\hat{y}$. In particular, solving the simple linear regression
problem
\[
y=\kappa\widehat{y}=\kappa X\widehat{\beta},\ \kappa\in\mathbb{R}
\]
yields an estimate $\widehat{\kappa}$ that can be used to rescale
the coefficients obtained from fitting the Elastic Net (Daniela Witten
and Robert Tibshirani, personal communication). The intuitive motivation
for this heuristic is that it will produce a $\widehat{\kappa}$ that
puts $\widehat{\beta}$ and $\widehat{y}$ on a reasonable scale for
fitting $y$.}
\textcolor{black}{Besides its simplicity, the principal advantage
of this approach is that it requires no analytical knowledge about
the amount of shrinkage that occurs as $\lambda_{G}$ is increased.
This is particularly appealing because the same strategy of regressing
$\hat{y}$ on $y$ can be used with more general problems with more
complicated forms, such as the Adaptive Robust GraphNet, where the
additional shrinkage caused by the graph penalty can be corrected
in this way.}
\end{singlespace}
Finally, we note that over-shrinking is not necessarily bad for classification
accuracy. Indeed it may improve accuracy due to the rather complicated
relationship between bias and variance in the of classification (for
an excellent discussion in the context of 0-1 loss see \citet{Friedman1997}).
The focus on recovering good estimates of coefficent magnitude in
this section is thus most relevant to regression and to situations
in which correct estimates of coefficient magnitude are important.
\textcolor{black}{\medskip{}
}
\begin{singlespace}
\subsection{\textcolor{black}{Interpreting GraphNet regression and classification
\medskip{}
}}
\end{singlespace}
\subsubsection{\textcolor{black}{Interpreting GraphNet parameters: dual variables
as prices\medskip{}
}}
\textcolor{black}{The GraphNet problem expressed in equation \eqref{eq:8}
derives from a constrained maximum likelihood problem, in which we
want to maximize the likelihood of the parameters given the data,
subject to some hard constraints on the solution---specifically, that
they are sparse and structured (in the sense that their $\ell_{1}$
and graph-weighted $\ell_{2}$ norms are less than or equal to some
constraint size). For concave likelihoods (as in generalized linear
models and the cases considered above), this is a constrained convex
optimization problem
\begin{eqnarray}
\underset{\beta}{\text{maximize}} & & \text{loglik}(\beta|X,y)\label{eq: constrained mle objective}\\
\text{subject to} & & \|\beta\|_{1}\leq c_{1}\label{eq: constrained mle l1}\\
& & \|\beta\|_{G}^{2}\leq c_{G},\label{eq: constrained mle graphpen}
\end{eqnarray}
where $c_{1}\in\mathbb{R}_{+}$ and $c_{G}\in\mathbb{R}_{+}$ set
hard bounds on the size of the coefficients in the $\ell_{1}$ and
$\ell_{G}$ norms, respectively. A standard approach for solving such
problems is to relax the hard constraints to linear penalties \citep{BoydVandenberghe2004}
and consider just those terms containing $\beta$, giving the {}``Lagrangian''
form of the GraphNet problem}
\textcolor{black}{
\begin{equation}
\widehat{\beta}\underset{\beta}{=\text{argmin}}\ -\text{loglik}(\beta|X,y)+\lambda_{1}\|\beta\|_{1}+\lambda_{G}\|\beta\|_{G}^{2},\ \lambda_{1},\lambda_{G}\in\mathbb{R}_{+},\label{eq:penalized loglikelihood}
\end{equation}
which contains a negative likelihood term that measures misfit to
the data as well as the two penalties characteristic of GraphNet estimators. }
\textcolor{black}{In this Lagrangian formulation, the dual variables
$\lambda_{1}$ and $\lambda_{G}$ represent (linear) costs in response
to a violation of the constraints. Since we solve problem \eqref{eq:penalized loglikelihood},
$c_{1}$ and $c_{G}$ are effectively zero, and we are penalized for
any deviation of the coefficients from zero. This leads to one interpretation
of $\lambda_{1}$ and $\lambda_{G}$: they are prices that we are
willing to pay to improve the likelihood at the expense of a less
sparse or less structured solution, respectively. For this reason,
examining fit sensitivity to different values of $\lambda_{1}$ and
$\lambda_{G}$ tells us about underlying structure in the data. For
example, if the task-related neural activity was very sparse and highly
localized in a few uncorrelated voxels, then we should be willing
to pay more for sparsity and less for smoothness (i.e., large $\lambda_{1}$,
small $\lambda_{G}$). In contrast, if large smooth and correlated
regions underlie the task, then tolerating a large $\lambda_{G}$
could substantially improve the fit. To explore such possibilities,
we can plot test rates from cross validations at different combinations
of parameters. Figure 6 shows plots of median test classification
rates as a function of $\lambda_{1}$ and $\lambda_{G}$ over the
parameter grid on which the various GraphNet classifiers were fit.
We see that there are regions in the $(\lambda_{1},\lambda_{G})$
parameter space that clearly result in better median classification
test rates, corresponding to fits with particular levels of smoothness
and sparsity.}
\textcolor{black}{\medskip{}
}
\subsubsection{\textcolor{black}{Interpreting GraphNet coefficients }}
\textcolor{black}{\medskip{}
}
Problem \eqref{eq:penalized loglikelihood} can also be arrived at
from a Bayesian perspective as a maximum a posteriori (MAP) estimator.
In this case, the form of the penalty \textbf{$\mathcal{P}(\beta)$
}is related to one's prior beliefs about the structure of the coefficients.
\textcolor{black}{For example, under the well-known equivalence of
penalized regression techniques and posterior modes, the Elastic Net
penalty corresponds to the prior
\[
p_{\lambda_{1},\lambda_{2}}(\beta)\propto\exp\left\{ -\left(\lambda_{1}\|\beta\|_{1}+\lambda_{2}\|\beta\|_{2}^{2}\right)\right\}
\]
\citep{ZouHastie}. The GraphNet penalty thus corresponds to the prior
distribution
\begin{eqnarray}
p_{\lambda_{1},\lambda_{G}}(\beta) & \propto & \exp\left\{ -\left(\lambda_{1}||\beta||_{1}+\lambda_{G}\beta^{T}G\beta\right)\right\} \nonumber \\
& \propto & \prod_{i=1}^{p}\exp\left\{ -\lambda_{1}|\beta_{j}|\right\} \prod_{i=1}^{p}\exp\left\{ -\lambda_{G}\sum_{i\sim j}\beta_{i}G_{ij}\beta_{j}\right\} ,\label{eq: bayesian interp}
\end{eqnarray}
where $i\sim j$ denotes that node $i$ in the graph $G$ is adjacent
to node $j$. Therefore, the GraphNet problems are also equivalent
to a MAP estimator of the coefficients with a prior consisting of
a convex combination of a global Laplacian (double-exponential) and
a local Markov Random Field (MRF) prior. In other words, GraphNet
methods explicitly take into account prior beliefs about coefficients
being globally sparse but locally structured. }
\begin{singlespace}
\subsection{\textcolor{black}{Optimization and computational considerations\medskip{}
}}
\end{singlespace}
\begin{singlespace}
\subsubsection{\textcolor{black}{Coordinate-wise descent and active set methods}}
\end{singlespace}
\textcolor{black}{\medskip{}
}
\begin{singlespace}
\textcolor{black}{Fitting regression methods to whole-brain fMRI data
requires efficient computational methods, particularly when they must
be cross-validated over a grid of possible parameter values. For instance,
in the shopping example described in greater detail below (section
2.5), 26,630 input features (voxels) at each of 7 time points are
used to classify future choices to purchase a product or not. Fitting
the Adaptive Robust GraphNet using leave-one-subject-out (LOSO) cross-validation
(i.e., 25 fits) for each realization of possible parameter values
over this $90\times5\times6\times10\times3$ grid of possible parameters
$\{\lambda_{1},G,\lambda_{G},\delta,\lambda_{1}^{*}\}$ requires $2,025,000$
model fits on $1,882$ observations of $186,410$ input features. }
\textcolor{black}{To efficiently fit GraphNet methods with millions
of parameter combinations over hundreds of thousands of input features,
we formulated the minimization problem (i.e., equations \eqref{eq:8},
\eqref{eq:11}, and \eqref{eq:12-1}) as a coordinate-wise optimization
procedure \citep{Tseng1988,Tseng2001} using active set methods \citep{Friedman2010}.
This approach fit one coefficient value at a time ({}``coordinate-wise''
descent), holding the rest constant, and kept an {}``active set''
of nonzero coefficients. Fitting was initiated with a large value
of $\lambda_{1}$ (corresponding to all coefficients being zero),
and then slowly decreased $\lambda_{1}$ to allow more and more coefficients
into the model fit. This procedure thus considered an {}``active
set'' of the model coefficients at each coordinate-wise update, rather
than all 186,410 inputs. Occasional sweeps though all the coefficients
were made to search for new variables to include, as in \citet{Friedman2010}.
Model fitting terminated before $\lambda_{1}$ reached zero, since
fitting a fully dense set of coefficients is computationally expensive
and known to produce poor estimates \citep{Hastie:2009p2681,Friedman2010}.
Various heuristics and model selection criteria may be used for choosing
a stopping point, for example, stopping once the AIC or BIC for the
model stops decreasing and starts increasing. AIC is known to be over-inclusive
in model selection, and is therefore a more conservative stopping
point.}
\textcolor{black}{Coordinate-wise descent is guaranteed to converge
for GraphNet methods because they are all of the form
\begin{equation}
\underset{\beta}{\text{argmin}}\ f(\beta_{1},...,\beta_{p})=\underset{\beta}{\text{argmin}}\ g(\beta_{1},..,\beta_{p})+\sum_{j=1}^{p}h(\beta_{j}),\label{eq:14}
\end{equation}
where $g(\beta_{1},..,\beta_{p})$ is a convex, differentiable function
(e.g., squared-error and Huber loss plus the quadratic penalty $||\beta||_{G}^{2}$),
and where each $h(\beta_{j})$ is a convex (but not necessarily differentiable)
function (e.g., the $\ell_{1}$ penalty). If the convex, non-differentiable
part of the penalty function is separable in coordinates $\beta_{j}$
(as is true of $||\beta||_{1}=\sum_{j=1}^{p}|\beta_{j}|$), then coordinate
descent converges to a global solution of the minimization problem
\citep{Tseng2001}. In the case of Huber loss or Huberized-hinge loss,
the two-part loss function can be written as a single quadratic loss
function using infimal convolution as described in section 2.2.3.
For instance, consider the coordinate-wise updates for the standard
GraphNet problem given in equation \eqref{eq:8}. Letting $\hat{y}=\tilde{X}\tilde{\beta}+X._{j}\beta_{j}$
(where $\tilde{X}=X._{\neq j}$ is the matrix $X$ with the $j$th
column removed, and $\tilde{\beta}=\beta_{\neq j}$ the coefficient
vector with the $j$th coefficient removed), the subdifferential of
the risk with respect to only the $j$th coefficient (holding the
others fixed) is
\begin{equation}
\partial_{\beta_{j}}\mathcal{R}_{p}=-X._{j}^{T}y+X._{j}^{T}\tilde{X}\tilde{\beta}+X._{j}^{T}X._{j}\beta_{j}+(\lambda_{2}/2)\tilde{\beta}^{T}(G_{\neq j}.)._{j}+\lambda_{2}G_{jj}\beta_{j}+(\lambda_{1}/2)\text{\ensuremath{\Gamma}}(\beta_{j}),\label{eq:15-1}
\end{equation}
where the set-valued function $\Gamma(\beta_{j})=-1$ if $\beta_{j}<0$,
$\Gamma(\beta_{j})=1$ if $\beta_{j}>0$ and $\Gamma(\beta_{j})\in[-1,1]$
if $\beta_{j}=0$. If we let $\Gamma(\beta_{j})=\text{sign}(\beta_{j}),$
in equation \eqref{eq:15-1} (which is always a particular subgradient
in the subdifferential of the risk), then the coordinate update iteration
for the $j$th coefficient estimate is
\begin{equation}
\hat{\beta}_{j}\leftarrow\frac{S\left(X._{j}^{T}(y-\tilde{X}\tilde{\beta})-(\lambda_{2}/2)\tilde{\beta}^{T}(G_{\neq j}.)._{j},\ \lambda_{1}/2\right)}{X._{j}^{T}X._{j}+\lambda_{2}G_{jj}},\label{eq:15-2}
\end{equation}
where
\begin{equation}
S(x,\gamma)=\text{sign}(x)(|x|-\gamma)_{+}\label{eq:soft-thresh}
\end{equation}
is the soft-thresholding function \citep{DonohoUST,Friedman:2007p36}.
Note that if graph $G=I$, and the columns of $X$ are standardized
to have unit norm, then the coordinate-wise Elastic Net update is
recovered \citep{VanDerKooij2007,FriedmanPWCO}.}
\end{singlespace}
\textcolor{black}{\medskip{}
}
\begin{singlespace}
\subsubsection{\textcolor{black}{Computational complexity}}
\end{singlespace}
\textcolor{black}{\medskip{}
}
\textcolor{black}{A closer look at equation \eqref{eq:15-2} reveals
that if the variables are standardized (such that $X._{j}^{T}X._{j}=1$)
then the $(c+1)$st coefficient update for the $j$th coordinate can
be rewritten
\begin{equation}
\hat{\beta}_{j}^{(c+1)}\leftarrow S\left(\sum_{i=1}^{N}x_{ij}r_{i}^{(c)}+\hat{\beta}_{j}^{(c)}-(\lambda_{2}/2)\sum_{k\neq j}\beta_{k}G_{kj},\ \lambda_{1}/2\right)/(1+\lambda_{2}G_{jj}),\label{15-3}
\end{equation}
where $r=y-\hat{y}$ is the vector of residuals. Letting $m$ be the
number of off-diagonal nonzero entries in $G$ and initializing with
$\hat{\beta}_{j}^{(0)}=0$ for all $j$ and $r^{(0)}=y$, the first
sweep through all $p$ coefficients will take $O(pn)+O(m)$ operations.
Once $a_{1}$ variables are included in the active set, $q$ iterations
are performed according to \eqref{15-3} until the new estimates converge,
at which point $\lambda_{1}$ is decreased incrementally and another
$O(pn)$ sweep is made through the coefficients to find the next active
set with $a_{2}$ variables (using the previous estimate as a warm
start to keep $q$ small). This procedure is repeated for $l$ values
of $\lambda_{1}$, until the fit stops improving or a pre-specified
coefficient density is reached. Let $a=\sum_{i=1}^{l}a_{i}$ denote
the total number of coefficients updates over all $l$ fits. The total
computational complexity is then $O(lpn)+O(lm)+O(aq)$. Thus if $G$
is relatively sparse (so $m$ is small) and if it requires few iterations
for coefficients in an active set to converge ($q$ small)---which
is true if the unpenalized loss function is quadratic---then the computational
complexity is dominated by the $O(lpn)$ term representing the sweep
through the coefficients necessary to find the next active set for
each new value of $\lambda_{1}$. We note that this suggests that
including a screening procedure such as the STRONG rules \citep{Tibs2012}
could further speed up fitting in this context. Either making $G$
dense or decreasing $\lambda_{1}$ until $a$ becomes large can cause
the other complexity terms to play a significant role and slow the
speed of the algorithm. For example, if $G$ is dense, then $m=p^{2}-p$
and the $O(lm)$ term will dominate.}
\textcolor{black}{\medskip{}
}
\begin{singlespace}
\subsubsection{\textcolor{black}{Cross validation, classification accuracy, and
parameter tuning}}
\end{singlespace}
\textcolor{black}{\medskip{}
}
For training and test data, trials for each subject were resampled
within-subject to consist of 80 trials with exactly 40 purchases.
If the subject originally had more than 40 purchases, sampling without
replacement was used to select 40. If the subject originally had fewer
than 40 purchases, sampling with replacement was used to select 40.
Similar sampling was used to select exactly 40 trials without purchases.
This resampling scheme ensured that the trials for each subject were
balanced between purchasing and not purchasing. Further, because our
cross-validation schemes defined folds on the subject level, this
ensured that every training and test set in the cross-validation was
also balanced.
\begin{singlespace}
\textcolor{black}{For the cross-validation, the range for these grid
values was chosen based off of a few preliminary fits. This grid is
very large, and with the refitting involved in cross-validation, resulted
in millions of fits. The smoothness of the rates as a function of
the parameters (see Figure 6) suggests that smaller grids are likely
better suited to most applications, and we anticipate that more efficient
adaptive approaches to parameter search---such as focused grid search
methods \citep{Jimenez2009} or sampling methods inspired by Bayesian
approaches to similar problems---will ultimately prove superior. We
leave these refinements to future work. The grid values used here
are given in the Appendix.}
\end{singlespace}
In order to choose a final set of coefficient estimates from multiple
fits across cross-validation folds, we took the element-wise median
of the coefficient vectors across the folds. Thus a feature corresponding
to a particular voxel at a particular TR would have to appear (be
nonzero) in more than half of the 25 cross-validation folds in order
to be included in the final coefficient estimate used in the out-of-sample
(OOS) analysis. There are several justifications for taking the median
across folds: (1) the median preserves sparsity, (2) the median is
the appropriate maximum likelihood estimator for the double-exponential
(Laplacian) distribution that corresponds to the $\ell_{1}$ sparsity
prior on the coefficients (see discussion in \citealt{Grosenick:2008p2789}),
(3) such a procedure is closely related to the Median Probability
Model, which is the model consisting of those variables that have
posterior probability $\geq0.5$ of being in a model, and which has
been shown to have optimal predictive performance for linear models
\citep{BarbieriBerger2004}, and (4) it is similar to other recently-developed
model selection procedures for sparse models such as Stability Selection
\citealt{MeinshausenBuhlmann2010} that use the number of times a
variable appears across multiple sparse fits to resampled data in
order to significantly improve model selection. Further, we have found
this approach to be quite effective in practice (see the out-of-sample
results that follow). Note that such inclusion of a variable only
if it appears in more than half of the 25 cross-validation folds is
a natural means of imposing some {}``reliability'' or {}``stability''
on the coefficients.
\textcolor{black}{\medskip{}
}
\subsection{\textcolor{black}{Application: Predicting buying behavior using fMRI\medskip{}
}}
\begin{singlespace}
\subsubsection{\textcolor{black}{Subjects and SHOP task}}
\end{singlespace}
\textcolor{black}{\medskip{}
}
\begin{singlespace}
\textcolor{black}{}
\begin{figure}[t]
\begin{centering}
\textcolor{black}{\includegraphics[scale=0.8]{NI_Figure3}}
\par\end{centering}
\textcolor{black}{\medskip{}
}
\begin{spacing}{0.5}
\raggedright{}\textbf{\textcolor{black}{\footnotesize Figure 3.}}\textcolor{black}{\footnotesize{}
Save Holdings, or Purchase (SHOP) task trial structure. Images represent
what the subject saw, bars represent 2 second TRs (T1-T7). Subjects
saw a labeled product (product period; 4 s, 2 TRs), saw the product\textquoteright{}s
price (price period; 4 s, 2 TRs), and then chose either to purchase
the product or not (by selecting either \textquoteleft{}\textquoteleft{}yes\textquoteright{}\textquoteright{}
or \textquoteleft{}\textquoteleft{}no\textquoteright{}\textquoteright{}
presented randomly on the right or left side of the screen; choice
period; 4 s, 2 TRs), before fixating on a crosshair (2 s, 1 TR) prior
to the onset of the next trial. }\end{spacing}
\end{figure}
\textcolor{black}{{} Data from 25 healthy right-handed subjects were
analyzed (Knutson et al., 2007). Along with the typical magnetic resonance
exclusions (e.g., metal in the body), subjects were screened for psychotropic
drugs, cardiac drugs, ibuprofen, substance abuse in the past month,
and history of psychiatric disorders (DSM IV Axis I) prior to collecting
informed consent. Subjects were paid \$20.00 per hour for participating
and also received \$40.00 in cash to spend on products. Of 40 total
subjects, 6 subjects who purchased fewer than four items per session
(i.e., $<10\%$) were excluded due to insufficient data to fit, 8
subjects who moved excessive amounts (i.e., $>2$ mm between whole
brain acquisitions) were excluded, and one subject's original fMRI
data could not be recovered and so was omitted, yielding the final
total of 25 subjects included in the analysis.}
\textcolor{black}{While being scanned, subjects participated in a
\textquotedbl{}Save Holdings Or Purchase\textquotedbl{} (SHOP) task
(Figure 3). During each task trial, subjects saw a labeled product
(product period; 4 sec), saw the product's price (price period; 4
sec), and then chose either to purchase the product or not (by selecting
either \textquotedbl{}yes\textquotedbl{} or \textquotedbl{}no\textquotedbl{}
presented randomly on the right or left side of the screen; choice
period; 4 s), before fixating on a crosshair (2 s) prior to the onset
of the next trial (see Figure 3). }
\textcolor{black}{Each of 80 trials featured a different product.
Products were pre-selected to have above-median attractiveness, as
rated by a similar sample in a pilot study. While products ranged
in retail price from \$8.00-\$80.00, the associated prices that subjects
saw in the scanner were discounted down to 25\% of retail value to
encourage purchasing. Therefore the cost of each product during the
experiment ranged from \$2.00 to \$20.00. Consistent with pilot findings,
this led subjects to purchase 30\% of the products on average, generating
sufficient instances of purchasing to fit. }
\textcolor{black}{To ensure subjects' engagement in the task, two
trials were randomly selected after scanning to count \textquotedbl{}for
real\textquotedbl{}. If subjects had chosen to purchase the product
presented during the randomly selected trial, they paid the price
that they had seen in the scanner from their \$40.00 endowment and
were shipped the product within two weeks. If not, subjects kept their
\$40.00 endowment. Based on these randomly drawn trials, seven of
twenty-five subjects (28\%) were actually shipped products. }
\textcolor{black}{Subjects were instructed in the task and tested
for comprehension prior to entering the scanner. During scanning,
subjects chose from 40 items twice and then chose from a second set
of 40 items twice (80 items total), with each set presented in the
same pseudo-random order (item sets were counterbalanced across subjects).
We consider only data from the first time each item was presented
here (see \citet{Grosenick:2008p2789} for a comparison between first
and second presentations). After scanning, subjects rated each product
in terms of how much they would like to own it and what percentage
of the retail price they would be willing to pay for it. Then, two
trials were randomly drawn to count \textquotedbl{}for real\textquotedbl{},
and subjects received the outcome of each of the drawn trials.}
\end{singlespace}
A second validation sample included 17 healthy right-handed subjects
\citep{Karmarkar:2012}. These subjects passed the same screening,
inclusion, and exclusion criteria. Of an original sample of 24, 6
subjects purchased fewer than four items per session, and one showed
excessive motion. These subjects were excluded from analyses, as before.
Subjects also received the same payment and underwent the same scanning
and experimental procedures. Importantly, however, subjects were different
individuals who were exposed to different products, and were scanned
more than three years after the original study.\textcolor{black}{\medskip{}
}
\begin{singlespace}
\subsubsection{\textcolor{black}{Image acquisition}}
\end{singlespace}
\textcolor{black}{\medskip{}
}
\begin{singlespace}
\textcolor{black}{Functional images were acquired with a 1.5 T General
Electric MRI scanner using a standard birdcage quadrature head coil.
Twenty-four 4-mm-thick slices (in-plane resolution 3.75 X 3.75 mm,
no gap) extended axially from the midpons to the top of the skull,
providing whole-brain coverage and adequate spatial resolution of
subcortical regions of interest (e.g., midbrain, NAcc, OFC). Whole-brain
functional scans were acquired with a T2{*}-sensitive spiral in-/out-
pulse sequence (TR=2 s, TE=40 ms, flip=90), which minimizes signal
dropout at the base of the brain \citep{GloverLaw}. High-resolution
structural scans were also acquired to facilitate localization and
coregistration of functional data, using a T1-weighted spoiled grass
sequence (TR=100 ms, TE=7 ms, flip=90).}
\end{singlespace}
\textcolor{black}{\medskip{}
}
\begin{singlespace}
\subsubsection{\textcolor{black}{Preprocessing}}
\end{singlespace}
\textcolor{black}{\medskip{}
}
\begin{singlespace}
\textcolor{black}{After reconstruction, preprocessing was conducted
using Analysis of Functional Neural Images (AFNI) software \citep{CoxAFNI}.
For all functional images, voxel time-series were sinc interpolated
to correct for non-simultaneous slice acquisition within each volume,
concatenated across runs, corrected for motion, and normalized to
percent signal change with respect to the voxel mean for the entire
task. For further preprocessing details see \citep{Grosenick:2008p2789}.
Given that spatial blur would artificially increase correlations between
variables for the voxel-wise analysis, we used data with no spatial
blur and a temporal high pass filter for all analyses. Note that in
general, smoothing before running analyses will compound the problems
with correlation mentioned above, resulting in {}``rougher'' (high-frequency)
coefficients overall (see discussion in \citep{Hastie:1995p2589}). }
\textcolor{black}{Spatiotemporal data were arranged as in previous
spatiotemporal analyses \citep{MouraoMiranda:2007p2565}. Specifically,
data was arranged as an $n\times p$ data matrix $X$ with $n$ corresponding
to the number of trial observations on the $p$ input variables, each
of which was a particular voxel at a particular time point. This yielded
26,630 voxels taken at 7 time points (each taken every 2 seconds),
yielding a total of $p=186,410$ input input features per trial. Altogether,
the data used for training and test from \citep{Knutson2007} included
$n=1,882$ trials across the 25 subjects. The validation sample from
\citep{Karmarkar:2012} included $n=322$ trials across the 17 subjects.
In the first case (training and testing on the \citet{Knutson2007}
data), the number of 'buy' trials were upsampled to match the number
of 'not buy' trials in order to efficiently use the data when fitting
the models. In the out-of-sample (OOS) validation on the \citet{Karmarkar:2012}
data, however, the number of 'not buy' trials were downsampled to
match the smaller number of 'buy' trials in order to be more conservative
in estimating the out-of-sample accuracy (and related $p$-values).}
\end{singlespace}
\textcolor{black}{\medskip{}
}
\begin{singlespace}
\textcolor{black}{}
\begin{table}[t]
\begin{centering}
\textcolor{black}{\caption{Median classification accuracy and parameters for SPDA and SVM classifiers
fit with Leave-5-Subjects-Out (L5SO) cross validation.}
}
\par\end{centering}
\textcolor{black}{\medskip{}
}
\begin{centering}
\textcolor{black}{}%
\begin{tabular}{lcccc||r@{\extracolsep{0pt}.}lr@{\extracolsep{0pt}.}lr@{\extracolsep{0pt}.}lr@{\extracolsep{0pt}.}lr@{\extracolsep{0pt}.}l}
\multicolumn{5}{c}{Classification Accuracy} & \multicolumn{10}{c}{Model Type}\tabularnewline
\hline
\hline
\multirow{2}{*}{\textcolor{black}{\scriptsize Method}} & \multirow{2}{*}{\textcolor{black}{\scriptsize Training}} & \multirow{2}{*}{\textcolor{black}{\scriptsize Test}} & \multirow{2}{*}{{\scriptsize OOS }} & \multirow{2}{*}{{\scriptsize p-value$^{\dagger}$}} & \multicolumn{2}{c}{\textcolor{black}{\scriptsize Sparse}} & \multicolumn{2}{c}{{\scriptsize Tikhonov}} & \multicolumn{2}{c}{\textcolor{black}{\scriptsize Structured}} & \multicolumn{2}{c}{\textcolor{black}{\scriptsize Robust}} & \multicolumn{2}{c}{\textcolor{black}{\scriptsize Adaptive}}\tabularnewline
& & & & & \multicolumn{2}{c}{\textcolor{black}{\scriptsize ($\lambda_{1}$)}} & \multicolumn{2}{c}{{\scriptsize ($\lambda_{2}$)}} & \multicolumn{2}{c}{\textcolor{black}{\scriptsize ($\lambda_{G}$)}} & \multicolumn{2}{c}{\textcolor{black}{\scriptsize ($\delta$)}} & \multicolumn{2}{c}{\textcolor{black}{\scriptsize ($\lambda_{1}^{*}$)}}\tabularnewline
\hline
\hline
& & & & & \multicolumn{2}{c}{} & \multicolumn{2}{c}{} & \multicolumn{2}{c}{} & \multicolumn{2}{c}{} & \multicolumn{2}{c}{}\tabularnewline
\textcolor{black}{\scriptsize Linear SVM$^{1}$} & \textcolor{black}{\scriptsize $97.9\%$} & \textcolor{black}{\scriptsize $\mbox{71.0\%}$} & {\scriptsize $65.8\%$} & {\scriptsize $2.7\times10^{-8}$} & \multicolumn{2}{c}{} & \multicolumn{2}{c}{\textcolor{black}{\scriptsize $^{\dagger\dagger}3.8\times10^{-6}$}} & \multicolumn{2}{c}{} & \multicolumn{2}{c}{\textcolor{black}{\scriptsize $\checkmark$}} & \multicolumn{2}{c}{}\tabularnewline
& & & & & \multicolumn{2}{c}{} & \multicolumn{2}{c}{} & \multicolumn{2}{c}{} & \multicolumn{2}{c}{} & \multicolumn{2}{c}{}\tabularnewline
\textcolor{black}{\scriptsize Lasso$^{2}$} & \textcolor{black}{\scriptsize $\mathbf{98.8}\%$} & \textcolor{black}{\scriptsize $68.5\%$} & {\scriptsize $58.4\%$} & {\scriptsize $0.003$} & \multicolumn{2}{c}{\textcolor{black}{\scriptsize $33$}} & \multicolumn{2}{c}{} & \multicolumn{2}{c}{} & \multicolumn{2}{c}{} & \multicolumn{2}{c}{}\tabularnewline
& & & & & \multicolumn{2}{c}{} & \multicolumn{2}{c}{} & \multicolumn{2}{c}{} & \multicolumn{2}{c}{} & \multicolumn{2}{c}{}\tabularnewline
\textcolor{black}{\scriptsize Elastic Net$^{3}$} & \textcolor{black}{\scriptsize $90.4\%$} & \textcolor{black}{\scriptsize $72.5\%$ } & {\scriptsize $64.3\%$} & {\scriptsize $3.3\times10^{-7}$} & \multicolumn{2}{c}{\textcolor{black}{\scriptsize $54$}} & \multicolumn{2}{c}{\textcolor{black}{\scriptsize $10000$}} & \multicolumn{2}{c}{} & \multicolumn{2}{c}{} & \multicolumn{2}{c}{}\tabularnewline
& & & & & \multicolumn{2}{c}{} & \multicolumn{2}{c}{} & \multicolumn{2}{c}{} & \multicolumn{2}{c}{} & \multicolumn{2}{c}{}\tabularnewline
\hline
& & & & & \multicolumn{2}{c}{} & \multicolumn{2}{c}{} & \multicolumn{2}{c}{} & \multicolumn{2}{c}{} & \multicolumn{2}{c}{}\tabularnewline
\textcolor{black}{\scriptsize GraphNet$^{4}$ (GN)} & \textcolor{black}{\scriptsize $86.9\%$} & \textcolor{black}{\scriptsize $73.7\%$ } & {\scriptsize $64.6\%$} & {\scriptsize $1.8\times10^{-7}$} & \multicolumn{2}{c}{\textcolor{black}{\scriptsize $68$}} & \multicolumn{2}{c}{\textcolor{black}{\scriptsize $1000$}} & \multicolumn{2}{c}{\textcolor{black}{\scriptsize $100$}} & \multicolumn{2}{c}{} & \multicolumn{2}{c}{}\tabularnewline
& & & & & \multicolumn{2}{c}{} & \multicolumn{2}{c}{} & \multicolumn{2}{c}{} & \multicolumn{2}{c}{} & \multicolumn{2}{c}{}\tabularnewline
\textcolor{black}{\scriptsize Robust GN (RGN)} & \textcolor{black}{\scriptsize $86.8\%$} & \textbf{\textcolor{black}{\scriptsize 74.5$\%$}} & {\scriptsize $64.9\%$} & {\scriptsize $1.8\times10^{-7}$} & \multicolumn{2}{c}{\textcolor{black}{\scriptsize $43$}} & \multicolumn{2}{c}{\textcolor{black}{\scriptsize $100$}} & \multicolumn{2}{c}{\textcolor{black}{\scriptsize $100$}} & \multicolumn{2}{c}{\textcolor{black}{\scriptsize $0.3$}} & \multicolumn{2}{c}{}\tabularnewline
& & & & & \multicolumn{2}{c}{} & \multicolumn{2}{c}{} & \multicolumn{2}{c}{} & \multicolumn{2}{c}{} & \multicolumn{2}{c}{}\tabularnewline
\textcolor{black}{\scriptsize RGN + temporal} & \textcolor{black}{\scriptsize $96.5\%$} & \textcolor{black}{\scriptsize $73.8\%$} & {\scriptsize $63.0\%$} & {\scriptsize $5.7\times10^{-6}$} & \multicolumn{2}{c}{\textcolor{black}{\scriptsize $42$}} & \multicolumn{2}{c}{\textcolor{black}{\scriptsize $1000$}} & \multicolumn{2}{c}{\textcolor{black}{\scriptsize $10$}} & \multicolumn{2}{c}{\textcolor{black}{\scriptsize $0.5$}} & \multicolumn{2}{c}{}\tabularnewline
& & & & & \multicolumn{2}{c}{} & \multicolumn{2}{c}{} & \multicolumn{2}{c}{} & \multicolumn{2}{c}{} & \multicolumn{2}{c}{}\tabularnewline
\textcolor{black}{\scriptsize Adaptive RGN } & \textcolor{black}{\scriptsize $91.4\%$} & \textcolor{black}{\scriptsize $73.8\%$} & {\scriptsize $\mathbf{67.1}\%$} & {\scriptsize $8.6\times10^{-10}$} & \multicolumn{2}{c}{\textcolor{black}{\scriptsize $50$}} & \multicolumn{2}{c}{\textcolor{black}{\scriptsize $10000$}} & \multicolumn{2}{c}{\textcolor{black}{\scriptsize $100$}} & \multicolumn{2}{c}{\textcolor{black}{\scriptsize $0.4$}} & \multicolumn{2}{c}{\textcolor{black}{\scriptsize $0.01$}}\tabularnewline
& & & & & \multicolumn{2}{c}{} & \multicolumn{2}{c}{} & \multicolumn{2}{c}{} & \multicolumn{2}{c}{} & \multicolumn{2}{c}{}\tabularnewline
\textcolor{black}{\scriptsize ARGN + temporal} & \textcolor{black}{\scriptsize $90.8\%$} & \textcolor{black}{\scriptsize $73.5\%$} & {\scriptsize $66.8\%$} & {\scriptsize $1.8\times10^{-9}$} & \multicolumn{2}{c}{\textcolor{black}{\scriptsize $40$}} & \multicolumn{2}{c}{\textcolor{black}{\scriptsize $1000$}} & \multicolumn{2}{c}{\textcolor{black}{\scriptsize $100$}} & \multicolumn{2}{c}{\textcolor{black}{\scriptsize $0.3$}} & \multicolumn{2}{c}{\textcolor{black}{\scriptsize $0.01$}}\tabularnewline
& & & & & \multicolumn{2}{c}{} & \multicolumn{2}{c}{} & \multicolumn{2}{c}{} & \multicolumn{2}{c}{} & \multicolumn{2}{c}{}\tabularnewline
\textcolor{black}{\scriptsize Support Vector GN} & \textcolor{black}{\scriptsize $85.3\%$} & \textcolor{black}{\scriptsize $73.0\%$} & {\scriptsize $62.4\%$} & {\scriptsize $1.6\times10^{-5}$} & \multicolumn{2}{c}{\textcolor{black}{\scriptsize $120$}} & \multicolumn{2}{c}{\textcolor{black}{\scriptsize $1000$}} & \multicolumn{2}{c}{\textcolor{black}{\scriptsize $10$}} & \multicolumn{2}{c}{\textcolor{black}{\scriptsize $0.5$}} & \multicolumn{2}{c}{}\tabularnewline
& & & & & \multicolumn{2}{c}{} & \multicolumn{2}{c}{} & \multicolumn{2}{c}{} & \multicolumn{2}{c}{} & \multicolumn{2}{c}{}\tabularnewline
\hline
\hline
& & & & \multicolumn{1}{c}{} & \multicolumn{2}{c}{} & \multicolumn{2}{c}{} & \multicolumn{2}{c}{} & \multicolumn{2}{c}{} & \multicolumn{2}{c}{}\tabularnewline
\end{tabular}
\par\end{centering}
\textcolor{black}{\scriptsize $^{1}$\citep{Cortes1995},$^{2}$\citep{Tibs1996},
$^{3}$\citep{ZouHastie}, $^{4}$\citep{HBM2009}. OOS is short for
{}``Out-Of-Sample''. Chance level is 50\%. $\dagger$ p-value is
calculated for the out-of-sample accuracy using an exact test for
the probability of success in a Bernoulli experiment with $n=322$
trials with success probability of 0.5. $\dagger\dagger$ This is
the $C$ parameter for the SVM. $\checkmark$ The linear SVM is robust
as a result of its hinge loss function, which does not have a parameter
$\delta$ associated with it.}
\end{table}
\end{singlespace}
\textcolor{black}{}
\begin{table}[t]
\begin{centering}
\textcolor{black}{\caption{Median classification accuracy and parameters for SPDA and SVM classifiers
fit with Leave-One-Subject-Out (LOSO) cross validation.}
}
\par\end{centering}
\textcolor{black}{\medskip{}
}
\begin{centering}
\textcolor{black}{}%
\begin{tabular}{lcccc||ccccc}
\multicolumn{5}{c}{Classification Accuracy} & \multicolumn{5}{c}{Model Type}\tabularnewline
\hline
\hline
\multirow{2}{*}{\textcolor{black}{\scriptsize Method}} & \multirow{2}{*}{\textcolor{black}{\scriptsize Training}} & \multirow{2}{*}{\textcolor{black}{\scriptsize Test}} & \multirow{2}{*}{{\scriptsize OOS }} & \multirow{2}{*}{{\scriptsize p-value$^{\dagger}$}} & \textcolor{black}{\scriptsize Sparse} & {\scriptsize Tikhonov} & \textcolor{black}{\scriptsize Structured} & \textcolor{black}{\scriptsize Robust} & \textcolor{black}{\scriptsize Adaptive}\tabularnewline
& & & & & \textcolor{black}{\scriptsize ($\lambda_{1}$)} & {\scriptsize ($\lambda_{2}$)} & \textcolor{black}{\scriptsize ($\lambda_{G}$)} & \textcolor{black}{\scriptsize ($\delta$)} & \textcolor{black}{\scriptsize ($\lambda_{1}^{*}$)}\tabularnewline
\hline
\hline
& & & & & & & & & \tabularnewline
\textcolor{black}{\scriptsize Linear SVM$^{1}$} & \textcolor{black}{\scriptsize $\mathbf{91.6}\%$} & \textcolor{black}{\scriptsize $\mbox{68.8\%}$} & {\scriptsize $65.2\%$} & {\scriptsize $9.7\times10^{-8}$} & & \textcolor{black}{\scriptsize $^{\dagger\dagger}7.6\times10^{-6}$} & & \textcolor{black}{\scriptsize $\checkmark$} & \tabularnewline
& & & & & & & & & \tabularnewline
\textcolor{black}{\scriptsize Lasso$^{2}$} & \textcolor{black}{\scriptsize $90.5\%$} & \textcolor{black}{\scriptsize $68.8\%$} & {\scriptsize $61.2\%$} & {\scriptsize $7.1\times10^{-5}$} & \textcolor{black}{\scriptsize $63$} & & & & \tabularnewline
& & & & & & & & & \tabularnewline
\textcolor{black}{\scriptsize Elastic Net$^{3}$} & \textcolor{black}{\scriptsize $90.8\%$} & \textcolor{black}{\scriptsize $70.0\%$ } & {\scriptsize $63.0\%$} & {\scriptsize $5.7\times10^{-6}$} & \textcolor{black}{\scriptsize $61$} & \textcolor{black}{\scriptsize $1000$} & & & \tabularnewline
& & & & & & & & & \tabularnewline
\hline
& & & & & & & & & \tabularnewline
\textcolor{black}{\scriptsize GraphNet$^{4}$ (GN)} & \textcolor{black}{\scriptsize $87.5\%$} & \textcolor{black}{\scriptsize $71.3\%$ } & {\scriptsize $67.7\%$} & {\scriptsize $4.1\times10^{-10}$} & \textcolor{black}{\scriptsize $54$} & \textcolor{black}{\scriptsize $10000$} & \textcolor{black}{\scriptsize $1000$} & & \tabularnewline
& & & & & & & & & \tabularnewline
\textcolor{black}{\scriptsize Robust GN (RGN)} & \textcolor{black}{\scriptsize $83.8\%$} & \textcolor{black}{\scriptsize $72.5\%$} & {\scriptsize $67.4\%$} & {\scriptsize $4.1\times10^{-10}$} & \textcolor{black}{\scriptsize $25$} & \textcolor{black}{\scriptsize $10$} & \textcolor{black}{\scriptsize $100$} & \textcolor{black}{\scriptsize $0.2$} & \tabularnewline
& & & & & & & & & \tabularnewline
\textcolor{black}{\scriptsize RGN + temporal} & \textcolor{black}{\scriptsize $83.8\%$} & \textcolor{black}{\scriptsize $72.5\%$} & {\scriptsize $67.1\%$} & {\scriptsize $8.6\times10^{-10}$} & \textcolor{black}{\scriptsize $55$} & \textcolor{black}{\scriptsize $100$} & \textcolor{black}{\scriptsize $1000$} & \textcolor{black}{\scriptsize $0.6$} & \tabularnewline
& & & & & & & & & \tabularnewline
\textcolor{black}{\scriptsize Adaptive RGN } & \textcolor{black}{\scriptsize $85.4\%$} & \textcolor{black}{\scriptsize $72.5\%$} & {\scriptsize $\mathbf{69.8}\%$} & {\scriptsize $1.7\times10^{-12}$} & \textcolor{black}{\scriptsize $20$} & \textcolor{black}{\scriptsize $10$} & \textcolor{black}{\scriptsize $1000$} & \textcolor{black}{\scriptsize $0.2$} & \textcolor{black}{\scriptsize $0.01$}\tabularnewline
& & & & & & & & & \tabularnewline
\textcolor{black}{\scriptsize ARGN + temporal} & \textcolor{black}{\scriptsize $88.3\%$} & \textbf{\textcolor{black}{\scriptsize 73.8$\%$}} & {\scriptsize $68.9\%$} & {\scriptsize $2.0\times10^{-11}$} & \textcolor{black}{\scriptsize $30$} & \textcolor{black}{\scriptsize $1000$} & \textcolor{black}{\scriptsize $100$} & \textcolor{black}{\scriptsize $0.2$} & \textcolor{black}{\scriptsize $0.01$}\tabularnewline
& & & & & & & & & \tabularnewline
\textcolor{black}{\scriptsize Support Vector GN} & \textcolor{black}{\scriptsize $89.5\%$} & \textbf{\textcolor{black}{\scriptsize 73.8$\%$}} & {\scriptsize $65.2\%$} & {\scriptsize $9.7\times10^{-8}$} & \textcolor{black}{\scriptsize $84$} & \textcolor{black}{\scriptsize $100$} & \textcolor{black}{\scriptsize $100$} & \textcolor{black}{\scriptsize $0.5$} & \tabularnewline
& & & & & & & & & \tabularnewline
\hline
\hline
& & & & \multicolumn{1}{c}{} & & & & & \tabularnewline
\end{tabular}
\par\end{centering}
\textcolor{black}{\scriptsize $^{1}$\citep{Cortes1995},$^{2}$\citep{Tibs1996},
$^{3}$\citep{ZouHastie}, $^{4}$\citep{HBM2009}. OOS is short for
{}``Out-Of-Sample''. Chance level is 50\%. $\dagger$ p-value is
calculated for the out-of-sample accuracy using an exact test for
the probability of success in a Bernoulli experiment with $n=322$
trials with chance level at 50\%. $\dagger\dagger$ This is the $C$
parameter for the SVM. $\checkmark$ The linear SVM is robust as a
result of its hinge loss function, which does not have a parameter
$\delta$ associated with it.}
\end{table}
\textcolor{black}{}
\begin{figure}[H]
\begin{spacing}{0.5}
\begin{raggedright}
\textcolor{black}{\includegraphics[scale=0.8]{NI_Figure4}\medskip{}
}
\par\end{raggedright}
\raggedright{}\textbf{\textcolor{black}{\footnotesize Figure 4. }}\textcolor{black}{\footnotesize (Left)
Smoothed histogram densities of leave-one-subject out (LOSO) accuracy
rates on test data. Models were fit to all subjects except one, and
then tested on the held-out subject. This was done for all subjects
and smoothed histograms of these rates were calculated for the best
fitting models. (Right) The same procedure was repeated, but leaving
5 subjects out at a time for a total of 25 cross-validation folds.
Both plots show some bi-modality suggestive of different underlying
groups.}\end{spacing}
\end{figure}
\pagebreak{}
\section{Results}
\begin{singlespace}
\subsection{\textcolor{black}{Classification rates }}
\end{singlespace}
\textcolor{black}{\medskip{}
}
\textcolor{black}{If neural substrates implicated in choice show invariance
across individuals, a method that successfully identifies and uses
these substrates to predict choice should generalize well across subjects.
We compared the GraphNet classifier accuracies with accuracies obtained
using linear SVM (where accuracy in this case is the ability to correctly
predict a subject's choices to purchase a product or not). In particular
we looked at generalization of fits to held-out {}``test'' sets
(consisting of subjects held out of a particular stage of the cross
validation procedure, but still present in other cross-validation
stages), and to out-of-sample (OOS) data (new data never used at any
stage of the model fitting) consisting of different subjects from
another study \citep{Karmarkar:2012}. Results and model parameters
for the GraphNet classifiers and linear SVM across the 25 subjects
from \citet{Knutson2007} ({}``Training'', {}``Test'') and 17
subjects from \citet{Karmarkar:2012} ({}``OOS'') are listed in
Tables 1 and 2, as well as a summary of each method's properties.
Models were fit using either leave-one-subject-out (LOSO, Table 1)
or leave-5-subjects-out (L5SO, Table 2) cross-validation, and both
training and test results are displayed to allow comparison of overfitting
on the training data versus the held-out test data. As cross-validation
is known to yield an overly optimistic estimate of the true classification
error rate \citep{Hastie:2009p2681}, model fits to the initial data
set ($n=25$; \citet{Knutson2007}) were tested on out-of-sample (OOS)
data ($n=17$; \citet{Karmarkar:2012}) collected more than three
years later using different subjects shown different products. These
out-of-sample results provide the most rigorous demonstration of fit
generalization to new data, adjusting for any over fitting by the
cross-validation procedure, and are the strongest evidence for invariance
in the neural representation of choice across subjects. The $p$-values
reported correspond to these out-of-sample accuracies on $n=322$
trials across the 17 new subjects.}
\begin{singlespace}
\subsubsection{\textcolor{black}{Generalization to held-out groups (L5SO cross-validation)}}
\end{singlespace}
\textcolor{black}{\medskip{}
}
\begin{singlespace}
\textcolor{black}{Best median training, median test, and out-of-sample
(OOS) rates are described for GraphNet classifiers fit over the grid
of parameters given in \eqref{eq:16} The linear SVM parameters are
also given in \eqref{eq:16}. Despite a more than 1000-fold increase
in the number of input features relative to earlier volume of interest
(VOI) analyses \citep{Grosenick:2008p2789}, whole-brain classifiers
performed significantly better than previous VOI-based predictions
fit to the same data \citep{Knutson2007,Grosenick:2008p2789}. Further,
among these whole-brain classifiers, adaptive and robust methods performed
best on out-of-sample data. SVGN performed similarly to the linear
SVM (but unlike linear SVM, yields structured, sparse coefficients
that aid interpretability). Further, Lasso and linear SVM tended to
overfit the training data more than the SPDA-GraphNet classifiers,
as evidenced by their higher training but lower test rates. Overall,
the Adaptive Robust GraphNet classifier showed the best out-of-sample
classification rate, with accuracy on new data of 67.1\% (for comparison,
the linear SVM accuracy was 65.8\%). Examining the distribution of
test classification rates across the 25 folds (25 sets leaving 5 subjects
out), Figure 4b shows that the linear SVM appears to have less variance
across test fits to held-out subjects. The marked non-normality of
these distributions is interesting, and motivated us to report median
rather than mean accuracy over cross-validation folds.}
\end{singlespace}
\textcolor{black}{\medskip{}
}
\begin{singlespace}
\subsubsection{\textcolor{black}{Generalization to held-out individuals (LOSO cross-validation)}}
\end{singlespace}
\textcolor{black}{\medskip{}
}
\begin{singlespace}
\textcolor{black}{In addition to the leave-5-subject out (L5SO) cross-validation,
we also ran leave-one-subject-out (LOSO) cross validation (i.e., using
the data from 24 subjects to predict results for each remaining subject).
Repeating this procedure for all subjects yielded one held-out classification
rate per subject, indicating how well the group fit generalized to
that subject. Repeating this for all subjects yielded one held-out
test rate per subject. This rate indicated how well the model fit
based on all but one subjects' data generalized to the held-out subject---a
measure of invariance across subjects as well as a quantity that may
be of interest in studies of individual differences. Figure 4a shows
smoothed histograms of the LOSO classification rates for the Robust
GraphNet, Adaptive Robust GraphNet, SVGN, and linear SVM classifiers.
Overall, the GraphNet classifiers outperform the linear SVM on LOSO
cross-validation across subjects. When the LOSO fits were used to
classify choice out-of-sample, the Adaptive Robust GraphNet classifier
again yielded the best performance, now at almost 70\% classification
accuracy. LOSO cross-validation appears to result in better OOS generalization
than L5SO cross-validation for this data. More important than the
improvement in classification performance, however, is the greater
interpretability of these methods. }
\end{singlespace}
\textcolor{black}{\medskip{}
}
\begin{singlespace}
\subsection{\textcolor{black}{Visualization and interpretation of coefficients
and parameters\medskip{}
}}
\end{singlespace}
\begin{singlespace}
\subsubsection{\textcolor{black}{Interpreting GraphNet coefficients}}
\end{singlespace}
\textcolor{black}{\medskip{}
}
\begin{singlespace}
\textcolor{black}{While GraphNet classifiers and linear SVM both classified
purchase choices successfully, the GraphNet-based classifiers produced
more interpretable results. Consistent with previous VOI-based analyses,
the GraphNet, Robust GraphNet classifier (Figure 5), and Adaptive
Robust GraphNet (Figure 5) classifiers all identified similar regions
to those chosen as VOIs \citep{Knutson2007}, with coefficients present
at the time points corresponding to peak discrimination in the VOI
time-series \citep{Knutson2007} and VOI classification \citep{Grosenick:2008p2789}.
In particular, nucleus accumbens (NAcc) activation began to positively
predict purchase choices at the time of product presentation, and
this prediction persisted throughout subsequent price presentation.
Medial prefrontal cortex (MPFC) and midbrain activation, on the other
hand, began to positively predict purchase choices at the onset of
price presentation (but not during previous product presentation).
Additionally---and not included in any previous findings---posterior
cingulate activation also began to robustly and positively predict
purchase choices during price presentation. Reassuringly, no regions'
activation predicted purchase choices during fixation presentation.
Interestingly, the best fits chose far more voxels that positively
predicted than negatively predicted purchasing.}
\end{singlespace}
\textcolor{black}{Together, these findings demonstrate that sparse,
structured, whole-brain methods like GraphNet can facilitate the discovery
of new behaviorally-relevant spatiotemporal neural activity patterns
that existing VOI-based methods miss, particularly when made robust
and adaptive. For example, given the temporal as well as spatial resolution
of the present design and data, it was possible to extend interpretation
of the model fit not only to where brain activity predicted purchasing
choices, but also to when and in what order, and to new regions not
chosen as VOIs in previous work emerged and improved the overall classification.
Thus, an investigator who knows when different events occurred (and
accounts for the lag and variation of the peak hemodynamic response)
can infer that different design components promoted eventual purchasing
choices by altering activity in specific regions. The ability of coefficient
vectors estimated from the \citet{Knutson2007} to accurately predict
choices of new subjects run on the SHOP task years later and shown
different products speaks both to the stability of the neural activity
related to the task across subjects and products, and to the quality
of the model. \medskip{}
}
\textcolor{black}{}
\begin{figure}
\begin{centering}
\textcolor{black}{\includegraphics[scale=0.75]{NI_Figure5}\medskip{}
}
\par\end{centering}
\begin{spacing}{0.5}
\raggedright{}\textbf{\textcolor{black}{\footnotesize Figure 5. }}\textcolor{black}{\footnotesize Whole-brain
classification results from the SHOP task (see Figure 3 for task structure).
(Top) Median coefficient maps from the best robust GraphNet classifier
(median test accuracy of 74.5\% over cross-validation folds and out-of-sample
accuracy of 64.9\%) fit using Leave-5-subject-out (L5SO) cross-validation
are shown at two time points for product, price, and choice periods,
as well as the fixation period. Warm colored coefficients denote areas
that predict purchasing a product, while cool-colored areas those
that predict not purchasing. The areas chosen by the robust GraphNet
classifier highlight regions suggested by previous studies including
the bilateral nucleus accumbens (NAcc) and the mesial prefrontal cortex
(MPFC) \citep{Knutson2007,Grosenick:2008p2789}, but also implicate
new regions including the anterior cingulate and and posterior cingulate
cortices. (Middle) Similar plots for the best adaptive robust GraphNet
classifier (median test accuracy of 72.5\% over cross-validation folds;
out-of-sample accuracy of 69.8\%) fit using leave-one-subject (LOSO)
cross-validation. Although the solution is sparser, the regions chosen
remain the same. (Bottom) Coefficients for the best linear SVM (median
test accuracy of 71\% over cross-validation folds; out-of-sample accuracy
of 65.8\%) fit using Leave-5-subject-out (L5SO) cross-validation for
comparison. }\end{spacing}
\end{figure}
\textcolor{black}{}
\begin{figure}
\begin{centering}
\textcolor{black}{\includegraphics[scale=0.8]{NI_Figure6}}
\par\end{centering}
\textcolor{black}{\medskip{}
}
\begin{spacing}{0.5}
\raggedright{}\textbf{\textcolor{black}{\footnotesize Figure 6. }}\textcolor{black}{\footnotesize Examples
of classification accuracy (test) plotted as a function of penalty
parameters. The blown up image on the left shows an image of the median
test accuracy rates for the GraphNet SPDA classifier (GN) as functions
of hyperparameters $\lambda_{1}$ and $\lambda_{G}$ (wit}\textbf{\textcolor{black}{\footnotesize h
$\lambda_{2}=0$}}\textcolor{black}{\footnotesize ). Warm colors indicate
median classification rates above 70\% (for L5SO cross-validation)
and cool colors median accuracy below 70\% (see color bar for scale).
The separate column (L) indicates the standard Lasso solution at }\textbf{\textcolor{black}{\footnotesize $\lambda_{G}=0$}}\textcolor{black}{\footnotesize .
There is a clear maxima at }\textbf{\textcolor{black}{\footnotesize $\lambda_{1}=40,\ \lambda_{G}=100$}}\textcolor{black}{\footnotesize .
The smaller images on the right show similar plots for the GraphNet
(GN), Robust GraphNet (RGN), Adaptive Robust GraphNet (ARGN), and
Support Vector GraphNet (SVGN) classifiers at four values of the graph
$G$ diagonal scale $\lambda_{2}$. Note the different scale on the
ARGN models. It is of some interest that all the plots are rather
slowly varying in the parameters and demonstrate significantly unimodal
peaks (neither of these need be the case).}\end{spacing}
\end{figure}
\begin{singlespace}
\subsubsection{\textcolor{black}{Interpreting GraphNet parameters}}
\end{singlespace}
\textcolor{black}{\medskip{}
}
\begin{singlespace}
\textcolor{black}{Figure 6 shows plots of median L5SO cross-validation
rates over the values $\{\lambda_{1},\lambda_{G},G\}$ \eqref{eq:15-1}
on which we fit the four GraphNet classifiers (other parameters are
set to the values shown in Table 1; plots for LOSO rates are similar).
In all cases, there is a region in the interior of the explored parameter
space $\{\lambda_{1},\lambda_{G},G\}$ on which the models empirically
perform best. In all cases this region involves both smoothing and
some level of sparsity, and the classifiers built with Lasso (L) and
Elastic Net (EN)---shown as separate bars for the GraphNet (GN) fits---underperform
relative to the sparse and smooth GraphNet classifiers on this data.
Comparison of the rates in Figure 6 suggests that a certain amount
of coefficient smoothness and inclusion of correlated variables in
the final fit is important for this data set, and that using a robust
loss function tightens the region of optimal parameter performance.}
\end{singlespace}
\textcolor{black}{\medskip{}
}
\begin{singlespace}
\section{\textcolor{black}{Discussion and Conclusions\medskip{}
}}
\end{singlespace}
\begin{singlespace}
\subsection{\textcolor{black}{Interpretable models for whole-brain spatiotemporal
fMRI}}
\end{singlespace}
\textcolor{black}{\medskip{}
}
\begin{singlespace}
\textcolor{black}{We sought to design and develop a novel classification
method for fMRI data that could fulfill several aims. First, the method
should deliver interpretable results for whole-brain data over multiple
time-points in the native data space. Second, the method should yield
classification accuracy (or goodness-of-fit) competitive with current
state-of-the-art multivariate methods. Third, the method should choose
relevant features in a principled and asymptotically consistent way
(i.e., it should include relevant features while excluding nuisance
parameters). Fourth, the method should accommodate flexible constraints
on model coefficients related to prior information (e.g., local smoothness,
connectivity). Fifth, the method should remain robust to outliers
in the data. Sixth, the method should generate coefficients with relatively
unbiased magnitudes (despite employing shrinkage methods to yield
sparsity). And seventh (and finally), the method should have the capacity
to detect a range of possible signals, from smooth and localized to
sparse and distributed. }
\textcolor{black}{The GraphNet-based methods presented here make a
first step toward meeting these desirable (and often competing) aims.
In particular, the Adaptive Robust GraphNet allows automatic variable
selection (\citet{Zou:2009p2991}), incorporation of prior information
in the form of a graph penalty, and yields minimally biased and asymptotically
consistent coefficient estimates as a result of adaptive reweighting.
Robust GraphNet methods can be applied to either regression or classification
settings (using Optimal Scoring), and generate classification rates
that compete favorably with state-of-the-art multivariate classifiers.
The tuning parameters $(\lambda_{1},\lambda_{G})$ and the graph $G$
allow for a diversity of sparse and smooth data, and the relationship
of model fits to these parameters provides information about the structure
of the detected signal.}
\textcolor{black}{Choice in the context of purchasing admittedly represents
only one application, and future validation on additional data sets
is necessary. However, in this context the GraphNet classifiers generalize
well to independent experiments involving purchasing (i.e. when fit
to new data collected years after the experiments originally used
to train the models, with different subjects and different products).
Adaptive Robust GraphNet methods showed the best out-of-sample generalization,
and generated parsimonious, interpretable models. It is worth noting
that the models that did best on the in-sample cross-validation test
folds were not best on out-of-sample data. This suggests that the
overfitting, or {}``optimism'', known to exist in cross-validation
\citep{Hastie:2009p2681} can effect models differently, and that
a true out-of-sample prediction is necessary to accurately assess
which models generalize best.}
\end{singlespace}
In summary, we have developed a family of robust, adaptive, and interpretable
methods that can be fit efficiently to large data sets over large
parameter grids. This method will allow investigators to search in
a data-driven fashion across the whole brain and multiple time points,
obviating the need for volume-of-interest based approaches in fMRI
classification and regression, and providing an effective alternative
to mass-univariate approaches for whole-brain analysis.
\textcolor{black}{\medskip{}
}
\begin{singlespace}
\subsection{\textcolor{black}{Application to SHOP task data}}
\end{singlespace}
\textcolor{black}{\medskip{}
}
\begin{singlespace}
\textcolor{black}{In the context of predicting human behavior from
brain data, the current whole brain methods offer clear advantages
over previous volume of interest based methods. In terms of classification
accuracy, previous work on the \citet{Knutson2007} data has resulted
in cross-validated test rates of 60\% (with a leave-one-out cross
validation using logistic regression on VOI-averaged data; see \citet{Knutson2007}
for details), and 67\% (with a $5\times2$ cross validation using
SPDA-Elastic Net on VOI voxel data; see \citet{Grosenick:2008p2789}
for details). Here, using the same preprocessing and data as in these
previous VOI-based approaches, but using GraphNet classifiers on whole-brain
data, we achieve test rates from $73.0-74.5\%$ for L5SO cross validation
and $71.3-73.8\%$ for LOSO cross validation. Further, out-of-sample
(OOS) rates for the GraphNet classifiers were $67.1\%$ (L5SO) and
and $69.8\%$ (LOSO). Thus, in this case, even out-of-sample rates
with GraphNet classifiers outperform in-sample cross validation test
rates on VOI-based classifiers---a considerable improvement. In taking
classification accuracy as a measure of goodness-of-fit, this indicates
that GraphNet classifiers result in better fits and improved generalization
relative to VOI methods, and suggests that the resulting coefficients
are a good representation of invariant features that discriminate
between choosing to purchase or not across subjects and products.}
\end{singlespace}
\textcolor{black}{Turning to examine the coefficients, we see that
the GraphNet classifiers reassuringly deliver findings consistent
with prior volume of interest based results \citep{Knutson2007,Grosenick:2008p2789},
replicating the observation that nucleus accumbens (NAcc) activation
begins to predict purchase choices during product presentation while
medial prefrontal cortical (MPFC) activation begins to predict purchase
choices during price presentation. It is also interesting to note
areas that were not included by previously applied methods, and might
not have been noticed if not for the whole-brain analysis (and which
might help account for the improved classification rates over previous
VOI analyses). }
\textcolor{black}{While one account posits that in the context of
fMRI, NAcc activation indexes gain predictions \citep{Knutson2001,Knutson2008},
an alternative account posits that NAcc activation instead indexes
gain prediction errors (e.g., \citealt{Hare2008}). To the extent
that gain predictions forecast future events while gain prediction
errors are adjustments of those forecasts after an error is detected,
the gain prediction account posits that NAcc activation in response
to products should predict subsequent purchase choices. Applied to
SHOP task data, the robust and adaptive robust GraphNet classifier
results clearly support the gain prediction functional account of
NAcc activity, since NAcc activation in response to products predicts
future choices to purchase, whereas MPFC activity does not. Instead,
MPFC activity predicts choice in response to later presented price
information, consistent with a value integration account (\citep{Knutson2005};
Figure 5). The GraphNet classifiers also revealed a previously unnoticed
result in which anterior and posterior cingulate activity clearly
predicts purchase choices at price presentation (Figure 5). Accounts
of cingulate function in the context of purchasing remain less developed
than similar accounts of NAcc and MPFC function. Nonetheless, this
result might be consistent with attentional and salience-based accounts
of posterior cingulate function \citep{McCoy2003}, and highlights
a region that deserves further investigation in the context of choice
prediction.}
\textcolor{black}{\medskip{}
}
\begin{singlespace}
\subsection{\textcolor{black}{Future directions}}
\end{singlespace}
\textcolor{black}{\medskip{}
}
\begin{singlespace}
\textcolor{black}{GraphNet methods can be further optimized, opening
new avenues for exploration. For instance, investigators might compare
graph constraints other than those related to just spatial-temporal
adjacency, including (1) weighted graphs derived from the data to
adapt to local smoothness, (2) cut-graphs derived from segmented brain
atlases that allow adjacent but functionally distinct regions to be
independently penalized, and (3) weighted graphs derived from structural
data, which would allow constraints on voxels adjacent on a connectivity
graph, rather than in space or time (see \citet{NgVaroquaux2012}
for a promising step in this direction). Further, investigators might
use the goodness-of-fit measure provided by GraphNet to infer which
of a set of structural graphs best relates to functional data, or
to adaptively alter graph weights to explore structure in functional
data (in a Variational Bayes framework, for example). }
\end{singlespace}
\textcolor{black}{All of the methods considered above assume linear
relationships between input features and target variables. While this
assumption suffices in many cases, signal saturation effects alone
suggest that it might not faithfully mirror underlying physiological
signals. Nonlinear methods based on scatterplot smoothers have recently
been developed and shown to work well in combination with coordinate-wise
methods \citep{Ravikumar2009}, and previous work applying sparse
regression to features derived using factor analysis have yielded
promising results \citep{ESS,Wager2011}. Investigators might thus
combine nonlinear methods with sparse structured feature selection
methods \citep{Allen2011} to generate more flexible and accurate,
yet still interpretable, models of brain dynamics. Finally, we note
that because we are operating directly on voxels data, we are working
in the {}``native'' reconstructed 3D data space rather than on factors
derived from this data or on a dictionary of basis functions that
approximate features of the data (e.g., wavelets). Certainly, the
optimization scheme described here would also extend to solving problems
using features derived from the data, and it is an interesting direction
for future research to explore GraphNet penalties in these other contexts
and to compare GraphNet methods to existing regression and classification
methods that operate on lower dimensional embeddings or dictionary
representations of the data. Whether operating directly on the data
with sparse structured methods or on derived features is more appropriate
will depend on the application. The methods presented here demonstrate
that the former approach can be quite effective, and provides results
that are easily interpreted in the native data space. }
\pagebreak{}
\section*{\textcolor{black}{Appendix }}
\subsection*{\textcolor{black}{Robust GraphNet: coordinate-wise coefficient updates
using infimal convolution}}
\textcolor{black}{\medskip{}
}
\textcolor{black}{}
\begin{algorithm}[H]
\textcolor{black}{\caption{Robust GraphNet update using infimal convolution}
}
\begin{enumerate}
\item \textcolor{black}{Given a set of data and parameters $\Omega=\{X,y,\lambda_{1},\lambda_{G}\}$,
previous coefficient estimates $\widehat{\alpha}^{(r)},\widehat{\beta}^{(r)}$,
and $p\times p$ positive semidefinite constraint graph $G\in S_{+}^{p\times p}$,
let
\begin{eqnarray*}
\widehat{\gamma}^{(r)} & = & [\widehat{\beta}^{(r)}\ \ \widehat{\alpha}^{(r)}]^{T}\\
Z & = & \left[X\ \ I_{n\times n}\right].
\end{eqnarray*}
}
\item \textcolor{black}{Choose coordinate $j$ using essentially cyclic
rule \citep{Tseng2001} and fix $\tilde{\gamma}=\{\gamma_{k}^{(r)}|k\neq j\}$,
$\tilde{Z}=Z._{\neq j},$$\tilde{\beta}=\{\beta_{k}^{(r)}|k\neq j\}$,
$\tilde{X}=X._{\neq j}$ .}
\item \textcolor{black}{Update $\widehat{\gamma}_{j}^{(r)}$ using
\[
\widehat{\gamma}_{j}^{(r+1)}\leftarrow\begin{cases}
\frac{S\left(Z._{j}^{T}(y-\tilde{Z}\tilde{\gamma})-(\lambda_{2}/2)\tilde{\gamma}^{T}(G'_{\neq j}.)._{j},\ \lambda_{1}/2\right)}{Z._{j}^{T}Z._{j}+\lambda_{G}G'_{jj}} & \text{if }j\in\{1,...,p\}\\
S\left((y-\tilde{Z}\tilde{\gamma})_{j},\ \lambda_{1}/2\right) & \text{if }j\in\{p+1,...,p+n\},
\end{cases}
\]
where $S(x,\lambda)$ is the element-wise soft-thresholding operator
in equation \eqref{eq:soft-thresh}. For adaptive version replace
$\lambda_{1}$ with $\lambda_{1}^{*}\widehat{w}_{j}$ in above update
(see section 2.1.5).}
\item \textcolor{black}{Repeat steps (1)-(3) cyclically for all $j\in\{1,\ldots,p+n\}$
until convergence (see discussion of convergence in \citet{Friedman:2007p36}). }
\item \textcolor{black}{Optional: rescale resulting estimates using method
from section 2.2.5.}\end{enumerate}
\end{algorithm}
\subsubsection*{\textcolor{black}{\medskip{}
}Derivation of updates in Algorithm 1\textcolor{black}{\medskip{}
}}
\textcolor{black}{For a particular coordinate $j$ , we are interested
in the estimates
\begin{eqnarray*}
\widehat{\gamma}_{j} & = & \underset{\gamma_{j}}{\text{argmin}}\ (1/2)\|y-\tilde{Z}\tilde{\gamma}-Z._{j}\gamma_{j}\|_{2}^{2}+\lambda_{G}\left(\tilde{\gamma}^{T}(G'_{\neq j}.)._{j}\gamma_{j}+G'_{jj}\gamma_{j}^{2}\right)+\lambda_{1}|\gamma_{j}|\ \text{if }j\in\{1,...,p\},\\
\widehat{\gamma}_{j} & = & \underset{\gamma_{j}}{\text{argmin}}\ (1/2)\|y-\tilde{Z}\tilde{\gamma}-Z._{j}\gamma_{j}\|_{2}^{2}+\delta|\gamma_{j}|\ \text{if }j\in\{p+1,...,p+n\}.
\end{eqnarray*}
By the arguments in section 2.4.1, this yields the coordinate-wise
updates
\begin{eqnarray*}
\widehat{\gamma}_{j} & \leftarrow & \frac{S\left(Z._{j}^{T}(y-\tilde{Z}\tilde{\gamma})-(\lambda_{2}/2)\tilde{\gamma}^{T}(G'_{-j}.)._{j},\ \lambda_{1}/2\right)}{Z._{j}^{T}Z._{j}+\lambda_{G}G'_{jj}}\ \ \text{if }j\in\{1,...,p\},\\
\widehat{\gamma}_{j} & \leftarrow & \frac{S\left(Z._{j}^{T}(y-\tilde{Z}\tilde{\gamma}),\ \lambda_{1}/2\right)}{Z._{j}^{T}Z._{j}}\ \ \text{if }j\in\{p+1,...,p+n\},
\end{eqnarray*}
where $S(x,\lambda)$ is the element-wise soft-thresholding operator
in equation \eqref{eq:soft-thresh}.}
\textcolor{black}{\medskip{}
}
\subsection*{\textcolor{black}{SVM-GraphNet classification: coordinate-wise coefficient
updates using infimal convolution}}
\textcolor{black}{\medskip{}
}
\textcolor{black}{}
\begin{algorithm}[H]
\textcolor{black}{\caption{SVM GraphNet classification update using infimal convolution}
}
\begin{enumerate}
\item \textcolor{black}{Given a set of data and parameters $\Omega=\{X,y,\lambda_{1},\lambda_{G}\}$,
previous coefficient estimates $\widehat{\alpha}^{(r)},\widehat{\beta}_{0}^{(r)},\widehat{\beta}^{(r)}$,
and $p\times p$ positive semidefinite constraint graph $G\in S_{+}^{p\times p}$,
let
\begin{eqnarray*}
\widehat{\gamma}^{(r)} & = & [\widehat{\beta}_{0}^{(r)}\ \ \widehat{\beta}^{(r)}\ \ \widehat{\alpha}^{(r)}]^{T}\\
Z & = & \left[y^{T}[1_{n\times1}\ \ X]\ \ I_{n\times n}\right].
\end{eqnarray*}
}
\item \textcolor{black}{Choose coordinate $j$ using essentially cyclic
rule \citep{Tseng2001} and fix $\tilde{\gamma}=\{\gamma_{k}^{(r)}|k\neq j\}$,
$\tilde{Z}=Z._{\neq j},$$\tilde{\beta}=\{\beta_{k}^{(r)}|k\neq j\}$,
$\tilde{X}=X._{\neq j}$ .}
\item \textcolor{black}{Update $\widehat{\gamma}_{j}^{(r)}$ using
\[
\widehat{\gamma}_{j}^{(r+1)}\leftarrow\begin{cases}
\tilde{\gamma}^{T}\tilde{Z}1_{n\times1}+N(\gamma_{j}-1) & \text{if }j=0\\
\\
\frac{S\left((\tilde{Z}^{T}\tilde{\gamma}-1_{n\times1})^{T}X._{j}-(\lambda_{2}/2)\tilde{\beta}^{T}(G{}_{\neq j}.)._{j},\ \lambda_{1}/2\right)}{X._{j}^{T}X._{j}+\lambda_{G}G{}_{jj}} & \text{if }j\in\{1,...,p\}\\
\\
H\left((\tilde{Z}\tilde{\gamma})_{j}-1,\ \delta\right) & \text{if }j\in\{p+1,...,p+n\},
\end{cases}
\]
where $S(x,\lambda)$ is the element-wise soft-thresholding operator
and $H(x,\delta)$ is given in equation \eqref{eq: H}.}
\item \textcolor{black}{Repeat (1) -(3) cyclically for all $j\in\{1,\ldots,p+n\}$
until convergence (see discussion of convergence in \citet{Friedman:2007p36}).}
\item \textcolor{black}{Optional: rescale resulting estimates using method
from section 2.2.5.}\end{enumerate}
\end{algorithm}
\subsubsection*{\textcolor{black}{\medskip{}
}Derivation of updates in Algorithm 2\textcolor{black}{\medskip{}
}}
\textcolor{black}{Following the description of the SVM given in section
2.1.7, we can take the same approach used to derive the Robust GraphNet
estimates with the Support Vector GraphNet estimates of section 2.2.3,
which we can write as
\begin{eqnarray*}
\widehat{\gamma} & = & \underset{\gamma}{\text{argmin}}\ (1/2\delta)\|1_{n\times1}-Z\gamma\|_{2}^{2}+\lambda_{G}\gamma_{\neq0}^{T}G'\gamma_{\neq0}+\sum_{j=0}^{p}w_{j}|\gamma_{j}|+\sum_{j=p+1}^{p+n}w_{j}\max(0,\gamma_{j})\\
\text{where} & & Z=\left[y^{T}[1_{n\times1}\ \ X]\ \ I_{n\times n}\right],\ \ \gamma=[\beta_{0}\ \ \beta\ \ \alpha],\ \ w_{j}=\begin{cases}
0 & \text{if }j=0\\
\lambda_{1} & \text{if }j=1,..,p\\
1 & \text{if }j=p+1,...,p+n
\end{cases}\\
& & G'=\left[\begin{array}{lll}
0 & 0_{1\times p} & 0_{1\times n}\\
0_{p\times1} & G & 0_{1\times n}\\
0_{n\times1} & 0_{n\times1} & 0_{n\times n}
\end{array}\right]\in S_{+}^{(p+n+1)\times(p+n+1)}.
\end{eqnarray*}
During coordinate wise descent, only one of the separable penalty
functions has an {}``active'' variable per descent step. Letting
$h(\gamma_{j})=\max(0,\gamma_{j})$, we thus have
\[
\widehat{\gamma}_{j}=\begin{cases}
\underset{\gamma_{j}}{\text{argmin}}\ (1/2\delta)\|1_{N\times1}-\tilde{Z}\tilde{\gamma}-Z._{j}\gamma_{j}\|_{2}^{2} & \text{if }j=0\\
\underset{\gamma_{j}}{\text{argmin}}\ (1/2\delta)\|1_{N\times1}-\tilde{Z}\tilde{\gamma}-Z._{j}\gamma_{j}\|_{2}^{2}+\lambda_{G}\left(\tilde{\gamma}^{T}(G'_{\neq j}.)._{j}\gamma_{j}+G'_{jj}\gamma_{j}^{2}\right)+\lambda_{1}|\gamma_{j}| & \text{if }j\in\{1,...,p\}\\
\underset{\gamma_{j}}{\text{argmin}}\ (1/2\delta)\|1_{N\times1}-\tilde{Z}\tilde{\gamma}-Z._{j}\gamma_{j}\|_{2}^{2}+h(\gamma_{j}) & \text{if }j\in\{p+1,...,p+n\}.
\end{cases}
\]
Then since
\[
Z._{j}=\begin{cases}
1_{N\times1} & \text{if }j=0\\
X._{j} & \text{if }j\in\{1,...,p\}\\
e_{j} & \text{if }j\in\{p+1,...,p+n\},
\end{cases}
\]
(where $e_{j}$ is the vector of all zeros except for the $j$th element,
which is 1) we have}
\textcolor{black}{
\[
\widehat{\gamma}_{j}\leftarrow\begin{cases}
-1_{N\times1}^{T}Z._{j}+(\tilde{Z}\tilde{\gamma})^{T}Z._{j}+\gamma_{j}Z._{j}^{T}Z._{j} & \text{if }j=0\\
\frac{S\left((\tilde{Z}\tilde{\gamma})^{T}Z._{j}-1_{N\times1}^{T}Z._{j}-(\lambda_{G}/2)\tilde{\gamma}^{T}(G'_{\neq j}.)._{j},\ \lambda_{1}/2\right)}{Z._{j}^{T}Z._{j}+\lambda_{G}G'_{jj}} & \text{if }j\in\{1,...,p\}\\
\frac{H\left((\tilde{Z}\tilde{\gamma})^{T}e_{j}-1_{N\times1}^{T}e_{j},\delta\right)}{e_{j}^{T}e_{j}} & \text{if }j\in\{p+1,...,p+n\},
\end{cases}
\]
yielding update:
\[
\hat{\gamma}_{j}\leftarrow\begin{cases}
\tilde{\gamma}^{T}\tilde{Z}1_{N\times1}+N(\gamma_{j}-1) & \text{if }j=0\\
\\
\frac{S\left((\tilde{Z}^{T}\tilde{\gamma}-1_{N\times1})^{T}X._{j}-(\lambda_{G}/2)\tilde{\beta}^{T}(G{}_{\neq j}.)._{j},\ \lambda_{1}/2\right)}{X._{j}^{T}X._{j}+\lambda_{G}G{}_{jj}} & \text{if }j\in\{1,...,p\}\\
\\
H\left((\tilde{Z}\tilde{\gamma})_{j}-1,\delta\right) & \text{if }j\in\{p+1,...,p+n\},
\end{cases}
\]
}where \textcolor{black}{where $S(x,\lambda)$ is the element-wise
soft-thresholding operator in equation \eqref{eq:soft-thresh} and}
\begin{equation}
H(x,\delta)=\begin{cases}
x-\delta & \text{if }x<1\\
x & \text{otherwise}.
\end{cases}\label{eq: H}
\end{equation}
\subsection*{\textcolor{black}{\medskip{}
Parameter grid used in cross-validation\medskip{}
}}
\begin{singlespace}
\textcolor{black}{Parameters $\{\lambda_{1},G,\lambda_{G},\delta,\lambda_{1}^{*}\}$
were taken over the following grid of values:
\begin{eqnarray*}
\lambda_{1} & \in & \{10,11,\ldots,99\}\\
G & \in & \left\{ L,\ L+\eta I,\ L+10^{2}\eta I,\ L+10^{3}\eta I,\ L+10^{4}\eta I\right\} \ \text{where }\eta=1/\lambda_{G}\ \text{for }\lambda_{G}>0\ \text{and }1\ \text{otherwise}\\
\lambda_{G} & \in & \{0,10^{1},10^{2},10^{3},10^{4},10^{5}\}\\
\delta & \in & \{0.2,0.3,0.4,0.5,0.6,0.7,1,2,10,100\}\\
\lambda_{1}^{*} & \in & \{1,10^{-1},10^{-2}\}.
\end{eqnarray*}
The linear SVM was fit over parameters
\begin{equation}
C\in\{10{}^{-6},10^{-5},10^{-4},10^{-3},10^{-2},10^{-1},10^{-0},2,3,4,5,6,7,10^{1},10^{2},10^{3}\}.\label{eq:16}
\end{equation}
\medskip{}
}
\end{singlespace}
\section*{\textcolor{black}{References}}
\begin{singlespace}
\textcolor{black}{\bibliographystyle{elsarticle-harv}
|
1,108,101,564,574 | arxiv | \section{Introduction}
In the $\Lambda$CDM paradigm, structure in the Universe arises from the initial density perturbations of an (almost) homogeneous dark matter distribution. Due to gravitational evolution, this leads to the appearance of collapsed structures, i.e. dark matter haloes. Some of the baryonic matter, following this process, cools down and settles at the centres of the gravitational potentials where it forms galaxies.
This mechanism has been studied through models of so-called spherical collapse \citep{Gunn1972, 1985ApJS...58...39B}, whose main prediction is the existence of a radius within which the material orbiting the halo is completely virialized. In general, this virial radius depends on cosmology and redshift, but both in numerical simulations and observations, fixed overdensity radii are widely used as proxies for this quantity. An example of this is $r_{200m}$, defined as the radius within which the average density is $200$ times the average matter density of the Universe, $\rho_\text{m}$. The corresponding enclosed mass is known as $M_{200m}$.
Halo mass functions constructed with these idealized definitions can capture the effects of cosmology \citep{1974ApJ...187..425P}, the nature of dark matter \citep{2013MNRAS.434.3337A}, and dark energy \citep{Mead_2016} on the growth of structure. In the real Universe, however, this picture is complicated by the triaxiality of haloes \citep{1991ApJ...378..496D, Monaco_1995} and the existence of clumpy (baryonic) substructure \citep{Bocquet_2015}.
Because the process of structure formation is hierarchical, massive haloes contain subhaloes, some of which host galaxies themselves. The resulting clusters of galaxies are the focus of this work. What makes these objects particularly unique is the fact that they are not fully virialized yet. To this day, they are still accreting both ambient material and subhaloes through filamentary structures surrounding them \citep{Bond_1996}. Because of their definition, however, traditional overdensity definitions of mass are not only affected by halo growth, but also by a pseudo-evolution due to the redshift dependence of $\rho_\text{m}$ \citep{2013ApJ...766...25D}.
\cite{Diemer_2014} and \cite{More_2015} were the first to note that this growth process leads to the formation of a sharp feature in the density profile that separates collapsed and infalling material. This feature, therefore, defines a natural boundary of the halo. The location of this edge, i.e. the splashback radius $r_\text{sp}$, has an obvious primary dependence on halo mass, but also a secondary dependence on accretion rate. While this behaviour can be qualitatively explained using simple semi-analytical models of spherical collapse, none of the analytical models currently proposed \citep{Adhikari_2014, Shi2016} can fully describe its dependency on mass and accretion rate \citep{Diemer_2017b}. Despite this, the corresponding definition of halo mass is particularly suited to define a universal mass function valid for a wide range of cosmologies \citep{diemer2020universal}.
In this paper, we try to bridge the gap between the theoretical understanding of the splashback feature and observational results, both past and future. The outer edge of clusters has already been extensively measured through different tracers: the radial distribution of galaxies from wide surveys \citep{2016ApJ...825...39M, Baxter_2017, Chang_2018}, but also their velocity distribution \citep{tomooka2020clusters, 2021MNRAS.503.4250F}, and in the weak-lensing signal of massive clusters \citep{Umetsu_2017, Chang_2018, Contigiani_2019b}. Furthermore, forecasts have already set expectations for what will be obtainable from near-future experiments \citep{Fong_2018, xhakaj2019accurately, 2020arXiv201011324W}. Despite the wealth of data and studies, however, not many \emph{splashback observables} have been proposed. The only robust application of this feature found in the literature is related to the study of quenching for newly accreted galaxies \citep{2020arXiv200811663A}.
To achieve our goal, we make use of hydrodynamical simulations of massive galaxy clusters, which we introduce in Section~\ref{sec:Hydrangea}. We focus mainly on $z=0$, but also make use of snapshots at redshifts $z=0.474$ and $z=1.017$. In Section \ref{sec:definition}, we start our discussion by introducing the physical interpretation of splashback and consider the connection between the galaxy and dark matter distributions. We then continue in Section \ref{sec:msrelation} and \ref{sec:redshift}, where we explain how galaxy profiles and weak-lensing mass measurements can be combined to construct a mass-size relationship for galaxy clusters. Finally, we summarize our conclusions and suggest future developments in Section \ref{sec:conclusions}.
\section{Hydrangea}
\label{sec:Hydrangea}
The Hydrangea simulations are a suite of $24$ zoom-in hydrodynamical simulations of massive galaxy clusters ($\log_{10}M_\text{200m}/\mathrm{M}_\odot$ between $14$ and $15.5$ at redshift $z = 0$) designed to study the relationship between galaxy formation and cluster environment \citep{Bahe2017}. They are part of the Cluster-EAGLE project \citep{Bahe2017,Barnes_2017} and have been run using the EAGLE galaxy formation model \citep{Schaye_2014}, which is known to reproduce galaxy observables such as colour distribution and star formation rates. To better reproduce the observed hot gas fractions in galaxy groups, the AGNdT9 variant of this model was used \citep{Schaye_2014}.
The zoom-in regions stretch to between 10 and 30 Mpc from the cluster centre, roughly corresponding to $\lesssim 10 r_{200m}$. For the definition of the cluster centre, in this work, we choose the minimum of the gravitational potential. We note, however, that this choice will not impact our conclusions since we will focus on locations around $r_\text{200m}$. The particle mass of $m \sim 10^{6}~\mathrm{M}_\odot$ for baryons and $m\sim10^{7}~\mathrm{M}_\odot$ for dark matter allows us to resolve galaxy positions down to stellar masses $M_\ast \geq 10^{8}~ \mathrm{M}_\odot$ and total masses $M_\text{sub} \geq 10^{9}~
\mathrm{M}_\odot$, respectively.
In Figure \ref{fig:comparison} we show the log-derivative of the stacked subhalo density $n_s(r)$ at large scales. This is the result of a fit obtained using the model of \cite{Diemer_2014}, and we refer the reader to the aforementioned paper and \cite{Contigiani_2019b} for a detailed explanation of the model and its components. \edit{The choice to employ this profile is based on its ability} to capture the sharp feature visible around $r_\text{200m}$, which is the focus of this work. We optimally sample its $8$-dimensional parameter space using an ensemble sampler \citep{Foreman-Mackey2013}.
In the same plot, we also include the stacked subhalo profile of the accompanying dark matter only (DMO) simulations, initialized with matching initial conditions. The two profiles match almost exactly, suggesting that baryonic effects do not alter this feature to a significant extent \citep[see also][]{2020arXiv201200025O}. While not shown, we report that the same conclusion can be reached by looking at the full matter distribution $\rho(r)$ in the two sets of simulations. Similarly, this feature is also visible in the number density of galaxies, $n_g(r)$. \edit{Due to our focus on all three of these profiles, we choose not to work with background subtracted quantities.}
For reference, we present a full list of the simulated clusters used in this paper and their relevant properties, some of them defined in the following sections, in Table~\ref{tab:clusterlist}.
\begin{table*}
\centering
\caption{The Hydrangea clusters used in this paper and their $z=0$ properties. $\Gamma_{0.3}$ is the accretion rate measured between $z=0$ and $z=0.297$. The three splashback radii $r_\text{sp}$, $r_\text{sp}^\text{g}$, $r_\text{sp}^\text{s}$ refer to the splashback radius measured, respectively, in the dark matter, galaxy, and subhalo distributions (see Section~\ref{sec:definition}). For two clusters, CE-28 and CE-18, the radius $r_\text{sp}$ is not used in this work because the dark matter distribution displays a featureless profile at large scales. All quantities are in physical units.}
\label{tab:clusterlist}
\begin{tabular}{lccccccc}
\hline
Name & $\Gamma_{0.3}$ & $M_\text{200m}$ & $r_\text{200m}$ & $r_\text{sp}$ & $r_\text{sp}^\text{g}$ & $r_\text{sp}^\text{sub}$\\
& & $[10^{14}~M_\odot]$ & [Mpc] & [Mpc] & [Mpc] & [Mpc]\\
\hline
CE-0 & 0.8 & 1.74 & 1.74 & 2.98 & 2.72 & 2.60 \\
CE-1 & 2.0 & 1.41 & 1.63 & 1.71 & 1.56 & 1.79 \\
CE-2 & 0.5 & 1.41 & 1.63 & 2.36 & 3.27 & 2.36 \\
CE-3 & 0.8 & 2.04 & 1.84 & 2.60 & 2.72 & 2.72 \\
CE-4 & 2.8 & 2.19 & 1.89 & 1.63 & 1.87 & 1.79 \\
CE-5 & 2.0 & 2.24 & 1.90 & 2.36 & 2.60 & 2.48 \\
CE-6 & 1.1 & 3.31 & 2.16 & 2.60 & 2.48 & 2.60 \\
CE-7 & 1.2 & 3.39 & 2.17 & 3.13 & 2.60 & 2.85 \\
CE-8 & 1.8 & 3.09 & 2.12 & 2.26 & 2.48 & 2.06 \\
CE-9 & 1.1 & 4.27 & 2.36 & 3.76 & 3.76 & 3.27 \\
CE-10 & 0.8 & 3.55 & 2.21 & 3.13 & 3.13 & 2.98 \\
CE-11 & 1.4 & 4.27 & 2.34 & 3.13 & 2.85 & 2.72 \\
CE-12 & 0.1 & 5.13 & 2.49 & 3.43 & 3.76 & 4.13 \\
CE-13 & 1.5 & 5.25 & 2.52 & 2.26 & 3.13 & 2.72 \\
CE-14 & 2.1 & 6.17 & 2.66 & 2.60 & 2.72 & 2.48 \\
CE-15 & 4.2 & 6.76 & 2.73 & 1.96 & 2.26 & 2.48 \\
CE-16 & 2.7 & 7.59 & 2.84 & 1.42 & 4.13 & 3.43 \\
CE-18 & 1.1 & 9.12 & 3.03 & - & 3.76 & 3.76 \\
CE-21 & 3.7 & 12.30 & 3.34 & 2.36 & 2.85 & 2.60 \\
CE-22 & 1.5 & 16.98 & 3.72 & 4.53 & 4.53 & 4.33 \\
CE-24 & 1.5 & 15.49 & 3.61 & 3.27 & 3.27 & 4.33 \\
CE-25 & 3.4 & 19.05 & 3.87 & 3.43 & 3.43 & 3.43 \\
CE-28 & 1.9 & 21.88 & 4.06 & - & 3.94 & 3.27 \\
CE-29 & 3.5 & 32.36 & 4.61 & 3.94 & 4.13 & 3.94 \\
\hline
\end{tabular}
\end{table*}
\begin{figure}
\centering
\includegraphics[width=0.46\textwidth]{1comparison.pdf}
\caption{The splashback feature visible in the average subhalo distribution of simulated high-mass clusters. We extract the logarithmic slope by fitting a smooth profile to the mean of the Hydrangea profiles rescaled by $r_\text{200m}$. We perform this operation both on the hydrodynamical simulations (Hydro) and their dark matter only counterparts (DMO). The minimum around $r_{200m}$ marks the halo boundary, and this figure highlights the lack of baryonic effects on the location or depth of this feature. The two logarithmic slope profiles are consistent with each other at the $1$ per cent level.}
\label{fig:comparison}
\end{figure}
\section{Splashback}
\label{sec:definition}
\subsection{Definition}
For haloes that continuously amass matter, material close to its first apocentre piles up next to the edge of the multi-stream region, where collapsed and infalling material meets \citep{Adhikari_2014}. A sudden drop in density, i.e. the feature visible in the profiles of Figure~\ref{fig:comparison}, is associated with this process.
This intuitive picture leads to three characterizations of the splashback radius, depending on the approach used to measure or model it:
\begin{enumerate}
\item The location of the outermost phase-space caustic.
\item The point of steepest slope in the density profile.
\item The apocentre of recently accreted material.
\end{enumerate}
While these definitions have all been previously hinted at in the introduction, in this section we explicitly present them and discuss the connections existing between them. This also justifies our adopted definition, based on the density profile.
The first definition is clearly motivated in the spherical case but fails once it is applied to realistic haloes. The presence of angular momentum and tidal streams from disrupted subhaloes \citep[see, e.g.,][]{2011MNRAS.413.1419V}, smooth out this feature and make its description murky. The second definition was the first suggested in the literature. Introduced by \cite{Diemer_2014}, it is based on the study of dark matter profiles in N-body simulations and has been linked to the first, more dynamical, definition \citep{Adhikari_2014, Shi2016}. The third was first suggested by \cite{Diemer_2017}, who showed that this location can be calibrated to the second one \citep{Diemer_2017b} by choosing specific percentiles of the apocentre distribution.
To clarify the relationship between the outermost caustic and apocentre, it is educational to use a self-similar toy model based on \cite{Adhikari_2014} to show the phase-space distribution of a constantly accreting halo with an NFW-like mass profile \citep{Navarro_1997}.
In the absence of dark energy, we follow the radial motion of particles,
\begin{equation}
\label{eq:motion}
\ddot{r} = \frac{GM(<r, t)}{r^2},
\end{equation}
between their first and second turnaround in the mass profile:
\begin{equation}
\label{eq:motion2}
M(r, t) = M(R, t) \frac{f_\text{NFW}(r/r_\text{s})}{f_\text{NFW}(R/r_s)}.
\end{equation}
We impose that the total mass evolves as $M(R, t) \propto t^{2\Gamma/3}$, $R\propto t^{2(1+\Gamma/3)/3}$, and the dimensionless NFW profile is defined as: $f(x) =\log(1+x)-x/(1+x)$. In this set of equations, $\Gamma$ is the dimensionless accretion rate, $R$ represents the turnaround radius, and the scale parameter $r_s$ is defined by the infall boundary condition
\begin{equation}
\label{eq:boundary}
\frac{d\log M}{d\log r}(R) = \frac{3\Gamma}{3+\Gamma}.
\end{equation}
This condition, combined with the turnaround dynamics, imposes that $M(R, t)\propto (1+z)^{-\Gamma}$ \citep{1984ApJ...281....1F}.
We point out that the dependence on the time-sensitive turnaround properties $M(R, t), R(t)$ can be factored out from the equations above, meaning that the entire phase-space at all times can be obtained with a single numerical integration.
In Figure~\ref{fig:gammavel} we show the result of this calculation, denoting the location of the outermost caustic as $r_\text{sp}^{c}$. The caustic is formed by the outermost radius at which shells at different velocities meet ($r/r_\text{sp}^c=1$ in the plot) and the location of shells at apocenter is defined by the intersection between the zero-velocity line and the phase-space distributions. From the figure, two things are noticeable: material at $r_\text{sp}^c$ has not reached its apocentre yet, and the ratio between these two locations depends on the accretion rate.
It is beyond the scope of this work to quantify this dependence since it depends heavily on the mass profile inside $r_\text{sp}^{c}$. Qualitatively, however, the difference between caustic and apocentre is easy to understand once the dynamical nature of this feature is considered: the halo is growing in size, and while some material is now reaching its apocentre, mass accreted more recently has the chance to overshoot it and form the actual caustic. In a static picture, this would not be the case.
In realistic haloes, this dependence on accretion rate is only one of many factors that biases and adds scatter to the relationship between the halo boundary and apocentres. Other factors include, e.g., non-spherical orbits and the presence of multiple accretion streams. Despite this, \cite{Diemer_2017} has shown that there is a clear link between the apocentre distribution and splashback. The percentile definition introduced there is particularly suited to theoretical investigations, but its usefulness in the very low-$\Gamma$ regime is still uncertain \citep{Mansfield_2017, xhakaj2019accurately}, and it has not been explored in the presence of modifications of gravity \citep{Adhikari_2018, Contigiani_2019}.
\begin{figure}
\centering
\includegraphics[width=0.46\textwidth]{2accretionratedep.pdf}
\caption{The phase-space structure of accreting dark matter haloes depends on the accretion rate $\Gamma$. We employ a toy model of spherical collapse to describe the multi-stream region of NFW-like haloes. The figure shows that the material at the outermost caustic, $r_\text{sp}^c$, is not necessarily at apocentre (where $v=0$) and that the ratio of these two radii is a function of accretion rate. For ease of readability, we have rescaled the coordinates by $r_\text{sp}^c$, and the velocity of collapsed material at this point. }
\label{fig:gammavel}
\end{figure}
For this work, we define the splashback radius as the location of the steepest slope as defined by a profile fit. In Table~\ref{tab:clusterlist} we report, for each cluster, this radius measured in the distribution of galaxies, subhaloes, and total matter ($r_\text{sp}^g, r_\text{sp}^s, r_\text{sp}$). The model is a modified Einasto profile \citep{Einasto1965} with the addition of a power-law to take into account infalling material \citep{Diemer_2014}. Regarding the goodness of fit, we find that up to and around $r_{200m}$ the standard deviation of the residuals is of order $10$ per cent. On the other hand, the presence of substructure superimposed on a shallow density profile results in normalized residuals of order $50$ per cent in the outer regions.
To further justify our approach, we show in Figure~\ref{fig:fit} how this simple definition of splashback radius is able to capture the phase-space boundary of different haloes, even when a sudden drop in density is absent. The main benefit of this definition is that it avoids the arbitrariness of the apocentre definition, or the bias induced by multiple caustics in the minimum slope definition \citep{Mansfield_2017}. Its main caveats, however, are that 1) it is computationally expensive since it requires high-resolution simulations and a multi-parameter fit procedure, and 2) it might not apply to low-mass clusters and galaxy groups. We leave this last question open for future investigations.
We wrap this subsection up by stressing that this definition of ``the'' splashback radius is, like any other, useful only to study its correlation with other properties, or quantify the impact of different physical processes. While the flexibility of the chosen model is not surprising given the number of free parameters, the clear connection between the phase-space and the log-derivative in individual haloes is a powerful and seemingly general result. Ultimately, however, the observational results focus on stacked projected density profiles, and so should the predictions.
\begin{figure}
\centering
\includegraphics[width=0.47\textwidth]{3fitexample.pdf}
\caption{Fitting simulated subhalo profiles with a smooth model. In the top panels, we show the radial subhalo distributions of two clusters (CE-16, left and CE-9, right), together with the best-fit profiles used to reconstruct the log-derivative. In the bottom panels, we show how the inferred location of the log-derivative minimum (vertical line) identifies the phase-space edge of relaxed (left) and perturbed (right) galaxy clusters. In the phase-space plots, the cluster on the left is formed by collapsed particles, while the stream visible on the lower right is infalling material. The right panels demonstrate how our approach is effective even in the presence of an on-going merger when the splashback feature is not visible as a sharp transition in the density profile.}
\label{fig:fit}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.46\textwidth]{4gammacorr.pdf}
\caption{The splashback radius and its correlation with the accretion rate. The ratio between the splashback radius and the $200$m overdensity radius correlates with the accretion rate. We show that this correlation exists for the clusters studied in this work and compare it to the relations obtained in three other studies (see text for references). }
\label{fig:gammarsp}
\end{figure}
\subsection{Accretion}
It is well established \citep{Diemer_2014, More_2015,Mansfield_2017,diemer2020part} that the location of the halo boundary correlates with the accretion rate
\begin{equation}
\Gamma_{0.3} = \frac{\Delta \log M_\text{200m}}{\Delta \log (1+z)}.
\end{equation}
In this work, this ratio is calculated in the redshift range $z=0$ to $z=0.293$, since this time interval roughly corresponds to one crossing time for all clusters considered here, i.e. how long ago the material currently at splashback has been accreted \citep{Diemer_2017}. Although this choice is partially arbitrary, we have investigated the dependence of our results on the redshift upper limit and we have verified that our main conclusions are not affected.
The archetypical relation demonstrating this idea is plotted in Figure~\ref{fig:gammarsp}, where we have also included the relations found in \cite{More_2015}, \cite{Diemer_2017b}, and \cite{diemer2020part}, to provide additional context. We find good agreement, even though a perfect match is not necessarily expected. The Hydrangea clusters represent a biased sample, selected to be mostly isolated at low redshift \citep{Bahe2017}. \edit{While the effect of this selection on the accretion rate distribution is not fully known, we} show below that a connection between cluster environment and this quantity exists, and the presence of mergers might therefore influence it. \edit{This is not surprising since a connection between accretion and large-scale bias is already known \citep[e.g., ][]{2010MNRAS.401.2245F}.}
We show this relationship explicitly in Figure~\ref{fig:slope} by using one of the parameters of the profile model. As visible in the figure, the power-law index of material outside of splashback correlates with the accretion rate. \edit{We find that this is true for both subhaloes and galaxies and that the difference between the two is consistent with sample variance}.
To try and explain this behaviour, we use a fully consistent model of spherical collapse introduced by \citet{1985ApJS...58...39B}, which was also used in \citet{Contigiani_2019}. The set-up of this toy model is the same as what is shown in Equation~\eqref{eq:motion}, but with a mass profile that also needs to be solved for. Starting from an initial guess for $M(r, t) = \mathcal{M}(r/R(t))$, orbits are integrated and their mass distribution is calculated. Iterating this process multiple times returns a self-similar density profile and orbits consistent with each other.
The result of this calculation is also shown in Figure~\ref{fig:slope}. Because the mass-profile prediction is not a power law, we plot a filled line displaying the range of logarithmic slopes allowed between $r_\text{sp}^\text{c}$ and $2r_\text{sp}^\text{c}$. The fact that this prediction is not a function of accretion rate implies that the correlation between the slope and the accretion rate seen in the simulations is not purely dynamical, and suggests a connection between the cluster environment and accretion rate.
We stress here that previous splashback works have mostly focused on stacked halo profiles, for which the expectation of the spherically symmetric calculation shown above is roughly verified, even in the presence of dark energy \citep{Shi2016}. We also recover this result for our sample (see the star symbol in Figure~\ref{fig:slope}), but we point out that this is a simple conclusion. Because Newtonian gravity is additive, stacking enough clusters should always recover the spherically symmetric result. Despite this, we also note that results from the literature do not always agree with this prediction. However, we do not linger on these discrepancies since 1) this was never the focus of previous articles, and 2) different methods to extract the power-law have been employed.
\subsection{Anisotropy}
This departure from the spherical case implies that anisotropies play a role in shaping the accretion rate $\Gamma$. To study the impact of accretion flows on the cluster boundary, we study $72$ sky projections of the Hydrangea clusters ($3$ each, perpendicular to the $x, y,$ and $z$ axes of the simulation boxes) and rotate them to align the preferred accretion axes in these planes. For each projection, we define this direction $\theta\in (-\pi/2, \pi/2)$ in two ways: 1) to capture the filamentary structure around the cluster between $r_\text{200m}$ and $5r_\text{200m}$, we divide the subhalo distribution in $20$ azimuthal bins and mark the direction of the most populated one, and 2) to capture the major axis of the BCG, we use unweighted quadrupole moments of the central galaxy's stellar profile within $10$ kpc from its centre. The mean projected distributions according to these two methods are presented in the left and right top panels of Figure~\ref{fig:filaments}, respectively.
\begin{figure}
\centering
\includegraphics[width=0.47\textwidth]{5power.pdf}
\caption{The distribution of subhaloes and galaxies outside the cluster edge as a function of accretion rate. Faster growing haloes display a more concentrated distribution of satellites outside of their boundary. This behaviour seen in individual clusters is not explained by simple models of spherical collapse (blue shaded area), but the average profile (marked by a star) matches the expectation. This suggests that non-isotropic processes shape this relation.}
\label{fig:slope}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.47\textwidth]{6filaments.pdf}
\caption{The impact of filaments and accretion flows on the cluster's edge. We rotate the $2$D subhalo distributions of different clusters to align their accretion axes. The top panels show the resulting mean distributions in a square region of size $5r_\text{200m}$ obtained with two definitions of this direction: one based on the presence of filaments outside $r_\text{200m}$ (left), and one based on the central galaxy's major axis (right). The first one better identifies the filamentary structures around the clusters, but the second one is closer to what can be observed. In the bottom panels, we show how the inferred $3$-dimensional logarithmic slope inside the quadrants aligned with the accretion direction (darker shade) differs from the profile outside (lighter shade). The results from the bottom panel imply that the central galaxy's major axis traces the direction of infalling material.}
\label{fig:filaments}
\end{figure}
Looking at the top-left panel of the figure, it is not surprising that filamentary structures of the cosmic web are visible around the central cluster -- this is by construction. \edit{Because of the higher contrast between outside and inside regions, the subhalo distribution exhibits a sharper feature in the directions pointing towards voids (see central panel of Figure~\ref{fig:filaments}).} More surprisingly, however, these same traits are also noticeable in the mean distributions aligned according to the central galaxy's axis (see lower panel).
This result implies that the distribution of stellar mass within the central $10$ kpc of the cluster contains information about the distribution of matter at radii which are a factor $10^2$ larger. In fact, the connection between the shape of the dark matter halo and the ellipticity of the brightest cluster galaxy (BCG, which is also the central galaxy for massive galaxy clusters) is known \citep{Okumura_2009, Herbonnet_2019, 2020MNRAS.495.2436R}. And, similarly to other results \citep{Conroy_2007, 2007MNRAS.375....2D}, the Hydrangea simulations predict that the stellar-mass buildup of the BCG is driven by the stripping of a few massive satellites after their first few pericentre passages (Bah\'{e} et al., in prep.). Because these galaxies quickly sink to the centre, the material they leave behind is, therefore, a tracer of their infalling direction.
\section{The mass--size relation}
\label{sec:msrelation}
In our sample, we find that the splashback feature seen in the galaxy, subhalo, and total matter profiles are all at the same location. The mean fractional difference between any two of $r_\text{sp}^g, r_\text{sp}^s$, or $r_\text{sp}$ is consistent with zero, with a mean standard deviation of $3$ per cent. We also verified that this statement is unaffected by cuts in subhalo mass or galaxy stellar mass. \edit{Due to the limited size of our sample, the effects of dynamical friction on the distribution of high-mass subhaloes are not visible \citep{2016JCAP...07..022A}.}
We emphasize, however that this does not mean that galaxy selection effects have no impact on these quantities. For example, it is an established result, both in the Hydrangea simulations \citep{oman2020homogeneous} and in observations \citep{2020arXiv200811663A}, that the location of a galaxy in projected phase-space correlates with its colour and star-formation rate. This is because a red colour preferentially selects quenched galaxies that have been orbiting the halo for a longer time.
Until their first apocentre after turnaround, galaxies act as test particles orbiting the overdensity as the halo grows in mass. In the standard cold dark matter paradigm, based on a non-interacting particle, it is not surprising then that the edge formed in their distribution is identical to the one seen in the dark matter profile. It should be noted, however, that this is not necessarily true in extended models in which dark matter does not act as a collisionless fluid. Due to their infalling trajectories, the distribution of galaxies will always display a splashback feature, even if the dark matter profile does not exhibit one.
In the cold dark matter scenario, our result implies that galaxies can be used to trace the edge of clusters. We note, in particular, that this measurement has already been performed several times using photometric surveys \citep{Baxter_2017, Nishizawa_2017, Chang_2018, Z_rcher_2019, Shin_2019}. Furthermore, due to the large number of objects detected, galaxy distributions obtained through this method offer the most precise measurements of splashback. The accuracy of the results, however, depends heavily on the details of the cluster finding algorithm \citep{Busch_2017, Shin_2019}.
With this in mind, we build an observational mass-size relation between the location of this feature in the galaxy distribution ($r_\text{sp}^g$) and the mass enclosed within it ($M_\text{sp}^g$). In Figure~\ref{fig:msscaling} we present the correlation between the two for the Hydrangea clusters. Because the splashback radius is roughly located at $r_\text{200m}$ (see Figure~\ref{fig:gammarsp}), this relationship can be understood as a generalization of the virial mass-radius relation, where we have introduced a dependence on accretion rate. Surprisingly, we find that the dependence on $\Gamma_{0.3}$ is well captured by a simple form:
\begin{equation}
\label{eq:scaling}
\frac{M_\text{sp}}{r_\text{sp}^3 } \propto (1+\Gamma_\text{0.3})^{\beta}.
\end{equation}
While we do not constrain $\beta$ precisely, we find that $\beta = 1.5$ provides an adequate fit by reducing the total scatter from $0.25$ dex to about half of this value. This choice of exponent and functional form is supported by the model of self-similar collapse used for Figure~\ref{fig:slope}, where we find that a power-law $\beta=1.45$ fits this relationship with the same precision as the exponential functions calibrated to numerical simulations shown in Figure~\ref{fig:gammarsp}. For a more extensive comparison with these predictions, we refer the reader to Section~\ref{sec:conclusions}.
The virial relation is a trivial connection between the mass and size of haloes based on an overdensity factor, but its observational power is limited by the fact that these masses are usually extracted from parametric fits to weak-lensing profiles that do not extend to the respective overdensity radii. Because of this, the overdensity masses have a strong dependence on the assumed mass-concentration relation \citep[see, e.g.][]{Umetsu_2020}. The splashback feature, on the other hand, naturally predicts a mass-size relation for galaxy clusters and does so without the need for external calibrations.
In Figure~\ref{fig:msscaling} we also plot the expected change in this relation to due modifications of gravity. We use the symmetron gravity model of \cite{Contigiani_2019} with parameters $f=1$ and $z_\mathrm{ssb}=1.5$, and assume that the change affects only the splashback radius and not the mass contained within it. The exact result depends on the theory parameters, but the expected change in this relation is around $0.15$ dex.
Experimentally, we argue that this relation can be probed using a combination of galaxy density profiles (to extract $r_\text{sp}^g$) and weak lensing measurements. Aperture masses \citep{2000ApJ...539..540C}, in particular, can be used to extract in a model-independent fashion the average projected mass within a large enough radius. If necessary, the aperture mass can also be deprojected to obtain a low-bias estimate \citep{Herbonnet_2020}.
\begin{figure}
\centering
\includegraphics[width=0.47\textwidth]{7scaling.pdf}
\caption{ The mass size relation of galaxy clusters. In the top panel, we show how the size of the cluster boundary seen in the galaxy distribution, $r_\text{sp}^{g}$, scales with its enclosed mass, $M_\text{sp}^g$. In the same panel we also show the median relation in Equation~\eqref{eq:scaling} obtained for $\beta = 0$ and how modifications of gravity are expected to affect this relation (blue dashed line, see text for more details). In this relation, a secondary dependence on the accretion rate $\Gamma_{0.3}$ is a source of scatter that can be captured if $\beta\neq 0$. As visible in the residuals in the bottom panel, a simple power-law form well reproduces this dependence. In the considered sample, we find that half of the total scatter ($0.25$ dex) is due to the mass accretion rate distribution.}
\label{fig:msscaling}
\end{figure}
\iffalse
\begin{figure}
\centering
\includegraphics[width=0.47\textwidth]{scaling.pdf}
\caption{The observed mass-size relation of galaxy clusters. For each cluster in our simulation, we take three projections and }
\label{fig:projection}
\end{figure}
\fi
\section{Redshift evolution}
\label{sec:redshift}
So far, we have only considered the simulation predictions at $z=0$. In this section, we extend our analysis to higher redshifts by exploring two other snapshots of the Hydrangea simulations at $z=0.474$ and $z=1.017$.
At these higher redshifts, we find that the scatter in the splashback relation for individual haloes is large. This is visible in Figure~\ref{fig:gammarspz}, where we plot the equivalent of Figure~\ref{fig:gammarsp} for these two snapshots. We recover the general result of \cite{diemer2020part} that the average values of $r_\mathrm{sp}/r_\mathrm{200m}$ and $\Gamma$ should be higher at early times, but the correlation between the two is completely washed out by $z=1$. We connect this to three causes: 1) The fixed time interval between the snapshots does not allow us to reliably estimate $\Gamma$ at higher redshift when the crossing times are smaller. 2) The lower number of resolved galaxies and subhaloes means that the residuals of the individual profile fits are larger around the virial radius. And finally, 3) the higher frequency of mergers at high redshift means that the numbers of haloes with profiles not displaying a clear splashback feature increases.
We find that Equation~\ref{eq:scaling} is still valid, even if our ability to constrain the scatter at high redshift is impeded by the sample variance. Furthermore, we report that the splashback overdensity $M_\text{sp}/r_\text{sp}^3$ has a redshift dependence. Or, in other words, that the logarithmic zero-point that was not specified in Equation~\ref{eq:scaling} is a function of redshift. Not accounting for the $\Gamma$ dependence, our best fit values for the logarithm of the average overdensity $\log_{10}(M_\text{sp}/\text{M}_\odot)-3\log_{10}( r_\text{sp}/\text{Mpc})$ are $[13.3, 13.8, 14.1]\pm 0.3$ at redshifts $[0, 0.5, 1]$.
Regarding the anisotropy in the splashback feature due to filamentary structures, we report that this phenomenon exists also at high redshift. In Figure~\ref{fig:filamentsz} we compare the sky-projected subhalo profiles $\Sigma_s(R)$ towards different directions, similarly to what we have done for Figure~\ref{fig:filaments}. In this case, however, we explicitly discuss the connection with observations by plotting directly the ratio of the density profiles inside quadrants oriented towards and perpendicular to the two accretion directions, instead of focusing on the result of the profile fits. The mean and variance of these ratios are calculated assuming that the different projections are independent. We find that the orientation of the major axis of the brightest cluster galaxy does not correlate with a splashback anisotropy at $z=1$. This is because, in most cases, the identification of a central, brightest galaxy is not straightforward at this redshift. At early times, the future central galaxy is still in the process of being created from the mergers of multiple bright satellites located close to the cluster's centre of potential.
To conclude this section, we point out that in the region around $r_\mathrm{200m}$, the difference between the profiles perpendicular and parallel to the central galaxy's major axis is about $10$ per cent at redshift $z\lesssim0.5$. This departure is well within the precision of galaxy profiles extracted from large surveys \cite[e.g.][]{2020arXiv200811663A}. Therefore, this measurement might already be possible using such catalogues.
\begin{figure}
\centering
\includegraphics[width=0.46\textwidth]{8gammacorrz.pdf}
\caption{The splashback radius and its correlation with the accretion rate as a function of redshift. This plot is an extension of Figure~\ref{fig:gammarsp} for redshifts $z=0.474$ (orange crosses) and $z=1.017$ (light blue plus symbols). The ratio between the splashback radius and the $200$m overdensity radius should correlate with the accretion rate $\Gamma$, but for the Hydrangea snapshot at $z=1.017$ the large sample variance washes out this correlation. Despite this, we still recover the expectation of previous results (plotted lines), a larger average $r_\mathrm{sp}/r_\mathrm{200m}$ at higher redshift. }
\label{fig:gammarspz}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.46\textwidth]{9filamentsz.pdf}
\caption{The impact of filaments and accretion flows on the outer density profile of massive haloes as a function of redshift. We plot the mean value and variance of the ratio between the $2$D subhalo distributions in quadrants perpendicular ($\Sigma_\perp$) and parallel ($\Sigma_\parallel$) to the accretion direction defined through two tracers. This ratio is closely related to what can be measured in observations. While the difference in profile towards and away from filamentary structures is visible at all redshifts, the orientation of the central galaxy is not a good tracer of the splashback anisotropy at high redshift. This is because the central BCG is still forming and its orientation is not yet finalized.}
\label{fig:filamentsz}
\end{figure}
\section{Discussion and conclusions}
\label{sec:conclusions}
On its largest scales, the cosmic web of the Universe is not formed by isolated objects, but by continuously flowing matter distributed in sheets, filaments and nodes. For accreting (and hence non-virialized) structures such as galaxy clusters, the splashback radius $r_
\text{sp}$ represents a physical boundary motivated by their phase-space distributions. \edit{To exploit the information content of this feature, in this paper we have introduced and studied two observable quantities related to it. }
\edit{First, we have shown that the full galaxy profile can be used to define a cluster mass, i.e. the mass within $r_\mathrm{sp}$. This is an extension of the traditional approach of using richness as mass proxy \citep[see, e.g., ][]{Simet_2016}.} Because of the dynamical nature of the equivalent feature in the dark matter profile, we conclude that, observationally, the splashback feature in the galaxy profile \emph{defines} the physical halo mass. Moreover, we have shown here that the natural relation between the mass and size of haloes according to this definition (see Fig.~\ref{fig:msscaling}) can be used to constrain new physics at cluster scales. Because this boundary is delimited by recently accreted material, we found that a majority of the scatter in this mass-size relation can be explained through a secondary dependence on accretion rate $\Gamma$.
\edit{Second, we have explored how this connection to the accretion rate might be interpreted as a connection between the geometry of the cosmic web and how clusters are embedded in it.} The relation between the two is made explicit in Figure~\ref{fig:slope}, Figure~\ref{fig:filaments}, and Figure~\ref{fig:filamentsz}. In these figures, we have investigated how the cluster environment affects both the halo growth and the stellar distribution of the central galaxy. This information, combined with the scatter of the mass-size relation, can therefore be used as a consistency check for any property that claims to select for accretion rate.
\subsection{The role of simulations}
\label{sec:sims}
In the last few years, the study of the splashback feature has evolved into a mature field both observationally and theoretically. We use this section to discuss explicitly the connection between the two, in light of this work and its connection to previous endeavours.
\edit{In the context of splashback, simulations have guided the formulation of theoretical principles and hypotheses. However, as more measurements become viable, it becomes necessary to provide clear and powerful observables.} Following this spirit, we used high-resolution hydrodynamical simulations to explore directly the connection between measurements based on sky-projected galaxy distributions and theoretical predictions.
Our conclusions regarding the mass-size relation and its redshift evolution are similar to the results of \cite{Diemer_2017b} and \cite{diemer2020part}, which are based on more extensive N-body simulations. For the sake of completeness, it important to note that, in the same papers, it was also found that the splashback overdensity is not universal, but has both a mild dependency on $M_\mathrm{200m}$ and a strong dependency on cosmology, especially at low redshift ($z\approx0.2$). Due to our limited sample, we are clearly unable to model these effects in this work. Nonetheless, we point out that our goal here is to construct a pure splashback scaling-relation based on galaxy profiles and weak lensing mass measurements. Every other dependency, if present, should be captured either as additional scatter or through different parameter values.
We also point out that that these previous works are based on the apocentre definition of splashback (see Section~\ref{sec:definition}). In contrast, we defined the splashback radius as the point of steepest slope according to a model fit to the density profiles of galaxy clusters. While we do not necessarily expect the two definitions to differ, our choice is based on its connection to observations, and the desire to highlight the fact that the splashback radius is not only some abstract halo property but can be defined as a characteristic of individual profiles, such as, e.g., the concentration parameter \citep{Navarro_1997}.
\edit{An alternative method employed by other studies \citep{Diemer_2014, Mansfield_2017, xhakaj2019accurately} to obtain a measure of $r_\mathrm{sp}$ makes use of the minimum of the logarithmic slope in smoothed profiles. While this approach is much faster than profile fitting when only $r_\mathrm{sp}$ is of interest, it does not describe the full shape visible in Figure~\ref{fig:comparison}. In particular, a model that captures the width of this feature is necessary to define the slope of the outer region without an arbitrary choice of which radial scales to consider. Because the model used here contains an asymptotic outer slope, this definition is unique.}
Our decision has, of course, its drawbacks. The versatility of the fitted model is necessary to capture the variance of the individual profiles, but the resulting intrinsic scatter is large and not the best suited to study tight splashback correlations (such as Figure~\ref{fig:gammarsp}). At the same time, the large parameter space might also be seen by some as a chance to study a multitude of correlations between different model parameters. However, we resist this temptation, as inferences based on such correlations might say more about the particular model employed than provide any physical information.
A subtler difference between our method to characterize splashback and other ones present in the literature is related to the definition of spherical density profiles. \cite{Mansfield_2017} and \cite{2020MNRAS.tmp.3386D} found that the most successful method to achieve a clear splashback feature for individual haloes is to measure the median profile along multiple angular directions. In light of the results of Figure~\ref{fig:filaments}, we argue that the distribution of splashback as a function of direction is skewed by the presence of a few dense filaments and hence the difference between a median and mean splashback can be substantial. Therefore, we stress that future works should exercise caution when employing such methods. The use of median profiles smooths substructure by focusing on the halo boundary in the proximity of voids, but because this process is itself correlated with the halo growth rates (see Figure~\ref{fig:slope}), the connection with observations is not as simple as one might expect.
\subsection{Next steps}
Because in this work we have focused only on high-mass objects ($M\sim10^{14.5}~M_\odot$), a natural future step is to investigate if the results apply also in other regimes. For example, a larger sample over a wide range of masses and redshifts is required to confirm the simple form of Equation~\eqref{eq:scaling} and verify if it applies to lower mass groups ($M\sim10^{13.5}~M_\odot$).
Exploring a wider range in mass, both in observations and simulations, can also be used to confirm a key prediction: because the median accretion rate is expected to be a function of mass and redshift \citep{More_2015}, we expect the mass-size relation for an observed halo sample to not necessarily follow a simple form.
Finally, we point out that our results encourage a concentrated effort towards understanding the relationship between cluster environment and splashback. What is discussed in Figure~\ref{fig:slope}, \ref{fig:filaments} and \ref{fig:filamentsz} suggests that the connection between accretion-flows, filaments, and cluster boundary is not a simple one. To better understand this process, it will become necessary to complement the usual \emph{inside-out} theoretical approaches to splashback, that look at haloes to define their boundaries, with \emph{outside-in} approaches, that connect the cosmic web to its nodes. In this context, the amount of splashback data gathered by projects such as the Kilo Degree Survey \citep{2013ExA....35...25D}, Dark Energy Survey \citep{2005astro.ph.10346T}, and, in the future, LSST \citep{2009arXiv0912.0201L} and \emph{Euclid} \citep{laureijs2011euclid} will provide a powerful probe for the study of structure formation.
\section*{Acknowledgements}
We thank Benedikt Diemer for providing valuable feedback on the manuscript.
OC is supported by a de Sitter Fellowship of the Netherlands Organization for Scientific Research (NWO). HH acknowledges support from the VICI grant
639.043.512 from NWO. YMB acknowledges funding from the EU Horizon 2020 research and innovation programme under Marie Sk{\l}odowska-Curie grant agreement 747645 (ClusterGal) and the NWO through VENI grant 639.041.751. The Hydrangea simulations were in part performed on the German federal maximum performance computer ``HazelHen'' at the maximum performance computing centre Stuttgart (HLRS), under project GCS-HYDA / ID 44067 financed through the large-scale project ``Hydrangea'' of the Gauss Center for Supercomputing. Further simulations were performed at the Max Planck Computing and Data Facility in Garching, Germany.
\iffalse
\section*{Data Availability}
The inclusion of a Data Availability Statement is a requirement for articles published in MNRAS. Data Availability Statements provide a standardised format for readers to understand the availability of data underlying the research results described in the article. The statement may refer to original data generated in the course of the study or to third-party data analysed in the article. The statement should describe and provide means of access, where possible, by linking to the data or providing the required accession numbers for the relevant databases or DOIs.
\fi
\bibliographystyle{mnras}
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1,108,101,564,575 | arxiv | \section{Introduction}
Plate tectonic theory provides a kinematic framework for describing crustal motion at the Earth's surface. Rates of motion for the largest tectonic plates have been inferred independently over million-year time-scales from geologic observations \citep{demets1990current} and decadal time-scales from geodetic measurements \citep{sella2002revel}. In many cases late-Cenozoic and decadal plate motion estimates agree to within 5\% \citep{demets1999new}, however $\sim$1 million year reorganization events have also been observed \citep{croon2008revised}. At smaller spatial scales ($1,000$ km) and shorter temporal scales ($<10,000$ years), earthquake and fault activity at some plate boundaries has also been found to be irregular in time \citep{bennett2004codependent, dolan2007long, dolan2016extreme}. A hypothesis that may provide an explanation for such short-term variations is that plate motions may be perturbed over short time scales due to changes in the crustal and upper mantle stress state caused by great earthquakes \citep{anderson1975accelerated, romanowicz1993spatiotemporal} or evolving earthquake sequences \citep{ismail1999numerical, ismail2007numerical}. A geodetically based inference of accelerated Pacific plate subduction following the 2011 Tohoku, Japan earthquake may provide an example of the direct observation of this type of behavior \citep{heki2013accelerated}. Though this interpretation is non-unique \citep{tomita2015first} it is consistent with the inference of accelerated subduction rates beneath Kanto inferred from repeating earthquakes \citep{uchida2016acceleration}. However, because stress changes are largest in the immediate vicinity of a great earthquake, possible changes in motion are likely to be observed in crustal blocks adjacent to the source. In particular, small tectonic plates or crustal blocks in the hanging wall above subduction zones may experience relatively large coseismic stress changes ($10^2-10^5$~Pa) along their edges with relatively little resistance to motion at the crust-mantle interface due to the fact that viscosities may be relatively low in subduction zone mantle wedges \citep{billen2001low, kelemen2003thermal}, resulting in a coseismically induced change in plate motion.
Here we consider this argument quantitatively by calculating a first-order approximation of the magnitude of the crustal block motion change induced by coseismic stress changes using a simple disk geometry for crustal plates and assuming a Newtonian mantle rheology. These first-order predictions can be tested against geodetically constrained estimates of velocity changes before and after both the 2003 $\mathrm{M}_{\mathrm{W}}=8.3$ Tokachi and 2011 $\mathrm{M}_{\mathrm{W}}=9.0$ Tohoku earthquakes in Japan.
A challenge to the direct observation of such motion changes near major plate boundaries is that geodetically observed surface motions reflect the combined, and correlated, contributions from both plate motions and earthquake cycle processes. In other words, interseismic strain accumulation, postseismic deformation, and plate rotations must be disentangled in the vicinity of large earthquakes. Here we utilize results from time-dependent block models of Japan that simultaneously consider these contributions in a way that minimizes temporal variations in nominally interseismic station velocity changes \citep{loveless2016two}.
\section{Coseismic Stress and Plate Motion Changes}
The motions of tectonic plates are in quasi-static force equilibrium if there is no angular acceleration and the sum of applied torques equals zero. The quasi-static torques may be partitioned into edge, $\mathbf{T}_\mathrm{E}$, and basal, $\mathbf{T}_\mathrm{B}$, components \citep{solomon1974some, forsyth1975relative, iaffaldano2015rapid} as,
\begin{equation}
\int \mathbf{T}_\mathrm{E} da_\mathrm{E} + \int \mathbf{T}_\mathrm{B} da_\mathrm{B} = 0,
\end{equation}
with $\mathbf{T}_\mathrm{E} = \mathbf{x} \times (\boldsymbol{\sigma}_\mathrm{E} \cdot \mathbf{n}_\mathrm{E})$ and $\mathbf{T}_\mathrm{B} = \mathbf{x} \times (\boldsymbol{\sigma}_\mathrm{B} \cdot \mathbf{n}_\mathrm{B})$. Here $\mathbf{x}$ is position, $\boldsymbol{\sigma}_\mathrm{E}$ is the stress at the seismogenic edges of the block, $\boldsymbol{\sigma}_\mathrm{B}$ is the viscous stress exerted by the mantle on the base of the block, $\mathbf{n}_\mathrm{E}$ is a unit vector normal to the fault at the block edge, and $\mathbf{n}_\mathrm{B}$ is a unit vector normal to the fault at the block base. The viscous stresses for an iso-viscous Newtonian mantle are $\boldsymbol{\sigma}_\mathrm{B} = \eta \boldsymbol{\epsilon}_\mathrm{B}$, where, $\eta$ is the viscosity of the mantle and $\boldsymbol{\epsilon}_\mathrm{B}$ is the strain rate at the base of the tectonic plate. We calculate the relative magnitudes of these torques considering the case of a plate with the shape of circular disk with radius $r$ and bounding faults extending from the surface to depth, $h$. A first-order approximation of the magnitude of the edge torque, $T_\mathrm{E}$, is given by,
\begin{equation}
T_\mathrm{E} = \left| \left| \int \mathbf{x} \times (\bm{\sigma}_\mathrm{E} \cdot \mathbf{n}_\mathrm{E}) da_\mathrm{E} \right| \right| = R \| \bm{\sigma}_\mathrm{E}\| A_\mathrm{E} = 2\pi R r h \sigma_\mathrm{E},
\end{equation}
where $R$ is the radius of the Earth, the area of the plate edge is $A_\mathrm{E}=2\pi r h$, and $\| \bm{\sigma}_\mathrm{E}\|$ gives the average stress on the edges of the plate. Similarly, a first-order approximation of the magnitude of the basal torque, $T_\mathrm{B}$, is given by,
\begin{equation}
T_\mathrm{B} = \left| \left| \int \mathbf{x} \times (\boldsymbol{\sigma}_\mathrm{B} \cdot \mathbf{n}_\mathrm{B}) da_\mathrm{B} \right| \right| = R \eta \| \dot{\boldsymbol{\epsilon}}_\mathrm{B}\| A_\mathrm{B} = \frac{\pi R r^2 \eta s}{D},
\end{equation}
where the basal area of the plate is $A_\mathrm{B} = \pi r^2$ and the strain rate in the upper mantle, $\dot{\boldsymbol{\epsilon}}_\mathrm{B}$, is given by the plate speed, $s$, divided by the thickness of an upper mantle-like layer, $D$, in the Couette flow approximation. Because we are concerned with the case where the quasi-static torque balance is satisfied, both before and after a coseismic stress change, we assume that any time variation is balanced and plate geometry fixed,
\begin{equation}
\frac{d}{dt} \left( T_\mathrm{E} + T_\mathrm{B} \right) = 2h \frac{d\sigma_\mathrm{E}}{dt} + \frac{\eta r}{D} \frac{ds}{dt}= 0.
\end{equation}
Discretizing over finite changes in coseismic stress, $d\sigma_\mathrm{E} / dt \approx \Delta \sigma_\mathrm{E} / \Delta t$, and plate speed, $ds / dt \approx \Delta s / \Delta t$, the change in plate speed as a function of coseismic stress changes is,
\begin{equation}
\Delta s = \frac{2 h D \Delta \sigma_\mathrm{E}}{\eta r},
\end{equation}
which may be interpreted as the change in plate speed, without regard to change in direction, as a result of coseismic stress changes in the upper crust. In this idealized quasi-static formulation the change in plate velocity occurs over seismic wave propagation time scales, temporally coincident with the change in coseismic stress. We can estimate the potential magnitude of this effect with characteristic parameters, $h = 15$ km, $D=100$ km. Upper mantle viscosities beneath Japan have been estimated independently from studies of both post-glacial and post-seismic rebound and range from $4\times10^{18}$ to $5\times10^{20}$~Pa$\cdot$s (Table~\ref{table1}) \citep{nur1974postseismic, nakada1986holocene, nakada1991late, suito1999simulation, ueda2003postseismic, nyst20061923, mavrommatis2014decadal}. In general, viscosity estimates at shorter forcing frequencies (100--1,000 year earthquake cycles) are $\sim 10\times$ lower than those inferred at longer duration forcing frequencies ($\sim100,000$ year glacial cycles). For intermediate field stress drops ($10^2-10^5$Pa) associated with the Tohoku earthquake \citep{toda2011using} and crustal blocks of radius 30 km, we estimate coseismically induced plate speed changes of ranging from $0.05-5.0$ mm/yr (Figure \ref{PredictedSpeedPerturbations}). A weakness of this scalar approach is that it only reflects changes in plate speed relative to a fixed point in the mantle. What this means is that there are cases where changes in plate velocity would not be captured with this theory. For example, if a plate were moving at $5$ mm/yr to the north before an earthquake and then $5$ mm/yr to the south after the earthquake, $\Delta s=0$, yet the change in plate velocity is $10$ mm/yr. The full tensor version of the theory addresses this issue but is beyond the scope of this paper.
\section{Block Speed Changes in Japan}
Due to the fact that GPS observations record not only plate motions but also earthquake cycle processes, the detection of plate motion changes requires an estimate of the relative contributions of these two effects. Previous work on the decomposition and block modeling of $~\sim19$~years of time-dependent GPS positions in Japan \citep{loveless2016two} has led to the development of minimally variable velocity fields over five temporally distinct year long epochs (Table~\ref{table2}). The multi-year duration of these epochs is longer than the sub-hourly time scale involved in the idealized quasi-static plate motion change theory described above. The reason for this is that these epochs are bracketed by specific seismic events, enabling comparison with coseismically imposed stress change, and are long enough that they can be decomposed into constitutive parts. In this work we assumed that, over each epoch, the GPS position time series at each station can be decomposed into four contributions: 1) linear trend (epoch velocity), 2) step function (coseismic displacements and equipment maintenance), 3) harmonic terms (annual and bi-annual signals), and 4) two exponentially decaying terms (accounting for short- and long-term post-seismic deformation). Epoch velocities estimated over these intervals are determined in way that is minimally variable in the sense that the relaxation timescale for the longer exponential term is determined so that the linear terms, which we take to represent nominally interseismic velocity fields, are as similar as possible from one epoch to the next.
These interseismic velocity fields are used as inputs into block models that simultaneously estimate block motions and earthquake cycle effects due to coupling on block-bounding faults and the subduction interfaces off the Pacific coast of Japan. It is the velocity component arising from block motions, rather than the raw GPS velocities, that we consider here for the identification of changes in plate speeds, $\Delta s$. In other words, changes in plate motions are determined by considering epoch-to-epoch differences in the estimates of block rotations derived from block model decomposition of the minimally variable GPS velocity fields. These changes in estimated rotation vectors (Euler poles) form the basis for this analysis. Instead of describing the change in motion of each block with a single scalar speed we consider a frequency distribution of velocities by calculating differential velocities at 1,000 randomly sampled geographic locations on each block (Figure~\ref{JapanDeltaS}). This approach allows us to simply see spatially complex behaviors such as those arising from local Euler poles, which are manifest as frequency distributions with large spreads (e.g., $>6$ mm/yr, Figure \ref{JapanDeltaS}).
Differential epoch-to-epoch block speeds are given by, $\Delta s_i = | \langle s(E_{i+1}) \rangle - \langle s(E_{i}) \rangle |$, where $\langle s(E_{i}) \rangle$ is the mean of the frequency distribution of location-by-location velocity during the epoch $E_i$ (Table~\ref{table2}). To determine if there are distinct changes in block motions associated with the geodetic observation epochs bracketing the 2003 Tokachi and 2011 Tohoku earthquakes, we define a block speed change index. This index evaluates as true for a given block if all three speed change conditions are met. First, the mean epoch-to-epoch speed change is greater than 5~mm/yr, $\langle \Delta s_i \rangle > 5$. This criterion identifies only the largest plate motion changes. Second, the variability of differential speeds calculated at points on a given block is less than 4~mm/yr, i.e., $SD(\Delta s_i) < 4$. These criteria limit the misidentification of block motion changes that are due to local rotation rate changes. We exclude these cases because the scalar theory only represents changes in plate speed. Third, the average epoch-to-epoch variability is less than 4~mm/yr, i.e., $\langle SD(\Delta s_i) < 4 \rangle$. This criterion precludes the misidentification of velocity changes in noisy regions. These three criteria are evaluated for each of the upper plate blocks in Japan.
Using these motion change detection criteria, we find no significant epoch-to-epoch speed changes other than between the intervals spanning the 2011 Tohoku earthquake. This includes a lack of identifiable plate speed changes in the epochs bracketing the 2003 Tokachi earthquake, which may be explained by the facts that there appear to be no small crustal blocks in the immediate vicinity of this event (Hokkaido and northern Honshu) where the coseismic stress changes were largest, and the magnitude of stress change was smaller than that due to the 2011 event. In contrast, we detect four blocks that appear to have discernable changes in plate speed in the epoch bracketing the 2011 Tohoku earthquake (Figure~\ref{JapanDeltaS}). Each of these blocks lies in central Honshu near the Itoigawa-Shizuoka tectonic line and Niigata-Kobe tectonic zone and has a characteristic length scale of $\sim 100$~km. The differential speed estimates for these blocks range from $2-5$~mm/yr, assuming upper mantle viscosities $>10^{18}$~Pa$\cdot$s (Figure~\ref{PredictedSpeedPerturbations}). Note that the change detection index does not identify plate motion changes in the block immediately above the rupture area of the Tohoku earthquake in eastern Honshu where geodetic observations suggest that postseismic deformation is greatest \citep{ozawa2011coseismic}.
\section{Discussion}
Slip on geometrically complex fault systems enables tectonic plates to move past one another. At these interfaces, fault system geometry may evolve in response to geometric \citep{gabrielov1996geometric} and kinematic \citep{mckenzie1969evolution} inconsistencies generated by repeated fault slip and cumulative fault offset. Similarly, fault slip rates are governed by the quasistatic equilibrium that sets the differential motion of bounding tectonic blocks. At longer time scales ($>1$~million year), changes in plate motion have been ascribed to changes in the relative buoyancy of subducted geometry/material as well as the collision and growth of tectonic terranes \citep{iaffaldano2006feedback, jagoutz2015anomalously}. At $1,000-100,000$ year time scales, emerging geological evidence suggests that rates of local fault system activity may vary significantly ($3-10\times$) in both strike-slip and thrust faulting environments \citep{dolan2016extreme, saint2016major}. Here we consider the possibility that short-term changes in fault slip rates may be the result of short-term changes in plate motions caused by coseismic stress changes. In other words, these observable changes in fault slip rates may be due to changes in the differential motion of bounding tectonic plates. This hypothesis provides an intrinsic source of variations in rates of fault system activity, with the implication that coseismically induced block motion changes persist sufficiently long to be captured in geologic records.
Here we’ve considered the case where coseismic stress changes modulate plate motions. However, this is an inherently incomplete description of potential changes in plate motions even in the narrow context earthquake cycle behaviors. For example, the time-varying effects of viscoelastic post-seismic deformation may spread over 100s--1000s of km \citep{pollitz1998viscosity}. Time-dependent viscous deformation of the upper mantle could therefore cause accelerated or decelerated surface velocities across an entire plate, blurring the conceptual distinction between classes of models other than the fact that the viscoelastic effect would decay with time and the coseismic stress-based model discussed here would be a more discrete change in plate motions. The viscoelastic explanation appears to be most in line with the previous suggestions of changing tectonic plate motions \citep{anderson1975accelerated, romanowicz1993spatiotemporal, pollitz1998viscosity} while the latter explanation is most consistent with ideas about fault and block systems \citep{gabrielov1996geometric, ismail1999numerical}. An additional aliasing is possible when considering both coseismic displacements and changes in block motion to be step-like in time. This may be considered a simplification of the idealized version of the theory presented here and a more temporally complex type of behavior may be plausible where evolving stresses throughout the earthquake cycle modulate block and plate motions more gradually.
For the Japan example considered here, the epoch-wise velocity fields used in the block models \citep{loveless2016two} were estimated after subtracting out annual and semi-annual harmonics, coseismic displacements of up to 200 earthquakes, and two exponential terms (short- and long-duration) following seven major earthquakes during the time span considered. If these exponentially decaying terms are interpreted as representing all postseismic deformation then the block motion changes estimated here can be interpreted as uniquely distinguished from crustal deformation that is driven by viscoelastic or afterslip processes. Similarly, the observation that the block motion change index used here does not identify the large blocks closest to the 2011 Tohoku earthquake may indicate that there is minimal aliasing between changes in block motion and more classical postseismic deformation processes. However, there is significant covariance in velocity field decomposition between the estimated linear velocity and duration of postseismic deformation even when temporal variations in the nominally interseismic velocity term are minimized \citep{loveless2016two}. The use of such velocity fields in the coseismic block motion change theory described here may therefore be considered to represent a parsimonious end-member interpretation.
In addition to this solid-earth focused hypothesis for rapid plate and fault motion evolution, climatically driven changes in surface loading have also been suggested to modulate fault slip rates \citep{hetzel2005slip, luttrell2007modulation}. Taken together there appears to be a growing suite of mechanisms that can modulate the rates of fault system behavior over relatively short time intervals including not only slip rate variations but also changes in interseismic loading rates \citep{sieh2008earthquake}. These processes may contribute to understanding of short-term fault and plate motion rate changes as well as contribute to an emerging view of tectonic motions as noisy trends through time (Figure~\ref{PlateVelocityTimeSeriesFigure}), where short-term (as low as $10^{-8}$~Hz) variations in fault system activity due to earthquakes and changes in surface loading lead to perturbations to long-term behavior. Whether or not a cluster of large earthquakes in a given region might change plate motions constructively or destructively over a large area remains an extant question.
\section{Conclusions}
The equilibrium torque balance on tectonic plates at short length scales ($<500$~km) suggests that plate speeds may be moderately perturbed ($<5$~mm/yr) by large coseismic stress changes ($10^3$~Pa) if the mantle beneath the crust is characterized by a relatively low viscosity ($<10^{19}$~Pa$\cdot$s). Following the 2011 $\mathrm{M}_\mathrm{W} = 9.0$ Tohoku earthquake in Japan we identify possible motion changes ($2-4$~mm/yr) in four small upper plate blocks across central and southern Honshu. Similar effects are not observed in blocks that are larger and/or more distant from the rupture source, nor following the smaller 2003 $\mathrm{M}_\mathrm{W} = 8.3$ Tokachi earthquake. This coseismic perturbation to block motion contributes to growing list of candidate processes that may modulate the temporal evolution of block and fault system activity at plate boundaries.
\begin{acknowledgments}
TBD
\end{acknowledgments}
\bibliographystyle{agufull08}
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1,108,101,564,576 | arxiv | \section{Introduction}
Three-dimensional General Relativity is one of the simplest gravitational systems \cite{Deser:1983tn,Deser:1983nh} and, in particular, solutions with negative cosmological constant (AdS${}_3$) have received special attention, due to their holographic nature \cite{Brown:1986nw, Witten:1998qj}. The absence of bulk propagating degrees of freedom makes this theory a privileged playground to better understand the role of boundary conditions in gravity. Indeed, the dynamics can be described by a pure boundary theory, as shown in the Chern--Simons formulation \cite{Achucarro:1987vz, Witten:1988hc,Banados:1994tn,Coussaert:1995zp,Henneaux:1999ib,Allemandi:2002sx,Banados:2006fe,Banados:2016zim}, (for a recent review, see also \cite{Donnay:2016iyk}).
Boundary conditions play a pivotal role in physics. Together with the choice of a bulk gauge for the metric, they fully determine the field content -- the solution space -- of the theory. Residual diffeomorphisms are those preserving the gauge choice. Among them, the ones respecting boundary conditions and carrying non-vanishing surface charges are the so-called asymptotic symmetry generators \cite{Regge:1974zd,Benguria:1976in,Brown:1986nw,Barnich:2001jy,Carlip:2005tz,Barnich:2007bf}.\footnote{For recent reviews, see \cite{Oblak:2016eij,Compere:2018aar,Ruzziconi:2019pzd}.} The surface charges are interesting quantities, for they encode observables of the system, such as its energy and momenta \cite{Wald:1993nt,Iyer:1994ys}.\footnote{Recently there has been a renewed interest in the charges structure of spacetime corners
\cite{Freidel:2020xyx,Freidel:2020svx,Freidel:2020ayo}.} The asymptotic symmetry generators form the asymptotic symmetry algebra, represented on the solution space by the projective charge algebra, trustworthy up to a universal central extension \cite{Brown:1986nw}. Probing various boundary conditions and their related surface charges is a natural question, the literature on the topic is extensive, see e.g. \cite{Barnich:2006av,Compere:2008us,Compere:2013bya,Troessaert:2013fma,Avery:2013dja,Grumiller:2016pqb,Grumiller:2017sjh,Poole:2018koa,Henneaux:2019sjx,Bergshoeff:2019rdb,Donnay:2020fof,Adami:2020ugu}. Along this line of thought, in this work we introduce a new set of boundary conditions, justified below, and study its consequences.
In the seminal work by Brown and Henneaux (BH) \cite{Brown:1986nw} it was shown that the asymptotic symmetry algebra of AdS$_3$, under certain boundary conditions encompassing BTZ black holes \cite{Banados:1992wn,Banados:1992gq,Carlip:1995qv}, consists in two commuting copies of the Virasoro algebra with central extensions $c^{\pm}=\frac{3\ell}{2G}$, $\ell$ being the AdS${}_3$ radius and $G$ the Newton constant. This result is considered as a precursor of the AdS/CFT correspondence \cite{tHooft:1993dmi, Susskind:1994vu, Maldacena:1997re, Gubser:1998bc, Witten:1998qj}, which, applied to three-dimensional General Relativity, conjectures the existence of a dual Confomal Field Theory (CFT) living on the two-dimensional boundary. Remarkably, the value of $c^{\pm}$ has been used to microscopically derive the Bekenstein--Hawking entropy of the BTZ black hole \cite{Strominger:1997eq}, using the Cardy formula \cite{Cardy:1986ie}. Moreover, by taking a suitable flat limit of asymptotically AdS${}_3$ gravity \cite{Barnich:2012aw}, it is possible to extend these considerations to asymptotically flat spacetimes \cite{Barnich:2010eb,Campoleoni:2018ltl}.
In the context of Penrose conformal compactification \cite{Penrose:1962ij,Penrose1964}, applied to the case of AdS$_3$ spacetime, the bulk metric induces a boundary conformal class $\left[g^{(0)}\right]$ of metrics rather than a metric \cite{FG1,Henningson:1998gx,Skenderis:1999nb,Rooman:2000zi,Rooman:2000ei,Fefferman:2007rka,Alessio:2017lps,Ciambelli:2019bzz}. The boundary conditions considered here are motivated by this approach. In BH, a particular representative of the equivalence class is picked up, namely the flat Minkowski metric $\eta$, and kept fixed under the action of the asymptotic symmetry algebra. This defines asymptotically (globally) AdS${}_3$ spacetimes (AAdS${}_3$).
In this manuscript we focus on asymptotically locally AdS${}_3$ (AlAdS${}_3$) spacetimes \cite{Graham:1991jqw,Graham:1999jg,Papadimitriou:2005ii,Fischetti:2012rd}, with no restriction on their boundary conformal structure. In general, the two-dimensional boundary metric $g^{(0)}$ is specified by three arbitrary functions in terms of which the Einstein equations can be exactly solved. We work in the Fefferman--Graham (FG) gauge \cite{FG1,Fefferman:2007rka} and we assume the boundary metric to be conformally flat, the conformal factor being an arbitrary smooth function independent of the radial coordinate.\footnote{The case in which the conformal factor admits a chiral splitting has been extensively analysed in previous works \cite{Troessaert:2013fma,Barnich2014TheDT}.} The resulting asymptotic symmetries contain the usual two copies of diffeomorphisms of the circle together with additional Weyl transformations of the boundary metric. These are often referred to in the literature as Penrose--Brown--Henneaux (PBH)\cite{Imbimbo:1999bj} transformations. Here we explicitly compute their associated surface charges,\footnote{For the surface charges we use the prescription given in \cite{Barnich:2001jy}.} and find that they are finite, integrable but non conserved, which is an interesting unusual combination (see \cite{Barnich:2007bf,Adami:2020ugu,Chandrasekaran:2020wwn} for related discussions).
Diffeomorphisms generating boundary Weyl rescalings are crucial in the context of holographic renormalization, pioneered by Skenderis and collaborators \cite{Henningson:1998gx,deHaro:2000vlm,Bianchi:2001kw,Skenderis:2002wp,Papadimitriou:2004ap,Papadimitriou:2005ii} (see also \cite{Emparan:1999pm,Kalkkinen:2001vg,Banados:2005rz,Compere:2008us,Anastasiou:2020zwc}). Regularizing the theory explicitly breaks Weyl invariance causing the emergence of a Weyl anomaly \cite{Henningson:1998gx,Imbimbo:1999bj,Bautier:1999ic,Schwimmer:2000cu,Ciambelli:2019bzz}.\footnote{For intrinsic field-theoretical studies of Weyl anomalies see \cite{Fradkin:1983tg,Duff:1993wm,Deser:1993yx,Deser:1996na,Boulanger:2007ab,Schwimmer:2008yh,Adam:2009gq,Schwimmer:2010za}.} The latter can be seen in the on-shell variational principle of the renormalized bulk action.
When specified to a variation of the conformal factor of the boundary metric, the corresponding variation of the on-shell action gives the Weyl anomaly, which is then interpreted as the trace anomaly of the boundary stress tensor \cite{DiFrancesco:1997nk,Balasubramanian:1999re,Skenderis:2000in}. Typically, in order to achieve a well-defined variational problem, Dirichlet boundary conditions are imposed on the metric \cite{Regge:1974zd,Papadimitriou:2005ii,Compere:2008us}. However, such a condition is too restrictive when working with a conformal class of boundary metrics \cite{Papadimitriou:2005ii}. Therefore we cannot insist that the variational problem be well defined and we impose only the cocycle condition on the second Weyl variation of the on-shell action \cite{Bonora:1985cq,Karakhanian:1994yd,Mazur:2001aa,Manvelyan:2001pv,Boulanger:2007st}. It turns out that the Weyl surface charges are finite and integrable, whereas their non conservation is accounted for by a symplectic flux through the boundary \cite{Wald:1993nt,Iyer:1994ys,Wald:1999wa,Harlow:2019yfa,Compere:2019bua,Compere:2020lrt}. The presence of an anomaly indicates that, in the dual theory, a current is not conserved at the quantum level, see \cite{Cheng:1985bj,Treiman:1986ep,Bilal:2008qx} for reviews. We construct new Weyl boundary currents compatible with the surface charges. Their non conservation translates into the anomalous Ward--Takahashi identity \cite{Ward:1950xp,Takahashi:1957xn} associated to Weyl symmetry of the putative holographic theory.
The paper is organized as follows. In Section \ref{S0}, we fix the FG gauge and introduce conformally flat boundary conditions. Correspondingly, we compute the asymptotic Killing vectors preserving these choices and their algebra. We show that the latter comprises, besides the usual left and right Witt sectors, a new abelian sector corresponding to Weyl rescalings of the boundary metric. We then solve Einstein equations and extract the action of the asymptotic symmetry algebra on solution space. In Section \ref{S2}, we compute the corresponding surface charges. Furthermore, we show that the charge algebra is centrally extended in both the Witt and the Weyl sector. In Section \ref{sec4}, we touch upon some features of the boundary holographic theory. We show in detail that, under our choice of boundary conditions, the variational problem is not well-defined due to the presence of the Weyl anomaly. We construct the boundary Weyl currents and show that their non-conservation can be interpreted in terms of an anomalous Ward-Takahashi identity for the boundary Weyl transformations. We close in Section \ref{S6} with a short summary and perspectives. Appendix \ref{AppA} contains a brief comparison of this work with \cite{Troessaert:2013fma}, whilst in Appendix \ref{AppB} we translate our results to the Chern--Simons formulation of the theory.
\section{New Boundary Conditions}
\label{S0}
The new boundary conditions considered in this work are motivated by the geometric approach originally introduced by Penrose \cite{Penrose:1962ij,Penrose1964} in the context of conformal compactification. In this framework, the boundary data for the full metric $g$ are located at infinite distance, due to the second order pole structure typical of AdS. Multiplying $g$ by $\Omega^2$, with $\Omega$ a positive function with a simple zero on the boundary, such a pole is eliminated and an induced metric on the boundary may be defined. There is however an ambiguity in the choice of $\Omega$. The replacement $\Omega\rightarrow \Omega'=e^{\omega}\Omega$, with $\omega$ a smooth function independent of the radial coordinate, induces a conformal transformation $g^{(0)}\rightarrow g^{(0)}_{{}_{\omega}}= e^{2\omega}g^{(0)}$ of the boundary metric. Such a freedom allows one to define only an equivalence class of conformally related boundary metrics, $\left[g^{(0)}\right]$, rather than a metric \cite{Rooman:2000zi,Rooman:2000ei,FG1,Fefferman:2007rka,Alessio:2017lps,Ciambelli:2019bzz}.
\subsection{Fefferman--Graham and Residual Diffeomorphisms}
\label{S1}
The FG gauge \cite{FG1, Fefferman:2007rka}\footnote{For a recent discussion see also \cite{Ruzziconi:2019pzd, Ciambelli:2020ftk, Ciambelli:2020eba}.} in three spacetime dimensions consists in choosing coordinates $x^{\mu}=(\rho,x^a)$, where $\rho\geq 0$ is a radial coordinate and $x^a=(t,\phi)$. The three gauge-fixing conditions for the metric are $g_{\rho\rho}=\frac{\ell^2}{\rho^2}$ and $g_{\rho a}=0$, where $\ell^2=-\frac{1}{\Lambda}$ is the AdS${}_3$ radius. The boundary is located at $\rho=0$. The line element takes the form
\begin{align}
\label{FG}
\text{d} s^2=g_{\mu\nu}\text dx^\mu \text dx^\nu=\frac{\ell^2}{\rho^2}\text{d}\rho^2+\gamma_{ab}(\rho,x)\text{d} x^a\text{d} x^b.
\end{align}
We solve Einstein equations with initial boundary condition $
\gamma_{ab}(\rho,x)=\mathcal{O}(\rho^{-2})$, so that $\Omega^2 g_{\mu\nu}$ is well defined at the boundary. Einstein equations for \eqref{FG} yield
\begin{align}
\label{FGExp}
\gamma_{ab}(\rho,x)=\rho^{-2}g^{(0)}_{ab}(x)+g^{(2)}_{ab}(x)+\rho^2g^{(4)}_{ab}(x),
\end{align}
with
\begin{align}
\label{Terms}
g^{(4)}_{ab}=\frac{1}{4}g_{ac}^{(2)}g^{cd}_{(0)}g_{db}^{(2)},\hspace{1cm}g_{(0)}^{ab}g^{(2)}_{ab}=-\frac{\ell^2}{2}R^{(0)},\hspace{1cm}D^a_{(0)}g_{ab}^{(2)}=-\frac{\ell^2}{2}\partial_bR^{(0)}.
\end{align}
We denote by $R^{(0)}$ and $D^{(0)}_a$ the Ricci scalar and the covariant derivative associated to $g_{ab}^{(0)}$, respectively. The leading term $g_{ab}^{(0)}$ of the expansion \eqref{FGExp} as $\rho\rightarrow 0$ is usually referred to as the boundary metric. From now on the indices will be raised and lowered using this metric.
Defining the holographic stress-energy tensor \cite{Henningson:1998gx, Balasubramanian:1999re} as
\begin{align}
\label{T}
T_{ab}=\frac{1}{8\pi G\ell}\left(g^{(2)}_{ab}+\frac{\ell^2}{2}g^{(0)}_{ab}R^{(0)}\right),
\end{align}
the last two equations of \eqref{Terms} imply
\begin{align}
\label{ST}
T_{a}{}^{a}=\frac{c}{24\pi}R^{(0)},\hspace{1cm}D_a^{(0)}T^{ab}=0,
\end{align}
where $c=\frac{3\ell}{2G}$ is the BH central charge \cite{Brown:1986nw}. The first equation in \eqref{ST} states that, for a general $g^{(0)}_{ab}$ whose Ricci scalar is non-vanishing, the trace of the tensor $T_{ab}$ defined in \eqref{T} is non-vanishing and proportional to the scalar curvature $R^{(0)}$, with a proportionality constant that is determined by the BH central charge. Hence the dual CFT living on the boundary must have a Weyl anomaly. We will further comment on this in Section \ref{sec4}. The full solution space $\chi$ is therefore characterized by five functions, three contained in $g^{(0)}_{ab}$ and two in $g^{(2)}_{ab}$ or, equivalently, in $T_{ab}$. Furthermore, these last two functions satisfy the dynamical constraints \eqref{Terms} or, equivalently, the second equation in \eqref{ST}. In the following, we will write $\chi=\{g^{(0)}_{ab},g^{(2)}_{ab}\}$.
The residual gauge diffeomorphisms are those preserving the FG gauge conditions. They are thus generated by the vector $\un\xi$ satisfying
\begin{align}
\label{2}
\mathcal{L}_{\un \xi}g_{\rho\rho}=0,\qquad \mathcal{L}_{\un \xi}g_{\rho a}=0, \qquad \mathcal{L}_{\un \xi}\gamma_{ab}=\mathcal{O}(\rho^{-2}).
\end{align}
The solution of these equations is
\begin{equation}
\un\xi:= \xi^\mu\partial_\mu=\xi^\rho\partial_\rho+\xi^a\partial_a,\label{AKV}
\end{equation}
with
\begin{align}
\label{3}
\xi^{\rho}=\rho\sigma(x),\hspace{1cm}\xi^a=Y^a(x)-\ell^2\partial_b\sigma(x)\int_{0}^{\rho}\frac{\text{d} \rho'}{\rho'}\gamma^{ab}(\rho',x).
\end{align}
In this expression, $\sigma(x)$ and $Y^a(x)$ are field-independent arbitrary functions and we note that $\xi^a$ depends on the metric field $\gamma^{ab}$. This motivates the introduction of the modified Lie bracket \cite{Barnich:2010eb}
\begin{align}
\label{4}
\big[\un \xi_1,\un\xi_2\big]_{M}:=\big[\un\xi_1,\un\xi_2\big]-\delta_{\un\xi_1}\un\xi_2+\delta_{\un\xi_2}\un\xi_1,
\end{align}
to study the asymptotic algebra. Here $\big[\cdot,\cdot\big]$ denotes the ordinary Lie bracket between vector fields and $\delta_{\un\xi_1}\un\xi_2[g]$ the variation of the vector field $\un\xi_2[g]$ due to the metric variation $\delta_{\un\xi}g=\mathcal{L}_{\un\xi}g$, i.e. $\delta_{\un\xi_1}\un\xi_2[g]=\un\xi_2[\delta_{\un\xi_1}g]$.
On defining
\begin{equation}
\label{6}
\hat{\xi}^{\rho}=\rho\hat{\sigma}(x), \qquad \hat{\sigma}(x)=Y_1^a(x)\partial_a\sigma_2(x)-Y_2^a(x)\partial_a\sigma_1(x),
\end{equation}
and
\begin{equation}
\hat{\xi}^{a}=\hat{Y}^a(x)-\ell^2\partial_{b}\hat\sigma(x)\int_{0}^{\rho}\frac{\text{d}\rho'}{\rho'}{\gamma}^{ab}(\rho',x),\qquad\hat{Y}^a(x)=Y_1^b(x)\partial_bY_2^a(x)-Y_2^b(x)\partial_bY_1^a(x),
\end{equation}
it is possible to show that our algebra is closed off shell \cite{Barnich:2010eb,Compere:2020lrt}
\begin{align}
\label{5}
\big[\un\xi_1,\un\xi_2\big]_{M}=\hat{\un\xi}.
\end{align}
To prove this we used that $\big[\xi_1,\xi_2\big]^a_{M}$, i.e. the $a$ component of $\big[\un\xi_1,\un\xi_2\big]_{M}=\big[\xi_1,\xi_2\big]^\rho_{M}\partial_\rho+\big[\xi_1,\xi_2\big]^a_{M}\partial_a$, satisfies the differential equation $\partial_{\rho}\big[\xi_1,\xi_2\big]^a_{M}=-\frac{\ell^2}{\rho^2}\gamma^{ab}\partial_b\big[\xi_1,\xi_2\big]^{\rho}_{M}$ with boundary condition $\lim_{\rho\to 0}\big[\xi_1,\xi_2\big]^a_{M}=\hat{Y}^a$.
On shell, the residual diffeomorphism generator \eqref{3} becomes
\begin{align}
\label{12}
\xi^a=Y^a-\frac{\rho^2}{2}\ell^2g^{ab}_{(0)}\partial_{b}\sigma+\frac{\rho^4}{4}\ell^2g^{ac}_{(0)}g^{(2)}_{cd}g^{db}_{(0)}\partial_b\sigma+\mathcal{O}(\rho^{6}).
\end{align}
Acting with the Lie derivative along $\un\xi$ on the on-shell line element \eqref{FG} we find the general variation of solution space
\begin{align}
\label{13}
\left(\mathcal{L}_{\un \xi}g_{\mu\nu}\right)\text{d} x^{\mu}\text{d} x^{\nu}=\frac{\ell^2}{\rho^2}\text{d}\rho^2+\left(\rho^{-2}\delta_{\un \xi}g^{(0)}_{ab}+\delta_{\un \xi}g^{(2)}_{ab}+\rho^2\delta_{\un \xi}g^{(4)}_{ab}\right)\text{d} x^a\text{d} x^b,
\end{align}
with
\begin{eqnarray}
\label{14}
&\delta_{\un \xi}g^{(0)}_{ab}=\mathcal{L}_{\un Y}g^{(0)}_{ab}-2\sigma g^{(0)}_{ab},\qquad \delta_{\un \xi}g^{(2)}_{ab}=\mathcal{L}_{\un Y}g^{(2)}_{ab}-\ell^2D^{(0)}_aD^{(0)}_b\sigma.
\end{eqnarray}
The first equation in \eqref{14} is telling us that a general variation of the boundary metric under the action of residual gauge diffeomorphisms has two independent contributions, one coming from $\sigma$ and the other from $Y^a$.
\subsection{Boundary Gauge Fixing}\label{SS4}
As stressed above, once a boundary metric $g^{(0)}_{ab}$ is assigned, the full solution space, comprising also the two functions in $g^{(2)}_{ab}$, is completely determined. That is, for every arbitrary choice of the boundary metric, solving \eqref{Terms} yields a complete solution of Einstein equations. However, as already advertised in the introduction, we do not leave $g^{(0)}_{ab}$ arbitrary, but impose the boundary condition
\begin{equation}
g_{ab}^{(0)}(x)=e^{2\varphi(x)}\eta_{ab},\label{confg}
\end{equation}
where $\eta_{ab}$ is the $2$-dimensional Minkowski metric in Lorentzian signature. Notice that every $2$-dimensional metric is conformally flat. That is, we can use boundary diffeomorphisms to fix $2$ components of the boundary metric in order to reach \eqref{confg}. This condition will constrain the form of the vector fields $Y^a$ appearing in \eqref{3}. Although \eqref{confg} is a restrictive boundary condition, it is a natural case to investigate. Note that an arbitrary variation of the boundary metric now reduces to an arbitrary variation of its conformal factor, i.e. $\delta g^{(0)}_{ab}=2(\delta\varphi )g^{(0)}_{ab}$.
Eq. \eqref{14} becomes then
\begin{align}
\label{16}
\delta_{\un \xi}g^{(0)}_{ab}=\mathcal{L}_{\un Y}g_{ab}^{(0)}-2\sigma g^{(0)}_{ab}=2(\delta_{\un\xi}\varphi)g_{ab}^{(0)}.
\end{align}
This means that $\un Y$ is a conformal Killing vector of $g^{(0)}_{ab}$
\begin{align}
\label{17}
\mathcal{L}_{\un Y}g^{(0)}_{ab}=D^{(0)}_{a}Y_b+D^{(0)}_{b}Y_a=2\Omega_{\un Y} g^{(0)}_{ab}, \qquad \Omega_{\un Y}=\frac{1}{2}D^{(0)}_aY^a.
\end{align}
where $\Omega_{\un Y}=\delta_{\un\xi}\varphi+\sigma$. Thence
\begin{align}
\label{16}
\delta_{\un \xi}g^{(0)}_{ab}=2(\Omega_{\un Y}-\sigma) g^{(0)}_{ab}.
\end{align}
Introducing light-cone coordinates $x^{\pm}=\frac{t}{\ell}\pm\phi$ we have $g^{(0)}_{ab}\text{d} x^a\text{d} x^b=-e^{2\varphi(x^+,x^-)}\text{d} x^+\text{d} x^-$ and \eqref{17} is solved by the usual chiral vectors
\begin{align}
\label{18}
Y^{+}=Y^+(x^+),\qquad Y^-=Y^-(x^-),\qquad \Omega_{\un Y}=\frac{1}{2}\left(\partial_-Y^-+\partial_+Y^+\right)+Y^+\partial_+\varphi+Y^-\partial_-\varphi.
\end{align}
Consistently, the only effect of the residual gauge symmetries on the boundary metric is to induce a Weyl transformation, i.e. a shift in its conformal factor, given by $\delta_{\un \xi}\varphi=\Omega_{\un Y}-\sigma$.
The standard Brown--Henneaux boundary conditions \cite{Brown:1986nw} $\delta_{\xi}\varphi=0$ are a subclass of our boundary conditions obtained by imposing $\sigma=\Omega_{\un Y}$. With this choice the effect of the conformal isometry generated by $\un Y$ exactly compensates the effect of the Weyl rescaling due to $\sigma$, as clear from \eqref{14}. Furthermore, also the boundary conditions studied in \cite{Troessaert:2013fma} are encompassed in our analysis, as we show in Appendix \ref{AppA}.
\subsection{Solution Space}\label{SS5}
In the conformally flat parametrization it is possible to explicitly solve Einstein equations for $g^{(2)}_{ab}$ given by the last two equations of \eqref{Terms}, \cite{Barnich:2010eb}. The first is an algebraic equation for $g^{(2)}_{+-}$ and yields
\begin{align}
\label{19}
g^{(2)}_{+-}=\ell^2\partial_{+}\partial_-\varphi,
\end{align}
where we used $R^{(0)}=8e^{-2\varphi}\partial_+\partial_-\varphi$. The second implies
\begin{align}
\label{20}
&\partial_{\mp}g^{(2)}_{\pm\pm}=-\ell^2\left(2\partial_{\pm}\varphi\partial_{\pm}\partial_{\mp}\varphi-\partial^2_{\pm}\partial_{\mp}\varphi\right),
\end{align}
with solutions
\begin{align}
\label{21}
g^{(2)}_{\pm\pm}=\ell^2\left[\Xi_{\pm\pm}(x^{\pm})+\partial^2_{\pm}\varphi-(\partial_{\pm}\varphi)^2\right],
\end{align}
where $\Xi_{\pm\pm}(x^{\pm})$ are two arbitrary functions of $x^{\pm}$. The holographic energy-momentum tensor \eqref{T} is
\begin{align}
\label{22}
T_{+-}=-\frac{\ell}{8\pi G}\partial_+\partial_-\varphi,\qquad T_{\pm\pm}=\frac{\ell}{8\pi G}\left[\Xi_{\pm\pm}(x^{\pm})+\partial_{\pm}^2\varphi-(\partial_{\pm}\varphi)^2\right].
\end{align}
While the general solution space is characterized by five independent functions of $x^+$ and $x^-$, the solution space in the conformally flat gauge is given by $\varphi(x^+,x^-)$ and the two chiral functions $\Xi_{\pm\pm}(x^{\pm})$. Thus, the solution space is $\chi=\{\Xi_{++}(x^+),\Xi_{--}(x^-),\varphi(x^+,x^-)\}$. Note that the presence of an arbitrary $\varphi(x^+,x^-)$ prevents a complete chiral splitting of the solution space and that, equivalently, the holographic stress-energy tensor components $T_{\pm\pm}$ in \eqref{22} are not chiral nor antichiral. This is one of the main differences with respect to \cite{Troessaert:2013fma}.
A generic variation of the solution space is generated by $\sigma$ and $Y^\pm$, so we symbolically write $\delta_{\un \xi}\chi=\delta_{(\sigma,Y^\pm)}\chi$. Using \eqref{14} with \eqref{17} we compute
\begin{align}
\label{23}
\delta_{(\sigma,0)}\varphi=-\sigma,\qquad\delta_{(\sigma,0)}\Xi_{\pm\pm}=0,
\end{align}
and
\begin{align}
\label{24}
\delta_{(0,Y^\pm)}\varphi=\partial_-Y^-+\partial_+Y^++2(Y^+\partial_+\varphi+Y^-\partial_-\varphi),\quad \delta_{(0,Y^\pm)}\Xi_{\pm\pm}=Y^{\pm}\partial_{\pm}\Xi_{\pm\pm}+2\Xi_{\pm\pm}\partial_{\pm}Y^{\pm}-\frac{1}{2}\partial^3_{\pm}Y^{\pm}.
\end{align}
Before proceeding to calculate the asymptotic symmetry algebra, it is convenient to trade $\sigma(x^+,x^-)$ for the new field dependent parameter $\omega(x^+,x^-)$ as
\begin{align}
\label{24.1}
\omega=\Omega_{\un Y}-\sigma.
\end{align}
Note that $\omega$ depends on the derivatives of $\varphi$. Using $\omega$, eqs. (\ref{23}-\ref{24}) can be more compactly written as
\begin{align}
\label{24.2}
\delta_{(\omega, Y^\pm)}\varphi=\omega,\qquad \delta_{(\omega, Y^\pm)}\Xi_{\pm\pm}=Y^{\pm}\partial_{\pm}\Xi_{\pm\pm}+2\Xi_{\pm\pm}\partial_{\pm}Y^{\pm}-\frac{1}{2}\partial^3_{\pm}Y^{\pm}.
\end{align}
The conformal factor $\varphi$ transforms only under $\omega$ while $\Xi_{\pm\pm}$ transform as the components of an anomalous $2$-dimensional CFT energy-momentum tensor \cite{Polchinski:1998rq}. Thanks to the redefinition of the residual diffeomorphisms generators \eqref{24.1} we have isolated the total part of the asymptotic symmetries that induces a Weyl rescaling of the boundary metric. This is in agreement with what is found in Appendix \ref{AppB}, where it is shown, using Chern--Simons formulation, that $\varphi$ completely decouples from the remaining dynamical content of the theory. Another more straightforward way to introduce $\omega$ is to require that the residual vector fields of \eqref{3} induce asymptotically a Weyl rescaling of the boundary metric
\begin{align}
\label{24.3}
\mathcal{L}_{\un\xi}\gamma_{ab}= 2\omega\rho^{-2}g_{ab}^{(0)}+O(\rho^0).
\end{align}
This equation leads to
\begin{align}
\label{24.4}
D^{(0)}_a Y_b+D^{(0)}_bY_b=2(\omega+\sigma)g^{(0)}_{ab},
\end{align}
which implies \eqref{24.1}.
Note that from the definition \eqref{T} of the holographic stress-energy tensor and from \eqref{14} it follows that, under a residual Weyl transformation, $T_{ab}$ transforms as
\begin{align}
\label{trT}
\delta_{(\omega,0)}T_{ab}=\frac{c}{12\pi}(D^{(0)}_aD^{(0)}_b\omega-g^{(0)}_{ab}\square{}^{(0)}\omega).
\end{align}
Hence, if we were to require that the vector field generating Weyl transformations satisfied
\begin{align}
\label{trTtr}
\delta_{(\omega,0)}T_{a}{}^a=-2\omega T_{a}{}^a-\frac{c}{12\pi}\square{}^{(0)}\omega\equiv-2\omega T_{a}{}^a ,
\end{align}
then the trace of $T_{ab}$, or equivalently $R^{(0)}$, would transform as a Weyl scalar of weight $-2$. This condition automatically implies that $\omega$ is an harmonic function
\begin{align}
\label{cond}
\square\omega=0,
\end{align}
whose general solution is
\begin{align}
\label{om}
\omega(x^+,x^-)=\omega^+(x^+)+\omega^-(x^-).
\end{align}
In the following, we will refer to this situation as the $\omega$-chiral case. Note that requiring the gauge parameter $\omega$ to satisfy \eqref{om} implies that $\varphi$ can vary under the action of the asymptotic symmetry group only as
\begin{align}
\label{varf}
\delta_{(\omega,Y^{\pm})}\varphi=\omega^+(x^+)+\omega^-(x^-).
\end{align}
We will return to the interpretation of \eqref{cond} in Section \ref{sec4}. For the moment let us just note that, under \eqref{cond}, even if the solution space does not admit a chiral splitting, its variation $\delta_{\un\xi}\chi$ can be decomposed into two sectors with definite chiralities, $\delta_{\un\xi^{\pm}}\chi=\{\delta_{(\omega^{\pm},Y^{\pm})}\Xi_{\pm\pm},\delta_{(\omega^{\pm},Y^{\pm})}\varphi\}$.
\subsection{Asymptotic Symmetry Algebra}
\label{SS7}
The on-shell residual diffeomorphisms generator in light-cone coordinates is
\begin{align}
\label{25}
\xi^{\rho}=\rho\sigma(x),\qquad \xi^{\pm}=Y^{\pm}(x^{\pm})+\rho^2\ell^2e^{-2\varphi}\partial_{\mp}\sigma+\rho^4\ell^2e^{-4\varphi}\left[\partial_{\mp}\sigma g^{(2)}_{+-}+\partial_{\pm}\sigma g^{(2)}_{\pm\pm}\right]+\mathcal{O}(\rho^6),
\end{align}
whereas the algebra is
\begin{eqnarray}
\label{27}
&\big[\xi_1,\xi_2\big]_{M}^{\rho}=\hat{\xi}^{\rho}=\rho\hat{\sigma},\qquad \hat{\sigma}=Y_1^{+}\partial_{+}\sigma_2+Y_1^{-}\partial_{-}\sigma_2-Y_2^{+}\partial_{+}\sigma_1-Y_2^{-}\partial_{-}\sigma_1,&\\
\label{28}
& \big[\xi_1,\xi_2\big]_{M}^{\pm}=\hat{\xi}^{\pm}=\hat{Y}^{\pm}+\rho^2\ell^2e^{-2\varphi}\partial_{\mp}\hat{\sigma}+\mathcal{O}(\rho^4),\qquad \hat{Y}^{\pm}=Y_1^{\pm}\partial_{\pm}Y_2^{\pm}-Y_2^{\pm}\partial_{\pm}Y_1^{\pm}.&
\end{eqnarray}
This algebra is a semi-direct sum: by denoting an element of the algebra as the pair $(\sigma,Y^\pm)$, the Lie bracket between two elements is $\big[(\sigma_1,Y_1^\pm),(\sigma_2,Y_2^\pm)\big]=(\hat{\sigma},\hat{Y}^\pm)$, where $\hat{\sigma}$ and $\hat{Y}^{\pm}$ are given in \eqref{27} and \eqref{28}.
We now reformulate the algebra in terms of the parameter $\omega$ introduced in \eqref{24.1}. The on-shell generator is
\begin{eqnarray}
\label{25.1}
&&\xi^{\rho}=\rho\left(\Omega_{\un Y}-\omega\right),\\
&& \xi^{\pm}=Y^{\pm}+\rho^2\ell^2e^{-2\varphi}\partial_{\mp}\left(\Omega_{\un Y}-\omega\right)+\rho^4\ell^2e^{-4\varphi}\left[\partial_{\mp}\left(\Omega_{\un Y}-\omega\right)g^{(2)}_{+-}+\partial_{\pm}\left(\Omega_{\un Y}-\omega\right)g^{(2)}_{\pm\pm}\right]+\mathcal{O}(\rho^6).
\end{eqnarray}
Notice that this reformulation introduces a field dependence in $\xi^\rho$, which was previously absent. This implies that we need to use the modified Lie bracket also for this component. We now obtain
\begin{align}
\label{27.1}
&\big[\xi_1,\xi_2\big]_{M}^{\rho}=\hat{\xi}^{\rho}=\rho\left(\Omega_{\hat{\un Y}}-\hat{\omega}\right),\qquad \hat{Y}^{\pm}=Y_1^{\pm}\partial_{\pm}Y_2^{\pm}-Y_2^{\pm}\partial_{\pm}Y_1^{\pm},\qquad \hat{\omega}=0,
\end{align}
and, as before,
\begin{align}
\label{28.1}
&\partial_{\rho}\left(\big[\xi_1,\xi_2\big]^{\pm}_M\right)=-\frac{\ell^2}{\rho^2}g^{ab}\partial_{b}\left(\big[\xi_1,\xi_2\big]^{\rho}_M\right),\qquad\lim_{\rho\to 0}\left(\big[\xi_1,\xi_2\big]^{\pm}_M\right)=\hat{Y}^{\pm}.
\end{align}
Integrating these equations leads to
\begin{equation}
\big[\xi_1,\xi_2\big]^{\pm}_M=\hat{\xi}^{\pm}=\hat{Y}^{\pm}+\rho^2\ell^2e^{-2\varphi}\partial_{\mp}\left(\Omega_{\hat{\un Y}}-\hat{\omega}\right)+\rho^4\ell^2e^{-4\varphi}\left[\partial_{\mp}\left(\Omega_{\hat{\un Y}}-\hat{\omega}\right)g^{(2)}_{+-}+\partial_{\pm}\left(\Omega_{\hat{\un Y}}-\hat{\omega}\right)g^{(2)}_{\pm\pm}\right]+\mathcal{O}(\rho^6),\label{Ypm}
\end{equation}
where $\hat{Y}^\pm$ and $\hat{\omega}$ are defined in \eqref{27.1}.
With this set of independent generators of variations in solution space $(\omega,Y^\pm)$, the asymptotic symmetry algebra is thus a direct sum of two copies of the Witt algebra with the abelian ideal of Weyl rescalings. In order to describe the asymptotic symmetry algebra in terms of a basis, we use the notation established in \eqref{AKV} with the subscript $(\omega,Y^\pm)$ such that
\begin{align}
\label{31.0}
\un\xi_{(\omega,Y^\pm)}=\xi^{\rho}_{(\omega,Y^\pm)}\partial_{\rho}+\xi^{+}_{(\omega,Y^\pm)}\partial_{+}+\xi^{-}_{(\omega,Y^\pm)}\partial_{-}.
\end{align}
Consider the vector field $\un\xi_{Y^\pm}:= \un\xi_{(0,Y^\pm)}=\xi^{\rho}_{Y^\pm}\partial_{\rho}+\xi^{+}_{Y^\pm}\partial_{+}+\xi^{-}_{Y^\pm}\partial_{-}$ and the mode expansions $Y_1^{\pm}\sim e^{inx^{\pm}}$ and $Y_2^{\pm}\sim e^{imx^{\pm}}$. Computing the mode decomposition of $\hat{Y}^{\pm}$ in \eqref{27.1}
\begin{align}
\label{31.3}
\hat{Y}^{+}=i(n-m)e^{i(n+m)x^+},\hspace{1cm}\hat{Y}^{-}=i(n-m)e^{i(n+m)x^-},
\end{align}
we gather
\begin{align}
\label{31.4}
\big[\un \xi^\pm_n,\un \xi^\pm_m\big]_{M}=i(n-m)\un \xi^\pm_{n+m}, \qquad \big[\un \xi^\pm_n,\un \xi^\mp_m\big]_{M}=0,
\end{align}
where we replaced the $Y^\pm$ subscript by the mode number $\un\xi_{Y^\pm}\mapsto \un\xi^\pm_{n}$.\footnote{Here the notation $\un\xi^\pm_{n}$ means that $\un\xi^+_{n}$ is the the full vector $\un\xi_{Y^+}$ with $Y^+\sim e^{inx^+}$ and $Y^-=0$, while $\un\xi^-_{n}$ is the vector $\un\xi_{Y^-}$ with $Y^+=0$ and $Y^-\sim e^{inx^-}$.} We thus have two copies of the Witt algebra, which is expected since for $\omega=0$ we reach BH boundary conditions, where this algebra has already been derived, \cite{Brown:1986nw}.
Set now $Y^\pm=0$ and consider $\un\zeta_{\omega}:= \un\xi_{(\omega,0)}=\zeta^{\rho}_{\omega}\partial_{\rho}+\zeta^{+}_{\omega}\partial_{+}+\zeta^{-}_{\omega}\partial_{-}$. Expanding $\omega_1\sim e^{ipx^+}e^{iqx^-}$ and $\omega_2\sim e^{irx^+}e^{isx^-}$ the algebra reads
\begin{align}
\label{31.5}
\big[\un\zeta_{pq},\un\zeta_{rs}\big]_{M}=0, \qquad
\big[\un\xi^\pm_n,\un\zeta_{rs}\big]_{M}=0.
\end{align}
where we replaced the $\omega$ subscript by the mode numbers $\un\zeta_{\omega}\mapsto \un\zeta_{pq}$. Denoting an element of the algebra as the pair $(\omega,Y^\pm)$, the Lie bracket between two elements is $\big[(\omega_1,Y^\pm_1),(\omega_2,Y^\pm_2)\big]=(0,\hat{Y}^\pm)$.
In the particular subclass of $\omega$ satisfying \eqref{om}, i.e. the $\omega$-chiral case, we can consider the algebra of the left and right Weyl sectors separately. Expanding $\omega^{\pm}\sim e^{ipx^{\pm}}$ we denote by $\un\zeta^{+}_{p}$ the vector $\un\zeta_{\omega}$ with $\omega^+\sim e^{ipx^+}$ and $\omega^-=0$ and by $\un\zeta^{-}_{p}$ the vector $\un\zeta_{\omega}$ with $\omega^-\sim e^{ipx^-}$ and $\omega^+=0$. The algebra now becomes
\begin{align}
\label{31.56}
\big[\un\zeta_p^{\pm},\un\zeta_{q}^{\pm}\big]_M=0,\qquad \big[\un\zeta_p^{\pm},\un\zeta_{q}^{\mp}\big]_M=0, \qquad \big[\un\xi^{\pm}_n,\un\zeta^{\pm}_p\big]_M=0,\qquad \big[\un\xi^{\pm}_n,\un\zeta^{\mp}_p\big]_M=0.
\end{align}
\section{Charges and Algebra}
\label{S2}
This Section is devoted to the study of asymptotic charges under the boundary conditions spelled above. We will discuss the charge algebra: we retrieve the usual Virasoro double copy, plus a Weyl sector with a non-trivial central extension.
\subsection{Surface Charges}
\label{S3}
Surface charges are computed using the prescription given in \cite{Barnich:2001jy}
\begin{align}
\label{32}
\cancel{\delta}Q_{\un\xi}[h,g]=\frac{1}{16\pi G}\int_{S^1_{\infty}} \frac{1}{2}\varepsilon_{\mu\nu\alpha}\text{d} x^{\alpha}K_{\un\xi}^{\mu\nu}[g,h]=\frac{1}{16\pi G}\int_0^{2\pi}\text{d} \phi K_{\un\xi}^{\rho t}[g,h].
\end{align}
Here $h_{\mu\nu}=\delta g_{\mu\nu}$ are the on-shell variations of the metric, the integration is on the circle at infinity spanned by $\phi$ and we use the convention $\varepsilon_{\rho t\phi}=1$. The antisymmetric tensor $K^{\mu\nu}_{\un\xi}[g,h]$ in \eqref{32} is explicitly given by
\begin{align}
\label{33}
&K^{\mu\nu}_{\un\xi}[g,h]=\sqrt{-g}\big[\xi^{\nu}D^{\mu}h-\xi^{\nu}D_{\sigma}h^{\mu\sigma}+\xi_{\sigma}D^{\nu}h^{\mu\sigma}+\frac{1}{2}hD^{\nu}\xi^{\mu}+\frac{1}{2}h^{\nu\sigma}(D^{\mu}\xi_{\sigma}-D_{\sigma}\xi^{\mu})-(\mu\leftrightarrow\nu)\big].
\end{align}
Charges are computed at fixed time $t$ and radial coordinate $\rho$ approaching the boundary and directly in the $\omega$ parametrization. The charges associated to $Y^\pm$ with $\omega=0$ are (as for the vector fields we define $\cancel{\delta} Q_{Y^\pm}[g,h]:=\cancel{\delta} Q_{(0,Y^\pm)}[g,h]$)
\begin{align}
\label{41}
\cancel{\delta}Q_{Y^\pm}[g,h]=\frac{\ell}{8\pi G}\int_0^{2\pi}\text{d}\phi\left(Y^-\delta\Xi_{--}+Y^+\delta\Xi_{++}\right).
\end{align}
To obtain these charges we used $\partial_{\phi}=\partial_+-\partial_-$ and integrated out total $\phi$ derivative terms. The $Y^\pm$ charges are thus integrable:
\begin{align}
\label{42}
Q_{Y^\pm}[g]=\frac{\ell}{8\pi G}\int_{0}^{2\pi}\text{d}\phi\left(Y^+\Xi_{++}+Y^-\Xi_{--}\right).
\end{align}
These are the usual conserved charges, found also with BH boundary conditions. The $Q_{Y^{\pm}}[g]$ in \eqref{42} are computed with respect to the background metric $\bar g$ defined by $\Xi_{\pm\pm}=0$, which is the BTZ black hole with vanishing mass and angular momentum. The ones computed with respect to the global AdS${}_3$ background can be obtained shifting $\Xi_{\pm\pm}\rightarrow \Xi_{\pm\pm}+\frac{1}{4}$ in \eqref{42}, \cite{Barnich:2012aw}.
The Weyl sector, found by setting $Y^\pm=0$, is also integrable and given by
\begin{align}
\label{43}
Q_{\omega}[g]=\frac{\ell^2}{8\pi G}\int_0^{2\pi}\text{d}\phi(\varphi\partial_t\omega-\omega\partial_t\varphi).
\end{align}
The same charges can be obtained using the Iyer--Wald prescription \cite{Wald:1993nt,Iyer:1994ys}. While these are the most general Weyl charges in our setup, we now restrict attention to the case \eqref{om}, i.e. $\omega=\omega^++\omega^-$. Correspondingly, the Weyl charges decompose as
\begin{align}
\label{Wc}
Q_{\omega}[g]=-\frac{\ell}{4\pi G}\int_{0}^{2\pi}\text{d}\phi\left(\omega^+\partial_+\varphi+\omega^-\partial_-\varphi\right)\equiv Q_{\omega^+}[g]+Q_{\omega^-}[g],
\end{align}
where we have integrated by parts. Note that these charges split into two pieces, generating the chiral and anti-chiral transformations of $\varphi$.
Contrarily to the $Y^\pm$ sector, $Q_{\omega}$ is not conserved. This is due to the presence of a non-vanishing symplectic flux through the boundary, as we will emphasize in Section \ref{sec4}.
We would like to stress that the main result of our paper is the computation of the surface charges including a new non-trivial Weyl sector. These additional interesting charges are finite, integrable but not conserved. These features make them special. Other examples of non-conserved integrable charges at finite boundaries are discussed in \cite{Adami:2020ugu}. We now proceed to compute their algebra.
\subsection{Charge Algebra}
\label{SS9}
We now verify that the surface charges, under the Poisson brackets, form a projective representation of the asymptotic symmetry algebra with modified Lie brackets
\begin{align}
\label{44}
\big\{Q_{\un\xi_1}[g],Q_{\un\xi_2}[g]\big\}=\delta_{\un \xi_2}Q_{\un\xi_1}[g]=Q_{\big[\un\xi_1,\un\xi_2\big]_M}[g]+\mathcal{K}_{\un\xi_1,\un\xi_2},
\end{align}
where $\mathcal{K}_{\un\xi_1,\un\xi_2}$ is the central extension.
We start by computing $\delta_{\un\xi_2}Q_{\un\xi_1}[g]$. Defining the integrand $K^{\rho t}_{\un\xi}[g]$ as $K^{\rho t}_{\un\xi}[g,h]=\delta K^{\rho t}_{\un\xi}[g]$, we have
\begin{align}
\label{45}
\delta_{(\omega_2,Y_2^\pm)}K^{\rho t}_{(\omega_1,Y_1^\pm)}[g]=\frac{\ell}{8\pi G}\left(Y_1^+\delta_{(\omega_2,Y_2^\pm)}\Xi_{++}+Y_1^-\delta_{(\omega_2,Y_2^\pm)}\Xi_{--}+\ell\delta_{(\omega_2,Y_2^\pm)}\varphi\partial_t\omega_1-\ell\omega_1\partial_t\delta_{(\omega_2,Y_2^\pm)}\varphi\right).
\end{align}
We work separately for the $Y^\pm$ and $\omega$ parts. This can be done because $\Xi_{\pm\pm}$ and $\varphi$ transform independently under $Y^\pm$ and $\omega$, respectively. For the $Y^\pm$ sector, after a straightforward computation, this yields the well-known BH central extension \cite{Brown:1986nw}
\begin{align}
\label{49}
\mathcal{K}_{\un\xi_{Y_1},\un\xi_{Y_2}}=\frac{1}{8\pi G}\int_{0}^{2\pi}\text{d} \phi\left(\partial_{\phi}Y_1^t\partial^2_{\phi}Y_2^{\phi}-\partial_{\phi}Y_2^t\partial^2_{\phi}Y_1^{\phi}\right).
\end{align}
Consider now the Weyl sector, obtained by setting $Y^\pm=0$ with non-vanishing $\omega$. We have
\begin{align}
\label{50}
\delta_{(\omega_2,0)} K^{\rho t}_{(\omega_1,0)}[g]=\frac{\ell^2}{8\pi G}\left(\omega_2\partial_t\omega_1-\omega_1\partial_t\omega_2\right).
\end{align}
Here, since the asymptotic symmetry algebra is abelian we have $Q_{\big[\un\zeta_1,\un\zeta_2\big]_M}[g]=0$. Thence
\begin{equation}
\label{WCEx}
\mathcal{K}_{\un\zeta_1,\un\zeta_2}=\frac{\ell^2}{8\pi G}\int_{0}^{2\pi}\text{d} \phi \left(\omega_2\partial_t\omega_1-\omega_1\partial_t\omega_2\right).
\end{equation}
The complete charge algebra is
$\big\{Q_{\un\xi_1}[g],Q_{\un\xi_2}[g]\big\}=\delta_{\un\xi_2}Q_{\un\xi_1}[g]=Q_{\hat{\un\xi}}[g]+\mathcal{K}_{\un\xi_1,\un\xi_2}$,
where $\hat{\un\xi}=\big[\un\xi_1,\un\xi_2\big]_M$ are gathered in \eqref{27.1} and \eqref{Ypm}. The total central extension is
$\mathcal{K}_{\un\xi_1,\un\xi_2}=\mathcal{K}_{\un\xi_{Y_1},\un\xi_{Y_2}}+\mathcal{K}_{\un\zeta_1,\un\zeta_2}$
To this expression contributes the ordinary BH central charge plus an additional term coming from the Weyl rescalings of the boundary metric.
The $Y^\pm$ sector of the central extension evaluated on the vector fields mode decomposition $\un\xi^\pm_n$ and $\un\xi^\pm_m$ (modes of $\un\xi_{(0,Y^\pm)}$) is
\begin{align}
\label{52.25}
\mathcal{K}_{\un\xi^\pm_n,\un\xi^\pm_m}=-im^3\frac{c^\pm}{12}\delta_{n+m,0},\qquad \mathcal{K}_{\un\xi^\pm_n,\un\xi^\mp_m}=0,\qquad c^{\pm}=c=\frac{3\ell}{2 G}.
\end{align}
On the other hand, the central extension for the modes decomposition of the Weyl sector yields
\begin{align}
\label{52.31}
\mathcal{K}_{\un\zeta_{pq},\un\zeta_{rs}}=-i(r-q)c_{_W}\omega_{q+s,q+s}\delta_{p+r,q+s}, \qquad c_{_W}=\frac{\ell}{2 G}.
\end{align}
The total charge algebra then reads
\begin{align}
\label{A10}
&\big\{Q_{\un\xi_n^\pm}[g],Q_{\un\xi_m^{\pm}}[g]\big\}=i(n-m)Q_{\un\xi^{\pm}_{n+m}}[g]-im^3\frac{c^{\pm}}{12}\delta_{n+m,0}, \\
&\big\{Q_{\un\xi_n^\pm}[g],Q_{\un\xi_m^{\mp}}[g]\big\}=0, \\
\label{mix}
&\big\{Q_{\un\zeta_{pq}}[g],Q_{\un\zeta_{rs}}[g]\big\}=-i(r-q)c_{_W}e^{2i(q+s)t/\ell}\delta_{p+r,q+s}, \\
&\big\{Q_{\un\xi_n^{\pm}}[g],Q_{\un\zeta_{pq}}[g]\big\}=0,
\end{align}
This algebra is the direct sum of two centrally extended Virasoro sectors and the centrally extended Weyl sector. We note that the Weyl central extension is explicitly time dependent. As such, we are dealing here with a one-parameter family of algebras, labelled by the time slice $t$ at which the charges are computed.
The total expression $\mathcal{K}_{\un\xi_1,\un\xi_2}$ is indeed a $2$-cocycle because it satisfies
\begin{align}
\label{52.1}
\mathcal{K}_{[\un\xi_1,\un\xi_2]_M,\un\xi_3}+\mathcal{K}_{[\un\xi_3,\un\xi_1]_M,\un\xi_2}+\mathcal{K}_{[\un\xi_2,\un\xi_3]_M,\un\xi_1}=0.
\end{align}
This equation is automatically satisfied for the Weyl sector and the mixed sector, while in the Witt sectors it is proved as usual. Furthermore, since the Virasoro central extension is non-trivial and any $2$-cocycles of an Abelian algebra cannot be a coboundary, \eqref{WCEx} is non-trivial.
Again, in the $\omega$-chiral case, the central extension for the Weyl left- and right-movers simplifies to
\begin{align}
\label{52.3}
\mathcal{K}_{\un\zeta^{\pm}_{p},\un\zeta^{\pm}_{q}}=ipc_{_W}^{\pm}\delta_{p+q,0},\qquad\mathcal{K}_{\un\zeta^{\pm}_{p},\un\zeta^{\mp}_{r}}=0
\end{align}
The total charge algebra then reads
\begin{align}
\label{chargealg1}
&\big\{Q_{\un\xi_n^\pm}[g],Q_{\un\xi_m^{\pm}}[g]\big\}=i(n-m)Q_{\un\xi^{\pm}_{n+m}}[g]-im^3\frac{c^{\pm}}{12}\delta_{n+m,0}, \\
\label{chargealg2}
&\big\{Q_{\un\xi_n^\pm}[g],Q_{\un\xi_m^{\mp}}[g]\big\}=0, \\
\label{chargealg3}
&\big\{Q_{\un\zeta_p^{\pm}}[g],Q_{\un\zeta_q^{\pm}}[g]\big\}=ipc^{\pm}_{_W}\delta_{p+q,0}, \\
\label{chargealg4}
&\big\{Q_{\un\zeta_p^{\pm}}[g],Q_{\un\zeta_q^{\mp}}[g]\big\}=0,\\
\label{chargealg5}
&\big\{Q_{\un\xi_n^{\pm}}[g],Q_{\un\zeta_p^{\pm}}[g]\big\}=0,\\
\label{chargealg6}
&\big\{Q_{\un\xi_n^{\pm}}[g],Q_{\un\zeta_p^{\mp}}[g]\big\}=0.
\end{align}
In this particular framework the Weyl central extension does not depend on time and therefore the one-parameter family of algebras reduces to a Kac-Moody current algebra. The algebra \eqref{chargealg1}-\eqref{chargealg6}, up to redefinition of generators, is the same as the one found in \cite{Troessaert:2013fma}, as reviewed in Appendix \ref{AppA}.
\section{Holographic Aspects}\label{sec4}
Thanks to the AdS/CFT dictionary \cite{tHooft:1993dmi, Susskind:1994vu, Maldacena:1997re, Gubser:1998bc, Witten:1998qj}, we know that the bulk gravity theory is dual to a boundary field theory. As long as the former is in the classical limit, the latter is strongly coupled. Therefore, little is known about it: we cannot construct its perturbative action but we still have access to non-perturbative features such as quantum symmetries expressed in terms of Ward--Takahashi identities of the path integral partition function \cite{Ward:1950xp, Takahashi:1957xn}. The goal of this Section is to show that there is a breaking in the conservation law of the Weyl current, which has a holographic dual counterpart as a boundary anomalous Ward--Takahashi identity \cite{Bilal:2008qx}. Before proceeding let us briefly review the emergence of the Weyl anomaly in the context of holographic renormalization.
The renormalized action for General Relativity in asymptotically locally AdS${}_3$ spacetimes is defined \cite{Henningson:1998gx,Skenderis:2002wp,Compere:2008us} as $S[g]=\lim_{\epsilon\to 0}S_{\epsilon}[g]$ where $S_{\epsilon}[g]$ is the regularized action, given by \\
\begin{align}
\label{renAc}
S_{\epsilon}[g]=\frac{1}{16\pi G}\int_{M_{\epsilon}}\text{d}^3x\sqrt{-g} \left(R-\frac{2}{\ell^2}\right)+\frac{1}{16\pi G}\int_{\partial M_{\epsilon}}\text{d}^2x \sqrt{-\gamma}\left(2K-\frac{2}{\ell}+\frac{\ell}{4}R^{(0)}\log\epsilon\right),
\end{align}\\
where $K$ is the trace of the extrinsic curvature of the constant $\rho$ hypersurface and the last two terms are the standards counterterms. The renormalized action $S[g]$ is therefore obtained by first introducing a cut-off at $\rho=\epsilon$ that allows the divergences to cancel and then by setting the limit $\epsilon\rightarrow 0$. Taking an on-shell variation of $S[g]$ yields\footnote{For the Chern-Simons reformulation of the variational problem, see Appendix \ref{AppB}.}
\begin{align}
\label{var}
\delta S[g]= \frac{1}{2}\int_{\partial M}\text{d}^2x\sqrt{-g^{(0)}}T^{ab}\delta g^{(0)}_{ab}=\frac{c}{24\pi}\int_{\partial M}\text{d}^2x\sqrt{-g^{(0)}}R^{(0)}\delta\varphi,
\end{align}
where in the last step we have used the conformally flat parametrization. Hence, with our choice of boundary conditions, the variational problem is not well-defined \cite{Papadimitriou:2005ii,Compere:2008us}.
Specifying $\delta$ to be the variation \eqref{24.2} induced by a Weyl diffeomorphism so that $\delta_{\omega}g^{(0)}_{ab}=2\omega g^{(0)}_{ab}$, we get
\begin{align}
\label{W An}
\delta_{\omega} S[g]=\frac{c}{24\pi}\int_{\partial M}\text{d}^2x\sqrt{-g^{(0)}} R^{(0)}\omega\equiv \int_{\partial M}\text{d}^2x \sqrt{-g^{(0)}} \mathcal{A}\, \omega,\qquad \mathcal{A}=\frac{c}{24\pi} R^{(0)},
\end{align}
which is the standard expression for the Weyl anomaly in AlAdS${}_3$ spacetimes. Note that we define $\mathcal{A}$ to be the integrand coefficient of $\omega$ in $\delta_{\omega}S[g]$ \cite{Deser:1993yx,Henningson:1998gx}.
Taking a variation of \eqref{var} yields the induced symplectic structure on the boundary $\partial M$ \cite{Crnkovic:1986ex,Compere:2018aar,Alessio:2019cae}
\begin{align}
\label{53}
\pmb{\omega}(\delta_1,\delta_2)= \frac{1}{2}\int_{\partial M} \text{d}^2x\delta_1 \left(\sqrt{-g^{(0)}}T^{ab}\right)\wedge\delta_2 g^{(0)}_{ab}=-\frac{1}{8\pi G}\int_0^{2\pi}\text{d}\phi\int_{t_1}^{t_2} \text{d} t\left(\square\delta_1\varphi\wedge\delta_2\varphi\right),
\end{align}
where we have used the conformally flat parametrization and where the flat laplacian is defined as $\square=-\ell^2\partial^2_t+\partial^2_{\phi}$. Integrating by parts in $t$ we have, discarding total $\phi$ derivatives, that
\begin{align}
\label{55}
\pmb{\omega}(\delta_1,\delta_2)=-\frac{\ell^2}{8\pi G}\int_0^{2\pi}\text{d}\phi\int_{t_1}^{t_2}\text{d} t\left[\partial_t\left(\delta_1\varphi\wedge\partial_t\delta_2\varphi\right)\right]=-\frac{\ell^2}{8\pi G}\int_0^{2\pi}\text{d}\phi\left[\delta_1\varphi\wedge\partial_t\delta_2\varphi\right]^{{}^{t_2}}_{{}_{t_1}}.
\end{align}
This shows that the difference in time of the Weyl charges is equal to the symplectic flux contracted with a Weyl generating vector field,\footnote{This result can be derived in first order formalism, \cite{Barnich:2019vzx}.}
\begin{align}
\label{56}
\pmb{\omega}(\delta_{\omega},\delta)=\frac{\ell^2}{8\pi G} \delta\int_0^{2\pi}\text{d}\phi\left[\varphi\partial_t\omega-\omega\partial_t\varphi\right]^{{}^{t_2}}_{{}_{t_1}}=\delta Q_{\omega}(t_2)-\delta Q_{\omega}(t_1).
\end{align}
Therefore the Weyl charges are not conserved but integrable, as mentioned above. We proceed now to reduce the theory to the $\omega$-chiral case.
We start by noticing that, taking a Weyl variation of equation \eqref{W An}, we obtain
\begin{align}
\label{varWan}
\delta_{\omega_1} \delta_{\omega_2}S[g]=-\frac{1}{8 G}\int_{\partial M}\text{d}^2x \omega_1 \square\omega_2.
\end{align}
We thus see that the effect of constraining the form of the asymptotic symmetry generators according to \eqref{cond} is to make $\delta_{\omega_1} \delta_{\omega_2}S[g]$ vanishing. In other words, \eqref{cond} means that we are not allowing the integrated Weyl anomaly to vary under the action of the asymptotic symmetry algebra.
This means that, under the above-spelled condition on $\omega(x)$, the operator $\delta_{\omega}$ on the functional $S[g]$ is a cocycle $\delta_{\omega_1}\delta_{\omega_2}S[g]=0$ \cite{Bonora:1985cq,Mazur:2001aa,Boulanger:2007ab,Boulanger:2007st}. From now on we will impose $\square\omega=0$ and comment on some holographic aspects in this framework.
We now proceed to construct a Weyl current \cite{Barnich:2013axa}. This procedure is well-known for the Virasoro sector, where the currents combine in the stress tensor of the boundary dual theory, and its conservation is interpreted in the bulk as Einstein equations, while in the boundary as the Ward--Takahashi identity for the transformations generated by $\xi^a_{(0,Y^{\pm})}$. In a similar fashion, given that the condition $\square \omega=0$ ensures we deal with chiral and anti-chiral Weyl charges generators \eqref{Wc}, we can define two Weyl currents. Starting from \eqref{33} we introduce
\begin{equation}
K_{(\omega,0)}^{\rho a}[g]=K_{(\omega^+,0)}^{\rho a}[g]+K_{(\omega^-,0)}^{\rho a}[g].
\end{equation}
Before giving their explicit expression, we first use the ambiguity in defining $K_{\un\xi}^{\mu\nu}[g,h]$
\begin{eqnarray}
\tilde K^{\rho a}_{(\omega^+,0)}[g] &=& K^{\rho a}_{(\omega^+,0)}[g]+\partial_b F^{[ba]}_{\omega^+},\\
\tilde K^{\rho a}_{(\omega^-,0)}[g] &=& K^{\rho a}_{(\omega^-,0)}[g]+\partial_b F^{[ba]}_{\omega^-}.
\end{eqnarray}
Choosing
\begin{equation}
F_{\omega^+}^{+-} =- {\ell\over 8\pi G}\varphi \omega^+,\qquad
F_{\omega^-}^{+-} ={\ell\over 8\pi G}\varphi \omega^-,
\end{equation}
we obtain
\begin{equation}
\tilde K^{\rho +}_{(\omega^+,0)}[g] = 0,\qquad
\tilde K^{\rho -}_{(\omega^+,0)}[g] = -{\ell \over 4\pi G}\omega^+\partial_+ \varphi,
\end{equation}
and
\begin{equation}
\tilde K^{\rho +}_{(\omega^-,0)}[g] =-{ \ell \over 4 \pi G}\omega^- \partial_- \varphi ,\qquad
\tilde K^{\rho -}_{(\omega^-,0)}[g] = 0.
\end{equation}
These are the integrands of the chiral and anti-chiral Weyl charges found in \eqref{Wc}. We can now introduce the two Weyl currents $\un{\tilde{J}}_{\omega^+}=\tilde J^a_{\omega^+}\partial_a=\tilde J^+_{\omega^+}\partial_++\tilde J^-_{\omega^+}\partial_-$ and $\un{\tilde{J}}_{\omega^-}=\tilde J^a_{\omega^-}\partial_a=\tilde J^+_{\omega^-}\partial_++\tilde J^-_{\omega^-}\partial_-$ for the two chirality sectors as
\begin{eqnarray}
\tilde K^{\rho a}_{(\omega^+,0)}[g] &=& \sqrt{-g^{(0)}}\omega^+ \tilde J^a_{\omega^+},\\
\tilde K^{\rho a}_{(\omega^-,0)}[g] &=& \sqrt{-g^{(0)}}\omega^- \tilde J^a_{\omega^-},
\end{eqnarray}
such that the currents are tensors ($K_{\un\xi}^{\mu\nu}[g,h]$ in \eqref{33} is a tensor density) and they do not depend by the gauge parameters $\omega^+$ and $\omega^-$. Their explicit expressions, using $\text{d} s^2_{\text{bdy}}=-e^{2\varphi} \text{d} x^+\text{d} x^-$, are
\begin{equation}
\tilde J^+_{\omega^+}= 0,\qquad
\tilde J^-_{\omega^+}= -{\ell e^{-2\phi} \over 2\pi G}\partial_+ \varphi,
\end{equation}
and
\begin{equation}
\tilde J^+_{\omega^-}=-{ \ell e^{-2\phi} \over 2 \pi G}\partial_- \varphi ,\qquad
\tilde J^-_{\omega^-}= 0.
\end{equation}
We eventually compute the boundary covariant divergence of these two currents and find:
\begin{eqnarray}
D_a^{(0)}\tilde J^a_{\omega^+}&=&-{\cal A} ,\\
D_a^{(0)}\tilde J^a_{\omega^-}&=&-{\cal A} ,
\end{eqnarray}
where ${\cal A}$ is the anomaly integrand coefficient defined in \eqref{W An}. We have thus shown that the Weyl currents are not conserved due to the presence of the anomaly \cite{Bilal:2008qx}. The boundary Weyl symmetry is broken, for the bulk counterpart Weyl charges are not conserved. This process is driven by the anomaly coefficient: for flat boundary metrics the current is conserved \cite{Troessaert:2013fma}, as we thoroughly review in Appendix \ref{AppA}.
\section{Conclusions}
\label{S6}
We summarize here our results and offer some possible outlooks of the present work. In the first part of the paper we have analyzed the asymptotic structure of 3-dimensional General Relativity for AlAdS${}_3$ spacetimes. In the spirit of keeping diffeomorphisms generating Weyl rescalings of the boundary metric disentangled from those generated by $Y^{a}$, we imposed a specific set of boundary conditions, namely the boundary metric being conformally flat, with only the conformal factor $\varphi$ free to vary within the solution space. Correspondingly, we have computed the asymptotic symmetry algebra of this setup. The boundary conditions adopted here do not lead to a well-defined variational principle. Nonetheless, we have found finite and integrable, although not conserved, surface charges associated to the bulk diffeomorphisms generating Weyl transformations. Integrability of the charges allows us to construct the charge algebra, which admits a new central extension in the Weyl sector.
Concerning the holographic interpretation, the AdS/CFT dictionary predicts that bulk asymptotic symmetries are dual to boundary global symmetries of a putative field theory. Although the holographic interpretation of the Weyl sector has been widely investigated, this has not been done explicitly in terms of asymptotic symmetries. That is, boundary currents built out of bulk asymptotic Weyl charges, such that their non-conservation results in the anomalous Ward--Takahashi identity, have not been previously constructed. In Section \ref{sec4} we filled this gap by explicitly deriving these currents in the $\omega$-chiral case.
In this manuscript we have not addressed the holographic interpretation corresponding to the most general variation of the boundary metric (i.e. $\omega$ not satisfying \eqref{cond}), which is certainly worth exploring. In this regard, a different choice of gauge in the bulk may be more suited, e.g. \cite{Ciambelli:2019bzz}. In particular, this raises the question on how the Weyl charges explicitly depend on the gauge condition \cite{Ciambelli:2020eba,Ciambelli:2020ftk,Compere:2019bua,Compere:2020lrt}. Another outlook is the extension of this work to higher dimensions. Specifically, we expect to unravel similar patterns in even-boundary dimensions, wherease it would also be interesting to investigate Weyl charges in odd-boundary dimensions. Furthermore, a suitable flat limit \cite{Barnich:2012aw,Ciambelli:2018wre} of these results might be relevant for the flat holography program \cite{Arcioni:2003xx,Dappiaggi:2005ci,Chandrasekaran:2020wwn,Laddha:2020kvp} and the recent developments in celestial CFT \cite{Strominger:2017zoo ,Donnay:2020guq,Ball:2019atb}. Eventually, on the macroscopic side of holography, i.e., in the fluid/gravity correspondence, it would be interesting to study the role of these boundary conditions from the fluid perspective \cite{Campoleoni:2018ltl}.
\section*{Acknowledgements}
We thank Andrea Campoleoni, Geoffrey Comp\`ere, Stephane Detournay, Adrien Fiorucci, Rob Leigh and C\'eline Zwikel for insightful discussions. The work of LC is supported by the ERC Advanced Grant ``High-Spin-Grav". The work of RR was supported by the FRIA (FNRS, Belgium) and the Austrian Science Fund (FWF), project P 32581-N. The work of PM is supported in part by the National Natural Science Foundation of China under Grant No. 11905156 and No. 11935009.
|
1,108,101,564,577 | arxiv | \section{Introduction}
\label{introduction}
Observation of the microwave sky reveals that the temperature of the cosmic microwave background (CMB) radiation is not exactly the same in all directions.
These small fluctuations in the temperature are imprinted on the entire sky, implying that the CMB is anisotropic.
These primordial anisotropies were first discovered in 1992 by the COsmic Background Explorer (COBE).
This was followed up by a remarkable series of ground-based and balloon-borne experiments,
and most recently by the Wilkinson Microwave Anisotropy Probe (WMAP).
These fluctuations are believed to have been generated within $10^{-35}$ seconds of the Big Bang.
CMB anisotropies are therefore a rich source of information about the early Universe, and have revolutionized the way we understand our Universe.
A study of CMB anisotropies also helps in probing the fundamental physics at energy scales much higher in magnitude compared to those accessible to particle accelerators.
CMB anisotropies are sensitive to the classical cosmology parameters such as expansion rate, curvature, cosmological constant, matter content, radiation content, and baryon fraction, and provide insights for modeling structure formation in the Universe \citep{1996LNP...470..207H}.
For example, measurements of the CMB anisotropies with ever-increasing precision have made it possible to establish a standard cosmological model that asserts that the Universe is nearly spatially flat \citep{Smooth_Nobel}.
The CMB contains a far greater wealth of information about the Universe through its (linear) polarization.
The CMB has acquired linear polarization through Thomson scattering during either decoupling or reionization,
sourced by the quadrupole anisotropy in the radiation distribution at that time \citep{Rees1968, Hu_White_1997}.
This quadrupole anisotropy could originate in different sources \citep{Hu_White_1997}:
the scalar or density fluctuations, vector or vortical fluctuations, and tensor or gravitational wave perturbations.
Scalar mode represents the fluctuations in energy density of the cosmological plasma at the last scattering
that causes a velocity distribution which leads to blue-shifted photons.
Vector mode represents the vorticity on the plasma which causes Doppler shifts that result into quadrupolar lobes.
However, such vorticity will be damped by expansion (as are all motions that are not enhanced by gravity),
and its effect are expected to be negligible.
Tensor perturbation are the effect of gravity waves which stretch and squeeze space in orthogonal directions.
This also stretches the photon wavelength, and hence produces a quadrupolar variation in temperature.
The dependence of CMB polarization on cosmological parameters differs from that of temperature anisotropies.
As such, it provides additional constraints on cosmological parameters that help break degeneracies.
Accurate measurements of CMB polarization will therefore enable us to affirm the validity/consistency of different cosmological models \citep{Zaldarriaga2004, Zaldarriaga_Spergel_Seljak_1997, Eisenstein1999}.
CMB polarization information is further expected to help determine initial conditions and evolution of structure in the Universe, origin of primordial fluctuations, existence of any topological defects, composition of the Universe, etc.
The Planck mission \citep{TPC2006} is a space-based full-sky probe for third-generation CMB experiments designed for this purpose.
Indeed, one of the main objectives of this mission is to measure the primordial fluctuations of the CMB with an accuracy prescribed by the fundamental astrophysical limits, through improvements in sensitivity and angular resolution, and through better control over noise and confounding foregrounds.
The higher angular resolution of Planck implies that higher-order peaks in the CMB angular spectra can be determined with better precision, which in turn translates to determination of cosmological parameters (such as baryon and dark matter densities) with improvement in statistical precision by an order of magnitude.
Furthermore, the Planck mission is designed to make extensive measurements of the $E$-mode polarization spectrum over multipoles up to $l \approx 1500$ with unprecedented precision, together with good control over polarized foreground noise.
These measurements are expected to provide insights into the physics of early Universe, epoch of recombination, structure formation,
allowable modes of primordial fluctuations (adiabatic vs.\ isocurvature modes), reionization history of the Universe, and help in establishing constraints on the primordial power spectrum.
Planck will also help constrain the fundamental physics at high energies that are not possible to probe through terrestrial experiments \citep{TPC2006}.
In our previous work \citep{AAS2012}, we estimated the CMB $TT$ power spectrum from four WMAP data realizations using a nonparametric function estimation methodology \citep{Beran2000,GMN+2004}.
This methodology does not impose any specific form or model for the power spectrum,
and determines the fit by optimizing a measure of smoothness that depends only on characteristics of the data.
This ensures that the fit and the subsequent analysis is approximately model-independent for sufficiently large data sizes.
Further, this methodology quantifies the uncertainties in the fit in the form of a high-dimensional ellipsoidal confidence set that is centered at this fit and captures the true but unknown power spectrum with a pre-specified probability.
This confidence set is the prime inferential object of this methodology which allows addressing complex inferential questions about the data meaningfully in a unified framework.
The Planck mission is expected to release high-accuracy CMB data sets in the near future.
In this paper, we therefore attempt to forecast the three CMB power spectra for the CMB to see what can be expected (and inferred) from real Planck data when it gets released, when analysed using this nonparametric regression methodology.
For this purpose,
we use synthetic Planck-like data conforming to the specifications and parameters of the Planck mission.
This synthetic data is based on the assumption that the best-fit $\Lambda$CDM model is the true model of the Universe.
In what follows, Sec.\ \ref{T_P_PS} briefly describes the three CMB angular power spectra, followed by a description (Sec.\ \ref{simulating}) of the synthetic data used in this work.
This is followed by the results (Sec.\ \ref{results}) and a conclusion (Sec.\ \ref{conclusion}).
\section{Temperature and polarization power spectra}
\label{T_P_PS}
Polarized radiation is typically characterized in terms of the Stokes parameters $I, Q, U$ and $V$ \citep{Jackson1998}.
The parameter $V$ describing circular polarization is not generated by Thomson scattering for the CMB.
The two Stokes parameters $Q$ and $U$ describing linear polarization form the components of a rank-2 symmetric trace-free tensor $\mathcal{P}_{ab}$.
Since any two-dimensional symmetric tensor can be represented using two scalar fields,
$\mathcal{P}_{ab}$ can be represented using to the \emph{electric} (i.e., gradient) $P_E$ and the \emph{magnetic} (i.e., curl) $P_B$ modes of polarization.
This decomposition is unique, and is similar to the decomposition of a vector field into a gradient and a divergence-free vector field \citep{Challinor2004}.
The $P_E$ and $P_B$ scalars are defined on a sphere, and can therefore be expanded in a spherical harmonics basis as
\begin{eqnarray}
P_E(\theta,\phi) & = & \sum_{l\geq2} \sum_{|m|\leq l} \sqrt{\frac{(l-2)!}{(l+2)!}} a_{lm}^E Y_{lm}(\theta,\phi) \\
P_B(\theta,\phi) & = & \sum_{l\geq2} \sum_{|m|\leq l} \sqrt{\frac{(l-2)!}{(l+2)!}} a_{lm}^B Y_{lm}(\theta,\phi),
\end{eqnarray}
which define the $E$- and $B$-mode multipoles $a_{lm}^E$ and $a_{lm}^B$ respectively \citep{Challinor2004, TPC2006, Hu_White_1997}.
Any CMB measurement can be decomposed into three maps ($T$, $E$ and $B$ respectively).
A total of six angular power spectra ($TT$, $EE$, $BB$, $TE$, $TB$, and $EB$) can be obtained from these three components \citep{Oliveira2005}.
These six power spectra are defined by expanding the $T$, $E$ and $B$ maps in terms of spherical harmonics,
resulting into the following correlation structure:
\begin{equation}
<a_{lm}^{Y*} a_{l^\prime m^\prime}^{Y^\prime}> = C_l^{YY^\prime} \delta_{ll^\prime} \delta_{mm^\prime},
\end{equation}
where $Y$, $Y^\prime$ are $E$, $B$ or $T$.
In the absence of parity violation, and under the assumption of Gaussian fluctuations,
the temperature and polarization anisotropies of the CMB are described by the
$C_l^{TT}, C_l^{EE},C_l^{BB},C_l^{TE}$ power spectra completely \citep{Challinor2004, Hu_White_1997, Zaldarriaga2004}.
Since the $B$-mode polarization is not expected to be detected by the Planck mission accurately \citep{TPC2006},
we focus only on the $C_l^{TT}$, $C_l^{EE}$ and $C_l^{TE}$ power spectra.
The standard deviation of the $C_l^{TT}, C_l^{EE},C_l^{TE}$ power spectra is approximately given by \citep{TPC2006, knox1995}
\begin{eqnarray}
\label{var_cltt}
(\Delta C_l^{TT}) &\simeq& \frac{2}{(2l+1)f_{sky}} (C_l^TT + \omega_T^{-1}W_l^{-2})^2 \\
\label{var_clee}
(\Delta C_l^{EE}) &\simeq& \frac{2}{(2l+1)f_{sky}} (C_l^EE + \omega_P^{-1}W_l^{-2})^2 \\
\label{var_clte}
(\Delta C_l^{TE}) &\simeq& \frac{2}{(2l+1)f_{sky}} \left((C_l^EE + \omega_P^{-1}W_l^{-2}) (C_l^TT + \omega_T^{-1}W_l^{-2}) + (C_l^TE)^2 \right),
\end{eqnarray}
where $\omega_T= (\sigma_{p,T} \theta_{FWHM})^{-2}$, and $\omega_P= (\sigma_{p,P} \theta_{FWHM})^{-2}$ are the weights per solid angle for temperature and polarization, and $f_{sky}$ is the fraction of observed sky in the experiment.
The $\sigma_{p,T}$ and $\sigma_{p,P}$ are noise standard deviations per resolution element ($\theta_{FWHM} \times \theta_{FWHM}$).
The window function for a Gaussian beam is
\begin{equation}
W_l =\exp\left(-l(l+1)/(2l_{beam}^2)\right),
\label{gaussian_beam}
\end{equation}
where $l_{beam}=\sqrt{8\ln 2}(\theta_{FWHM})^{-1}$.
The importance of decomposing the $\mathcal{P}_{ab}$ tensor in terms of the $E$ and $B$ modes comes from the fact that linear scalar perturbations do not generate any $B$-mode polarization \citep{Hu_White_1997}.
The tensor mode contributes to both $E$ as well as $B$ modes, whereas the vector mode contributes only to the $B$-mode polarization.
Therefore the $E$ part of the decomposition stems from the scalar and tensor modes, and the $B$ part originates in the vector and tensor modes \citep{TPC2006, Challinor2004, Hu_White_1997}.
Cosmological relevance of these polarization modes is as follows.
The $E$-mode mainly follows the velocity of the cosmological plasma at decoupling.
Compared to the temperature anisotropies, which originate in photon density fluctuations at the last scattering,
the $E$ mode therefore contains more information about some of the cosmological parameters
\citep{Zaldarriaga_Spergel_Seljak_1997, Challinor2004}, and can lead to better estimates of parameters
such as the baryon and cold dark matter densities.
Polarization power spectra have an oscillatory structure that is analogous to that of the $TT$ power spectrum.
For example, peaks in $EE$ power spectrum are out of phase with those in the $TT$ power spectrum due to anisotropy generated at the last scattering.
The $TE$ power spectrum, which has a higher amplitude compared to the $EE$ power spectrum, is a measure of the correlations (positive or negative) between density and velocity fluctuations \citep{Scott_Smoot2006, Challinor2004}.
The phase difference between acoustic peaks in the $TT$, $EE$ and $TE$ power spectra can be used as a model-independent check for the physics of acoustic oscillations \citep{Challinor2004}.
Adiabatic and isocurvature perturbations also have different effects on the phase of the polarization spectra:
Predicted polarization power spectra for isocurvature perturbations show out-of-phase peaks and dips
compared to those for adiabatic perturbations such that the power spectra from isocurvature perturbations appear to be $l$-shifted versions of those for adiabatic perturbations \citep{Sievers2007}.
A meaningful estimation of the polarization power spectra can therefore be used to determine which of the two scenarios is closer to truth.
Another cosmological phenomenon that affects the polarization spectra is reionization:
Reionization of Universe started when the first generation of stars started producing a flux photons.
The resulting free electrons started re-scattering the CMB radiation.
Although only a small fraction of CMB photons got scattered this way during the reionization era,
the imprints of reionization are expected to be seen as distortions in the polarization power spectra at large angular scales of the order of 10 degrees.
The height and location of the reionization bump \citep{Zaldarriaga1997, Kaplinghat2003, Holder2003} expected at low multipoles ($l\lesssim 20$) has information
related to total optical depth and the reionization epoch redshift \citep{Zaldarriaga_Spergel_Seljak_1997, Kaplinghat2003}.
Although the precision of reionization bump detection is limited by cosmic variance at low $l$ \citep{Hu_Holder_2003},
constraining it will help understand the reionization history better,
and break degeneracies between several cosmological parameters by constraining the optical depth better \citep{Zaldarriaga_Spergel_Seljak_1997, Eisenstein1999}.
\begin{figure}
\centerline{\includegraphics[height=0.4\textheight]{5-TT_simulated_planck}}
\caption{\label{fig_TT_sim}A realization of simulated $TT$ power spectrum data for the Planck mission, generated using FuturCMB \citep{futurcmb}. Black points: data including noise; red points: simulated data after subtracting noise.}
\end{figure}
\begin{figure}
\centerline{\includegraphics[height=0.4\textheight]{5-EE_simulated_planck}}
\caption{\label{fig_EE_sim}A realization of simulated $EE$ power spectrum data for the Planck mission, generated using FuturCMB \citep{futurcmb}. Black points: data including noise; red points: simulated data after subtracting noise.}
\end{figure}
\begin{figure}[h]
\centerline{\includegraphics[height=0.4\textheight]{5-TE_simulated_planck}}
\caption{\label{fig_TE_sim}A realization of simulated $TE$ power spectrum data for the Planck mission, generated using FuturCMB \citep{futurcmb}. FuturCMB assumes the noise to be zero for the $TE$ data.}
\end{figure}
\section{Synthetic data for the Planck mission}
\label{simulating}
For forecasting the CMB angular power spectra for the Planck mission,
we generate the synthetic Planck-like data using the FuturCMB code \citep{futurcmb}.
FuturCMB generates a simulated angular power spectrum using a user-provided theoretical power spectrum $C_l^{true}$ (which is assumed to be the true spectrum)
for frequency channels representing Planck measurements, and generates the corresponding noise power spectrum $N_l$ conforming to the Planck characteristics.
This is done by generating a random realization of the spherical harmonics coefficients $a_{lm}$, assumed to be Gaussian random variables with
mean zero and variance
\begin{equation}
\label{var_alm}
\text{Var}(a_{lm}) = C_l^{true} + N_l,
\end{equation}
where $C_l^{true}$ is a proxy for the true but otherwise unknown angular power spectrum.
For $C_l^{true}$, we use spectra generated using CAMB \citep{Lewis:1999bs} for the best-fit $\Lambda$CDM cosmological parameters
obtained from the WMAP 7-year data.
We also limit FuturCMB to $l\leq 2500$, a range that corresponds to the three Planck frequency channels (100, 142 and 217 GHz).
$N_l$ is the noise power spectrum given by
\begin{equation}
N_l = \omega^{-1} W_l^{-2},
\end{equation}
where $\omega$ is $\omega_T$ and $\omega_P$ for temperature and polarization respectively,
and $W_l$ is the window function for a Gaussian beam (Eq.\ \ref{gaussian_beam}).
The noise in the $TE$ power spectrum is taken to be zero because noise contributions from different maps are uncorrelated \citep{futurcmb}.
FuturCMB then calculates the power spectra data $C_l^{map}$ as
\begin{equation}
\label{cl_map}
C_l^{map} = \frac{1}{2(l+1)} \sum_{m=-l}^{+l} |a_{lm}|^2.
\end{equation}
Eq.\ \ref{cl_map} is an unbiased estimator of Eq.\ \ref{var_alm}; its expected value is therefore equal to $C_l^{true} + N_l$.
The noise spectra are expected to dominate over the true spectrum for sufficiently high values of $l$.
This is seen in figures \ref{fig_TT_sim}--\ref{fig_TE_sim},
where the black points represent the FuturCMB output $C_l^{map}$ of the $TT$, $EE$ and $TE$ spectra:
the upward trends in the tail of $TT$ and $EE$ spectra are the result of noise dominating the data at high $l$s.
The $TE$ power spectrum, on the other hand, does not show any such upward trend because the noise in the $TE$ spectrum is assumed zero.
To obtain synthetic $TT$ and $EE$ data, we therefore subtract the corresponding noise spectra from the $C_l^{map}$ output of FutureCMB (Figures \ref{fig_TT_sim}--\ref{fig_EE_sim}, red points).
The covariance matrix of the simulated angular power spectra is taken to be a diagonal matrix with diagonal elements defined in Eq.\ \ref{var_cltt}, \ref{var_clee} and \ref{var_clte} for $TT$, $EE$, and $TE$ respectively.
\section{Results and discussion}
\label{results}
\begin{figure}
\centerline{\includegraphics[height=0.38\textheight]{5-TT_simulated_Planck_comparison}}
\caption{\label{fig_TT_comp} $TT$ nonparametric fits. Blue, full-freedom fit (EDoF$\approx$72); red, restricted-freedom fit (EDoF$=$27); black: best-fit $\Lambda$CDM spectrum; grey: simulated data realization.}
\end{figure}
\begin{figure}
\centerline{\includegraphics[height=0.38\textheight]{5-EE_simulated_planck_comparison}}
\caption{\label{fig_EE_comp} $EE$ nonparametric fits. Blue, full-freedom fit (EDoF$\approx$190); red, restricted-freedom fit (EDoF$=$24); black: best-fit $\Lambda$CDM spectrum; grey: simulated data realization.}
\end{figure}
\begin{figure}[h]
\centerline{\includegraphics[height=0.38\textheight]{5-TE_simulated_planck_comparison}}
\caption{\label{fig_TE_comp} $TE$ nonparametric fits. Blue, full-freedom fit (EDoF$\approx$95); red, restricted-freedom fit (EDoF$=$40); black: best-fit $\Lambda$CDM spectrum; grey: simulated data realization.}
\end{figure}
\paragraph{Nonparametric fits to synthetic Planck data.}
For estimating the power spectra from synthetic Planck data, we use the nonparametric regression method described in \citep{AAS2012}.
While our synthetic data is generated under the assumption that the $\Lambda$CDM model as estimated from the WMAP 7-year data is the true model of the Universe,
this formalism for nonparametric regression and inference itself does not make any assumptions about the shape of the true regression function underlying the data; it is asymptotically model-independent.
A nonparametric fit, under this methodology, can be characterized by its effective degrees of freedom (EDoF), which can be thought of as the equivalent of the number of parameters in a parametric regression problem.
Using this methodology, we obtain nonparametric fits to the synthetic $TT$, $EE$ and $TE$ data by appropriately constraining the EDoF of the fits.
(Additional details about our nonparametric fits can be found in Sec.\ \ref{further_details_1} and \ref{further_details_2}.)
Figures \ref{fig_TT_comp}, \ref{fig_EE_comp} and \ref{fig_TE_comp} show nonparametric fits (red curves) to the $TT$, $EE$, and $TE$ data respectively,
which are in good agreement with the underlying $\Lambda$CDM spectra ($C_l^{true}$, black curves) used to generate the synthetic data.
This shows that this nonparametric regression methodology, which does not assume any specific form of the true (but generally unknown) regression function, can recover the underlying true spectrum with high accuracy especially where noise levels are not too high.
\paragraph{How well will the Planck fits be determined by data alone?}
To see how noise in the data affects local uncertainties in a fitted spectrum,
we compute approximate 95\% confidence intervals for each fitted $C_l$ using 5000 randomly sampled spectra from the corresponding confidence set.
The ratio of this confidence interval to the absolute value of the fitted $|C_l|$ (assumed to be nonzero) is a relative measure of how well each fitted $C_l$ is determined \citep{GMN+2004,AAS2012}:
A value $\ll 1$ implies that the fit is well determined by the data, and a value $\gtrsim$ 1 implies that the data contain very little information about height of the power spectrum at that $l$.
\begin{figure}[t]
\centerline{
\includegraphics[height=0.4\textheight]{5-boxcar_planck}
}
\caption{\label{fig_boxcar_planck} The results of a probe of the confidence sets for the $TT$ (red), $EE$ (blue), and $TE$ (green) nonparametric restricted-freedom fits to the synthetic Planck data, to determine how well the fits are expected to be determined by the data alone.
The quantity plotted for each data realization is the total vertical variation at each $l$ within the respective 95\% ($2 \sigma$) confidence set, divided by the absolute value of the fit (assumed nonzero):
Values $\ll 1$ indicate that the fit is tightly determined by the data, whereas values $\gtrsim$ 1 indicate that the data contain very little information about the height of the angular power spectrum at that $l$.
Disregarding the low-$l$ region for the $EE$ fit, and spikes for the $TE$ fit (which arise from nearly zero fitted $C_l$ values), the marked vertical lines indicate the approximate $l$-value at which each curve rises above 1.}
\end{figure}
In Figure \ref{fig_boxcar_planck}, we plot this relative error (95\%) for all three spectra as a function of the multipole index $l$.
We see that, by this criterion, the Planck power spectra are expected to be well-determined up to $l \approx 2462 (TT), 1377 (EE)$ and $1727 (TE)$.
Since the $TE$ fit oscillates around zero (Figure \ref{fig_TE_comp}), this quantity takes very high values at $l$s where the $TE$ spectrum has a nearly zero value.
This results in multiple spikes in Figure \ref{fig_boxcar_planck} (green dash curve), but this does not imply that the fit is ill-determined at these $l$s.
Ignoring these spikes, we see that the relative error in the $TE$ fit is below unity up to $l \approx 1727$, which indicates the range over which this fit is expected to be well-determined by data alone.
\paragraph{Uncertainties on the locations and heights of peaks and dips.}
Locations and heights of peaks and dips in the CMB power spectra contain information about cosmological models and parameters.
Uncertainties in the location and height of a peak or a dip in a fitted spectrum can thus help assess uncertainties in the values of related parameter.
Following the procedure outlined in \citep{AAS2012}, we sampled the 95\% confidence set of each fitted spectrum uniformly to generate spectral variations while ensuring that at least 5000 of these are acceptable (see Sec.\ \ref{further_details_2} for details).
Figures \ref{fig_TT_box}, \ref{fig_EE_box}, and \ref{fig_TE_box} show the results of this exercise, together with tabulated values in Tables \ref{table:TT}, \ref{table:EE}, and \ref{table:TE} respectively.
The box around a peak or a dip represents the largest horizontal and vertical variations in the scatter; these represent the 95\% confidence intervals on the location and height of a peak or a dip.
\begin{figure}[h]
\centerline{\includegraphics[height=0.4\textheight]{5-Simulated_Planck_TT_withOUT_fsky_27D_boxes_5k}}
\caption{\label{fig_TT_box} 95\% confidence boxes the locations and heights of peaks and dips in the $TT$ fit. Black curve is the restricted-freedom monotone fit to the synthetic Planck $TT$ data (grey points). The number of acceptable spectral variations sampled from the 95\% confidence set is 5000. These uncertainties are tabulated in Table \ref{table:TT}.}
\end{figure}
\begin{figure}[h]
\centerline{\includegraphics[height=0.4\textheight]{5-Simulated_Planck_EE_withOUT_fsky_2379D_boxes_5k_30D_our_Simulation}}
\caption{\label{fig_EE_box} 95\% confidence boxes the locations and heights of peaks and dips in the $EE$ fit. Black curve is the restricted-freedom monotone fit to the synthetic Planck $TT$ data (grey points). The number of acceptable spectral variations sampled from the 95\% confidence set is 5000. These uncertainties are tabulated in Table \ref{table:EE}.}
\end{figure}
\begin{figure}[h]
\centerline{\includegraphics[height=0.4\textheight]{5-Simulated_Planck_TE_withOUT_fsky_40D_boxes_5k_2}}
\caption{\label{fig_TE_box} 95\% confidence boxes the locations and heights of peaks and dips in the $TE$ fit. Black curve is the restricted-freedom monotone fit to the synthetic Planck $TT$ data (grey points). The number of acceptable spectral variations sampled from the 95\% confidence set is 5000. These uncertainties are tabulated in Table \ref{table:TE}.}
\end{figure}
For the $TT$ fit, these boxes around peaks and dips are tiny (Figure \ref{fig_TT_box}), which is a reflection of the accuracy of the Planck $TT$ data.
Such precise determination of peaks and dips should lead to more robust estimates of related cosmological parameters than what is currently available.
The theoretical $TT$ power spectrum (Figure \ref{fig_TT_comp}, black curve) shows a small upturn at low $l$.
This upturn is primarily the result of the integrated Sachs-Wolf (ISW) effect.
In Figure \ref{fig_TT_box}, this upturn corresponds to the first dip in the spectral variations sampled from the 95\% confidence set.
Peaks and dips in the $EE$ power spectrum show reasonably low uncertainties up to $l \approx 1200$ (Figure \ref{fig_EE_box}).
Beyond this, the data contain high levels of noise, and therefore all uncertainties become much larger.
This is in agreement with the behavior of the $EE$ curve in Figure \ref{fig_boxcar_planck} (blue curve).
We also expect a small bump in the $EE$ power spectrum which is related to the epoch of reionization.
This bump is indeed seen in both nonparametric $EE$ fits in Figure \ref{fig_EE_comp}.
Figure \ref{fig_EE_box} shows a (tiny) 95\% uncertainty box for this bump.
A precise determination of the height and location of this peak (Table \ref{table:EE}) should reveal useful information about the epoch of reionization.
The uncertainty boxes on peaks and dips in the $TE$ fit (Figure \ref{fig_TE_box}) are reasonably small till $l\approx 1800$, again in agreement with the result depicted in Figure \ref{fig_boxcar_planck} (green dashed curve).
The somewhat peculiar uncertainty boxes on the last three peaks in the $TE$ fit
are due to the fact that there are two spurious peaks at $l = 1920$ and $l = 2070$ in the restricted-freedom fit (see Sec.\ \ref{further_details_1}) which are a result of the high noise levels at high $l$s.
At low multipoles, the $TE$ fit also shows a bump (Figure \ref{fig_TE_comp}) which is related to the epoch of reionization.
Although a similar bump in the $EE$ fit is known to be more informative \citep{TPC2006}, a determination of this peak in the $TE$ spectrum should also lead to useful information about reionization.
\paragraph{Are the acoustic peaks in the $TT$ and $EE$ spectra out of phase with respected to each other?}
From the fundamental physics of the CMB anisotropies, we expect the acoustic peaks in the $TT$ and $EE$ power spectra to be out phase with respect to each other.
One way of establishing this is by considering the ratio of peak locations in the $EE$ fit to the corresponding ones in the $TT$ fit, which should be
$(m+0.5)/m$ for the $m$th peak \citep{Sievers2007}.
We depict the 95\% peaks locations of TT power spectra versus EE one in
Figure \ref{fig_EE_TT_ratio_Planck} shows the 95\% confidence intervals on peak locations in the $EE$ fit (red) plotted against the corresponding uncertainties in the $TT$ fit (blue).
Also plotted are the peak location pairs corresponding to the best-fit $\Lambda$CDM model (black dots),
and points (green) based on the theoretical expectation $(m+0.5)/m$.
All the plotted quantities are by and large consistent with each other,
indicating that the expected behavior of out-of-phase peak locations is indeed vindicated by the data.
\begin{figure}[th]
\centerline{
\includegraphics[width=0.65\textwidth]{5-EE_TT_ratio_Planck}
}
\caption{\label{fig_EE_TT_ratio_Planck} The 95\% confidence intervals on peak locations in the $EE$ fit (red) plotted against the corresponding confidence intervals in the $TT$ fit (blue). Also plotted are the peak location pairs corresponding to the best-fit $\Lambda$CDM model (black dots), and points (green) based on the theoretical expectation $(m+0.5)/m$. All the plotted quantities are by and large consistent with each other, indicating that the expected behavior of out-of-phase peak locations is indeed vindicated by the data.}
\end{figure}
\paragraph{An estimate of the acoustic scale parameter $l_A$.}
Table \ref{table:TT} lists the 95\% confidence intervals on peak and dip locations and heights for the $TT$ power spectrum fit.
As a way of illustrating the role of these uncertainties in the estimation of cosmological parameters, we consider the following relationship \citep{HFZ+2001,DL2002} between the location $l_m$ of the $m$th peak, the acoustic scale $l_A$, and the shift parameter $\phi_m$ for TT power spectrum: $l_m = l_A ( m - \phi_m )$.
If we substitute the end-points of the 95\% confidence interval for the $m$th peak location, then
this relationship results into hyperbolic confidence bands in the $l_A-\phi_m$ plane (Figure \ref{fig_phi_vs_lA_8_peak}).
The intersection of these bands (for the first 8 peaks in the $TT$ fit) determine an estimated confidence interval for the acoustic scale $300\leq l_A \leq 305$ which is in agreement with the reported value $l_A = 300$ by \citep{PNB+2003}.
In comparison with our previous estimate based on the WMAP 7-year data \citep{AAS2012}, the current estimate has improved remarkably.
Furthermore, any additional information about the phase shifts $\phi_m$ can lead to an even more refined estimate for acoustic scale.
\begin{figure}[h]
\centerline{\includegraphics[width=\textwidth]{5-phi_vs_lA_8_peak}}
\caption{\label{fig_phi_vs_lA_8_peak} Confidence ``bands'' for the acoustic scale $l_A$ and the shift $\phi_m$ for the $m$th peak, as derived from the 95\% confidence intervals on the first eight peak locations of estimated TT power spectrum.}
\end{figure}
\section{Conclusion}
\label{conclusion}
In this paper, we have addressed the question of what could be the expected outcome of a nonparametric analysis of the $TT$, $EE$ and $TE$ power spectrum data sets from the Planck space mission when they get released.
For this purpose, we have used synthetic/simulated Planck-like data sets based on the best-fit $\Lambda$CDM model,
and have analysed them using a nonparametric regression and inference methodology.
Our results show that the $TT$ power spectrum can be estimated with such a high accuracy that all peaks are resolved up to $l\leq2500$.
We expect that the $EE$ and $TE$ power spectra will be reasonably well-determined and can help, e.g., better understand reionization history, address the issue of adiabatic versus isocurvature perturbations, and estimate of some of the cosmological parameters precisely.
As a result, we expect to have a better understanding of the Universe via the Planck data even from an agnostic, nonparametric approach.
\begin{table}[h]
\centering
\scalebox{0.9}{
\begin{tabular}{l|l|l|l}
Peak Location & Peak Height & Dip Location & Dip Height \\
\hline
\hline
$l_1: (204, 234)$ & $h_1: (5401.543, 5925.611)$ & $l_{1+{1 \over 2}}: (406, 421)$ & $h_{1+{1 \over 2}}: (1638.521, 1769.197)$ \\
$l_2: (525, 549)$ & $h_2: (2527.018, 2691.862)$ & $l_{2+{1 \over 2}}: (666, 688)$ & $h_{1+{1 \over 2}}: (1689.753, 1793.608)$ \\
$l_3: (805, 827)$ & $h_3: (2465.483, 2596.654)$ & $l_{3+{1 \over 2}}: (1009, 1028)$ & $h_{1+{1 \over 2}}: (969.1924, 1019.3422)$ \\
$l_4: (1120, 1141)$ & $h_4: (1215.164, 1267.128)$ & $l_{4+{1 \over 2}}: (1301, 1325)$ & $h_{1+{1 \over 2}}: (663.6103, 695.9041)$ \\
$l_5: (1418, 1439)$ & $h_5: (806.5287, 845.0737)$ & $l_{5+{1 \over 2}}: (1630, 1666)$ & $h_{1+{1 \over 2}}: (348.6765, 370.1998)$ \\
$l_6: (1714, 1761)$ & $h_6: (387.0516, 412.0869)$ & $l_{6+{1 \over 2}}: (1922, 2007)$ & $h_{1+{1 \over 2}}: (199.8328, 225.3753)$ \\
$l_7: (1989, 2085)$ & $h_7: (219.8173, 251.4042)$ & $l_{7+{1 \over 2}}: (2220, 2442)$ & $h_{1+{1 \over 2}}: (64.88648, 112.57429)$ \\
$l_8: (2282, 2498)$ & $h_8: (71.91322, 141.91509)$ & $l_{8+{1 \over 2}}: (2394, 2500)$ & $h_{1+{1 \over 2}}: (15.27434, 111.02498)$ \\
\hline
\end{tabular}
}
\caption{95\% Confidence Interval on Several Features of TT Angular Power Spectrum}
\label{table:TT}
\end{table}
\begin{table}[h]
\centering
\scalebox{0.9}{
\begin{tabular}{l|l|l|l}
Peak Location & Peak Height & Dip Location & Dip Height \\
\hline
\hline
$l_1: (7, 26)$ & $h_1: (-0.0025,0.0235)$ & $l_{1+{1 \over 2}}: ( 17, 27)$ & $h_{1+{1 \over 2}}: (-0.0029, 0.0181)$ \\
$l_2: (133, 144)$ & $h_2: ( 0.9234, 1.2542)$ & $l_{2+{1 \over 2}}: ( 193, 205)$ & $h_{2+{1 \over 2}}: (0.4620, 0.8600)$\\
$l_3: (390, 401)$ & $h_3: ( 20.4654, 23.2528)$ & $l_{3+{1 \over 2}}: ( 521, 532)$ & $h_{3+{1 \over 2}}: (5.4424, 7.5745)$\\
$l_4: (681, 697)$ & $h_4: ( 35.2102, 39.9650)$ & $l_{4+{1 \over 2}}: ( 825, 844)$ & $h_{4+{1 \over 2}}: (8.6629, 14.2180)$\\
$l_5: (978, 1009)$ & $h_5: ( 38.5544, 46.5531)$ & $l_{5+{1 \over 2}}: ( 1135, 1184)$ & $h_{5+{1 \over 2}}: (4.9459, 15.6909)$\\
$l_6: (1258, 1360)$ & $h_6: ( 24.2445, 40.3917)$ & $l_{6+{1 \over 2}}: ( 1401, 1668)$ & $h_{6+{1 \over 2}}: (-4.0435, 20.6010)$\\
$l_7: (1442, 1802)$ & $h_7: ( 7.3139, 44.8427)$ & $l_{7+{1 \over 2}}: ( 1461, 1969)$ & $h_{7+{1 \over 2}}: (-33.4637, 28.6720)$\\
$l_8: (1626, 1999)$ & $h_8: ( -12.4126, 73.5849)$ & $l_{8+{1 \over 2}}: ( 1755, 2000)$ & $h_{8+{1 \over 2}}: (-46.4440, 65.6556)$\\
\hline
\end{tabular}
}
\caption{95\% Confidence Interval on Several Features of EE Angular Power Spectrum}
\label{table:EE}
\end{table}
\begin{table}[h]
\centering
\scalebox{0.9}{
\begin{tabular}{l|l|l|l}
Peak Location & Peak Height & Dip Location & Dip Height \\
\hline
\hline
$l_1: (3, 38)$ & $h_1: (0.6092, 2.5822)$ & $l_{1+{1 \over 2}}: ( 142, 158)$ & $h_{1+{1 \over 2}}: (-49.8581, -37.7906)$ \\
$l_2: (301, 316)$ & $h_2: (113.1289, 136.2571)$ & $l_{2+{1 \over 2}}: (461, 477)$ & $h_{2+{1 \over 2}}: (-83.9852, -68.6596)$\\
$l_3: (588, 605)$ & $h_3: (26.2370, 44.0873)$ & $l_{3+{1 \over 2}}: (741, 758)$ & $h_{3+{1 \over 2}}: (-145.3450, -124.9381)$\\
$l_4: (903, 923)$ & $h_4: (56.4278, 76.1789)$ & $l_{4+{1 \over 2}}: (1061, 1086)$ & $h_{4+{1 \over 2}}: (-94.7879, -76.2354)$\\
$l_5: (1199, 1243)$ & $h_5: (4.2335, 23.1510)$ & $l_{5+{1 \over 2}}: (1353, 1400)$ & $h_{5+{1 \over 2}}: (-76.0768, -55.1005)$\\
$l_6: (1511, 1579)$ & $h_6: (1.9257, 25.1787)$ & $l_{6+{1 \over 2}}: (1650, 1750)$ & $h_{6+{1 \over 2}}: (-49.6416, -25.6796)$\\
$l_7: (1772, 1954)$ & $h_7: (-14.4294, 13.7468)$ & $l_{7+{1 \over 2}}: (1822, 2081)$ & $h_{7+{1 \over 2}}: (-42.0202, -2.8288)$\\
$l_8: (1884, 2208)$ & $h_8: (-20.8813, 37.7615)$ & $l_{8+{1 \over 2}}: (1959, 2282)$ & $h_{8+{1 \over 2}}: (-50.6428, 11.4636)$\\
$l_9: (2023, 2354)$ & $h_9: (-21.1805, 40.5083)$ & $l_{9+{1 \over 2}}: (2073, 2419)$ & $h_{9+{1 \over 2}}: (-83.8615, 10.2379)$\\
$l_10: (2155, 2499)$ & $h_10: (-48.9460, 75.1592)$ & $l_{10+{1 \over 2}}: (2257, 2500)$ & $h_{10+{1 \over 2}}: (-116.2237, 60.3390)$\\
\hline
\end{tabular}
}
\caption{95\% Confidence Interval on Several Features of TE Angular Power Spectrum}
\label{table:TE}
\end{table}
|
1,108,101,564,578 | arxiv | \section{Introduction}
Central to the theory of quantum integrable systems is
quantum $R$-matrix satisfying the celebrated Yang-Baxter equation.
General $R$-matrices
with additive spectral parameter are
parametrized via elliptic functions.
The simplest elliptic $R$-matrix is
\begin{equation}
\label{02}
R(\lambda )=\sum_{a=0}^{3}
\frac{\theta_{a+1}(2\lambda +\eta |\tau )}
{\theta_{a+1}(\eta |\tau )}\,
\sigma_a \otimes \sigma_a \,,
\end{equation}
where $\theta_a(z|\tau)$ are Jacobi $\theta$-functions,
$\sigma_a$ are Pauli matrices, and $\sigma_0$ is the unit
matrix.
This $R$-matrix is associated with the celebrated
8-vertex model solved by Baxter \cite{Baxter}, being the
matrix of local Boltzmann weights at the vertex.
The transfer matrix of this model is the generating function
of conserved quantities for
the integrable anisotropic (of $XYZ$ type)
spin-$\frac{1}{2}$ chain.
Integrable spin chains of $XYZ$-type and their
higher spin generalizations can be solved
by the generalized algebraic Bethe
ansatz \cite{FT,Takebe}.
In lattice integrable models with elliptic $R$-matrix (\ref{02}),
the algebra of local observables
is the Sklyanin algebra \cite{Skl1,Skl2}
which is a special 2-parametric deformation of the
universal enveloping algebra $U(gl(2))$. A concrete
model is defined by fixing a particular representation
of this algebra. Such representations can be realized by difference
operators.
Similar to the $sl(2)$-case
(models of $XXX$-type), the representations are labeled
by a continuous parameter which is called
{\it spin}, and for positive half-integer values of this
parameter the operators representing the Sklyanin algebra
generators are known to have a finite-dimensional invariant space.
However, we allow the
spin to take any complex value, so we are going to
work in a general infinite-dimensional representation of the
algebra of observables.
Integrable spin chains of XXX-type with infinite-dimensional representations
of symmetry algebra at the sites were first
studied in the seminal papers \cite{Lipatov,FK} in the context of
high energy QCD, see also \cite{noncompactXXX}.
Later, lattice models with trigonometric $R$-matrix (of XXZ-type)
with non-compact quantum group symmetry were considered
\cite{noncompactXXZ}. A representation-theoretical approach
to models with elliptic $R$-matrix and
``non-compact" Sklyanin algebra symmetry
is presently not available but there is no doubt that it should
exist.
In this paper we present a direct construction of the
elliptic $R$-matrix intertwining the tensor product of two
arbitrary infinite-dimensional representations of the
Sklyanin algebra. It can be realized
as a difference operator in two variables, in general of
infinite order, so we often call this object {\it $R$-operator}
rather than $R$-matrix. Another important object is a
face type $R$-matrix related to
the $R$-operator via a functional version
of the vertex-face correspondence.
The latter $R$-matrix provides an elliptic
analog of $6j$-symbols.
Our method closely follows
the similar construction in the chiral Potts model
\cite{Korepanov,BS,KMN}
and the broken $\mbox{\Bbb Z} _N$-symmetric model \cite{HasYam,brokenZ}.
It is based on the observation that the elementary
$L$-operator is in fact a composite object built from
simpler entities called ``intertwining vectors" \cite{brokenZ,Has}.
Then the proof of the Yang-Baxter equation and other
properties of the $L$-operator can be reduced
to simple manipulations with the intertwining vectors using
basic relations between them. Remarkably, all elements of this procedure
have a nice graphic interpretation which makes them
rather clear and greatly
simplifies the arguments. It provides simultaneously a very good
illustration and an important heuristic tool.
This graphical technique
resembles both the one developed for
the Chiral Potts and
broken $\mbox{\Bbb Z} _N$-symmetric models and the one
known in the representation theory of $q$-deformed
algebras, in particular in connection with $q$-deformation
of $6j$-symbols \cite{KR88}.
However, practical realization of these ideas in the
infinite-dimensional setting is by no means obvious.
Technically, it is rather different
from what is customary in the 8-vertex model and its relatives.
Our construction goes along the lines
of our earlier work \cite{Z00} devoted to the $Q$-operator
for spin chains with infinite-dimensional representations
of the Sklyanin algebra
at each site and extensively uses such really special functions as
elliptic gamma-function and elliptic generalization
of hypergeometric series. The theory of elliptic hypergeometric
functions originated by Frenkel and Turaev
in \cite{FrTur} is now an actively
developing new branch of mathematics (see, e.g.,
\cite{Spir,Warnaar,Rosengren03}
and references therein).
The elliptic $R$-operator appears to be a composite object whose
building blocks are operators which intertwine
representations of the Sklyanin algebra with spins
$\ell$ and $-\ell -1$ (the $W$-operator). They can be
expressed through the elliptic hypergeometric series ${}_4 \omega_{3}$
with an operator argument. The kernels of the $W$-operators are
expressed through ratios of the elliptic gamma-function.
These intertwining operators were found
in our earlier paper \cite{Z00} as a by-product
of the general elliptic $Q$-operator construction.
Here we re-derive this result with the help of the
intertwining vectors using
much more direct arguments.
We also give a construction of vacuum vectors
for the elliptic $L$-operator using the graphic technique
and show how they are related to the kernel of the $W$-operator.
It should be remarked that similar results,
in one or another form,
can be found in the existing literature.
In particular, the elliptic $R$-operator has been
found \cite{DKK07} in terms of
operators which implement elementary permutations of
parameters entering the $RLL=LLR$ relation.
A solution to the star-triangle equation
built from ratios of the elliptic gamma-functions
was recently suggested in \cite{BS10}.
Some closely related matters are discussed in the
recent paper \cite{Spir10}.
It seems to us that our approach
may be of independent interest since it emphasizes
the connection with the Sklyanin algebra
and allows one
to obtain more detailed results
in a uniform way.
The paper is organized as follows.
Section 2 contains the necessary things related to
the Sklyanin algebra, its realization by
difference operators and representations.
In section 3 we describe a space of discontinuous functions
of special form, where the Sklyanin algebra acts, and which
are identified with kernels of difference
operators.
Here we follow \cite{Z00}.
The technique of intertwining vectors developed
in Section 4 is used in
Section 5 to construct operators which intertwine
representations of the Sklyanin algebra with spins
$\ell$ and $-\ell -1$. They
appear to be the most important constituents of
the elliptic $R$-operator. In section 6 we show how
the vacuum vectors for the $L$-operator constructed
in \cite{Z00} emerge within
the approach of the present paper.
The construction of the elliptic $R$-operator
and related objects for arbitrary
spin is presented in Section 7, where the Yang-Baxter and
star-triangle relations are also discussed.
Some concluding remarks are given in Section 8.
Appendix A contains necessary information on the special
functions involved in the main part of the paper. In Appendix B
some details of the calculations with elliptic hypergeometric
series are presented.
\section{Representations of the Sklyanin algebra}
The aim of this section is to give the necessary
preliminaries on representations of the Sklyanin algebra.
We begin with a few formulas related to
the quantum $L$-operator
with elliptic dependence on the spectral parameter.
The elliptic quantum $L$-operator is
the matrix
\begin{equation}
\mbox{{\sf L}}(\lambda)=\frac{1}{2}
\left ( \begin{array}{cc}
\theta _{1}(2\lambda){\bf s}_0 +
\theta _{4}(2\lambda){\bf s}_3 &
\theta _{2}(2\lambda){\bf s}_1 +
\theta _{3}(2\lambda){\bf s}_2
\\& \\
\theta _{2}(2\lambda){\bf s}_1 -
\theta _{3}(2\lambda){\bf s}_2 &
\theta _{1}(2\lambda){\bf s}_0 -
\theta _{4}(2\lambda){\bf s}_3
\end{array} \right )
\label{L}
\end{equation}
with non-commutative matrix elements.
Specifically,
${\bf s}_a$ are difference operators in a
complex variable $z$:
\begin{equation}
{\bf s}_{a} = \frac{\theta _{a+1}(2z -2\ell \eta)}
{\theta _{1}(2z)}\, e^{\eta \partial _z}
-\frac{\theta _{a+1}(-2z -2\ell \eta)}
{\theta _{1}(2z)}\, e^{-\eta \partial _z}
\label{Sa}
\end{equation}
introduced by Sklyanin \cite{Skl2}.
Here $\theta_a(z)\equiv \theta_a(z|\tau)$ are
Jacobi $\theta$-functions
with the elliptic module $\tau$,
$\mbox{Im}\,\tau >0$, $\ell$ is a complex number (the spin),
and $\eta \in \mbox{\Bbb C}$ is a parameter which is assumed to
belong to the fundamental parallelogram
with vertices $0$, $1$, $\tau$, $1+\tau$, and to be
incommensurate with $1, \tau$.
Definitions and transformation properties
of the $\theta$-functions are listed in Appendix A.
The four operators ${\bf s}_a$ obey the commutation
relations of the Sklyanin algebra\footnote{The standard generators
of the Sklyanin algebra \cite{Skl1}, $S_a$,
are related to ours as follows:
$S_{a}=(i)^{\delta _{a,2}}\theta _{a+1}(\eta ){\bf s}_a$.}:
\begin{equation}
\begin{array}{l}
(-1)^{\alpha +1}I_{\alpha 0}{\bf s}_{\alpha}{\bf s}_{0}=
I_{\beta \gamma}{\bf s}_{\beta}{\bf s}_{\gamma}
-I_{\gamma \beta}{\bf s}_{\gamma}{\bf s}_{\beta}\,,
\\ \\
(-1)^{\alpha +1}I_{\alpha 0}{\bf s}_0 {\bf s}_{\alpha}=
I_{\gamma \beta}{\bf s}_{\beta}{\bf s}_{\gamma}
-I_{\beta \gamma}{\bf s}_{\gamma}{\bf s}_{\beta}
\end{array}
\label{skl6}
\end{equation}
with the structure constants
$I_{ab}=\theta _{a+1}(0)\theta _{b+1}(2\eta)$.
Here $a,b =0, \ldots , 3$ and
$\{\alpha ,\beta , \gamma \}$ stands for any cyclic
permutation of $\{1 ,2,3\}$.
The relations of the Sklyanin algebra
are equivalent to the condition that the $\mbox{{\sf L}}$-operator
satisfies the
``$R{\sf L}{\sf L}={\sf L}{\sf L}R$"
relation with the elliptic $R$-matrix (\ref{02}).
The parameter
$\ell$ in (\ref{Sa}) is called the spin of the representation.
If necessary, we write ${\bf s}_a ={\bf s}_{a}^{(\ell )}$
or ${\sf L}^{(\ell )}(\lambda )$
to indicate the dependence on $\ell$.
When $\ell \in
\frac{1}{2} \mbox{\Bbb Z}_{+}$, these operators
have a finite-dimensional invariant subspace, namely,
the space
$\Theta_{4\ell}^{+}$ of {\it even} $\theta$-functions of
order $4\ell$ (see Appendix A). This
is the representation space of the $(2\ell +1)$-dimensional
irreducible representation (of series a))
of the Sklyanin algebra. For example,
at $\ell =\frac{1}{2}$ the functions
$\bar \theta_4 (z)$, $\bar \theta_3 (z)$
(hereafter we use the notation $\bar \theta_a (z)\equiv
\theta _a (z|\frac{\tau}{2})$)
form a basis
in $\Theta_{2}^{+}$, and the generators
${\bf s}_a$, with respect to this basis,
are represented by $2\times 2$
matrices $(-i)^{\delta_{a,2}}
(\theta_{a+1}(\eta ))^{-1}\sigma_a$. In this case,
${\sf L}(\lambda )=R(\lambda -\frac{1}{2}\eta )$,
where $R$ is the 8-vertex model $R$-matrix (\ref{02}).
In general, the representation space of the Sklyanin
algebra where the operators ${\bf s}_a$ act is called
{\it quantum space} while the two-dimensional space
in which the $L$-operator is the 2$\times$2 matrix
is called {\it auxiliary space}.
As is proved in \cite{z},
the space $\Theta_{4\ell}^{+}$ for
$\ell \in \frac{1}{2}\mbox{\Bbb Z}_{+}$ is annihilated by the operator
\begin{equation}
\label{W}
{\bf W}_{\ell} = c\sum_{k=0}^{2\ell +1} (-1)^k
\left [ \begin{array}{c} 2\ell +1 \\ k \end{array} \right ]
\, \frac{ \theta _{1}(2z+
2(2\ell -2k +1)\eta )}{\prod_{j=0}^{2\ell +1}
\theta_1(2z+2(j-k)\eta )} \,
e^{(2\ell -2k +1)\eta \partial_z }.
\end{equation}
where $c$ is a normalization constant to be
fixed below.
Hereafter, we use the
``elliptic factorial" and ``elliptic binomial" notation:
\begin{equation}
\label{binom} [j]\equiv \theta_1(2j\eta)\,,
\;\;\;\;\;\; [n]!=\prod_{j=1}^{n}[j]\,,
\;\;\;\;\;\;
\left [ \begin{array}{c}n\\m\end{array}\right ]
\equiv \displaystyle{\frac{[n]!}{[m]![n-m]!}}\,.
\end{equation}
The defining property of the operator ${\bf W}_{\ell}$
established in \cite{z} is that ${\bf W}_{\ell}$
intertwines representations
of spin $\ell$ and of spin $-(\ell +1)$:
\begin{equation}
\label{S3}
{\bf W}_{\ell}\,
{\bf s}_{a}^{(\ell )}=
{\bf s}_{a}^{(-\ell -1)} {\bf W}_{\ell}\,,
\;\;\;\;\;\; a=0,\ldots , 3\,.
\end{equation}
The same intertwining relation can be written
for the quantum $L$-operator (\ref{L}):
\begin{equation}\label{S3a}
{\bf W}_{\ell}\,
{\sf L}^{(\ell )}\, (\lambda )=
{\sf L}^{(-\ell -1)}(\lambda ) {\bf W}_{\ell}.
\end{equation}
Note that the operator ${\bf W}_{\ell}$ serves
as an elliptic analog of
$(d/dz)^{2\ell +1}$ in the following sense.
In the case of the algebra $sl(2)$, the intertwining
operator between representations of spins $\ell$ and
$-\ell -1$ (realized by differential operators in $z$) is
just $(d/dz)^{2\ell +1}$. It annihilates the linear space
of polynomials of degree $\leq 2\ell$ (which results in
the rational degeneration of the elliptic space
$\Theta_{4\ell}^{+}$).
For us it is very
important to note that ${\bf W}_{\ell}$ can be extended to
arbitrary complex values of $\ell$
in which case it is represented by a half-infinite series in the
shift operator $e^{2\eta \partial_z}$ \cite{Z00}.
The series is an elliptic analog of the very-well-poised
basic hypergeometric series with an operator argument.
The explicit form
is given below in this paper.
The intertwining relations
(\ref{S3}) hold true in this more general case, too.
Very little is known about
infinite-dimensional representations of the Sklyanin
algebra. The difference operators (\ref{Sa})
do provide such a representation but any
characterization of the space of functions
where they are going to act is not available at the moment, at least
for continuous functions.
On the other hand, the difference character of the operators
(\ref{Sa}) suggests to consider their action on a space
of discontinuous functions of a special form. The latter are
naturally identified with kernels of difference operators.
This formalism was used in our earlier paper \cite{Z00}.
It is reviewed in the next section.
\section{Kernels of difference operators}
Let $\delta (z)$ be the function equal to zero everywhere
but at $z=0$, where it equals $1$:
$\delta (z)=0$, $z\neq 0$, $\delta (0)=1$.
(We hope that the same notation as for
the conventional delta-function will cause no confusion because
the latter will not appear in what follows.)
Clearly, $z\delta (z)=0$ and $\delta ^2 (z)=\delta (z)$.
Consider the space ${\cal C}$ of functions of the form
\begin{equation}
\label{inf1}
f(z)=\sum_{k\in\raise-1pt\hbox{$\mbox{\Bbbb Z}$}}f_k \delta (z-\nu +2k\eta )\,,
\;\;\;\;\;\;\; f_k \in \mbox{\Bbb C} \,,
\end{equation}
where $\nu \in \mbox{\Bbb C}$.
This space is isomorphic to the direct product of $\mbox{\Bbb C}$ and the
linear space of
sequences $\{ f_k \}_{k\in {\bf \raise-1pt\hbox{$\mbox{\Bbbb Z}$}}}$. We call functions of the
form (\ref{inf1}) {\it combs}.
Clearly, the Sklyanin algebra realized as in (\ref{Sa})
acts in this space (shifting $\nu \to \nu \pm \eta$).
A comb is said to be finite from the right
(respectively, from the left)
if there exists $M\in \mbox{\Bbb Z}$ such that $f_k =0$ as $k>M$
(respectively, $k<M$). Let ${\cal C}^{\vdash}$ (respectively,
${\cal C}^{\dashv}$) be the space of combs finite from the left
(respectively, from the right).
We define the pairing
\begin{equation}
\label{inf2}
(F(z),\, \delta (z-a))=F(a)
\end{equation}
for any function $F(z)$, not necessarily of the form (\ref{inf1}).
In particular,
\begin{equation}
\label{inf3}
(\delta (z-a),\, \delta (z-b))=
\delta (a-b)\,.
\end{equation}
Formally, this pairing can be written as an integral:
\begin{equation}\label{pairing}
(F(z),\, \delta (z-a))=
\int dz F(z)\delta (z-a)
\end{equation}
(perhaps a $q$-integral symbol would be more appropriate).
We stress that the integral here means nothing more
than another notation for the pairing,
especially convenient in case of many variables.
By linearity, the pairing can be extended to the whole
space of combs. We note that the pairing between the spaces
${\cal C}^{\vdash}$ and ${\cal C}^{\dashv}$ is well defined
since the sum is always finite.
Combs are to be thought of as kernels of difference operators.
By a difference operator in one variable we mean any
expression of the form
\begin{equation}
\label{D1}
{\bf D}=\sum_{k\in {\raise-1pt\hbox{$\mbox{\Bbbb Z}$}}}c_k(z)e^{(\mu +2k\eta )\partial _{z}}\,,
\;\;\;\;\;\; \mu \in \mbox{\Bbb C}\,.
\end{equation}
The comb
\begin{equation}
\label{D2}
D(z, \zeta )=
\sum_{k\in {\raise-1pt\hbox{$\mbox{\Bbbb Z}$}}}c_k(z)
\delta (z-\zeta +\mu +2k\eta )\,,
\end{equation}
regarded as a function of any one of the variables $z$, $\zeta$,
is the kernel of this difference operator in the following sense.
Using the pairing introduced above, we can write:
\begin{equation}
\label{D3}
({\bf D}f)(z)=
\int D(z,\zeta )f(\zeta )d\zeta =
\sum_{k\in {\raise-1pt\hbox{$\mbox{\Bbbb Z}$}}}c_k(z)f(z+\mu +2k\eta )\,.
\end{equation}
The kernel $D(z, \zeta )$ can be viewed as an infinite matrix
with continuously numbered rows ($z$) and columns ($\zeta$).
Then the convolution with respect to the second
argument of the kernel, as in (\ref{D3}),
defines action of the operator from the left.
The convolution with respect to the first argument
defines the action from the right,
\begin{equation}
\label{D4}
(f{\bf D})(z)=
\int f(\zeta )D(\zeta ,z )d\zeta \,,
\end{equation}
equivalent to the action of the transposed
difference operator from the left:
\begin{equation}
\label{D5}
{\bf D}^{{\sf t}}=
\sum_{k\in {\raise-1pt\hbox{$\mbox{\Bbbb Z}$}}}
e^{-(\mu +2k\eta )\partial _{z}}
c_k(z) =
\sum_{k\in {\raise-1pt\hbox{$\mbox{\Bbbb Z}$}}}
c_k(z-\mu -2k\eta )
e^{-(\mu +2k\eta )\partial _{z}}\,.
\end{equation}
The transposition ${\sf t}$ is the anti-automorphism of the
algebra of difference operators such that
$\bigl ( c(z)e^{\alpha \partial_{z}}\bigr )^{{\sf t}}=
e^{-\alpha \partial_{z}}c(z)$.
In terms of the above pairing we can write
$(f, {\bf D}g)=({\bf D}^{{\sf t}}f, g)$.
The following simple remarks
will be useful in what follows. Let $F(z), G(z)$ be any functions,
then $F(z)D(z, \zeta )G(\zeta )$,
with $D(z, \zeta )$ as above, is the kernel of the difference
operator
$$
F{\bf D}G=\sum_{k\in {\raise-1pt\hbox{$\mbox{\Bbbb Z}$}}}c_k(z)
F(z)G(z+\mu +2k\eta ) e^{(\mu +2k\eta )\partial _{z}}
$$
which is the composition of the multiplication by $G$,
action of the operator ${\bf D}$ and subsequent multiplication by $F$.
Let $D^{(1)}(z, \zeta )$, $D^{(2)}(z, \zeta )$ be kernels of
difference operators ${\bf D}^{(1)}$, ${\bf D}^{(2)}$
respectively, then the convolution
$$
\int d\xi D^{(2)}(z, \xi )D^{(1)}(\xi , \zeta )
$$
is the kernel of the difference operator ${\bf D}^{(2)}{\bf D}^{(1)}$.
If the kernels $D^{(1)}(z, \zeta )$, $D^{(2)}(z, \zeta )$
are combs finite from the left (right)
as functions of $z$, then the convolution is always well defined
and the resulting kernel belongs to the same space of combs.
The kernels of Sklyanin's operators (\ref{Sa}) are:
\begin{equation}
\label{D7}
s_a(z,z')=
\frac{\theta _{a+1}(2z -2\ell \eta)}
{\theta _{1}(2z)}\,\delta (z-z' +\eta)
-\frac{\theta _{a+1}(-2z -2\ell \eta)}
{\theta _{1}(2z)}\, \delta (z-z' -\eta )\,.
\end{equation}
Note that $s_a(-z,-z')=s_a(z,z')$.
Let us find the kernel of the $L$-operator (\ref{L}).
Using identities for theta-functions, it is easy to see that
\begin{equation}\label{D7a}
L_{\zeta}^{z}(\lambda )=\theta_{1}(2\lambda +2\ell \eta )
V^{-1}(\lambda + \ell \eta ,z)
\left ( \begin{array}{cc}
\delta (z\! -\! \zeta \! +\! \eta ) &0 \\
0 & \delta (z\! -\! \zeta \! -\! \eta ) \end{array} \right )
V(\lambda - \ell \eta ,z),
\end{equation}
where $V(\lambda , z)$ is the matrix
$$
V(\lambda , z)=\left ( \begin{array}{cc}
\bar \theta_{4}(z+\lambda )&
\bar \theta_{3}(z+\lambda ) \\
\bar \theta_{4}(z-\lambda )&
\bar \theta_{3}(z-\lambda ) \end{array} \right )
$$
and $V^{-1}(\lambda , z)$ is its inverse:
$$
V^{-1}(\lambda , z)=\frac{1}{2\theta _1(2z)}
\left ( \begin{array}{rr}
\bar \theta_{3}(z-\lambda )&
-\bar \theta_{3}(z+\lambda ) \\
-\bar \theta_{4}(z-\lambda )&
\bar \theta_{4}(z+\lambda ) \end{array} \right )
$$
(recall that $\bar \theta_{a}(z)\equiv \theta_{a}(z|\frac{\tau}{2})$).
A crucial point is that the diagonal matrix with delta-functions
factorizes into the product of column and row vectors:
$$
\left ( \begin{array}{cc}
\delta (z\! -\! \zeta \! +\! \eta ) &0 \\
0 & \delta (z\! -\! \zeta \! -\! \eta ) \end{array} \right )
=\left ( \begin{array}{c}
\delta (z\! -\! \zeta \! +\! \eta )\\
\delta (z\! -\! \zeta \! -\! \eta )
\end{array} \right )
\Bigl (\delta (z\! -\! \zeta \! +\! \eta ),
\delta (z\! -\! \zeta \! -\! \eta ) \Bigr )
$$
and thus so does $L^{z}_{\zeta}(\lambda )$.
The vectors which represent the factorized
kernel of the $L$-operator are
``intertwining vectors" introduced
in the next section.
\section{Intertwining vectors}
We introduce the 2-component (co)vector
\begin{equation}\label{vectors}
\bigl |\zeta \bigr >=
\left (
\begin{array}{l}\bar \theta_{4}(\zeta)\\
\bar \theta_{3}(\zeta)
\end{array}
\right ),
\quad \quad
\bigl < \zeta \bigr |= \bigl (
\bar \theta_{4}(\zeta),\,
\bar \theta_{3}(\zeta) \bigr ).
\end{equation}
The vector orthogonal to $\bigl < \zeta \bigr |$ is
$\bigl |\zeta \bigr >^{\bot}=
\left (
\begin{array}{r}\bar \theta_{3}(\zeta)\\
-\bar \theta_{4}(\zeta)
\end{array}
\right )$,
the covector orthogonal to $\bigl | \zeta \bigr >$ is
${}^{\bot}\! \bigl <\zeta \bigr |=\bigl ( \bar \theta_{3}(\zeta),
-\bar \theta_{4}(\zeta)\bigr )$, so
$\bigl < \zeta \bigr | \zeta \bigr >^{\bot}=
{}^{\bot}\!\bigl < \zeta \bigr | \zeta \bigr >=0$.
More generally, we have:
\begin{equation}
\label{scprod}
\bigl < \xi \bigr |\zeta \bigr >^{\bot}=
2\theta_{1}(\xi +\zeta )\theta_{1}(\xi -\zeta )=-
{}^{\bot}\! \bigl < \xi \bigr |\zeta \bigr > \,.
\end{equation}
Note also that
\begin{equation}
\label{orth}
\bigl | \zeta +{\scriptstyle \frac{1}{2} }
(1+\tau )\bigr >=
e^{-\frac{\pi i \tau}{2} -2\pi i \zeta}
\bigl | \zeta \bigr >^{\bot}.
\end{equation}
\begin{figure}[t]
\centering
\includegraphics[angle=-00,scale=0.50]{intwvec.eps}
\caption{\it Intertwining vectors.}
\label{fig:intwvec}
\end{figure}
Introduce now the {\it intertwining vectors}
\begin{equation}\label{intw1}
\Bigl | \phi_{z'}^{z}(\lambda )\Bigr > =
\frac{1}{\sqrt{2\theta _1 (2z)}}\Bigl (
\bigl |z+\lambda \bigr > \delta (z-z'+\eta )
+\bigl |z-\lambda \bigr > \, \delta (z-z'-\eta )\Bigr ),
\end{equation}
\begin{equation}\label{intw1a}
\Bigl | \bar \phi_{z'}^{z}(\lambda )\Bigr > =
\frac{1}{\sqrt{2\theta _1 (2z)}}\Bigl (
\bigl |z-\lambda \bigr >^{\bot} \delta (z-z'+\eta )
-\bigl |z+\lambda \bigr >^{\bot}\delta (z-z'-\eta )\Bigr )
\end{equation}
and the corresponding covectors
\begin{equation}\label{intw2}
\Bigl < \phi_{z'}^{z}(\lambda )\Bigr | =
\frac{1}{\sqrt{2\theta _1 (2z)}}\Bigl (
\bigl < z+\lambda \bigr | \delta (z-z'+\eta )
+\bigl <z-\lambda \bigr | \, \delta (z-z'-\eta )\Bigr ),
\end{equation}
\begin{equation}\label{intw2a}
\Bigl < \bar \phi_{z'}^{z}(\lambda )\Bigr | =
\frac{1}{\sqrt{2\theta _1 (2z)}}\Bigl (
{}^{\bot}\! \bigl < z-\lambda \bigr | \delta (z-z'+\eta )
-{}^{\bot} \! \bigl <z+\lambda \bigr |\delta (z-z'-\eta )\Bigr ).
\end{equation}
It is easy to check that
$$
\Bigl | \phi_{z'}^{z}(-\lambda )\Bigr >
=\sqrt{\frac{\theta_1 (2z')}{\theta_1 (2z)}}
\,\,\Bigl |\phi_{z}^{z'}(\lambda +\eta )\Bigr > \,,
$$
$$
\Bigl | \bar \phi_{z'}^{z}(-\lambda )\Bigr >
=-\sqrt{\frac{\theta_1 (2z')}{\theta_1 (2z)}}
\,\,\Bigl | \bar \phi_{z}^{z'}(\lambda -\eta )\Bigr > \,.
$$
The intertwining vectors satisfy the following orthogonality
relations:
\begin{equation}\label{orth1}
\Bigl < \phi_{z'}^{z}(\lambda )\Bigr | \,
\bar \phi_{z ''}^{z}(\lambda )\Bigr > =
\theta_1(2\lambda )\delta (z'-z'') \Bigl (
\delta (z\! - \! z'\! +\! \eta )+
\delta (z\! - \! z'\! -\! \eta )\Bigr ),
\end{equation}
\begin{equation}\label{orth2}
\Bigl < \phi_{z}^{z'}(\lambda +\eta )\Bigr | \,
\bar \phi_{z}^{z''}(\lambda -\eta )\Bigr > =
\theta_1(2\lambda )\frac{\theta_1(2z)}{\theta_1(2z')}
\, \delta (z'-z'') \Bigl (
\delta (z\! - \! z'\! +\! \eta )+
\delta (z\! - \! z'\! -\! \eta )\Bigr ),
\end{equation}
\begin{equation}\label{orth3}
\int d\zeta \, \Bigr | \,
\bar \phi_{\zeta}^{z}(\lambda )\Bigr >
\Bigl < \phi_{\zeta}^{z}(\lambda )\Bigr | =
\theta_1(2\lambda )
\left (\begin{array}{cc}1 & 0 \\ 0& 1 \end{array}\right ),
\end{equation}
\begin{equation}\label{orth4}
\int d\zeta \, \frac{\theta_1(2\zeta )}{\theta_1(2z)}\,
\Bigr | \, \bar \phi_{z}^{\zeta}(\lambda -\eta )\Bigr >
\Bigl < \phi_{z}^{\zeta}(\lambda +\eta )\Bigr | =
\theta_1(2\lambda )
\left (\begin{array}{cc}1 & 0 \\ 0& 1 \end{array}\right ).
\end{equation}
\begin{figure}[t]
\centering
\includegraphics[angle=-00,scale=0.45]{scalarprod.eps}
\caption{\it The graphic representation of the relation
$W^{z,\zeta}(\lambda - \mu)
\Bigl < \phi_{z'}^{z}(\lambda +\frac{\eta}{2})\Bigr | \,
\bar \phi_{\zeta '}^{\zeta}(\mu - \frac{\eta}{2})\Bigr >
=W^{z',\zeta '}(\lambda -\mu )
\Bigl < \phi_{z'}^{z}(\mu+\frac{\eta}{2})\Bigr | \,
\bar \phi_{\zeta '}^{\zeta}(\lambda -\frac{\eta}{2} )\Bigr > .$
The horizontal bold line segment common for the covector
to the left and the vector to the right means taking scalar
product of the two-dimensional (co)vectors.
The intersection point of the spectral parameter lines corresponds to
the ``vertex" $W^{z,\zeta}(\lambda - \mu)$.}
\label{fig:scalarprod}
\end{figure}
The general scalar product of two intertwining vectors is
$$
\begin{array}{lll}
\Bigl < \phi_{z'}^{z}(\lambda )\Bigr | \,
\bar \phi_{\zeta '}^{\zeta}(\mu )\Bigr > \!\! & = &
\displaystyle{
\frac{1}{\sqrt{4\theta_1(2z)\theta_1(2\zeta )}}}
\\ && \\
&\times &\Bigl \{\theta_1(z\! +\! \zeta \! +\! \lambda \! -\! \mu )
\theta_1(z\! -\! \zeta \! +\! \lambda \! +\! \mu )
\delta (z\! - \! z'\! +\! \eta )
\delta (\zeta \! - \! \zeta '\! +\! \eta )
\\ && \\
&& -\,\,
\theta_1(z\! +\! \zeta \! +\! \lambda \! +\! \mu )
\theta_1(z\! -\! \zeta \! +\! \lambda \! -\! \mu )
\delta (z\! - \! z'\! +\! \eta )
\delta (\zeta \! - \! \zeta '\! -\! \eta )
\\ && \\
&& +\,\,
\theta_1(z\! +\! \zeta \! -\! \lambda \! -\! \mu )
\theta_1(z\! -\! \zeta \! -\! \lambda \! +\! \mu )
\delta (z\! - \! z'\! -\! \eta )
\delta (\zeta \! - \! \zeta '\! +\! \eta )
\\ && \\
&& -\,\,
\theta_1(z\! +\! \zeta \! -\! \lambda \! +\! \mu )
\theta_1(z\! -\! \zeta \! -\! \lambda \! -\! \mu )
\delta (z\! - \! z'\! -\! \eta )
\delta (\zeta \! - \! \zeta '\! -\! \eta )\Bigr \} .
\end{array}
$$
It is a matter of direct verification to see that such scalar
products satisfy the ``intertwining relation":
\begin{equation}\label{intw3}
W^{z,\zeta}(\lambda - \mu)
\Bigl < \phi_{z'}^{z}(\lambda +\frac{\eta}{2})\Bigr | \,
\bar \phi_{\zeta '}^{\zeta}(\mu - \frac{\eta}{2})\Bigr >
=W^{z',\zeta '}(\lambda -\mu )
\Bigl < \phi_{z'}^{z}(\mu+\frac{\eta}{2})\Bigr | \,
\bar \phi_{\zeta '}^{\zeta}(\lambda -\frac{\eta}{2} )\Bigr > \,,
\end{equation}
where the quantities $W^{z,\zeta}(\lambda )$ solve
the following difference equations in $z,\zeta $:
\begin{equation}\label{intw3a}
\begin{array}{l}
\displaystyle{
W^{z+\eta, \zeta +\eta}(\lambda )
=\frac{\theta_1 (z+\zeta +\lambda +\eta )}{\theta_1
(z+\zeta +\lambda +\eta )}\, W^{z, \zeta}(\lambda )},
\\ \\
\displaystyle{
W^{z+\eta, \zeta -\eta}(\lambda )
=\frac{\theta_1 (z-\zeta +\lambda +\eta )}{\theta_1
(z+\zeta +\lambda +\eta )}\, W^{z, \zeta}(\lambda )}.
\end{array}
\end{equation}
These equations can be solved in terms of the elliptic
gamma-function $\Gamma (z|\tau , 2\eta):={\sf \Gamma} (z)$
\cite{R3,FV3}
(see Appendix A):
\begin{equation}\label{intw4}
W^{z, \zeta}(\lambda )=e^{-2\pi i \lambda z/\eta}\,\,
\frac{{\sf \Gamma} (z+\zeta +\lambda +\eta )
{\sf \Gamma} (z-\zeta +\lambda +\eta )}{{\sf \Gamma} (z+\zeta -\lambda +\eta )
{\sf \Gamma} (z-\zeta -\lambda +\eta )}\,.
\end{equation}
There is a freedom to multiply the solution by
an arbitrary $2\eta$-periodic function of
$z+\zeta$ and $z-\zeta$. We put this function
equal to $1$. (However, this does not mean that this is the best
normalization; other possibilities will be discussed elsewhere.)
In our normalization
\begin{equation}\label{WW}
W^{z, \zeta}(\lambda )W^{z, \zeta}(-\lambda )=1
\end{equation}
but $W^{z, \zeta}(\lambda )$ is not symmetric under
permutation of $z$ and $\zeta$.
\begin{figure}[t]
\centering
\includegraphics[angle=-00,scale=0.45]{L.eps}
\caption{\it The kernel of the $L$-operator
$L^{z}_{\zeta }(\lambda , \mu )=
\Bigr | \, \bar \phi_{\zeta}^{z}(\lambda -
\frac{\eta}{2} )\Bigr >
\Bigl < \phi_{\zeta}^{z}(\mu +\frac{\eta}{2})\Bigr |$.}
\label{fig:L}
\end{figure}
The intertwining vectors can be represented graphically
as shown in Fig. \ref{fig:intwvec}. The vertical line carries the
spectral parameter and serves as a line of demarcation
between the ``real" (transparent) world and the ``shadow" world.
Then the relation (\ref{intw3})
means that the horizontal line in Fig. \ref{fig:scalarprod}
can be moved through the intersection point of the two
spectral parameter lines. This intersection point is a new
graphic element which corresponds to
$W^{z, \zeta}(\lambda -\mu )$.
The kernel of the $L$-operator for the
representation of spin $\ell$ can be written
in the factorized form as the product of intertwining
vectors:
\begin{equation}\label{intw5}
L^{\!(\ell )} {}^{z}_{\zeta }(\lambda )=
\Bigr | \, \bar \phi_{\zeta}^{z}(\lambda_{+} -\frac{\eta}{2} )\Bigr >
\Bigl < \phi_{\zeta}^{z}(\lambda_{-}+\frac{\eta}{2})\Bigr |\,,
\quad \quad \lambda_{\pm}=\lambda \pm (\ell +\frac{1}{2})\eta \,.
\end{equation}
It clear that the spectral parameter $\lambda$
and the representation parameter $\ell \eta$
enter here on equal footing, so the
notation $L^{\!(\ell )} {}^{z}_{\zeta }(\lambda )=
L^{z}_{\zeta }(\lambda _{+}, \lambda_{-})$
is sometimes also convenient. Graphically, the
kernel of the $L$-operator is shown in Fig.
\ref{fig:L}.
\section{Intertwining operators for arbitrary spin}
\begin{figure}[t]
\centering
\includegraphics[angle=-00,scale=0.45]{vertices.eps}
\caption{\it The vertices $W^{z, \zeta}(\lambda -\mu )$
and $W^{z}_{\zeta}(\lambda -\mu )$.}
\label{fig:vertices}
\end{figure}
There is a relation which is ``dual" to (\ref{intw3})
(see also Fig. \ref{fig:scalarprod}) meaning that it can be
read from the same configuration of lines in the figure
by exchanging the real and shadow world pieces of the plane
(see Fig. \ref{fig:dualscalarprod}).
Two new elements appear: first, the
vertex $W_{\zeta}^{z}(\lambda -\mu )$ is different from the one
in Fig. \ref{fig:scalarprod} and, second, one should take
convolution ($\int d\zeta$) with respect to the ``intermediate"
variable $\zeta$ associated to the finite triangle in the
shadow world. The two vertices, $W^{z, \zeta}(\lambda -\mu )$ and
$W_{\zeta}^{z}(\lambda -\mu )$, are shown separately in
Fig. \ref{fig:vertices}.
According to Fig. \ref{fig:dualscalarprod},
the dual relation has the form
\begin{equation}\label{intw6}
\int \! d\zeta \, W_{\zeta}^{z}(\lambda -\mu )
\Bigr | \, \bar \phi_{z'}^{\zeta}(\lambda -\frac{\eta}{2} )\Bigr >
\Bigl < \phi_{z'}^{\zeta}(\mu +\frac{\eta}{2})\Bigr | =\!
\int \! d\zeta \, W_{z'}^{\zeta}(\lambda -\mu )
\Bigr | \, \bar \phi_{\zeta}^{z}(\mu -\frac{\eta}{2} )\Bigr >
\Bigl < \phi_{\zeta}^{z}(\lambda +\frac{\eta}{2})\Bigr |.
\end{equation}
Changing the notation $\lambda \to \lambda_{+}$,
$\mu \to \lambda_{-}$, one can write it as
$$
\int \! d\zeta \, W_{\zeta}^{z}(\lambda_{+}-\lambda_{-})
L_{z'}^{\zeta}(\lambda_{+}, \lambda_{-}) =\!
\int \! d\zeta \, W_{\zeta}^{z}(\lambda_{+}-\lambda_{-})
L_{\zeta}^{z}(\lambda_{-}, \lambda_{+})
$$
which is just the intertwining relation for the
$L$-operator ${\sf L}^{(\ell )}(\lambda )=
{\sf L}(\lambda_{+}, \lambda_{-})$ (\ref{S3a}),
with $W_{\zeta}^{z}(\lambda_{+}-\lambda_{-})$ being the
kernel of the difference operator ${\bf W}_{\ell}$.
Taking this into account, we are going to find solutions
for the $W_{\zeta}^{z}$ in the space of combs finite
either from the right or from the left.
\begin{figure}[t]
\centering
\includegraphics[angle=-00,scale=0.45]{dualscalarprod.eps}
\caption{\it The graphical representation of equation
(\ref{intw6}).}
\label{fig:dualscalarprod}
\end{figure}
Let us take the scalar product of both sides of equation
(\ref{intw6}) with the covector
$\Bigl < \phi_{z'}^{z''}(\lambda +\frac{3\eta}{2})\Bigr |$
from the left and the vector
$\Bigl |\bar \phi_{\zeta '}^{z}(\lambda +\frac{\eta}{2})\Bigr >$
from the right. Using the orthogonality relations
(\ref{orth1}), (\ref{orth2}), we obtain:
\begin{equation}\label{intw7}
\frac{W_{z''}^{z}(\lambda -\mu )}{\theta_1(2z'')}
\Bigl < \phi_{z'}^{z''}(\mu +\frac{\eta}{2})\Bigr | \,
\bar \phi_{\zeta '}^{z}(\lambda + \frac{\eta}{2})\Bigr > =
\frac{W_{z'}^{\zeta '}(\lambda -\mu )}{\theta_1(2z')}
\Bigl < \phi_{z'}^{z''}(\lambda +\frac{3\eta}{2})\Bigr | \,
\bar \phi_{\zeta '}^{z}(\mu - \frac{\eta}{2})\Bigr > .
\end{equation}
This functional relation for $W_{\zeta}^{z}$
can be solved in terms of $W^{z, \zeta}$ with the help of
(\ref{intw3}): $W_{\zeta}^{z}(\lambda )
W^{\zeta , z}(\lambda +\eta )=\theta_1 (2\zeta )$.
However, this solution is not exactly what we need
because it is not a comb-like function. Proceeding
in a slightly different way, one can rewrite (\ref{intw7})
as a system of difference equations for $W_{\zeta}^{z}(\lambda )$:
\begin{equation}\label{intw7a}
\begin{array}{l}
\displaystyle{
W_{\zeta +\eta}^{z+\eta}(\lambda )=
\frac{\theta_1(2\zeta +2\eta )}{\theta_1 (2\zeta )}\,
\frac{\theta_1(z+\zeta -\lambda )}{\theta_1
(z+\zeta +\lambda +2\eta )}\, W_{\zeta}^{z}(\lambda )},
\\ \\
\displaystyle{
W_{\zeta +\eta}^{z-\eta}(\lambda )=
\frac{\theta_1(2\zeta +2\eta )}{\theta_1 (2\zeta )}\,
\frac{\theta_1(\zeta -z -\lambda )}{\theta_1
(\zeta -z +\lambda +2\eta )}\, W_{\zeta}^{z}(\lambda )}.
\end{array}
\end{equation}
Comparing with (\ref{intw3a}), one immediately
finds a solution in the space of combs:
$$
W_{\zeta}^{z}(\lambda )=
\frac{c(\lambda )\theta_1 (2\zeta )}{W^{\zeta , z}(\lambda +\eta )}
\sum_{k\in \raise-1pt\hbox{$\mbox{\Bbbb Z}$}}\delta (z-\zeta -\nu +2k\eta )
$$
with $W^{z, \zeta}$ given by (\ref{intw4}) and arbitrary $\nu$.
(The factor in front of the sum is also a solution
but in the space of meromorphic functions.)
The function $c(\lambda )$ introduced here
for the proper normalization is not determined from the
difference equations. It will be fixed below.
One may truncate the comb from the left choosing
$\nu =\lambda$; then the coefficients in front of
$\delta (z-\zeta -\lambda +2k\eta )$ with $k<0$ vanish
because the function
$W^{\zeta , z}(\lambda +\eta )$ has poles at
$\zeta =z -\lambda +2k\eta$, $k\leq -1$.
Another possibility is to truncate the comb from the right
choosing $\nu =-\lambda$; then the arguments of the delta-functions
at $k \leq 0$ exactly coincide with the half-infinite
lattice of zeros of
the function $W^{\zeta , z}(\lambda +\eta )$, and so one
can make the truncated comb by taking residues.
Below we use the first possibility and consider the solution
\begin{equation}\label{intw8}
\begin{array}{lll}
W_{\zeta}^{z}(\lambda )&=&\displaystyle{
\frac{c(\lambda )\theta_1 (2\zeta )}{W^{\zeta , z}(\lambda +\eta )}
\sum_{k\geq 0}\delta (z-\zeta -\lambda +2k\eta )}
\\ && \\
&=&\displaystyle{c(\lambda )\sum_{k\geq 0}\frac{\theta_1
(2z-2\lambda +4k\eta )}{W^{z-\lambda +2k\eta , z}(\lambda +\eta )}\,
\delta (z-\zeta -\lambda +2k\eta )}
\end{array}
\end{equation}
which is the kernel of the difference operator
$$
\begin{array}{lll}
{\bf W}(\lambda )&=&\displaystyle{c(\lambda )
\sum_{k\geq 0}\frac{\theta_1
(2z-2\lambda +4k\eta )}{W^{z-
\lambda +2k\eta , z}(\lambda +\eta )}\,
e^{(-\lambda +2k\eta )\partial_z}}
\\ && \\
&=&\displaystyle{c(\lambda )\sum_{k\geq 0}
e^{2\pi i (\lambda +\eta )(z-\lambda +2k\eta )/\eta }
\theta_1 (2z-2\lambda +4k\eta )}
\\ && \\
&& \displaystyle{\,\,\,\,\,\,\,\, \times
\frac{{\sf \Gamma} (2z-2\lambda +2k\eta )
{\sf \Gamma} ( -2\lambda +2k\eta )}{{\sf \Gamma} ( 2z+2\eta +2k\eta )
{\sf \Gamma} ( 2\eta + 2k\eta )}\, e^{(-\lambda +2k\eta )\partial_z}}.
\end{array}
$$
Rewriting the coefficients in terms of the elliptic
Pochhammer symbols with the help of (\ref{gam6}), (\ref{gam6a})
and extracting a common multiplier,
we obtain
$$
{\bf W}(\lambda )=
\tilde c(\lambda )
\frac{e^{2\pi i\lambda (z-\lambda )/\eta}
{\sf \Gamma} (-2\lambda ){\sf \Gamma} (2z-2\lambda +2\eta )}{{\sf \Gamma} (2\eta )\,
{\sf \Gamma} (2z+2\eta )}
\sum_{k\geq 0}\frac{[\frac{z-\lambda}{\eta}+
2k]}{[\frac{z-\lambda}{\eta}]\, [1]_k}\,
\frac{[\frac{z-\lambda}{\eta}]_k \,
[-\frac{\lambda}{\eta}]_k}{[\frac{z}{\eta}+1]_k }
\, e^{(-\lambda +2k\eta )\partial_z},
$$
where $\tilde c(\lambda )=ie^{\frac{\pi i \tau}{6}}\eta _{D}(\tau )
c(\lambda )$.
The infinite sum can be written in terms of the
elliptic hypergeometric series ${}_{4}\omega_{3}$ (see Appendix A
for the definition) with operator argument:
\begin{equation}\label{intw9}
{\bf W}(\lambda )= \tilde c(\lambda )
\frac{e^{2\pi i\lambda (z-\lambda )/\eta}
{\sf \Gamma} (-2\lambda ){\sf \Gamma} (2z-2\lambda +2\eta )}{{\sf \Gamma} (2\eta )\,
{\sf \Gamma} (2z+2\eta )}\,
{\scriptstyle {{\bullet}\atop{\bullet}}} {}_{4}\omega_{3}\left (\frac{z-\lambda}{\eta}; \,
-\frac{\lambda}{\eta}; \, e^{2\eta \partial_z}\right )\! {\scriptstyle {{\bullet}\atop{\bullet}}}
\, e^{-\lambda \partial_z}.
\end{equation}
Here the double dots mean normal ordering such that the shift
operator $e^{2k\eta \partial_z}$ is moved to the right.
By construction, this operator satisfies the intertwining
relation
\begin{equation}\label{intw10}
{\bf W}(\lambda -\mu ){\sf L}(\lambda , \mu )=
{\sf L}(\mu , \lambda ) {\bf W}(\lambda -\mu ).
\end{equation}
The intertwining property (\ref{intw10})
suggests that
${\bf W}(\lambda ){\bf W}(-\lambda )=\mbox{id}$ or, equivalently,
\begin{equation}\label{intw12}
\int \! d\zeta \, W^{z}_{\zeta}(\lambda )
W^{\zeta}_{z'}(-\lambda )=\delta (z-z')
\end{equation}
which is a shadow world analog of (\ref{WW}).
This is indeed true
provided that the function $c(\lambda )$ is fixed to be
\begin{equation}\label{cfixed}
c(\lambda )=
\frac{\rho_0 \, e^{\pi i \lambda ^2/\eta}}{{\sf \Gamma} (-2\lambda )},
\end{equation}
where the constant $\rho_0$ is
\begin{equation}\label{rho0a}
\rho_0 = \frac{{\sf \Gamma} (2\eta )}{ie^{\frac{\pi i \tau}{6}}\eta_D (\tau )}=
\frac{e^{\frac{\pi i}{12}(2\eta -3\tau )}}{i\eta_D (2\eta )}
\end{equation}
(clearly, there is still
a freedom to multiply $c(\lambda )$ by
a function $\varphi (\lambda )$ such that
$\varphi (\lambda )\varphi (-\lambda )=1$).
It should be noted that the very fact that the
product ${\bf W}(\lambda ){\bf W}(-\lambda )$
is proportional to the
identity operator is by no means obvious from
the infinite series representation
(\ref{intw9}). This fact was explicitly proved
in \cite{DKK07} with the
help of the Frenkel-Turaev summation formula.
For completeness, we present some details of this
calculation in Appendix B. It is this calculation that
allows one
to find $c(\lambda )$ explicitly.
We thus conclude that the
properly normalized intertwining
operator ${\bf W}(\lambda )$ reads
\begin{equation}\label{intw9a}
{\bf W}(\lambda )= e^{-\frac{\pi i\lambda ^2}{\eta} +
\frac{2\pi i \lambda z}{\eta}}\,
\frac{{\sf \Gamma} (2z\! -\! 2\lambda \! +\! 2\eta )}{{\sf \Gamma} (2z+2\eta )}\,
{\scriptstyle {{\bullet}\atop{\bullet}}} {}_{4}\omega_{3}\left (\frac{z-\lambda}{\eta}; \,
-\frac{\lambda}{\eta}; \, e^{2\eta \partial_z}\right )\! {\scriptstyle {{\bullet}\atop{\bullet}}}
\, e^{-\lambda \partial_z},
\end{equation}
or, in terms of the parameter $d\equiv 2\ell +1\in \mbox{\Bbb C}$
related to the spin $\ell$ of the representation,
\begin{equation}\label{intw11}
{\bf W}_{\ell}\equiv {\bf W}(d\eta )=
e^{-\pi i d^2\eta +2\pi i dz}
\frac{{\sf \Gamma} (2z -2(d-1)\eta )}{{\sf \Gamma} (2z+2\eta )}\,
{\scriptstyle {{\bullet}\atop{\bullet}}} {}_{4}\omega_{3}\left (\frac{z}{\eta} -d; \,
- d; \, e^{2\eta \partial_z}\right )\! {\scriptstyle {{\bullet}\atop{\bullet}}}
\, e^{-d\eta \partial_z}.
\end{equation}
It is not difficult to see that the change of sign
$z\to -z$ transforms ${\bf W}(\lambda )$ to
another intertwining operator for the
Sklyanin algebra, which is an infinite series in shifts
in the opposite direction. (It is this latter
operator which was constructed in the paper \cite{Z00}.)
It can be obtained
within the same approach if one uses the other possibility
to truncate the comb which has been discussed above.
If $\ell \in \frac{1}{2}\mbox{\Bbb Z} _+$ (i.e., $d\in \mbox{\Bbb Z} _+$), then the
elliptic hypergeometric series is terminating and both operators
are represented by finite sums (containing $d+1$ terms).
Moreover, they coincide with each other and are explicitly given
by the formula
\begin{equation}\label{intw13}
{\bf W}_{\ell}=\left (ie^{\pi i (-\eta + \frac{\tau}{6})}
\eta _{D}(\tau )\right )^d
\sum_{k=0}^{d} (-1)^k
\left [ \begin{array}{c} d \\ k \end{array} \right ]
\, \frac{ \theta _{1}(2z-
2(d-2k)\eta )}{\prod_{j=0}^{d}
\theta_1(2z+2(k-j)\eta )} \,
e^{(-d +2k)\eta \partial_z }
\end{equation}
which coincides with
equation (\ref{W}).
Let us conclude this section by summarizing the
graphic elements of the diagrams and rules of their
composing. The plane is divided
into ``transparent" and ``shadow" pieces
by a number of straight dashed lines in such a way that
each segment of any line is a border between pieces
of the different kind.
Each dashed line carries a spectral parameter denoted by
$\lambda$, $\mu$, etc.
There may be also bold straight lines which become
dotted when they go through shadow pieces of the plane.
Each shadow piece (bounded by dashed or dotted lines
or by infinity) carries a complex variable denoted by
$z$, $\zeta$, etc. Those which sit on infinite pieces
are fixed while those which sit on finite pieces bounded
by lines of any type should be ``integrated" in the sense
of the pairing (\ref{pairing}). The intersection points
of the dashed lines are of two types depending on
the way how the transparent and shadow parts are
adjacent to it. Correspondingly, there are
two types of vertex functions shown in Fig. \ref{fig:vertices}.
The intersection of a dashed line with a bold one
corresponds to an intertwining (co)vector as shown in
Fig. \ref{fig:intwvec}. Finite bold segments mean
taking scalar products of (co)vectors associated with their
endpoints.
\section{Vacuum vectors}
In order to make a closer contact with our earlier work
\cite{Z00}, it is useful to demonstrate how the vacuum vectors for
the $L$-operator can be constructed within the approach
developed in the previous sections. Let us recall the general
definition of the vacuum vectors.
Consider an arbitrary $L$-operator ${\sf L}$
with two-dimensional auxiliary space $\mbox{\Bbb C}^2$, i.e., an arbitrary
$2\times 2$
operator-valued matrix
$$
{\sf L}=\left ( \begin{array}{cc} {\bf L}_{11}& {\bf L}_{12}\\
{\bf L}_{21}& {\bf L}_{22} \end{array} \right ).
$$
The operators ${\bf L}_{ij}$ act in
a linear space ${\cal H}$ which is called the quantum space of
the $L$-operator.
For the moment, let $\phi$, $\psi$, etc denote vectors from
$\mbox{\Bbb C}^2$ and $X, X_1$, etc vectors from ${\cal H}$,
then acting by the quantum
$L$-operator on the tensor product
$X\otimes \phi$, we, generally speaking, obtain a mixed state
in the quantum space:
${\sf L}X\otimes \phi =X_1\otimes \phi_1+X_2\otimes \phi_2$.
The special case of a pure state,
\begin{equation}\label{V1}
{\sf L}X\otimes \phi =X'\otimes \psi \,,
\end{equation}
is of prime importance.
The relation (\ref{V1}) (in the particular case ${\cal H}\cong
\mbox{\Bbb C}^2$) was the key point for Baxter in his solution of
the 8-vertex model \cite{Baxter}.
(This is what he called the ``pair-propagation through a vertex"
property.) Taking the scalar product
with the vector $\psi^{\bot}$ orthogonal to $\psi$, we get:
\begin{equation}
\label{V2}
\bigl (\psi^{\bot}{\sf L}\phi \bigr )X=0\,,
\end{equation}
i.e., the operator
${\bf K}=\bigl (\psi^{\bot}{\sf L}\phi \bigr )$ (acting in the
quantum space only) has a zero mode
$X\in {\cal H}$.
Suppose (\ref{V1}) (or (\ref{V2})) holds with some
vectors $\phi$, $\psi$;
then the vector $X$ is called
a {\it vacuum vector} of the $L$-operator.
An algebro-geometric
approach to the equation (\ref{V1}) for finite-dimensional
matrices ${\bf L}_{ik}$ was suggested by Krichever \cite{krivac}
and further developed in \cite{kz95,kz99}. In our paper \cite{Z00}
the Baxter's method of vacuum vectors was adopted to the
infinite-dimensional representations of the Sklyanin algebra.
For $L$-operators with elliptic spectral parameter it is
convenient to pass to the elliptic parametrization of the
components of the vectors $\phi$, $\psi$ as is given by
(\ref{vectors}). Writing ${\sf L}(\lambda )\bigl |\zeta \bigr >$
(respectively, $\bigl <\zeta \bigr |{\sf L}(\lambda)$)
we mean that the 2$\times$2 matrix ${\sf L}$ acts on the
2-component vector from the left (respectively, on the
2-component covector from the right).
Similarly, we introduce right and left vacuum vectors
$X_R$, $X_L$ according to the relations
\begin{equation}
\label{V3}
\bigl < \zeta \bigr | {\sf L}(\lambda)X_R =
\bigl < \xi \bigr | X'_R \,,
\;\;\;\;\;\;
X_L\bigl < \zeta \bigr | {\sf L}(\lambda) =
X'_L \bigl < \xi \bigr |\,.
\end{equation}
In the latter formula the matrix elements of
${\sf L}$ act on $X_L$ from the right.
Introducing the operator
\begin{equation}
\label{V5}
{\bf K}= {\bf K}(\zeta , \xi )=
\bigl < \zeta \bigr | {\sf L}(\lambda)\bigl |\xi \bigr >^{\bot},
\end{equation}
we can rewrite (\ref{V3}) as
${\bf K}X_R =X_L{\bf K}=0$.
The explicit form of the operator ${\bf K}$ can be
found from (\ref{L}),(\ref{Sa}):
\begin{equation}
\label{E1}
{\bf K}={\bf K}(\zeta, \xi)=\rho (z) e^{\eta \partial_{z}}+
\rho (-z) e^{-\eta \partial_{z}}\,,
\end{equation}
where
$$
\rho (z)=\frac{1}{\theta_{1}(2z)}
\prod_{\epsilon =\pm}\theta_{1}\Bigl (z+\epsilon \zeta
-\lambda_{+}+\frac{\eta}{2} \Bigr )
\theta_{1}\Bigl (z+\epsilon \xi
+\lambda_{-} +\frac{\eta}{2} \Bigr )\,.
$$
These difference operators appeared in \cite{kz99,Z00} and later were
independently introduced in \cite{Rosengren04,Rains}.
So, the equations for the right and left vacuum vectors
read
\begin{equation}
\label{R1a}
\rho (z)X_{R}(z+\eta)=
-\rho (-z)X_{R}(z-\eta)\,,
\end{equation}
\begin{equation}
\label{L1a}
\rho (-z-\eta)X_{L}(z+\eta)=
-\rho (z-\eta)X_{L}(z-\eta)\,.
\end{equation}
\begin{figure}[t]
\centering
\includegraphics[angle=-00,scale=0.50]{vacuum.eps}
\caption{\it The graphic representation of equation
(\ref{vac1}): action of the $L$-operator to right
vacuum vectors.}
\label{fig:vacuum}
\end{figure}
Instead of solving these equations explicitly, below we
show how the vacuum vectors emerge within the approach of the
present paper.
The key relation is
(see Fig. \ref{fig:vacuum})
\begin{equation}\label{vac1}
\begin{array}{ll}
&\displaystyle{
\int \! d\zeta \, \Bigl < \phi^{z'}_{\xi}(\mu +\frac{\eta}{2})
\Bigr | \bar \phi ^{z}_{\zeta}(\lambda_{+}-\frac{\eta}{2})\Bigr >
\Bigl < \phi^{z}_{\zeta}(\lambda_{-} +\frac{\eta}{2})\Bigr | \,
W^{\xi , \zeta}(\lambda_{+}-\mu)W^{\zeta}_{\xi '}(\lambda_{-}-\mu )}
\\ &\\
=&\displaystyle{
\int \! d\zeta \, \Bigl < \phi^{z'}_{\xi}(\lambda_{+} +\frac{\eta}{2})
\Bigr |\bar \phi ^{\zeta}_{\xi '}(\lambda_{-}-\frac{\eta}{2})\Bigr >
\Bigl < \phi^{\zeta}_{\xi '}(\mu +\frac{\eta}{2})\Bigr | \,
W^{z' , z}(\lambda_{+}-\mu)W^{z}_{\zeta}(\lambda_{-}-\mu )}.
\end{array}
\end{equation}
The left-hand side represents the action of the
$L$-operator ${\sf L}^{(\ell )}(\lambda )=
{\sf L}(\lambda_{+}, \lambda_{-})$ to the covector
$\Bigl < \phi^{z'}_{\xi}(\lambda_{+} \! +\! \frac{\eta}{2})
\Bigr |$ in the auxiliary space from the right and to the vector
$W^{\xi , \zeta}(\lambda_{+}-\mu)W^{\zeta}_{\xi '}(\lambda_{-}-\mu )$
in the quantum space from the left. It is convenient to denote
\begin{equation}\label{X}
X_{R}^{\xi, \xi '}(z|\lambda _{+}, \lambda _{-})=
W^{\xi , z}(\lambda_{+}-\frac{\eta}{2})\,
W^{z}_{\xi '}(\lambda_{-}-\frac{\eta}{2}),
\end{equation}
then relation (\ref{vac1}) can be rewritten (after setting
$z'=\xi \pm \eta$ and some transformations) as the system of equations
\begin{equation}\label{vac2}
\begin{array}{l}
\displaystyle{
\Bigl < \xi -\mu \Bigr |{\sf L}(\lambda_{+} +\mu , \lambda_{-}+\mu )
X_{R}^{\xi, \xi '}=a\Bigl < \xi ' -\mu \Bigr |\,
X_{R}^{\xi +\eta, \xi '+\eta}+b
\Bigl < \xi ' +\mu \Bigr | \,X_{R}^{\xi +\eta, \xi '-\eta}},
\\ \\
\displaystyle{
\Bigl < \xi +\mu \Bigr |{\sf L}(\lambda_{+}+\mu , \lambda_{-}+\mu )
X_{R}^{\xi, \xi '} =
c\Bigl < \xi ' -\mu \Bigr | \,X_{R}^{\xi -\eta, \xi '+\eta}
+d \Bigl < \xi ' +\mu \Bigr | \,X_{R}^{\xi -\eta, \xi '-\eta}},
\end{array}
\end{equation}
where $X_{R}^{\xi, \xi '} =X_{R}^{\xi, \xi '}
(z|\lambda_{+}, \,
\lambda_{-})$ and
$$
a=-\frac{\theta_1 (\xi \! +\! \xi '\! -\!
\lambda_{+}\! +\! \lambda_{-}\! +\! \eta )
\theta_1 (\xi \! -\! \xi '\! -\!
\lambda_{+}\! -\! \lambda_{-}\! -\! 2\mu )}{\theta_1
(2\xi ' +2\eta )}\,,
$$
$$
b=\frac{\theta_1 (\xi \! -\! \xi '\! -\!
\lambda_{+}\! +\! \lambda_{-}\! +\! \eta )
\theta_1 (\xi \! +\! \xi '\! -\!
\lambda_{+}\! -\! \lambda_{-}\! -\! 2\mu)}{\theta_1
(2\xi ' -2\eta )}\,,
$$
$$
c=-\frac{\theta_1 (\xi \! -\! \xi '\! +\!
\lambda_{+}\! -\! \lambda_{-}\! -\! \eta )
\theta_1 (\xi \! +\! \xi '\! +\!
\lambda_{+}\! +\! \lambda_{-}\! +\! 2\mu)}{\theta_1
(2\xi ' +2\eta )}\,,
$$
$$
d=\frac{\theta_1 (\xi \! +\! \xi '\! +\!
\lambda_{+}\! -\! \lambda_{-}\! -\! \eta )
\theta_1 (\xi \! -\! \xi '\! +\!
\lambda_{+}\! +\! \lambda_{-}\! +\! 2\mu)}{\theta_1 (2\xi '-2\eta )}.
$$
We note that setting $\mu =0$ one obtains from
(\ref{vac2})
$$
\Bigl < \xi \Bigr |{\sf L}(\lambda_{+}, \lambda_{-})
X_{R}^{\xi, \xi '}=\Bigl < \xi ' \Bigr |
\left (a_0 X_{R}^{\xi +\eta , \xi '+\eta}+b_0
X_{R}^{\xi +\eta , \xi '-\eta}\right )=\Bigl < \xi ' \Bigr |
\left (c_0 X_{R}^{\xi -\eta , \xi '+\eta}+d_0
X_{R}^{\xi -\eta , \xi '-\eta}\right ),
$$
where $a_0 =a(\mu =0)$, etc.
This means that $X_{R}^{\xi, \xi '}(z|\lambda_{+}\! -\!
\frac{\eta}{2},
\lambda_{-}\! -\! \frac{\eta}{2})$ is the right vacuum vector
for the $L$-operator (see the first equation
in (\ref{V3}). Moreover, we conclude that
\begin{equation}\label{vac3}
a_0 X_{R}^{\xi +\eta , \xi '+\eta}+b_0
X_{R}^{\xi +\eta , \xi '-\eta}=
c_0 X_{R}^{\xi -\eta , \xi '+\eta}+d_0
X_{R}^{\xi -\eta , \xi '-\eta}.
\end{equation}
One can that the vacuum vector is in fact a composite
object. It is a product
of two $W$-functions.
Equations (\ref{intw4}), (\ref{intw8}) together with
the 3-term identity for the Jacobi theta-function imply the relation
\begin{equation}\label{vac4}
\Bigl < \xi \Bigr |{\sf L}(\lambda_{+}, \lambda_{-})
X_{R}^{\xi, \xi '}\bigl (z\bigr |\lambda_{+},
\lambda_{-}\bigr )=
\theta_{1}(2\lambda_{-}+\eta )
\Bigl < \xi ' \Bigr |
X_{R}^{\xi, \xi '}\bigl (z\bigr |\lambda_{+}\! +\! \eta ,
\lambda_{-}\! +\! \eta \bigr )
\end{equation}
which is equation (4.22) from our paper \cite{Z00}
written in the slightly different notation.
The left vacuum vectors can be considered in a similar way.
\section{The $R$-operator and related objects}
\begin{figure}[t]
\centering
\includegraphics[angle=-00,scale=0.50]{RLL.eps}
\caption{\it The intertwining relation
$\check {\sf R}\, {\sf L}\otimes {\sf L}=
{\sf L}\otimes {\sf L}\, \check {\sf R}$.}
\label{fig:RLL}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[angle=-00,scale=0.50]{R.eps}
\caption{\it The kernel of the $R$-operator
$R^{zz'}_{\zeta \zeta '}
(\lambda _{+}, \lambda _{-}|\mu _{+}, \mu _{-})$.}
\label{fig:R}
\end{figure}
The $R$-operator $\check {\sf R}=
\check {\sf R}(\lambda _{+}, \lambda _{-}|
\mu _{+}, \mu _{-})$
intertwines the product of two
$L$-operators:
\begin{equation}\label{R1}
\check {\sf R}(\lambda _{+}, \lambda _{-}|
\mu _{+}, \mu _{-}){\sf L}(\lambda _{+}, \lambda _{-})
\otimes {\sf L}(\mu _{+}, \mu _{-})=
{\sf L}(\mu _{+}, \mu _{-})\otimes
{\sf L}(\lambda _{+}, \lambda _{-})
\check {\sf R}(\lambda _{+}, \lambda _{-}|
\mu _{+}, \mu _{-}).
\end{equation}
Passing to the different notation,
$\lambda _{\pm}=\lambda \pm (\ell +\frac{1}{2})\eta$,
$\mu _{\pm}=\mu \pm (\ell ' +\frac{1}{2})\eta$,
we can rewrite (\ref{R1}) in a more conventional form:
\begin{equation}\label{R2}
\check {\sf R}^{(\ell \ell ')}(\lambda , \mu )\,
{\sf L}^{(\ell )}(\lambda )\otimes {\sf L}^{(\ell ')}(\mu )=
{\sf L}^{(\ell ')}(\mu )\otimes
{\sf L}^{(\ell )}(\lambda )\,
\check {\sf R}^{(\ell \ell ')}(\lambda , \mu ).
\end{equation}
Here $\check {\sf R}^{(\ell \ell ')}(\lambda , \mu )=
\check {\sf R}(\lambda _{+}, \lambda _{-}|
\mu _{+}, \mu _{-})$ is a difference operator in
two variables acting in the tensor product of
the quantum spaces for the two $L$-operators.
In terms of the kernels equation (\ref{R1}) reads:
\begin{equation}\label{R3}
\begin{array}{ll}
&\displaystyle{\int \! d\zeta \int \! d\zeta '\,
R^{zz'}_{\zeta \zeta '}
(\lambda _{+}, \lambda _{-}|\mu _{+}, \mu _{-})
L^{\zeta}_{\xi}(\lambda _{+}, \lambda _{-})
L^{\zeta '}_{\xi '}(\mu _{+}, \mu _{-})}
\\ & \\
=&\displaystyle{\int \! d\zeta \int \! d\zeta '\,
L^{z}_{\zeta}(\mu _{+}, \mu _{-})
L^{z'}_{\zeta '}(\lambda _{+}, \lambda _{-})
R^{\zeta \zeta '}_{\xi \xi '}
(\lambda _{+}, \lambda _{-}|\mu _{+}, \mu _{-})}.
\end{array}
\end{equation}
Graphically it is shown in Fig. \ref{fig:RLL}.
The figure clarifies the structure of the kernel
of the $R$-operator which is shown in more detail
in Fig. \ref{fig:R}. It is clear that the kernel is
the product of four $W$-vertices: two of them are of
the $W^{z, \zeta}$-type (meromorphic functions) and
the other two are of the $W_{\zeta}^{z}$-type
(comb-like functions). Specifically, we can write:
$$
\begin{array}{ll}
&R^{zz'}_{\zeta \zeta '}
(\lambda _{+}, \lambda _{-}|\mu _{+}, \mu _{-})
\\ &\\
=&W^{z,z'}(\lambda _{+}-\mu_{-})
W_{\zeta '}^{z'}(\lambda_{-}-\mu_{-})
W_{\zeta}^{z}(\lambda_{+}-\mu_{+})
W^{\zeta ,\zeta '}(\lambda _{-}-\mu_{+})
\\ &\\
=&\displaystyle{W^{z,z'}(\lambda _{+}-\mu_{-})
\left [
\frac{c(\lambda _{-}-\mu _{-})
\theta_1(2\zeta ')}{W^{\zeta ',z'}(\lambda _{-}-
\mu_{-}+\eta )}\sum_{k'\geq 0}\delta (z'-\zeta ' -\lambda_{-}
+\mu_{-}+2k'\eta )\right ]}
\\ &\\
& \,\,\,\, \times \displaystyle{\left [
\frac{c(\lambda _{+}-\mu _{+})\theta_1(2\zeta )}{W^{\zeta ,z}
(\lambda _{+}-\mu_{+}+\eta )}
\sum_{k\geq 0}\delta (z-\zeta -\lambda_{+}
+\mu_{+}+2k\eta )\right ]
W^{\zeta ,\zeta '}(\lambda _{-}-\mu_{+})}
\end{array}
$$
which is the kernel of the difference
operator
\begin{equation}\label{R4a}
\check {\sf R}(\lambda _{+}, \lambda _{-}|\mu _{+}, \mu _{-})=
W^{z,z'}(\lambda_{+}-\mu_{-})
{\bf W}^{(z')}(\lambda_{-}\! -\! \mu_{-})
{\bf W}^{(z)}(\lambda_{+}\! -\! \mu_{+})
W^{z,z'}(\lambda_{-}-\mu_{+})
\end{equation}
(here the notation ${\bf W}^{(z)}$ means that the
operator ${\bf W}$ acts to the variable $z$.
In full, the $R$-operator reads
\begin{equation}\label{R4}
\begin{array}{ll}
&\check {\sf R}(\lambda _{+}, \lambda _{-}|\mu _{+}, \mu _{-})\, =\,
e^{-\frac{\pi i}{\eta}(\lambda_{+}-\mu _{+})^2
-\frac{\pi i}{\eta}(\lambda_{-}-\mu _{-})^2
+\frac{2\pi i}{\eta}(\lambda_{+}-\mu _{+})z
+\frac{2\pi i}{\eta}(\lambda_{-}-\mu_{-})z'}
\\ &\\
\times &\, \displaystyle{
e^{-\frac{2\pi i}{\eta}(\lambda_{+}-\mu_{-})z}
\, \frac{{\sf \Gamma} (z+z' +\lambda_{+} -\mu_{-}+\eta )
{\sf \Gamma} (z-z' +\lambda_{+}
-\mu_{-}+\eta )}{{\sf \Gamma} (z+z' -\lambda_{+} +\mu_{-}+\eta )
{\sf \Gamma} (z-z' -\lambda_{+} +\mu_{-}+\eta )}}
\\ &\\
\times &\, \displaystyle{
\frac{{\sf \Gamma} (2z' \! -\! 2(\lambda _{-}\!
-\! \mu _{-}) \! +\! 2\eta )}{{\sf \Gamma} (2z' +2\eta )}
\, {\scriptstyle {{\bullet}\atop{\bullet}}} {}_{4}\omega_{3}\left (
\frac{z'+\mu _{-} -\lambda _{-}}{\eta};\,
\frac{\mu _{-}-\lambda _{-}}{\eta};\,
e^{2\eta \partial_{z'}}\right )\! {\scriptstyle {{\bullet}\atop{\bullet}}} \,
e^{-(\lambda_{-}-\mu_{-})\partial_{z'}}}
\\ &\\
\times &\, \displaystyle{
\frac{{\sf \Gamma} (2z\! -\! 2(\lambda_{+}\! -\!
\mu_{+})\! +\! 2\eta )}{{\sf \Gamma} (2z +2\eta )}
\, {\scriptstyle {{\bullet}\atop{\bullet}}} {}_{4}\omega_{3}\left (
\frac{z+\mu _{+}-\lambda _{+}}{\eta};\,
\frac{\mu _{+}-\lambda _{+}}{\eta};\,
e^{2\eta \partial_{z}}\right )\! {\scriptstyle {{\bullet}\atop{\bullet}}} \,
e^{-(\lambda_{+}-\mu_{+})\partial_{z}}}
\\ &\\
\times &\,
\displaystyle{e^{-\frac{2\pi i}{\eta}(\lambda_{-}-\mu_{+})z}
\, \frac{{\sf \Gamma} (z+z' +\lambda_{-} -\mu_{+}+\eta )
{\sf \Gamma} (z-z' +\lambda_{-}
-\mu_{+}+\eta )}{{\sf \Gamma} (z+z' -\lambda_{-} +\mu_{+}+\eta )
{\sf \Gamma} (z-z' -\lambda_{-} +\mu_{+}+\eta )}}.
\end{array}
\end{equation}
The difference operators in the third and the fourth lines
of the r.h.s. commute because they act in different variables
but both of them do not commute with the operator of
multiplication by the function
$W^{z ,z'}(\lambda _{-}-\mu_{+})$. Note that the $R$-operator
can be also written in terms of the ${}_{6}\omega_{5}$ series
due to the identity
\begin{equation}\label{R4id}
\begin{array}{ll}
&\displaystyle{ {\scriptstyle {{\bullet}\atop{\bullet}}}
{}_{4}\omega_{3}\left (
\frac{z-\lambda}{\eta}; \, -\frac{\lambda}{\eta};\,
e^{2\eta \partial_z}\right ) {\scriptstyle {{\bullet}\atop{\bullet}}} e^{-\lambda \partial_z}
W^{\zeta , z}(\mu )}
\\ & \\
=&\displaystyle{W^{\zeta , z-\lambda}(\mu )
{\scriptstyle {{\bullet}\atop{\bullet}}}
{}_{6}\omega_{5}\left (
\frac{z-\lambda}{\eta}; \, -\frac{\lambda}{\eta},\,
\frac{z+\zeta +\mu -\lambda +\eta}{2\eta}, \,
\frac{z-\zeta +\mu -\lambda +\eta}{2\eta};\,
e^{2\eta \partial_z}\right ) {\scriptstyle {{\bullet}\atop{\bullet}}} e^{-\lambda \partial_z}}.
\end{array}
\end{equation}
\begin{figure}[t]
\centering
\includegraphics[angle=-00,scale=0.50]{RRR.eps}
\caption{\it The Yang-Baxter equation for the
$R$-operator.}
\label{fig:RRR}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[angle=-00,scale=0.50]{WWW.eps}
\caption{\it The star-triangle equation (\ref{st1a}) for the
$W$-operators. Equation (\ref{st1b}) corresponds to the same
configuration of lines with complimentary shadow parts
of the plane.}
\label{fig:WWW}
\end{figure}
The Yang-Baxter equation for the $R$-operator is schematically
shown in the self-\-exp\-la\-na\-to\-ry Fig. \ref{fig:RRR}.
One can see that
as soon as the $R$-operator is a composite object,
the Yang-Baxter equation can be reduced to simpler equations
for its elementary constituents. The latter are the
$W$-vertices of the two types. For them one can prove
a sort of the star-triangle relations
\begin{equation}\label{st1a}
W^{z',z}(\mu -\nu )W^{z', z''}(\lambda -\mu )
W^{z}_{z''}(\lambda -\nu )=
\int \! d\zeta W^{z}_{\zeta}(\lambda -\mu )
W^{z',\zeta}(\lambda -\nu )W^{\zeta}_{z''}(\mu -\nu )
\end{equation}
\begin{equation}\label{st1b}
W^{z,z'}(\lambda -\mu )W^{z'', z'}(\mu -\nu )
W^{z}_{z''}(\lambda -\nu )=
\int \! d\zeta W^{z}_{\zeta}(\mu -\nu )
W^{\zeta , z'}(\lambda -\nu )W^{\zeta}_{z''}(\lambda -\mu )
\end{equation}
schematically shown
in Fig. \ref{fig:WWW}. The proof is given in Appendix B.
As is seen from Fig. \ref{fig:RRR}, the proof of the
Yang-Baxter equation is reduced to sequential transferring
of vertical lines from the left to the right through intersection
points of the other lines with the use of
the star-triangle relations (\ref{st1a}) and (\ref{st1b})
at each step.
Let us note that the both sides of the
star-triangle relations (\ref{st1a}) and
(\ref{st1b}) represent the kernels of
the difference operators explicitly written
in Appendix B ((\ref{st2a}) and (\ref{st2b})
respectively).
\begin{figure}[t]
\centering
\includegraphics[angle=-00,scale=0.50]{S.eps}
\caption{\it The kernel $S^{z}_{\xi}(z',z'')$
``dual" to the kernel of the $R$-operator
(cf. Fig. \ref{fig:R}).}
\label{fig:S}
\end{figure}
There is an object ``dual" to the
$R$-operator $\check {\sf R}$
in the sense that its kernel is graphically represented by the
same pattern, with shadow parts of the plane being complimentary to those
in Fig. \ref{fig:R}. This duality provides a transformation
which is an infinite-dimensional version of the vertex-face
correspondence.
It sends the $R$-operator to a difference operator
in one variable rather than two. We call it the $S$-operator.
It acts in the variable $z$ and depends on $z'$ and $z''$
as parameters.
Its kernel, $S^{z}_{\xi}(z',z''| \lambda_{+}, \lambda_{-};
\mu _{+}, \mu_{-})$, or simply
$S^{z}_{\xi}(z',z'')$ in short, is shown in Fig. \ref{fig:S}.
This kernel is to be regarded
as an $R$-matrix for a face-type model
with complex variables associated to shadow parts of the plane.
It generalizes the fused Boltzmann weights of the SOS-type
8-vertex model \cite{DJMO}.
According to Fig. \ref{fig:S}
it reads
\begin{equation}\label{S1}
S^{z}_{\xi}(z',z'')=\int \! d\zeta
W^{z}_{\zeta}(\lambda_{+}-\mu_{-})
W^{z', \zeta}(\lambda_{+}-\mu_{+})
W^{\zeta , z''}(\lambda_{-}-\mu_{-})
W^{\zeta}_{\xi}(\lambda_{-}-\mu_{+}).
\end{equation}
The convolution is taken with respect to the variable
sitting in the finite parallelogram at the center of
Fig. \ref{fig:S}. Since each of the two $W^{z}_{\zeta}$-vertices
is represented by a half-infinite sum of the type (\ref{intw8}),
the whole expression (\ref{S1}) is a double sum.
Performing the convolution and re-arranging the double sum, we can
write
$$
S^{z}_{\xi}(z',z'')=c(\lambda_{+}\! -\! \mu_{-})c(\lambda_{-}\! -\! \mu_{+})
\theta_1(2\xi)\sum_{n\geq 0} A_n(z, z', z'')\,
\delta (z\! -\! \xi \! -\! \lambda_{+}\! -\!
\lambda_{-}\! +\! \mu_{+}\! +\! \mu_{-}\! +\! 2n\eta ),
$$
where
$$
\begin{array}{ll}
& A_n(z, z', z'')
\\ &\\
=& \displaystyle{\sum_{k=0}^{n}
\frac{\theta_1(2z\! -\! 2\lambda_{+}\! +\! 2\mu_{-}\! +\! 4k\eta )\,
W^{z', z-\lambda_{+} +\mu_{-}+2k\eta}(\lambda_{+}\! -\! \mu_{+})\,
W^{z-\lambda_{+} +\mu_{-}+2k\eta, z''}
(\lambda_{-}\! -\! \mu_{-})}{W^{z-\lambda_{+} +\mu_{-}+2k\eta, z}
(\lambda_{+}\! -\! \mu_{-}\! +\! \eta )\,
W^{z-\lambda_{+}-\lambda_{-}+\mu_{+}+\mu_{-}+2n\eta ,\,
z-\lambda_{+} +\mu_{-}+2k\eta}
(\lambda_{-}\! -\! \mu_{+}\! +\! \eta )}}.
\end{array}
$$
Using the explicit form of the $W$-functions
it is straightforward to show
that the kernel $S^{z}_{\xi}(z',z'')$ is expressed in terms of the elliptic
hypergeometric series ${}_{10}\omega _{9}$ as follows:
\begin{equation}\label{S2}
\begin{array}{lll}
S^{z}_{\xi}(z',z'')&=&
\displaystyle{C \theta_1(2\xi)
e^{2\pi i \xi +\frac{2\pi i}{\eta}((\lambda_{+}-\lambda_{-})z
+(\lambda_{-}-\mu_{+})\xi +(\mu_{+}-\lambda_{+})z')}}
\\ &&\\
&\times &\displaystyle{
\frac{\tilde \theta_1 (z-z' +\mu_{-}-\mu_{+}+\eta )\,
\tilde \theta_1 (z-\xi +\mu_{+}+\mu_{-}-
\lambda_{+}-\lambda_{-})}{\tilde \theta_1
(z\! -\! z' \! +\! \mu_{-}\! +\! \mu_{+}\! -\!
2\lambda_{+} \! +\! \eta )\,
\tilde \theta_1 (z\! -\! \xi \! -\! \mu_{+}\! +\!
\mu_{-} \! -\! \lambda_{+}\! +\! \lambda_{-})}}
\\ &&\\
&\times &\displaystyle{
\frac{{\sf \Gamma} (2z+2\mu_{-}-2\lambda_{+}+2\eta )}{{\sf \Gamma}
(2z+2\eta )}
\prod_{j=5}^{10} \frac{{\sf \Gamma} (2 \alpha_j \eta)}{{\sf \Gamma}
(2 (\alpha_1 \! -\! \alpha_j \! +\! 1)\eta)}}
\\ &&\\
&\times &\displaystyle{
{}_{10}\omega _{9}(\alpha_1; \alpha_4 , \ldots , \alpha_{10})
\sum_{n\geq 0}\delta (z\! -\! \xi \! -\! \lambda_{+}\! -\!
\lambda_{-}\! +\! \mu_{+}\! +\! \mu_{-}\! +\! 2n\eta )}.
\end{array}
\end{equation}
Here $C$ is a constant which depends on the spectral
parameters, $\tilde \theta_1(x)\equiv
\theta_1 (x|2\eta )$ and the values of the $\alpha_j$'s are
$$
\alpha_1 = \frac{z+\mu_{-}-\lambda_{+}}{\eta},
\quad \alpha_4 =\frac{\mu_{-}-\lambda_{+}}{\eta},
\quad \alpha_{5,6}=\frac{z\pm \xi +\mu_{+}+\mu_{-}-
\lambda_{+}-\lambda_{-}}{2\eta},
$$
$$
\alpha_{7,8}=\frac{z\pm z' +\mu_{-}-\mu_{+}+\eta}{2\eta},
\quad
\alpha_{9,10}=\frac{z\pm z'' +\lambda_{-}-\lambda_{+}+\eta}{2\eta}\,.
$$
One can see that the series ${}_{10}\omega _{9}$ is balanced
(the balancing condition (\ref{mh2}) is
satisfied) and terminating ($\alpha_6 =-n$ because
of the $\delta$-function). Equation (\ref{S2}) is a version
of the Frenkel-Turaev result \cite{FrTur} adopted to
continuous values of parameters and obtained
by a different method.
The $S$-operator satisfies a sort of the Yang-Baxter
equation which can be graphically represented like in
Fig. \ref{fig:RRR} with transparent pieces of the plane
being changed to the
shadow ones and vice versa.
Another object closely related to the $R$-operator
is the ``transfer matrix on 1 site"
\begin{equation}\label{S33}
{\sf T}(\lambda_{+}, \lambda_{-}|
\mu_{+}, \mu_{-})=\mbox{tr}_{\mu}
\left ( \check {\sf R}(\lambda_{+}, \lambda_{-}|
\mu_{+}, \mu_{-}){\sf P}\right ),
\end{equation}
where ${\sf P}$ is the permutation operator
of the two quantum spaces and the trace is taken in
the space associated with the spectral parameters
$\mu_{\pm}$. The kernel of this transfer matrix is
\begin{equation}\label{S4}
T^{z}_{\xi}(\lambda_{+}, \lambda_{-}|
\mu_{+}, \mu_{-})= \int d\xi R^{\zeta z}_{\xi \zeta}
(\lambda_{+}, \lambda_{-}|
\mu_{+}, \mu_{-}).
\end{equation}
It is not difficult to see that this kernel
is expressed
through the kernel $S^{z}_{\xi}(z',z'')$ given by
(\ref{S2}) as follows:
\begin{equation}\label{S5}
T^{z}_{\xi}(\lambda_{+}, \lambda_{-}|
\mu_{+}, \mu_{-})= S^{z}_{\xi}(\xi ,z)
(\lambda_{-}, \lambda_{+}|
\mu_{+}, \mu_{-}).
\end{equation}
(Note the exchange
of the spectral parameters $\lambda_{+}
\leftrightarrow \lambda_{-}$ in the right-hand side.)
The easiest way to see this is to draw the
corresponding pictures.
\section{Concluding remarks}
In this paper we have presented a unified approach to
intertwining operators for quantum integrable models
with elliptic $R$-matrix associated with the Sklyanin algebra.
We work in the most general setting of
infinite-dimensional representations (with
a complex spin parameter $\ell$) realized by difference
operators in the space of functions of a complex variable $z$.
The elementary building blocks are so-called intertwining
vectors and $W$-functions which are defined in terms of their
scalar products. These elements have a nice graphic
representation as diagrams in the transparent/shadow plane
which allows one to easily construct more complicated objects
like $L$-operators, their vacuum vectors and different
kinds of $R$-matrices and to prove relations between them.
An important constituent of the construction is the
intertwining operator for representations with spins
$\ell$ and $-\ell -1$. For general values of $\ell$, it is
given by the elliptic hypergeometric series ${}_{4}\omega_{3}$
with operator argument.
In fact the material presented here is only the very beginning
of the theory of integrable ``spin chains" with elliptic
$R$-matrices and infinite-dimensional space of states at
each site. Indeed, our discussion has been focused on
a single $L$ or $R$ operator which is relevant to a spin chain
of just one site. The next step is to construct the
transfer matrix, i.e., to consider a chain
of the $R$-operators and to take trace in the auxiliary space.
We plan to address this problem elsewhere.
It would be also very desirable to find a direct connection
of our approach with elliptic beta integrals
\cite{Spir10,Spir01}. Presumably, the pairing
(\ref{inf2}) or (\ref{pairing}) should be replaced by a sum
of residues.
Among other things, the results presented in this paper indicate
convincingly that there should exist a meaningful theory of
infinite-dimensional representations of the Sklyanin algebra.
Such a theory is still to be developed
and this paper may provide some background
in reaching this ambitious goal.
At last, one should keep in mind that the Sklyanin algebra
is just a very particular representative of a wide family
of elliptic algebras \cite{OF} and, moreover, integrable
systems associated to algebras from
this class can be constructed \cite{OR}.
It would be very interesting to investigate to what extent
the methods developped in the present paper can be extended
to other elliptic algebras and corresponding integrable models.
Such an extension will probably require a further generalization
of elliptic hypergeometric series.
\section*{Acknowledgments}
The author is grateful to S.\-Der\-ka\-chov for a discussion
of the work \cite{DKK07}.
This work was supported in part by RFBR grant 08-02-00287,
by joint RFBR grants 09-01-92437-CEa,
09-01-93106-CNRS, 10-01-92104-JSPS
and by
Federal Agency for Science and Innovations of Russian Federation
under contract 14.740.11.0081.
\section*{Appendix A}
\defB\arabic{equation}{A\arabic{equation}}
\setcounter{equation}{0}
\subsubsection*{Theta-functions}
We use the following definition of the
Jacobi $\theta$-functions:
\begin{equation}
\begin{array}{l}
\theta _1(z|\tau)=-\displaystyle{\sum _{k\in \raise-1pt\hbox{$\mbox{\Bbbb Z}$}}}
\exp \left (
\pi i \tau (k+\frac{1}{2})^2 +2\pi i
(z+\frac{1}{2})(k+\frac{1}{2})\right ),
\\ \\
\theta _2(z|\tau)=\displaystyle{\sum _{k\in \raise-1pt\hbox{$\mbox{\Bbbb Z}$}}}
\exp \left (
\pi i \tau (k+\frac{1}{2})^2 +2\pi i
z(k+\frac{1}{2})\right ),
\\ \\
\theta _3(z|\tau)=\displaystyle{\sum _{k\in \raise-1pt\hbox{$\mbox{\Bbbb Z}$}}}
\exp \left (
\pi i \tau k^2 +2\pi i
zk \right ),
\\ \\
\theta _4(z|\tau)=\displaystyle{\sum _{k\in \raise-1pt\hbox{$\mbox{\Bbbb Z}$}}}
\exp \left (
\pi i \tau k^2 +2\pi i
(z+\frac{1}{2})k\right ).
\end{array}
\label{theta}
\end{equation}
They also can be represented as infinite products.
The infinite product representation for the $\theta_1(z|\tau)$
reads:
\begin{equation}
\label{infprod}
\theta_1(z|\tau)=i\,\mbox{exp}\, \Bigl (
\frac{i\pi \tau}{4}-i\pi z\Bigr )
\prod_{k=1}^{\infty}
\Bigl ( 1-e^{2\pi i k\tau }\Bigr )
\Bigl ( 1-e^{2\pi i ((k-1)\tau +z)}\Bigr )
\Bigl ( 1-e^{2\pi i (k\tau -z)}\Bigr ).
\end{equation}
Throughout the paper we write
$\theta _a(x|\tau)=\theta _a(x)$,
$\theta (z|\frac{\tau}{2})=\bar \theta (z)$.
The transformation properties for shifts by the periods are:
\begin{equation}
\label{periods}
\theta_a (x\pm 1)=(-1)^{\delta _{a,1}+\delta _{a,2}}
\theta_a (x)\,,
\;\;\;\;\;
\theta_a (x\pm \tau )=(-1)^{\delta _{a,1}+\delta _{a,4}}
e^{-\pi i \tau \mp 2\pi i x}
\theta_a (x)\,.
\end{equation}
Under the modular transformation $\tau \to -1/\tau$
the $\theta$-functions behave as follows:
\begin{equation}
\label{mod}
\begin{array}{l}
\theta_{1}(z|\tau )=i \,\sqrt{i/\tau} \,
e^{-\pi i z^2/\tau }\theta_{1}(z/\tau |-1/\tau )\,,
\\ \\
\theta_{2}(z|\tau )= \sqrt{i/\tau} \,
e^{-\pi i z^2/\tau }\theta_{4}(z/\tau |-1/\tau )\,,
\\ \\
\theta_{3}(z|\tau )= \sqrt{i/\tau}\,
e^{-\pi i z^2/\tau }\theta_{3}(z/\tau |-1/\tau )\,,
\\ \\
\theta_{4}(z|\tau )= \sqrt{i/\tau} \,
e^{-\pi i z^2/\tau }\theta_{2}(z/\tau |-1/\tau )\,.
\end{array}
\end{equation}
The identities often used in the computations are
\begin{equation}
\begin{array}{l}
\bar \theta _4 (x)\bar \theta _3(y)+\bar
\theta _4 (y)\bar \theta _3(x)=
2\theta _4 (x+y)\theta_4 (x-y),
\\ \\
\bar \theta _4 (x)\bar \theta _3(y)-
\bar \theta _4 (y)\bar \theta _3(x)=
2\theta _1 (x+y)\theta_1 (x-y),
\\ \\
\bar \theta _3 (x)\bar \theta _3(y)+
\bar \theta _4 (y)\bar \theta _4(x)=
2\theta _3 (x+y)\theta_3 (x-y),
\\ \\
\bar \theta _3 (x)\bar \theta _3(y)-
\bar \theta _4 (y)\bar \theta _4(x)=
2\theta _2 (x+y)\theta_2 (x-y),
\end{array}
\label{theta34}
\end{equation}
\begin{equation}\label{Fay}
\begin{array}{ll}
& \theta_1(z-a-d)\theta_1(z-b-c)
\theta_1(a-d)\theta_1(c-b)
\\ &\\
+&\theta_1(z-b-d))\theta_1(z-a-c)\theta_1(b-d)
\theta_1(a-c)
\\ &\\
=&\theta_1(z-c-d)\theta_1(z-a-b)\theta_1(a-b)
\theta_1(c-d).
\end{array}
\end{equation}
By $\Theta_n$ we denote the space of $\theta$-functions
of order $n$, i.e., entire functions
$F(x)$, $x\in \mbox{\Bbb C}$, such that
\begin{equation}
F(x+1)=F(x)\,,
\;\;\;\;\;\;
F(x+\tau)=(-1)^n e^{-\pi i n\tau -2\pi i nx}F(x)\,.
\label{8}
\end{equation}
It is easy to see that $\mbox{dim} \,\Theta_n =n$.
Let $F(x)\in \Theta_n$, then $F(x)$ has a multiplicative
representation of the form
$F(x)=c\prod _{i=1}^{n}\theta_1(x-x_i)$,
$\sum _{i=1}^{n}x_i =0$,
where $c$ is a constant. Imposing, in addition to (\ref{8}),
the condition $F(-x)=F(x)$, we define the space
$\Theta_{n}^{+}\subset \Theta_{n}$ of {\it even}
$\theta$-functions of order $n$, which
plays the important role in representations
of the Sklyanin algebra. If $n$ is an even number,
then $\mbox{dim}\, \Theta_{n}^{+} =\frac{1}{2}n +1$.
\subsubsection*{Elliptic gamma-function}
Here we collect the main formulas on the elliptic
gamma-function \cite{R3,FV3}. We use the (slightly modified)
notation of \cite{FV3}.
The elliptic gamma-function
is defined by the double-infinite product
\begin{equation}
\label{gamma}
\Gamma(z|\tau, \tau ')=\prod_{k,k'=0}^{\infty}
\frac{1-e^{2\pi i ((k+1)\tau +(k'+1)\tau ' -z)}}
{1-e^{2\pi i (k\tau +k'\tau ' +z)}}.
\end{equation}
A sufficient condition for the product to be convergent is
$\mbox{Im}\,\tau >0$, $\mbox{Im}\,\tau' >0$.
We need the following properties of the elliptic gamma-function:
\begin{equation}
\label{gamma1}
\Gamma (z+1|\tau , \tau ')=
\Gamma (z|\tau , \tau ')\,,
\end{equation}
\begin{equation}
\label{gamma2}
\Gamma (z+\tau |\tau , \tau ')=
-ie^{-\frac{\pi i \tau'}{6}}\eta_{D}^{-1}(\tau')e^{\pi i z}
\theta_{1}(z|\tau')
\Gamma (z|\tau , \tau ')\,,
\end{equation}
\begin{equation}
\label{gamma3}
\Gamma (z+\tau' |\tau , \tau ')=
-ie^{-\frac{\pi i \tau}{6}}\eta_{D}^{-1}(\tau)e^{\pi i z}
\theta_{1}(z|\tau)
\Gamma (z|\tau , \tau ')\,,
\end{equation}
where
$$
\eta_{D}(\tau)=e^{\frac{\pi i \tau}{12}}
\prod_{k=1}^{\infty}\Bigl ( 1-e^{2\pi i k\tau}\Bigr )
$$
is the Dedekind function. Another useful property is
\begin{equation}
\label{gamma5}
\Gamma (z|\tau , \tau ')
\Gamma (\tau ' -z|\tau , \tau ')=
\frac{ie^{\pi i \tau '/6}\eta_{D}(\tau ')}{e^{\pi i z}
\theta_{1}(z|\tau ')}\,.
\end{equation}
Note also that $\Gamma (z|\tau , \tau ')
\Gamma (\tau +\tau ' -z|\tau , \tau ')=1$.
Under the modular transformation $\tau \to -1/\tau$ the
elliptic gamma-function behaves as follows \cite{FV3}:
\begin{equation}
\label{modg}
\Gamma (z|\tau , \tau ')=e^{i \pi P(z)} \,
\frac{\Gamma (z/\tau \,|-1/\tau , \tau '/\tau )}
{\Gamma ((z-\tau )/\tau '\,|-\tau /\tau ', -1/\tau ')}\,,
\end{equation}
where
\begin{equation}
\label{polP}
\begin{array}{lll}
P(z)&=& \displaystyle{-\frac{1}{3\tau \tau '}\,z^3
+\frac{\tau +\tau ' -1}{2\tau \tau '}\,z^2
-\frac{\tau^2 +\tau '^2 +3\tau \tau ' -3\tau -3\tau ' +1}
{6\tau \tau '}\, z\,-} \\ &&\\
&-&\displaystyle{\frac{(\tau +\tau ' -1)(\tau +\tau ' -\tau \tau ')}
{12 \tau \tau '}}\,.
\end{array}
\end{equation}
Let us list the most frequently used
formulas for
${\sf \Gamma} (z) \equiv \Gamma (z|\tau , 2\eta )$.
Using (\ref{gamma3}) several times, we obtain:
\begin{equation}
\label{gam6}
\frac{{\sf \Gamma} (x+2k\eta )}{{\sf \Gamma} (x)}=
e^{\pi i \eta k^2}R^{-k}e^{\pi i kx}
\prod_{j=0}^{k-1}\theta_1 (x+2j\eta )\,,
\end{equation}
\begin{equation}
\label{gam7}
\frac{{\sf \Gamma} (x-2k\eta )}{{\sf \Gamma} (x)}=
(-1)^k e^{\pi i \eta k^2}R^{k}e^{-\pi i kx}
\prod_{j=0}^{k-1}
\Bigl ( \theta_1 (-x+2\eta +2j\eta )\Bigr )^{-1}\,,
\end{equation}
where $R=ie^{\pi i (\eta +\tau /6)}\eta_{D}(\tau )$.
In particular, ratios of such functions are expressed
through the elliptic Pochhammer symbols as
\begin{equation}\label{gam6a}
\begin{array}{l}
\displaystyle{
\frac{{\sf \Gamma} (2\alpha \eta + 2 k\eta )}{{\sf \Gamma} (2\beta \eta + 2 k\eta )}
=e^{2\pi i (\alpha -\beta )k\eta}\,
\frac{{\sf \Gamma} (2\alpha \eta )}{{\sf \Gamma} (2\beta \eta )}\,
\frac{[\alpha ]_k}{[\beta ]_k}},
\\ \\
\displaystyle{
\frac{{\sf \Gamma} (2\alpha \eta - 2 k\eta )}{{\sf \Gamma} (2\beta \eta - 2 k\eta )}
=e^{-2\pi i (\alpha -\beta )k\eta}\,
\frac{{\sf \Gamma} (2\alpha \eta )}{{\sf \Gamma} (2\beta \eta )}\,
\frac{[1-\beta ]_k}{[1-\alpha ]_k}}\,.
\end{array}
\end{equation}
As is seen from (\ref{gamma}), the function
$\Gamma (z|\tau , 2\eta )$ has zeros at the points
$z= 2(k+1)\eta + (m+1)\tau +n$,
and simple poles at the points
$z=-2k\eta -m\tau +n$,
where $k,m$ run over non-negative integers
and $n$ over all integers.
The residues of the elliptic gamma-function at the poles
at $z=-2k\eta$, $k=0,1,2, \ldots$ are:
\begin{equation}
\label{residue}
\mbox{res}\,\Bigl |_{z=-2k\eta}
{\sf \Gamma} (z)=(-1)^k
e^{\pi i \eta k^2}R^{k}r_0
\prod_{j=1}^{k}
\Bigl ( \theta_1 (2j\eta )\Bigr )^{-1}\,,
\end{equation}
where
$$
r_0=
\mbox{res}\,\Bigl |_{z=0}
{\sf \Gamma} (z)=
-\frac{e^{\pi i(\tau +2\eta )/12}}{2\pi i
\eta_D(\tau)\eta_D(2\eta )}\,.
$$
\subsubsection*{Elliptic hypergeometric series}
Here we follow \cite{FrTur}.
We define the elliptic Pochhammer symbol (the
shifted elliptic factorial) by
\begin{equation}
\label{Pohg} [x]_k \equiv [x] [x+1] \ldots [x+k-1]\,,
\end{equation}
where $[x]=\theta_1(2x\eta )$ (cf. (\ref{binom})).
By definition, the elliptic hypergeometric series is
\begin{equation}
\label{mh1}
{}_{r+1}\omega_{r}(\alpha_1 ; \alpha_4 , \alpha_5 , \ldots ,
\alpha_{r+1};z|2\eta , \tau )=
\! \sum_{k=0}^{\infty}z^k
\frac{[\alpha_1 +2k][\alpha_1 ]_k}{[\alpha_1][k]!}
\prod_{m=1}^{r-2}
\frac{[\alpha_{m+3}]_k}{[\alpha_1 \!-\!\alpha_{m+3}\!+\!1]_k}\,.
\end{equation}
This is an elliptic analog of the very-well-poised basic
hypergeometric series \cite{GR}. The series is said to be
{\it balanced} if $z=1$ and
\begin{equation}
\label{mh2}
r-5+(r-3)\alpha_1 =2\sum_{m=1}^{r-2}\alpha_{m+3}\,.
\end{equation}
For a series $\sum_{k\geq 0}c_k$
of the form (\ref{mh1}), the balancing condition
(\ref{mh2}) means that the ratio
$c_{k+1}/c_k$ of the coefficients
is an elliptic function of $k$.
For balanced series (\ref{mh1}), we drop the
argument $z=1$ and the parameters $\eta , \tau$ writing
it simply as
${}_{r+1}\omega_{r}(\alpha_1 ; \alpha_4 , \ldots ,
\alpha_{r+1})$. For instance,
\begin{equation}
\label{mh3}
{}_{8}\omega_{7}(\alpha_1 ; \alpha_4 , \alpha_5 , \alpha_6 ,
\alpha_7, \alpha_{8})=
\! \sum_{k=0}^{\infty}
\frac{[\alpha_1 +2k][\alpha_1 ]_k}{[\alpha_1][k]!}
\prod_{m=1}^{5}
\frac{[\alpha_{m+3}]_k}{[\alpha_1 \!-\!\alpha_{m+3}\!+\!1]_k}\,.
\end{equation}
The series is called
{\it terminating} if at least one of the parameters
$\alpha_4 , \ldots , \alpha_{r+1}$ is equal to a negative
integer number. In this case the sum is
finite and there is no problem of convergence.
If, say $\alpha_{r+1}=-n$, then the series terminates at $k=n$.
The terminating balanced
series were shown \cite{FrTur} to possess nice
modular properties.
That is why they were called {\it modular hypergeometric series}.
The modular hypergeometric series obey a number of
impressive identities. One of them is the elliptic analog
of the Jackson summation formula:
\begin{equation}\label{Jackson}
\!\! \!\!{}_{8}\omega_{7}(\alpha _1; \alpha_4, \ldots ,
\alpha_7, -n)=\! \frac{[\alpha_1 \! +\! 1]_n
[\alpha_1 \! -\! \alpha_4 \! -\! \alpha_5 \! +\! 1]_n
[\alpha_1 \! -\! \alpha_4 \! -\! \alpha_6 \! +\! 1]_n
[\alpha_1 \! -\! \alpha_5 \!
-\! \alpha_6 \! +\! 1]_n}{[\alpha_1 \! -\! \alpha_4 \! +\! 1]_n
[\alpha_1 \! -\! \alpha_5 \! +\! 1]_n
[\alpha_1 \! -\! \alpha_6 \! +\! 1]_n
[\alpha_1 \! -\! \alpha_4 \! -\! \alpha_5
\! -\! \alpha_6 \! +\! 1]_n}
\end{equation}
which is valid provided that the balancing condition
$2\alpha_1 +1 = \alpha_4 + \alpha_5 + \alpha_6 + \alpha_7 -n$ is satisfied
(the Frenkel-Turaev summation formula \cite{FrTur}).
A remark on the notation is in order. In the modern notation
\cite{Spir}, what we call ${}_{r+1}\omega _{r}(\alpha_1 ;
\alpha_4 , \ldots , \alpha_{r+1}|\eta , \tau )$
(following \cite{FrTur}), would be
${}_{r+3}V_{r+1}(a_1 ;
a_6, \ldots , a_{r+3}|q^2, p)$ with
$q=e^{2\pi i\eta}$, $p=e^{2\pi i\tau}$,
$a_j =e^{4\pi i\eta\alpha_{j-2}}$. In particular,
our ${}_{4}\omega _{3}$ would be ${}_{6}V_{5}$.
We understand that the modern notation is better justified
by the meaning of the elliptic
very-well-poisedness condition than the old one
and is really convenient in many cases.
However, we decided to use the old Frenkel-Turaev
notation for the reason that the additive parameters
$\alpha_j$ are more convenient for us than
their exponentiated counterparts.
We think that it is simpler than
to introduce a version of
${}_{r+1}V_{r}$ with additive parameters.
\section*{Appendix B}
\defB\arabic{equation}{B\arabic{equation}}
\setcounter{equation}{0}
In this appendix we give some details of calculations
which involve modular hypergeometric series.
\subsubsection*{The normalization of the $W_{\zeta}^{z}$-kernel}
Let us consider convolution of the kernels
$W_{\zeta}^{z}(\lambda )$ and $W_{z'}^{\zeta}(-\lambda )$
given by equation (\ref{intw8}):
$$
\begin{array}{ll}
&\displaystyle{\int d\zeta W_{\zeta}^{z}(\lambda )
W_{z'}^{\zeta}(-\lambda )}
\\ &\\
=& \displaystyle{\int d\zeta \frac{
c(\lambda )c(-\lambda )\theta_1 (2\zeta )
\theta_1 (2z')}{W^{\zeta ,z}(\lambda +\eta )
W^{z', \zeta}(-\lambda +\eta )}
\sum_{k,k'\geq 0}\delta (z-\zeta -\lambda +2k\eta )
\delta (\zeta -z' + \lambda +2k'\eta )}
\\ &\\
=& \displaystyle{\sum_{k,k'\geq 0}
\frac{c(\lambda )c(-\lambda )\theta_1 (2z-2\lambda +4k\eta )
\theta_1 (2z')}{W^{z-\lambda +2k\eta ,z}(\lambda +\eta )
W^{z', z-\lambda +2k\eta}(-\lambda +\eta )}\,
\delta (z -z' + \lambda +2(k+k')\eta )}
\\ &\\
=& \displaystyle{c(\lambda )c(-\lambda )\sum_{n\geq 0}
\left (\sum_{k=0}^{n} \frac{\theta_1 (2z-2\lambda +4k\eta )
\theta_1 (2z+4n\eta )}{W^{z-\lambda +2k\eta ,z}(\lambda +\eta )
W^{z+2n\eta , z-\lambda +2k\eta}(-\lambda +\eta )}\right )
\delta (z -z' +2n\eta )}\,.
\end{array}
$$
In order to calculate it explicitly, consider the sum
\begin{equation}\label{B1}
S_n(z)=\sum_{k=0}^{n} \frac{\theta_1
(2z-2\lambda +4k\eta )}{W^{z-\lambda +2k\eta ,z}(\lambda +\eta )
W^{z+2n\eta , z-\lambda +2k\eta}(\eta -\lambda)},
\end{equation}
where the $W$-functions are given by (\ref{intw4}):
$$
W^{z-\lambda +2k\eta ,z}(\lambda +\eta )
=e^{-\frac{2\pi i}{\eta}(\lambda +\eta )(z-\lambda +2k\eta )}
\frac{{\sf \Gamma} (2z+2\eta +2k\eta )
{\sf \Gamma} (2\eta +2k\eta )}{{\sf \Gamma} (2z-2\lambda +2k\eta )
{\sf \Gamma} (-2\lambda +2k\eta )},
$$
$$
W^{z+2n\eta ,z-\lambda +2k\eta}(\eta -\lambda)
=e^{\frac{2\pi i}{\eta}(\lambda -\eta )(z+2n\eta )}
\frac{{\sf \Gamma} (2z \! -\! 2\lambda \! +\! 2\eta \! +\! 2n\eta \! +\! 2k\eta )
{\sf \Gamma} (2\eta +2n\eta -2k\eta )}{{\sf \Gamma} (2z +2n\eta +2k\eta )
{\sf \Gamma} (2\lambda +2n\eta -2k\eta )}.
$$
Plugging this into (\ref{B1}) and representing ratios of elliptic
gamma-functions through elliptic Pochhammer symbols
with the help of (\ref{gam6}), (\ref{gam7}), we obtain:
$$
\begin{array}{lll}
S_n (z)&=& \displaystyle{
e^{4\pi i z -\frac{2\pi i}{\eta}\lambda (\lambda +\eta )
-4\pi i (\lambda -\eta )n}}
\\ &&\\
&\times & \displaystyle{\theta_1(2z -2\lambda )\,
\frac{{\sf \Gamma} (-2\lambda ){\sf \Gamma} (2z-2\lambda ) {\sf \Gamma} (2z+2n\eta )
{\sf \Gamma} (2\lambda +2n\eta )}{{\sf \Gamma} (2\eta ) {\sf \Gamma} (2\eta +2n\eta )
{\sf \Gamma} (2z+2\eta )
{\sf \Gamma} (2z-2\lambda +2n\eta +2\eta )}}
\\ &&\\
&\times & \displaystyle{
\sum_{k=0}^{n} \frac{[\frac{z-\lambda}{\eta}
+2k][\frac{z-\lambda}{\eta}]_k}{[\frac{z-\lambda}{\eta}][1]_k}\,
\frac{[-\frac{\lambda}{\eta}]_k \,\,
[\frac{z}{\eta}+n]_k [-n]_k}{[\frac{z}{\eta}+1]_k \,\,
[\frac{z-\lambda}{\eta}+n+1]_k [-\frac{\lambda}{\eta}-n+1]_k}}.
\end{array}
$$
The sum in the last line is the terminating balanced elliptic
hypergeometric series
$${}_{8}\omega_{7}\left (\frac{z-\lambda}{\eta};
-\frac{\lambda}{\eta}, \, \frac{z}{\eta}+n,\,
\frac{z-\lambda +\eta}{2\eta},\, \frac{z-\lambda +\eta}{2\eta}, \,
-n\right )$$
which is equal to
$$
\frac{[\frac{z-\lambda}{\eta}+1]_n \,
[1-n]_n \, [\frac{z+\lambda +\eta}{2\eta}]_n \,
[-\frac{z+\lambda + \eta}{2\eta}-n+1]_n}{[\frac{z}{\eta}+1]_n
[-\frac{\lambda}{\eta}-n+1]_n
[\frac{z-\lambda +\eta}{2\eta}]_n
[-\frac{z-\lambda +\eta}{2\eta}-n+1]_n}
$$
(see (\ref{Jackson})). Because of the factor
$[1-n]_n$ this is zero unless $n=0$. Therefore,
$S_n(z)=0$ if $n\geq 1$ and
$$
S_0(z)= e^{4\pi iz -\frac{2\pi i}{\eta}\lambda (\lambda +\eta )}
\, \frac{{\sf \Gamma} (2\lambda ){\sf \Gamma} (-2\lambda )
{\sf \Gamma} (2z){\sf \Gamma} (2z-2\lambda ) \theta_1 (2z-2\lambda )}{{\sf \Gamma} ^2(2\eta )
{\sf \Gamma} (2z + 2\eta ){\sf \Gamma} (2z-2\lambda +2\eta )}.
$$
We thus have
$$
\int \! d\zeta \, W_{\zeta}^{z}(\lambda )
W_{z'}^{\zeta}(-\lambda )=
c(\lambda )c(-\lambda )\theta_1(2z)S_0(z)\delta (z-z').
$$
Using identities for the elliptic gamma-function the
product $\theta_1(2z)S_0(z)$ can be simplified to
$$
\theta_1(2z)S_0(z)=\rho_{0}^{-1}e^{-2\pi i \lambda ^2/\eta}
{\sf \Gamma} (2\lambda ){\sf \Gamma} (-2\lambda ),
$$
where
\begin{equation}\label{rho0}
\rho_0 = \frac{{\sf \Gamma} (2\eta )}{ie^{\frac{\pi i \tau}{6}}\eta_D (\tau )}=
\frac{e^{\frac{\pi i}{12}(2\eta -3\tau )}}{i\eta_D (2\eta )}.
\end{equation}
So, setting
\begin{equation}\label{B3}
c(\lambda )=
\frac{\rho_0 \, e^{\pi i \lambda ^2/\eta}}{{\sf \Gamma} (-2\lambda )}
\end{equation}
we obtain the relation (\ref{intw12}):
$\int \! d\zeta \, W^{z}_{\zeta}(\lambda )
W^{\zeta}_{z'}(-\lambda )=\delta (z-z')$.
\subsubsection*{The star-triangle relations}
Let us verify the
star-triangle relation (\ref{st1a})
\begin{equation}\label{st1}
W^{z',z}(\mu -\nu )W^{z', z''}(\lambda -\mu )
W^{z}_{z''}(\lambda -\nu )=
\int \! d\zeta W^{z}_{\zeta}(\lambda -\mu )
W^{z',\zeta}(\lambda -\nu )W^{\zeta}_{z''}(\mu -\nu )
\end{equation}
(see Fig. \ref{fig:WWW}). We use formulas (\ref{intw4}),
(\ref{intw8}).
The left hand side is
$$
\begin{array}{ll}
&\displaystyle{c(\lambda -\nu )\theta_1 (2z'')
\frac{W^{z',z}(\mu -\nu )
W^{z', z''}(\lambda -\mu )}{W^{z'', z}(\lambda -\nu +\eta )}
\sum_{n\geq 0}\delta (z-z'' -\lambda +\nu +2n\eta )}
\\ &\\
=&\displaystyle{c(\lambda -\nu )\sum_{n\geq 0}
\theta_1(2z'')
\frac{W^{z',z}(\mu -\nu )
W^{z', z-\lambda +\nu +
2n\eta}(\lambda -\mu )}{W^{z-\lambda +\nu +
2n\eta, z}(\lambda -\nu +\eta )}\,
\delta (z-z'' -\lambda +\nu +2n\eta )}
\\ &\\
= &\displaystyle{c(\lambda -\nu )\sum_{n\geq 0}
C_n (z',z)\delta (z-z'' -\lambda +\nu +2n\eta )},
\end{array}
$$
where $c(\lambda )$ is given by (\ref{B3}) and
$$
\begin{array}{lll}
C_n(z',z)&=&\displaystyle{
e^{-\frac{2\pi i}{\eta}\left [
(\lambda -\nu )z' -(\lambda -\nu +\eta )z+
(\lambda -\nu )(\lambda -\nu +\eta )\right ]}\,
\theta_1(2z-2\lambda +2\nu +4n\eta )}
\\ &&\\
&\times &\displaystyle{
\frac{{\sf \Gamma} (2\nu -2\lambda ){\sf \Gamma} (2z-2\lambda +2\nu )
{\sf \Gamma} (z \! +\! z' \! +\! \mu \! -\! \nu \! +\! \eta )
{\sf \Gamma} (z'\! -\! z \! +\! 2\lambda \! -\! \mu \! -\! \nu
\! +\! \eta )}{{\sf \Gamma} (2\eta )\,
{\sf \Gamma} (2z+2\eta )\, {\sf \Gamma} (z+z' -2\lambda +\mu +\nu +\eta )\,
{\sf \Gamma} (z'-z -\mu +\nu +\eta )}}
\\ &&\\
&\times &\displaystyle{
\frac{[\frac{z-\lambda +\nu}{\eta}]_n \,
[\frac{\nu -\lambda}{\eta}]_n \,
[\frac{z+z'+\nu -\mu +\eta}{2\eta}]_n \,
[\frac{z-z'+\nu -\mu +\eta}{2\eta}]_n}{[1]_n
\, [\frac{z}{\eta}+1]_n \,
[\frac{z+z'-2\lambda +\nu +\mu +\eta}{2\eta}]_n \,
[\frac{z-z'-2\lambda +\nu +\mu +\eta}{2\eta}]_n}}.
\end{array}
$$
One can see from this expression that the left hand side of
(\ref{st1}) is the kernel of the difference operator
\begin{equation}\label{st2a}
\begin{array}{ll}
&\displaystyle{e^{\frac{2\pi i}{\eta}(\lambda -\nu )(z-z')
-\frac{\pi i}{\eta}(\lambda -\nu )^2 }
\frac{{\sf \Gamma} (2z\! -\! 2\lambda \! +\!
2\nu \! +\! 2\eta )
{\sf \Gamma} (z \! +\! z' \! +\! \mu \! -\! \nu \! +\! \eta )
{\sf \Gamma} (z'\! -\! z \! +\! 2\lambda \! -\! \mu \! -\! \nu
\! +\! \eta )}{{\sf \Gamma} (2z+2\eta )\,
{\sf \Gamma} (z+z' -2\lambda +\mu +\nu +\eta )\,
{\sf \Gamma} (z'-z -\mu +\nu +\eta )}}
\\ &\\
\times &\displaystyle{
{\scriptstyle {{\bullet}\atop{\bullet}}} {}_{6}\omega_{5}\left (
\frac{z-\lambda +\nu}{\eta}; \, \frac{\nu -\lambda}{\eta}, \,
\frac{z+z'+\nu -\mu +\eta}{2\eta},\,
\frac{z-z'+\nu -\mu +\eta}{2\eta}; \,
e^{2\eta \partial_z} \right ) {\scriptstyle {{\bullet}\atop{\bullet}}} e^{(\nu -\lambda )\partial_z}}.
\end{array}
\end{equation}
Let us turn to the right hand side of (\ref{st1}).
It is
$$
\begin{array}{ll}
&\displaystyle{c(\lambda -\mu ) c(\mu -\nu )
\int \! d\zeta \frac{W^{z',\zeta}(\lambda -\nu )
\theta_1 (2\zeta )\theta_1 (2z'')}{W^{\zeta , z}
(\lambda -\mu +\eta )W^{z'',\zeta}(\mu -\nu +\eta )}}
\\&\\
& \displaystyle{\quad \quad \quad \quad \times
\sum_{k,k'\geq 0}
\delta (z\! -\! \zeta \! -\! \lambda \! +\!
\mu \! +\! 2k\eta )\delta (\zeta \! -\! z'' \! -\! \mu \! +\!
\nu \! +\! 2k'\eta )}
\\&\\
=& \displaystyle{c(\lambda -\mu ) c(\mu -\nu )
\sum_{n\geq 0}B_n (z',z)\,
\delta (z\! -\! z'' \! -\! \lambda \! +\!
\nu \! +\! 2n\eta )},
\end{array}
$$
where
$$
B_n(z',z)=\sum_{k=0}^{n}
\frac{\theta_1(2z\! -\! 2\lambda \! +\! 2\mu \! +4k\eta )
\theta_1(2z\! -\! 2\lambda \! +\! 2\nu \! +4n\eta )\,
W^{z', z-\lambda +\mu +
2k\eta}(\lambda \! -\! \nu )}{W^{z-\lambda +\mu +
2k\eta , z}(\lambda -\mu +\eta )
W^{z-\lambda +\nu +2n\eta , z-\lambda +\mu +
2k\eta}(\mu -\nu +\eta)}.
$$
The next step is to identify this sum with the
terminating elliptic hypergeometric series with a
pre-factor. The latter is essentially a product of ratios
of the ${\sf \Gamma}$-functions. Specifically, we have:
$$
\begin{array}{lll}
B_n(z',z)&=&\displaystyle{
e^{\frac{2\pi i}{\eta}\left [
(\lambda -\nu )(z'-z)+(\lambda -\mu )(\lambda -\mu +\eta )
+(\lambda -\nu )(\mu -\nu +\eta )\right ]
+4\pi i z +4\pi i (\mu -\nu +\eta )n}}
\\ &&\\
&\times & \displaystyle{
\frac{{\sf \Gamma} (2\mu \! -\! 2\lambda ){\sf \Gamma} (2z\! -\! 2\lambda \! +\! 2\mu )
{\sf \Gamma} (z\! +\! z' \! +\! \mu \! -\! \nu \! +\!
\eta ){\sf \Gamma} (z'\! -\! z \! +\!
2\lambda \! -\! \mu \! -\! \nu
\! +\! \eta )}{{\sf \Gamma} (2\eta )\, {\sf \Gamma} (2z +2\eta )
{\sf \Gamma} (z\! +\! z' \! -\! 2\lambda \! +\! \mu \! +\! \nu
\! +\! \eta )
{\sf \Gamma} (z' \! -\! z \! -\! \mu \! +\! \nu \! +\! \eta )}}
\\ &&\\
&\times & \displaystyle{
\frac{{\sf \Gamma} (2z \! -\! 2\lambda \! +\! 2\nu \! +\! 2n\eta )\,
{\sf \Gamma} (2\nu \! -\! 2\mu \! +\! 2n\eta )}{{\sf \Gamma} (2\eta \! +\! 2n\eta )
{\sf \Gamma} (2z \! -\! 2\lambda \! +\! 2\mu \! +\! 2\eta \! +\! 2n\eta )}\,
\theta_1 (2z -2\lambda +2\mu )}
\\ &&\\
&\times & \displaystyle{
{}_{8}\omega_{7} \left (\alpha _1 ; \alpha _4, \ldots ,
\alpha _7, -n\right )},
\end{array}
$$
where the parameters $\alpha_i$ are:
$$
\alpha_1 = \frac{z-\lambda +\mu}{\eta},\quad
\alpha_4 = \frac{\mu-\lambda}{\eta}, \quad
\alpha_5 = \frac{z-\lambda +\nu}{\eta}+n,
$$
$$
\alpha_6 = \frac{z+z' +\mu -\nu +\eta}{2\eta}, \quad
\alpha_7 = \frac{z-z' +\mu -\nu +\eta}{2\eta}, \quad
\alpha_8 = -n.
$$
The series with these parameters is balanced, so one can
apply the Frenkel-Turaev summation formula (\ref{Jackson}).
The result is
$$
\begin{array}{ll}
&{}_{8}\omega_{7} \left (\alpha _1 ; \alpha _4, \ldots ,
\alpha _7, -n\right )
\\ &\\
=& \displaystyle{\frac{[\frac{z-\lambda +\mu}{\eta}+1]_n \,
[\frac{\lambda -\nu}{\eta}+1-n]_n \,
[\frac{z-z'-\mu +\nu +\eta}{2\eta}]_n \,
[-\frac{z+z'-\mu +\nu +\eta}{2\eta}+1-n]_n}{[\frac{z}{\eta}+1]_n \,
[\frac{\mu -\nu}{\eta}+1-n]_n \,
[\frac{z-z'-2\lambda +\mu +\nu +\eta}{2\eta}]_n \,
[-\frac{z+z'-2\lambda +\mu +\nu +\eta}{2\eta}+1-n]_n}}.
\end{array}
$$
Now it is straightforward to calculate
the ratio $C_n(z', z)/B_n(z', z)$.
One can see that all $z,z'$ and $n$ dependent factors cancel
in the ratio and one is left with
$$
\frac{C_n(z', z)}{B_n(z', z)}
=\frac{{\sf \Gamma} (2\eta )}{ie^{\frac{\pi i \tau}{6}}\eta _D (\tau )}
\,
\, \frac{e^{\frac{2\pi i}{\eta}(\lambda -\mu )(\nu -\mu )}
{\sf \Gamma} (2\nu -2\lambda )}{{\sf \Gamma} (2\mu -2\lambda )
{\sf \Gamma} (2\nu -2\mu )}=
\frac{c(\lambda -\mu )c(\mu -\nu )}{c(\lambda -\nu )},
$$
where $c(\lambda )$ is given by (\ref{B3}).
This means that the left and right hand sides of
(\ref{st1}) are indeed equal to each other.
The other star-triangle relation, (\ref{st1b}),
is proved in a similar way. We note that its both sides
are kernels of the difference operator
\begin{equation}\label{st2b}
\begin{array}{ll}
&\displaystyle{e^{\frac{\pi i}{\eta}(\lambda -\nu )
(2\mu -\lambda -\nu )}
\frac{{\sf \Gamma} (2z\! -\! 2\lambda \! +\!
2\nu \! +\! 2\eta )
{\sf \Gamma} (z \! +\! z' \! +\! \lambda \! -\! \mu \! +\! \eta )
{\sf \Gamma} (z\! -\! z' \! +\! \lambda \! -\! \mu
\! +\! \eta )}{{\sf \Gamma} (2z+2\eta )\,
{\sf \Gamma} (z\! +\! z' \! +\! 2\nu \! -\!
\lambda \! -\! \mu \! +\! \eta )\,
{\sf \Gamma} (z\! -\! z' \! +\! 2\nu \! -\!
\lambda \! -\! \mu \! +\! \eta )}}
\\ &\\
\times &\displaystyle{
{\scriptstyle {{\bullet}\atop{\bullet}}} {}_{6}\omega_{5}\left (
\frac{z-\lambda +\nu}{\eta}; \, \frac{\nu -\lambda}{\eta}, \,
\frac{z+z'-\lambda +\mu +\eta}{2\eta},\,
\frac{z-z'-\lambda +\mu +\eta}{2\eta}; \,
e^{2\eta \partial_z} \right ) {\scriptstyle {{\bullet}\atop{\bullet}}} e^{(\nu -\lambda )\partial_z}}.
\end{array}
\end{equation}
|
1,108,101,564,579 | arxiv |
\section{Introduction}
Redundant robots---which have more \gls{DoF} than required for a task---have been widely studied and deployed due to their intrinsic flexibility.
The higher dimensionality of the joint configuration space \gls{wrt} the task space makes these systems more adaptable as multiple solutions can be found.
\input{figures/fig-intro.tex}
However, this flexibility introduces a higher complexity for both planning and control that rapidly increases with the system and task dimensionality.
For example, computing the joint configuration from a task-space pose, i.e. \gls{IK}, becomes increasingly more challenging with the increase of the redundancy dimension \cite{siciliano,d2001learning}.
This problem is also encountered when dealing with the inverse dynamics problem, which is used to derive the control laws used in interaction control \cite{siciliano,xin2020,keppler2018}.
To address these inverse problems in rigid systems, multiple optimization frameworks and approaches to deal with challenges arising from numerical conditioning have been developed.
Currently, inverse problems for soft robots\footnote{Here, we use soft robots to refer both to systems made from non-rigid materials as well as those with compliant control, e.g., collaborative robots.} and optimization of the task-space dynamics are still open problems \cite{bruder2019}.
A recent approach to robustness for achieving task-space compliance behaviors include systems for increasing the robustness of projections through software by modulating/adapting the references \cite{xin2020}. This robustness can also be achieved by exploiting more complex hardware design that embeds variable mechanical compliance directly into the robot structure and its actuation \cite{keppler2018,braun2013robots}. As an example, \textit{Keppler et al.} have proposed a control architecture that enables to retain both robustness and accurate tracking \cite{keppler2018} while related work has algorithmically optimised spatiotemporal modulation of impedance to achieve tasks more efficientl \cite{nakanishi2011stiffness}. These approaches takes advantages of hardware equipped with Variable Stiffness Actuators (VSAs) --which allows better dynamic performances from the hardware, but they also increase the complexity and cost of motion planning with these system besides the need for very accurate modelling of the VSA structures.
\input{figures/fig-conceptual-overview.tex}
Inverse kinematics solutions and task-space dynamics projections are required for controlling redundant robots and they share similar challenges, as analyzed in depth in \cite{dietrich2015,khatib1993,Vijayakumar-RSS-19}.
In summary, both problems rely on the inversion of the Jacobian matrix, which is non-square in redundant manipulator due to the different task and joint space dimensions \cite{siciliano}.
The pseudo-inverse is a transformation that solves such a problem.
It separates the information regarding robot states into two orthogonal sub-spaces (task-space and null-space), which are not expected to exchange information (i.e., energy).
Therefore, retaining the orthogonality between these two sub-spaces is paramount for the algorithms' stability \cite{dietrich2015,khatib1993,Vijayakumar-RSS-19}.
Maintaining this orthogonality depends on both the robot kinematics and the task -- which can be quite difficult to achieve and maintain, especially during highly variable situations, such as sudden changes in contacts and dynamic interactions (\autoref{fig:intro}).
As one example, the dynamically consistent inverse obtains orthogonality via the minimization of the kinetic energy projected by the null-space into the task-space \cite{Vijayakumar-RSS-19, khatib1987unified}.
Passive controllers have been proposed to theoretically guarantee interaction stability under uncertain interaction conditions, using for instance virtual tanks as energy storage (i.e., path integral) for the non-conservative energy of the controller.
However, their passive behavior trades-off tracking performances to retain safety of interaction, making this framework difficult to deploy in highly variable environments \cite{Dietrich2016}. The result presented in their manuscript focuses on verifying passivity and safety of interaction and does not provide a clear quantitative analysis of task-space tracking performance. However, based on the figure of the Cartesian tracking error reported for the simulation results for a 4-\gls{DoF} planar manipulator, it seems that we can expect about \SI{1}{\centi\meter} residual pose error in the best case scenario (i.e., there is energy in tank).
Moreover, virtual tank impedance controllers are only passive if there is energy left in their virtual tanks.
Due to passivity constraints, the tanks' energy can be only charged from external energy sources \cite{Dietrich2016}.
Realising passive control is made even more difficult when dealing with null-space and task-space controllers.
In fact, as they are orthogonal to each other, tracking the total energy exchanged by the manipulator is challenging \cite{babarahmati2019}.
Another challenge to stability of virtual tank controllers is to maintain the orthogonality between null-space and task-spaces during highly variable tasks.
Higher non-linearity in the dynamics reduces the accuracy in the computation of the orthogonal projection that, consequentially, generates unaccounted energy transfer between the two sub-spaces \cite{Vijayakumar-RSS-19,babarahmati2019}.
This work investigates the possibility of using superimposition of passive task-space controllers to drive redundant manipulators rather than relying on null-space controllers, cf. \autoref{fig:conceptual_overview}.
Conceptually, this follows the idea to generate task-space wrenches at multiple links and map them back to joint-level torques.
To do so, the proposed solution will not need to rely on any mathematical projections (and implicitly, matrix inversions) required by the null-space projections.
Virtual mechanical constraints are instead generated using a superimposition of task-space controllers to control task-space and the redundant degrees of freedom of the manipulators.
However, implementing such a solution will require a controller framework that is intrinsically stable.
The recently proposed \gls{FIC} \cite{babarahmati2019} is a passive controller meeting this requirement.
It relies upon a non-linear stiffness behavior in the task-space to track the energy exchanged between the robot and the environment, and treats the unexpected energy flow from the null-space as an external perturbation.
The controller uses the concept of fractal impedance for the implementation of a passive controller that can provide good performances in both trajectory and force tracking.
Thereby, it detaches the robot stability from the postural optimization, which are currently bounded for interaction controllers relying on \gls{QP} optimization \cite{xin2020}. Furthermore, our proposed method is independent of any specific type of actuation and thus can be used in any torque/force controlled robot.
In summary, our contributions are:
\begin{enumerate}
\item \emph{Superimposition of Passive Task-Space Controllers} to preserve the primary task by sacrificing a secondary tasks through exploitation of the mechanical redundancy. The priority of the controller is determined by the maximum force exertable by the controller, as will be explained in \autoref{sec:method}. As the framework only relies on the forward computation of kinematics and Jacobian, it is numerically stable and computationally inexpensive. Further, it can be used with uncertain and imprecise dynamics models as it only relies on the kinematics model (\ref{sec:StackOfFIC}).
\item Proposal of a new force profile for a Fractal Attractor which enables a smooth transition between convergence and divergence phases (\ref{sec:SigFractImp}).
\item Validation of our approach both in simulation to test contact interaction with unknown obstacles and in real hardware experiments (\ref{sec:experiments}) to evaluate reference tracking performance for fast reference motion (\ref{sec:results}).
As all open parameters have a physically tractable meaning and the controller is intrinsically stable, online tuning can safely be performed.
\end{enumerate}
We intend to open source our implementation for simulation and hardware experiments with the publication of this manuscript.
\section{Method} \label{sec:method}
The null-space of a redundant manipulator is a set of joint-space configurations having the same end-effector pose. Therefore, null-space optimization frameworks identify the optimal joint-space configuration for a given task. Stack of Task optimization methods are iterative algorithms applying Null-Space optimization to a hierarchy of tasks. Their main limitation is that null-space projections are inserted in the control loop, rendering the controllers susceptible to numerical instability connected with the null-space projections.
The proposed method, shown in \autoref{fig:conceptual_overview}, aims to remove null-space projections from the control loop. Null-space projections are used to account for the external interaction in the whole-body control optimization problem \cite{xin2020}. This type of formulation requires not only to make \textit{a priori} assumptions on the environmental interaction, but it also renders the controller stability dependent on their accuracy. Thus, the controller stability is highly susceptible to erroneous assumptions, which lead only to sub-optimal behaviour in the best case scenario \cite{xin2020}.
The proposed method unravels the co-dependency between stability and assumptions made on the external environment by using a superimposition of task-space controllers that generates virtual force field (i.e., soft mechanical constraints) that pull the robot towards the desired configuration. This is a different approach to handling redundancy compared to the null-space approach. The superimposition of the controller will bias the robot to move towards a certain preferred posture, without guaranteeing that this particular configuration will be reached. In fact, the controller will continuously maintain a mechanical equilibrium between the virtual forces and the environmental interaction without requiring any assumption on the environmental interaction. This implies that the controller is robust to unknown environmental interaction, but is not guaranteed to be in a global optimum. However, it is likely to settle in the closest minimum in the system energetic manifold (i.e., the closest state with mechanical equilibrium).
In synthesis, the relative strength of these virtual constraints will determine the trade-off between the tasks assigned to the controllers and, consequentially, the order in which the task accuracy will be sacrificed. While this method can be applied with any type of task-space controllers, using passive controllers guarantees stability by independently verifying that all the superimposed controllers are stable. Among the different passive controllers, we have chosen the Fractal Impedance Controller due to its explicit formulation of the tasks in terms of virtual mechanical constraints (i.e., desired force/displacement behavior), enabling direct control of the controllers' trade-off policies. For the scope of this paper, an optimization-based inverse kinematics algorithm was used to obtain the reference configuration (\emph{postural optimization}). Forward kinematics is subsequently used to extract the task-space references for the individual superimposed controllers. In place of this postural optimization, more comprehensive planners and frameworks could be used to provide and update the reference configurations.
\subsection{Inverse Problem and Kineto-Static Duality}
\label{sec:IPM}
The generalized inverse of $A \in \mathbb{R}^{n \times m}$ is defined as any matrix $G\in \mathbb{R}^{m \times n}$ that satisfies the following equations:
\begin{equation}
\label{Ginverse}
\begin{cases}
\vec{a}=G \vec{b}+ (I_{n}-GA)\vec{a}_\epsilon=G \vec{b}+ P \vec{a}_\epsilon\\
AGA-A=0
\end{cases}
\end{equation}
where $\vec{a} \in \mathbb{R}^{n}$, $\vec{b} \in \mathbb{R}^{m}$, $\vec{a}_\epsilon \in \mathbb{R}^{n}$ and $I_n \in \mathbb{R}^{n \times n}$ is the identity matrix. $P$ is a projection matrix that projects a generic vector $a_\epsilon$ into the null-space of $A$, $\mathcal{N}(A)$.
Redundant robots are more flexible than non-redundant systems, however, they do not have a bijective transformation between generalized coordinates and task-space.
Thus, control algorithms rely on numerical optimization to solve the inverse problem and identify viable strategies. This is task dependent and degenerates when $A$ drops rank (i.e., $det(A)=0$) \cite{Vijayakumar-RSS-19,xin2020}.
Specifically, the rank of the inverse projection matrix drops if the robot is in a singular configuration or the task constraints are violated (e.g., unexpected sudden loss of contact) \cite{Vijayakumar-RSS-19}.
The idea of taking advantage of the kineto-static duality to address the inverse problem has been introduced with the concept of Port-Hamiltonian control in \cite{hogan1985impedanceP1,hogan1985impedanceP2}. In fact, the kinematic joint-space information can be used to derive task-space behavior and task-space force interaction can be used to relate back to joint-space torques:
\begin{equation}
\label{kinetostatic}
\begin{cases}
\vec{\nu}=J \dot{\vec{q}} \\
\vec{\tau}=J^\text{T} \vec{h}
\end{cases}
\end{equation}
where $J\in \mathbb{R}^{n \times m}$ is the geometric Jacobian matrix, $\vec{\nu} \in \mathbb{R}^n$ is the end-effector twist, $\vec{\dot{q}} \in \mathbb{R}^m$ is the joint velocities' vector and $\vec{h} \in \mathbb{R}^n$ is the end-effector wrench, $\vec{\tau} \in \mathbb{R}^m$ is the joint torques' vector.
\subsection{Fractal Impedance Controller} \label{sec:FractImp}
\input{figures/fig-fractal-impedance.tex}
The FIC controls the robot as a non-linear mass-spring system, and generates the attractor in \autoref{fig:2b} around the desired state. The equivalent mechanical system equation is:
\begin{equation*}
\Lambda_c (\vec{q}) \vec{\Ddot{x}} + n(q,\dot{q}) + K(\vec{\tilde{x}}) \vec{\tilde{x}}=F_{Ext}
\end{equation*}
\noindent where $\Lambda_c (\vec{q})$ is the projection of the task-space inertia matrix at the end-effector, $n(q,\dot{q})$ the non-linear robot dynamics and $F_{Ext}$ is the external force.
The state-dependent stiffness gain $K(\vec{\tilde{x}})$ is derived from the desired end-effector interaction properties (i.e., force/displacement), which can be regulated online without affecting stability \cite{babarahmati2019}. For completeness, we provide a proof of stability in \ref{sec:stabilityAnalysis}.
The attractor is implemented using a switching behavior that introduces an additional nonlinear spring which triggers when the system starts converging (i.e., zero crossing of $\vec{\dot{x}}$).
The updated impedance conserves the energy accumulated in the controller while diverging and redistributes the energy altering the trajectory during the convergence, as shown in \autoref{fig:fractal_attractor}.
Therefore, the stability of the controller is guaranteed by the fractal attractor (\autoref{fig:fractal_attractor}). This determines the passivity of the controller and the online adaptability; it is independent of the chosen impedance.
For each \gls{DoF} in the task-space, the \gls{FIC} is given in \autoref{alg:FIC}. The control torques ($\vec{\tau}_{\text{ctr}}$) can be calculated from $\vec{h}_e \in \mathbb{R}^6$ using \eqref{kinetostatic}. Differently from the \gls{FIC} control scheme introduced in \cite{babarahmati2019} on a sharp force/torque saturation, this manuscript introduces a more flexible force profile. The new force profile allows to independently tune the linear, non-linear and saturation behaviors of the controller wrench, making it easier to tune the controller for different tasks.
\begin{algorithm}
\SetAlgoLined
\SetKwData{Left}{left}
\SetKwData{Up}{up}
\SetKwFunction{FindCompress}{FindCompress}
\SetKwInOut{Input}{input}
\SetKwInOut{Output}{output}
\Input{Convergence$/$Divergence, $\tilde{x}$, $\tilde{x}_{\text{max}}$}
\Output{$h_{\text{e}}$}
\eIf{diverging from ${x_{\text{d}}}$}{
$h_{\text{e}}=f(\tilde{x})= K(\tilde{x})\tilde{x}$\\
}{
$h_{\text{e}} = \frac{4 E(\tilde{x}_{\text{max}})}{\tilde{x}_{\text{max}}^2} (0.5\tilde{x}_{\text{max}} -\tilde{x})$\\
}
\renewcommand{\nl}{\let\nl\oldnl} where:\\
\renewcommand{\nl}{\let\nl\oldnl} ${x_{\text{d}}}$ is the desired position \\
\renewcommand{\nl}{\let\nl\oldnl} $h_{\text{e}}$ is the desired force at the end-effector \\
\renewcommand{\nl}{\let\nl\oldnl} $\tilde{x}$ is the pose error \\
\renewcommand{\nl}{\let\nl\oldnl} $K(\tilde{x})$ is the nonlinear stiffness\\
\renewcommand{\nl}{\let\nl\oldnl} $x_\text{e}$ is the end-effector position\\
\renewcommand{\nl}{\let\nl\oldnl} $\tilde{x}=x_\text{d}-x_\text{e}$ is the position error \\
\renewcommand{\nl}{\let\nl\oldnl} $\tilde{x}_{\text{max}}$ is the maximum displacement reached at the end of the divergence phase \\
\renewcommand{\nl}{\let\nl\oldnl} $E$ is the energy associated with the divergence profile of the impedance controller \\
\caption{Mono-dimensional \gls{FIC}}
\label{alg:FIC}
\end{algorithm}
\subsection{Sigmoidal Force Profile for Fractal Impedance} \label{sec:SigFractImp}
The \gls{FIC} relies on a stiffness profile.
The profile proposed in \cite{babarahmati2019} results in fast changes in stiffness, and only allows limited task-dependent tuning of the profile.
Therefore, we propose a sigmoidal force profile for an easier definition of the stiffness profile, allowing to better adapt the robot impedance behavior to the different task.
Similarly to the profile proposed by \cite{babarahmati2019}, the sigmoidal profile is fully determined based on the maximum force ($F_{\text{Max}}$) to be exerted at a chosen position error ($|\tilde{x}|=\tilde{x}_{\text{b}}$).
Here, the position error is defined as the difference between the desired end-effector pose and the current pose ($\tilde{x}=x_d-x$).
We introduce an additional displacement parameter ($|\tilde{x}|=\tilde{x}_{0}$) which describes the minimum displacement to activate the nonlinear impedance, as shown in \autoref{fig:2c}.
The proposed force profile thus becomes:
\begin{equation}
\label{ForceProf}
F_{\text{K}}=\left\{
\begin{array}{ll}
K_0 \tilde{x}, & |\tilde{x}|<\tilde{x}_{0}\\\\
\text{sgn} (\tilde{x}) (\Delta F (1-e^{-\frac{|\tilde{x}| - \tilde{x}_0}{b}})+& \\
+ K_0 \tilde{x}_0), & \tilde{x}_{0}\le |\tilde{x}|<\tilde{x}_{b}\\\\
\text{sgn}(\tilde{x}) F_{\text{Max}}, & \text{Otherwise}
\end{array}\right.
\end{equation}
where $b=(\tilde{x}_\text{b}-\tilde{x}_\text{0})/S$ is the characteristic length, $S$ determines the shape of the sigmoid curve and $\Delta F =(F_{\text{Max}} - K_0.\tilde{x}_0)$. In this work, we use $S=20$ to ensure force saturation before $\tilde{x}_\text{b}$.
The proposed force profile can further be associated to an energy (\autoref{fig:2d}) that is an unbounded Lipschitz function.
It therefore respects the requirement for Lyapunov's stability by the fractal attractor controller \cite{babarahmati2019}.
For the proposed force profile, this becomes:
\begin{equation}
\label{EnergyProf}
E_{\text{K}}=\left\{
\begin{array}{ll}
0.5 K_0 \tilde{x}^2, & |\tilde{x}|<\tilde{x}_{0}\\\\
F_{\text{Max}}|\tilde{x}| -&\\
+F_{\text{Max}}\tilde{x}_0 + (K_0\tilde{x}_0^2)/2 -&\\
+(1-e^{-\frac{|\tilde{x}| - \tilde{x}_0}{b}})b\Delta F,
& \tilde{x}_{0}\le |\tilde{x}|<\tilde{x}_{b}\\\\
F_{\text{Max}}|\tilde{x}|-&\\
+F_{\text{Max}}\tilde{x}_0 + (K_0\tilde{x}_0^2)/2 -&\\
+(1-e^{-\frac{\tilde{x}_b - \tilde{x}_0}{b}})b\Delta F, & \text{Otherwise}
\end{array}\right.
\end{equation}
\subsection{Controller Superimposition for the Control of Redundant Robots} \label{sec:StackOfFIC}
We propose to implement the same solution using virtual soft mechanical constraints generated by a superimposition of task-space controllers that drive the robot to assume a commanded reference posture.
The benefit of using impedance controllers based on fractal impedance is that their passivity allows for superimposition without compromising overall system stability.
Therefore, the total torque vector ($\vec{\tau_{\text{tot}}}$) can be computed by the superimposition of controllers as:
\begin{equation}
\label{TControllerStack}
\begin{array}{ll}
\vec{\tau}_{\text{tot}}=\sum_{i=1}^n J_i^\text{T}h_{ei}
\end{array}
\end{equation}
where $J_i$ and $h_{ei}$ are the Jacobian and the wrench generated by the impedance controller of the $i^{th}$-link, as depicted in \autoref{fig:conceptual_overview}.
\section{Evaluation} \label{sec:experiments}
We evaluate our proposed method using a 7-\gls{DoF} torque-controlled Kuka LWR3+ manipulator in both simulation and hardware experiments. We apply a superimposition of two task-space controllers: A 6-\gls{DoF} \gls{FIC} controller at the end-effector ($7^{th}$ link) and a 3-\gls{DoF} \gls{FIC} controller at the elbow ($4^{th}$ link of the KuKA URDF) for postural control.
\input{figures/fig-simulation-overview.tex}
\input{figures/TableI.tex}
To generate pose references for each of the controllers, we perform an optimization to obtain a configuration satisfying the end-effector reference. Here, we use a one-step variant of \gls{AICO} \cite{toussaint2009robot}.
Note, while the end-effector pose reference can be passed in directly to the end-effector controller, a postural optimization is used in this case to obtain a pose reference for the null-space or additional superimposed controllers. We extract the reference pose for each of the controllers using forward kinematics.
\subsection{Reference Trajectories}
The figure-of-8 (i.e., lemniscate) trajectory has been selected to show the dynamic behavior of the robot. The trajectory is composed of two orthogonal sinusoidal trajectories. The vertical trajectory has an amplitude of \SI{0.2}{\meter} and the transverse trajectory amplitude is \SI{0.1}{\meter}. The figure-of-8 trajectory is particularly demanding due to its multiple velocity inversions and wide joint movements range. Thus, introducing high variability of both the Jacobian and the inertial behavior of the robot.
We test the figure-of-8 reference motion in both simulation and hardware experiments.
In hardware experiments, we further test a sinusoidal trajectory with an amplitude of \SI{0.5}{\meter} and velocities up to about \SI{0.7}{\meter\per\second}. The straight-line experiment enabled us to test interaction and robustness at higher speeds.
\subsection{Simulation Experiments}
We simulate the robot using the Gazebo physics simulator and apply the Superimposition of Passive Task-Space Controllers control scheme directly without compensating for gravity, Coriolis, or other dynamic effects (in contrast to \cite{babarahmati2019}), i.e., as a model-free compliant controller.
For the simulation experiments, we compare nominal tracking performance with an interaction scenario where an unsensed environment obstacle has been introduced, cf. \autoref{fig:gazebo_unsensed_force_interaction}.
\subsection{Hardware Experiments}
In our hardware experiments, we use a Kuka LWR3+ robot. We control the manipulator using the \gls{FRI} at \SI{333.3}{\hertz} in \emph{joint impedance} mode with all gains set to zero to enable feed-forward torque control. Note, unlike our simulation experiments the Kuka's built-in controller compensates for dynamic effects and gravity.
On the real robot the tracking of the figure-of-8 trajectory has also been tested with and without a human operator applying random perturbations. The values used in the controller for the simulation and the experiments are reported in \autoref{CtrParam}. It shall also be noted that during the experiment we have kept the minimum set of controlled \gls{DoF} required to fully control the 3 \gls{DoF} of redundancy for the assigned tasks, being the task invariant to the configuration of the $7^{th}$ \gls{DoF} due to the symmetric geometry of the end-effector in the manipulator (\autoref{fig:intro}).
\section{Results}
\label{sec:results}
\input{figures/fig-simulation-results.tex}
\input{figures/fig-hw-results.tex}
To complement the plots in this section, the reader is recommended to watch the supplementary video demonstrating the tracking and interaction both in simulation and hardware experiments. We also include a sequence demonstrating the safe behavior of the controller during calibration of the \gls{FIC} parameters given in \autoref{CtrParam}.
The simulation results are shown in \autoref{fig:Sim_8-NOInt} for the free motion, and in \autoref{fig:Sim_8-WInt} for the interaction behavior. They show that the robot can be successfully controlled without dynamic compensation, and that it can achieve dexterous dynamic behaviors.
The tracking \glspl{RMSE} at the end-effector are recorded without interaction as RMSE$_\text{x}=\SI{5.6}{\milli\meter}$, RMSE$_\text{y}=\SI{4.6}{\milli\meter}$, and RMSE$_\text{z}=\SI{6.1}{\milli\meter}$. For simulation with interaction with an obstacle: RMSE$_\text{x}=\SI{13.2}{\milli\meter}$, RMSE$_\text{y}=\SI{4.2}{\milli\meter}$, and RMSE$_\text{z}=\SI{5.5}{\milli\meter}$.
I.e., the tracking performance degrades in one dimension impacted by the obstacle ($x$), while being virtually unaffected in $y$ and $z$.
The experimental data for the hardware experiments of the figure-of-8 trajectory are shown in \autoref{8-NOInt} and \autoref{8-Int}.
The errors recorded during free motion are: RMSE$_\text{x}=7.6$ \si{\milli\meter}, RMSE$_\text{y}=1.5$ \si{\milli\meter}, and RMSE$_\text{z}=8.6$ \si{\milli\meter}.
The perturbations do not affect the tracking performance at the end-effector task, but they are fully compensated by the deflection from the secondary task target at the elbow joint, as shown in \autoref{8-Int}b.
The \gls{RMSE} during interaction are RMSE$_\text{x}=20.3$ \si{\milli\meter}, RMSE$_\text{y}=7.9$ \si{\milli\meter}, and RMSE$_\text{z}=9.1$ \si{\milli \meter}.
The results for the straight-line trajectory experiment (\autoref{Line-NoInt} and \autoref{line-Int}) show an ability of the controller to complete the task and reject perturbations by reducing tracking on the secondary task.
The errors are RMSE$_\text{x}=6.1$ \si{\milli\meter}, RMSE$_\text{y}=4.6$ \si{\milli\meter}, and RMSE$_\text{z}=6.7$ \si{\milli \meter}.
The RMSE for interaction are RMSE$_\text{x}=9.6$ \si{\milli\meter}, RMSE$_\text{y}=7.7$ \si{\milli\meter}, and RMSE$_\text{z}=7.3$ \si{\milli\meter}.
It shall also be remarked how the robot remained safe to interact with despite the high joint feed-forward torques involved in the motions, which reached $\approx30$ \si{\newton\meter} for both the 8 trajectory and the linear trajectory.
\section{Discussion}
\label{sec:discussion}
The results show the proposed method enables an intrinsically stable control framework for redundant robots which does not rely on inverse dynamics and projection matrices.
The proposed method is robust to unknown environmental interactions and singularities, where safe means that the robot does not show erratic behaviors even while perturbed or when there is a sudden change in the desired task (e.g., sudden acceleration/deceleration). The RMSE data show how the robot keeps the minimum tracking accuracy (i.e., maximum error) contained under \SI{1}{\centi\meter} for the unperturbed experiments, which is in line with the task requirement set in the controller parameters $x_\text{b}=\SI{1.1}{\centi\meter}$. This results are in line with the results obtained in \cite{babarahmati2019}, and they are lower than other impedance controller frameworks that usually have error of few centimeters \cite{dietrich2015,Dietrich2016}.
The introduction of significant perturbation degrades the minimum tracking accuracy to \SI{2}{\centi\meter}, but the controller remains stable and it is able to recover once the perturbation ends. It shall be remembered that the trade-off between accuracy and robustness is a known trade-off in interaction control frameworks \cite{ott2010unified}. While admittance control may provide better accuracy, it requires accurate knowledge of interaction force intensity and direction in all the points of contact with the environment. On the other hand, impedance controllers provide better safety of interaction but sacrifice tracking accuracy in favor of compliance \cite{ott2010unified,xin2020}. Variable impedance controllers have been proposed as a solution to this dilemma, but the stability requirements on the impedance updates are often very stringent and difficult to retain under highly variable environmental conditions \cite{angelini2019online,li2018force}. The proposed framework provides the better of both worlds providing good tracking accuracy while retaining the robustness typical of impedance controllers; furthermore, it enables online adjustment of the impedance profiles \cite{babarahmati2019}.
The data also confirm our hypothesis that redundant robot interaction behavior can be accurately defined without any \textit{a priori} knowledge of the system dynamics model, being \autoref{TControllerStack} the control command.
In our simulation experiments, we show that the tracking performance in this work is similar to the results reported in \cite{babarahmati2019} that relied on a compensation of the robot dynamics and the use of a null-space controller. This latest result is particularly important because robots' mechanical properties such as inertia and joints friction matrices are often difficult to retrieve and highly unreliable \cite{bruder2019, vasudevan2017}.
The knowledge of the actuation characteristics and the kinematic structure are still necessary for the implementation of the proposed method. However, they are both normally accurate, and easier to obtain if not available.
\input{figures/fig-hw-results-straight-line.tex}
The FIC controller generates an asymptotically stable potential field around the target state that enables direct superimposition of multiple controllers without compromising the system stability.
The controller superimposition generates a force field that acts as a trade-off cost function determining the preferred path of motion in robot's configuration space.
The force upper-bound of the controllers guarantees that the loss of accuracy in the main task is contained $x_\text{b}$ until the condition are compatible with the mechanical characteristics of the system.
Especially, if we consider that the proposed method is a compliant postural controller, where accurate tracking is subordinate to robustness of interaction.
In other words, the controller stabilizes the robot around a desired posture relying on the non-linear stiffness profile to compensate for its non-linear and environmental interaction, sacrificing the redundancy task before degrading the end-effector task beyond the selected accuracy.
This is confirmed by both the simulation and the experimental data, showing how the FIC controller tries and successfully keeps the accuracy under \SI{0.11}{\centi\meter} in unperturbed conditions.
The data also describe that the controller fully sacrifices the redundancy task in the attempt of retaining the same accuracy while experiencing external perturbation that exceed its mechanical limits, as shown in \autoref{fig:Sim_8-WInt}, \autoref{8-Int} and \autoref{line-Int}.
The data show that the proposed method can achieve a highly dynamic interaction using variable impedance at the controller level. The FIC also enables online tuning of the impedance behavior and is robust to reduce bandwidth in the feedback signals \cite{babarahmati2019}, allowing to switch from rigid to soft behaviors seemingly. Nevertheless, the performances are strictly related to the physical hardware capabilities, and a higher band-pass in the mechanics of the robot implies a higher stiffness to mass ratio. Therefore, it will be interesting to study these capabilities by deploying in hardware equipped with VSA to conduct a systematic experiment on these properties. Furthermore, it may also enable the switching from a model-based (e.g., the one proposed in \cite{keppler2018}) to a data-driven control of their non-linear actuators dynamics. It is worth also noting that, at the current stage, it is impossible to compare the results of these types of architectures due to their different hardware requirements.
Nevertheless, we can say that both of them achieve non-linear impedance behavior and robustness to highly dynamic interactions. The fractal impedance controller hardware requirements are less stringent, and it has a simpler formulation. The results presented in \cite{keppler2018} indicate that the DLR robot can achieve a stiffer behavior. However, it is impossible to discriminate if they are connected solely to a hardware superiority or there is also a controller component in play.
The superimposition of task-space controllers also opens new possibilities for improving controllability and dexterity for compliant robots, developing human motor control theory, and robust control architecture for learning algorithms.
In fact, the dynamics of soft robots are even more challenging to model than rigid dynamics \cite{bruder2019}, as the dynamic modelling of robots is founded on the assumption of rigidity \cite{siciliano}. In regards to human motor control, having a framework that enables robustness and dexterity of interaction will enable to overcome the current limitation of the \gls{PMP} model, which still relies on inverse matrices for trajectory optimization \cite{tommasino2017,tiseo2018}.
Finally, the learning algorithms are currently facing the challenges of performing a system identification to guarantee the stability of the learned behavior.
The proposed method removes this challenge and the learning component can focus on learning how to synchronize the task-space controller to maximize the efficiency and the dexterity of the robot.
\section{Conclusions} \label{sec:conclusion}
The experimental results confirmed our hypothesis that it is possible to control a redundant robot with a superimposition of task-space controllers.
This approach renders the architecture intrinsically robust to singularity and fully passive which guarantees stability.
It is important to properly balance the strength of the controllers to guarantee that the secondary tasks do not interfere with the end-effector controller, which may result challenging under certain conditions. Nevertheless, unbalanced controllers may interfere with the action efficacy, but not with the robustness and stability of interaction.
The proposed framework does not require any \textit{a priori} knowledge of the system dynamics parameters (i.e., Inertia, Friction, and Gravity).
It suited for applications where the stability of interaction to unpredictable environments is more important than the tracking accuracy.
Future work will focus on improving the coordination among the secondary task-space controllers to improve the tracking accuracy of the end-effector task as well as systems were coupling effects may be introduced.
\section*{To do list}
\begin{itemize}
\item Finalise Simulations Method
\item Add a figure with task-space controllers representation
\item Revise Introduction to be aligned with final message
\item Run Simulations and Write Results down results
\item Write Discussion, Conclusion and Abstract
\item (Gazebo) Check if we should add the viscous dissipation as in \cite{babarahmati2019} to account for model errors
\end{itemize}
\section*{Potential contributions}
\begin{itemize}
\item \cite{babarahmati2019} uses a fixed null-space resolution term in Alg. 1. Here, we use a Stack of FICs to get around the null-space resolution.
\item "Claim increase robustness to unpredicted/unknown obstacles. Obviously, the target has to be within the reach of the terminal part of the manipulator"
\item tbc: Dealing with orientations - we may want to look into this closer
\item Model-free impedance controller - to the best of our knowledge the first implementation/demonstration of this [comparison to previous: Previous used a model-based gravity compensation; our work does not. It can also be used to play arbitrary trajectories]
\end{itemize} |
1,108,101,564,580 | arxiv | \section{\bf Statement of the problem with motivation}
Extreme value laws are limit laws of linearly normalized partial maxima $M_n=$ \\ $\max\{X_1, \ldots, X_n\}$ of independent and identically distributed (iid) random variables (rvs) $X_1,X_2,\ldots,$ with common distribution function (df) $F,$ namely,
\begin{equation}\label{Introduction_e1}
\lim_{n\to\infty}P(M_n\leq a_nx+b_n)=\lim_{n\to\infty}F^n(a_nx+b_n)=G(x),\;\;x\in \mathcal{C}(G),
\end{equation}
where $a_n>0,$ $b_n\in\mathbb R,$ are norming constants, $G$ is a non-degenerate df, $\mathcal{C}(G)$ is the set of all continuity points of $G.$ If, for some non-degenerate df $G,$ a df $F$ satisfies (\ref{Introduction_e1}) for some norming constants $a_n>0,$ $b_n\in\mathbb R,$ then the df $F$ is said to belong to the max domain of attraction of $G$ under linear normalization and we denote it by $F\in \mathcal{D}_l(G).$ Extreme value laws $G$ satisfying (\ref{Introduction_e1}) are types of the following distributions, namely, the Fr\'{e}chet law, $\; \Phi_\alpha(x) = \exp(-x^{-\alpha}),$ $0\leq x;\;$ the Weibull law, $\; \Psi_\alpha(x) =$ $\exp(- |x|^{\alpha}),$ $ x<0;\;$ and the Gumbel law, $\; \Lambda(x)$ $ = \exp(-\exp(-x)),$ $ x\in\mathbb R;$ $\alpha>0\;$ being a parameter. Here and elsewhere in this article dfs are specified only for $\;x\;$ values for which they belong to $\;(0,1)\;$ and probability density functions (pdfs) are specified for values wherein they are positive. Two dfs $G$ and $H$ are said to be of the same type if $G(x) = H(Ax + B), x \in \mathbb R,$ for some constants $A > 0, B \in \mathbb R.$ Extreme value laws $G$ satisfy the stability relation $G^n(A_n x + B_n) = G(x), x \in \mathbb R, n \geq 1,$ where $A_n > 0, B_n \in \mathbb R$ are constants as given below and hence they are also called as max stable laws. The stability relation means that df of linearly normalized maxima of a random sample of size $n$ from df $G$ is equal to $G$ for every $n.$ We have, for $n \geq 1$ and $x \in \mathbb R,$
\begin{equation} \label{stable}
\Phi_\alpha^n\left(n^{1/ \alpha}x\right) = \Phi_\alpha(x); \;\; \Psi_\alpha^n\left(n^{-1 /\alpha} x\right) = \Psi_\alpha(x); \;\; \text{and}\;\; \Lambda^n(x + \log n) = \Lambda(x). \end{equation}
Note that (\ref{Introduction_e1}) is equivalent to
\begin{eqnarray}\label{Introduction_e1.2}
\lim_{n\to\infty}n\{1-F(a_nx+b_n)\}=-\log G(x), \; x \in \{y\in \mathbb R: G(y) > 0\}.
\end{eqnarray}
Criteria for $F\in \mathcal{D}_l(G)$ are well known (see, for example, Galambos, 1987; Resnick, 1987; Embrechts et al., 1997). These are repeated in the Appendix for ease of reference.
Let $X_{1:n} \leq X_{2:n} \leq \ldots \leq X_{n:n}$ denote the order statistics from the random sample $\;\{X_1, \ldots, X_n\} \;$ from the df $\;F\;$ and $F \in \mathcal{D}_l(G)$ for some non-degenerate df $G$ so that $F$ satisfies (\ref{Introduction_e1}) for some norming constants $a_n > 0, b_n \in \mathbb R.$ It is well known (see, for example, Galambos, 1987) that the df of the $k$-th upper order statistic $\;X_{n-k+1:n},\;$ for a fixed positive integer $k,$ is given by $\;F_{k:n}(x) = P\left( X_{n-k+1:n} \leq x \right) = \sum_{i=0}^{k-1} \binom ni F^{n-i}(x) (1 - F(x))^{i}, x \in \mathbb R,$ and the limit $G_k(x) = \lim_{n\to\infty} F_{k:n}(a_n x + b_n) $ is given by
\begin{equation}\label{eqn:r_max_df}
G_k(x) = G(x) \sum_{i=0}^{k-1} \frac{(- \log G(x))^i}{i!},\;\; x \in \{y: G(y) > 0\}.
\end{equation}
An obvious question is whether $G_k$ satisfies some stability relation like (\ref{stable}). Though we are unable to give a satisfactory answer to this question, we discuss implication of such a stability condition in this article. We show that $\;G_k\;$ is tail equivalent to $\;1 - (1-F(\cdot))^k\;$ and look at the consequences of this, one of which, for example, is that $\;G_k \in \mathcal{D}_l(\Phi_{k\alpha})\;$ if $F\in \mathcal{D}_l(\Phi_{\alpha}).\;$ These lead to generation of an hierarchy of Fr\'{e}chet laws with different integer exponents. Similar questions are also discussed when the sample size $\;n\;$ is a rv. A few of the results obtained here are available in Peng et al. (2010) and Barakat and Nigm (2002) but the proofs given here are elementary using tail equivalence unlike the proofs in these two articles. Results given here also generalize results given in Peng et al. (2010) and hold under power norming also. Analogous results for lower order statistics can be written down using results proved in this article.
This article is organized as follows. The next section discusses tail equivalence of $\;G_k \;$ and its consequences when sample size $\;n\;$ is fixed. The limiting distribution of normalized $\;k$-th upper order statistic $\;X_{n-k+1:n}\;$ when the sample size $\;n\;$ is discrete uniform rv is discussed in Section 3 and in Section 4 when the random sample size is binomial, Poisson, logarithmic, geometric, and negative binomial. Tail behaviour of the limit law obtained in Barakat and Nigm (2002) is also discussed in this section. Discrete uniform random maxima appears for the first time in the literature in this article though this was announced by the first author in the International Congress of Mathematicians 2006 held at Madrid, Spain (see Ravi, 2006). Some examples are given in Section 5 and stochastic orderings are discussed in Section 6. Proofs of some of the results are given after the statements and proofs of some main results along with some preliminary results are given in Section 7. Section 8 has some known results used in the article which are repeated here for ease of reading.
Throughout this article we use the following notations. For a df $\;F,\;$ its left extremity is denoted by $\;l(F)= \inf \set{x: F(x)>0},\;$ its right extremity by $\;r(F) = \sup\{x \in \mathbb R: F(x) < 1\},$ and $\;F^{-}(y) = \inf \left\{ x : F(x) \geq y \right\}, y \in \mathbb R. \;$
The symbol $\; \stackrel{d}\rightarrow \;$ denotes convergence in distribution, for two functions $\;g(.)\;$ and $\;h(.),\;$ $g(x) \sim h(x)\;$ as $\;x \rightarrow a\;$ if $\;\lim\limits_{x\rightarrow a} \dfrac {g(x)}{h(x)}=1\;$ so that $\;-\ln x \sim 1-x\;$ as $\;x\rightarrow 1\;$ and $\;-\ln F(x) \sim 1-F(x)\;$ as $\;x\rightarrow r(F)\;$ for any df $\;F.\;$ Further, $\;g(x) \sim o( h(x))\;$ as $\;x \rightarrow a\;$ if $\;\lim\limits_{x\rightarrow a} \dfrac {g(x)}{h(x)}=0.$
\section{\bf Limiting behaviour of $\;k$-th upper order statistic when sample size is fixed}
We study a possible stability relation for the df $\;F_{k:n}\;$ of the $\;k$-th upper order statistic from a random sample of size $\;n\;$ from a df $\;F,\;$ where $\;k\;$ is a fixed positive integer. For convenience, wherever necessary, we assume that the df $\;F\;$ is absolutely continuous with pdf $\;f.$
\subsection{On stability relation for df of $\;k$-th upper order statistic}
\begin{thm} If $\;F\in D_l(G)\;$ for some max stable df $\;G\;$ so that $\;F\;$ satisfies (\ref{Introduction_e1}) for some norming constants $\;a_n>0,b_n \in \mathbb R,\;$ and $\;F_{k:n}\;$ satisfies the stability relation $\;F_{k:n}(a_n x+b_n)=F(x),\;$ $ x\in R,\;$ $n\ge 1,\;$ then $\;F(x) = 0\;$ if $\;G(x) = 0\;$ and
\begin{eqnarray} \label{Stability-k}
F(x)=G(x) \sum_{i=0}^{k-1} \frac {(-\ln G(x))^i}{i!}, \;\;\;\;x \in \{y: G(y) > 0\}, \;\;G =\Phi_\alpha\; \mbox{or} \; \Psi_\alpha \; \mbox{or} \; \Lambda.
\end{eqnarray}
\end{thm}
\begin{proof}[\textbf {Proof:}] We have $\;F_{k:n}(x) = P(X_{n-k+1:n} \leq x)$ $= \sum_{i=0}^{k-1} \binom ni F^{n-i}(x) (1-F(x))^{i}, \; x \in \mathbb R.\;$ If
\begin{equation} F_{k:n}(a_n x+b_n) = \sum_{i=0}^{k-1} \binom ni F^{n-i}(a_nx+b_n) (1-F(a_nx+b_n))^{i} = F(x), \label{eqn:r}\end{equation}
then taking limit as $\;n \rightarrow \infty\;$ in (\ref{eqn:r}) we have $\;F(x) = \lim_{n \rightarrow \infty} F_{k:n}(a_nx+b_n) $
\begin{eqnarray}
& = & \lim_{n \rightarrow \infty} \sum_{i=0}^{k-1} \frac 1{i!} \left(F^n(a_nx+b_n)\right)^{1-\frac in} \prod_{j=0}^{i-1} \left( \frac {n-j}n n(1-F(a_nx+b_n))\right) \nonumber \\
&=& \sum_{i=0}^{k-1}\frac 1{i!} G(x) \prod_{j=0}^{i-1} (-\ln G(x)) =G(x) \sum_{i=0}^{k-1}\frac{(-\ln G(x))^i}{i!} \label{nonrandomlimit}
\end{eqnarray} so that (\ref{Stability-k}) holds.
\end{proof}
\begin{rem} For $\;k=1,$ $F_{1:n}\;$ is the df of the partial maxima $\;X_{n:n}\;$ and in this case if $\;F=G,\;$ then $\;G\;$ satisfies the stability relation (\ref{stable}). However, though $\;G \in D_l(G),\;$ note that, with $\;F = G\;$ in (\ref{eqn:r}), we end up with $\;G(x) \sum_{i=0}^{k-1} \frac {(-\ln G(x))^i}{i!} = G(x),\;$ and hence $\;G\;$ has to be a degenerate df, a contradiction. This means that the $\;k$-th upper order statistic from a random sample from one of the extreme value distributions will not satisfy a property like (\ref{eqn:r}). However, we show later that if a df $\;F \in D_l(G) \;$ for some $\;G,\;$ then a function of the type $\;F(x) \sum_{i=0}^{k-1} \dfrac {(-\ln F(x))^i}{i!} \;$ is a df and belongs to $\;D_l(G)\;$ with possibly a different exponent. \label{rem:r}
\end{rem}
\begin{defn} \label{Def-Fk} For any df $\;F\;$ and fixed integer $\;k \ge 1,\;$ define $F_k(x)=F(x)\sum\limits_{i=0}^{k-1} \dfrac {(-\ln F(x))^i}{i!},\;$ $x \in \{y: F(y)>0\}.\;$ \end{defn}
\begin{thm}
Let $\;X\;$ have absolutely continuous df $\;F \;$ with pdf $\;f.\;$ Then the df $\;F_k \;$ satisfies the recurrence relation
\begin{equation} F_{k+1}(x)=F_k(x)+\dfrac {F(x)}{k!} (-\ln F(x))^k, \quad k\ge 1, \;\;x\in \{y: F(y)>0\}. \label{eqn:recdf} \end{equation}
Further, the pdf of $\;F_{k+1}\;$ is given by
\begin{equation} f_{k+1}(x)=\dfrac{f(x)}{k!} (-\ln F(x))^k, \quad k\ge 1, \;\;x\in \{y: F(y)>0\}.\label{eqn:recpdf} \end{equation} \label{thm:r-max-df}
\end{thm}
\begin{proof}[\textbf{Proof:}] The proof of the relation (\ref{eqn:recdf}) is evident from the definition of $F_k\;$ since
\[ F_{k+1}(x)-F_k(x)=F(x)\sum_{i=0}^k \dfrac{(-\ln F(x))^i}{i!}-F(x)\sum_{i=0}^{k-1}\dfrac{(-\ln F(x))^i}{i!}=\dfrac{F(x)}{k!}(-\ln F(x))^k.\]
We use induction to prove (\ref{eqn:recpdf}).
For $k=1$, we have $\;F_2(x)=F(x)(1-\ln F(x)).\;$ By differentiation, $\;f_2(x)=$ $f(x)(1-\ln F(x))$ $ + F(x) \left(-\dfrac {f(x)}{F(x)}\right)$ $=f(x)(-\ln F(x))\ge 0,\;$ which is (\ref{eqn:recpdf}) for $\;k=1.\;$ Assuming that (\ref{eqn:recpdf}) is true for $\;k=m,\;$ we have $ F_{m+2} (x)=F_{m+1}(x)+\dfrac {F(x)}{(m+1)!} (-\ln F(x))^{m+1},$ and upon differentiating with respect to $\;x,\;$ we get $\;f_{m+2} (x) = $ $ f_{m+1}(x)+\frac{f(x)}{(m+1)!} \set{-(m+1)(-\ln F(x))^m + (-\ln F(x))^{m+1}}$ $ = \dfrac {f(x)}{(m+1)!} (-\ln F(x))^{m+1},\;$ which is (\ref{eqn:recpdf}) with $\;k = m+1.\;$ Finally since $\;f(x)\ge 0\;$ and $\;0<F(x)<1,\;$ it follows that $\;f_{k+1}(x)\ge 0,\;$ completing the proof.
\end{proof}
The following theorem whose proof is given in Section 7 shows that any positive power of tail of a df is also a tail and looks at the tail behaviour of the tail $\;(1-F(.))^k \;$ and this will be used to study the tail behaviour of $\;F_k \;$ in the next result.
\begin{thm} If $\;F\;$ is a df with pdf $\;f,\;$ then for every positive integer $\;k,\;$ $\;\left(1-F(x)\right)^k\;$ is tail of the df $\;H_k(x)=1-\left(1-F(x)\right)^k,\ x\in \mathbb R, \;$ and $\;H_k\;$ is also absolutely continuous with pdf $\; H_k^\prime(x)=k\set{1-F(x)}^{k-1} f(x), x \in \mathbb R.$ Further, the following are true.
\begin{enumerate}
\item[(a)] If $\;F\in D_l(\Phi_\alpha),\;$ then $\;r(H_k)=r(F)=\infty,\;$ and $\;H_k \in D_l(\Phi_{k\alpha})\;$ so that (\ref{Introduction_e1}) holds with $\;F = H_k, G = \Phi_{k\alpha},\;$ $a_n =F^-(1 - (1/n)^{1/k}), b_n=0.$
\item[(b)] If $\;F\in D_l(\Psi_\alpha)\;$ then $\;r(H_k)=r(F)<\infty,\;$ and $\;H_k \in D_l(\Psi_{k\alpha})\;$ so that (\ref{Introduction_e1}) holds with $\;F = H_k, G = \Psi_{k\alpha},\;$ $a_n =r(F)-F^- (1 - (1/n)^{1/k}), b_n=r(F).$
\item[(c)] If $\;F\in D_l(\Lambda),$ $\;a_n = v(b_n)\;$ and
$\;b_n = F^{-}\left(1 - \frac{1}{n}\right) \;$ as in (c) of Theorem $\ref{T-l-max}$ so that (\ref{Introduction_e1}) holds with $\;G = \Lambda,\;$ then $\;r(H_k)=r(F),\;$ and $\;H_k \in D_l(\Lambda)\;$ so that (\ref{Introduction_e1}) holds with $\;F = H_k, G = \Lambda,\;$ $\;a_n = \dfrac{v(b_n)}{k}, b_n = H_k^-( 1 - 1/n).\;$
\end{enumerate} \label{lem:1} \end{thm}
\begin{rem}If $\;F\;$ is a df, for every positive integer $\;k,$ $F^k\;$ is a df and if $\;1-F\;$ is tail of a df then $\;(1-F)^k\;$ is also a tail. \label{rem:1}\end{rem}
The next result is for fixed sample size and its proof is given in Section 7.
\begin{thm} Let rv $\;X\;$ have absolutely continuous df $\;F\;$ with pdf $\;f\;$ and $\;k\;$ be a positive integer. Then for $\; F_k(x)=F(x)\sum_{i=0}^{k-1} \dfrac {(-\ln F(x))^i}{i!},$ $\;x \in \{y: F(y) > 0\},\;$ the following results are true.
\begin{enumerate}
\item[(a)] $F_k\;$ is a df with $\;r(F_k) = r(F),\;$ pdf $\;f_k(x)=\dfrac{f(x)}{(k-1)!} (-\ln F(x))^{k-1},\;$ $x \in \{y \in \mathbb R: F(y) > 0\};$ and $\;\lim_{x \rightarrow r(F)} \dfrac{1-F_k(x)}{(1-F(x))^k} = \dfrac{1}{k!},\;$ so that $\;F_k\;$ is tail equivalent to $\;H_k.$
\item[(b)] If $\;F\in D_l(\Phi_\alpha),\;$ then $\;r(F_k)=r(F)=\infty,\;$ and $\;F_k \in D_l(\Phi_{k\alpha})\;$ so that (\ref{Introduction_e1}) holds with $\;F = F_k, G = \Phi_{k\alpha},\;$ $ \;a_n =F^- (1 - (k!/n)^{1/k}), b_n=0.\;$
\item[(c)] If $\;F\in D_l(\Psi_\alpha)\;$ then $\;r(F_k)=r(F)<\infty,\;$ and $\;F_k \in D_l(\Psi_{k\alpha})\;$ so that (\ref{Introduction_e1}) holds with $\;F = F_k, G = \Psi_{k\alpha},\;$ $\;a_n =r(F) - F^-(1 - (k!/n)^{1/k}), b_n=r(F).\;$
\item[(d)] If $\;F\in D_l(\Lambda),$ $\;a_n = v(b_n)\;$ and
$\;b_n = F^{-}\left(1 - \frac{1}{n}\right) \;$ as in (c) of Theorem $\ref{T-l-max}$ so that (\ref{Introduction_e1}) holds with $\;G = \Lambda,\;$ then $\;r(F_k)=r(F),\;$ and $\;F_k \in D_l(\Lambda)\;$ so that (\ref{Introduction_e1}) holds with $\;F = F_k, G = \Lambda,\;$ $\;a_n = \dfrac{v(b_n)}{k}, b_n = F_k^-( 1 - 1/n).\;$
\end{enumerate} \label{Thm-Fk}
\end{thm}
\section{\bf Limiting behaviour of $k$th upper order statistics when sample size is discrete uniform rv}
In this section, we assume that the sample size $\;n\;$ in the previous section is a discrete uniform rv $\;N_n\;$ with probability mass function (pmf) $\;P(N_n=r)=\frac 1n,$ $ r=m+1,m+2,\ldots,m+n,\;$ $N_n\;$ independent of the iid rvs $\;X_1, X_2, \ldots,\;$ where $\;m\ge 1\;$ is a fixed integer. The tail behaviour of the limit of linearly normalized $\;k$-th upper order statistic $\;X_{N_n-k+1:N_n}\;$ which is the $\;k$-th maximum among $\; \set{X_1,X_2,\ldots, X_{N_n}}\;$ is discussed here. Observe that $\;X_{N_n-k+1:N_n}$ is well defined for $\;1\le k \le m.\;$
For fixed integer $\;k, \, 1 \le k \le m,\;$ the df of the $\;k$-th upper order statistic of a sequence of iid rvs with random sample size $\;N_n\;$ is given by
\begin{eqnarray}
F_{k:N_n}(x)&=&P(X_{N_n-k+1:N_n} \le x ) = \sum_{r=m}^\infty P(X_{N_n-k+1:N_n} \le x , N_n=r) \nonumber
\\&=& \sum_{r=m}^\infty \sum_{i=0}^{k-1} \binom ri F^{r-i}(x) (1-F(x))^i P(N_n=r), \; x \in \mathbb R. \label{eqn:gr-max1}
\end{eqnarray}
\begin{thm} If $\;F\in D_l(G)\;$ for some max stable df $\;G\;$ so that $\;F\;$ satisfies (\ref{Introduction_e1}) for some norming constants $\;a_n>0,b_n \in \mathbb R,\;$
then for fixed integer $\;k, 1 \le k \le m,\;$ the limiting distribution $\;\lim_{n \rightarrow \infty} F_{k:N_n}(a_n x + b_n)\;$ is equal to
\begin{equation} J_k(x) = k\Set{\frac {1-G(x)}{-\ln G(x)}}-G(x) \sum_{l=1}^{k-1} (k-l) \frac {(-\ln G(x))^{l-1}}{l!}, \; x \in \{y \in \mathbb R: G(y) > 0\}, \label{eqn:dr-max} \end{equation} where $\; G =\Phi_\alpha \;\mbox{or}\; \Psi_\alpha \;\mbox{or}\; \Lambda.$ \label{Thm_DU1}
\end{thm}
\begin{defn} \label{Def-Fk} For any df $\;F,\;$ and fixed integer $\;k \ge 1,\;$ we define $U_k(x)=k\Set{\frac {1-F(x)}{-\ln F(x)}}-F(x) \sum_{l=1}^{k-1} (k-l) \frac {(-\ln F(x))^{l-1}}{l!},\;$ $x \in \{y: F(y)>0\}.\;$ \end{defn}
\begin{thm} If rv $\;X\;$ has absolutely continuous df $\;F\;$ with pdf $\;f,\;$ $\;k\;$ is a fixed positive integer, and $\;U_1(x)=\dfrac {1-F(x)}{-\ln F(x)},\;$ $x \in \{y: F(y) > 0\},\;$ then $\;U_1\;$ is a df with $\;r(U_1) = r(F),\;$ pdf $\;u_1(x)=\dfrac {f(x)}{F(x)}\dfrac {U_1(x)-F(x)}{-\ln F(x)}=\dfrac {f(x)}{F(x)}\Set{\dfrac {1-F(x)+F(x) \ln F(x)}{(-\ln F(x))^2}},\; x \in \{y \in \mathbb R: F(y) > 0\};\;$ and $\;\lim_{x \rightarrow r(F)} \dfrac{1-U_1(x)}{1-F(x)} = \dfrac 12\;$ so that $\;U_1\;$ is tail equivalent to $\;F.$
\label{Thm-FkW}
\end{thm}
\begin{rem} $U_1\;$ in the above theorem was derived first and shown to be a df in Ravi (2006) and this and the next result generalize the results obtained in Ravi (2006).
\end{rem}
\begin{thm} Let rv $\;X\;$ have absolutely continuous df $\;F\;$ with pdf $\;f\;$ and $\;k\;$ be a fixed positive integer. Then for $\;U_k\;$ as in Definition \ref{Def-Fk} and $\;H_k(x) = 1 - (1-F(x))^k, x \in \mathbb R,\;$ the following results are true.
\begin{enumerate}
\item[(a)] $U_k\;$ is a df with $\;r(U_k) = r(F),\;$ pdf $\;u_k(x)=\dfrac {kf(x)}{(-\ln F(x))^2}\Set{\dfrac 1{F(x)} - \sum_{l=0}^k \dfrac {(-\ln F(x))^l}{l!}},\;$ $x \in \{y \in \mathbb R: F(y) > 0\}.$
\item[(b)] $\;\lim_{x \rightarrow r(F)} \dfrac{1-U_k(x)}{(1-F(x))^k} = \dfrac{1}{(k+1)!},\;$ so that $\;U_k\;$ is tail equivalent to $\;H_k.$
\item[(c)] If $\;F\in D_l(\Phi_\alpha),\;$ then $\;r(U_k)=r(F)=\infty,\;$ $\;U_k \in D_l(\Phi_{k\alpha})\;$ so that (\ref{Introduction_e1}) holds with $\;F = U_k, G = \Phi_{k\alpha},\;$ $\;a_n =F^- (1 - ((k+1)!/n)^{1/k}), b_n=0.$
\item[(d)] If $\;F\in D_l(\Psi_\alpha)\;$ then $\;r(U_k)=r(F)<\infty,\;$ $\;U_k \in D_l(\Psi_{k\alpha})\;$ so that (\ref{Introduction_e1}) holds with $\;F = U_k, G = \Psi_{k\alpha},\;$ $\;a_n =r(F) - F^-(1 - ((k+1)!/n)^{1/k}), b_n=r(F).$
\item[(e)] If $\;F\in D_l(\Lambda),$ $\;a_n = v(b_n)\;$ and
$\;b_n = F^{-}\left(1 - \frac{1}{n}\right) \;$ as in (c) of Theorem $\ref{T-l-max}$ so that (\ref{Introduction_e1}) holds with $G = \Lambda,$ then $\;r(U_k)=r(F),\;$ $\;U_k \in D_l(\Lambda)\;$ so that (\ref{Introduction_e1}) holds with $\;F = U_k, G = \Lambda,\;$ $\;a_n = \dfrac{v(b_n)}{k}, b_n = U_k^-( 1 - 1/n).$
\end{enumerate} \label{Thm-FkDU}
\end{thm}
\begin{rem} Note that df $\;U_k\;$ satisfies the recurrence relation $\displaystyle U_{k+1}(x)= U_k(x)+U_1(x)-F(x) \sum_{l=1}^k \frac {(-\ln F(x))^{l-1}}{l!}.$ For, $\;U_{k+1}(x)-U_k(x)=$
\begin{eqnarray*}
&=& (k+1)U_1(x)-F(x) \sum_{l=1}^k (k+1-l) \frac {(-\ln F(x))^{l-1}}{l!} - kU_1(x)+F(x) \sum_{l=1}^{k-1} (k-l) \frac {(-\ln F(x))^{l-1}}{l!}
\\&=& U_1(x) - F(x) \sum_{l=1}^{k-1} (k+1-l-k+l) \frac {(-\ln F(x))^{l-1}}{l!} -F(x) \frac {(-\ln F(x))^{k-1}}{k!}
\\&=& U_1(x) - F(x) \sum_{l=1}^k \frac {(-\ln F(x))^{l-1}}{l!}.
\end{eqnarray*}
\end{rem}
\section{\bf Limiting behaviour of $k$th upper order statistic when sample size is random}
In the first subsection we consider the cases when the sample size $\;N_n\;$ follows shifted binomial, Poisson, and logarithmic distributions and show that the limit distribution in all these cases is the same as that in the fixed sample size case.
After this we consider the cases when the sample size follows shifted geometric and negative binomial and derive non-trivial results.
\subsection{Binomial, Poisson and Logarithmic $\;k$-th upper order statistic}
\begin{thm} \label{Thm_FkB}
\begin{itemize}
\item[(a)] Assume that $\;N_n\;$ is a shifted binomial rv with $\;P(N_n=r)=\binom n{r-m}p_n^{r-m}q_n^{m+n-r},$ $ r=m,m+1,m+2,\ldots,m+n,\;$ for some integer $\;m \geq 1,\;$ with $\;\lim_{n \rightarrow \infty} p_n = 1. \;$ If $\;F\in D_l(G)\;$ for some max stable df $\;G\;$ so that $\;F\;$ satisfies (\ref{Introduction_e1}) for some norming constants $\;a_n>0,b_n \in \mathbb R,\;$
then for fixed integer $\;k, 1 \le k \le m,\;$ the limiting distribution $\;\lim_{n \rightarrow \infty} F_{k:N_n}(a_n x + b_n)\;$ is same as in (\ref{nonrandomlimit}).
\item[(b)] Assume that $\;N_n\;$ is a shifted Poisson rv with $\;P(N_n=r)=\dfrac {e^{-\lambda_n} \lambda_n^{r-m}}{(r-m)!},$ $ r=m,m+1,m+2,\ldots,\;$ for some integer $\;m \geq 1,\;$ with $\;\lim_{n \rightarrow \infty} \dfrac{\lambda_n}{n} = 1.$
If $\;F\in D_l(G)\;$ for some max stable df $\;G\;$ so that $\;F\;$ satisfies (\ref{Introduction_e1}) for some norming constants $\;a_n>0,b_n \in \mathbb R,\;$
then for fixed integer $\;k, 1 \le k \le m,\;$ the limiting distribution $\;\lim_{n \rightarrow \infty} F_{k:N_n}(a_n x + b_n)\;$ is same as in (\ref{nonrandomlimit}).
\item[(c)] Assume that $\;N_n\;$ is a shifted logarithmic rv with $\;P(N_n=r)=\dfrac1{-\ln (1-\theta_n)} \dfrac{\theta_n^{r-n}}{r-n},$ $ r=n+1,n+2,\ldots,\;$ for some $\;\theta_n, 0 < \theta_n < 1,\;$ with $\;\lim_{n \rightarrow \infty} \dfrac{\theta_n}{-\ln (1-\theta_n)} = 1.$ If $\;F\in D_l(G)\;$ for some max stable df $\;G\;$ so that $\;F\;$ satisfies (\ref{Introduction_e1}) for some norming constants $\;a_n>0,b_n \in \mathbb R,\;$
then for fixed integer $\;k, \,k \ge 1,\;$ the limiting distribution $\;\lim_{n \rightarrow \infty} F_{k:N_n}(a_n x + b_n)\;$ is same as in (\ref{nonrandomlimit}).
\end{itemize}
\end{thm}
\subsection{Geometric $\;k$-th upper order statistic}
Assume that $\;N_n\;$ is a shifted geometric rv with pmf $\;P(N_n=r)=p_nq_n^{r-m}, r=m,m+1,m+2,\ldots,\;$ where $\;0< p_n < 1, q_n = 1-p_n\;$ and $\;\lim_{n \rightarrow \infty} np_n = 1.$
\begin{thm} If $\;F \in D_l(G)$ for some max stable law $\;G,\;$ then for fixed integer $\;k, 1 \le k \le m,\;$ the limiting distribution $\;\lim_{n \rightarrow \infty} F_{k:N_n}(a_n x + b_n)\;$ is equal to
\begin{equation} L_k(x)=1-\left(\frac{-\ln G(x)}{1-\ln G(x)}\right)^k, \; x \in \{y \in \mathbb R: G(y) > 0\}. \label{eqn:gr-max} \end{equation} \label{Thm_FkG}
\end{thm}
\begin{rem} Note that $\;L_k(x)= \begin{cases} 1-\left(\frac 1{1+x^\alpha}\right)^k & \;\text{if } G(x)=\Phi_\alpha(x), \\ 1-\left(\dfrac {(-x)^\alpha}{1+(-x)^\alpha}\right)^k & \; \text{if } G(x)=\Psi_\alpha(x), \;\;\;\; \\ 1-\left(\frac {e^{-x}}{1+e^{-x}}\right)^k & \;\text{if } G(x)=\Lambda(x), \end{cases}\;$ so that the first two are Burr distributions of XII kind which is discussed later, and the last is the logistic distribution.
\end{rem}
Limit distributions of extremes of a random number of random variables has been obtained for three extreme value distributions $\Phi_\alpha,\Psi_\alpha,\Lambda$ in Vasantalakshmi, M.S.,(2010) using result of Barakat et al(2002). The Burr-family is also introduced in the same. In the thesis, only case when $N_n$ follow geometric law was considered. In this paper an attempt is made to study the tail behaviour of $k-$th upper order statistic when the df $F\in D_l(G)$ and the results is considered for different distributions of $N_n$. \\
As we did in Definition \ref{Def-Fk}, we now define a new function and show that it is a df and study its properties similar to those in Theorem \ref{Thm-Fk}.
\begin{defn} For any df $\;F,\;$ and fixed integer $\;k \ge 1,\;$ we define $\;R_k(x)=1-\left( \dfrac {-\ln F(x)}{1-\ln F(x)}\right)^k,\;$ $x \in \{y: F(y)>0\}.\;$ \label{Def-Fk2} \end{defn}
\begin{thm} If rv $\;X\;$ has absolutely continuous df $\;F\;$ with pdf $\;f\;$ $\;k\;$ is a positive integer and $\; R_k\;$ is as defined above, then the following results are true.
\begin{enumerate}
\item[(a)] $R_k\;$ is a df with pdf $\;r_k(x)=\dfrac{kf(x) (-\ln F(x))^{k-1}}{F(x) (1-\ln F(x))^{k+1}},\;$ $x \in \{y \in \mathbb R: F(y) > 0\},$ $\;r(R_k) = r(F),\;$ and $\lim_{x \rightarrow r(F)} \dfrac{1-R_k(x)}{(1-F(x))^k} = 1,\;$ so that $\;R_k\;$ is tail equivalent to $\;H_k.$
\item[(b)] If $\;F\in D_l(\Phi_\alpha),\;$ then $\;r(R_k)=r(F)=\infty,\;$ and $\;R_k \in D_l(\Phi_{k\alpha})\;$ so that (\ref{Introduction_e1}) holds with $\;F = R_k, G = \Phi_{k\alpha},$ $ \;a_n =F^- (1 - (1/n)^{1/k}), b_n=0.$
\item[(c)] If $\;F\in D_l(\Psi_\alpha)\;$ then $\;r(R_k)=r(F)<\infty,\;$ and
$\;R_k \in D_l(\Psi_{k\alpha})\;$ so that (\ref{Introduction_e1}) holds with $\;F = R_k, G = \Psi_{k\alpha},\;$ $a_n =r(F) - F^-(1 - (1/n)^{1/k}), b_n=r(F).$
\item[(d)] If $\;F\in D_l(\Lambda),\;$ $a_n = v(b_n)\;$ and
$\;b_n = F^{-}\left(1 - \dfrac{1}{n}\right) \;$ as in (c) of Theorem $\ref{T-l-max}$ so that (\ref{Introduction_e1}) holds with $G = \Lambda,$ then $\;r(R_k)=r(F),\;$ and
$\;R_k \in D_l(\Lambda)\;$ so that (\ref{Introduction_e1}) holds with $\;F = R_k, G = \Lambda,\;$ $a_n = \dfrac{v(b_n)}{k}, b_n = R_k^-( 1 - 1/n).$
\end{enumerate} \label{Thm-FkG2}
\end{thm}
\begin{rem} Burr (1942) proposed twelve explicit forms of dfs which have since come to be known as the Burr system of distributions. These have been studied quite extensively in the literature. A number of well-known distributions such as the uniform, Rayleigh, logistic, and log-logistic are special cases of Burr dfs. A df $\;W\;$ is said to belong to the Burr family if it satisfies the differential equation
\begin{equation} \dfrac {dW(x)}{dx}=W(x) (1-W(x)) h(x,W(x)) \label{eqn:burr} \end{equation}
where $\;h(x,W(x))\;$ is a non-negative function for $\;x\;$ for which the function is increasing. One of the forms of $\;h(x,W(x))\;$ is $\;h(x,W(x))=\dfrac {h_1(x)}{W(x)}\;$ where $\;h_1(x)\;$ is a non-negative function. Then (\ref{eqn:burr}) takes the form $\;\dfrac {dW(x)}{dx}=(1-W(x)) h_1(x).\;$ We now show that $\;R_k\;$ is a member of the Burr family of dfs.
\end{rem}
\begin{thm} The dfs $R_k$ in Definition \ref{Def-Fk2} belong to the Burr family. \label{Thm-FkG3} \end{thm}
\subsection{Negative Binomial $\;k$-th upper order statistic}
We assume that $\;N_n\;$ is a shifted negative binomial rv with pmf $\;P(N_n=l)=\binom {l-m+r-1}{l-m} p_n^r q_n^{l-m}, r=m,m+1,m+2,\ldots,\;$ where $\;0< p_n < 1, q_n = 1-p_n\;$ and $\;\lim_{n \rightarrow \infty} np_n = 1.$
\begin{thm} If $\;F \in D_l(G)$ for some max stable law $\;G,\;$ then for fixed integer $\;k, 1 \le k \le m,\;$ the limiting distribution $\;\lim_{n \rightarrow \infty} F_{k:N_n}(a_n x + b_n)\;$ is equal to
\begin{equation} S_k(x)=\sum_{l=0}^{k-1} \binom {l+r-1}l \dfrac 1{(1-\ln G(x))^{r}} \left( \dfrac {-\ln G(x)}{1-\ln G(x)}\right)^l, \; x \in \{y \in \mathbb R: G(y) > 0\}. \label{eqn:nbr-max} \end{equation} \label{Thm-FkNB1}
\end{thm}
As we did in Definition \ref{Def-Fk2}, we now define a new function and show that it is a df and study its properties similar to those in Theorems \ref{Thm-Fk} and \ref{Thm-FkG2}.
\begin{defn} For any df $\;F,\;$ and fixed integer $\;k \ge 1,\;$ we now define $\;T_k(x)=\sum_{l=0}^{k-1} \binom {l+r-1}l \dfrac {(-\ln F(x))^l}{(1-\ln F(x))^{l+r}},\;$ $x \in \{y: F(y)>0\}.\;$ \label{Def-Fk3} \end{defn}
The following theorem contains a recurrence relation.
\begin{thm}
The df $\;T_k\;$ in (\ref{eqn:nbr-max}) satisfies the recurrence relation
\begin{equation} T_{k+1}(x)= T_k(x)+ \binom {k+r-1}k \dfrac {(-\ln F(x))^k}{(1-\ln F(x))^{k+r}}, \;\; k\ge 1, x \in \mathbb R. \label{eqn:recdf1} \end{equation}
Further its pdf is given by
\begin{equation} t_{k+1}(x)= \dfrac 1{B(r,k+1)} \dfrac {f(x)}{F(x)} \dfrac {(-\ln F(x))^k}{(1-\ln F(x))^{r+k+1}}, \;\; k\ge 1, x\in \mathbb R. \label{eqn:recpdf1} \end{equation} \label{Thm_FkNB2}
\end{thm}
\begin{thm} Let rv $\;X\;$ have absolutely continuous df $\;F\;$ with pdf $\;f\;$ and $\;k\;$ be a fixed positive integer. Then for $\; T_k\;$ as in Definition \ref{Def-Fk2}, the following results are true.
\begin{enumerate}
\item[(a)] $T_k\;$ is a df with pdf $\;t_k(x)=\dfrac 1{B(r,k)} \dfrac {f(x)}{F(x)} \dfrac {(-\ln F(x))^{k-1}}{(1-\ln F(x))^{r+k}},\;$ $x \in \{y \in \mathbb R: F(y) > 0\},$ right extremity $\;r(T_k) = r(F),\;$ and
$\lim_{x \rightarrow r(F)} \dfrac{1-T_k(x)}{(1-F(x))^k} = \dfrac{k}{B(r,k)},\;$ so that $\;T_k\;$ is tail equivalent to $\;H_k.$
\item[(b)] If $\;F\in D_l(\Phi_\alpha),\;$ then $\;r(T_k)=r(F)=\infty,\;$ and
$\;T_k \in D_l(\Phi_{k\alpha})\;$ so that (\ref{Introduction_e1}) holds with $\;F = T_k,\;$ $\;G = \Phi_{k\alpha},\;$ $a_n =F^- (1 - (1/n)^{1/k}), b_n=0.$
\item[(c)] If $\;F\in D_l(\Psi_\alpha)\;$ then $\;r(T_k)=r(F)<\infty,\;$ and
$\;T_k \in D_l(\Psi_{k\alpha})\;$ so that (\ref{Introduction_e1}) holds with $\;F = T_k,\;$ $\;G = \Psi_{k\alpha},\;$ $a_n =r(F) - F^-(1 - (1/n)^{1/k}), b_n=r(F).$
\item[(d)] If $\;F\in D_l(\Lambda),$ $\;a_n = v(b_n)\;$ and
$\;b_n = F^{-}\left(1 - \dfrac{1}{n}\right) \;$ as in (c) of Theorem $\ref{T-l-max}$ so that (\ref{Introduction_e1}) holds with $\;F = T_k,\;$ $G = \Lambda,$ then $\;r(T_k)=r(F),\;$ and
$\;T_k \in D_l(\Lambda)\;$ so that (\ref{Introduction_e1}) holds with $\;F = T_k,\;$ $\;G = \Lambda,\;$ $a_n = \dfrac{v(b_n)}{k}, b_n = T_k^-( 1 - 1/n).$
\end{enumerate} \label{Thm_FkNB3}
\end{thm}
\subsection{A general result on tail behaviour of random $k$-th upper order statistics}
The following result, proved in Barakat and Nigm (2002) for order statistics under power normalization, can be proved just by replacing the power normalization there by linear normalization. Here the tail behaviour of the limit law is studied.
\begin{thm} If $\;\{X_n, n \geq 1\}\;$ is a sequence of iid rvs with df $\;F,\;$ $F \in D_l(G),\;$ $G = \Phi_\alpha \;$ or $\;\Psi_\alpha\;$ or $\;\Lambda,\;$ $k\;$ is a fixed positive integer and the positive-integer valued rv $\;N_n\;$ is such that $\;\frac {N_n}n\;$ converges in probability to $\;\tau,\;$ a positive valued rv, then
\[ \lim_{n\to \infty} P(X_{N_n-k+1:N_n} \leq a_nx+b_n) = \sum_{i=0}^{k-1}\dfrac{(-\log G(x))^i}{i!} \int_{0}^{\infty} y^i G^y(x) dP(\tau \leq y), \;\;x \in \mathbb R. \]
\end{thm}
\begin{thm} If $\displaystyle B_k(x)= \sum_{i=0}^{k-1}\dfrac{(-\log F(x))^i}{i!} E_{\tau}(\tau^i F^\tau(x))\;$ with $\;F \;$ an absolutely continuous df, then
\begin{enumerate}
\item[(a)] $B_k\;$ is a df with $\;r(B_k) = r(F),\;$ pdf $\displaystyle b_{k}(x)=\dfrac{(-\log F(x))^{k-1}}{(k-1)!} E_{\tau}(\tau^{k} F^{\tau-1}(x)),\;$ and satisfying the recurrence relation $\displaystyle B_{k+1}(x) = B_k(x) +\dfrac{(-\log F(x))^k}{k!} E_{\tau}(\tau^k F^\tau(x)), x \in \{y \in \mathbb R: F(y) > 0\} ;$
\item[(b)] $B_k$ is tail equivalent to $\;H_k.$
\end{enumerate}\label{Thm_FkBN}
\end{thm}
\section{Examples}
In this section we demonstrate the results through some examples. We consider the standard Pareto, uniform and standard normal distributions. We also consider exponential distribution as a special case of gamma distribution. For the purpose of specifying the norming constants, we use the following notations: $\;\delta_k=\dfrac 1{k!},\,$ $\theta_k=\frac 1{kB(r,k)}\;$ and $\gamma_k=\frac 1{k!}E_\tau(\tau^k);\;$ and
\begin{eqnarray*} \eta_k = \left\{ \begin{array}{cll} \delta_k & \;\; \text{if} & \;\;V_k = F_k, \\ \delta_{k+1} & \; \; \text{if} & \;\;V_k = U_k, \\ \delta_1 & \;\; \text{if} & \;\; V_k = R_k, \\ \theta_k & \;\; \text{if} & \;\; V_k = T_k, \\ \gamma_k & \;\; \text{if} & \;\; V_k = B_k. \end{array} \right. \end{eqnarray*}
\begin{enumerate}
\item[(a)] If $F(x)=\Phi_\alpha(x)=\exp \Set{-x^{-\alpha}}, \quad x>0,$ the Frechet law, then $\;V_k \in D_l( \Phi_{k\alpha})$ with norming constants $a_n=(n \eta_k)^{1\over k \alpha},\;$ and $\;b_n = 0.$
\item[(b)] For standard Pareto df with $1-F(x) \sim cx^{-\alpha}, \quad x>1,$ for some constants $c>0,\alpha>0,$ we have $F \in D_l(\Phi_\alpha)$ with the norming constants $b_n=0$ and $a_n=(cn)^{\frac 1\alpha}$ and with $\;V_k \in D_l( \Phi_{k\alpha})$ with norming constants $a_n=(c^k n\eta_k)^{-1\over k \alpha},\;$ and $\;b_n = 0.$
\item[(c)] If $F(x)=\Psi_\alpha(x)=\exp \Set{-(-x)^\alpha}, \quad x<0,$ the Weibull law, $\;V_k \in D_l( \Psi_{k\alpha})$ with norming constants $\;a_n=(n\eta_k)^{1\over k \alpha}\;$ and $\;b_n = 0.$
\item[(d)] If $F(x)=x, \quad 0<x<1\;$ is the standard uniform distribution, then $F \in D_l(\Psi_\alpha)$ with norming constants $a_n=n^{-1}\;$ and $\;b_n = 1,\;$ and $\;V_k \in D_l( \Psi_{k\alpha})$ with norming constants $a_n=(n\eta_k)^{1\over k \alpha}$ and $b_n = 1.$
\item[(e)] If $\;F(x)= \dfrac 1{\sqrt{2\pi}} \int_{-\infty}^x e^{-x^2/2} dx, x \in \mathbb R, \;$ the standard normal distribution, then using Mills' ratio as in Feller (1971), $1-F(x) \sim \frac 1x F^\prime(x)=\frac 1x \frac 1{\sqrt{2\pi}} \exp \Set{-\frac {x^2}2}$. Further $\;F\;$ is a von-Mises function as in (\ref{von-Mises}) and $\;F \in D_l(\Lambda)\;$ with auxiliary function $\;v(x) = \frac {1-F(x)}{F^\prime(x)}\sim \frac 1x.\;$ Then the norming constants can be chosen as $\;a_n=(2\ln n)^{-1/2}, b_n=\sqrt{2\ln n}-\frac 12 \left(\frac {\ln 4\pi +\ln \ln n}{\sqrt{2\ln n}}\right).\;$ Further, $\;V_k \in D_l(\Lambda)\;$ with auxiliary function $\frac 1k v(x)={1\over kx}\;$ and the norming constants can be chosen as $\;a_n=\Set{\frac 2k (\ln n+\ln \delta_1)}^{-1/2}, b_n=\sqrt{\frac {2 (\ln n+\ln \delta)}k}-\frac {\sqrt k} 2 \Set{\frac {\ln 4\pi +\ln (\ln n+\ln \delta) -\ln k}{\sqrt {2\ln n+\ln \delta}}}.$
\item[(f)] Consider a DF $F(x)=1-\exp \Set{-\frac x{1-x}}\; \mbox{if} \; 0 \le x < 1 \;.$ Then simple computations show that $ \lim_{x \to1} \frac {\set{1-F(x)}F^{\prime\prime}(x)}{\set{F^\prime (x)}^2}=\lim_{x \rightarrow 1} \frac {\set{1-F(x)}^2(1-2x)}{(1-x)^4} \times \frac {(1-x)^4}{\set{1-F(x)}^2}=-1.$ Further $F^{\prime\prime}(x) < 0$ if for $x>\frac 12.$ Then $F\in D_l(\Lambda) $ with the auxiliary function $\;v(x) = \frac {1-F(x)}{F^\prime(x)}=(1-x)^2\;$ and the norming constants $\; b_n= \frac {\ln n }{1+\ln n}\;$ and $\;a_n= \frac 1{\set{1+\ln n}^2}.$ Further $V_k\in D_l(\Lambda) $ with the auxiliary function $\;\frac 1k v(x)=\frac 1k (1-x)^2\;$ and the norming constants $\; b_n=\frac {\ln n+\ln \delta_1 }{k+\ln n+\ln \delta_1}\;$ and $\;a_n= \frac 1{\set{1+\ln n+\ln \delta_1}^2}.$
\item[(g)] Consider the Gamma distribution with DF satisfying $F^\prime(x)=\dfrac {x^\alpha e^{-x}}{\Gamma(\alpha+1)}, \quad x>0, \alpha>0.$ The $F\in D_l(\Lambda)$ with a constant auxiliary function $v(x)=1$ and norming constants $a_n=1, b_n= \ln n + \alpha \ln \ln n - \ln \Gamma(\alpha+1).$ Further $V_k\in D_l(\Lambda) $ with the auxiliary function $\;\frac 1k v(x)=\frac 1k\;$ and the norming constants $a_n=\dfrac 1k, b_n= \dfrac {\ln n+\ln \delta_1 }k + \alpha \ln( \ln n+\ln \delta_1) - \alpha\ln k- \ln \Gamma(\alpha+1).$\\ In particular when $\alpha=0$ we get exponential distribution with PDF $f(x)=e^{-x}, \quad x>0.$ The corresponding norming constants are $a_n=1,b_n=\ln n$ for $F$,and $a_n=\frac 1k,b_n=\frac {\ln n + \ln \delta_1} k$ for $V_k.$
\item[(h)] For Log-Gamma distribution with df $F^\prime(x)=\frac {\alpha^\beta x^{-\alpha-1} (\ln x)^{\beta-1}}{\Gamma(\beta)},$ $\;F \in D_l(\Phi_\alpha)\;$ with norming constants $b_n=0$ and $\;a_n=((\Gamma(\beta))^{-1}(\ln n)^{\beta-1}n^{1/\alpha}.$ Further $\;V_k \in D_l(\Phi_{k\alpha})\;$ with norming constants $b_n=0$ and $b_n=0$ and $\;a_n=\set{\frac {(\ln n +\ln \delta_1)^{\beta-1} n^{\frac 1k}}{k^{\beta-1}\Gamma(\beta)(k!)^{\frac 1k}}}^{\frac 1\alpha}.$
\item[(i)] For Cauchy distribution with df $F^\prime(x)=\frac 1{\pi(1+x^2)},$ $\;F \in D_l(\Phi_\alpha)\;$ with norming constants $b_n=0$ and $\;a_n=n/\pi.$ Further $\;V_k \in D_l(\Phi_{k\alpha})\;$ with norming constants $b_n=0$ and $\;a_n=\frac 1\pi \Set { n\delta_1}^{\frac 1k}.$
\item[(j)] For Beta distribution with df $F(x)=\frac {x^{\alpha-1}(1-x)^{\beta-1}}{B(\alpha,\beta)},$ $\;F \in D_l(\Psi_\alpha)\;$ with norming constants $b_n=1$ and $\;a_n=\Set{n \frac {\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta+1)}}^{-1/\beta}.$ Further $\;V_k \in D_l(\Phi_{k\alpha})\;$ with norming constants $b_n=1$ and $\;\Set{ \Set{n\delta_1 }^{-\frac 1k} \frac {\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta+1)} }^{-\frac 1\beta}.$
\end{enumerate}
\section{Stochastic orderings}
In this section, we study stochastic orderings between the rvs having dfs $\;F, \, H_k\;$ and the limit dfs for the $\;k$-th random upper order statistic. We denote the rvs, by abuse of notation, by the dfs themselves. A rv $X$ is said to be stochastically smaller than $Y,$ denoted by $X \le_{\mbox{st}} Y$ if for all real $x,$ the df of $X,$ $P(X\ge x) \le P(Y\ge x),$ the df of $Y.$
\begin{thm} The following are true.
\begin{enumerate}
\item[(i)] $U_1 \le_{\mbox{st}} F.$
\item[(ii)] $H_k \le_{\mbox{st}} H_{k+1}, F_k(x) \le_{\mbox{st}} F_{k+1}(x), R_k(x) \le_{\mbox{st}} R_{k+1}(x), T_k(x) \le_{\mbox{st}} T_{k+1}(x).$
\item[(iii)] $ U_k(x) \le_{\mbox{st}} U_{k+1}(x)$ and $ U_k(x) \le_{\mbox{st}} F(x).$
\item[(iv)] $R_k(x) \le_{\mbox{st}} H_k(x) \le_{\mbox{st}} F(x).$
\item[(v)] $ F_k(x) \le_{\mbox{st}} F(x)$ and $T_k(x) \le_{\mbox{st}} F^k(x).$
\item[(vi)] $R_k(x) \le_{\mbox{st}} T_k(x) $
\item[(vii)] $F_k(x) \le_{\mbox{st}} U_k(x) $
\end{enumerate}\label{Thm_StOr}
\end{thm}
\section{Some preliminary results and proofs of some main results}
The following lemma is useful for computations involving derivatives.
\begin{lem} \label{lem:lr} Let $\;S,\;h\;$ be two real valued functions of a real variable and $\;S^{(r)}\;$ denote the $\;r$-th derivative of $\;S,\,r \geq 1,\;$ an integer. Whenever the derivatives exist, the following are true.
\begin{enumerate}
\item[(a)] If $\; S(x) =\dfrac {h(x)}{1-x}, x \ne 1,\;$ $\;S^{(r)}(x)=r! \sum_{i=0}^r \dfrac {h^{(i)}(x)}{i!(1-x)^{r-i+1}}.\;$
\item[(b)] If $\;S(x)=h(x)e^x,\;$ $\;S^{(r)}(x)=e^x \sum\limits_{i=0}^r \binom ri h^{(i)}(x).$
\item[(c)] If $\;S(x)=\dfrac {x^m}{(1-x)^n},$ $x\ne 1,\;$ for some integers $\;m \ge n \ge 1,\;$ $\;S^{(r)}(x)=r!\sum_{i=0}^r \binom m{r-i} \binom {n+i-1}i \dfrac {x^{m-r+i}}{(1-x)^{n+i}}.$ \end{enumerate}
\end{lem}
\begin{proof}
(a) By differentiating $\;(1-x)S(x)=h(x)\;$ repeatedly $\;r\;$ times with respect to $\;x,\;$ we get $\;(1-x)S^{(r)}(x)- r S^{(r-1)}(x)=h^{(r)}(x) \Rightarrow S^{(r)}(x) = \frac {h^{(r)}(x)}{1-x}+\frac{r}{1-x}S^{(r-1)}(x).$ Hence
\begin{eqnarray*}
S^{(r)}(x)&=&\frac {h^{(r)}(x)}{1-x}+\dfrac{r}{1-x}\left(\dfrac{h^{(r-1)}(x)}{1-x}+\dfrac{r-1}{1-x}S^{(r-2)}(x)\right),
\\&=&\dfrac {h^{(r)}(x)}{1-x}+\dfrac{rh^{(r-1)}(x)}{(1-x)^2}+\dfrac{r(r-1)}{(1-x)^2}\left(\frac{h^{(r-2)}(x)}{1-x}+\dfrac{r-2}{1-x}S^{(r-3)}(x)\right),
\\&=&\dfrac {r!h^{(r)}(x)}{r!(1-x)}+\dfrac{r!h^{(r-1)}(x)}{(r-1)!(1-x)^2}+\dfrac{r!h^{(r-2)}(x)}{(r-2)!(1-x)^3}+\ldots + \dfrac{r!h^{(1)}(x)}{1!(1-x)^r}+\dfrac{r!h(x)}{0!(1-x)^{r+1}}, \\ & & \text{proving (a). }
\end{eqnarray*}
(b) The proof is by induction on $\;r.\;$ For $\;r=1,$ observe that $\; S^{(1)}(x)=e^x(h(x)+ h^{(1)}(x))=e^x \sum_{i=0}^1 \binom 1i h^{(i)}(x), \;\; \mbox{where} \; h^{(0)}(x) = h(x),$ which is true.
Assuming that the result is true for $\;r=s,\;$ that is, $\;S^{(s)}(x)=e^x \sum\limits_{i=0}^s \binom si h^{(i)}(x),\;$ and differentiating again with respect to $\;x,\;$ we have
\begin{eqnarray*}
S^{(s+1)}(x)&=&e^x \sum_{i=0}^s \binom si h^{(i)}(x)+e^x \sum_{i=0}^s \binom si h^{(i+1)}(x),
\\&=&e^x \sum_{i=0}^s \binom {s+1}i \left(\dfrac {s+1-i}{s+1}\right) h^{(i)}(x)+e^x \sum_{i=0}^s \binom {s+1}{i+1}\left( \dfrac {i+1}{s+1}\right)h^{(i+1)}(x),
\\&=&e^x \sum_{j=0}^{s+1} \binom {s+1}j \left(\dfrac {s+1-j}{s+1}\right) h^{(j)}(x)+e^x \sum_{j=0}^{s+1} \binom {s+1}j\left( \dfrac j{s+1}\right) h^{(j)}(x),
\\&=&e^x \sum_{j=0}^{s+1} \binom {s+1}j h^{(j)}(x), \;\;\text{which completes the proof of (b).}
\end{eqnarray*}
(c) The proof is by induction on $\;r.\;$ For $\;r=1,\;$ observe that $\; S^{(1)}(x)=\dfrac{mx^{m-1}}{(1-x)^n}+\dfrac {nx^m}{(1-x)^{n+1}} = \sum_{i=0}^1 \binom m{1-i} \binom {n+i-1}i \dfrac {x^{m-1+i}}{(1-x)^{n+i}},$ which is true.
Assuming that the result is true for $\;r=s,\;$ that is, $\;S^{(s)}(x)=s!\sum\limits_{i=0}^s \binom m{s-i} \binom {n+i-1}i \dfrac {x^{m-s+i}}{(1-x)^{n+i}},\;$ and differentiating again with respect to $\;x,\;$ we have
\begin{eqnarray*}
&& S^{(s+1)}(x)= s!\sum_{i=0}^s \binom m{s-i} \binom {n+i-1}i \left( \dfrac {(m-s+i) x^{m-s+i-1}}{(1-x)^{n+i}}+\dfrac {(n+i) x^{m-s+i}}{(1-x)^{n+i+1}}\right),
\\&=& s!\sum_{i=0}^s \binom m{s-i+1} \binom {n+i-1}i \dfrac {(s+1-i) x^{m-s-1+i}}{(1-x)^{n+i}}+ s!\sum_{i=0}^s \binom m{s-i} \binom {n+i}{i+1} \dfrac {(i+1) x^{m-s+i}}{(1-x)^{n+i+1}},
\\&=& s!\sum_{j=0}^{s+1} \binom m{s+1-j} \binom {n-1+j}j \dfrac {(s+1-j) x^{m-s-1+j}}{(1-x)^{n+j}}+ s!\sum_{j=0}^{s+1} \binom m{s+1-j} \binom {n-1+j}j \dfrac {j x^{m-s-1+j}}{(1-x)^{n+j}},
\\&=& (s+1)! \sum_{j=0}^{s+1} \binom m{s+1-j} \binom {n-1+j}j \dfrac {j x^{m-s-1+j}}{(1-x)^{n+j}}, \;\;\text{proving (c).}
\end{eqnarray*}
\end{proof}
\begin{rem} We have the following particular cases.
\begin{enumerate}
\item[(i)] In (a) of Lemma \ref{lem:lr}, if $\;h(x)=x^m\;$ for some fixed positive integer $\;m\ge 1,\;$ we have $\; h^{(l)}(x)=\frac{m!}{(m-l)!} x^{m-l}\;$ and $\; S^{(r)}(x)=r! \sum_{l=0}^r \frac{m! x^{m-l}}{l!(m-l)!(1-x)^{r+1-l}}=r! \sum_{l=0}^r \binom mr\frac {x^{m-l}}{(1-x)^{r+1-l}}.$
\item[(ii)] In (b) of Lemma \ref{lem:lr}, if $\;h(x)=x^m\;$ then $\;S^{(r)}(x)=e^x \sum\limits_{l=0}^r \binom rl\frac {m!}{(m-l)!} x^{m-l}.$
\end{enumerate}
\end{rem}
\begin{proof}[\textbf {Proof of Theorem \ref{lem:1}:}] For real $\;x < y,$ $F(x)\le F(y)\;$ implies that $\;\set{1-F(x)}^k \ge \set{1-F(y)}^k,\;$ so that $\;H_k(x) \le H_k(y).\;$ Hence $\;H_k\;$ is non-decreasing.
Since $\;F\;$ is right-continuous, $\;H_k\;$ is right-continuous. Also, $\; \lim\limits_{x\rightarrow-\infty} H_k(x)=0,\;$ and $\;\lim\limits_{x\rightarrow \infty} H_k(x)=1. \;$
Further, $\;H_k\;$ is also absolutely continuous with pdf $\; h_k(x)=H_k^\prime(x)=k\set{1-F(x)}^{k-1} f(x), x \in \mathbb R.$
Observe that $\;H_k(r(F))=1-(1-F(r(F)))^k=1\;$ so that $\;r(H_k)\le r(F).$ If possible let $\;r(H_k)<r(F).\;$ Then $1=H_k(r(H_k))=1-(1-F(r(H_k)))^k<1\;$ a contradiction so that $\;r(H_k)= r(F).$ Now we prove (a), (b) and (c).
\begin{enumerate}
\item[(a)] If $\;F\in D_l(\Phi_\alpha),\;$ then by (a) of Theorem $\ref{T-l-max},$ $\; 1-F\in RV_{-\alpha} \Rightarrow \lim_{t \rightarrow \infty}\dfrac {1-F(tx)}{1-F(t)} = x^{-\alpha}.\;$ Further, $\;\lim_{t \rightarrow \infty}\dfrac {1-H_k(tx)}{1-H_k(t)}=$ $\lim_{t \rightarrow \infty}\left(\dfrac {1-F(tx)}{1-F(t)}\right)^k$ $ = x^{-k\alpha}. \;$ Hence $\;1-H_k\in RV_{-k\alpha}\;$ which implies that $\;H_k\;$ belongs to $\;D_l(\Phi_{k\alpha}).\;$ Observing that $\;1-H_k=(1-F)^k,\;$
$\; \left(\dfrac 1{1-H_k}\right)^- (n)= \left(\dfrac 1{(1-F)^k}\right)^-(n)=\left(\dfrac 1{1-F}\right)^- \left(n^{\frac 1k}\right),\;$ we obtain the norming constants, proving (a).
\item[(b)] If $\;F\in D_l(\Psi_\alpha),\;$ then $\;\displaystyle 1-F\left(r(F)-\frac 1x\right) \in RV_{-\alpha} \Rightarrow \lim_{t \rightarrow \infty}\dfrac{1-F\left(r(F)-\dfrac 1{tx}\right)}{1-F\left(r(F)-\dfrac 1t\right)} = x^{-\alpha}.\;$ Further, $\;r(H_k)=r(F).\;$ It follows that
$\;\lim_{t \rightarrow \infty}\frac {1-H_k\left(r(H_k)-\frac 1{tx}\right)}{1-H_k\left(r(H_k)-\frac 1t\right)}$ \\ $=\lim_{t \rightarrow \infty} \left(\frac {1-F\left(r(F)-\frac 1{tx}\right)}{1-F\left(r(F)-\frac 1t\right)}\right)^k= x^{-k\alpha}.$
The rest of the proof is on lines similar to the proof in (a) and is omitted.
\item[(c)] Since $\;F\in D_l(\Lambda),\;$ there exist a constant $\;c>0,\;$ continuous, positive function $\;v,\;$ with $\;\lim\limits_{t\uparrow r(F) } v^\prime(t)=0\;$ such that for all $\;t \in (z, r(F)),\;$ with $\;z < r(F),\;$ $\;F\;$ is a von-Mises function which has a representation $\displaystyle 1-F(x)=c\exp \left(-\int_z^x \frac 1{v(t)} dt\right),\;$ and $\;F\;$ satisfies the condition $\;\displaystyle \lim_{x\rightarrow \infty}\dfrac{\left(1-F(x)\right)F^{\prime\prime}(x)}{\left(F^\prime (x)\right)^2}=-1.\;$ We now show that $\;H_k\;$ is also a von-Mises function and use (e) of Theorem $\ref{T-l-max}$ to conclude that $\;H_k \in D_l(\Lambda).\;$ \\
Since $H_k^\prime(x)=k\left(1-F(x)\right)^{k-1}F^\prime(x),$ and $H_k^{\prime\prime}(x)=k\left(1-F(x)\right)^{k-1}F^{\prime\prime}(x)-k(k-1)\left(1-F(x)\right)^{k-2} \left(F^\prime(x)\right)^2,$ it follows that \[\lim_{x\rightarrow \infty}\dfrac{(1-H_k(x))H_k^{\prime\prime}(x)}{(H_k^\prime (x))^2}=\lim_{x\rightarrow \infty} \frac 1k \left(\frac{(1-F(x))F^{\prime\prime}(x)}{(F^\prime (x))^2}-k+1\right) = -1,\] and hence from (\ref{von-Mises}), $\;H_k\;$ is a von-Mises function and belongs to $\;D_l(\Lambda)\;$ with norming constants $\;a_n=\dfrac 1k v(b_n)\;$ and $\;b_n=\left(\dfrac 1{1-H_k}\right)^-(n),\;$ proving (c).
\end{enumerate}
\end{proof}
\begin{proof}[\textbf{Proof of Theorem \ref{Thm-Fk}:}]
\begin{enumerate}
\item[(a)] By Theorem \ref{thm:r-max-df}, $\;F_k\;$ is a df with pdf $\;f_k(x)=$ \\ $\frac {f(x)}{(k-1)!} (-\ln F(x))^{k-1}.\;$ To show that $\;r(F) = r(F_k),\;$ we consider two cases. First, if $\;r(F)=\infty,\;$ then $\;\lim_{x\rightarrow r(F)} F_k(x)=\lim_{x\rightarrow r(F)} F(x)\sum_{r=0}^{k-1} \frac {(-\ln F(x))^r}{r!}=1 \Rightarrow r(F_k) \le r(F).\;$ If possible, let $\;r(F_k)<\infty.\;$ Then $\;1 =F_k(r(F_k))$ \\ $=F(r(F_k))\sum_{i=0}^{k-1} \frac {(-\ln F(r(F_k)))^i}{i!}$ $ <F(r(F_k))\sum_{i=0}^\infty \frac {(-\ln F(r(F_k)))^i}{i!}$ \\ $=F(r(F_k))\exp \set{-\ln F(r(F_k))}=1,\;$ a contradiction proving that $\; r(F_k)=\infty.\;$ If $\;r(F)<\infty,\;$ arguing as above, if possible, let $\;r(F_k)<r(F).\;$ Since $\;F(r(F_k))=1,\;$ we must have $\;r(F_k)=r(F)-\epsilon,\;$ for some $\;\epsilon >0.\;$ Repeating steps as earlier we get a contradiction proving that $\;\epsilon \;$ must be equal to 0 and hence $\;r(F_k) = r(F).\;$
Defining $\; h(x)=\dfrac {-\ln F(x) }{1-F(x)},\;$ we have $\; \lim_{x\rightarrow r(F)} h(x)=1\;$ by L'Hospital's rule, and $\; \lim_{x \rightarrow r(F)}\frac {1-F_k(x)}{(1-F(x))^k}$ $=\lim_{x \rightarrow r(F)}\frac {-f_k(x)}{-k(1-F(x))^{k-1}f(x)}$ $
= \lim_{x \rightarrow r(F)}\frac {-\frac {f(x)}{(k-1)!} (-\ln F(x))^{k-1}}{-k(1-F(x))^{k-1}f(x)} =$ $ \lim_{x \rightarrow r(F)} \frac {h^{k-1}(x)}{k!}=$ $\frac 1{k!},\;$ showing that $\;F_k\;$ is tail equivalent to $\;H_k.$
\item[(b)] If $\;F\in D_l(\Phi_\alpha),\;$ then by (a) of Theorem $\ref{T-l-max},$ $\; 1-F\in RV_{-\alpha} \Rightarrow \lim_{t \rightarrow \infty}\dfrac {1-F(tx)}{1-F(t)} = x^{-\alpha}.\;$ Further, $\;r(F)=r(F_k)=\infty,\;$ and using (a), and multiplying and dividing by $\;\left(\dfrac {1-F(tx)}{1-F(t)}\right)^k,\;$ we get $\;\lim_{t \rightarrow \infty}\dfrac {1-F_k(tx)}{1-F_k(t)}=$ $\lim_{t \rightarrow \infty}\left(\dfrac {1-F(tx)}{1-F(t)}\right)^k =$ $ x^{-k\alpha}.\;$
Hence $\;1-F_k \in RV_{-k\alpha},\;$ which implies that $\;F_k\;$ belongs to $\;D_l(\Phi_{k\alpha}).\;$ Observing that $\;1-H_k=(1-F)^k\;$ and $\;1-F_k \sim \dfrac 1{k!} (1-H_k)=\dfrac 1{k!} (1-F)^k,\;$ and
$\; \left(\dfrac 1{1-H_k}\right)^- (n)= \left(\dfrac 1{(1-F)^k}\right)^-(n)=\left(\dfrac 1{1-F}\right)^- \left(n^{\frac 1k}\right),\;$ and $\;\left(\dfrac 1{1-F_k}\right)^- (n)=\left(\dfrac {k!}{(1-F)^k}\right)^- (n)=\left(\dfrac 1{1-F}\right)^- \left(\dfrac n{k!}\right)^{\frac 1k}, $ we obtain the norming constants, proving (b).
\item[(c)] If $\;F\in D_l(\Psi_\alpha),\;$ then by (b) of Theorem $\ref{T-l-max},$ $\;\displaystyle 1-F\left(r(F)-\frac 1x\right) \in RV_{-\alpha} \Rightarrow \lim_{t \rightarrow \infty}\dfrac{1-F\left(r(F)-\dfrac 1{tx}\right)}{1-F\left(r(F)-\dfrac 1t\right)} = x^{-\alpha}.\;$ Further, $\;r(F_k)=r(F)\;$ and the rest of the proof is on lines similar to the proof in (b) and is omitted.
\item[(d)] We have $\;\lim_{x\rightarrow \infty} \dfrac{1-F_k(x)}{1-H_k(x)} = \lim_{x\rightarrow \infty} \dfrac {1-F_k(x)}{\set{1-F(x)}^k} = \frac 1{k!}.$
Thus $\;F_k\;$ and $\;H_k\;$ are tail equivalent and hence by proof of (c) of Theorem \ref{lem:1}, $\;F_k \in D_l(\Lambda)\;$ and the function $\;\dfrac 1k v(t)\;$ satisfies all the conditions for it to be an auxiliary function, proving (d).
\end{enumerate}
\end{proof}
\begin{proof}[\textbf{Proof of Theorem \ref{Thm_DU1}:}]
Setting the operator $D^r=\dfrac{d^r}{dF^r(x)}, r=1,2,3,\ldots,\;$ and considering the term corresponding to $\;i=0\;$ in (\ref{eqn:gr-max1}), we have $\; \frac 1n \sum_{r=m+1}^{m+n} F^r(x) = \frac {F^{m+1}(x)-F^{m+n+1}(x)}{n(1-F(x))}.\;$ Applying Lemma \ref{lem:lr}, we get
$\;D^i \Set{\frac {F^{m+1}(x)-F^{m+n+1}(x)}{1-F(x)}} =$ \\ $i! \sum_{l=0}^i\set{\binom {m+1}l \frac {F^{m+1-l}(x)}{(1-F(x))^{i+1-l}}- \binom {m+n+1}l \frac {F^{m+n+1-l}(x)}{(1-F(x))^{i+1-l}}}.\;$ For the general $\;i$th term, for $\;i\ge 1,\;$ we have $\;\frac 1n \sum_{r=m+1}^{m+n} \binom ri F^{r-i}(x) \set{1-F(x)}^i = \frac {\set{1-F(x)}^i}{i!n} \sum_{r=m+1}^{m+n} \frac {r!}{(r-i)!} F^{r-i}(x) $
\begin{eqnarray*}
&=& \frac {\set{1-F(x)}^i}{i!n} \sum_{r=m+1}^{m+n} D^i F^r(x) =\frac {\set{1-F(x)}^i}{i!n} D^i \set{\sum_{r=m+1}^{m+n} F^r(x)}
\\&=& \frac {\set{1-F(x)}^i}{i!n} D^i \Set{\frac {F^{m+1}(x)-F^{m+n+1}(x)}{1-F(x)}}
\\&=&\frac {\set{1-F(x)}^i}{i!n} i! \sum_{l=0}^i\set{\binom {m+1}l \frac {F^{m+1-l}(x)}{(1-F(x))^{i+1-l}}- \binom {m+n+1}l \frac {F^{m+n+1-l}(x)}{(1-F(x))^{i+1-l}}}
\end{eqnarray*}
\begin{eqnarray*}
&=&\frac 1n \sum_{l=0}^i \set{ \binom {m+1}l F^{m+1-l}(x)- \binom {m+n+1}l F^{m+n+1-l}(x)}\set{1-F(x)}^{l-1}
\\&=& \frac {F^{m+1}(x)-F^{m+n+1}(x)}{n(1-F(x))}+ \sum_{l=1}^i \binom {m+1}l F^{m+1-l}(x) \frac {\set{n(1-F(x))}^{l-1}}{n^l} -\\ && \sum_{l=1}^i \frac {(m+n+1)(m+n)\cdots(m+n+2-l)}{n^l l!} F^{m+n+1-l}(x) \set{n(1-F(x))}^{l-1}.
\end{eqnarray*}
Substituting these in (\ref{eqn:gr-max1}), we get
\begin{eqnarray*}
F_{k:N_n}&=& \frac {F^{m+1}(x)-F^{m+n+1}(x)}{n(1-F(x))}+\sum_{i=1}^{k-1} \frac {F^{m+1}(x)-F^{m+n+1}(x)}{n(1-F(x))}+\\&& \sum_{i=1}^{k-1} \sum_{l=1}^i \binom {m+1}l F^{m+1-l}(x) \frac {\set{n(1-F(x))}^{l-1}}{n^l} -\\&&\sum_{i=1}^{k-1} \sum_{l=1}^i \frac 1{l!} \prod_{j=-1}^{l-2} \frac {m+n-j}n F^{m+n+1-l}(x) \set{n(1-F(x))}^{l-1}.
\end{eqnarray*}
Replacing $\;x\;$ by $\;a_n x+b_n\;$ and taking limit as $\;n \rightarrow \infty\;$ we have
\begin{eqnarray*}
J_k(x)&=&\lim_{n\to \infty} F_{k:N_n}(a_nx+b_n)
\\&=&\frac {1-G(x)}{-\ln G(x)}+\sum_{i=1}^{k-1} \frac {1-G(x)}{-\ln G(x)}+\sum_{i=1}^{k-1}\sum_{l=1}^i 0+\sum_{i=1}^{k-1} \sum_{l=1}^i \frac 1{l!} G(x)
(-\ln G(x))^{l-1}
\\&=& k\Set{\frac {1-G(x)}{-\ln G(x)}}-(k-1)G(x)-G(x) \sum_{l=2}^{k-1} (k-l) \frac {(-\ln G(x))^{l-1}}{l!},
\end{eqnarray*}
completing the proof.
\end{proof}
\begin{proof}[\textbf{Proof of Theorem \ref{Thm-FkW}:}]
Clearly $\lim_{x\to +\infty} U_1(x)=$ $\lim_{x\to +\infty} \frac {1-F(x)}{-\ln F(x)}=
$ $\frac {-f(x)}{-\frac {f(x)}{F(x)}} =1$ and \\
$ \lim_{x\to -\infty} U_1(x)=\lim_{x\to -\infty} \frac {1-F(x)}{-\ln F(x)}=0.$ Further $U_1$ is right continuous since $F$ is. Differentiating with respect to $x$ we get
$u_1(x)=U_1^\prime(x)=\frac {-\ln F(x) (-f(x))-(1-F(x))\frac {-f(x)}{F(x)}}{(-\ln F(x))^2}=$ $\frac {-f(x)F(x)+f(x)U_1(x)}{-F(x) \ln F(x)}$ $=\frac {f(x)}{F(x)}\frac {U_1(x)-F(x)}{-\ln F(x)}.$
Now we claim that $U_1(x) \ge F(x).$ If the claim is true, then $\;\frac {1-F(x)}{F(x)} \ge -\ln F(x) \Rightarrow \frac 1{F(x)} -1 \ge \ln \frac 1{F(x)}.$ Setting $\;y=\frac 1{F(x)}-1,\;$ since $F(x)\le 1$ it follows that $y\ge 0.$ Then from the preceding expression it follows that $\; y\ge \ln (1+y) \Rightarrow e^y \ge 1+y \;$ which is true and hence the claim is true. Thus $U_1$ is non-decreasing and so is a df.
Observe that $U_1(r(F))=\lim_{x\to r(F)} \frac {1-F(x)}{-\ln F(x)}=1$ so that $r(U_1)\le r(F).$ If possible, let $r(U_1)< r(F).$ Then $0<F(r(U_1))<1$ and then using Taylor's expansion we have $\;-\ln (F(r(U_1)))=$ $ -\ln (1-(1-F(r(U_1))))=$ $(1-F(r(U_1)))+\frac {(1-F(r(U_1)))^2}{2}+\ldots > (1-F(r(U_1))).\;$
Hence $U_1(r(U_1))<1$ which is a contradiction, proving $\;r(U_1)=r(F).$ We then have $\; \lim_{x\to \infty} \frac {1-U_1(x)}{1- F(x)}=$ $ \lim_{x\to \infty} \frac {U_1^\prime(x)}{f(x)}=$ $\lim_{x\to \infty} \frac 1{F(x)} \Set{\frac {1-F(x)+F(x) \ln F(x)}{(1-F(x))^2}}$ $=\frac 12,\;$ proving the theorem.
\end{proof}
\begin{proof}[\textbf{Proof of Theorem \ref{Thm-FkDU}:}]
\begin{enumerate}
\item[(a)] Clearly $\lim_{x\to +\infty}U_k(x)=1$ and $\lim_{x\to -\infty}U_k(x)=0.$ Further $U_k$ is right continuous since $F$ is. Differentiating, we get
\begin{eqnarray*}
u_k(x) &=& ku_1 (x)-f(x) \sum_{l=1}^{k-1} (k-l) \frac {(-\ln F(x))^{l-1}}{l!}+f(x)\sum_{l=1}^{k-1} (k-l) \frac {(l-1)(-\ln F(x))^{l-2}}{l!}
\\&=& ku_1 (x)-f(x) \sum_{l=2}^k (k-l+1) \frac {(-\ln F(x))^{l-2}}{(l-1)!}+f(x)\sum_{l=1}^{k-1} (k-l) \frac {(l-1)(-\ln F(x))^{l-2}}{l!}
\\&=& ku_1 (x)-f(x) \sum_{l=2}^k l(k-l+1) \frac {(-\ln F(x))^{l-2}}{l!}+f(x)\sum_{l=2}^k (k-l)(l-1) \frac {(-\ln F(x))^{l-2}}{l!}
\\&=& \frac {kf(x)}{F(x)}\Set{\frac {1-F(x)+F(x) \ln F(x)}{(-\ln F(x))^2}} - k f(x) \sum_{l=2}^k \frac {(-\ln F(x))^{l-2}}{l!}
\\&=& \frac {kf(x)}{(-\ln F(x))^2}\Set{\frac 1{F(x)} - \sum_{l=0}^k \frac {(-\ln F(x))^l}{l!}} >0, \mbox{since} \; \sum_{l=0}^k \frac {(-\ln F(x))^l}{l!}<e^{-\ln F(x)}=\frac 1{F(x)}
\end{eqnarray*}
Hence $U_k$ is non-decreasing and hence is a df. Observe that $\; \lim_{x\to r(F)} U_k(x)=\lim_{x\to r(F)} \set{ kU_1(x)- (k-1)F(x) -F(x) \sum_{l=2}^{k-1} (k-l) \frac {(-\ln F(x))^{l-1}}{l!}}=1,\;$ so that $r(U_k)\le r(F).$ If possible, let $\;r(U_k)<r(F).\;$ For convenience set $\xi=-\ln F(r(U_k)).$ Then it follows that $e^{-\xi}= F(r(U_k))$ and for $F(r(U_k))<1$ we have $\xi>0,$ and
\begin{eqnarray*}
1 = U_k(r(U_k)) &=& k\Set{\frac {1-F((r(U_k)))}{-\ln F((r(U_k)))}} -F(r(U_k)) \sum_{l=1}^{k-1} (k-l) \frac {(-\ln F(r(U_k)))^{l-1}}{l!}
\\&=& k\Set{\frac {1-e^{-\xi}}{\xi}}-\sum_{l=1}^{k-1} (k-l) \frac {e^{-\xi} \xi^{l-1}}{l!}
= k\sum_{j=1}^\infty \frac{e^{-\xi} \xi^{j-1}}{j!}-\sum_{l=1}^{k-1} (k-l) \frac {e^{-\xi} \xi^{l-1}}{l!}
\\&=& k\sum_{j=k}^\infty \frac{e^{-\xi} \xi^{j-1}}{j!}+\sum_{l=1}^{k-1} \frac {e^{-\xi} \xi^{l-1}}{(l-1)!}
=k\sum_{j=k}^\infty \frac{e^{-\xi} \xi^{j-1}}{j!}+1-\sum_{l=k}^\infty \frac {e^{-\xi} \xi^{l-1}}{(l-1)!}
\\&=& k\sum_{j=k+1}^\infty \frac{e^{-\xi} \xi^{j-1}}{j!}+\frac{e^{-\xi} \xi^{k-1}}{(k-1)!} +1-\sum_{l=k+1}^\infty \frac {e^{-\xi} \xi^{l-1}}{(l-1)!}- \frac {e^{-\xi} \xi^{k-1}}{(k-1)!} \\
&=& 1+ k\sum_{j=k+1}^\infty \frac{\xi^j}{j!} - \sum_{j=k+1}^\infty \frac {\xi^j}{(j-1)!} =1+ \sum_{j=k+1}^\infty \frac{\xi^j}{(j-1)!}\Set{\frac kj-1} <1,
\end{eqnarray*} a contradiction as $j>k,$ proving that $r(U_k) = r(F). $
\item[(b)] Observe that $\lim_{x\to \infty} \frac {1-U_k(x)}{1-H_k(x)} $
\begin{eqnarray*}
&=&\lim_{x\to \infty} \frac {1-kU_1(x)+F(x) \sum\limits_{l=1}^{k-1} (k-l) \dfrac {(-\ln F(x))^{l-1}}{l!}}{(1-F(x))^k}
\\&=& \lim_{x\to \infty} \frac {1-k\Set{\frac {1-F(x)}{-\ln F(x)}}+F(x) \sum\limits_{l=1}^{k-1} (k-l) \dfrac {(-\ln F(x))^{l-1}}{l!}}{(1-F(x))^k}
\\&=& \lim_{x\to \infty}\Set{\frac {1-F(x)}{-\ln F(x)}} \frac {-\ln F(x) -k(1-F(x)) +F(x) \sum\limits_{l=1}^{k-1} (k-l) \dfrac {(-\ln F(x))^l}{l!}}{(1-F(x))^{k+1}}
\\&=& \lim_{x\to \infty} \frac {-\frac {f(x)}{ F(x)} +k f(x) + f(x) \sum\limits_{l=1}^{k-1} (k-l) \dfrac {(-\ln F(x))^l}{l!}- f(x) \sum\limits_{l=1}^{k-1} (k-l) \dfrac {(-\ln F(x))^{l-1}}{(l-1)!}}{-(k+1)f(x) (1-F(x))^k}
\\&=& \lim_{x\to \infty} \frac {-\frac 1{ F(x)}+\sum\limits_{l=0}^{k-1} (k-l) \dfrac {(-\ln F(x))^l}{l!}- \sum\limits_{l=0}^{k-1} (k-l-1) \dfrac {(-\ln F(x))^l}{l!}}{-(k+1) (1-F(x))^k}
\\&=& \lim_{x\to \infty} \frac 1{F(x)} \frac {1-F(x)\sum\limits_{l=0}^{k-1} \dfrac {(-\ln F(x))^l}{l!}}{(k+1) (1-F(x))^k}
= \lim_{x\to \infty} \frac 1{(k+1) F(x)} \lim_{x\to \infty} \frac {1-F_k(x)}{(1-F(x))^k}
=\frac 1{(k+1)!}
\end{eqnarray*}
\end{enumerate}
The rest of proof is same as in Theorem \ref{Thm-Fk} except that $\;k!\;$ is replaced by $\;(k+1)!\;$
\end{proof}
\begin{proof}[\textbf{Proof of Theorem \ref{Thm_FkB}:}] \begin{itemize}
\item[(a)]
Let $\;D^r=\dfrac{d^r}{d(p_n F(x))^r},\, r=1,2,3,\ldots.\;$ Considering the term corresponding to $\;i=0\;$ in (\ref{eqn:gr-max1}), we have
\begin{eqnarray*} \sum_{r=m}^{m+n} F^r(x) \binom n{r-m}p_n^{r-m}q_n^{m+n-r} & = & F^m(x) \sum_{r=m}^{m+n} \binom n{r-m}(p_nF(x))^{r-m}q_n^{m+n-r}, \\ & = & F^m(x) (p_nF(x)+q_n)^n. \end{eqnarray*}
By Leibnitz rule for derivative of the product, we have
\[ D^i \left((p_n F (x))^m (p_nF(x)+q_n)^n\right) = i! \sum_{l=0}^i \binom ml \binom n{i-l}(p_n F (x))^{m-l} (p_nF(x)+q_n)^{n-i+l}. \]
For $\;1 \leq i \leq k-1, \;$ in (\ref{eqn:gr-max1}), we now have
\begin{eqnarray*}
& &\sum_{r=m}^{m+n}\binom ri F^{r-i}(x) (1-F(x))^i \binom n{r-m}p_n^{r-m}q_n^{m+n-r}
\\&=& \dfrac{F^{m-i}(x)(1-F(x))^i}{i!} \sum_{r=m}^{m+n} \binom n{r-m} \dfrac {r!}{(r-i)!} (p_n F (x))^{r-m}q_n^{m+n-r},
\\&=& \dfrac{F^{m-i}(x)(1-F(x))^i}{i!} (p_n F (x))^{i-m}\sum_{r=m}^{m+n} \binom n{r-m} \dfrac {r!}{(r-i)!} (p_n F (x))^{r-i}q_n^{m+n-r},
\\&=& \dfrac{F^{m-i}(x)(1-F(x))^i}{i!} (p_n F (x))^{i-m}\sum_{r=m}^{m+n} \binom n{r-m} D^i (p_n F (x))^r q_n^{m+n-r},
\\&=& \dfrac{F^{m-i}(x)(1-F(x))^i}{i!} (p_n F (x))^{i-m} D^i \sum_{r=m}^{m+n} \binom n{r-m} (p_n F (x))^r q_n^{m+n-r},
\end{eqnarray*}
\begin{eqnarray*}
&=& \dfrac{F^{m-i}(x)(1-F(x))^i}{i!} (p_n F (x))^{i-m} D^i \left( (p_n F (x))^m (p_nF(x)+q_n)^n\right).
\\&& F^{m-i}(x)(1-F(x))^i \sum_{l=0}^i \binom ml \binom n{i-l}( p_n F (x))^{i-l} (p_nF(x)+q_n)^{n-i+l}
\end{eqnarray*}
Substituting all these expressions, we get
\begin{eqnarray*}
F_{k:N_n}(x) &=& F^m(x) (p_nF(x)+q_n)^n+\sum_{i=1}^{k-1} (1-F(x))^i \binom ni p_n^i F^m(x) (p_nF(x)+q_n)^{n-i}\\&&+\sum_{i=1}^{k-1} (1-F(x))^i \sum_{l=1}^i \binom ml \binom n{i-l}p_n^{i-l}F^{m-l}(x)(p_nF(x)+q_n)^{n-i+l},
\end{eqnarray*}
\begin{eqnarray*}
\\&=& F^m(x) (p_nF(x)+q_n)^n+\sum_{i=1} ^{k-1} (1-F(x))^i \dfrac {n!}{(n-i)!i!} p_n^i F^m(x) (p_nF(x)+q_n)^{n-i}
\\&& +\sum_{i=1}^{k-1} (1-F(x))^i \sum_{l=1}^i \binom ml \dfrac {n!}{(i-l)!(n-i+l)!} p_n^{i-l}F^{m-l}(x)(p_nF(x)+q_n)^{n-i+l},
\\&=& F^m(x) (p_nF(x)+q_n)^n+\sum_{i=1}^{k-1} \dfrac {(n(1-F(x)))^i}{i!} \left(\prod_{j=0}^{l-1} \dfrac {n-j}n\right) p_n^i F^m(x) (p_nF(x)+q_n)^{n-i}
\\&& +\sum_{i=1}^{k-1} \sum_{l=1}^i \dfrac {(n(1-F(x)))^i}{n^l(i-l)!} \binom ml \left(\prod_{j=0}^{i-l-1} \dfrac {n-j}{n}\right) p_n^{i-l}F^{m-l}(x)(p_nF(x)+q_n)^{n-i+l}.
\end{eqnarray*}
Replacing $\;x\;$ by $\;a_nx + b_n\;$ above and using the facts that $\; \lim_{n \rightarrow \infty}p_n=1$ and
\[ \lim_{n \rightarrow \infty} (q_n+p_nF(a_n x + b_n))^n=\lim_{n \rightarrow \infty} \left(1-\dfrac {np_n (1-F(a_n x + b_n))}{n}\right)^n=e^{-(-\ln G(x))}=G(x), \] we get
$\; F_k(x) = G(x) +G(x) \sum_{i=1}^{k-1} \frac {(-\ln G(x))^i}{i!} +\sum_{i=1}^{k-1} \sum_{l=1}^i 0 = G(x) \sum_{i=0}^{k-1} \dfrac {(-\ln G(x))^i}{i!}.$
This completes the proof of (a).
\item[(b)] Let $\;D^r=\dfrac{d^r}{d(\lambda_n F(x))^r},\, r=1,2,3,\ldots.\;$ Considering the term corresponding to $\;i=0\;$ in (\ref{eqn:gr-max1}), we have
\[ \sum_{r=m}^\infty F^r(x)\frac {e^{-\lambda_n} \lambda_n^{r-m}}{(r-m)!} = e^{-\lambda_n} F^m(x) \sum_{r=m}^\infty \frac{(\lambda_n F (x))^{r-m}}{(r-m)!} =e^{-\lambda_n(1-F(x)} F^m(x). \]
For $\;1 \leq i \leq k-1, \;$ in (\ref{eqn:gr-max1}), we now have
\begin{eqnarray*}
& &\sum_{r=m}^\infty \binom ri F^{r-i}(x)(1-F(x))^i \dfrac {e^{-\lambda_n} \lambda_n^{r-m}}{(r-m)!},
\\&=& \dfrac{e^{-\lambda_n}F^{m-i}(x)(1-F(x))^i}{i!} \sum_{r=m}^\infty \dfrac {r!}{(r-i)!} \dfrac{(\lambda_n F (x))^{r-m}}{(r-m)!},
\\&=& \dfrac{e^{-\lambda_n}F^{m-i}(x)(1-F(x))^i}{i!} (\lambda_n F (x))^{i-m} \sum_{r=m}^\infty \dfrac {r!}{(r-i)!} \dfrac{( \lambda_n F (x))^{r-i}}{(r-m)!},
\\&=& \dfrac{e^{-\lambda_n}F^{m-i}(x)(1-F(x))^i}{i!} (\lambda_n F (x))^{i-m} \sum_{r=m}^\infty \dfrac 1{(r-m)!}D^i (\lambda_n F (x))^r,
\\&=& \dfrac{e^{-\lambda_n}F^{m-i}(x)(1-F(x))^i}{i!} (\lambda_n F (x))^{i-m} D^i\left(\left( \lambda_n F (x) \right)^m e^{\lambda_nF(x)}\right),
\end{eqnarray*}
\begin{eqnarray*}
&=& \dfrac{e^{-\lambda_n}F^{m-i}(x)(1-F(x))^i}{i!} (\lambda_n F (x))^{i-m} e^{\lambda_n F (x) }\sum_{l=0}^i \binom il \dfrac {m!}{(m-l)!} ({\lambda_n F (x) })^{m-l},\\&=& \dfrac{e^{-\lambda_n(1-F(x)}F^{m-i}(x)(1-F(x))^i}{i!} \sum_{l=0}^i \binom il \dfrac {m!}{(m-l)!} (\lambda_n F (x))^{i-l},
\\&=& \dfrac{e^{-\lambda_n(1-F(x)}F^{m-i}(x)(n(1-F(x)))^i}{i!} \left({\dfrac {\lambda_n}n F (x) }\right)^i + \\&& \dfrac{e^{-\lambda_n(1-F(x)}F^{m-i}(x)(n(1-F(x)))^i}{i!} \sum_{l=1}^i \binom il \dfrac {m!}{(m-l)!} \dfrac {(\lambda_n F (x))^{i-l}}{n^i}.
\end{eqnarray*}
Substituting all these expressions in equation (\ref{eqn:gr-max1}), we get
\begin{eqnarray*}
F_{k:N_n}(x)&=& e^{-\lambda_n(1-F(x)} F^m(x) + \sum_{i=1}^{k-1} \dfrac{e^{-\lambda_n(1-F(x)}F^{m-i}(x)(n(1-F(x)))^i}{i!} \left(\dfrac {\lambda_n}n F (x) \right)^i +\\&& \sum_{i=1}^{k-1} \dfrac{e^{-\lambda_n(1-F(x)}F^{m-i}(x)(n(1-F(x)))^i}{i!} \sum_{l=1}^i \binom il \frac {m!}{(m-l)!} \dfrac {(\lambda_n F (x))^{i-l}}{n^i}.
\end{eqnarray*}
Replacing $\;x\;$ by $\;a_n x + b_n\;$ above and using the facts and $\;\lim\limits_{n \rightarrow \infty} e^{-\lambda_n(1-F(a_nx+b_n)} = G(x),\;$ we get
\begin{eqnarray*}
F_{k}(x)&=& G(x)+\sum_{i=1}^{k-1} \frac {G(x) \set{-\ln G(x)}^i}{i!} (1) + \sum_{i=1}^{k-1} \frac {G(x) \set{-\ln G(x)}^i}{i!} \sum_{l=1}^i (0)
\\ &=& G(x)\sum_{r=0}^{k-1} \left(\dfrac {(-\ln G(x))^r}{r!}\right), \;\;\text{completing the proof of (b). }
\end{eqnarray*}
\item[(c)] Let $\;D^r=\dfrac{d^r}{d(F(x))^r},\, r=1,2,3,\ldots.\;$ Considering the term corresponding to $\;i=0\;$ in (\ref{eqn:gr-max1}), we have
\[ \sum_{r=m}^\infty F^r(x)\frac {e^{-\lambda_n} \lambda_n^{r-m}}{(r-m)!} = e^{-\lambda_n} F^m(x) \sum_{r=m}^\infty \frac{(\lambda_n F (x))^{r-m}}{(r-m)!} =e^{-\lambda_n(1-F(x)} F^m(x). \]
For $\;1 \leq i \leq k-1, \;$ in (\ref{eqn:gr-max1}), we now have
\begin{eqnarray*}
\sum_{r=n+1}^\infty F^r(x) \Set{\dfrac 1{-\ln (1-\theta_n)}} \dfrac {\theta_n^{r-n}}{r-n}
&=& \Set{\dfrac 1{-\ln (1-\theta_n)}} F^n(x) \sum_{r=n+1}^\infty \frac {(\theta_nF(x))^{r-n}} {r-n} \\&=& \Set{\dfrac 1{-\ln (1-\theta_n)}} F^n(x) \Set{-\ln (1-\theta_nF(x))}
\end{eqnarray*}
By Leibnitz rule we have
\begin{eqnarray*} && D^i \set{ - F^n(x) \ln(1-\theta_nF(x))}
\\&=& -\frac {n!}{(n-i)!} F^{n-i}(x) \ln(1-\theta_nF(x)) -\sum_{l=0}^{i-1} \binom il \frac {n!}{(n-l)!} F^{n-l}(x) \frac {(-1)^{i-l-1}(i-l-1)!(-\theta_n)^{i-l}}{(1-\theta_nF(x))^{i-l}}
\end{eqnarray*}
Now consider the general $i$th term for $i\ge 1.$
\begin{eqnarray*}
&& \sum_{r=n+1}^\infty \binom ri F^{r-i}(x) \set{1-F(x)}^i \Set{\dfrac 1{-\ln (1-\theta_n)}} \dfrac {\theta_n^{r-n}}{r-n}
\\&=& \set{1-F(x)}^i \Set{\dfrac 1{-\ln (1-\theta_n)}} \sum_{r=n+1}^\infty \frac {r!}{i!(r-i)!} F^{r-i}(x) \frac {\theta_n^{r-n}}{r-n}
\end{eqnarray*}
\begin{eqnarray*}
\\&=& \frac {\set{1-F(x)}^i}{i!} \Set{\dfrac 1{-\ln (1-\theta_n)}} D^i \set{ - F^n(x) \ln(1-\theta_nF(x))}
\\&=& -\frac {\set{1-F(x)}^i}{i!} \Set{\dfrac 1{-\ln (1-\theta_n)}} \frac {n!}{(n-i)!} F^{n-i}(x) \ln(1-\theta_nF(x)) \\&& -\frac {\set{1-F(x)}^i}{i!} \Set{\dfrac 1{-\ln (1-\theta_n)}} \sum_{l=0}^{i-1} \binom il \frac {n!}{(n-l)!} F^{n-l}(x) \frac {(-1)^{i-l-1}(i-l-1)!(-\theta_n)^{i-l}}{(1-\theta_nF(x))^{i-l}}
\end{eqnarray*}
Substituting all these expressions in equation (\ref{eqn:gr-max1}), we get
\begin{eqnarray*}
F_{k:N_n}&=& \Set{\dfrac {-\ln (1-\theta_nF(x))}{-F(x) \ln (1-\theta_n)}} F^{n+1}(x) + \sum_{i=1}^{k-1}\frac {\set{1-F(x)}^i}{i!} \Set{\dfrac {-\ln(1-\theta_nF(x)}{-F(x) \ln (1-\theta_n)}} \frac {n!}{(n-i)!} F^{n-i+1}(x) -\\&& \sum_{i=1}^{k-1} \frac {\set{1-F(x)}^i}{i!} \Set{\dfrac 1{-\ln (1-\theta_n)}} \sum_{l=0}^{i-1} \binom il \frac {n!}{(n-l)!} F^{n-l}(x) \frac {(-1)^{i-l-1}(i-l-1)!(-\theta_n)^{i-l}}{(1-\theta_nF(x))^{i-l}}
\end{eqnarray*}
Since $\lim_{n\rightarrow \infty } \Set{\dfrac {-\ln (1-\theta_nF(a_nx+b_n))}{-F(a_nx+b_n) \ln (1-\theta_n)}}=1,$ we get
\[ F_k(x) = \lim_{n\rightarrow\infty} F_{k:N_n}( a_n x + b_n ) = G(x)+\sum_{i=1}^{k-1} \frac {(-\ln G(x))^i}{i!} -\sum_{i=1}^{k-1} 0=G(x)+\sum_{i=1}^{k-1} \frac {(-\ln G(x))^i}{i!}\]
This completes the proof of (c).
\end{itemize}
\end{proof}
\begin{proof} [\textbf{Proof of Theorem \ref{Thm_FkG}:}]
Let $\;D^r=\dfrac{d^r}{d(q_n F(x))^r},\, r=1,2,3,\ldots.\;$ Considering the term corresponding to $\;i=0\;$ in (\ref{eqn:gr-max1}), we have
\[ \sum_{r=m}^\infty F^r(x) p_nq_n^{r-m}=p_n F^m(x) \sum_{r=m}^\infty (q_n F (x))^{r-m}= \left( \frac{p_nF^m(x)}{1-q_n F(x)}\right).\]
Applying (a) of Lemma \ref{lem:lr}, we get $\;D^i\left(\frac {(q_nF(x))^m}{1-q_n F(x)}\right) = i! \sum_{l=0}^i \binom ml \frac {(q_nF(x))^{m-l}}{(1-q_nF(x))^{i+1-l}}.$
For $\;1 \leq i \leq k-1, \;$ in (\ref{eqn:gr-max1}), we now have $\sum_{r=m}^\infty \binom ri F^{r-i}(x) (1-F(x))^i p_n q_n^{r-m}$
\begin{eqnarray*}
&=& \frac{p_n F^{m-i}(x)(1-F(x))^i}{i!} (q_n F (x))^{i-m}\sum_{r=m}^\infty \frac {r!}{(r-i)!} (q_n F (x))^{r-i},
\\&=& \frac{p_n F^{m-i}(x)(1-F(x))^i}{i!} ( q_n F (x))^{i-m}\sum_{r=m}^\infty D^i ( q_n F (x))^r,
\\ &=& \frac{p_n F^{m-i}(x)(1-F(x))^i}{i!} (q_n F (x))^{i-m} D^i \sum_{r=m}^\infty (q_n F (x))^r,
\\&=& \frac{p_n F^{m-i}(x)(1-F(x))^i}{i!} (q_n F (x))^{i-m} D^i \left( \frac{(q_n F (x))^m}{1-q_nF(x)}\right),
\\&=& \frac {p_n F^{m-i}(x)(1-F(x))^i}{i!} (q_n F (x))^{i-m} i! \sum_{l=0}^i \binom ml \frac {(q_nF(x))^{m-l}}{(1-q_nF(x))^{i+1-l}} \;\mbox{ by Lemma \ref{lem:lr}, }
\\&=& p_n F^{m-i}(x)(1-F(x))^i \sum_{l=0}^i \binom ml \frac { (q_nF(x))^{i-l}}{(1-q_nF(x))^{i+1-l}},
\\&=& \frac{p_nq_n^i F^m(x)(1-F(x))^i}{(1-q_nF(x))^{i+1}} +\sum_{l=1}^i \binom ml \frac {p_n q_n^{i-l}F^{m-l}(x)(1-F(x))^i}{(1-q_nF(x))^{i+1-l}}.
\end{eqnarray*}
Substituting all these expressions in equation (\ref{eqn:gr-max1}), we get
\begin{eqnarray*}
F_{k:N_n}(x) &=& P(X_{k:N_n} \le x ) \\&=&
\frac {p_n F^m(x) }{1-q_n F(x)}+ \sum_{i=1}^{k-1}\left(\frac{p_nq_n^i F^m(x)(1-F(x))^i}{(1-q_nF(x))^{i+1}} +\sum_{l=1}^i \binom ml \frac {p_n q_n^{i-l}F^{m-l}(x)(1-F(x))^i}{(1-q_nF(x))^{i+1-l}}\right).
\end{eqnarray*}
Replacing $\;x\;$ by $\;a_n x + b_n\;$ above and using the facts
\begin{eqnarray*} \lim_{n \rightarrow \infty} np_n& = &1, \;\; \lim_{n \rightarrow \infty} q_n^k= \lim_{n \rightarrow \infty} \left(1-\dfrac {np_n}n\right)^k =1, \\ \lim_{n \rightarrow \infty} n(1-q_nF(a_n x + b_n)) & = &\lim_{n \rightarrow \infty} n(p_nF(a_n x + b_n)+1-F(a_n x + b_n))= 1-\ln G(x), \end{eqnarray*} we get
\begin{eqnarray*}
L_k(x)&=& \lim_{n\rightarrow\infty} \frac {np_n F^m(a_n x + b_n) }{n(1-q_n F(a_n x + b_n))}+\lim_{n\rightarrow\infty} \sum_{i=1}^{k-1} \frac{np_nq_n^i F^m(a_n x + b_n)(n(1-F(a_n x + b_n)))^i}{n^{i+1}(1-q_nF(a_n x + b_n))^{i+1}}+\\&&\lim_{n\rightarrow\infty} \sum_{i=1}^{k-1} \sum_{l=1}^i \binom ml \frac {np_n q_n^{i-l}F^{m-l}(a_n x + b_n)(n(1-F(a_n x + b_n)))^i}{n^{i+1}(1-q_nF(x))^{i+1-l}},
\\&=& \frac 1{1-\ln G(x)} + \sum_{i=1}^{k-1} \frac {(-\ln G(x))^i}{(1-\ln G(x))^{i+1}}+ \sum_{i=1}^{k-1} 0
= \frac 1{1-\ln G(x)} \sum_{i=1}^k \left( \frac {-\ln G(x)}{1-\ln G(x)}\right)^{i-1},
\end{eqnarray*}
$\;=1-\left( \frac {-\ln G(x)}{1-\ln G(x)}\right)^k,\;$ completing the proof.
\end{proof}
\begin{proof} [\textbf{Proof of Theorem \ref{Thm-FkG2}:}]
For $\;-\infty < x<y < \infty,\;$ we have $\;F(x) \le F(y),\;$ so that
\begin{eqnarray*}
&&1-\ln F(x) \ge 1-\ln F(y) \Rightarrow 1-\dfrac 1{1-\ln F(x)} \ge 1-\dfrac 1{1-\ln F(y)} \\ &\Rightarrow&
\left(\frac {-\ln F(x)} {1-\ln F(x)}\right)^k \ge \left(\frac {-\ln F(y)} {1-\ln F(y)}\right)^k \Rightarrow R_k(x) \le R_k(y),
\end{eqnarray*}
showing that $\;R_k\;$ is non-decreasing. Since $\;F\;$ is right continuous, we have
\[ \lim_{\epsilon \rightarrow 0} R_k(x+\epsilon)=\lim_{\epsilon \rightarrow 0} \left(1-\left(\dfrac {-\ln F(x+\epsilon)} {1-\ln F(x+\epsilon)}\right)^k\right)=R_k(x),\] so that $\;R_k\;$ is right continuous. Further $\;R_k(+\infty)=1\;$ and $\;R_k(-\infty)=0,\;$ and hence $\;R_k\;$ is a df. And, if $\;F\;$ is absolutely continuous with pdf $\;f,\;$ then $\;R_k\;$ is absolutely continuous with pdf $\;r_k(x)=\dfrac {kf(x) (-\ln F(x))^{k-1}}{F(x) (1-\ln F(x))^{k+1}}.\;$
Note that $\; \lim_{x\rightarrow r(F)} R_k(x)=1,\;$ and hence $\;r(R_k)\le r(F) \le \infty.\;$ If possible let $\;r(R_k)<r(F).\;$ Then
\begin{eqnarray*} & & F(r(R_k)) < 1 \Rightarrow 1 -\ln F(r(R_k)) >1 \Rightarrow \dfrac 1{1 -\ln F(r(R_k))} <1 \Rightarrow 1-\dfrac 1{1 -\ln F(r(R_k))} >0
\\&\Rightarrow& \dfrac {-\ln F(r(R_k))}{1 -\ln F(r(R_k))} >0 \Rightarrow 1-\left( \dfrac {-\ln F(r(R_k))}{1 -\ln F(r(R_k))}\right)^k <1 \Rightarrow R_k(r(R_k))<1,
\end{eqnarray*}
which is a contradiction, proving $\;r(R_k)=r(F).$
We have \[ \lim_{x\rightarrow \infty} \frac {1-R_k(x)}{1-H_k(x)}= \lim_{x\rightarrow \infty} \left( \dfrac {-\ln F(x)}{1-F(x)}\right)^k \dfrac 1{(1-\ln F(x))^k} = 1,\] so that $\;1-R_k \sim 1-H_k,\;$ or $\;R_k\;$ is tail equivalent to $\;H_k,\;$ proving (a).
Because of the tail equivalence of $\;R_k\;$ and $\;H_k,\;$ (b), (c) and (d) follow from the corresponding results in Theorem \ref{Thm-Fk}.
\end{proof}
\begin{proof} [\textbf{Proof of Theorem \ref{Thm-FkG3}:}]
With $\;W(x)=R_k(x),\;$ we have
\[ \dfrac {dR_k(x)}{dx}=r_k(x)=\dfrac {kf(x) (-\ln F(x))^{k-1}}{F(x) (1-\ln F(x))^{k+1}}=(1-R_k(x)) h_1(x),\] where $\;h_1(x)=\dfrac {-k f(x)}{F(x) (1-\ln F(x)) \ln F(x)}>0,$ so that (\ref{eqn:burr}) is satisfied and hence $\;R_k\;$ belongs to the Burr family.
\end{proof}
\begin{proof} [\textbf{Proof of Theorem \ref{Thm-FkNB1}:}]
Let $\;D^r=\dfrac{d^r}{d(q_n F(x))^r},\, r=1,2,3,\ldots.\;$ Note that
\[ \sum_{l=m}^\infty \binom {l-m+r-1}{l-m} q_n^{l-m}=(1-q_n)^{-r} \text{ and } \sum_{l=m}^\infty \binom {l-m+r-1}{l-m} q_n^l=q_n^m (1-q_n)^{-r}.\]
Considering the term corresponding to $\;i=0\;$ in (\ref{eqn:gr-max1}), we have
\[ \sum_{l=m}^\infty F^l(x) \binom {l-m+r-1}{l-m} p_n^r q_n^{l-m}= p_n^r F^m(x) \sum_{l=m}^\infty \binom {l-m+r-1}{l-m} (q_n F (x))^{l-m}=\dfrac {p_n^rF^m(x)} {(1-q_n F(x))^r}.\]
For $\;1 \leq i \leq k-1, \;$ in (\ref{eqn:gr-max1}), we now have
\begin{eqnarray*}
& &\sum_{l=m}^\infty \binom li F^{l-i}(x) (1-F(x))^i \binom {l-m+r-1}{l-m} p_n^r q_n^{l-m}
\\&=& \dfrac{p_n^r F^{m-i}(x)(1-F(x))^i}{i!} (q_n F (x))^{i-m}\sum_{l=m}^\infty \binom {l-m+r-1}{l-m} \dfrac {l!} {(l-i)!} (q_n F (x))^{l-i},
\\&=& \dfrac{p_n^r F^{m-i}(x)(1-F(x))^i}{i!} (q_n F (x))^{i-m}\sum_{l=m}^\infty \binom {l-m+r-1}{l-m} D^i (q_n F (x))^l,
\\&=& \dfrac{p_n^r F^{m-i}(x)(1-F(x))^i}{i!} (q_n F (x))^{i-m} D^i\left(\dfrac {(q_nF(x))^m}{(1-q_n F(x))^r}\right),
\\&=& \dfrac{p_n^r F^{m-i}(x)(1-F(x))^i}{i!} i! (q_n F (x))^{i-m}\sum_{l=0}^i \binom m{i-l} \binom {r-1+l}l \dfrac {(q_nF(x))^{m-i+l}}{(1-q_nF(x))^{r+l}},
\\&=& p_n^r (1-F(x))^i \sum_{l=0}^i \binom m{i-l} \binom {r-1+l}l \dfrac {(q_nF(x))^l F^{m-i}(x)}{(1-q_nF(x))^{r+l}},
\\&=& p_n^r (1-F(x))^i \sum_{l=0}^i \binom m{i-l} \binom {r-1+l}l \dfrac {q_n^l F^{m-i+l}(x)}{(1-q_nF(x))^{r+l}},
\\&=& p_n^r (1-F(x))^i \sum_{l=0}^{i-1} \binom m{i-l} \binom {r-1+l}l \dfrac {q_n^l F^{m-i+l}(x)}{(1-q_nF(x))^{r+l}} + \\&& (np_n)^r (n(1-F(x)))^i \binom {r+i-1}i\dfrac {q_n^iF^m(x)}{(n(1-q_nF(x)))^{r+i}}.
\end{eqnarray*}
Substituting we have
\begin{eqnarray*}
F_{k:N_n}(x) &=&\dfrac {p_n^rF^m(x)} {(1-q_n F(x))^r}+\sum_{i=1}^{k-1} p_n^r (1-F(x))^i \sum_{l=0}^{i-1} \binom m{i-l} \binom {r-1+l}l \dfrac {q_n^l F^{m-i+l}(x)}{(1-q_nF(x))^{r+l}} + \\&&\sum_{i=1}^{k-1} (np_n)^r (n(1-F(x)))^i \binom {r+i-1}i\dfrac {q_n^iF^m(x)}{(n(1-q_nF(x)))^{r+i}}.
\end{eqnarray*}
Replacing $\;x\;$ by $\;a_n x + b_n\;$ above and using the facts
$\;\lim_{n \rightarrow \infty} np_n =1, \;$ $\lim_{n \rightarrow \infty} F(a_n x + b_n) = 1,\;$ $\;\lim_{n \rightarrow \infty} n(1-F(a_n x + b_n))= -\ln G(x),\;$ $\;\lim_{n\rightarrow \infty} n(1-q_nF(a_n x + b_n))= 1-\ln G(x),\;$ and $\;\lim_{n\rightarrow \infty} q_n^k = \lim_{n\rightarrow \infty} (1-\dfrac {np_n}n)^k = 1,\;$ we get
\begin{eqnarray*}
S_k(x)&=& \dfrac 1{(1-\ln G(x))^{r}}+\sum_{i=1}^{k-1} 0 + \sum_{i=1}^{k-1} \binom {r+i-1}i \dfrac {(-\ln G(x))^i}{(1-\ln G(x))^{r+i}},
\\&=& \sum_{i=0}^{k-1} \binom {r+i-1}i \dfrac {(-\ln G(x))^i}{(1-\ln G(x))^{r+i}},
\end{eqnarray*}
completing the proof.
\end{proof}
\begin{proof} [\textbf{Proof of Theorem \ref{Thm_FkNB2}:}]
The proof for the relation (\ref{eqn:recdf1}) is evident from the definition of $\;T_k\;$ since $\; T_{k+1}(x)- T_k(x)=\binom {k+r-1}k \dfrac {(-\ln F(x))^k}{(1-\ln F(x))^{k+r}}.$ The proof of (\ref{eqn:recpdf1}) is by induction.\\ For $\;k=1,\;$ the df $\;T_2(x)=\dfrac 1{(1-\ln F(x))^r}+\binom r1 \dfrac {(-\ln F(x))}{(1-\ln F(x))^{r+1}}. \;$ Hence the pdf becomes
\begin{eqnarray*}
t_2(x)&=&\dfrac {r f(x)}{F(x)} \dfrac 1{(1-\ln F(x))^{r+1}} +\binom r1 (-\frac {f(x)}{F(x)}) \left(\dfrac 1{(1-\ln F(x))^{r+1}}-\dfrac {(r+1)(-\ln F(x))}{(1-\ln F(x))^{r+2}}\right),
\\&=&\binom r1 \left(\dfrac {f(x)}{F(x)}\right)\dfrac {(r+1)(-\ln F(x))}{(1-\ln F(x))^{r+2}} =\dfrac 1{B(r,2)} \dfrac {f(x)}{F(x)}\dfrac {(-\ln F(x))}{(1-\ln F(x))^{r+2}}
\end{eqnarray*}
which agrees with (\ref{eqn:recpdf1}) for $\;k=1.\;$
Assuming that the relation (\ref{eqn:recpdf1}) is true for $\;k=s,\;$
we have $\;T_{s+1}(x)=T_s(x)+\binom {s+r-1}s \dfrac {(-\ln F(x))^s}{(1-\ln F(x))^{s+r}}\;$ with $\;t_{s+1}(x)=\dfrac 1{B(r,s+1)} \dfrac {f(x)}{F(x)} \dfrac {(-\ln F(x))^s}{(1-\ln F(x))^{r+s+1}}.\;$ Now consider $\;T_{s+2}(x)=T_{s+1}(x)+\binom {s+r}{s+1} \dfrac {(-\ln F(x))^{s+1}}{(1-\ln F(x))^{s+1+r}}.$\\
Differentiating with respect to $\;x\;$ we get
\begin{eqnarray*}
&& t_{s+2}(x)-t_{s+1}(x)= \binom {s+r}{s+1} \left(-\dfrac {f(x)}{F(x)}\right) \left(\dfrac {(s+1)(-\ln F(x))^s}{(1-\ln F(x))^{s+1+r}}-\dfrac {(s+r+1)(-\ln F(x))^{s+1}}{(1-\ln F(x))^{s+r+2}}\right),
\\&=& - (s+1) \binom {s+r}{s+1} \dfrac {f(x)}{F(x)}\dfrac {(-\ln F(x))^s}{(1-\ln F(x))^{s+1+r}} +(s+2)\dfrac {s+r+1}{s+2}\binom {s+r}{s+1} \dfrac {f(x)}{F(x)} \dfrac {(-\ln F(x))^{s+1}}{(1-\ln F(x))^{s+r+2}},
\\&=& -\dfrac 1{B(r,s+1)}\dfrac {f(x)}{F(x)}\dfrac {(-\ln F(x))^s}{(1-\ln F(x))^{s+1+r}}+ (s+2)\binom {s+r+1}{s+2} \dfrac {f(x)}{F(x)} \dfrac {(-\ln F(x))^{s+1}}{(1-\ln F(x))^{s+r+2}},
\\&=&\dfrac 1{B(r,s+2)}\dfrac {f(x)}{F(x)} \dfrac {(-\ln F(x))^{s+1}}{(1-\ln F(x))^{s+r+2}}.
\end{eqnarray*}
Thus the relation (\ref{eqn:recpdf1}) follows. Note that $\;t_{k+1}(x)>0.\;$ Hence the proof.
\end{proof}
\begin{proof} [\textbf{Proof of Theorem \ref{Thm_FkNB3}:}]
\begin{enumerate}
\item[(a)] By Theorem \ref{Thm_FkNB2}, $\;T_k\;$ is a df with pdf $\;t_k(x)= $ \\ $ \dfrac 1{B(r,k)} \dfrac {f(x)}{F(x)} \dfrac {(-\ln F(x))^{k-1}}{(1-\ln F(x))^{r+k}}.\;$ Note that $\;\lim_{x\rightarrow r(F)} T_k(x)=$ \\
$\lim_{x\rightarrow r(F)}\left( \dfrac 1{(1-\ln F(x))^r} + \sum_{l=1}^{k-1} \binom {l+r-1}l \dfrac {(-\ln F(x))^l}{(1-\ln F(x))^{l+r}} \right)=1,\;$ so that $\;r(T_k) \le r(F) \le \infty.\;$ If possible, let $\;r(T_k)<r(F).\;$
Since $\;0<F(r(T_k))<F(r(F))=1,\;$ it follows that $\;0<\dfrac 1{1-\ln F(r(T_k))}<1\;$ and $\;0<\dfrac {-\ln F(r(T_k))} {1-\ln F(r(T_k))}<1.\;$ Then
\[T_k(r(T_k))=\sum_{l=0}^{k-1} \binom {l+r-1}l \dfrac {(-\ln F(r(T_k)))^l}{(1-\ln F(r(T_k)))^{l+r}} <\sum_{l=0}^\infty \binom {l+r-1}l \dfrac {(-\ln F(r(T_k)))^l}{(1-\ln F(r(T_k)))^{l+r}}=1,\]
which is a contradiction, proving that $\;r(T_k)=r(F).$ We have
\[ \lim_{x\rightarrow \infty} \dfrac {1-T_k(x)}{1-H_k(x)}=\lim_{x\rightarrow \infty} \dfrac {-t_k(x)}{-k(1-F(x))^{k-1}f(x)}=\lim_{x\rightarrow \infty} \dfrac 1{B(r,k)} \dfrac 1{kF(x)} \dfrac {u^{k-1}(x)}{(1-\ln F(x))^{r+k}}= \dfrac 1{kB(r,k)}.\]
Thus $\;1-T_k \sim 1-H_k,\;$ or $\;T_k\;$ is tail equivalent to $\;H_k.\;$
Because of the tail equivalence of $\;T_k\;$ and $\;H_k,\;$ (b), (c) and (d) follow from the corresponding results in Theorem \ref{Thm-Fk}.
\end{enumerate}
\end{proof}
\begin{proof} [\textbf{Proof of Theorem \ref{Thm_FkBN}:}]
\begin{enumerate}
\item[(i)] Clearly $B_k$ is right continuous with $\;B_k(+\infty)=1\;$ and $\;B_k(-\infty)=0.$ We have $\;B_k(r(F))= E_{\tau}(F^\tau(r(F))) + \sum_{i=1}^{k-1}\frac{(-\log F(r(F)))^i}{i!}E_{\tau}(\tau^i F^\tau(r(F)))=$ $E_\tau(1)+\sum_{i=1}^{k-1}0=1.\;$ Hence $r(B_k)\le r(F).$ If possible, let $r(B_k) < r(F).$ Then $F(r(B_k))<1$ and
\begin{eqnarray*}
B_k(r(B_k))&=& \sum_{i=0}^{k-1}\frac{(-\log F(r(B_k)))^i}{i!} E_{\tau}(\tau^i F^\tau(r(B_k))) \\
&=& E_\tau\set{ \sum_{i=0}^{k-1}\dfrac{(-\tau \log F(r(B_k)))^i}{i!} F^\tau(r(B_k))} <E_\tau\set{ \sum_{i=0}^\infty \dfrac{(-\tau \log F(r(B_k)))^i}{i!} F^\tau(r(B_k))}
\\&=& E_\tau \set{ \exp {(-\tau \log F(r(B_k)))} F^y(r(B_k))} = E_{\tau}(\tau^0) =1,
\end{eqnarray*} a contradiction proving that $\;r(B_k) = r(F).\;$ The recurrence for $B_{k+1}$ follows trivially by definition of $B_k.$
The pdf $b_{k+1}(x)$ is proved by induction on $k.$ Considering the case of $k=1,2,$ we have
$B_1(x)=$ $ E_{\tau}(F^\tau(x)),\;$ $B_2(x)=$ $V_1(x)+\frac {(-\log F(x))^1}{1!} E_{\tau}(\tau F^\tau(x)).\;$ Differentiating with respect to $x$ and taking derivative under the expectation sign
\begin{eqnarray*}
b_1(x)&=& E_{\tau}(\tau F^{\tau-1} (x)) f(x),
\\b_2(x)&=&v_1(x)+\frac {-f(x)}{F(x)} E_{\tau}(\tau F^\tau (x)) +\frac {(-\log F(x))^1}{1!} E_{\tau}(\tau^2 F^{\tau-1} (x)) f(x),
\\&=& \frac {(-\log F(x))^1}{1!} E_{\tau}(\tau^2 F^{\tau-1} (x)) f(x).
\end{eqnarray*}
Assuming the result to be true for $k,$ we have $b_k(x) = \dfrac{(-\log F(x))^{k-1}}{(k-1)!}E_{\tau}(\tau^k F^{\tau-1} (x)) f(x)$ and
\begin{eqnarray*}
B_{k+1}(x)&=& B_k(x)+\dfrac{(-\log F(x))^k}{k!} E_{\tau}(\tau^k F^\tau(x)),\\
b_{k+1}(x)&=&b_k(x)+\frac {-(-\log F(x))^{k-1}f(x)}{(k-1)! F(x)} E_{\tau}(\tau^k F^\tau(x)) + \frac {(-\log F(x))^k}{k!} E_{\tau}(\tau^{k+1} F^{\tau-1} (x))f(x),
\\&=& \frac {(-\log F(x))^k}{k!} E_{\tau}(\tau^{k+1} F^{\tau-1} (x))f(x).
\end{eqnarray*}
Hence the result.
\item[(ii)] We have
\begin{eqnarray*}
\lim_{x\to \infty} \frac {1-B_k(x)}{(1-F(x))^k}&=&\lim_{x\to \infty} \frac {-b_k(x)}{-kf(x) (1-F(x))^{k-1}},
\\&=& \lim_{x\to \infty} \dfrac{(-\log F(x))^{k-1}}{(k-1)!k (1-F(x))^{k-1}} E_{\tau}(\tau^k F^{\tau-1} (x))f(x),
\\&=& \lim_{x\to \infty} \dfrac{u^{k-1}(x)}{k!} E_{\tau}(\tau^k F^{\tau-1} (x))f(x)= \frac 1{k!} E_\tau(\tau^k \times 1) = \frac 1{k!}E_\tau(\tau^k).
\end{eqnarray*}
\end{enumerate}
\end{proof}
\begin{proof} [\textbf{Proof of Theorem \ref{Thm_StOr}:}]
\begin{enumerate}
\item[(i)] Proved in theorem (\ref{Thm-FkW}).
\item[(ii)] Since $0\le F(x)\le 1$ it also follows that $0\le 1-F(x)\le 1.$ The results follow by observing that $\;\frac {1- H_{k+1}(x)}{1-H_k(x)}<1\;$, $\;\frac {1- R_{k+1}(x)}{1-R_k(x)}<1\;$, $\;F_{k+1}(x)-F_k(x)\ge0\;$ and $\;T_{k+1}(x)-T_k(x)\ge0.$
\item[(iii)] By recurrence relation we have $\displaystyle U_{k+1}(x)-U_k(x) = U_1(x) - F(x) \sum_{l=1}^k \frac {(-\ln F(x))^{l-1}}{l!}.$
Observe that
\begin{eqnarray*}
&& F(x) \sum_{l=1}^{k-1} \frac {(-\ln F(x))^{l-1}}{l!} = \frac {F(x)}{-\ln F(x)} \sum_{l=1}^k \frac {(-\ln F(x))^l}{l!} < \frac {F(x)}{-\ln F(x)} \sum_{l=1}^\infty \frac {(-\ln F(x))^l}{l!} \\&=& \frac {F(x)}{-\ln F(x)} \Set{e^{-ln F(x)}-1}=\frac {F(x)}{-\ln F(x)} \Set{\frac 1{ F(x)}-1} = \frac {1-F(x)}{-\ln F(x)}=U_1(x)
\end{eqnarray*}
Hence $\;U_{k+1}(x)-U_k(x) >0\;$ and result follows.\\ From (i) $U_1(x) \le_{\mbox{st}} F(x) \Rightarrow U_1(x) \ge F(x).$ Assume that $U_k(x) \ge F(x).$ Then by recurrence relation we have $U_{k+1}(x) > U_k(x) \ge F(x).$
\item[(iv)] From (i) $\;U_1(x)>F(x)\ \Rightarrow \frac 1{F(x)} -1 \ge -\ln F(x)$ it follows that
\begin{eqnarray*}
&& 1- \ln F(x) \le \frac 1{F(x)} \Rightarrow \frac 1{1- \ln F(x) } \ge F(x) \Rightarrow 1- \frac 1{1- \ln F(x) } \le 1-F(x) \\
&\Rightarrow& \Set{\frac {-\ln F(x)}{1- \ln F(x)}}^k \le (1-F(x))^k \le 1-F(x) \Rightarrow 1-R_k(x) \le 1-H_k(x) \le 1-F(x)
\end{eqnarray*}
Hence $R_k(x) \le_{\mbox{st}} H_k(x) \le_{\mbox{st}} F(x).$
\item[(v)] Since $\;1-\ln F(x) < \frac 1{F(x)}\;$
\begin{eqnarray*}
1-F_k(x)&=&1-F(x)\sum_{r=0}^{k-1}\Set{\frac {(-\ln F(x))^r}{r!}}=(1-F(x))-F(x)\sum_{r=1}^{k-1}\Set{\frac {(-\ln F(x))^r}{r!}}<1-F(x)
\\1-T_k(x)&=&1-\sum_{l=0}^{k-1} \binom {l+r-1}l \frac {\set{-\ln F(x)}^l}{(1-\ln F(x))^{r+l}}
\\&=& 1-\frac 1{(1-\ln F(x))^r}- \sum_{l=0}^{k-1} \binom {l+r-1}l \frac {\set{-\ln F(x)}^l}{(1-\ln F(x))^{r+l}} < 1-\frac 1{(1-\ln F(x))^r} < 1-F^r(x)
\end{eqnarray*}
\item[(vi)] For $0<p<1$ and $r\ge 1$ such that $p+q=1,$ setting $p=\dfrac 1{1-\ln F(x)}$ it follows that $R_k(x)=1-q^k$ and $T_k(x)=\sum\limits_{l=0}^{k-1} \binom {l+r-1}l p^r q^l.$\\ Let $X\sim R_k(x)=1-q^k$ and $Y\sim T_k(x)=\sum\limits_{l=0}^{k-1} \binom {l+r-1}l p^r q^l.$\\
Since the sum of iid geometric rvs is a negative binomial rv it follows that $\set{X>x} \subset \set{Y>x}$ and hence $P(X>x) \le P(Y>x) \Rightarrow 1-R_k(x) \le 1-T_k(x)$
\item[(vii)] Setting $\xi=-\ln F(x)$ one gets $\;U_k(x)\;$
\begin{eqnarray*}
&=& k\Set{\frac {1-F(x)}{-\ln F(x)}} -F(x) \sum_{l=1}^{k-1} (k-l) \frac {(-\ln F(x))^{l-1}}{l!}
= k\Set{\frac {1-e^{-\xi}}{\xi}}-\sum_{l=1}^{k-1} (k-l) \frac {e^{-\xi} \xi^{l-1}}{l!}
\\&=& k\sum_{j=1}^\infty \frac{e^{-\xi} \xi^{j-1}}{j!}-\sum_{l=1}^{k-1} (k-l) \frac {e^{-\xi} \xi^{l-1}}{l!} = k\sum_{j=k}^\infty \frac{e^{-\xi} \xi^{j-1}}{j!}+\sum_{l=1}^{k-1} \frac {e^{-\xi} \xi^{l-1}}{(l-1)!}
\end{eqnarray*}
\begin{eqnarray*}
=k\sum_{j=k+1}^\infty \frac{e^{-\xi} \xi^{j-1}}{j!}+\sum_{l=1}^k \frac {e^{-\xi} \xi^{l-1}}{(l-1)!} >\sum_{l=1}^k \frac {e^{-\xi} \xi^{l-1}}{(l-1)!} =\sum_{l=0}^{k-1} \frac {F(x) \set{-\ln F((x)}^l}{l!} =F_k(x)
\end{eqnarray*}
\end{enumerate}
\end{proof}
\section{Appendix}
\begin{thm} \label{T-l-max}
\begin{enumerate}
\item[(a)] $F \in {\mathcal D}_{l}(\Phi_{\alpha})\;$ for some $\;\alpha > 0\;$ iff
$\;1 - F\;$ is regularly varying with exponent $\;- \alpha,\;$ that is, $\;\lim_{t \rightarrow \infty} \dfrac{1 - F(tx)}{1 - F(t)} = x^{- \alpha}, \, x > 0.\;$ In this case, one may choose $\;a_n = F^{-}\left(1 - \dfrac{1}{n}\right)\;$ and $\;b_n = 0\;$ so that (\ref{Introduction_e1}) holds with $G = \Phi_{\alpha}.$
\item[(b)] $F \in {\mathcal D}_{l}(\Psi_{\alpha})\;$ for some $\;\alpha > 0\;$ iff
$\;r(F) < \infty\;$ and $\;1 - F\left( r(F) - \frac{1}{.}\right)\;$ is regularly varying with exponent $\;- \alpha.\;$ In this case, one may choose $\;a_n = F^{-}\left(r(F) - \dfrac{1}{n}\right)\;$ and $\;b_n = r(F)\;$ so that (\ref{Introduction_e1}) holds with $G = \Psi_{\alpha}.$
\item[(c)] $F \in {\mathcal D}_{l}(\Lambda)\;$ iff there exists a positive function
$\;v\;$ such that $\;\lim_{t \uparrow r(F)} \dfrac{1 - F\left(t + v(t)x\right)}{1 - F(t)} = e^{-x}, \, x \geq 0, \, r(F) \leq \infty.\;$ If this condition holds for some $\;v,\;$ then \\$\;\int_{a}^{r(F)}\left(1 - F(s)\right) ds < \infty, \, a < r(F),\;$ and the condition holds with the choice $\;v(t) = \dfrac{\int_{t}^{r(F)}\left(1 - F(s)\right) ds}{\left(1 -
F(t)\right)}\;$ and one may choose $\;a_n = v(b_n)\;$ and $\;b_n = F^{-}\left(1 - \dfrac{1}{n}\right) \;$ so that (\ref{Introduction_e1}) holds with $G = \Lambda.$ One may also choose $\;a_n = F^{-}\left(1 - \dfrac{1}{ne}\right) - b_n, \, b_n = F^{-}\left(1 -
\dfrac{1}{n}\right). \;$ This condition was by Worm (1998). Also, $\;v\;$ may be taken as the mean residual life time of a rv $\;X\;$ given $\;X > t\;$ where $\;X\;$ has df $\;F.\;$
\item[(d)] (Proposition 1.1(a), Resnick, 1987) A df $\;F\;$ is called a von-Mises function if there exists $\;z < r(F)\;$ such that for constant $\;c > 0,\;$ and $\;z < x < r(F),\;$ $\;1-F(x)=$ $c \exp\left( -\int_{z}^{x}\dfrac{1}{f(u)}du\right), \;$ where $\;f(u) > 0,$ $z<u<r(F),\;$ and $\;f\;$ is absolutely continuous on $\;(z,r(F))\;$ with density $\;f^{\prime}(u)\;$ and $\;\lim_{x\uparrow r(F)} f^{\prime}(u) = 0,\;$ and $\;f\;$ is called the auxiliary function. If df $\;F\;$ is a von-Mises function, then $\;F \in D_l(\Lambda)\;$ with norming constants $\;a_n = f(b_n),$ $b_n = F^-\left(1-\dfrac{1}{n}\right),\;$ so that (\ref{Introduction_e1}) holds with $G = \Lambda.$
\item[(e)] (Proposition 1.1(b), Resnick, 1987) Suppose df $\;F\;$ is absolutely continuous with $\;F^{\prime\prime}(x) < 0, x \in (z, r(F)),\;$ and if
\begin{equation} \lim_{x\uparrow r(F)} \dfrac{(1-F(x))F^{\prime\prime}(x)}{(F^{\prime}(x))^2} = -1, \label{von-Mises} \end{equation} then $\;F\;$ is a von-Mises function and $\;F \in D_l(\Lambda),\;$ and we can choose auxiliary function as $\;f = \dfrac{1-F}{F^{\prime}}.\;$ Conversely, a twice differentiable von-Mises function satisfies $\;(\ref{von-Mises}).$
\end{enumerate}
\end{thm}
Acknowledgement: The first author thanks Prof. R Vasudeva for bringing the thesis Vasantalakshmi M.S. (2010) and a particular case of the problem discussed to his notice.
\bibliographystyle{amsplain}
|
1,108,101,564,581 | arxiv | \section{Introduction}
In general, the performance of a device is intimately connected through fundamental physical laws to the properties of the materials or the sub-elements employed in its realization, and these connections may have far reaching implications to whole branches of engineering. For instance, the energy conversion efficiency of a solar cell is limited by several fundamental limits. For what concerns the photoexcited carrier exploitation, the Shockley-Queisser limit applies \cite{ShockleyJAP1961}; the photon trapping inside the absorber is instead ruled, in the ray-optics regime, by the Yablonovitch limit \cite{Yablonovitch} or by more general formulas recently proposed by Fan \textit{et al.} \cite{FanPRL2012} for wavelength-size patterned cells.
Focusing back on the optical science, and more specifically on the integrated optical devices framework, recent developments are moving towards reconfigurable systems constituted by several elements, in order to implement complex operations on classical or quantum signals \cite{MillerPhotRes2013, PeruzzoNatComm2014}. As basic building blocks operating on the amplitude or on the phase of the wave, besides traditional switching elements -- like those relying on thermic, electric, or plasma dispersion effects -- devices involving novel materials are under investigation in the present years. Among them, it can be cited $\mathrm{VO}_2$ \cite{PoonAPL2013, PoonOE2012, AtwaterOE2010, WeissOE2012, OoiNanophotonics2013, JoushaghaniOE2015}, GST ($\mathrm{Ge}_2 \mathrm{Sb}_2 \mathrm{Te}_5$) \cite{TsudaOE2012, PruneriAPL2013, PerniceAPL2012, PerniceAdvMat2013, RudeACSPhot2015, AKUMNanoLett2013}, ITO (Indium Tin Oxide) \cite{Volker, AtwaterNanoLett2010}, polymeric materials \cite{LeutholdNatPhot2014}, and resistive switches \cite{LeutholdOptica2014}. With these materials, and in connection to other concepts like plasmonic waveguides, it is expected that certain device metrics like miniaturization, speed, energy consumption, and state retention, will be improved\cite{Volker_Review}. However, advantages usually come at a price, and in the present case this can be globally summarized as large losses.
\begin{figure*}[ht!]
\centering
\includegraphics[width=\linewidth]{fig1}
\caption{Fundamental limit for a phase-switching optical element. (a) Schematic of the switching action. (b) Minimum transmission for a $\pi$-switch as a function of the material figure of merit $\gamma_{mat}$. The points describe the action of a simple device consisting of a waveguide loaded by the active material; different points correspond to different parameters $n$, $\Delta n$, $\kappa$ and $\Delta \kappa$. All the devices lie in the allowed region of the chart; however, certain devices are strongly sub-optimal. (c) Possible implementation of an optimized \textit{phase} switch based on a material which has \textit{intensity} switching properties.}
\end{figure*}
For instance, plasmonic waveguides systematically suffer from large losses, especially in the visible- and near-infrared spectral range, which is of interest for communications \cite{BoltassevaJOSAB2015}. This does not occur by chance, since the field confinement and the propagation losses are connected by a fundamental relation involving the sole properties of the plasmonic material, and hence of the noble metal optical constants \cite{ArbabiArxiv2014}. The presence of fundamental limits in optics, however, does not only concern guiding elements: considering intensity modulators, it has been recently highlighted that, when graphene is the active material, the insertion loss of the overall device is substantially governed by the graphene conductivity tensor, according to an inequality proved for planar, multilayered devices embedding conducting sheets \cite{TamagnoneNatPhot2014}. In this article we generalize that result, proving the existence of a lower limit also on the insertion losses introduced by a phase actuator. Moreover, our result applies in general to every two-port device with arbitrary geometry, like realistic structures in integrated optics. Data reported in the literature are critically analyzed in view of the present theory, and the role of resonance in switching devices is highlighted. A material figure of merit, depending on the sole dielectric constants of the switching material, turns out to be the central quantity for both amplitude and phase switches.
\section{Fundamental limit on the losses of phase actuators}
The first problem we address is to determine a fundamental limit on the insertion loss of a phase modulator. In its simplest implementation, its schematic is given in Fig.~1 (a).
It is a two-port linear device that, when passing from state $I$ to state $II$, switches the phase of the output beam by a certain amount. Here we focus on the case of a $\pi$ switch, which is of relevance in most applications. While an ideal phase switch does not act on the amplitude, a real device possibly does that. Such a loss may be due to back-reflection, to scattering into other channels, or to absorption inside the switching region. Following the theory outlined in \cite{TamagnoneNatPhot2014, SchaugPettersenIRE1959}, and proved in the Supplementary Material for a device of arbitrary geometry, it turns out that the insertion losses are ultimately determined by the sole complex permittivity of the switching material employed in the device. In formulas,
\begin{equation}
\frac{4\ \mathrm{min}[T_I,T_{II}]}{(1-\mathrm{min}[T_I,T_{II}])^2} \le \smash{\displaystyle\max_{\mathbf r \in V} \frac{|\varepsilon_I (\mathbf{r}) - \varepsilon_{II} (\mathbf{r})|^2}{4\ \varepsilon_I'' (\mathbf{r}) \ \varepsilon_{II}'' (\mathbf{r})} } \equiv \gamma_{mat}
\label{limit_pi_modulator}
\end{equation}
where $T_{I,II}$ are the intensity transmittance of the device in states $I$ and $II$, $\varepsilon_{I,II} (\mathbf r )$ are the (complex) permittivities inside the volume $V$ where the switching action takes place, and $\varepsilon''$ denotes the imaginary part of the permittivity. In most of the cases, the difference $\varepsilon_I (\mathbf{r}) - \varepsilon_{II} (\mathbf{r})$ is non-zero and constant at the sole spatial locations corresponding to the switching material. Hence, the second member of Eq.~1 only depends on its permittivity, defining a \textit{material figure of merit} independent of the specific device shape. Solving the inequality for $\mathrm{min}[T_I,T_{II}]$, the diagram reported in Fig.~1 (b) is obtained. Here exists a forbidden region which is inaccessible by any device built out of a material which has a given $\gamma_{mat}$; in other words, it is the switching material that ultimately dictates the minimum amount of losses introduced by the device into the optical path\footnote{It should be highlighted that the limit expressed by Eq.~1 is reached when the main contribution to the total losses is that originating from the absorption in the switching material. Hence, the reduction of losses such as reflection and scattering, or dissipation in opaque components other than the switching material, is always beneficial.}. A material with a small $\gamma_{mat}$ will necessarily behave as a ``bad'' actuator, while a material with a large $\gamma_{mat}$ can potentially be at the base of a well performing device. A trivial case is that of a transparent material which only changes the refractive index; in this case, $\gamma_{mat} \rightarrow \infty$, and it is clearly possible to build an ideal phase switch by simply placing the material itself into the optical path. The reverse is more subtle: given a material with $\gamma_{mat} \rightarrow \infty$, a design effort is in general needed to approach the fundamental limit.
To clarify this point, and to check the validity of the general inequality Eq.~1, we analyze the device schematized in the right part of Fig.~1 (b). It simply consists of a waveguide loaded with the switching material; the overlap of the latter with the modal field is $\Gamma$. For a sufficiently weak perturbation\footnote{The weak perturbation approximation can be safely applied to low-contrast structures; however, finite-element simulations showed that it can be applied with a good accuracy also to silicon-on-insulator waveguides, provided that the loading material does not introduce a very large perturbation to the cladding index, or that the field overlap with the loading material is small enough.}, the waveguide effective index is modified by $(n + i \kappa) \cdot \Gamma$ in state $I$, and by $(n + \Delta n + i \kappa + i \Delta \kappa) \cdot \Gamma$ in state $II$ \cite{SynderLove}. Since the length of the loaded section must be $L_{\pi} = \lambda_0 / 2 \Gamma \Delta n$, the transmittances in states $I$ and $II$ are given by the formulas reported in the Figure; notice that in these expressions the dependence on $\Gamma$ cancels out. By extracting a random set of $n$, $\Delta n$, $\kappa$ and $\Delta \kappa$, the blue dots in Fig.~1 (b) are obtained. All these points lie in the allowed region of the graph. A detailed observation reveals that there is a narrow area between the forbidden region and the cloud of blue points which is not filled, and two possible causes for this effect have been identified. First, the waveguide perturbation approximation has been assumed here, and this may result weaker in certain areas of the parameter space. Second, the blue dots follow from the analysis of a specific device geometry, i.e., the loaded waveguide; this choice may result in devices which do not reach the optimality boundary in the small $\gamma_{mat}$ region. A similar behaviour will be also observed in Sect.~3 about amplitude actuators, and a general solution to that will be discussed in detail in Sect.~4.
Here instead we focus on two cases of special interest, which have been referred to in the above. One is that of a material which is nearly transparent in both states; its representative point is labeled (i) on the graph. Specifically, its parameters are $n = 2$, $\Delta n = 1$, $\kappa = 1.5 \times 10^{-3}$, $\Delta \kappa = 0$. This leads to $\gamma_{mat} \simeq 10^5$ and $T_{I} = T_{II} = 0.99$: that is, a nearly-ideal phase delay device with negligible insertion losses.
Consider instead a material characterized by $n = 2$, $\Delta n = 1$, $\kappa = 5 \times 10^{-6}$, $\Delta \kappa = 0.5$. Again, the figure of merit is $\gamma_{mat} \simeq 10^5$, but the insertion loss in state $II$ is large: $T_{II} = 0.04$ [point (ii)]. In essence, when attempting to realize a loaded-waveguide phase actuator device which relies on this material, a very poor performance is obtained. This is because $\Delta \kappa$ is large compared to $\Delta n$, and the loaded waveguide section mostly works as an amplitude switch.
However, even relying on such a material, it is possible to design a phase switch that approaches the limit given by Eq.~1. Consider for instance the device sketched in Fig.~1 (c): it consists of a ring resonator filter loaded by the switching material. While the switching material itself essentially acts as an amplitude switch, the global device implements a $\pi$ phase shift actuator. Indeed, in the transparent state, and for resonant wavelengths, the ring behaves as an all-pass filter which shifts the output phase by $\pi$ (state $I$). In the opaque state, instead, the ring is ``broken'' and no phase shift appears at the output port (state $II$). This is an example which shows the potentiality of the concept of material figure of merit $\gamma_{mat}$ and of Eq.~1: by a proper device design, it is possible to obtain a quasi-ideal phase switch even though at a first glance the material itself is not suited for that purpose. The distance from the zero-insertion loss condition ($IL \simeq 0 \leftrightarrow T \simeq 1$) is here tuned by a device parameter, the coupling efficiency $K$ [see Fig.~1 (c)]; small $K$'s mean less $IL$'s. It should however be noticed that a small $K$, and hence a small $IL$, is accompanied by a narrow bandwidth, a known tradeoff encountered in optical devices based on resonance.
\section{Fundamental limit on the losses of amplitude actuators}
The second problem we address is that of evaluating the performance of an amplitude switch. Its working principle is schematized in Fig.~2 (a): state $I$ is the ``on'' of the device, in the sense that light is not blocked; conversely, state $II$ is the ``off''. An ideal amplitude switch would leave all the radiation pass in state $I$, while completely blocking it in state $II$. Nonidealities are hence described by the insertion loss $IL$ and by the extinction ratio $ER$, usually expressed in dB scale: $IL = -10 \log_{10} T_I$, $ER = -10 \log_{10} T_{II}/T_I$. As for the phase switch, by generalizing the theory reported in Ref.~\cite{TamagnoneNatPhot2014} it can be shown that the following inequality holds:
\begin{equation}
\frac{T_I \left( \sqrt{T_I/T_{II}} - 1 \right)^2 }
{ \left( 1 - T_I \right)
\left( T_I/T_{II} - T_I \right) } \le \gamma_{mat}
\end{equation}
where the material figure of merit $\gamma_{mat}$ only depends on the switching material permittivities in states $I$ and $II$ (see Eq.~1).
\begin{figure}[tb]
\centering
\includegraphics[width=\linewidth]{fig2}
\caption{Fundamental limit for an amplitude-switching optical element. (a) Schematic of the switching action. (b) Minimum insertion loss as a function of the material figure of merit when an extinction ratio of 20 dB is required. The points represent the loss of a loaded-waveguide intensity switching device, where the refractive index and attenuation coefficient of the material in states $I$ and $II$ is randomly chosen. All the points lie in the allowed region. (c) Validation of the theory based on the analysis of literature data about $\mathrm{VO}_2$. (d) Same as in panel (c), here about GST ($\mathrm{Ge}_2 \mathrm{Sb}_2 \mathrm{Te}_5$). In panels (c) and (d) empty symbols correspond to experimental works, while filled symbols to theoretical ones.}
\end{figure}
Similarly to the result concerning phase actuators, an intensity actuator relying on a material with small $\gamma_{mat}$ will have large insertion losses; conversely, if a material with large $\gamma_{mat}$ is employed, small insertion losses can be obtained. If, for instance, an extinction ratio of 20 dB is required, the limiting curve reported in Fig.~2 (b) applies. Again, the validity of the limit is confirmed by analyzing the performance of the loaded-waveguide device, now designed to act as an intensity switch, in the weak perturbation approximation. Assuming that the complex refractive index of the switching material is $(n + i \kappa)$ in state $I$ and $(n + \Delta n + i \kappa + i \Delta \kappa)$ in state $II$, under this approximation it is straightforward to show that, to achieve an extinction ratio $ER$, the insertion loss is $IL = ER\cdot \kappa/ \Delta \kappa$, independent of the overlap factor $\Gamma$ between the guided mode and the switching material. We extracted a random set of quartets ($n$, $\Delta n$, $\kappa$, $\Delta \kappa$), and represented as a blue dot in Fig.~2 (b) the corresponding pair $(\gamma_{mat},IL)$. All the dots lie in the allowed region, thus confirming the validity of Eq.~2 over a large span of $\gamma_{mat}$.
The support to Eq.~2 reported above however relies on a quite special device geometry and on the weak perturbation approximation; these are also the reasons why the allowed region is not completely filled by the blue points. The discussion about how to get closer to the forbidden region will be systematically addressed in the next Section; here we instead gain further confidence into Eq.~2 by relying on theoretical and experimental results reported in the literature. We have chosen two cases of study, the phase-change materials vanadium dioxide ($\mathrm{VO}_2$) and GST ($\mathrm{Ge}_2 \mathrm{Sb}_2 \mathrm{Te}_5$). These materials attracted much attention in the last years, since the huge contrast which characterizes the optical responses of the two states allows to implement extremely compact devices, with footprints down to submicrometer size. In addition, devices based on these materials are interesting thanks to low energy consumption, to self-holding operation (in the case of GST), and thanks to the integrability of the switching material into existing platforms; most remarkably, into silicon photonics or in connection with surface plasmons. However, most of them suffer from quite large insertion losses, and it naturally arises the question if these losses can be eliminated through a careful design of the devices and technology improvement, or if they are inherent in employing phase change materials.
In Fig.~2 (c) we plot as dots the insertion losses \textit{vs} the extinction ratios of several $\mathrm{VO}_2$-based devices reported in the literature. Empty marks correspond to experimental works, and filled marks to theoretical ones. All the representative points lie in the allowed region of the graph. It is worth noticing that the results of theoretical works, and especially that of \cite{PoonOE2012}, lie very close to the forbidden region: by relying on vanadium dioxide, no further improvements are possible. Here we employed the value $\gamma_{mat} = 3$, which follows from the complex refractive indices reported in \cite{PoonOE2012}; the values reported in the other articles lead to slightly different $\gamma_{mat}$, but we systematically checked that the corresponding ($IL$, $ER$) values were lying outside the related forbidden region. Similarly, in Fig. 2 (d) we report a set of IL-ER pairs taken from the literature about GST. Here the forbidden region is narrower, in consequence of the fact that GST has a larger $\gamma_{mat}$ with respect to $\mathrm{VO}_2$. Consistently, there are reports in the literature of device performances close to the fundamental limit \cite{PerniceAPL2012}.
Far from being a complete review of the switching materials employed in integrated optics and nanophotonics, the analyses detailed above show the potentials and limitations of two relevant phase change materials at telecom wavelengths, and provide further confirmation of the validity of Eq.~2.
\section{Resonant versus non-resonant amplitude actuators}
It will now be shown that a switching device whose working principle is non-resonant wave propagation through a region loaded by the absorbing material may be quite far from optimality. Consider, for instance, the family of devices whose representative points are highlighted by a straight line in Fig.~2 (c). These points lie on a straight line since they follow from insertion losses and extinction ratio given per unit length, being the device a plasmonic waveguide loaded by the switching material. Despite the waveguide itself is well optimized (the points are essentially tangent to the curve which delimits the forbidden region), when devices with larger and larger extinction ratio are desired, they turn out to deviate more and more from the fundamental limit. Clearly, this problem is not limited to the VO$_2$-based device of Ref.~\cite{PoonOE2012}; rather, it concerns every switching device based on light propagation through the switching region. While this is not an issue as far as single actuators with low extinction ratios are involved, it may pose a problem in applications where a cascade of actuators or large extinction ratios are needed.
However, following the limit theory, there are no first-principle limitations to realize a device with insertion losses smaller than those inherent to a component based on wave propagation. Again, as observed above for phase actuators, the key is to base the switch on a resonant element. In Fig.~3 we compare a device based on wave propagation through a simple loaded waveguide with a ring resonator where a section of the loop is replaced by the loaded waveguide.
\begin{figure}[tb]
\centering
\includegraphics[width=\linewidth]{fig3}
\caption{Performance of propagation-based and resonance-based amplitude actuators in comparison with the fundamental limit. The resonance-based device can perform better than the propagation-based one, especially in the large $ER$ region. Filled and empty dots are obtained by randomly choosing the key parameters for the two geometries (see text). The tick marked (i) represents the minimum $IL$ achievable at arbitrarily large $ER$ with the ring-based device. The tick marked (ii) represents the minimum $IL$ achievable at arbitrarily large $ER$ for the most general switching device relying on a material with $\gamma_{mat} = 3$.}
\end{figure}
The points describe realistic devices based on a rib Silicon waveguide loaded with VO$_2$, whose geometry is taken from \cite{AtwaterOE2010}. This waveguide is characterized by two complex effective indices, corresponding to the two states of the vanadium oxide: $n_{\mathrm{eff},I} = 2.92$, $n_{\mathrm{eff},II} = 2.68$, $\kappa_{\mathrm{eff},I} = 0.025$, $\kappa_{\mathrm{eff},II} = 0.112$. For a fixed waveguide geometry, and consequently for a given pair of propagation constants $\beta_{I,II} = 2 \pi (n_{\mathrm{eff};I,II} + i \kappa_{\mathrm{eff};I,II}) / \lambda_0$, the only relevant device parameter in the propagation configuration is the length. In the resonant configuration, instead, there are two relevant parameters\footnote{A more refined model would include, for instance, reflection and scattering at the unloaded/loaded waveguide interface, and distributed backscattering. However, since these losses mechanisms can be reduced by a proper engineering, and since the aim is here to analyze the \textit{intrinsic} limits of actuators, these losses are not included in the model.}: the loaded section length $L$ and the intensity coupling coefficient $K$. The total ring length is fixed by imposing the resonance condition either in state $I$ or in state $II$. From the point distribution -- which follows from a random set of the key parameters $L$ and $K$ -- it turns out that, in the large extinction ratio region, the device based on resonance may perform much better than that based on propagation, and that performances very close to the fundamental limit can be obtained. This resonance-mediated approach to the fundamental limit occurs even in the case that the loaded waveguide design by itself is not optimal, which may occur, for instance, due to fabrication constraints. Consider again the data in Fig.~3. Here, the line corresponding to the propagation-based device is not tangent to the forbidden region (the line tangent to the forbidden region, given by $IL = ER \cdot \left(\sqrt{1+1/\gamma_{mat}}-1\right)/2$, is highlighted as a dashed line close to the origin of axes in Fig.~3). Nevertheless, by embedding such a waveguide into a resonant ring, performance much closer to the fundamental limit could be obtained.
Although for illustrative purposes here we analyzed a VO$_2$-based device, the hint that a resonant device is closer to the fundamental limit than a device based on light propagation will be demonstrated in a general form in the following. To this end, we notice that, in the resonant device, the large extinction ratio regime is reached under the critical coupling condition. Neglecting the bare waveguide transmission losses, one has $ER \rightarrow \infty$ when the coupling between the bus waveguide and the ring is matched with the transmission loss through the loaded section. Consistently with the notation of Fig.~ 2, the material state $II$ has to be chosen as the device ``off'' state; thus, the critical coupling condition is written $K = 1-e^{-2 \beta''_{II} L}$. Given this constraint, the insertion loss at critical coupling is readily obtained in closed form:
\begin{widetext}
\begin{equation}
IL_{\mathrm{ring}, ER \rightarrow \infty} = -10 \log_{10} \frac{e^{-2\beta''_{I}L} + e^{-2\beta''_{II}L} - 2 e^{-(\beta''_{I} + \beta''_{II}) L} \cos{[(\beta'_{I} - \beta'_{II}) L]}}{1 + e^{-2(\beta''_{I} + \beta''_{II}) L} - 2 e^{-(\beta''_{I} + \beta''_{II}) L} \cos{[(\beta'_{I} - \beta'_{II}) L]}}.
\end{equation}
\end{widetext}
It can be shown (see Supplementary Material) that this expression is minimized when $L \rightarrow 0$, i.e., when $K \rightarrow 0$, and that the limit value is
\begin{eqnarray}
\lefteqn { IL_{\mathrm{ring}, ER \rightarrow \infty, \mathrm{min}} = } \\ \nonumber
&& = -10 \log_{10} \frac{ (\beta''_{I} - \beta''_{II})^2 + (\beta'_{I} - \beta'_{II})^2 }
{ (\beta''_{I} + \beta''_{II})^2 + (\beta'_{I} - \beta'_{II})^2 }.
\end{eqnarray}
The existence of this limit, and the fact that it is finite, is a proof that a critically coupled ring resonator device always outperforms the propagating-wave device, as long as large extinction ratios are considered. The proof given in the Supplementary Material also supports that this conclusion is independent of the specific material under consideration. For the specific case of the VO$_2$-based device analyzed above, this limit is reported as a tick marked (i) in Fig.~3. Consistently, this limit lies below all the points representing the resonant devices at large extinction ratios, while it is above the fundamental limit
\begin{equation}
IL_{\mathrm{fund}, ER \rightarrow \infty} = -10 \log_{10} \frac{\gamma_{mat}}{1 + \gamma_{mat} }
\end{equation}
obtained from Eq.~2 and labelled (ii) in Fig.~3.
It should be noticed that the limit in Eq.~(5) involves the bulk material permittivity (in the case of Fig.~3, VO$_2$), while that in Eq.~(4) involves the propagation constant of the considered waveguide design (in the case of Fig.~3, that of Ref.~\cite{AtwaterOE2010}). However, it is proved in the Supplementary Materials that the limit in Eq.~(4) is always larger than that in Eq.~(5), independently of the specific choice of the switching material and of the waveguide geometry. As it was already noticed in section 2 about phase actuators, the use of resonant components has the drawback that the bandwidth is in general reduced with respect to the case of propagation based devices. Anyway, as far as the optimality with respect to insertion losses are concerned, the results given above together with those given in Sect.~2 support the conclusion that the concept of resonance may play a crucial role in the optimization of optical actuator. Although the discussion in the present article deals with integrated optical waveguides and ring resonators, the generality of the resonance and critical coupling concepts allows to extrapolate the present results also to other photonic platforms such as photonic crystals and metamaterials \cite{TunablePhotonicCrystals, ShalaevLPR2011}.
We conclude this section by noticing that the above analysis does not depend on the choice of the material ``transparent'' state as state $I$ and of the ``opaque'' state as state $II$, or vice-versa. In the deduction of Eq.~2, indeed, this assumption has not been made, and the designer is free to choose the switching material ``opaque'' state for the device ``on'' state (i.e., the device state which does not block the light flow), or the opposite. This fact may be exploited in view of energy saving. Suppose that the need is to design a device intended for normally-on operation, and that the switching material has one of the two states which is power-hungry. The device can be engineered to use the power-hungry material state for the device ``off'' state, hence reducing the overall energy consumption. While this conclusion is general and holds for arbitrary device geometry, it can be read out directly in the framework of the critically coupled ring resonator by noticing that Eqs~3-4 are invariant for the exchange $I \leftrightarrow II$.
\section{Comparison of different materials employed in actuators}
The power of the limits expressed by the inequalities given in Eqs.~1-2 is that it is sufficient to know the figure of merit $\gamma_{mat}$ of the (bulk) switching material to have significant insights into the potentiality of a new material, prior to directly designing specific devices. Furthermore, the limits may be of help as far as an optimization is concerned, when the decision whether to proceed with further optimization steps has to be taken. It is clear that the inequalities given above and the material figure of merit only provide information on a single metric on the device performance, while other issues like bandwidth, footprint, switching energy, state retention, switching time etc.\ are not grasped by $\gamma_{mat}$. Nevertheless, the knowledge of $\gamma_{mat}$ could be of help, for instance, in choosing the material which is best suited for operation in a certain wavelength range. Indeed, $\gamma_{mat}$ only depends on the permittivities, which are often known from optical reflectometry or ellipsometry, from first-principle structural calculations, or from other models.
In Fig.~4 we propose this spectral comparison, regarding two phase change materials ($\mathrm{Ge}_2 \mathrm{Sb}_2 \mathrm{Te}_5$, referred to as GST, and $\mathrm{VO}_2$), a transparent conductive oxide (Indium Tin Oxide, ITO), and a semiconductor (Silicon).
\begin{figure}[htb!]
\centering
\includegraphics[width=\linewidth]{fig4}
\caption{Spectral dependence of the figure of merit for four materials employed in nanophotonics, whose working principle is different. Dielectric modulation in $\mathrm{VO}_2$ and GST ($\mathrm{Ge}_2 \mathrm{Sb}_2 \mathrm{Te}_5$) is due to a phase transition, while in ITO and Silicon the plasma effect due to free charge is considered. In Silicon, for wavelengths longer than the bandgap, extrinsic losses due to waveguide scattering are included.}
\end{figure}
In the phase change materials, the permittivity change is induced by a structural transition -- amorphous-crystalline in the case of GST, and from a monoclinic to a rutile structure in the case of $\mathrm{VO}_2$. The dielectric functions are retrieved from \cite{TazawaJJAP2007, OravaAPL2008}. In the case of ITO, the plasma dispersion effect modeled by a Drude contribution to the permittivity is responsible for the modulation effect. Here, typical parameters for the dielectric response are taken from \cite{Volker, AtwaterNanoLett2010, DobrowolskiAO1983, MichelottiOL2009, Jung} and correspond to a mobility of $15\ \mathrm{cm}^2/\mathrm{Vs}$. As opposed to the phase change materials, whose response is intrinsic to their structure (a change in certain optical matrix elements for GST \cite{Parrinello}, and a semiconductor-insulator Mott transition for $\mathrm{VO}_2$ \cite{Basov}), the plasma effect in ITO can be tuned through the electron population injected or accumulated in the active region. It turns out that the material figure of merit significantly depends on that parameter, gaining more than one order of magnitude over a wide spectral range for an order-of-magnitude change in the electron density.
The plasma dispersion effect is also at the origin of the response of Silicon \cite{SorefIEEE1987}, and is here quantified assuming a mobility of $1500\ \mathrm{cm}^2/\mathrm{Vs}$, and an injected electron density of $10^{18}$ or $10^{19}\ \mathrm{cm}^{-3}$. By introducing also the effect of holes the figure of merit is increased by a factor $\sim 2$. As opposed to the other materials, which have a flat response in a wide spectral range, Silicon strongly feels the effect of a bandgap. If the bulk Si permittivity is employed, in the ``undoped'' state the material is well transparent, implying values of $\gamma_{mat}$ larger than $10^6$ above the $1.2\ \mu$m wavelength. However, when Silicon is employed for optical waveguides, extrinsic losses due to roughness scattering and surface state absorption always occur. These losses, despite being extrinsic to the bulk material, and rather connected to the device itself, can however be accounted for in the material figure of merit, defining a $\gamma_{mat}$ for an effective ``waveguide-Silicon'' material. Assuming a loss of $1\ \mathrm{dB/cm}$ \cite{MorichettiPRL2010}, values of $\gamma_{mat} = 10^{4} - 10^5$, flat in the whole near-infrared spectral range, are obtained. If instead a low-loss $0.1\ \mathrm{dB/cm}$ Si waveguide is considered \cite{BibermanOL2012}, the material figure of merit increases by an order of magnitude. As for ITO, also in Silicon the figure of merit depends significantly on the injected charge density. Hence, provided that the mobility is not reduced when a large charge density is involved, it is convenient to work in this regime. This is a consequence of the balance between the real and imaginary part of the permittivity given by the Drude model, and applies to every material whose switching action relies on this mechanism.
\section{Conclusions and perspectives}
In conclusion, we derived fundamental limits on the losses of arbitrarily shaped two-port amplitude and phase optical actuators. Finding their roots into a simple manipulation of Maxwell equations for linear and reciprocal dielectrics, the validity of these limits extends to a wealth of linear switching devices, and in particular to integrated optics devices regardless of the specific geometric configuration. The key role is played by the switching material, whose effectiveness is quantified by a material figure of merit simply defined in terms of the permittivities. While the introduced figure of merit does not give insights into certain metrics like switching time, footprint, state retention, switching energy etc., it sets clear limits on the optical performances of any device which relies on a given material. Further, we observed a peculiar connection between the ability to reach the fundamental limit and the presence of resonance and of critical coupling in the operation principle of the device. We believe that the present theory provides an important metric tool which will direct researchears towards highly performing optical devices and materials.
\section*{Acknowledgements}
The research leading to these results has received funding from the EU Seventh Framework Programme (FP7/2007-2013) under grant agreement number 323734 ``Breaking the Barrier on Optical Integration'' (BBOI). Fruitful discussions with Daniele Melati and Marco Morandotti are also gratefully acknowledged.
\section*{Journal reference}
This article is published in \textit{Laser and Photonics Reviews} \textbf{9}, No.~6, 666 (2015). DOI: 10.1002/lpor.201500101
\section{General scattering inequality for a multiport, dielectrically-driven optical actuator}
Consider a generic two-state optical system like that represented in Fig.~S1.
\begin{figure}[tb]
\centering
\includegraphics[width=\linewidth]{figs1}
\caption{Schematic of a two-state switching optical element. In the linear optical response regime it is fully characterized by the scattering matrix $\mathbf S$, which connects the input amplitudes with the output amplitudes. The switching action is eventually ascribed to a modulation of the dielectric constant spatial distribution $\varepsilon (\mathbf{r})$.}
\end{figure}
An ``active region'' is connected to the exterior via a number of waveguides, where ingoing and outgoing fields are quantified by the complex amplitudes $\mathbf A = (a_1 \ldots a_i \ldots)$ and $(\mathbf B = b_1 \ldots b_i \ldots)$, respectively. The harmonic time dependence $e^{i \omega t}$ is assumed; disregarding non-linear phenomena, the system's response is fully described by the scattering matrix $\mathbf S$ that connects $\mathbf A$ with $\mathbf B$. Actually, the amplitudes $a_i$ does not necessarily correspond to spatially separated waveguide channels, the only requirement is modal orthogonality between the (discrete) scattering channels. Hence, the present theory applies to more general systems, like periodic arrays (where plane waves corresponding to open diffraction channels are involved), systems with spherical or cylindrical symmetry (where Bessel spatial harmonics are involved), or mode-division multiplexing devices.
It will be now shown that the changes in the system as seen from the exterior, i.e., the device switching performance described by a change in the $S$-parameters from state $I$ to state $II$, are fundamentally connected to the values that the permittivity of the materials contained in the ``active region'' assume in the two states.
Starting point is the expansion into normal modes of the transverse electromagnetic fields corresponding to the scattering channels. Referring again to Fig.~S1, we define a surface $\Omega$ which encloses the device. The position of $\Omega$ is chosen to be sufficiently far away from the ``active region'': in this way, on $\Omega$ the electromagnetic fields are only those corresponding to the guided modes of the waveguides, which are supposed to orthogonally enter $\Omega$. The electromagnetic field of each mode $\mu$ at $\Omega$ is hence fully described by the tangential components of the fields\cite{WhatIs}:
\begin{eqnarray}
\mathbf E_{T,\mu} (x,y,z) & = & (a_{\mu} e^{-i \beta_{\mu} z} + b_{\mu} e^{i \beta_{\mu} z})\ \mathbf e_{T,\mu}(x,y) \nonumber \\
\mathbf H_{T,\mu} (x,y,z) & = & (a_{\mu} e^{-i \beta_{\mu} z} - b_{\mu} e^{i \beta_{\mu} z})\ \mathbf h_{T,\mu}(x,y)
\label{fields_expansion}
\end{eqnarray}
where $z$ is a local coordinate pointing into the $\Omega$ surface, and $\mathbf e_{T,\mu}(x,y)$, $\mathbf h_{T,\mu}(x,y)$ are the modal field profiles. Here, $\mu$ is an index which may refer to the modes of different waveguides, or of the same waveguide; in both cases, the orthogonality relation
\begin{equation}
\int_{\Omega} \left( \mathbf e_{\mu} \times \mathbf h^*_{\nu} \right) \cdot \mathrm{d} \mathbf n= 2 \delta_{\mu, \nu}
\end{equation}
holds. With $\mathrm{d} \mathbf n$ we identify the surface element with normal pointing into $\Omega$. We also assume that the waveguides have real and isotropic permeability and permittivity, and hence that the modal transverse field profiles can be chosen as real\cite{Tamir}.
Consider the sourceless Maxwell equations for the situation \textit{I}:
\begin{eqnarray}
\nabla \times \mathbf E_I & = & -i \omega \mu_0 \mathbf H_I \label{maxw1} \\
\nabla \times \mathbf H_I & = & i \omega \varepsilon_I \mathbf E_I \label{maxw2}
\end{eqnarray}
where $\mathbf E_I$ and $\mathbf H_I$ are the fields corresponding to the excitation vector $\mathbf A_I$, and to the internal device configuration described by the permittivity $\varepsilon_I$. We also suppose that all the materials do not have a magnetic response.
By dot-multiplying on the left Eq.~(\ref{maxw1}) with $\mathbf H_{II}$ and Eq.~(\ref{maxw2}) with $\mathbf E_{II}$, and summing, we get:
\begin{equation}
\mathbf H_{II} \cdot (\nabla \times \mathbf E_I) + \mathbf E_{II} \cdot (\nabla \times \mathbf H_I) = i \omega (\mathbf E_{II} \varepsilon_I \mathbf E_I - \mathbf H_{II} \mu_0 \mathbf H_I)
\label{intermedio1}
\end{equation}
Rewriting Eqs.~(\ref{maxw1}-\ref{maxw2}) for ``\textit{II}'', and multiplying for the fields ``\textit{I}'', an expression similar to Eq.~(\ref{intermedio1}) is obtained. Summing up these intermediate results, the following lemma is obtained:
\begin{equation}
\nabla \cdot \left( \mathbf E_I \times \mathbf H_{II} - \mathbf E_{II} \times \mathbf H_I \right) = i \omega \left( \mathbf E_{II} ( \varepsilon_I - \varepsilon_{II} ) \mathbf E_I \right)
\end{equation}
This expression has now to be integrated over the volume $V$ enclosed in $\Omega$. The integral of the left-hand side is connected with the scattered amplitudes via the equivalence
\begin{eqnarray}
\int_V \nabla \cdot \left( \mathbf E_I \times \mathbf H_{II} - \mathbf E_{II} \times \mathbf H_I \right) \mathrm{d} \mathbf r = \nonumber \\
= - \int_{\Omega} \left( \mathbf E_I \times \mathbf H_{II} - \mathbf E_{II} \times \mathbf H_I \right) \cdot \mathrm{d} \mathbf n = \nonumber \\
= - 4 \sum_{\mu} \left( b_{\mu, I} a_{\mu, II} - a_{\mu, I} b_{\mu, II} \right) = \nonumber \\
- 4 \mathbf A_{II}^T ( \mathbf S_I - \mathbf S_{II}^T ) \mathbf A_I
\end{eqnarray}
hence resulting in
\begin{equation}
4 \mathbf A_{II}^T ( \mathbf S_I - \mathbf S_{II}^T ) \mathbf A_I = -i \omega \int_{V} \mathbf E_{II} ( \varepsilon_I - \varepsilon_{II} ) \mathbf E_I\ \mathrm{d} \mathbf r \label{uno}.
\end{equation}
This result connects the input/output amplitudes for a given excitation configuration in the system states \textit{I} and \textit{II} with the local field distribution inside the interaction region and the material properties described by the permittivity. Similarly, we need to link the device losses as seen from the scattering channels with the power dissipation caused by the presence of a lossy dielectric. From the definitions of the scattering amplitudes $\mathbf A$ and $\mathbf B$, and the dipolar dissipation formula \cite{Jackson}, one obtains
\begin{equation}
\mathbf A_I^H (\mathbf I - \mathbf S_I^H \mathbf S_I ) \mathbf A_I = - \frac{i \omega}{4} \int_V \mathbf E_I^* (\varepsilon_I - \varepsilon^*_I) \mathbf E_I \label{due}
\end{equation}
and similarly for state $II$, where \textit{H} stands for the Hermitian conjugate and $\mathbf I$ is the identity matrix.
Equations \ref{uno} and \ref{due} can be introduced in a chain of inequalities proved in\cite{TamagnoneNatPhot2014}, giving rise to a device and a material figure of merit linked by an inequality:
\begin{equation}
\gamma_{dev} \equiv \frac{|\mathbf A_{II}^T ( \mathbf S_I - \mathbf S_{II}^T ) \mathbf A_I|^2}
{\left[ A_I^H (\mathbf I - \mathbf S_I^H \mathbf S_I ) \mathbf A_I \right] \left[ A_{II}^H (\mathbf I - \mathbf S_{II}^H \mathbf S_{II} ) \mathbf A_{II} \right]} \label{gammadev}
\end{equation}
\begin{equation}
\gamma_{mat} \equiv \smash{\displaystyle\max_{\mathbf r \in V} \frac{|\varepsilon_I (\mathbf{r}) - \varepsilon_{II} (\mathbf{r})|^2}{4\ \varepsilon_I'' (\mathbf{r}) \ \varepsilon_{II}'' (\mathbf{r})} } \label {gammamat}
\end{equation}
\begin{equation}
\gamma_{dev} \le \gamma_{mat}. \label{fundamental}
\end{equation}
In Eq.~(\ref{gammamat}) the maximum is taken over the whole active region, and $\varepsilon'' = (\varepsilon - \varepsilon^*)/2i$. The equations above generalize those reported in \cite{TamagnoneNatPhot2014} to an arbitrarily-shaped, dielectrically driven multiport switching device.
\section{Fundamental limit for a phase optical actuator}
Starting from Eqs.~(\ref{gammadev}-\ref{fundamental}), and setting $\mathbf A_I = (1, 0)^T$, $\mathbf A_{II} = (0, 1)^T$ one has
\begin{eqnarray}
\gamma_{dev} = \frac{\left| |t_I|e^{i \phi_I} - |t_{II}| e^{i \phi_{II}} \right|^2}{\left( 1 - |r_I|^2 - |t_{I}|^2 \right)
\left( 1 - |r_{II}|^2 - |t_{II}|^2 \right)} \\
= \frac{\left| |t_I| + |t_{II}| \right|^2}{\left( 1 - |r_I|^2 - |t_{I}|^2 \right)
\left( 1 - |r_{II}|^2 - |t_{II}|^2 \right)}
\end{eqnarray}
if a $\pi$ phase switch, operating in transmission, is required. Here, $t_{I,II}$ and $r_{I,II}$ are the complex amplitude transmission and reflection coefficients, and $\phi_{I,II}$ the transmission phases. Introducing the intensity transmittance $T_{I,II} = |t_{I,II}|^2$, the following inequalities hold:
\begin{eqnarray}
\gamma_{dev} \ge \frac{\left| \sqrt{T_I} + \sqrt{T_{II}} \right|^2}{\left( 1 - T_I \right) \left( 1 - T_{II} \right)} \\
\ge \frac{\left| \sqrt{T_I} + \sqrt{T_{II}} \right|^2}{(1-\mathrm{min}[T_I,T_{II}])^2} \\
\ge \frac{\left| 2 \sqrt{ \mathrm{min}[T_I,T_{II}] } \right|^2}{(1-\mathrm{min}[T_I,T_{II}])^2} \\
= \frac{4\ \mathrm{min}[T_I,T_{II}]}{(1-\mathrm{min}[T_I,T_{II}])^2}
\end{eqnarray}
from which, in conjunction with (\ref{fundamental}), Eq.~1 of the main article is obtained.
\section{Fundamental limit for an amplitude actuator}
The fundamental limit for an amplitude actuator, Eq.~2 of the main article, is obtained from Eqs.~(\ref{gammadev}-\ref{fundamental}). This generalizes the results reported in \cite{TamagnoneNatPhot2014} to an arbitrarily-shaped, dielectrically driven optical amplitude actuator.
\section{Limits for a critically-coupled ring-resonator amplitude actuator}
In the following we will provide a numerical proof of the statements reported in the discussion about Equations 3, 4 and 5 in the main text.
To the sake of clarity, we recall here the relevant quantities:
\begin{eqnarray}
\lefteqn {IL_{\mathrm{ring}} =} \\ \nonumber
&&-10 \log_{10} \frac{e^{-2\beta''_{I}L} + e^{-2\beta''_{II}L} - 2 e^{-(\beta''_{I} + \beta''_{II}) L} \cos{((\beta'_{I} - \beta'_{II}) L)}}{1 + e^{-2(\beta''_{I} + \beta''_{II}) L} - 2 e^{-(\beta''_{I} + \beta''_{II}) L} \cos{((\beta'_{I} - \beta'_{II}) L)}}. \label{ring1}
\end{eqnarray}
\begin{equation}
IL_{\mathrm{ring}, \mathrm{min}} =
-10 \log_{10} \frac{ (\beta''_{I} - \beta''_{II})^2 + (\beta'_{I} - \beta'_{II})^2 }
{ (\beta''_{I} + \beta''_{II})^2 + (\beta'_{I} - \beta'_{II})^2 }. \label{ring2}
\end{equation}
\begin{equation}
IL_{\mathrm{fund} } = -10 \log_{10} \frac{\gamma_{mat}}{1 + \gamma_{mat} }
\label{ring2}
\end{equation}
where the notation $ER \rightarrow \infty$ reported in the main text is not repeated for short.
\begin{figure*}[t]
\centering
\includegraphics[width = 11 cm]{figs2}
\caption{Numerical proof of Eq.~\ref{ring_ineq} concerning the limits on critically coupled ring resonator amplitude actuators.}
\end{figure*}
We will now prove that
\begin{equation}
IL_{\mathrm{ring}} > IL_{\mathrm{ring}, \mathrm{min}} > IL_{\mathrm{fund} }
\label{ring_ineq}
\end{equation}
for a wide range of values of $\beta'_{I}$, $\beta'_{II}$, $\beta''_{I}$, $\beta''_{II}$, and $L$. To this extent, we first restate the problem in terms of a set of normalized variables. In the weak perturbation approximation for a loaded waveguide, one has $\beta_{I,II} L = 2 \pi \left( n_{\mathrm{eff}} + \Gamma (n_{I,II} + i \kappa_{I,II}) \right) L/\lambda$, where $n_{\mathrm{eff}}$ is the unloaded waveguide effective index, $\Gamma$ is the overlap factor between the modal field and the switching material, and $n_{I,II} + i \kappa_{I,II}$ is the complex refractive index of the switching material. If the propagation constants are rewritten as follows
\[
\beta_{I,II} L = 2 \pi \left( 1 + \tilde{n}_{I,II} + i \tilde{\kappa}_{I,II}) \right) \tilde{L} ,
\]
with $\tilde{n}_{I,II} = \frac{\Gamma}{n_{eff}} n_{I,II} $, $\tilde{\kappa}_{I,II} = \frac{\Gamma}{n_{eff}} \kappa_{I,II} $, $ \tilde{L} = \frac{L n_{eff}}{\lambda}$, the material figure of merit results
\[
\gamma_{mat} = \frac{\left| (\tilde{n}_I + i \tilde{\kappa}_I)^2 - (\tilde{n}_{II} + i \tilde{\kappa}_{II})^2 \right|^2}
{4\ \mathrm{Im} \left[ (\tilde{n}_I + i \tilde{\kappa}_I)^2 \right] \mathrm{Im} \left[ (\tilde{n}_{II} + i \tilde{\kappa}_{II})^2 \right]}.
\]
In essence, the independent variables are the five real numbers $\tilde{n}_{I,II}$, $\tilde{\kappa}_{I,II}$, and $\tilde{L}$. By allowing these variables to randomly and independently sweep the interval $[0.01,1]$, we obtained a set of values of $IL_{\mathrm{ring}}$, $IL_{\mathrm{ring}, \mathrm{min}}$, and $ IL_{\mathrm{fund} }$. By plotting the values $IL_{\mathrm{ring}} - IL_{\mathrm{ring}, \mathrm{min}}$ and $ IL_{\mathrm{fund} } - IL_{\mathrm{ring}, \mathrm{min}}$ versus $\gamma_{mat}$ (see Fig.~S2), we get strong numerical evidence for Eq.~\ref{ring_ineq}.
|
1,108,101,564,582 | arxiv | \section{Introduction}
\begin{comment}
Recommender systems help users mitigate the information overload problem by suggesting relevant items. To generate personalized recommendations they apply machine learning algorithms, regretfully making them vulnerable to attacks that can be carried out by injecting malicious input. Thus far research on attacks focused on tampering with usage patterns, commonly known as shilling attacks.
In this work we identify side-information as another possible input type that can be manipulated, focusing on images associated with the items.
We propose several models that differ by the amount of knowledge and capabilities required by the adversary. Using two benchmark datasets we prove the effectiveness of these approaches. We show that even when the visual features are of low contribution to the recommender system, they can be used by the adversary to completely alter the recommended lists.
Our hope is this pioneering study will contribute to better understanding this new type of adversarial attacks and thus contribute to AI safety.
\end{comment}
During the last decades, with the increasing importance of the Web, Recommender Systems (RS) have gradually taken a more significant place in our lives. Such systems are used in various domains, from e-commerce to content consumption, where they provide value both for consumers and recommendation service providers. From a consumer perspective, RS for example help consumers deal with information overload. At the same time, recommender systems can create various types of economic value, e.g., in the form of increased sales or customer retention \cite{jannachjugovactmis2019,schafer1999recommender,fleder2009blockbuster, pathak2010empirical,jannach2019towards}.
The economic value associated RS makes them a natural target of \emph{attacks}. The goal of attackers usually is to \emph{push} certain items from which they have an economic benefit, assuming that items that are ranked higher in the recommendation list are seen or purchased more often. Various types of attack models for RS have been proposed in the literature \cite{si2020shilling, gunes2014shilling}. The large majority of these attack models is based on injecting fake profiles into the RS. These profiles are designed in a way that they are able to mislead the RS
to recommend certain items with a higher probability.
Nowadays, many deployed systems are however not purely collaborative anymore and do not base their recommendations solely on user preference profiles. Instead, oftentimes hybrid approaches are used that combine collaborative signals (e.g., explicit ratings or implicit feedback) with side information about the items.
In the most recent years, it was moreover found that item images, i.e., the visual appearance of items, can represent an important piece of information that can be leveraged in the recommendation process. These developments led to the class of \emph{visually-aware} recommender systems. Fueled by the advances in deep learning, a myriad of such approaches were proposed in recent years \cite{he2016vbpr, bostandjiev2012tasteweights,kim2004viscors,linaza2011image, bruns2015should,iwata2011fashion,deldjoo2016content,he2016sherlock}.
In many domains, catalog sizes can be in the millions. Hence, from a practical point of view, it can be challenging or impractical for recommendation service providers, e.g., an e-commerce shop or media streaming site, to capture and maintain item images (e.g., product photos) for the entire catalog by themselves. Therefore, a common practice is to rely on the \emph{item providers} to make their own images available to recommendation service providers by a dedicated API. This is a cross-domain practice, where e-commerce companies like Amazon\footnote{\url{https://vendorcentral.amazon.com/}}
enable sellers to upload photos and content platform like YouTube\footnote{\url{https://support.google.com/youtube/answer/72431}} allow creators to upload their thumbnails. With the term \emph{item providers}, we here refer to those entities (individuals or organizations) who act as suppliers of the items that are eventually recommended and who benefit economically from their items being listed in recommendations.
In modern visually-aware recommender systems, the features of the uploaded item photos, as discussed above, can influence the ranking of the items. Therefore, the item provider might be an \emph{attacker} and try to manipulate the images in a way that the underlying machine learning model ranks the item to be pushed higher in the recommendation lists of users. We call this novel way of manipulating a recommender system a \emph{visual attack}.
While the literature on adversarial examples is rich, only few works exist on their use in the context of RS \cite{di2020taamr,tang2019adversarial}. Moreover, these works rely on impractical assumptions and require the attacker to have access to the internal representation of the RS. In contrast, the proposed algorithm assumes milder and more realistic assumptions.
The main idea of the approach is to systematically manipulate the provided item images in a human-imperceptible way so that the algorithm used by the recommender predicts a higher relevance score for the item itself. Technically, we accomplish this by devising appropriate gradient approximation techniques. The contributions of our work are as follows:
\begin{itemize}
\item We identify a new vulnerability of RS, which emerges in situations where item providers are in control of the images that are used in visually-aware recommenders;
\item We propose novel technical attack models that exploit this vulnerability under different levels of resources available to the adversary;
\item We evaluate these models on two datasets, showing that they are effective even in cases where the visual component only plays a minor role for the ranking process;
\end{itemize}
To make our results reproducible, we share all data and code used in our experiments online\footnote{https://github.com/vis-rs-attack/code}.
\begin{comment}
During the last couple of decades with the increasing importance of the web, recommender systems (RS) have gradually taken a more significant place in our lives. In various domains, from e-commerce to content consumption, recommender systems create value by suggesting relevant items to users.
The economic value of recommenders is unquestionable \cite{schafer1999recommender}, as recommending an item may increase its sales \cite{fleder2009blockbuster, pathak2010empirical}, increase customer retention \cite{kwon2012design} and in general it positively impacts various business goals \cite{jannach2019towards}.
Throughout the paper, we term a participant who gains personal benefit from recommending an item as a \emph{producer}. This is a general term, where in e-commerce a producer can be a vendor or manufacturer and in a vacation rental platform it can refer to a hotel owner.
The established merits of RS may cause some unscrupulous producers to go astray and maliciously interfere with the genuine results of the recommender in order to have their products recommended more often.
The most motivated attack, referred to as the \emph{push attack}, is invoked by a malicious attacker and aims to increase the probability of certain items to be recommended to various users. The opposite attack, \emph{nuke attack}, with the aim of deliberately decreasing the visibility of certain items through the recommender, is out of scope for this paper.
Formally, an attack on a recommender system has three key factors: \emph{adversary}, \emph{pushed items} and \emph{target users}. The adversary is the participant who exerts the attack; the pushed items refer to the items the adversary wishes to promote; and the target users are those whose recommendation lists are manipulated by the attack. In this paper we consider only motivated attacks, where the adversary is the producer of the pushed items.
The vulnerability of recommenders arise from the open nature of their data acquisition process. Hitherto, research on attacking recommender systems has focused on exploiting only a single type of data: the user feedback. This input is typically provided through open platforms, where almost any user can leave implicit or explicit feedback.
The most common attack, widely known as the ``shilling attack'', is based on injecting user feedback so as to promote the pushed items. Alas, since users tend to trust recommenders \cite{zimmerman2002exposing}, as they seem objective compared to other alternatives like advertisements, this attack might be fruitful.
However, there are more input sources that the recommender relies on. One such input, which has recently gained lots of attention in the context of recommenders, is the visual appearance of items. The underlying assumption is that images reveal information about the associated items and therefore should not be ignored. Since the item catalog's size in a typical recommender may be in the millions, it is impractical or even impossible for the RS to capture by itself a photo of each item \footnote{Throughout the paper we assume each item is associated with exactly one image and defer a variable number of images per item to later work.}. Hence, a common practice is to rely on the item producers to provide their own images by a dedicated API. This is a cross domain practice, where e-commerce companies like Amazon\footnote{\url{https://vendorcentral.amazon.com/}} or eBay \footnote{\url{http://developer.ebay.com/DevZone/XML/docs/Reference/eBay/UploadSiteHostedPictures.html}} enable sellers to upload photos and content platform like YouTube\footnote{\url{https://support.google.com/youtube/answer/72431}} allow creators to upload their thumbnails. This situation, where item producers control a factor used by the recommendation engine, may create a conflict of interest. Some producers might be tempted to manipulate their own images in a way that promotes the items in the recommender while remains unnoticed by users.
Historically, images did not pose any threat, since they were not used as part of the modeling process, but were only used for presentation purposes by visualizing the recommended list. After all, a recommendation is more persuasive if it allows the user to literally see the recommended item \cite{al2014visualization, swearingen2001beyond, nanou2010effects}. However, in recent years, images are no longer a safe data source. With the advent of deep learning, the vitality of images for recommendation has been discovered and currently images play an integral role in the modeling process \cite{he2016vbpr, bostandjiev2012tasteweights, kim2004viscors, linaza2011image, bruns2015should, iwata2011fashion, deldjoo2016content, he2016sherlock}.
Having a new modality entails complete new potential types of attacks on RS and the purpose of this paper is to probe them. We emphasize that understanding attack strategies is key to designing robust algorithms, which are resistant to such attacks.
We first present a highly effective attack where the attacker has absolute knowledge on the underlying RS. We then show how the knowledge requirements can be gradually relaxed, at the cost of attack effectiveness. To the best of our knowledge, this is the first work to propose algorithms for attacking recommender systems by exploiting the visual appearances of the target items. We dub this type of attack as the \emph{``visual attack''}.
This paper is organized as follows. Section \ref{sec:relared} reviews related work and gives necessary background. In Section \ref{sec:attacks} we elaborate on the key factors that comprise visual attacks, in Sections \ref{sec:wb_attack} and \ref{sec:bb_attacks} we describe two types of such attacks and report their effectiveness in Section \ref{sec:experiments}. We conclude the paper in Section \ref{sec:discussion}.
The main contributions of this paper are as follows.
\begin{itemize}
\item We identify a new vulnerability in RS.
\item We present attacks that exploit this vulnerability, under different levels of resources available to the adversary.
\item Experiments on real-life datasets prove the effectiveness of these attacks.
\item The code is made available at: \url{https://github.com/kdd2020-rs/vis-attack}.
\end{itemize}
\end{comment}
\section{Background and Related Work}\label{sec:relared}
We review three prerequisite research areas: visually-aware RS, attack models, and the use of adversarial examples in image classification.
\subsection{Architecture of Visually-Aware RS}\label{ssec:vrec}
With respect to their technical architecture, state-of-the-art visually-aware recommender systems commonly incorporate high-level features of item images that were previously extracted from fixed pre-trained deep models. The system therefore invokes a pre-trained Image Classification model (IC), e.g., ResNet \cite{he2016deep} or VGG \cite{simonyan2014very}, which returns the output of the penultimate layer of the model, dubbed the \emph{feature vector} of the image. The advantage of using such pre-trained IC model is that they were trained on millions of images, allowing them to extract feature vectors that hold essential information regarding the visualization of the image.
However, the feature vector is \emph{not} optimized for generating relevant
recommendations. It is thus left to the recommender to learn a transformation layer
that extracts valuable information with respect to recommendations. Usually this transformation is accomplished by multiplying the feature vector by a learned matrix $E$. Since the image is passed through a deterministic transformation, it defines a revised input layer having the feature vector replacing the pixels.
A generic schematic visualization of this approach is depicted in Figure \ref{fig:vrs}.
\begin{figure}[t!]
\setlength{\belowcaptionskip}{-10pt}
\includegraphics[scale=0.5]{architecture-b.pdf}
\caption{Common Architecture of a Visually-aware RS}
\label{fig:vrs}
\vspace{0.4cm}
\end{figure}
VBPR \cite{he2016vbpr} was probably the first work to follow this approach and extends BPR \cite{rendle2012bpr} to incorporate images. It maintains two embedding spaces, derived from the usage patterns and the item visualization. The final predicted score results from the sum of the scores of these two spaces. Sherlock \cite{he2016sherlock} adds the ability to model the feature vector with respect to the item category, by learning a dedicated transformation matrix $E_i$ per each category $i$.
Later, DeepStyle \cite{liu2017deepstyle} improved the category-aware approach by learning a single embedding matrix which maps the categories to the latent space of the images. Thus, a single transformation matrix $E$ is sufficient. The authors of this paper showed that the reduced number of parameters leads to enhanced performance.
\subsection{Attacks on Recommender Systems}
The best explored attack models in the literature are based on the injection of fake users into the system, called \emph{shilling}. These fake users leave carefully designed feedback that is intended to both seem genuine and to disrupt the output of the RS such that the pushed items will be recommended to the target users \cite{si2020shilling,gunes2014shilling}. This shilling attack can be highly effective for cold items, a ubiquitous phenomenon in real-life datasets \cite{sar2015data}.
The effectiveness of an attack is usually measured by the amount of target users that get the pushed item as a recommendation. In the case of shilling attacks, there is a positive correlation between effectiveness and adversary resources, where resources are usually composed of knowledge and capacity. The relevant knowledge includes several aspects, for example, knowing the type of the deployed model,
observing the values of the parameter set, or even having read access to the rating dataset that empowers the model. The capacity refers to the amount of shilling users and fake feedback the adversary can inject. Trivially,
with more knowledge and capacity
the effectiveness of the attacks increases \cite{mobasher2007attacks}. As we will show, the trade-off between resources and effectiveness also exists in visual attacks. However, the relevant resources in these types of attacks are entirely different from shilling attacks.
There are only very few existing works that rely on image information to attack an RS, as mentioned above, \cite{di2020taamr} and \cite{tang2019adversarial}.
However, these works assume that the attacker can either manipulate the \emph{internal representation} of the images as used by the RS, or has complete knowledge about the system.
Moreover, in the case of \cite{di2020taamr}, the goal is to change the predicted category classification of an item, whereas in our approach we aim to push items to the top-k lists of target users.
\subsection{Adversarial Examples in Image Classification}
\label{subsec:adversarial-examples}
Adversarial examples in machine learning are inputs that seem almost identical to examples from the true distribution
of data that the algorithm works on. However, they are designed to deceive the model and cause it to output incorrect
predictions. In the context of image classification, adversarial examples are constructed by adding imperceptible changes
to genuine images, with the aim of crossing the decision boundary of the classifier (e.g., \cite{szegedy2013intriguing}).
Adversarial examples can be generated either through \emph{white-box} (WB) or \emph{black-box} (BB) approaches.
\paragraph{White-box:} This attack assumes the adversary has complete knowledge of the model and can compute gradients using backpropagation. Unlike regular stochastic gradient descent (SGD), which updates the model \emph{parameters} in the \emph{opposite} direction of the gradient, the adversary updates the \emph{input} in the direction of the gradient. Let $\eta$ be the adversarial direction. Then an adversarial example is generated as $\Tilde{x}=x + \epsilon\eta$.
\paragraph{Black-box:}
Such an attack requires a milder assumption: the ability to query the model \cite{chakraborty2018adversarial}. In the absence of gradients to guide the adversary, it invokes the model multiple times with various perturbations within a small volume around a target image and uses the associated feedback to infer the right direction. However, due to the high-dimensionality of the \emph{pixel space}, even efficient attacks need at least tens of thousands of requests to the classifier, limiting their applicability \cite{ilyas2018black,cheng2018query,tu2019autozoom, guo2019simple}.
Our proposed approach does not operate on the pixel space, which in turn limits the number of queries to the model.
To assure the updated image does not deviate too much from the original one, some metric $d(\Tilde{x}, x)$ is usually defined to quantify the visual dissimilarity attributed to the adversarial change. Commonly, $d$ is computed by either $\ell_2$ or $\ell_\infty$, and the goal is to deceive the model while keeping a low dissimilarity score to keep the change undistinguishable by humans.
Various approaches were proposed to control the degree of visual dissimilarity. Examples include the ``fast gradient sign method'' \cite{goodfellow2014explaining}, optimized for $L_\infty$, which approximates this search problem by a binarization of the gradient, computed as $\eta\leftarrow sign(\eta)$. Early stopping was suggested by \cite{kurakin2016adversarial} and Madry et al. \cite{madry2017towards} used the ``projected gradient descent'' to limit the $\ell_2$-norm.
Both white-box and black-box attacks can improve effectiveness by working iteratively and taking multiple small \emph{steps} in the direction of the gradient. For instance, the ``Iterative Gradient Sign'' technique \cite{kurakin2016adversarialb} applies multiple steps of the ``fast gradient sign method'' \cite{goodfellow2014explaining} and reports boost in performance.
\section{Proposed Technical Approach}
\label{sec:technical-approach}
In this section we demonstrate how an attacker can exploit the identified vulnerability in visually-aware RS that are based on an architecture as sketched in Figure \ref{fig:vrs}. We will first elaborate how a white-box attack can be mounted and then move on to more realistic black-box attacks. In particular we will show how to attack both predicted scores and rankings.
\subsection{Preliminaries and Assumptions}
The proposed visual attacks solely modify the visualizations of the pushed items. Therefore, pushing a certain item would not change the score of any other item. As a corollary, gradient computations in attacks with several pushed items are independent of each other. Hence, in this paper we assume there is a single pushed item.
There are three types of target users: specific users, segments or general population. Focusing on specific users is beneficial as it is equivalent to personal advertising and may exploit the influence of opinion leaders. However, in order to attack a specific user, the adversary has to know the consumption history of that user. Then, that user can be cloned by introducing a new user to the RS with the exact same usage patterns. The attack is carried out on the cloned user and the target user will, as a result, be affected by the attack.
Approaching general population dispenses with the need to have prior knowledge about individual users, but it may result in huge exposure of the pushed items to irrelevant users.
A segmented attack is an in-between approach, where the adversary targets a set of users with similar tastes.
As we will explain later, even in the general population and segmented attacks we can safely assume that the adversary attacks a \emph{single} mock user that was \emph{injected by the attacker} to the system.
\subsection{An Upper Bound for Visual Dissimilarity}
In the presented attack models, the item images are systematically perturbed in a way that the related item's score or ranking is improved. This process is done step-wise and at each step more noise is added to the image. In order to prevent the attack from being detected and to ensure good image quality, it is important to limit the number of steps.
To assess the perceptual visual distortion caused by the adversarial changes, we use the widely adopted \emph{objective measure} Structural SIMilarity (SSIM) index \cite{wang2004image}. This index considers image degradation as perceived change in structural information. The idea behind structural information is that pixels have stronger dependencies when they are spatially close.
The SSIM index also incorporates, with equal importance, luminance masking and contrast masking terms. These terms reflect the phenomena where image distortions are more visible in bright areas and in the background, respectively.
The average SSIM indices after 10, 20 and 30 steps are: 0.965, 0.942 and 0.914, respectively (higher is better), which entails a noticeable degradation after 20 steps. Most images have a solid white background, however beyond 20 steps the attack might be revealed as some noise is noticed mostly in the background, which shows some unnatural texture.
Therefore we set the maximal allowed number of steps to 20 in our experiments.
\subsection{A White-Box Attack}\label{sec:wb_attack}
To define an upper bound for the effectiveness of visual attacks as investigated in our paper, we show how a white-box attack can be designed. This attack assumes the adversary has a \emph{read} access to the parameters of the model, which is in general not realistic.
At inference time, to allow personalized ranking, the score of user $u$ for item $i$ is computed by a visually-aware recommender as $s_{u,i}=f(u, i, p_i)$, where $p_i$ is the image associated with item $i$. Note that the only assumption we make on $f(\cdot)$ is that it is differentiable. To avoid clutter, we omit the subscripts whenever their existence is clear from the context. To manipulate the image, the adversary computes the partial derivatives $\frac{\partial s}{\partial p}$ and updates the pixels in that direction, while keeping the distorted image as close as possible to the genuine one. Following \cite{goodfellow2014explaining}, we binarize these partial derivatives by taking their sign, as our preliminary experiments asserted it leads to faster convergence. We note that any future improvements in adversarial examples for images may contribute to the effectiveness of this proposed approach.
The visual attack, as mentioned, is performed in small \emph{steps}. Each step $t+1$ aims to increase the predicted score of the pushed item by updating the image as follows:
\begin{equation}\label{eq:step}
p^{t+1}=p^t + \epsilon\cdot sign(\frac{\partial s}{\partial p^t})
\end{equation}
where $p^0$ is the genuine image and $\epsilon$ is a small constant that controls the step size so the adversarial changes will be overlooked by the human eye.
\subsection{A Black-Box Attack on Scores}
As in the white-box attack, the goal of the adversary is to compute the gradient of the score $s$ with respect to all pixels in the image $p$ in the corresponding image, i.e., $\frac{\partial s}{\partial p}$. However, as explained in Section \ref{subsec:adversarial-examples}, black-box attacks in image classification usually require an enormous amount of requests to the classifier because they work in the \emph{pixel space}. Applying similar techniques in visual attacks would therefore be impractical.
Remember that the RS subnetwork is unknown to the attacker (neither its parameters or even its architecture).
However, remember that in our target architecture for visually-aware RS (Figure \ref{fig:vrs}), we assume that the system leverages
a pretrained IC model. As discussed in Section \ref{ssec:vrec}, such IC models are publicly available and held constant during training, while the rest of the network is optimized for the specific task and data. In our case, given a new image, the RS extracts a feature vector $f_i$ from image $p_i$ by feed forwarding it through the IC and outputting the penultimate layer.
This situation now allows us to apply the chain rule and the multiplication between the partial derivatives of the score $s$ with respect to $f$ and the Jacobian of $f$ with respect to $p$ can be computed. Formally:
$\frac{\partial s}{\partial p}=\frac{\partial s}{\partial f} \cdot \frac{\partial f}{\partial p}$. Upon computing this value, a step is made as defined in Eq. \ref{eq:step}.
Since we assume the parameter set of the image processing unit is known, $\frac{\partial f}{\partial p}$ can be analytically computed using backpropagation.
\subsection{Computation of $\frac{\partial s}{\partial f}$}
Decoupling the partial derivative into two quantities allows to avert numerical computations in the pixel space. However, the computation of the partial derivatives $\frac{\partial s}{\partial f}$ under the black-box model of the RS is rather complicated because $f$ is not the actual input layer.
To illustrate the challenge, consider the following approach for numerical computation of the partial derivatives, where $d$ denotes the dimensionality of $f$. Create $d$ perturbations of $p$, such that each perturbation $p^k$, for $0\leq k < d$, under the constraint $\left\| p-p^k \right\| \leq \delta$, yields a feature vector $f^k=f + \epsilon\cdot \mathcal{I}(k)$ for some predefined $\delta$ and arbitrary small $\epsilon$, where $\mathcal{I}(k)$ is the one-hot vector with $1$ for $k$-th coordinate and $0$ elsewhere. That is, each perturbation isolates a different dimension in $f$. Then, replace the genuine image $d$ times, each time with a different perturbation, and obtain from the RS a new personalized score $s^k$. Now the numerical partial derivative $\frac{\partial s}{\partial f^k}$ is given by $\frac{s^k - s}{\epsilon}$. However, since each pixel in $p$ holds a discrete value and affects all dimensions in $f$, finding such a set of perturbed images is intractable.
Furthermore, in the absence of the ability to directly control $f$, it is not clear how to apply existing methods of gradient estimation. For instance, the widely used NES \cite{wierstra2008natural} requires to sample perturbations from a Gaussian distribution centered around the input $f$; even the seminal SPSA algorithm \cite{spall1992multivariate} demands a \emph{symmetric} set of perturbations.
As such methods are unattainable, we devise a novel approach to estimate the gradients. This method does not have any constraints on the structure of the perturbations. Moreover, the number of perturbations required by this method is \emph{sublinear} in $d$.
To this end, we first present a method that requires exactly $d$ perturbations and then show how this amount can be reduced.
The attacker creates $d$ random perturbations of the image uniformly drawn from the $L_\infty$-ball centered around $p$ with radius $\delta$, for some arbitrary small $\delta$, and obtains their corresponding feature vectors $f^k$ together with the associated personalized scores $s^k$.
Zou et al.~\cite{zou2019lipschitz} proved that CNN-based architectures are Lipschitz continuous, and empirically showed that their Lipschitz constant is bounded by $1$.
Consequently, since each perturbation is similar to the original image, up to a tiny difference $\delta$, also the deltas in the feature vectors $\left\| f^k-f \right\|$ are expected to be epsilonic, making them suitable for numeric computation of $\frac{\partial s}{\partial f}$.
Using matrix notation, the set of vectors $\{f^k\}$ and scalars $\{s^k\}$ are referred to as matrix $\bm{F}\in \mathbb{R}^{d\times d}$ and vector $\bm{s}\in \mathbb{R}^d$, correspondingly.
This translates into a linear system $\bm{F}x=\bm{s}$ such that its solution $x$ is the numerical computation of $\frac{\partial s}{\partial f}$. Keep in mind that a linear system is shift invariant. That is, the same solution $x$ is obtained upon subtracting $f^T$ from each row of $\bm{F}$ and subtracting $s$ from each component in $\bm{s}$. Now the coefficient matrix is the change in $f$ and the dependent variable is the resulting change in score:
\begin{equation}\label{eq:derivative}
x = (\bm{F}-f^T)^{-1}(\bm{s}-s) = \Delta\bm{F}^{-1}\Delta\bm{s}=\frac{\partial s}{\partial f}
\end{equation}
Note that a single solution exists if and only if $\bm{F}$ is nonsingular. Since it is obtained by random perturbations, it is almost surely the case \cite{kahn1995probability}; otherwise a new set of perturbations can be drawn.
\paragraph{Reducing the Number of Perturbations}
Each perturbation entails uploading a new image to the server. In ResNet, for example, $d=2,048$, which results in a large amount of updates. We next address this problem and suggest a trade-off between effectiveness of each step and amount of required image updates.
Note that although Eq.~\ref{eq:derivative} requires $d$ perturbations for finding a single solution, an approximation can be found with fewer perturbations. Formally, let $d'$ be the number of perturbations the adversary generates, for some $d'<d$. This defines an underdetermined system of linear equations, and among the infinite solutions as the estimated gradients, the one with the minimal $\ell_2$ norm can be found \cite{cline1976l_2}.
Another advantage of reducing the number of perturbations is the ability to generalize. There is an unlimited number of possible perturbations, and every subset of $d$ perturbations defines another linear system, with another exact solution. Those exact solutions vary in their distances from the true, unobserved derivatives. However, since the adversary randomly select a single subset of perturbations, the exact solution may ``overfit'' to this instantiation of the linear system, and not generalize to other solutions that could be yielded from other subsets.
Such behavior might be highly problematic, and must be addressed correspondingly. To remedy this problem, regularization comes in useful. Reducing the number of perturbations, and finding a solution with minimal $\ell_2$ norm can be thought of as a means of regularization, and can potentially improve performance. Indeed, as we show in Section \ref{sec:experiments}, our approach not only reduces the required amount of perturbations also leads to improved performance.
\subsection{A Black-Box Attack on Rankings}
The attack described in the previous section is suited to target RS that reveal the scores, e.g., in the form of predicted ratings. A more common situation, however, is that we can only observe the personalized rankings of the items. Here, we therefore describe a novel black-box method that operates on the basis of such rankings.
In this attack, the adversary generates $d$ perturbations as well, and the scores $s_k$ should be recovered by using only their ranking $r_k$. If the score distribution of the $d$ perturbations was known, the inverse of the CDF could accurately recover the scores. A common assumption is that item scores follow normal distribution \cite{steck2015gaussian}. However, even if it is true in practice, its mean and variance remain unknown. As observed from Eq. \ref{eq:derivative}, the solution is shift invariant, so the mean of the distribution is irrelevant, but the variance does play a role.
\paragraph{Proposed Approach}
To bypass this problem we note that perhaps the entire catalog's scores follow a Gaussian distribution, where various items are involved. However, the differences in the perturbations' scores stem only from small changes to the image, while the pushed item itself is fixed. Therefore, we assume the scores reside within a very small segment $[s_{min},s_{max}]$ of the entire distribution. If the segment is small enough, it can be approximated by a uniform distribution. Hence, a mapping between ranking and score is given by $s_k=\frac{N-r_k}{N}(s_{max}-s_{min})$, where $N$ is the number of items in the catalog.
The remaining question is how to compute these values without knowledge of $s_{min}$ or $s_{max}$. First, as Eq. \ref{eq:derivative} is shift invariant, we can assume $s_{min}=0$. Second, because we solve a linear system, multiplying the dependent variable (scores) by a scalar, also scales the solution (derivatives) by the same factor. Since we take the sign of the gradient, scaling has no effect on the outcome. Thereby, we can assume $s_{max}=1$, and therefore $s_k=1-\frac{r_k}{N}$.
Remarkably, this simple approach performed well in our initial experiments. In these experiments, we created a \emph{privileged baseline} that receives the empirical standard deviation as input (information that is not accessible to the adversary), which allows it to recover the scores using the inverse of the CDF of the appropriate normal distribution.
Nevertheless, assuming uniform distribution outperformed that baseline. We explain this phenomenon by the fact that the perturbations do not perfectly follow a Gaussian distribution, which leads to inconsistency in the inferred scores.
\paragraph{Dealing with Partial Ranking Knowledge:}
One possible limitation of our algorithm is the assumption that the complete ranking of all items in the catalog is revealed by the RS.
However, in some cases not all catalog items are shown even when the user scrolls down the recommendation list.
Therefore, the pushed item might not be shown to the attacker, which averts the proposed attack since no gradients could be computed.
To overcome this obstacle, a surrogate user $u'$ is introduced by the attacker and set as the attacked one. $u'$ maintains two properties: 1) receives the pushed item in the top of the list, so gradients can be computed; 2) is similar to $u$, so the computed gradients are useful for the attacker.
Specifically, $u'$ is identical to the target user $u$, but has an added item $i'$ in the consumption history. This item is carefully selected, such that it makes the pushed item penetrate to the top items of user $u'$, thus allowing to compute $\frac{\partial s_{u',i}}{\partial f_i}$. Note that this partial derivative is close to $\frac{\partial s_{u,i}}{\partial f_i}$ but is not identical, as it is affected by $i'$.
Let $n$ denote the number of items in the history of $u$ and $u(i)$ a user with only item $i$ in the consumption history. Naturally the relative contribution of $i'$ to the gradient decreases with $n$.
As the attacker has no prior knowledge on the underlying RS, by simply assuming a linear correlation, the contribution of $i'$ is counteracted as follows: $\frac{\partial s_{u,i}}{\partial f_i} = \frac{(n+1)\cdot \frac{\partial s_{u',i}}{\partial f_i} - \frac{\partial s_{u(i'),i}}{\partial f_i}}{n}$. Once $i$ appears in the top of $u$'s recommendations, the attack is performed directly on $u$.
Item $i'$ is selected from the top recommendations for $u(i)$. In most cases, adding item $i'$ to the consumption history of $u$ is not enough to place $i$ in the top of the list of $u'$. Therefore, several attack steps are performed on $u(i')$, to increase $s_{u(i'),i}$ until $i$ appears at the top of the list of $u'$. Overall, in order to push item $i$, the algorithm requires to create 2 additional users: $u(i)$ and $u(i')$, which is an attainable request in general.
\begin{comment}
\section{Background and Related Work}\label{sec:relared}
A noteworthy paper, which studies the concept of adversarial examples in RS is \cite{tang2019adversarial}.
That work shows the effect on recommendations when the adversary can directly manipulate the \emph{internal representation} of the images, rather than the images directly.
While that paper contributes to raising awareness of adversarial examples in RS, it is unreasonable to assume that the adversary has \emph{write} access to internal representations.
Notably visual attacks are out of scope of that paper, as it does not conceive to alter images or to operate at the pixel space in general. A recent paper \cite{di2020taamr} suggests to tamper with the actual images. However, this attack assumes the adversary has complete knowledge of the system. Additionally, it does not suggest push attacks, but aims to increase the probability of some categories to be recommended.
While our paper introduces the new concept of visual attack, the proposed algorithms build on advances in related fields. Namely, this work resides at the intersection of three fields: classic shilling attacks, visually-aware recommenders and adversarial examples in image classification. The purpose of this section is to review these fields, while focusing on concepts that are relevant to the proposed approaches.
\subsection{Shilling Attacks}
The basic intuition behind the shilling attack is that recommender systems are only as good as their input data. In this attack, the adversary introduces bot-users to the system. These users leave carefully designed feedback that is intended to seem both genuine and to disrupt the output of the recommender such that the target items will be recommended to the target users. See \cite{si2020shilling, gunes2014shilling} for comprehensive reviews on shilling attack techniques.
Broadly speaking, the effectiveness of an attack is measured by the amount of target users that indeed get the pushed item as a recommendation. In the case of a shilling attack, there is a positive correlation between effectiveness and adversary resources, where resources are usually composed of knowledge and capacity. The relevant knowledge includes several aspects, for example, knowing the type of the deployed model (e.g., whether the engine is item-based k-nearest neighbors or Matrix Factorization), observing the values of the parameter set or even having reading access to the rating dataset that empowers the model. The capacity refers to the amount of shilling users and fake feedback the adversary can inject. Trivially, as the adversary possesses more knowledge and has greater capacity, the effectiveness of the attacks increases \cite{mobasher2007attacks}. As we will share, the trade-off between resources and effectiveness also exists in visual attacks. However, the relevant resources in these types of attacks are completely different from the shilling attacks.
The shilling attack has been well studied, thereby increasing awareness of this attack and significantly contributing to the effort towards robust recommenders. Recent research helped to design attack-resistant models that do not dramatically alter their recommendations due to fake feedback \cite{resnick2007influence, mehta2008attack, mehta2008survey}. Additionally, a noticeable leap was made in detecting such attacks \cite{burke2006classification, williams2007defending, chirita2005preventing, williams2007defending}. All in all, the flourishing research in this area forces malevolent adversaries to seek new opportunities. This paper anticipates the next major type of attack and we hope it will encourage research towards more robust recommenders.
\subsection{Visually-aware Recommender Systems}\label{ssec:vrec}
\begin{figure}
\setlength{\belowcaptionskip}{-10pt}
\includegraphics[ scale=0.65]{architecture.pdf}
\caption{Visually-aware RS: Architecture}
\label{fig:vrs}
\end{figure}
The two most dominant paradigms in RS are Collaborative Filtering (CF) and Content-Based Filtering (CB). While CF algorithms model users and items based on information gathered from many users, CB recommends items based on a comparison between the content of the items and a user profile. Nowadays, to achieve optimal performance most recommenders apply a hybrid approach, which combines more than a single paradigm. Arguably, the most prevalent unstructured side information on items is their visual appearance, and its effectiveness has been proven in a myriad of works: \cite{he2016vbpr, bostandjiev2012tasteweights, kim2004viscors, linaza2011image, bruns2015should, iwata2011fashion, deldjoo2016content, he2016sherlock}.
State-of-the-art visually-aware recommenders incorporate high-level features of item images extracted from fixed pre-trained deep models. Specifically, the recommender invokes a pre-trained Image Classification model (IC), e.g., ResNet \cite{he2016deep} or VGG \cite{simonyan2014very} and extracts the output of the penultimate layer, dubbed as the \emph{feature vector}.
On the one hand, a state-of-the-art IC model is trained on millions of images, allowing it to extract feature vectors that hold essential information regarding the visualization of the image. On the other hand, the feature vector is \emph{not} optimized for generating relevant recommendations. It is thus left to the recommender to learn a transformation layer that extracts valuable information with respect to recommendations.
Usually this transformation is accomplished by multiplying the feature vector by a learned matrix $E$.
Since the image is passed through a deterministic transformation, it defines a revised input layer having the feature vector replacing the pixels.
A generic schematic visualization of this approach is depicted in figure \ref{fig:vrs}.
VBPR \cite{he2016vbpr} is the first work to follow this approach and extends BPR \cite{rendle2012bpr} to incorporate images. It maintains two embedding spaces, derived from the usage patterns and the item visualization. The final predicted score results from the sum of the scores of these two spaces.
Sherlock \cite{he2016sherlock} adds the ability to model the feature vector with respect to the item category, by learning a dedicated transformation matrix $E_i$ per each category $i$.
Later, DeepStyle \cite{liu2017deepstyle} improved the category-aware approach by learning a single embedding matrix which maps the categories to the latent space of the images. Thus, a single transformation matrix $E$ is sufficient. The authors of this paper showed that the reduced number of parameters leads to enhanced performance.
Since images are an integral part of the modeling process of recommenders, an adversarial producer may tamper with the image of a target item. The tampered image should increase the probability of recommending the item to the target users while remaining indistinguishable from the genuine image.
To the best of our knowledge, this is the first work that manipulates the pixel intensities of images in order to enable push attacks.
\subsection{Adversarial Examples in Image Classification}
Adversarial examples in machine learning are inputs that seem almost identical to examples from the true distribution of data that the algorithm works on. However, they are designed to deceive the model and cause it to output incorrect predictions. In the context of image classification, adversarial examples are constructed by adding changes to genuine images, with the aim of crossing the decision boundary of the classifier (e.g., \cite{szegedy2013intriguing}) and yet remain undistinguished to the human eye.
Adversarial examples are divided into two categories, \emph{white-box} and \emph{black-box} attacks. They differ in the knowledge of the adversary, which dictates the means to guide the adversary in the search for an adversarial example.
\subsubsection{White-box}
This attack assumes the adversary has complete knowledge of the model and can compute gradients using backpropagation.
Unlike regular stochastic gradient descent (SGD), that updates the \emph{parameters} of the model in the \emph{opposite} direction of the gradient, the adversary updates the \emph{input} in the \emph{same} direction of the gradient. Let $\eta$ be the adversarial direction. Then an adversarial example is generated as $\Tilde{x}=x + \epsilon\eta$.
The step size $\epsilon$ has a dual role. As in regular SGD, it ensures the update is in the right direction even in non-linear problems. But in this setting it also has another role, to retain the updated image close to the original one. Formally, some metric $d(\Tilde{x}, x)$ is defined to quantify the visual dissimilarity attributed to the adversarial change. Commonly, $d$ is computed by either $\ell_2$ or $\ell_\infty$, and the aim of the adversary is to deceive the model while keeping a low dissimilarity score to keep the adversarial change undistinguished by the human eye.
Many approaches have been proposed to control the degree of visual dissimilarity. Examples include the ``fast gradient sign method'' \cite{goodfellow2014explaining}, optimized for $L_\infty$, which approximates this search problem by a binarization of the gradient, computed as $\eta\leftarrow sign(\eta)$. Early stopping was suggested by \cite{kurakin2016adversarial} and Madry et al. \cite{madry2017towards} used the ``projected gradient descent'', to limit the $\ell_2$-norm.
\subsubsection{Black-box}\label{ssub:bb}
This attack requires a milder assumption: the ability to query the model. In the absence of gradients to guide the adversary, it invokes the model multiple times with various perturbations within a small volume around a target image and uses the associated feedback to infer the right direction.
However, due to the high-dimensionality of the \emph{pixel space}, even efficient attacks need at least tens of thousands of requests to the classifier, limiting their applicability\cite{ilyas2018black, cheng2018query, tu2019autozoom, guo2019simple}.
The black-box threat is further subdivided according to the specification of the attacked model, which may return the \emph{score} of each predicted category, or only its \emph{ranking}. Once again, there is a trade-off between the effectiveness of the attack and the specification level of the model.
Both the white-box and black-box attacks can improve effectiveness by working iteratively and taking multiple small \emph{steps} in the direction of the gradient. For instance, the ``Iterative Gradient Sign'' \cite{kurakin2016adversarialb} applies multiple steps of the ``fast gradient sign method'' \cite{goodfellow2014explaining} and reports boost in performance.
\end{comment}
\begin{comment}
\section{Attacks on Visually-aware Recommenders}\label{sec:attacks}
As mentioned above, an attack has three key factors: \emph{adversary}, \emph{pushed items} and \emph{target users}. In this section we elaborate on these factors and specify the assumptions on them. This allows us to describe in the next sections several attacks, spanning all plausible embodiments of the aforementioned factors.
\paragraph{Pushed items} Visual attacks work by solely modifying the visualization of the pushed items. Therefore, pushing a certain item would not change the score of any other item one iota. As a corollary, attacks with several pushed items are independent of each other. Hence, throughout the paper we assume there is a single pushed item.
\paragraph{Target users} There are three types of target users: specific users, segments or general population.
Focusing on specific users is beneficial as it is equivalent to personal advertising and may exploit the influence of opinion leaders. However, in order to attack a specific user, the adversary has to know the consumption history of that user. Then, that user can be cloned by introducing to the RS a new user with the exact same usage patterns, and the attack is carried out on the cloned user. We note that in many practical scenarios, such information is practically unattainable and we demonstrate this attack mainly as a proof of concept. Furthermore, since the impact of attacking a certain user is limited, potential adversaries would probably prefer to focus on other types of target users.
Approaching general population dispenses with the need to have prior knowledge, but it results in exposure of the pushed items to irrelevant users. Increasing the visibility of the target item to irrelevant users is unlikely to impact the promotional activity and even might lead to reputational risk.
Segmented attack is an in-between approach, where the adversary targets a set of users with similar tastes.
As we will explain later, even in the general population and segmented attacks we can safely assume that the adversary attacks a \emph{single} mock user that was \emph{injected by the attacker} to the system.
\paragraph{Adversary} We start off by describing an attack that assumes the adversary has read access to the model parameters, commonly known as the White-box attack. We then gradually loosen the knowledge requirements, in different variants of Black-box attacks.
\section{White-Box Attack}\label{sec:wb_attack}
This attack assumes the adversary has a \emph{read} access to the parameters of the model. At inference time, to allow personalized ranking, the score of user $u$ given for item $i$ is computed by a visually-aware recommender as $s_{u,i}=f(u, i, p_i)$, where $p_i$ is the image associated with item $i$. Note that the only assumption we make on $f(\cdot)$ is it is differentiable. To avoid clutter, we omit the subscripts whenever their existence is clear from the context. To manipulate the image, the adversary computes the partial derivatives $\frac{\partial s}{\partial p}$ and updates the pixels in that direction, while keeping the distorted image as close as possible to the genuine one. Following \cite{goodfellow2014explaining}, we binarize these partial derivatives by taking their sign, as our preliminary experiments asserted it leads to faster convergence. We note that any future improvements in adversarial examples for images may contribute to the effectiveness of this proposes approach.
The visual attack is performed in small \emph{steps}. Each step $t+1$ aims to increase the predicted score of the pushed item by updating the image as follows: $p^{t+1}=p^t + \epsilon\cdot sign(\frac{\partial s}{\partial p^t})$, where $p^0$ is the genuine image and $\epsilon$ is a small constant that controls the step size so the adversarial changes will be overlooked by the human eye.
This type of attack mainly serves as an upper bound for attack effectiveness since assuming the adversary has access to the parameter set of the model might be unrealistic. Therefore, potential attackers need to resort to black-box approaches, detailed in the next section.
\subsection{Visual Similarity}
Since every update step adds some noise to the image, taking too many steps might expose the attack. It is therefore of primary importance to limit the number of steps. In Section \ref{sec:experiments} we present a trade-off between the number of steps and the performance of the attack.
To assess the perceptual visual distortion caused by the adversarial changes, we use the widely adopted \emph{objective measure} Structural SIMilarity (SSIM) index \cite{wang2004image}.
This index considers image degradation as perceived change in structural information. The idea behind structural information is that pixels have stronger dependencies when they are spatially close. The SSIM index also incorporates, with equal importance, luminance masking and contrast masking terms. These terms reflect the phenomena where image distortions are more visible in bright areas and in the background, respectively.
To compute the SSIM index between two images, local SSIM indices are computed over small corresponding windows and their average is returned. Formally, Let $x$ and $y$ be windows of a clean image and of an adversarial perturbation of it. The SSIM index is computed as:
$S(x, y)=\frac{(2\mu_x\mu_y + C_1)(2\sigma_{xy} + C_2)}{(\mu_x^2 + \mu_y^2 + C_1)(\sigma_x^2 + \sigma_y^2 + C_2)}$
where $\mu$, $\sigma^2$ and $\sigma_{xy}$ stand for mean, variance and covariance, respectively, and $C_1,C_2$ are small constants to avoid devision by zero. The index ranges between -1 and 1, and value 1 is obtainable only for identical images.
We computed the index values using the scikit-image library\footnote{\url{https://scikit-image.org/docs/dev/auto_examples/transform/plot_ssim.html}}.
Figure \ref{fig:vis_steps} illustrates the aggregate changes along with their associated SSIM scores after a various number of update steps on two random images. As can be seen, until about 20 steps the changes are fairly imperceptible. Afterwards the attack might be revealed as some noise is noticed mostly in the background, which is no longer solid. This phenomenon repeats in the dataset, as the average SSIM indices after 10, 20 and 30 steps are: 0.965, 0.942 and 0.914, respectively. Since there is a noticeable degradation after 20 steps, throughout the paper we set the maximal allowed number of steps to 20.
\begin{figure}
\includegraphics[scale=0.6]{vis-steps-ssim.pdf}
\caption{Illustration of the visual change in images with increasing number of update steps}
\label{fig:vis_steps}
\end{figure}
\section{Black-Box Attacks}\label{sec:bb_attacks}
We describe two variants of black-box attacks: score-feedback and rank-feedback. In the latter, we assume the RS outputs the ranking of the items; while in the former it also reports the personalized scores of each item.
\subsection{Score-feedback}
Just like in the white-box attack, the adversary would like to compute the gradient of the score $s$ with respect to all pixels in the image $p$ in the corresponding image, i.e., $\frac{\partial s}{\partial p}$. However, as explained in Subsection \ref{ssub:bb}, black-box attacks in image classification require an enormous amount of requests to the classifier because they work in the \emph{pixel space}. Applying similar techniques in visual attacks would be impractical.
To mitigate this problem, we reiterate that visually-aware recommenders usually leverage the capabilities of a pretrained IC model. As depicted in Figure \ref{fig:vrs} and explained subsection \ref{ssec:vrec}, the IC is publicly available and held fixed during training, while the rest of the network is optimized for the specific task and data.
The system extracts a feature vector $f_i$ from image $p_i$ by feed forwarding it through the IC and outputting the penultimate layer.
This allows us to apply the chain rule and the multiplication between the partial derivatives of the score $s$ with respect to $f$ and the Jacobian of $f$ with respect to $p$ is computed. Formally:
$\frac{\partial s}{\partial p}=\frac{\partial s}{\partial f} \cdot \frac{\partial f}{\partial p}$.
Since we assume the parameter set of the image processing unit is known, then $\frac{\partial f}{\partial p}$ can be analytically computed using backpropagation.
Decoupling the partial derivative into two quantities allows to avert numerical computations in the pixel space. However, the computation of the partial derivatives $\frac{\partial s}{\partial f}$ under the black-box model of the RS is rather complicated because $f$ is not the actual input layer. To illustrate the challenge, consider the following approach for numerical computation of the partial derivatives, where $d$ denotes the dimensionality of $f$. Create $d$ perturbations of $p$, such that each perturbation $p^k$, for $0\leq k < d$, under the constraint $\left\| p-p^k \right\| \leq \delta$, yields a feature vector $f^k=f + \epsilon\cdot \mathcal{I}(k)$ for some predefined $\delta$ and arbitrary small $\epsilon$, where $\mathcal{I}(k)$ is the one-hot vector with $1$ for $k$-th coordinate and $0$ elsewhere. That is, each perturbation isolates a different dimension in $f$. Then, replace the genuine image $d$ times, each time with a different perturbation, and obtain from the RS a new personalized score $s^k$. Now the numerical partial derivative $\frac{\partial s}{\partial f^k}$ is given by $\frac{s^k - s}{\epsilon}$. However, since each pixel in $p$ holds a discrete value and affects all dimensions in $f$, finding such a set of perturbed images is intractable.
Furthermore, in the absence of the ability to directly control $f$, it is not clear how to apply existing methods of gradient estimation. For instance, the widely used NES \cite{wierstra2008natural} requires to sample perturbations from a Gaussian distribution centered around the input $f$; even the seminal SPSA algorithm \cite{spall1992multivariate} demands a \emph{symmetric} set of perturbations. As such methods are unattainable, we devise a novel approach to estimate the gradients.
\oren{Oren - write some motivation}To this end, we create $d$ random perturbations of the image uniformly drawn from the $L_\infty$-ball centered around $p$ with radius $\delta$, for some arbitrary small $\delta$, and obtain their corresponding feature vectors $f^k$ together with the associated personalized scores $s^k$.
\cite{zou2019lipschitz} proved that CNN-based architectures are Lipschitz continuous, and empirically showed that their Lipschitz constant is bounded by $1$.
Consequently, since each perturbation is similar to the original image, up to a tiny difference $\delta$, also the deltas in the feature vectors $\left\| f^k-f \right\|$ are expected to be epsilonic, making them suitable for numeric computation of $\frac{\partial s}{\partial f}$.
Using matrix notation, the set of vectors $\{f^k\}$ and scalars $\{s^k\}$ are referred to as matrix $\bm{F}\in \mathbb{R}^{d\times d}$ and vector $\bm{s}\in \mathbb{R}^d$, correspondingly.
This translates into a linear system $\bm{F}x=\bm{s}$ such that its solution $x$ is the numerical computation of $\frac{\partial s}{\partial f}$. To see it, keep in mind that a linear system is shift invariant. That is, the same solution $x$ is obtained upon subtracting $f^T$ from each row of $\bm{F}$ and subtracting $s$ from each component in $\bm{s}$. Now the coefficient matrix is the change in $f$ and the dependent variable is the resulting change in score:
\begin{equation}\label{eq:derivative}
x = (\bm{F}-f^T)^{-1}(\bm{s}-s) = \Delta\bm{F}^{-1}\Delta\bm{s}=\frac{\partial s}{\partial f}
\end{equation}
Note that a single solution exists if and only if $\bm{F}$ is nonsingular. Since it is obtained by random perturbations, it is almost surely the case \cite{kahn1995probability}, and otherwise a new set of perturbation can be drawn.
After each step, an actual image has to be created and to be uploaded to the RS, so the next step will be computed on the updated one. This operation has two consequences:
\begin{itemize}
\item The values at $p_{t+1}$ are float numbers, which should be rounded to integers in order to make an actual image. Adversarial examples for images exploit delicate nuances, and rounding all pixels after each step might diminish them. This is yet another reason to limit the number of steps.
\item Updating the corresponding image might consume some time. Since it does not require to retrain the model, and it is common to have an API for item producers to update their images, its overhead is bearable.
\end{itemize}
\subsection{Rank-feedback}
Recommenders usually do not reveal the actual scores, but only present the items according to their personalized rank, which make the rank-feedback attack more plausible. In this attack, the adversary generates $d$ perturbations as well, and the scores $s_k$ should be recovered by using only their ranking $r_k$. If the score distribution of the $d$ perturbations was known, the inverse of the CDF could accurately recover the scores. A common assumption is that item scores follow normal distribution \cite{steck2015gaussian}. However, even if it is true in practice, its mean and variance remain unknown. As observed from Eq. \ref{eq:derivative}, the solution is shift invariant, so the mean of the distribution is irrelevant, but the variance does play a role. To bypass this problem we note that perhaps the entire catalog's scores follow a Gaussian distribution, where various items are involved. However, the differences in the perturbations' scores stem only from small changes to the image, while the pushed item itself is fixed. Therefore, we assume the scores reside within a very small segment $[s_{min},s_{max}]$ of the entire distribution. If the segment is small enough, it can be approximated by a uniform distribution. Hence, a mapping between ranking and score is given by $s_k=\frac{N-r_k}{N}(s_{max}-s_{min})$, where $N$ is the number of items in the catalog. The remaining question is how to compute these values without knowledge of $s_{min}$ or $s_{max}$. First, as Eq. \ref{eq:derivative} is shift invariant, we can assume $s_{min}=0$. Second, because we solve a linear system, multiplying the dependent variable (scores) by a scalar, also scales the solution (derivatives) by the same factor. Since we take the sign of the gradient, scaling has no effect on the outcome. Thereby, we can assume $s_{max}=1$, and therefore $s_k=1-\frac{r_k}{N}$.
Remarkably, this simple approach performed well in our initial experiments - we created a \emph{privileged baseline} that receives the empirical standard deviation as input (information that is not accessible to the adversary), which allows it to recover the scores using the inverse of the CDF of the appropriate normal distribution.
Nevertheless, assuming uniform distribution outperformed that baseline. We explain this phenomenon by the fact that the perturbations do not perfectly follow a Gaussian distribution, which leads to inconsistency in the inferred scores.
One limitation of this algorithm is the assumption that the complete ranking of all items in the catalog is revealed by the RS. However, in many cases only the top items appear, and even scrolling-down the recommendation list is limited to at most hundreds of items. Relieving this limitation could be a further extension of this paper.
As explained above, we assume a negligible overhead due to uploading a single image to the server. However, the rank-feedback attack requires to upload $d$ perturbations of the image per each step. In ResNet for example there are 2,048 latent dimensions, resulting in an overwhelming amount of updates. We next address this problem and suggest a trade-off between effectiveness of each step and amount of required image updates.
\subsection{Reduced Amount of Perturbations}
Although Eq. \ref{eq:derivative} requires $d$ perturbations for finding a single solution, an approximation can be found with fewer perturbations. Formally, let $d'$ be the number of perturbations the adversary generates, for some $d'<d$. This defines an underdetermined system of linear equations, and among the infinite solutions, the one with the minimal $\ell_2$ norm can be found \cite{cline1976l_2}.
Another advantage of reducing the number of perturbations is the ability to generalize.
There is an unlimited number of possible perturbations, and every subset of $d$ perturbations defines another linear system, with another exact solution. Those exact solutions vary in their distances from the true, unobserved derivatives. However, since the adversary randomly select a single subset of perturbations, the exact solution may ``overfit'' to this instantiation of linear system, and not generalize to other solutions that could be yielded from other subsets. Such behavior might be \emph{fatal}, and must be addressed. To remedy this problem, regularization comes in useful. Reducing the number of perturbations, and finding a solution with minimal $\ell_2$ norm can be thought of as a means regularization, and can potentially improve performance. Indeed, as we show in Section \ref{sec:experiments}, additionally to reducing the required amount of perturbations this approach can lead to improved performance.
\end{comment}
\section{Experimental Evaluation} \label{sec:experiments}
We demonstrate the effectiveness of the attack models by conducting experiments on several combinations of RSs and datasets.
\subsection{Experiment Setup}
\subsubsection{Datasets}
We relied on two real-world datasets that were derived from the Amazon product data \cite{mcauley2015image}.
The first dataset is the Clothing, Shoes and Jewelry dataset (named ``Clothing'' for short) and the second one is the Electronics dataset.
These datasets differ in all major properties, like domain, sparsity and size, which shows visual attacks are a ubiquitous problem. Arguably chief among their distinctive traits is the importance of visual features, quantitatively approximated by the gain in performance attributed to the ability to model images.
To compute this trait, dubbed as ``visual gain'', we use AUC \cite{rendle2012bpr} as a proxy for performance. Since VBPR \cite{he2016vbpr} extends BPR \cite{rendle2012bpr} to model images, the visual gain is computed as the relative increase in the AUC obtained by VBPR over BPR.
Table \ref{tab:datasets} summarizes statistics of these datasets.
It is noticed that the visual gain in the electronics dataset ($2\%=\frac{0.85}{0.83}-1$) is much lower than in the clothing dataset ($23\%=\frac{0.79}{0.64}-1$).
We therefore expect that visual features in the electronics domain would be of lesser importance, and as a consequence the RS will only moderately consider them.
Nevertheless, we show that visual attacks are effective even in this domain.
\begin{table}[t!
\centering
\begin{tabular}{|l|c|c|c|c|}
\hline
Dataset & \#users & \#items & \#feedback & Visual gain \\
\hline
Clothing & 127,055 & 455,412 & 1,042,097 & 23\% \\
Electronics & 176,607 & 224,852 & 1,551,960 & 2\% \\
\hline
\end{tabular}
\caption{Dataset statistics after preprocessing}
\label{tab:datasets}
\end{table}
\subsubsection{Attacked Models}
We experiment with two visually-aware RS as the underlying attacked models.
The first is the seminal algorithm VBPR \cite{he2016vbpr}.
To assess the effect of visual attacks in a common setting where \emph{structured} side information is available, the second model is DeepStyle \cite{liu2017deepstyle}, which also considers the item categories.
We train a model for each combination of dataset and underlying RS, yielding 4 models in total. We randomly split each dataset into training/validation sets, in order to perform hyperparameter tuning. Both algorithms are very competitive and the obtained AUC of VBPR (DeepStyle) on the validation set of the Clothing dataset is 0.79 (0.80) and 0.85 (0.86) in the Electronics domain.
\subsubsection{Visual Features}
The underlying image classification model is ResNet \cite{he2016deep}. This is a 50 layer model, trained on 1.2 million ImageNet \cite{deng2009imagenet} images. For each item to be modeled by the RS, its associated image is first fed into the IC model and the penultimate layer of size 2,048 is extracted as the feature vector.
\subsubsection{Evaluation Metric}
The purpose of attacks is to increase the visibility of the pushed items in top-k recommendations. Therefore, we follow \cite{sarwar2001item,mehta2008attack} and evaluate the effectiveness of an attack using the Hit-Ratio (HR), which measures the fraction of affected users by the attack. Formally, let $\mathcal{S}$ be the set of user-item pairs subject to an attack. For each $(u,i)\in\mathcal{S}$, the Boolean indicator $H^k_{u,i}$ is true if and only if item $i$ is in the top-k recommendation to user $u$. Then the Hit Ratio is defined as follows:
$$HR@k = \frac{1}{|\mathcal{S}|}\sum_{(u,i)\in\mathcal{S}} H^k_{u,i}$$
In real-life datasets with many items, the probability of a random pushed item to appear in the top of the recommendation list is extremely low. Therefore HR directly measures the impact of the attack, as the number of pre-attack hits is negligible or even zero.
\subsection{Varying the Target Population}
We design dedicated experiments for each type of target populations to reflect the different assumed capabilities of the adversary.
\subsubsection{Specific-users attack}
In this threat, the adversary knows the consumption list of the attacked user and hence can impersonate that user by introducing to the RS a new user with the same list.
To simulate these attacks, we randomly select 100 user-item pairs to serve as the attacked users and pushed items.
\subsubsection{Segmented attack}
Following \cite{burke2005segment}, a segment is a group of users with similar tastes, approximated by the set of users with a common specific item in their interaction histories. For instance, a segment can be defined as all users who purchased a certain Adidas shoe. We name the item that defines the segment as the \emph{segment item}\footnote{Supporting multiple segment items can be trivially implemented as well.}.
To perform this attack, the adversary creates a single mock user who serves as the target user, whose interaction history includes only the segment item.
We do not assume that the adversary possesses information about usage patterns in general or knows which item defines the optimal segment item. Therefore, the segment item is simply chosen by random among all items in the same category of the pushed item.
In our experiments we chose 100 random pairs of pushed and target items. We emphasize that the actual target users are those who interacted with the segment item, but we do not assume the adversary knows their identities.
We report the effectiveness of the attack on the actual set of target users, which is obscured from the adversary.
\subsubsection{General population}
This attack assumes that the adversary wishes to associate the pushed item with every user in the dataset.
As this threat model does not assume the adversary has any knowledge of usage behavior, preferences of mass population need to be approximated. To this end, an adversary might pick $N$ random items, naturally without knowing which users have interacted with them. The assumption is if $N$ is large enough, then these items span diverse preferences.
Then the adversary adds these items by injecting $N$ users, each with a single item in their interaction history, to function as the set of target users. The \emph{active target} user is chosen in a round-robin manner, and an adversarial step is made to maximize the score for that user.
However, we noticed that this method fails, as maximizing the score for the active target user also deteriorates the scores of the pushed item for many other target users. This is an expected behavior, as it is infeasible to find a direction that increases the scores of an entire set of randomly selected users.
Keeping in mind that the objective of the adversary is not to increase the average score of the pushed item, but to bubble it up to the \emph{top} of many users' lists, we modify this approach by dynamically choosing the next active user. Preceding to each step, the adversary ranks the target users by their personalized score of the pushed item\footnote{Note that this operation adds $N$ parallelable queries to the recommender and does not require to alter the image.}. Then it concentrates on target users who are more likely to receive the pushed item in the top of their recommendation list. This is done by setting the active target user as the one in the $p^{th}$ percentile. As $p$ increases from 0 to $100\%$, the adversary concentrates on a larger portion of the target users. we tuned $p$ on 5 equally distanced values in the range $[0,1]$ and obtained optimal performance at $p=25\%$.
In this type of experiment we pick at random 100 pairs of items, each of which serves as the pushed item and set $N=200$ as the number of random items. Once again, the adversary cannot know the actual effectiveness of this attack on the general population, since their identities and usage patterns are not disclosed.
We report the performance of the attack on $1,000$ randomly chosen users who serve as a representative sample of the general population.
\subsection{Baselines}
Although the main purpose of this paper is to empirically demonstrate the existence of a threat on AI safety posed by visual attacks, it is still important to compare the effectiveness of our
methods with relevant baselines.
To empirically demonstrate the existence of the identified vulnerability and to judge the relative effectiveness of the proposed attack models, we include two baseline attacks in our experiments. Since, to the best of our knowledge, this is the first work to investigate these types of visual attacks in the context of recommenders, there are no prior baselines.
Nevertheless, to show the importance of carefully designed perturbations, we propose two simple baseline methods that perform push attacks by altering genuine images.
\begin{itemize}
\item The first is an easy-to-detect attack, which replaces the original image $p$ of the pushed item with the image $p_{pop}$, associated with the most popular item from the same category. The rationale behind this baseline is that some of the success of the most popular items is attributed to their images. A visually-aware recommender should capture the visual features that correlate with consumption, and reflect it in superior scores for items with these images.
\item The second baseline is a softer version of the former, making it harder to be detected. In this baseline, $p\leftarrow p+\epsilon\cdot p_{pop}$, where $\epsilon$ controls the amount of change added to the image. We experimented values in the range $[0.01, 0.1]$ in steps of $0.01$.
\end{itemize}
However, while in all configurations both baselines systematically increased the scores of the pushed items and improved their ranking in thousands of places on average, they failed to penetrate to the top of the lists and to improve $HR@K$. The resulting Hit Ratios were therefore consistently very close to zero in all experiments, which is why we omit the results in the following sections. Ultimately,
this evidence reinforces our preliminary assumption on the importance of well-guided adversarial examples.
\subsection{Results}
Here, we report the results of the experiments and present multiple trade-offs between adversarial resources and effectiveness. Across all experiments, when partial ranking knowledge is available, we assume that only the top 1\% of the ranking is known.
\subsubsection{Specific-user Attack}
Table \ref{tab:specific_users} details the HR$@k$ of different attacks for various values of $k$. We fixed the number of perturbations (32) and steps (20), leading to a total of $640=32\cdot 20$ image updates.
Attack types denoted by WB, BB-Score, BB-Rank and BB-Partial stand for white-box, black-box on scores, black-box on rankings and black-box on partial ranking knowledge, respectively.
The results demonstrate the effectiveness of all proposed attack models. In terms of the absolute numbers for the Hit Ratio we can observe that after the attack the probability of a randomly pushed item to appear in the top-20 list is substantial, even when using a black-box approach. Generally, the results also validate the existence of trade-off between knowledge on the attacked system and effectiveness. We notice that the white-box attack, which assumes absolute knowledge on the underlying system, as expected outperforms the rest of the attacks. However, the differences between the black-box approaches on rankings and scores are usually small and depend on the metric and dataset.
It is worthwhile to mention that even in the Electronics dataset, where visual features play a minor role, with attributed gain of only $2\%$ to the AUC, the attack is still effective. For a considerable amount of users, it can bubble up random items to the top of their lists. This proves that under attack, even seemingly less important side-information can be used to manipulate the results of collaborative filtering models.
\begin{table
\begin{center}
\begin{tabular}{ |m{4.2em}|m{3.8em}|c||m{2.3em}|m{2.7em}|m{2.7em}| }
\hline
Dataset & RS & Attack & $HR@1$ & $HR@10$ & $HR@20$ \\
\hline
\multirow{8}{*}{Clothing} &
\multirow{4}{*}{VBPR} & WB & 0.79 & 0.89 &0.91 \\
\cline{3-6}
& & BB-Score & 0.43 & 0.54 & 0.58 \\
\cline{3-6}
& & BB-Rank & 0.41 & 0.51 & 0.53 \\
\cline{3-6}
& & BB-Partial & 0.23 & 0.23 & 0.23 \\
\cline{2-6}
& \multirow{4}{*}{DeepStyle} & WB & 0.73 & 0.80 &0.83 \\
\cline{3-6}
& & BB-Score & 0.39 & 0.50 & 0.52 \\
\cline{3-6}
& & BB-Rank & 0.42 & 0.50 & 0.54 \\
\cline{3-6}
& & BB-Partial & 0.10 & 0.10 & 0.10 \\
\hline
\multirow{8}{*}{Electronics} &
\multirow{4}{*}{VBPR} & WB & 0.45 & 0.57 & 0.60 \\
\cline{3-6}
& & BB-Score & 0.15 & 0.22 & 0.27 \\
\cline{3-6}
& & BB-Rank & 0.14 & 0.24 & 0.27 \\
\cline{3-6}
& & BB-Partial & 0.47 & 0.50 & 0.50 \\
\cline{2-6}
& \multirow{4}{*}{DeepStyle} & WB & 0.53 & 0.62 & 0.66 \\
\cline{3-6}
& & BB-Score & 0.18 & 0.20 & 0.29 \\
\cline{3-6}
& & BB-Rank & 0.20 & 0.26 & 0.29 \\
\cline{3-6}
& & BB-Partial & 0.20 & 0.20 & 0.20 \\
\hline
\end{tabular}
\end{center}
\caption{Efficacy of specific-user attack}
\label{tab:specific_users}
\end{table}
Due to space limitations, in the rest of the paper we report the results only of a subset of the configurations.
First, as we did not find any significant difference in terms of the attack effects for the different algorithms (VBPR and DeepStyle) we focus on the former. Second, while the HR of Clothing and Electronics are different, their trends across the experiments remain the same. Hence, we report the results of the Clothing dataset. Third, the default value of $k$ is 20 in the $HR@k$ measure.
Moreover, in the experimented datasets across the three color channels, pixels take values in the range $(0,255)$. We report the results obtained by setting the minimal step size $\epsilon$, which bounds the size of each update to at most $1$, as it led to superior performance comparing to higher values.
Figures \ref{fig:perturbations} and \ref{fig:steps} extend Table \ref{tab:specific_users} and probe the effects of isolated factors.
Figure \ref{fig:perturbations} visualizes the trade-offs between effectiveness, number of steps, and number of perturbations under the BB-Rank attack. It plots HR as a function of the number of steps, for several values of perturbations per step.
\begin{figure}[h!t]
\includegraphics[trim={1.3cm 0.4cm 2cm 1cm}, clip, scale=0.41, center]{perturbations.pdf}
\caption{Effect of number of perturbations and update steps}
\label{fig:perturbations}
\end{figure}
\begin{figure}[h!t]
\includegraphics[trim={1.3cm 0.4cm 2.cm 1cm}, clip, scale=0.41, center]{steps.pdf}
\caption{Effectiveness of each attack over number of steps}
\label{fig:steps}
\end{figure}
As can be seen, if the adversary employs 64 perturbations per step and wants to obtain HR of roughly 0.5, then 8 steps are required, which means a total of $512=64\cdot 8$ image updates. In comparison, obtaining on-par performance with 16 perturbations requires to almost double the number of steps, and to take 15 steps, which might deteriorate the visualization of the image. However, that alternative requires only $240=16\cdot 15$ image updates.
We note that using 2,048 perturbations leads to a single solution of the gradient, without ability to generalize. Its effect is detrimental and these attacks fail to increase the HR.
In the rest of the experiments we fix the number of perturbations to 64, as it achieves solid performance with a moderate budget of image updates. Figure \ref{fig:steps} shows the effectiveness of each attack at various numbers of steps.
We first observe that the effectiveness of the four types of attacks monotonically increases with the number of steps. This demonstrates the effectiveness of our computations,
as each additional step directs the adversary to achieve more impact.
The BB-Score and BB-Rank behave very similarly throughout the attack, which shows our proposed method of estimating scores by ranking is consistent.
Ranking based on partial knowledge also leads to solid performance, which shows the potential of this attack in real-life scenarios.
Finally, we hypothesize that the effectiveness of an attack depends on the relevance of the pushed item to the target user, approximated by the pre-attack ranking. Figure \ref{fig:from_rank} shows the effectiveness of the BB-Rank attack where the pushed items are grouped by their initial ranking. As it shows, attacks on relevant items converge faster and achieve higher HR. Remarkably, even when the pushed item is irrelevant to the user, i.e., the initial ranking is in the hundreds of thousands, within a plausible number of 20 steps, the adversary succeeds in pushing the item to the top of the lists of the vast majority of the target users.
\begin{figure}[h!t]
\includegraphics[trim={1.37cm 0.4cm 2.1cm 1cm}, clip, scale=0.41, center]{from-rank.pdf}
\caption{Effectiveness over different rank ranges}
\label{fig:from_rank}
\end{figure}
\subsubsection{Segmented and General Population Attacks}
Figure \ref{fig:segment_general} shows the performance of the segmented and general population attacks side by side. Again, we can observe a correlation between the adversary's knowledge about the target users and performance. Compared to the general population attack, the Hit Ratio is higher for the segmented attack, where the adversary knows a single item in the history of all users (although does not know the identity of them). Furthermore, both of these attacks are outperformed by the specific-user attacks (not shown in the figure), in which the adversary possesses vast knowledge on the interaction history of the users. This comes as no surprise, since deep knowledge on users entails more accurate guidance for the adversary.
When given only \emph{partial} ranking knowledge, the attack is still successful for a significant amount of users, e.g., around 1.6\% of the entire user base for the general population attack.
\begin{figure}[h!t]
\includegraphics[trim={0.5cm 0.4cm 2cm 1cm}, clip, scale=0.41, center]{segment-general.pdf}
\caption{Segmented and general population attacks}
\label{fig:segment_general}
\end{figure}
\section{Discussion and Outlook}\label{sec:discussion}
In this paper we identified and investigated a new type of vulnerability of recommender systems that results from the use of externally-provided images. We devised different white-box and, more importantly, black-box attack models to exploit the vulnerabilities. Experiments on two datasets and showed that the proposed attack models are effective in terms of manipulating the scores or rankings of two visually-aware RS.
While the proposed approaches rely on images to perform attacks, the same techniques can be applied on any continuous side information. Consequently, a possible future work is to investigate the vulnerabilities of audio-based \cite{van2013deep}, video-based \cite{covington2016deep} or textual review-based \cite{shalom2019generative} recommenders .
We hope this paper will raise awareness about image-based attacks and spur research on means to combat such attacks.
In the context of defending RS against shilling attacks, we are encouraged by the impressive effectiveness of detection algorithms \cite{burke2006classification,williams2007defending,li2016shilling,zhang2018ud,tong2018shilling}, and the results of robust RS \cite{turk2018robust,alonso2019robust}. Also, existing works on increasing the robustness of RS against adversarial examples \cite{he2018adversarial,du2018enhancing} may ignite research on robustness against visual-attacks. Additionally, research against adversarial examples in general is fruitful and several algorithms were proposed for detection \cite{carlini2017adversarial,xu2017feature} and new robust models were devised \cite{burke2006classification,gu2014towards}.
We anticipate a plethora of work in these veins towards more robust recommenders, which
are more robust to tiny visual changes.
\newpage
\bibliographystyle{ACM-Reference-Format}
|
1,108,101,564,583 | arxiv | \section{Introduction}
Our goal in this paper is to prove the following two main results.
For a graph $G$, we write $\Delta(G)$, $\omega(G)$, and $\chi(G)$ to denote the
maximum degree, clique number, and chromatic number of $G$. When the context is
clear, we simply write $\Delta$, $\omega$, and $\chi$.
\begin{thm}
\label{main1}
If $G$ is a graph with $\chi\ge\Delta\ge 13$, then $\omega\ge \Delta-3$.
\end{thm}
\begin{thm}
\label{main2}
Let $G$ be a graph and let $\mathcal{H}(G)$ denote the subgraph of $G$ induced by
$\Delta$-vertices. If $\chi\ge \Delta$, then $\omega\ge \Delta$ or
$\omega(\mathcal{H}(G))\ge \Delta-5$.
\end{thm}
The proofs of Theorems~\ref{main1} and \ref{main2}, are
both somewhat detailed, so we first prove Theorem~\ref{easy}, which is a
simpler result that plays a central role in proving our two main theorems.
(Recall from
Brooks' Theorem, that if $\chi > \Delta\ge 3$, then $G$ contains
$K_{\Delta+1}$, so the interesting case of these theorems is when
$\chi=\Delta$.)
\begin{thm}
\label{easy}
If $G$ is a graph with $\chi\ge \Delta= 13$, then $G$ contains $K_{10}$.
\end{thm}
Borodin and Kostochka conjectured in 1977 that if $G$ is a graph with
$\Delta\ge 9$ and $\omega\le\Delta-1$, then $\chi\le\Delta-1$. The hypothesis
$\Delta\ge9$ is needed, as witnessed by the following example. Form $G$ from
five disjoint copies of $K_3$, say $D_1,\ldots,D_5$, by adding edges between $u$
and $v$ if $u\in D_i$, $v\in D_j$, and $i-j\equiv1\bmod 5$. This graph is
8-regular with $\omega=6$ and $\chi=\ceil{15/2}$, since each color is used on at
most 2 of the 15 vertices. Various other examples with $\chi=\Delta$ and
$\omega<\Delta$ are known for $\Delta\le 8$ (see for example~\cite{CR2}).
The Borodin-Kostochka Conjecture has been proved for various families of graphs.
Reed~\cite{Reed} used probabilistic arguments to prove it for graphs with
$\Delta\ge 10^{14}$. The present authors~\cite{CR2} proved it for claw-free
graphs (those with no induced $K_{1,3}$).
The contrapositive of the conjecture states that if $\Delta\ge 9$ and
$\chi\ge\Delta$, then $\omega\ge\Delta$. The first result in this direction was
due to Borodin and Kostochka~\cite{BK}, who proved that
$\omega\ge\floor{\frac{\Delta+1}{2}}$. Subsequently, Mozhan~\cite{Moz1} proved that
$\omega\ge\floor{\frac{2\Delta+1}{3}}$ and Kostochka~\cite{Kostochka} proved
that $\omega\ge\Delta-28$. Finally, Mozhan proved that $\omega\ge\Delta-3$
when $\Delta\ge 31$ (this result was in his Ph.D. thesis, which unfortunately is
not readily accessible). Theorem~\ref{main1} strengthens Mozhan's result, by
weakening the condition to $\Delta\ge 13$.
Work in the direction of Theorem~\ref{main2} began in~\cite{kierstead2009ore},
where Kierstead and Kostochka proved that if $\chi\ge\Delta\ge 7$ and
$\omega\le\Delta-1$, then $\omega(\mathcal{H}(G))\ge 2$. This was strengthened
in~\cite{KRS} to the conclusion $\omega(\mathcal{H}(G))\ge\floor{\frac{\Delta-1}2}$. We
further strengthen the conclusion to $\omega(\mathcal{H}(G))\ge \Delta-5$.
We give more background
in the introduction to Section~\ref{sectionMain1}.
Most of our notation is standard, as in~\cite{IGT}.
We write $K_t$ and $E_t$ to denote the complete and empty graphs on $t$
vertices, respectively. We write $[n]$ to denote $\{1,\ldots,n\}$.
The \textit{join} of disjoint graphs $G$ and $H$,
denoted $\join{G}{H}$, is formed from $G+H$ by adding all edges with one
endpoint in each of $G$ and $H$. For a vertex $v$ and a set $S$ (containing
$v$ or not) we write $d_S(v)$ to denote $|S\cap N(v)|$. When vertices $x$ and
$y$ are adjacent, we write $x\leftrightarrow y$; otherwise $x\not\leftrightarrow y$. Note that in a
$\Delta$-critical graph, every vertex has degree $\Delta$ or $\Delta-1$.
A vertex $v$ is \textit{high} if $d(v)=\Delta$ and \textit{low} otherwise.
\section{Mozhan Partitions}
All of our proofs rely heavily on the concept of Mozhan Partitions. Informally,
for a $\Delta$-critical graph $G$, such a partition is a $(\Delta-1)$-coloring
(with useful properties) of all but one vertex of $G$. These ideas were
implicit in~\cite{Rabern2} and~\cite{KRS}, and much earlier in~\cite{Moz1}. We
find that making them explicit makes the proofs easier.
\begin{defn}
A \emph{Mozhan $(3,3,3,3)$-partition} of a graph $G$ with
$\Delta=13$ is a partition $(V_1,\ldots,V_4)$ of
$V(G)$ such that:
\begin{enumerate}
\item[(1)] There exists $j\in[4]$ such that $\chi(G[V_i])=3$ for all $i\in
[4]-j$; and
\item[(2)] $G[V_j]$ has a component $R$, called the \textit{active} component,
that is $K_4$ and $\chi(G[V_j]-R)\le 3$; and
\item[(3)] for each $v\in V(R)$ and $i\in[4]-j$ with $d_{V_i}(v)=3$, the
graph $G[V_i+v]$ has a $K_4$ component; and
\item[(4)] for each $v\in V(R)$ and $i\in[4]-j$, if $v$ has 2 neighbors in the
same component $D$ of $G[V_i]$,
then $\chi(D+v)=4$.
\end{enumerate}
\end{defn}
\begin{lemma}
\label{Mozhan3PartitionsExist}
Every $\Delta$-critical graph $G$ with $\Delta=13$ has a Mozhan $(3,3,3,3)$-partition.
\end{lemma}
\begin{proof}
To form a Mozhan $(3,3,3,3)$-partition of $G$, choose $v$, a 12-coloring of $G-v$,
and a partition of the set of 12 color classes into 4 parts each of size 3
so as to minimize the total number of edges within parts; finally, add $v$ to
some part $V_j$ where it has 3 neighbors.
(Note that edges incident to $v$ don't count toward the total within parts.)
We show that this partition satisfies the four properties required by Defintion~1.
By construction, (1) holds. Let $R$ be the component of $G[V_j]$ containing
$v$. To prove (2), it suffices to show that $R$ is $K_4$. By Brooks'
Theorem, it is enough to show that $\Delta(R)\le 3$. Suppose instead that
there exists $u\in V(R)$ with $d_R(u)>3$; choose $u$ to minimize the distance
in $R$ from $u$ to $v$. Uncolor the vertices on a shortest path $P$ in $R$ from
$u$ to $v$; move $u$ to some $V_k$ where it has at most 3 neighbors (and make
$u$'s new club active). Color the vertices of $P$, starting at $v$ and working
along $P$; this is possible since each vertex of $P$ has at most 2 colored
neighbors in $R$ when we color it. The resulting new partition has fewer edges
within color classes, since we lost at least 4 edges incident to $u$ and gained
at most 3 incident to $v$.
This contradiction implies that $\Delta(R)\le 3$, so $R$ must be $K_4$ by
Brooks' Theorem. Thus (2) holds.
Now we prove (3). Choose a vertex $u\in V(R)$ and $i\in[4]-j$.
Delete $u$ from $V_j$ and add it to $V_i$ (making $u$'s new club active); this
maintains the total number of edges within parts, so this gives another Mozhan
3-partition. By the above proof of (2), $v$ lies in a component of $G[V_i]$
that is $K_{4}$. Thus, (3) holds. If (4) is false, then $v$ has at most 2
neighbors in each component of $G[V_i]$; 3-color $D+v$ so that $v$ avoids the
colors on its neighbors in $V_i\setminus D$. Now 3-color $G[X_j-v]$, giving a
12-coloring of $G$. This contradiction implies (4).
\end{proof}
Given a Mozhan $(3,3,3,3)$-partition,
we call the $V_i$ \emph{clubhouses}, we call each
component of $G[V_i]$ a \emph{club} meeting in that clubhouse, and we call
the vertices \emph{members}.
Roughly speaking, we choose a member $v$ of the active club and send it to some
clubhouse where it has 3 neighbors. This creates a new Mozhan $(3,3,3,3)$-partition, and
so $v$ must form a $K_4$ with its neighbors in the new clubhouse. We repeat
this ``sending out'' process many times. If a club $R$ sends two members to
another club $S$ (either in succession or with one other member sent out
between them), and after this $R$ again becomes active, then (4) implies that
$R$ must be complete to $S$. By repeating this process, we find lots of edges,
and eventually show that $G$ must contain a large clique.
\begin{named3}
If $G$ is a graph with $\chi\ge\Delta=13$, then $G$ contains $K_{10}$.
\label{Delta13}
\end{named3}
\begin{proof}
It suffices to consider when $G$ is $\Delta$-critical.
We begin with a Mozhan $(3,3,3,3)$-partition.
By (2) in the definiton of Mozhan partition, let $R$ be the active $K_{4}$. By
symmetry, assume that $R\in A$. We move some member $v\in R$ into some
other clubhouse $E\in\{A, B, C,D\}$ for which $d_{E}(v)=3$. We move each
vertex to a new club at most once. We never move a member $v\in R$ to a
club $S$ if $R$ is complete to $S$. Note that $d_{E}(v)\ge 3$ for every
clubhouse $E$, since otherwise we move $v$ to $E$ and get a 12-coloring of $G$.
Since $d_G(v)\le 13$, we have $d_{E}(v)>3$ for at most one clubhouse $E$;
thus, we can move each unmoved $v\in R$ to at least 2 of the 3 other clubhouses
unless $R$ is complete to a copy of $K_3$ in that clubhouse.
Subject to these constraints, if $R$ can send an unmoved member to a club where
it's already sent one member, then it does so; otherwise $R$ chooses
arbitrarily a (valid) clubhouse to send a member to. We repeat this process
until either we find a $K_{10}$ or the active $K_4$ has no valid club to which
to send a member. (Clearly, the process terminates, since each vertex is moved
at most once.) We show that when the process terminates, the active component
is contained in a $K_{10}$.
We prove the theorem via four claims.
\begin{claim}
Each pair of clubs $R$ and $S$ are either (i) always complete to each other
or (ii) never complete to each other, i.e., the set of pairs of clubs that are
complete to each other does not change.
\label{claim1}
\end{claim}
\noindent
{\bf Proof of Claim 1.}
By symmetry, assume that $R\in V_1$ and $S\in V_2$. Suppose that either $R$
and $S$ were complete to each other and have just become non-complete or they
are non-complete to each other and when the next vertex moves they will become
complete to each other.
By symmetry $R$ is active, we have $u\in R$ and $v\in S$ with $u\not\leftrightarrow v$, and
$R-u$ is complete to $S$. To get a good 12-coloring of $G$, move
some $w\in R$ to $S$, then move $v\in S$ to $R$. Now since $u\not\leftrightarrow v$, we
get a 12-coloring of $G$. This point is somewhat subtle, for the resulting
partition may fail to be a Mozhan partition. In particular, moving $w$ to $S$
or $v$ to $R$ may \emph{increase} the total number of edges within parts.
However $w$ has at most one neighbor in $V_2$ other than vertices originally in
$S$ and $u$ has at most one neighbor in $V_1$ other than vertices originally in
$R$; this allows the 12-coloring. This contradiction completes the claim.
\begin{claim}
If $G$ contains a $K_4$ in one clubhouse joined to disjoint copies of
$K_3$ in two other clubhouses, then the $K_3$'s are joined to each
other, i.e., $G$ contains $K_{10}$.
\label{claim2}
\end{claim}
\noindent
{\bf Proof of Claim 2.}
Let $R\in V_1$ be the $K_4$ and let $S\in V_2$ and $T\in V_3$ be copies of
$K_3$ joined to $R$. Let $V(R)=\{w_1,\ldots,w_4\}$. Suppose $u \in S$ and $v
\in T$ and $u\not\leftrightarrow v$. As in Claim~\ref{claim1}, the partition we now
construct may fail to be a Mozhan partition; however, we will conclude that $G$
has a 12-coloring. (Recall that $d_{V_2}(v)\ge 3$, for otherwise we can move
$w_1$ to $T$ and $v$ to $V_2$ and get a good 12-coloring.) Move $w_1$ to $S$
and $u$ back to $R$. Clearly $u$ has at most one neighbor in $V_1$ outside of
$R$. Also $d_{V_2}(v)\ge 4$, which implies that $d_{V_1}(v)\le 3$. Now move
$w_2$ to $T$ and $v$ back to $R$. Since $w_1$ has at most one neighbor in
$V_2$ outside of $S$ and $w_2$ has at most one neighbor in $V_3$ outside of
$T$, each of $G[V_2]$ and $G[V_3]$ has a 3-coloring. Furthermore
$G[\{u,v,w_3,w_4\}]=K_4-e$ with $u\not\leftrightarrow v$ and each of $u$ and
$v$ has at most one additional neighbor in $V_1$. Thus, we can 3-color
$G[V_1\setminus\{u,v,w_3,w_4\}]$, then find a common color for $u$ and $v$, then
finally color $w_3$ and $w_4$. This gives a 12-coloring of $G$.
This contradiction implies that $u \leftrightarrow v$, and by symmetry yields a $K_{10}$.
\begin{claim}
No club becomes active four times.
\label{claim3}
\end{claim}
\noindent
{\bf Proof of Claim~\ref{claim3}.}
Suppose the contrary.
We continue until the first time some club $R$ has sent two members to the same
club $S$ and $R$ is again active (this happens no later than the fourth time $R$
becomes active). By symmetry, assume that $R \in A$ has sent two members to $S \in
B$; either two in a row or else with one other member between them (these are
the only possibilities). We show that $R$ is complete to $S$. This yields
a contradiction, since $R$ never sends members to a club which is complete to $R$.
We have two cases: (i) $R$ sent two successive members to $S$ and (ii) $R$ sent
a member to $C$ or $D$ between the members it sent to $S$.
By possibly assuming that we started at the second time that $R$ became active
(rather than the first), in Case (i) we can assume that $R$ sent its first and
second members to $S$.
{\bf Case (i)} We consider 5 key steps in the process; these are each just
after $R$ becomes active or just after $R$ sends a member to $S$.
We write $(a_1,a_2,a_3,a_4; b_1,b_2,b_3)$ to denote the members of $R$ and
$S$ at step 1 (the first time $R$ is active).
For steps 2--5, write
$(a_2,a_3,a_4;a_1,b_1,b_2,b_3)$;
$(a_2,a_3,a_4,a_5;a_1,b_2',b_3')$; $(a_3,a_4,a_5;a_1,a_2,b_2',b_3')$;
and $(a_3,a_4,a_5,a_6;a_1,a_2,b_3'')$.
We show that $\{a_1,a_2,a_3,a_4,a_5,a_6,b_3''\}$ induces $K_7$. From
step 4, $a_1\leftrightarrow a_5$ (otherwise we move $a_1$ to $R$ and get a
good coloring).
From step 5, $a_6\leftrightarrow\{a_1,a_2\}$, since otherwise we move $a_3$ to
$S$, then move $a_1$ (or $a_2$) to $R$ and get a good coloring.
Similarly, from step 5, $\{a_3,a_4,a_5,a_6\}\leftrightarrow b_3''$; otherwise we
move, say, $a_6$ to $S$ and get a good coloring. Thus
$\{a_1,a_2,a_3,a_4,a_5,a_6,b_3''\}$ induces $K_7$, as desired.
However, now $R$ is complete to $S$, and by Claim~1, it always has been.
This contradicts our rule for sending vertices out from the active club.
Thus Case (i) can never happen.
{\bf Case (ii)} This case is similar to Case (i), but
now we consider 7 key steps. Even though at one point $R$ sends a member to
clubhouse $C$, we only consider the members of $R$ and $S$ (no club in $C$).
Now at steps 1-7 we have
$(a_1,a_2,a_3,a_4;b_1,b_2,b_3)$;
$(a_2,a_3,a_4;a_1,b_1,b_2,b_3)$;
$(a_2,a_3,a_4,a_5;a_1,b_2,b_3')$;
$(a_3,a_4,a_5;a_1,b_2,b_3')$;
$(a_3,a_4,a_5,a_6;a_1,b_2,b_3'')$;
$(a_4,a_5,a_6;a_1,a_3,b_2,b_3'')$;
and $(a_4,a_5,a_6,a_7;a_1,a_3,b_3''')$.
We show that $\{a_1,a_3,a_4,a_5,a_6,a_7,b_3'''\}$ induces $K_7$.
This implies that $R$ and $S$ are complete to each other, and yields a
contradiction as above. Recall that in each of the steps above, each of the
clubs is a $K_3$ or $K_4$, so we get a lot of edges that way.
We can assume that $b_2$ doesn't change until it gets replaced by $a_3$, since
each club has at most 3 members replaced (by the extremality of $R$). In fact,
at least two of $b_3, b_3', b_3''$ are the same (but we don't use this fact).
By step 3, $a_3\leftrightarrow
b_3'$, since $a_3\leftrightarrow \{a_1,b_2\}$ by step 6. Also, $a_1\leftrightarrow a_5$ by step
3; otherwise we send $a_3$ to $S$, then send $a_1$ to $R$ and get a good
coloring.
By step 6, we have $a_1\leftrightarrow a_6$; otherwise, we send $a_1$ to $R$.
Also by step 5, we have $a_3\leftrightarrow \{a_4,a_5,a_6\}$.
Now by step 7, we have $\{a_1,a_3\}\leftrightarrow a_7$; otherwise, send $a_4$ to $S$
and (say) $a_3$ to $R$. Now by step 7, we have $\{a_4,a_5,a_6,a_7\}\leftrightarrow
b_3'''$;
otherwise, send (say) $a_4$ to $S$ and get a good coloring.
This implies that $\{a_1, a_3, a_4, a_5, a_6, a_7, b_3'''\}$ is a clique.
Now $R$ and $S$ are complete to each other, which implies that Case (ii)
never happens. Thus, before any club becomes active four times, the process
must terminate. This proves Claim~3.
\begin{claim}
\label{claim4}
$G$ contains $K_{10}$.
\end{claim}
\noindent
{\bf Proof of Claim~\ref{claim4}.}
Let $R$ be the active $K_4$.
The process continues if there exists an unmoved vertex $u\in R$ and a vertex
$v\in S$, where $S$ is a club in some clubhouse $E$, with $u\leftrightarrow v$ such that
$d_{E}(u)\le 3$ and $R$ is not complete to $S$. For each $u\in R$, we have
$d_{E}(u)>3$ for at most one clubhouse $E$. So if some unmoved vertex $u\in R$
has no clubhouse available to move to, then at least two of the other three
clubhouses are forbidden for $u$ because $R$ is complete to some $K_3$ within them.
(Furthermore, $u$ must be high. We won't use this fact now, but will need it in a
later proof.)
By Claim~\ref{claim3}, no clubhouse becomes active four times, so each club $E$
will contain an unmoved vertex.
Thus, the active $K_4$ is joined to $K_3$'s in two other clubhouses.
By Claim~\ref{claim2}, $G$ contains $K_{10}$. This completes the proof.
\end{proof}
\begin{remark}
The proof of Lemma~\ref{Delta13} gives much more. First, it works equally
well with any number $k$ of clubhouses. So for any $G$ with $\Delta=3k+1$,
if $G$ has no $(\Delta-1)$-coloring, then $G$ contains a clique of size
$\Delta-3$. In fact, the proof works for any number of clubhouses, when each
consists of at least 3 color classes. However, now we get the weaker conclusion
that $G$ contains a clique of size at least $\Delta-t$, where $t$ is the
largest number of color classes in a single clubhouse. Using at most two
clubhouses of size 4, we see that every graph $G$ with no
$(\Delta-1)$-coloring contains a clique of size $\Delta-4$.
Furthermore, if the process terminates, then the active club (and all the clubs
joined to it) must contain no unmoved low vertex. These observations
yield the following two results.
\end{remark}
\begin{cor}
\label{weak1}
If $G$ is a graph with $\chi\ge\Delta$, then $\omega\ge \Delta-4$.
\end{cor}
\begin{cor}
\label{weak2}
Let $G$ be a graph and let $\mathcal{H}(G)$ denote the subgraph of
$G$ induced by $\Delta$-vertices.
If $\chi\ge \Delta$, then $\omega\ge \Delta$ or $\omega(\mathcal{H}(G))\ge \Delta-7$.
\end{cor}
In the remainder of the paper, we improve the lower bound in
Corollary~\ref{weak1} by 1 (when $\Delta\ge 13$) and we improve the lower
bound in Corollary~\ref{weak2} by 2.
\section{The First Main Theorem}
\label{sectionMain1}
A \emph{hitting set} is an independent set that intersects every maximum
clique. If $I$ is a hitting set and also a maximal independent set, then
$\Delta(G-I)\le \Delta(G)-1$ and $\chi(G-I)\ge \chi(G)-1$. (In our
applications, we can typically assume that $\Delta(G-I)=\Delta(G)-1$, since
otherwise we get a good coloring or a big clique from Brooks' Theorem. We
give more details in the proof of Theorem~\ref{main1}.) So if
$G-I$ has a clique of size $\Delta(G-I)-t$, for some constant $t$, then
also $G$ has a clique of size $\Delta(G)-t$.
We repeatedly remove hitting sets to reduce a graph with $\Delta\ge 13$
to one with $\Delta=13$. Since we proved that every graph with
$\chi\ge\Delta=13$ contains $K_{10}$, this proves that every $G$ with
$\chi\ge\Delta\ge13$ contains $K_{\Delta-3}$.
This idea is not new. Kostochka~\cite{Kostochka} proved
that every graph with $\omega\ge\Delta-\sqrt{\Delta}+\frac32$ has a hitting set.
Rabern~\cite{Rabern} extended this result to the case $\omega\ge\frac34(\Delta+1)$,
and King~\cite{King} strengthened his argument to prove
that $G$ has a hitting set if $\omega>\frac23(\Delta+1)$. This condition is
optimal, as illustrated by the lexicographic product of an odd cycle and a clique.
Finally, King's argument was refined by Christofides, Edwards, and
King~\cite{CEK} to show that these lexicographic products of odd cycles and
cliques are the only tightness examples; that is, $G$ has a hitting set if
$\omega\ge\frac23(\Delta+1)$ and $G$ is not the lexicographic product of an odd
cycle and a clique. Hitting set reductions have application to other vertex
coloring problems.
Using this idea (and others), King and Reed~\cite{KR} gave a
short proof that there exists $\epsilon > 0$ such that
$\chi\le\ceil{(1-\epsilon)(\Delta+1)+\epsilon\omega}$.
To keep this paper largely self-contained, we prove our own hitting set lemma.
In the present context, it suffices to find a hitting set when $G$ is a minimal
counterexample to the Borodin-Kostochka conjecture with $\Delta\ge14$.
Such a $G$ is $\Delta$-critical, which facilitates a shorter proof.
In~\cite{CR1}, we proved a number of results about so called $d_1$-choosable
graphs (defined below), which are certain graphs that cannot appear as induced
subgraphs in a $\Delta$-critical graph. We leverage these $d_1$-choosability
results to prove our hitting set lemma, then use the hitting set lemma to
reduce to the case $\Delta=13$, which we proved in Lemma~\ref{Delta13}. Since
the proofs of the $d_1$-choosability results in~\cite{CR1} are lengthy, we give
a short proof of the special case that we need here.
A \textit{list assignment} $L$ is an assignment $L(v)$ of a set of allowable
colors to each vertex $v\in V(G)$. An $L$-coloring is a proper coloring such
that each vertex $v$ is colored from $L(v)$. A graph $G$ is
\textit{$d_1$-choosable} if $G$ has an $L$-coloring for every list assignment
$L$ such that $|L(v)|\ge d(v)-1$. No $\Delta$-critical graph contains an induced
$d_1$-choosable subgraph $H$ (by criticality, color $G-H$, then extend the
coloring to $H$, since it is $d_1$-choosable). For a list assignment $L$, let
$Pot(L)=\cup_{v\in V(G)}L(v)$. The following lemma is central in proving each
of our $d_1$-choosability results.
\begin{lemma}[Small Pot Lemma, \cite{Kierstead,RS}]
\label{SPL}
For a list size function $f:V(G) \rightarrow \{0,\ldots,|G|-1\}$, a graph $G$ is
$f$-choosable iff $G$ is $L$-colorable for each list assignment $L$ such that
$|L(v)|=f(v)$ for all $v\in V(G)$ and $|\cup_{v\in V(G)}L(v)| < |G|$.
\end{lemma}
\begin{proof}
Fix $G$ and $f$. For any $U \subset V$ and any list assignment $L$, let $L(U)$
denote $\cup_{v\in U}L(v)$. Let $L$ be an $f$-assignment such that $|L(V)| \ge
|G|$ and $G$ is not $L$-colorable. For each $U \subseteq V$, let $g(U) = |U| -
|L(U)|$. Since $G$ is not $L$-colorable, Hall's Theorem implies there exists
$U$ with $g(U)>0$. Choose $U$ to maximize $g(U)$. Let $A$ be an arbitrary set
of $|G|-1$ colors containing $L(U)$. Construct $L'$ as follows. For $v\in U$,
let $L'(v)=L(v)$. Otherwise, let $L'(v)$ be an arbitrary subset of $A$ of size
$f(v)$. Now $|L'(V)| < |G|$, so by hypothesis, $G$ has an $L'$-coloring. This
gives an $L$-coloring of $U$. By the maximality of $g(U)$, for $W \subseteq
(V\setminus U)$, we have $|L(W)\setminus L(U)| \ge |W|$. Thus, by Hall's
Theorem, we can extend the $L$-coloring of $U$ to all of $V$.
\end{proof}
\begin{lemma}[\cite{CR1}
\label{K_tClassification}
For $t \geq 4$, $\join{K_t}{B}$ is not $d_1$-choosable iff $\omega(B) \geq
\card{B} - 1$; or $t = 4$ and $B$ is $E_3$ or $K_{1,3}$; or $t = 5$ and $B$ is $E_3$.
\end{lemma}
\begin{proof
If $\omega(B)\ge |B|-1$, then assign each $v\in V(\join{K_t}{B})$ a subset of
$\{1,\ldots,t+|B|-2\}$; since $\omega(\join{K_t}{B})\ge t+|B|-1$, clearly $G$ is not colorable
from this list assignment. This proves one direction of the lemma; now we
consider the other.
Suppose the lemma is false, and let $G$ be a counterexample.
If $\omega(B)\le|B|-2$, then $B$ contains either
(i) an independent set $S=\{x_1,x_2,x_3\}$ or
(ii) a set $S=\{x_1,x_2,x_3,x_4\}$ with $x_1x_2,x_3x_4\notin E(B)$.
If $B$ contains only (i), then $S = E_3$ and $t \ge 6$
(by moving any dominating vertices from $B$ to $K_t$).
Let $T=V(K_t)$ and denote $T$ by $\{y_1,\ldots,y_t\}$.
In Cases (i) and (ii) we assume that $t=6$ and $t=4$, respectively.
Color $G\setminus (S\cup T)$ from its lists (this is possible since each vertex
has at least two neighbors in $S\cup T$).
Let $R=G[S\cup T]$. For each $v\in V(R)$, let $L(v)$ denote the list of colors
now available for $v$. Note that $\card{L(v)}\ge d_R(v)-1$; specifically,
$|L(x_i)|\ge d_S(x_i)+t-1$ and $|L(y_j)|\ge |S|+t-2$ for all $x_i\in S$ and
$y_j\in T$. We assume these inequalities hold with equality. When we have $i,
j, k$ with $x_i\not\leftrightarrow x_j$ and $|L(x_i)|+|L(x_j)|>|L(y_k)|$, we often use the
following technique, called \emph{saving a color} on $y_k$ via $x_i$ and $x_j$.
If there exists $c\in L(x_i)\cap L(x_j)$, then use $c$ on $x_i$ and $x_j$.
Otherwise, color just one of $x_i$ and $x_j$ with some $c\in (L(x_i)\cup
L(x_j))\setminus L(y_k)$. For a set $U$, let $L(U)=\cup_{v\in U}L(v)$.
Case (i)
By the Small Pot Lemma, assume that $|L(G)|\le 8$. This implies $|L(x_i)\cap
L(x_j)|\ge 2$ for all $i,j\in[3]$. If there exist $x_i$ and $y_k$
with $L(x_i)\not\subseteq L(y_k)$, then color $x_i$ to save a color on
$y_k$. Color the remaining $x$'s with a common color; this saves an
additional color on each $y$. Now finish greedily, ending with $y_k$.
Thus, we have $L(x_i)\subset L(y_k)$ for all $i\in [3]$ and $k\in [6]$. This
gives $\card{\cup_{i=1}^3L(x_i)}\le 7$. Since
$\sum_{i=1}^3|L(x_i)|=15>2|\cup_{k=1}^3L(x_k)|$, we have a color $c\in
\cap_{i=1}^3L(x_i)$. Use $c$ on every $x_i$ and finish greedily.
Case (ii)
By the Small Pot Lemma, assume that $|L(G)|\le 7$. If $S$ induces at least two edges, then
$|L(x_1)|+|L(x_2)|\ge 8$ and $|L(x_3)|+|L(x_4)|\ge 8$. So $L(x_1)\cap L(x_2)\ne
\emptyset$.
Color $x_1$ and $x_2$ with a
common color $c$. If $|L(y_1)-c|\le 5$, then save a color on $y_1$ via $x_3$
and $x_4$. Now finish greedily, ending with $y_1$.
Suppose $S$ induces exactly one edge $x_1x_3$. Similar to the previous
argument, $L(x_1)\cap L(x_2)=\emptyset$; otherwise, use a common color on $x_1$
and $x_2$, possibly save on $y_1$ via $x_3$ and $x_4$, then
finish greedily. By symmetry, $L(x_1)=L(x_3)=\{a,b,c,d\}$ and
$L(x_2)=L(x_4)=\{e,f,g\}$. Also by symmetry, $a$ or $e$ is missing from
$L(y_1)$. So color $x_1$ with $a$ and $x_2$ and $x_4$ with $e$ and $x_3$
arbitrarily; now finish greedily, ending with $y_1$. So instead $G[S]=E_4$. If a common
color appears on 3 vertices of $S$, use it there, then finish greedily.
If not, then by pigeonhole, at least 5 colors appear on pairs of vertices.
Color two disjoint pairs, each with a common color. Now finish the coloring
greedily.
\end{proof}
The following lemma of King enables us to find an independent transversal.
\begin{lemma}[Lopsided Transversal Lemma~\cite{King}]\label{Lopsided}
Let $H$ be a graph and $V_1 \cup \cdots \cup V_r$ a partition of $V(H)$.
If there exists $s \geq 1$ such that for each $i \in \irange{r}$ and each
$v \in V_i$ we have $d(v) \leq \min\set{s, \card{V_i}-s}$, then $H$ has an
independent transversal $I$ of $V_1, \ldots, V_r$.
\end{lemma}
Now we have all the tools to prove our hitting set lemma.
\begin{lemma}
\label{ourHittingLemma}
Every $\Delta$-critical graph with $\chi \geq \Delta \geq 14$ and
$\omega=\Delta-4$ has a hitting set.
\end{lemma}
\begin{proof}
Suppose the lemma is false, and let $G$ be a counterexample minimizing $\card{G}$.
Consider distinct
intersecting maximum cliques $A$ and $B$. Since a vertex in their intersection
has degree at most $\Delta$, we have $\card{A \cap B} \geq \card{A} + \card{B}
- (\Delta + 1) = \Delta - 9 \geq 5$. Since $G$ contains no $d_1$-choosable
subgraph, letting $A\cap B=K_t$ in Lemma
\ref{K_tClassification} implies that $\omega(G[A\cup B])\ge|A\cup B|-1$.
Hence $\card{A \cap B} = \Delta - 5$.
Suppose $C$ is another maximum clique intersecting $A$ and let $U = A \cup B
\cup C$. Now for $J = A\cap B \cap C$, we have $\card{J} = \Delta - 5 + \Delta
- 4 - \card{U} = 2\Delta - 9 - \card{U}$. Since $J \neq \emptyset$, we have $\card{U}
\leq \Delta + 1$, giving $\card{J} \geq 4$. If $\omega(G[U]) \geq \card{U} -
1$, then $C=A$ or $C=B$, a contradiction. Hence, by Lemma
\ref{K_tClassification} we must have $\card{J} < 6$ and $\card{U} = \card{J} +
3$. But now $6 > \card{J} = 2\Delta - 9 - \card{U} > 28 - 18$, a
contradiction. Thus, every maximum clique intersects at most one other maximum
clique. Hence we can partition the union of the maximum cliques into sets
$D_1, \ldots, D_r$ such that either $D_i$ is a $(\Delta-4)$-clique $C_i$ or
$D_i = C_i \cup \set{x_i}$ for a $(\Delta-4)$-clique $C_i$, where $x_i$ is
adjacent to all but one vertex of $C_i$.
For each $D_i$, if $D_i=C_i$, then let $K_i=C_i$. If $D_i=C_i\cup\{x_i\}$, then
let $K_i=C_i\cap N(x_i)$.
Consider the subgraph $F$ of $G$ formed by taking the subgraph induced on the
union of the $K_i$ and then making each $K_i$ independent. We
apply Lemma \ref{Lopsided} to $F$ with $s = \frac{\Delta}{2} - 2$. We have two
cases to check, when $K_i = C_i$ and when $K_i = C_i \cap N(x_i)$. In the
former case, $\card{K_i} = \Delta-4$ and for each $v \in K_i$ we have $d_F(v)
\leq \Delta(G) + 1 - (\Delta-4) = 5$. Hence $d_F(v) \leq \frac{\Delta}{2} - 2
= \min\set{\frac{\Delta}{2} - 2, \Delta - 4 - (\frac{\Delta}{2} - 2)}$ since
$\Delta \geq 14$. In the latter case, we have $\card{K_i} = \Delta - 5$ and
since every $v \in K_i$ is adjacent to $x_i$ and to the vertex in $C_i\setminus
K_i$, neither of which is in $F$, we have $d_F(v) \leq \Delta - (\Delta-4) =
4$. This gives $d_F(v) \leq \frac{\Delta}{2} - 3 = \min\set{\frac{\Delta}{2}
- 2, \Delta-5 - (\frac{\Delta}{2} - 2)}$ since $\Delta \geq 14$. Now Lemma
\ref{Lopsided} gives an independent transversal $I$ of the $K_i$, which is a
hitting set.
\end{proof}
Now we can prove the first of our two main results. For convenience, we restate it.
\begin{named1}
Every graph with $\chi \geq \Delta \geq 13$ contains $K_{\Delta-3}$.
\end{named1}
\begin{proof}
Let $G$ be a counterexample minimizing $|G|$; note that $G$ is vertex critical.
By Theorem~\ref{easy}, we have $\Delta \ge 14$. If $\omega<\Delta-4$, let $I$ be any maximal independent set;
otherwise let $I$ be a hitting set given by Lemma~\ref{ourHittingLemma} expanded to a maximal independent set.
Now $\omega(G-I)<\Delta(G) - 4$, $\Delta(G-I)\le \Delta(G)-1$ and $\chi(G-I) = \chi(G)-1$. In fact, we have $\Delta(G-I) = \Delta(G) - 1$ for
otherwise applying Brooks' theorem to $G-I$ yields $\omega(G) \geq \chi(G-I) \geq \Delta(G) - 1$, a contradiction.
Hence $\chi(G-I) \geq \Delta(G-I) \geq 13$ and $\omega(G-I)< \Delta(G-I) - 3$ contradicting minimality of $|G|$.
\end{proof}
As we note (and generalize) in the following section, the proof of
Theorem~\ref{easy} works equally well when $\Delta=10$. In that case, $G$
contains a $K_7$, which extends Theorem~\ref{main1} to the case $\Delta=10$. We
suspect that this theorem holds also for $\Delta\in\{11,12\}$.
\section{The Second Main Theorem}
In this section, we prove our second main theorem. First, we prove two
lemmas that follow from~\cite{CR1} about list coloring (which we use to
forbid certain subgraphs in a $\Delta$-critical graph). We also extend
the definition of Mozhan partitions to a broader setting.
\begin{lemma}[\cite{CR1}]\label{mixedLemmaK4}
Let $G=\join{K_4}{E_2}$. If $L$ is a list assignment with
$\card{L(v)} \geq d(v) - 1$ for all $v \in V(G)$ and
$\card{L(w)} \geq d(w)$ for some $w\in V(K_4)$, then $G$ has an $L$-coloring.
\end{lemma}
\begin{proof}
Denote $V(E_2)$ by $\set{x, y}$.
By the Small Pot Lemma, $\card{Pot(L)} \leq 5<6\le \card{L(x)} + \card{L(y)}$.
After coloring $x$ and $y$ the same, finish greedily, ending with $w$.
\end{proof}
\begin{lemma}[\cite{CR1}]\label{mixedLemmaK3}
Let $G=\join{K_3}{E_2}$. If $L$ is a list assignment such that $\card{L(v)}
\geq d(v) - 1$ for all $v \in V(G)$ and for some $w\in V(K_3)$ and
some $x\in V(E_2)$ we have $\card{L(w)} \geq d(w)$ and $\card{L(x)}\ge d(x)$,
then $G$ has an $L$-coloring.
\end{lemma}
\begin{proof}
Denote $V(E_2)$ by $\set{x, y}$.
By the Small Pot Lemma, $\card{Pot(L)} \leq 4<5\le \card{L(x)} + \card{L(y)}$.
After coloring $x$ and $y$ the same, finish greedily, ending with $w$.
\end{proof}
Now we extend the definition of Mozhan partitions to a more general context.
Defintion~1 is the special case of the following definition when
$\vec{r}=(3,3,3,3)$.
\begin{defn}
Let $G$ be a graph. For $\vec{r}$ a vector of length $k$ of positive integers
a \emph{Mozhan
$\vec{r}$-partition} of $G$ is a partition $\parens{V_1, \ldots, V_k}$
of $G$ such that:
\begin{enumerate}
\item There is $j \in \irange{k}$ such that $\chi(G[V_i]) = r_i$ for all $i \in \irange{k} - j$; and
\item $G[V_j]$ has a $K_{r_j+1}$ component $R$, called the \emph{active}
component, such that $\chi(G[V_j] - R) \leq r_j$; and
\item for each $v \in V(R)$ and $i \in \irange{k}-j$ with $d_{V_i}(v) = r_i$,
the graph $G[V_i + v]$ has a $K_{r_i+1}$ component; and
\item if $v \in V(R)$ has at least $d_{V_i}(v) + 1 - r_i$ neighbors in a
component $K$ of $G[V_i]$ for some $i \in \irange{k}-j$, then $K+v$ is $(r_i+1)$-chromatic.
\end{enumerate}
\end{defn}
\begin{lemma}
\label{MozhanPartitionsExist}
Let $k \geq 2$ and $G$ be a $(t+1)$-vertex-critical graph with $\Delta <
t + k$. If $\vec{r}$ is a length $k$ vector of positive integers
with $\sum_{i \in \irange{k}} r_i = t$, then $G$ has a Mozhan $\vec{r}$-partition.
\end{lemma}
\noindent
The proof is the same as for Lemma~\ref{Mozhan3PartitionsExist}, so we omit
it. More details are in~\cite{Rabern2} and~\cite{KRS}.
Now we prove our second main result.
\begin{named2}
Let $G$ be a graph and let $\mathcal{H}(G)$ denote the subgraph of $G$ induced by
$\Delta$-vertices. If $\chi\ge \Delta$, then $\omega\ge \Delta$ or
$\omega(\mathcal{H}(G))\ge \Delta-5$.
\end{named2}
\begin{proof}
The proof is similar to what we outlined to prove Corollary~\ref{weak1}, but now
we refine the process and analyze it more carefully to get a stronger result.
Essentially we run the process we used to prove Theorem~\ref{easy}, and show
that it works equally well in the more general context of a Mozhan
$\vec{r}$-partition.
For $\Delta\le 6$, the Theorem is trivial. For $\Delta \ge 7$, we write
$\Delta-1$ as a sum of 3's and 4's. Let $\vec{r}$ be a vector consisting of
these 3's and 4's (in arbitrary order); by Lemma~\ref{MozhanPartitionsExist},
we get a Mozhan $\vec{r}$-partition.
Running the same process as before implies that $G$ contains
$K_{\Delta-4}$. If $\omega(\mathcal{H}(G))\ge \Delta-5$, then we are done; otherwise, this
$K_{\Delta-4}$ contains at least two low vertices. If any of them has not yet
moved, then we move some vertex of the active club to the club containing this low
vertex and send the low vertex out. This continues the process.
So the process can terminate only when the active club is active for a second
or third time, it lies in a $K_{\Delta-4}$, and all low vertices in this
$K_{\Delta-4}$ have already moved. We now slightly refine the process and
show that we can avoid this problem.
The proof is akin to that of Theorem~\ref{easy}. Because the proofs of Claims 1,
2, and 3 here are the same as in that theorem, we do not reprove them, but only
state them in their more general form. The main difference here is the proof of
Claim~4.
Consider a club that is a clique and all of the clubs to which it is joined that
are also cliques. By Claim~2 these clubs are also joined to each other. We
call such a maximal group of clubs a \textit{clubgroup}. In effect we treat the
vertices of a clubgroup like one big club. If any club in the clubgroup is
active and another club in the clubgroup has an unmoved vertex $v$ with
$d_{V_i}(v)=r_i$ for some club $V_i$ not in the clubgroup, then we can make
$v$'s club active and send $v$ out to club $V_i$. We call such a vertex $v$
\textit{available}.
If a clubgroup spans all clubhouses, then $\omega\ge \Delta$, so we are done.
If a clubgroup spans all but one clubhouse, then $\omega\ge \Delta-4$. We
call such a clubgroup \textit{big}; otherwise a clubgroup is \textit{small}.
Each big clubgroup becomes active at most twice.
Suppose otherwise.
Each of the first two times it becomes active, it sends out a member to
the same club. Now if it becomes active a third time, it must be joined
to that club. By Claim~2, $G$ contains a $K_{\Delta}$.
So to prove the theorem, it suffices to show that if a big clubgroup becomes
active a second time, and induces no clique on $\Delta-5$ high vertices, then
it has an unmoved low vertex that it can send out.
We refine the process as follows. When a club becomes active and is choosing
among available members to send out, it gives preference to high vertices over
low vertices. This refinement allows us to prove the theorem.
\begin{claimNew}
\label{claim1New}
Each pair of clubs $R$ and $S$ are either (i) always complete to each other
or (ii) never complete to each other, i.e., the set of pairs of clubs that are
complete to each other does not change.
\end{claimNew}
\begin{claimNew}
If $G$ contains $K_{r_i+1}$ in the active clubhouse $V_i$ joined to
$K_{r_j}$ and $K_{r_k}$ in clubhouses $V_j$ and $V_k$, then the
cliques in $V_j$ and $V_k$ are also joined to each other.
\label{claim2New}
\end{claimNew}
\begin{claimNew}
No club becomes active four times.
\label{claim3New}
\end{claimNew}
\begin{claimNew}
$\omega\ge \Delta$ or $\omega(\mathcal{H}(G))\ge \Delta-5$.
\label{claim4New}
\end{claimNew}
\noindent
{\bf Proof of Claim~\ref{claim4New}.}
The process must terminate by the finiteness of the graph.
By Claim~\ref{claim3New}, it must terminate at a big clubgroup, since each small
clubgroup that has become active at most three times will have an available vertex
to send out.
When a big clubgroup becomes active for the first time, it induces a clique $K$ on
at least $\Delta-4$ vertices. If $\omega(\mathcal{H}(G))\le\Delta-6$, then $K$ contains two
low vertices; at most one has been moved, so $K$ has an available low vertex.
So suppose the process terminates at a big clubgroup the second time it
becomes active. Let $A$ denote the big clubgroup and let $v$ denote
the first vertex moved into $A$. If $v$ is high, then $A$ contained two unmoved
low vertices (since $\omega(\mathcal{H}(G))\le \Delta-6$), so when $A$ became active a second
time, it still had an available (low) vertex to send out; hence the process did
not terminate. So $v$ must be low. Let $B$ denote the clubgroup
which sent $v$ to $A$. We conclude that $B$ had at most one high
vertex. If $B$ had an unmoved high vertex, then $B$ would have sent it
out (rather than $v$); if $B$ had more than one moved vertex, then $B$ was
previously active, so it would have sent $v$ to the same club that it first
sent a member to.
Let $u$ denote the first member sent out from $A$ and let $C$ denote the club to
which $u$ was sent. Finally, let $w$ denote the second member sent into $A$.
If $w$ is high, then again $A$ has an available low vertex; hence $w$ is
low. Similarly, if $u$ is high, then $A$ now still has an available low
vertex; hence $u$ is low. Since $v$ is low, $v$ is adjacent to all of $C$;
otherwise we could move $v$ to $C$ and get a $(\Delta-1)$-coloring. Now since
$v$ has only $\Delta-1$ neighbors, $C=B$.
Since $B$ is in the only clubhouse not spanned by $A$, we get that $|A\cup
B|=\Delta$. Also, $u, v, w$ are all low and all adjacent to each vertex in
$A\cup B\setminus\{u,v,w\}$. Now by Lemma~\ref{mixedLemmaK3},
each low vertex in $B$ is joined to $A$. Recall that $B$ contains at most one
high vertex, and let $x$ be some low vertex in $B$. Now since $\{u,v,w,x\}$ are
all adjacent to all of $A$, Lemma~\ref{mixedLemmaK4} implies that the final
vertex of $B$ is also adjacent to all of $A$. Thus $A\cup B$ induces $K_{\Delta}$,
which completes the proof.
\end{proof}
If $G$ is a graph with $\Delta=3k+1$ for some $k\ge 2$ and $\chi\ge \Delta$,
then the above proof needs only clubhouses of size 3. Now an active big
clubgroup induces a clique $K$ of size $\Delta-3$ (rather than only size at
least $\Delta-4$, as above). So if $\omega(\mathcal{H}(G))\le \Delta-5$, then $K$ contains
at least two low vertices and all of the arguments above still apply. This
yields the following corollary.
\begin{cor}
\label{cor3}
Let $G$ be a graph with $\Delta=3k+1$ for some $k\ge 3$. If $\chi\ge\Delta$,
then $\omega\ge \Delta$ or $\omega(\mathcal{H}(G))\ge \Delta-4$.
\end{cor}
We conjecture that the previous theorem actually holds with $\omega(\mathcal{H}(G)) \geq \Delta
- 5$ replaced by $\omega(\mathcal{H}(G)) \geq \Delta - 4$. The case $\Delta = 3k + 1$ (and
$k\ge 3$) is Corollary~\ref{cor3}.
In \cite{Rabern2}, the second author proved this result for $\Delta = 6$;
later in \cite{KRS} it was proved for $\Delta=7$. The condition
$\omega(\mathcal{H}(G)) \geq \Delta - 4$ would be tight since the graph $O_5$ in Figure
\ref{fig:O5} is a counterexample to $\omega(\mathcal{H}(G)) \geq \Delta - 3$ when
$\Delta=5$. In fact, it was shown in \cite{KRS} that $O_5$ is the only
counterexample to $\omega(\mathcal{H}(G)) \geq \Delta - 3$ when $\Delta=5$.
\begin{conj}
Let $G$ be a graph.
If $\chi\ge\Delta$, then $\omega\ge\Delta$ or
$\omega(\mathcal{H}(G))\ge \Delta-4$.
\end{conj}
\begin{figure}[hbt]
\centering
\begin{tikzpicture}[scale = 10]
\tikzstyle{VertexStyle}=[shape = circle, minimum size = 1pt, inner sep = 1.2pt, draw]
\Vertex[x = 0.270266681909561, y = 0.890800006687641, L = \tiny {L}]{v0}
\Vertex[x = 0.336266696453094, y = 0.962799992412329, L = \tiny {L}]{v1}
\Vertex[x = 0.334666579961777, y = 0.821199983358383, L = \tiny {L}]{v2}
\Vertex[x = 0.56306654214859, y = 0.890800006687641, L = \tiny {H}]{v3}
\Vertex[x = 0.244666695594788, y = 0.731600046157837, L = \tiny {L}]{v4}
\Vertex[x = 0.417866677045822, y = 0.732000052928925, L = \tiny {L}]{v5}
\Vertex[x = 0.243866696953773, y = 0.543200016021729, L = \tiny {L}]{v6}
\Vertex[x = 0.415866762399673, y = 0.542800068855286, L = \tiny {L}]{v7}
\Vertex[x = 0.0926666706800461, y = 0.890000000596046, L = \tiny {H}]{v8}
\tikzstyle{EdgeStyle}=[]
\Edge[](v1)(v0)
\tikzstyle{EdgeStyle}=[]
\Edge[](v2)(v0)
\tikzstyle{EdgeStyle}=[]
\Edge[](v3)(v0)
\tikzstyle{EdgeStyle}=[]
\Edge[](v2)(v1)
\tikzstyle{EdgeStyle}=[]
\Edge[](v3)(v1)
\tikzstyle{EdgeStyle}=[]
\Edge[](v2)(v3)
\tikzstyle{EdgeStyle}=[]
\Edge[](v5)(v4)
\tikzstyle{EdgeStyle}=[]
\Edge[](v6)(v4)
\tikzstyle{EdgeStyle}=[]
\Edge[](v6)(v5)
\tikzstyle{EdgeStyle}=[]
\Edge[](v6)(v7)
\tikzstyle{EdgeStyle}=[]
\Edge[](v7)(v4)
\tikzstyle{EdgeStyle}=[]
\Edge[](v7)(v5)
\tikzstyle{EdgeStyle}=[]
\Edge[](v4)(v8)
\tikzstyle{EdgeStyle}=[]
\Edge[](v6)(v8)
\tikzstyle{EdgeStyle}=[]
\Edge[](v5)(v3)
\tikzstyle{EdgeStyle}=[]
\Edge[](v7)(v3)
\tikzstyle{EdgeStyle}=[]
\Edge[](v0)(v8)
\tikzstyle{EdgeStyle}=[]
\Edge[](v1)(v8)
\tikzstyle{EdgeStyle}=[]
\Edge[](v2)(v8)
\end{tikzpicture}
\caption{The graph $O_5$ is a $\Delta$-critical graph with $\Delta=5$ and $\omega(\mathcal{H}(G))=1$.}
\label{fig:O5}
\end{figure}
|
1,108,101,564,584 | arxiv |
\section{Transfer functions in
{\fontfamily{pcr}\fontshape{tt}\selectfont KBHtablesNN.fits}}
\label{appendix3a}
The transfer functions are stored in the file {\tt KBHtablesNN.fits} as binary
extensions and parametrized by the value of the observer inclination angle
$\theta_{\rm o}$ and the horizon of the black hole $r_{\rm h}$. We found
parametrization by $r_{\rm h}$ more convenient than using the rotational
parameter $a$, although the latter choice may be more common. Each
extension provides values of a particular transfer function for different
radii, which are given in terms of $r-r_{\rm h}$, and for the Kerr
ingoing axial coordinates $\varphi_{\rm K}$. Values of the horizon
$r_{\rm h}$, inclination $\theta_{\rm o}$, radius $r-r_{\rm h}$ and
angle $\varphi_{\rm K}$, at which the functions are evaluated, are defined as
vectors at the beginning of the FITS file.
The definition of the file {\tt KBHtablesNN.fits}:%
\begin{enumerate} \itemsep -0.4em
\vspace*{-1.5em}
\item[0.] All of the extensions defined below are binary.
\item The first extension contains six integers defining which of the
functions is present in the tables. The integers correspond to the delay,
$g$-factor, cosine of the local emission angle, lensing, change of the
polarization angle
and azimuthal emission angle, respectively. Value $0$
means that the function is not present in the tables, value $1$ means it is.
\item The second extension contains a vector of the horizon values in $GM/c^2$
($1.00 \le r_{\rm h} \le 2.00$).
\item The third extension contains a vector of the values of the observer's
inclination angle $\theta_{\rm o}$ in degrees
($0^\circ \le \theta_{\rm o} \le 90^\circ$, $0^\circ$ -- axis, $90^\circ$
-- equatorial plane).
\item The fourth extension contains a vector of the values of the radius
relative to the horizon $r-r_{\rm h}$ in $GM/c^2$.
\item The fifth extension contains a vector of the values of the azimuthal
angle $\varphi_{\rm K}$ in radians ($0 \le \varphi_{\rm K} \le 2\pi$).
Note that $\varphi_{\rm K}$ is a Kerr ingoing axial coordinate, not the
Boyer-Lindquist one!
\item All the previous vectors have to have values sorted in an increasing
order.
\item In the following extensions the transfer functions are defined, each
extension is for a particular value of $r_{\rm h}$ and $\theta_{\rm o}$.
The values of $r_{\rm h}$ and $\theta_{\rm o}$ are changing with each
extension in the following order:\\ \hspace*{-2.6em}
\parbox{\textwidth}{
{\begin{centering}
$r_{\rm h}[1] \times \theta_{\rm o}[1]$,\\
$r_{\rm h}[1] \times \theta_{\rm o}[2]$,\\
$r_{\rm h}[1] \times \theta_{\rm o}[3]$,\\
\dots \\
\dots \\
$r_{\rm h}[2] \times \theta_{\rm o}[1]$,\\
$r_{\rm h}[2] \times \theta_{\rm o}[2]$,\\
$r_{\rm h}[2] \times \theta_{\rm o}[3]$,\\
\dots \\
\dots \\
\end{centering}}}\\
\item Each of these extensions has the same number of columns (up to six).
In each column, a particular transfer function is stored -- the delay,
$g$-factor, cosine of the local emission angle, lensing, change of the
polarization angle and azimuthal emission angle, respectively.
The order of the functions is important
but some of the functions may be missing as defined in the first extension
(see 1.\ above). The functions are:
\begin{description} \itemsep -2pt
\vspace*{-0.2em}
\item[\rm delay] -- the Boyer-Lindquist time in $GM/c^3$ that elapses
between the emission of a photon from the disc and absorption of the
photon by the observer's eye at infinity plus a constant,
\item[\rm $g$-factor] -- the ratio of the energy of a photon received by the
observer at infinity to the local energy of the same photon when emitted
from an accretion disc,
\item[\rm cosine of the emission angle] -- the cosine of the local emission
angle between the emitted light ray and local disc normal,
\item[\rm lensing] -- the ratio of the area at infinity perpendicular to the
light rays through which photons come to the proper area at the disc
perpendicular to the light rays and corresponding to the same flux tube,
\item[\rm change of the polarization angle in radians] -- if the light
emitted from the disc is linearly polarized then the direction of
polarization will be changed by this angle at infinity --
counter-clockwise if positive, clockwise if negative (we are looking
towards the coming emitted beam); on the disc we measure the angle of
polarization with respect to the ``up'' direction perpendicular to the
disc with respect to the local rest frame; at infinity we also measure the
angle of polarization with respect to the ``up'' direction perpendicular to
the disc -- the change of polarization angle is the difference between
these two angles,
\item[\rm azimuthal emission angle in radians] -- the angle between the
projection of the three-momentum of an emitted photon into the disc (in the
local rest frame co-moving with the disc) and the radial tetrad vector.
\end{description}
For mathematical formulae defining the
functions see eqs.~(\ref{gfac})--(\ref{lensing}),
(\ref{delay1})--(\ref{polar}) and (\ref{azim_angle}) in
Chapter~\ref{transfer_functions}.
\item Each row corresponds to a particular value of $r-r_{\rm h}$
(see 4.\ above).
\item Each element corresponding to a particular column and row is a vector.
Each element of this vector corresponds to a particular value of
$\varphi_{\rm K}$ (see 5.\ above).
\end{enumerate}
We have pre-calculated three sets of tables -- {\tt KBHtables00.fits},
{\tt KBHtables50.fits} and {\tt KBHtables99.fits}.
All of these tables were computed for an accretion disc near a Kerr black
hole with no disc corona present. Therefore, ray-tracing in the vacuum Kerr
space-time could be used for calculating the transfer functions.
When computing the transfer functions, it was supposed that the matter in
the disc rotates on stable circular (free) orbits above the marginally stable
orbit. The matter below this orbit is freely falling and has the same energy and
angular momentum as the matter which is on the marginally stable orbit.
The observer is placed in the direction $\varphi = \pi/2$. The black hole
rotates counter-clockwise. All six functions are present in these tables.
Tables are calculated for these values of the black-hole horizon:\\
-- {\tt KBHtables00.fits}: 1.00, 1.05, 1.10, 1.15, \dots, 1.90, 1.95, 2.00
(21 elements),\\
-- {\tt KBHtables50.fits}: 1.00, 1.10, 1.20, \dots, 1.90, 2.00 (11 elements),\\
-- {\tt KBHtables99.fits}: 1.05 (1 element),\\
and for these values of the observer's inclination:\\
-- {\tt KBHtables00.fits}: 0.1, 1, 5, 10, 15, 20, \dots, 80, 85, 89
(20 elements),\\
-- {\tt KBHtables50.fits}: 0.1, 1, 10, 20, \dots, 80, 89 (11 elements),\\
-- {\tt KBHtables99.fits}: 0.1, 1, 5, 10, 15, 20, \dots, 80, 85, 89
(20 elements).
The radii and azimuths at which the functions are evaluated are same for all
three tables:\\
-- radii $r-r_{\rm h}$ are exponentially increasing from 0 to 999
(150 elements),\\
-- values of the azimuthal angle $\varphi_{\rm K}$ are equidistantly spread
from 0 to $2\pi$ radians with a much denser cover ``behind'' the black hole,
i.e.\ near $\varphi_{\rm K} = 1.5\pi$
(because some of the functions are changing heavily in this area for higher
inclination angles, $\theta_{\rm o} > 70^\circ$) (200 elements).
\section{Tables in
{\fontfamily{pcr}\fontshape{tt}\selectfont KBHlineNN.fits}}
\label{appendix3b}
Pre-calculated functions ${\rm d}F(g)\equiv{\rm d}g\,F(g)$ defined in
the eq.~(\ref{conv_function}) are stored in FITS files
{\tt KBHlineNN.fits}. These functions are used by all axisymmetric models.
They are stored as binary extensions and they are parametrized by the value of
the observer inclination angle
$\theta_{\rm o}$ and the horizon of the black hole $r_{\rm h}$. Each
extension provides values for different
radii, which are given in terms of $r-r_{\rm h}$, and for different
$g$-factors. Values of the $g$-factor, radius $r-r_{\rm h}$, horizon
$r_{\rm h}$, and inclination $\theta_{\rm o}$, at which the functions are
evaluated, are defined as vectors at the beginning of the FITS file.
The definition of the file {\tt KBHlineNN.fits}:%
\begin{enumerate} \itemsep -2pt
\vspace*{-0.5em}
\item[0.] All of the extensions defined below are binary.
\item The first extension contains one row with three columns that define
bins in the $g$-factor:
\begin{list}{--}{\setlength{\topsep}{-2pt}\setlength{\itemsep}{-2pt}
\setlength{\leftmargin}{1em}}
\item integer in the first column defines the width of the bins
(0 -- constant, 1 -- exponentially growing),
\item real number in the second column defines the lower boundary of the
first bin (minimum of the $g$-factor),
\item real number in the third column defines the upper boundary of the
last bin (maximum of the $g$-factor).
\end{list}
\item The second extension contains a vector of the values of the radius
relative to the horizon $r-r_{\rm h}$ in $GM/c^2$.
\item The third extension contains a vector of the horizon values in $GM/c^2$
($1.00 \le r_{\rm h} \le 2.00$).
\item The fourth extension contains a vector of the values of the observer's
inclination angle $\theta_{\rm o}$ in degrees
($0^\circ \le \theta_{\rm o} \le 90^\circ$, $0^\circ$ -- axis, $90^\circ$
-- equatorial plane).
\item All the previous vectors have to have values sorted in an increasing
order.
\item In the following extensions the functions ${\rm d}F(g)$ are defined, each
extension is for a particular value of $r_{\rm h}$ and $\theta_{\rm o}$.
The values of $r_{\rm h}$ and $\theta_{\rm o}$ are changing with each
extension in the same order as in tables in the {\tt KBHtablesNN.fits} file
(see the previous section, point 7.). Each extension has one column.
\item Each row corresponds to a particular value of $r-r_{\rm h}$
(see 2.\ above).
\item Each element corresponding to a particular column and row is a vector.
Each element of this vector corresponds to a value of the function
for a particular bin in the $g$-factor.
This bin can be calculated from number of elements of the vector and data
from the first extension (see 1.\ above).
\end{enumerate}
We have pre-calculated several sets of tables for different limb
darkening/brightening laws and with different resolutions. All of them were
calculated from tables in the {\tt KBHtables00.fits} file (see the previous
section for details) and therefore these tables are calculated for the same
values of the black-hole horizon and observer's inclination. All of these tables
have equidistant bins in the $g$-factor which fall in the interval
$\langle0.001,1.7 \rangle$. Several sets of tables are available:\\
-- {\tt KBHline00.fits} for isotropic emission, see eq.~(\ref{isotropic}),\\
-- {\tt KBHline01.fits} for Laor's limb darkening, see eq.~(\ref{laor}),\\
-- {\tt KBHline02.fits} for Haardt's limb brightening, see eq.~(\ref{haardt}).\\
All of these tables have 300 bins in the $g$-factor and 500 values of the radius
$r-r_{\rm h}$ which are exponentially increasing from 0 to 999.
We have produced also tables
with a lower resolution -- {\tt KBHline50.fits}, {\tt KBHline51.fits}, and
{\tt KBHline52.fits} with 200 bins in the $g$-factor and 300 values of the
radius.
\section{Lamp-post tables in
{\fontfamily{pcr}\fontshape{tt}\selectfont lamp.fits}}
\label{appendix3c}
This file contains pre-calculated values of the functions needed for the
lamp-post model. It is supposed that a primary source of emission is placed
on the axis at a height $h$ above
the Kerr black hole. The matter in the disc rotates on stable circular (free)
orbits above the marginally stable orbit and it is freely falling below this
orbit where it has the same energy and angular momentum as the matter which
is on the marginally stable orbit.
It is assumed that the corona between the source and the disc is optically
thin, therefore ray-tracing in the vacuum Kerr space-time could be used for
computing the functions.
There are five functions stored in the {\tt lamp.fits} file as binary
extensions. They are parametrized by the value of the horizon of the black hole
$r_{\rm h}$, and height $h$, which are defined as vectors at the beginning of
the FITS file. Currently only tables for $r_{\rm h}=1.05$
(i.e.\ $a\doteq0.9987492$)
and $h=2,\,3,\,4,\,5,\,6,\,8,\,10,\,12,\,15,\,20,\,30,\,50,\,75$ and $100$ are
available.
The functions included are:
\begin{list}{}{\setlength{\topsep}{-2pt}\setlength{\itemsep}{-2pt}}
\item[\rm -- angle of emission in degrees] -- the angle under which a photon
is emitted from a primary source placed at a height $h$ on the axis above the
black hole measured by a local stationary observer ($0^\circ$ -- a photon is
emitted downwards, $180^\circ$
-- a photon is emitted upwards),
\item[\rm -- radius] -- the radius in $GM/c^2$ at which a photon strikes the
disc,
\item[\rm -- $g$-factor] -- the ratio of the energy of a photon hitting the
disc to the energy of the same photon when emitted from a primary source,
\item[\rm -- cosine of the incident angle] -- an absolute value of the cosine
of the local incident angle between the incident light ray and local disc
normal,
\item[\rm -- azimuthal incident angle in radians] -- the angle between the
projection of the three-mo\-mentum of the incident photon into the disc (in
the local rest frame co-moving with the disc) and the radial tetrad vector.
\end{list}
For mathematical formulae defining the functions see
eqs.~(\ref{gfac_lamp})--(\ref{azim_angle_inc}) in Section~\ref{lamp-post}.
The definition of the file {\tt lamp.fits}:
\begin{enumerate} \itemsep -0.1em
\vspace*{-0.5em}
\item[0.] All of the extensions defined below are binary.
\item The first extension contains a vector of the horizon values in $GM/c^2$,
though currently only FITS files with tables for one value of the black-hole
horizon are accepted ($1.00 \le r_{\rm h} \le 2.00$).
\item The second extension contains a vector of the values of heights $h$
of a primary source in $GM/c^2$.
\item In the following extensions the functions are defined, each extension is
for a particular value of $r_{\rm h}$ and $h$. The values of $r_{\rm h}$ and
$h$ are changing with each extension in the following order:\\[1em] \hspace*{-2.6em}
\parbox{\textwidth}{
{\begin{centering}
$r_{\rm h}[1] \times h[1]$,\\
$r_{\rm h}[1] \times h[2]$,\\
$r_{\rm h}[1] \times h[3]$,\\
\dots \\
\dots \\
$r_{\rm h}[2] \times h[1]$,\\
$r_{\rm h}[2] \times h[2]$,\\
$r_{\rm h}[2] \times h[3]$,\\
\dots \\
\dots \\
\end{centering}}}\\[0.3em]
\item Each of these extensions has five columns.
In each column, a particular function is stored -- the angle of emission,
radius, $g$-factor, cosine of the local incident angle and azimuthal incident
angle, respectively. The extensions may have a different number of rows.
\end{enumerate}
\section{Coefficient of reflection in
{\fontfamily{pcr}\fontshape{tt}\selectfont
fluorescent\_line.fits}}
\label{appendix3d}
Values of the coefficient of reflection $f(\mu_{\rm i},\mu_{\rm e})$ for
a fluorescent
line are stored for different incident and reflection angles in this file.
For details on the model of scattering used for computations see
\cite{matt1991}.
It is assumed that the incident radiation is a power law with the photon index
$\Gamma=1.7$. The coefficient does not change its angular dependences for other
photon indices, only its normalization changes (see Fig.~14 in\break
\citealt{george1991}). The FITS file consists of three binary extensions:
\begin{list}{}{\setlength{\topsep}{-2pt}\setlength{\itemsep}{-2pt}}
\item[--] the first extension contains absolute values of the cosine of incident
angles,
\item[--] the second extension contains values of the cosine of reflection
angles,
\item[--] the third extension contains one column with vector elements, here
values of the coefficient of reflection are stored for different incident angles
(rows) and for different reflection angles (elements of a vector).
\end{list}
\section{Tables in
{\fontfamily{pcr}\fontshape{tt}\selectfont refspectra.fits}}
\label{appendix3e}
The function $f(E_{\rm l};\mu_{\rm i},\mu_{\rm e})$ which gives dependence of
a locally emitted spectrum on the angle of incidence and angle of emission
is stored in this FITS file. The emission is induced by a power-law incident
radiation. Values of this function were
computed by the Monte Carlo simulations of Compton scattering, for details see
\cite{matt1991}. The reflected radiation depends on the photon index $\Gamma$
of the incident radiation.
There are several binary extensions in this fits file:
\begin{list}{}{\setlength{\topsep}{-2pt}\setlength{\itemsep}{-2pt}}
\item[--] the first extension contains energy values in keV where
the function $f(E_{\rm l};\mu_{\rm i},\mu_{\rm e})$ is computed, currently the
interval from 2 to 300~keV is covered,
\item[--] the second extension contains the absolute values of the cosine of
the incident angles,
\item[--] the third extension contains the values of the cosine of the
emission angles,
\item[--] the fourth extension contains the values of the photon indices
$\Gamma$ of the incident power law,
currently tables for $\Gamma=1.5,\,1.6,\,\dots,\,2.9$ and $3.0$ are computed,
\item[--] in the following extensions the function
$f(E_{\rm l};\mu_{\rm i},\mu_{\rm e})$ is defined, each extension is
for a particular value of $\Gamma$; here values of the function are stored as a
vector for different incident angles (rows) and for different angles of
emission (columns), each element of this vector corresponds to a value of
the function for a certain value of energy.
\end{list}
\section{Transfer functions}
Photons emitted from an accretion disc are affected by gravitation in
several ways. They change their energy due to the gravitational and
Doppler shifts because photons are emitted by matter moving close to the source of
strong gravitation.
The cross-section of light tube changes as the photons propagate in the curved
space-time. This effect is strongest for high inclinations of the observer
when trajectories of photons coming from behind the black hole are bent very
much. Bending of light also influences the effective area from which photons
arrive due to different emission angles. Here, aberration caused by motion of
the disc matter plays its role as well. All three of these effects, the
$g$-factor, lensing and emission angle, influence the intensity of light that
the observer at infinity measures.
If the local emission from the disc is non-stationary one
must take into account the relative delay with which photons from different
parts of the disc arrive to the observer.
A part of the local emission may be due to the reflection of the light incident
on the disc from the corona above. In this case the disc radiation may be
partially polarized. The polarization vector changes as the light propagates
through the curved space-time. The change of polarization angle that occurs
between the emission of light at the disc and its reception by the observer at
infinity is needed for calculation of the overall polarization that the
observer measures. When calculating the local polarization we need to know the
geometry of incident and emitted light rays. Therefore we need to know also the
azimuthal emission angle of the emitted photons.
Thus six functions are necessary to transfer the local flux to the observer at
infinity. The exact definitions for all of them will be given in next
sections. For simplicity we will refer to them as transfer functions.
Various authors have computed radiation from matter moving around a black hole
in different approximations and in different parameter space.
De Felice, Nobili \& Calvani \citeyearpar{felice1974}
computed the effects of gravitational dragging on the electromagnetic radiation
emitted by particles moving on bound orbits around a Kerr black hole.
\mbox{\cite{cunningham1975}} studied the combined effects of the gravitational and
Doppler shifts together with the gravitational lensing effect on the X-ray
radiation from an accretion disc in the strong gravitational field of a black
hole. He introduced a concept of a transfer function where he includes all of
the above-mentioned effects into a single function that describes the overall
influence of the gravitational field on light rays emerging from the disc. He
also studied self-irradiation of the disc and its impact on the disc emission
(Cunningham \citeyear{cunningham1976}). Since then numerous authors have
investigated emission from disc (line, blackbody, reflected, etc.) in strong
gravity -- \cite{gerbal1981,asaoka1989};\break\cite{fabian1989,kojima1991,
laor1991,karas1992};\break\cite{viergutz1993};\hspace*{1em}\cite{bao1994};
\hspace*{1em}\cite{hameury1994};
\cite{zakharov1994,speith1995,bromley1997};\break\cite{martocchia2000,
cadez2003,beckwith2004}. In the situation when
full spectral resolution is not available, power spectra of light curves provide
us with partial information on the emission from accretion discs
\citep[see e.g.][]{abramowicz1991,schnittman2004}.
Other works also included the studies of net polarization from accretion discs,
here
we should mention\break\cite{connors1980,laor1990,chen1991};\break
\cite{matt1993b,matt1993,agol1997,bao1997};\break
\cite{ogura2000}. All of the above-mentioned authors used various
methods
for computing the transfer of photons from the accretion disc to the observer
far away from the central black hole. Some of them assumed the black hole to be
non-rotating, others performed their calculations for a rotating Kerr black
hole. \cite{karas1995} and \cite{usui1998} explored a substantially more
complex case when self-gravity of the disc is not neglected and photons move
in a space-time, the metric of which is itself derived from a numerical solution
of Einstein's equations. Similar approach is appropriate also for ray-tracing
in the field of a fast rotating neutron star, and in this way these techniques
are pertinent to the study of sources of quasi-periodic oscillations in X-ray
binaries.
Our numerical codes for computing the emission from accretion discs
(see Chapter~\ref{chapter3}) have to be fast, so
that they are suitable for fitting data. That is the reason for which
we have decided to pre-calculate the transfer functions and store them in the
form of tables in a
FITS\footnote{Flexible Image Transport System. See e.g.\ \cite{hanisch2001}
or \href{http://fits.gsfc.nasa.gov/}{\tt http://fits.gsfc.nasa.gov/} for
specifications.} file (see Appendix~\ref{fits}). We have chosen to
compute the transfer of photons from the infinity (represented by setting
the initial radius $r_{\rm i}=10^{11}$ in our numerical calculations) to the
disc numerically by
solving the equation of the geodesic. For this purpose the Bulirsch-Stoer method
of integration has been used (alternatively, the semi-analytical method of
elliptical integrals could be used, see e.g.\ \citealt{rauch1994}).
An optically thick and geometrically thin flat disc has been assumed. Only null
geodesics starting at the observer at infinity and ending at the equatorial
plane of the black hole, without crossing this plane, has been taken into
account (higher order images of the disc have not been computed).
The Kerr space-time has been assumed and the medium between the disc and the
observer has been supposed to be optically thin
for the wavelengths that we are interested in (X-rays in our case). Special Kerr
ingoing coordinates, which are non-singular on the
horizon and bring spatial infinity to a finite value (to zero),
have been used (see Appendix~\ref{kerr}). This has enabled us
to calculate the transfer functions with a high precision even very close to the
horizon of the black hole. By using these coordinates the
Boyer-Lindquist azimuthal coordinate $\varphi$ has been unfolded to the Kerr
coordinate $\varphi_{\rm K}$, which is more precise when interpolating between
the computed values on the disc. Some of the transfer functions depend on the
motion of the matter in the disc, which has been assumed to be Keplerian above
the marginally stable orbit $r_{\rm ms}$ and freely falling below it, where it
has the same energy and angular momentum as the matter which is on the marginally
stable orbit. All of the transfer functions
except lensing have been actually calculated analytically (see formulae in the
next sections) but with the numerical mapping of the impact parameters $\alpha$
and $\beta$ to the disc coordinates $r$ and $\varphi_{\rm K}$.
The lensing has been computed
by numerical integration of the equation of the geodesic deviation by the same
method as described above. The initial conditions for the numerical integration
are summarized in Appendix~\ref{initial}. In numerical computations,
approximately $10^5$
geodesics have been integrated. They have covered the disc from the horizon to
an outer radius of $r_{\rm out}\sim 1000$ in such a way that the region closest
to the horizon has been most densely covered. Thus we have obtained values of
transfer functions on a non-regular grid over the disc which have been
interpolated to a regular grid in $r$ and $\varphi_{\rm K}$ coordinates
using Delaunay triangulation.
Graphical representation of the transfer functions can be seen in
Appendix~\ref{atlas}.
In the following sections we summarize the equations used for the evaluation of
particular transfer functions over the disc. In this chapter as well as
everywhere else in the thesis we use units where
$GM_{\bullet}=c=1$ with $M_{\bullet}$ being the mass of the central black hole.
\section{Gravitational and Doppler shift}
The four-momentum of the photons emitted from the disc
is given by (see e.g.\ \citealt{carter1968} and \citealt{misner1973})
\begin{eqnarray}
\label{eq1}
p_{\rm e}^t & = & [a(l-a)+(r^2+a^2)(r^2+a^2-al)/\Delta]/r^2\, ,\\
p_{\rm e}^r & = & {\rm R}_{\rm sgn}\{(r^2+a^2-al)^2-
\Delta[(l-a)^2+q^2]\}^{1/2}/r^2\, ,\\
p_{\rm e}^{\theta} & = & -q/r^2\, ,\\
\label{eq4}
p_{\rm e}^{\varphi} & = & [l-a+a(r^2+a^2-al)/\Delta]/r^2\, ,
\end{eqnarray}
where $a$ is the dimensionless angular momentum of the black hole
($0\leq a \leq 1$).
In eqs.~(\ref{eq1})--(\ref{eq4}) we employed usual notation
(see Appendix~\ref{kerr} for the exact meaning of the quantities).
The combined gravitational and Doppler shift ($g$-factor) is defined as the
ratio of the energy of a photon received by an observer at infinity
to the local energy of the same photon when emitted from an accretion disc
\begin{equation}
\label{gfac}
g=\frac{\nu_{\rm o}}{\nu_{\rm e}}=\frac{{p_{{\rm o}\,t}}}{{p_{{\rm e}\,\mu}}\,
U^{\mu}}=-\frac{1}{{p_{{\rm e}\,\mu}}\,
U^{\mu}}\, .
\end{equation}
Here $\nu_{\rm o}$ and $\nu_{\rm e}$ denote the frequency of the observed and
emitted photons respectively, and $U^\mu$ is a four-velocity of the matter
in the disc.
\begin{figure}[tb]
\dummycaption\label{mm_gfac_angle}
\includegraphics[width=0.5\textwidth]{mm_gfac_diz}
\includegraphics[width=0.5\textwidth]{mm_Angle_diz}
\mycaption{Maximum and minimum values of energy shift (left) and emission angle
(right) of photons originating from different radii in the disc. The dependence
on the black-hole angular
momentum, $a$, is shown for three different inclination angles,
$\theta_{\rm o}=0.1^\circ,\,45^\circ$ and $85^\circ$.
Each curve corresponds to a fixed value of $a$ in the range $\langle0,1\rangle$,
encoded by line colours.}
\vspace*{1.3em}
\end{figure}
\begin{figure}[tb]
\vspace*{1.9em}
\dummycaption\label{mm_gfac_NN}
\includegraphics[width=0.5\textwidth]{mm_gfac_45}
\includegraphics[width=0.5\textwidth]{mm_gfac_85}
\vspace*{0.5em}
\mycaption{Contour lines of extremal energy shift factors in the plane of the
rotation parameter $a$ versus radius $r$ for observer inclination
$\theta_{\rm o}=45^\circ$ (left) and $85^\circ$ (right). Maximum $g$-factor
is shown at the top, minimum $g$-factor at the bottom. Radius is in units of $GM/c^2$.
The horizon is shown in black. The curve $a(r)_{|r=r_{\rm{}ms}}$ (dashed) is also
plotted across the contour lines. Notice that, in the traditional
disc-line scheme, no radiation is supposed to originate from radii below marginally
stable orbit $r_{\rm ms}$. If this is the case then
one must assume that all photons originate outside the dashed curve
and so the effect of frame-dragging is further reduced.}
\vspace*{1.7em}
\end{figure}
Contour graphs of the $g$-factor in Appendix~\ref{atlas} show that the effect of
rotation of the black hole is visible only in its vicinity
($r\lesssim10$). Near the horizon the $g$-factor decreases down to zero due to
the gravitational redshift, whereas far from the black hole the Doppler shift
prevails. Very far from the black hole, where matter of the disc rotates slowly,
the $g$-factor goes to unity.
Preliminary considerations on the line emission (its energy shift and its width)
can be based on the extremal values of the energy shift, $g_{+}$ and
$g_{-}$, which photons experience when arriving at the observer's
location from different parts of the disc. This is particularly relevant
for some narrow lines whose redshift can be determined more accurately
than if the line is broad (e.g.\ \citealt{turner2002,guainazzi2003,
yaqoob2003}). Careful discussion of $g_{\pm}$ can, in principle,
circumvent the uncertainties which are introduced by the uncertain form of
the intrinsic emissivity of the disc and yet still constrain some of
parameters. Advantages of this technique were pointed out already by
\cite{cunningham1975} and it was further developed by \cite{pariev2001}.
Fig.~\ref{mm_gfac_angle} shows the extremal values of the
redshift factor for the observer inclinations
$\theta_{\rm{}o}=0.1^{\circ},45^{\circ}$ and
$85^{\circ}$. These were computed along circles with radius $r$
in the disc plane. Corresponding contour lines of constant
values of $g_{\pm}$ in the plane $a$ versus $r$ can be seen
in Fig.~\ref{mm_gfac_NN}.
In other words, radiation is supposed to originate from radius
$r$ in the disc, but it experiences a different redshift depending
on the polar angle. Contours of the redshift factor provide a very useful
and straightforward technique to determine the position of a flare or a
spot, provided that a narrow spectral line is produced
and measured with sufficient accuracy. The most direct
use of extremal $g$-values would be if one were able to
measure variations of the line profile from its lowest-energy
excursion (for $g_{-}$) to its highest-energy excursion
(for $g_{+}$), i.e.\ over a complete cycle.
Even partial information can help to constrain models (for example,
the count rate is expected to dominate the observed line at the
time that $g=g_{+}$, and so the high-energy peak is
easier to detect). While
it is very difficult to achieve sufficient precision on the highly
shifted and damped red wing of a broad line,
prospects for using narrow lines are indeed interesting,
provided that they originate close enough to the black hole and that
sufficient resolution is achieved both in the energy and time domains.
Fig.~\ref{mm_gfac_NN} also makes it clear that it is possible in
principle (but intricate in practice) to deduce the $a$-parameter value
from spectra. It can be seen from the redshift factor that
the dependence on $a$ is rather small and it quickly becomes
negligible if the light from the source is dominated by contributions
from $r\gtrsim10$.
\section{Emission angle}
Here we examine the angle of emission with respect to
the disc normal. We assume the local frame co-moving with the
medium of the disc. The cosine of the local emission angle is
\begin{equation}
\label{cosine}
\mu_{\rm e}=\cos{\delta_{\rm e}}=
-\displaystyle\frac{{p_{{\rm e}\,\alpha}\,n^{\alpha}}}
{{p_{{\rm e}\,\mu}\,U^{\mu}}}\, ,
\end{equation}
where $n^\alpha=-e_{(\theta)}^\alpha$ are components of the disc normal.
One can see from contours of $\mu_{\rm e}$ plotted in the disc plane
(Appendix~\ref{atlas}) and from graphs of its maximum and minimum values
(Fig.~\ref{mm_gfac_angle}) that gravitation influences visibly only those
light rays that pass close to the black hole on their way to the observer,
and especially for large inclination angles. Due to the effect of light
bending and aberration there exists a point on the disc where the emission
angle is zero, i.e.\ the local direction of the emission is perpendicular to the
disc surface. On the other hand, near the horizon only the light rays emitted
almost parallel to the disc plane can reach the observer.
Very far from the black hole, where matter rotates slowly, the
emission angle is determined mainly by special-relativistic aberration
and it gradually approaches the value of the observer inclination.
\begin{figure}[tb]
\dummycaption\label{mm_lens}
\includegraphics[width=0.5\textwidth]{mm_lens_diz}
\includegraphics[width=0.5\textwidth]{mm_transf_diz}
\mycaption{Same as in Fig.~\ref{mm_gfac_angle} but for lensing (left) and
overall transfer function (right).}
\end{figure}
\section{Gravitational lensing}
We define lensing as the ratio of the cross-section ${\rm d}S_{\rm f}$ of the
light tube at infinity
to the cross-section ${\rm d}S_\perp$ of the same light tube at the disc:
\begin{equation}
\label{lensing}
l = \frac{{\rm d}S_{\rm f}}{{\rm d}S_\perp}=\frac{1}{\sqrt{\|Y_{\rm e1}\|^2
\|Y_{\rm e2}\|^2 - <Y_{\rm e1},Y_{\rm e2}>^2}}\, .
\end{equation}
The four-vectors $Y_{\rm e1}$ and $Y_{\rm e2}$ are transported
along the geodesic according to the equation of the geodesic deviation from
infinity where they are unit, space-like and perpendicular
to each other and to the four-momentum of light (for the exact definition of
these vectors very far from the black hole see Appendix~\ref{initial}).
In eq.~(\ref{lensing}) we have denoted the magnitude of a four-vector by
$\|\ \|$ and scalar product of two four-vectors by $<\ ,\ >$.
The cross section of light tube is constant for all observers
(see \citealt{schneider1992}).
Lensing can significantly amplify the emission from some parts of the disc
(located behind the black hole from the point of view of the observer and in
Kerr ingoing coordinates). This is true mainly for observers with large
inclination angles (see figures in Appendix~\ref{atlas} and Fig.~\ref{mm_lens}).
When one wants to estimate the total effect that gravitation has on the
intensity of light coming from different parts of the disc, one has to take into
account all the three effects -- the \mbox{$g$-factor}, the emission angle and
the lensing. These effects combine into a single function defined by
(see~eq.~(\ref{emission}))
\begin{equation}
\label{F}
F = g^2\mu_{\rm e}\,l\, .
\end{equation}
We call it the overall transfer function.
See contour graphs in Appendix~\ref{atlas} and
plots in Fig.~\ref{mm_lens} for the total gravitational amplification of photon
flux emitted from a Keplerian accretion disc.
\begin{figure}[tb]
\begin{center}
\dummycaption\label{mm_delay}
\includegraphics[width=0.5\textwidth]{mm_delay_diz}
\mycaption{Same as in Fig.~\ref{mm_gfac_angle} but for maximum and minimum
relative time delay.}
\end{center}
\end{figure}
\section{Relative time delay}
The relative time delay $\Delta t$ is the Boyer-Lindquist time which elapses
between the emission of a photon from the disc and its reception by an observer
(plus a certain constant so that the
delay is finite close to the black hole but not too close). We integrate
the equation of the geodesic in Kerr ingoing coordinates and thus we calculate
the delay in the Kerr ingoing time coordinate $\Delta t_{\rm K}$. The
Boyer-Lindquist time can be obtained from $\Delta t_{\rm K}$
using the following equation:
\begin{figure}[tbh]
\begin{center}
\dummycaption\label{pol_angle}
\begin{tabular}{c@{\quad}l}
\parbox[t]{5cm}{\vspace*{1cm}\includegraphics[height=5cm]{pol_angle}} &
\parbox[t]{9cm}{\vspace*{-0.3cm}
\begin{itemize} \itemsep -2pt
\item [\rm(i)] Let three-vectors $\vec{p_A}$, $\vec{n_A}$, $\vec{n'_A}$
and $\vec{f_A}$ be the momentum of a photon, normal to
the disc, projection of the normal to the plane perpendicular to the momentum
and a vector which is parallelly transported along the geodesic (as four-vector),
respectively;
\item [\rm (ii)] let $\Psi_A$ be an angle between $\vec{n'_A}$
and $\vec{f_A}$;
\item [\rm (iii)] let the quantities in (i) and (ii) be evaluated at
the disc for $A=1$ with respect to the local rest frame co-moving with the
disc, and at infinity for \hbox{$A=2$} with respect to the stationary observer
at the same light geodesic;
\item [\rm (iv)] then the change of polarization angle is defined
as $\Psi=\Psi_2-\Psi_1$.
\end{itemize}}
\end{tabular}
\mycaption{Definition of the change of polarization angle $\Psi$.}
\end{center}
\end{figure}
\begin{equation}
{\rm d}t = {\rm d}t_{\rm K}-\left [ 1+ \frac{2r}{(r-r_+)(r-r_-)}\right ]
{\rm d}r\, ,
\end{equation}
with $r_{\pm}=1\pm\sqrt{1-a^2}$ being inner ($-$) and outer ($+$) horizon of the
black hole.
Then we integrate the above equation and define the time delay as
\begin{myeqnarray}
\label{delay1}
\Delta t & = & \Delta t_{\rm K}-\left [r+\frac{2}{r_+ - r_-}\,
\ln{\frac{r-r_+}{r-r_-}}+\ln{[(r-r_+)(r-r_-)]}\right ]
& & {\rm for}\ \ a < 1\, ,\hspace*{2em}\\
\label{delay2}
\Delta t & = & \Delta t_{\rm K}-\left [r-\frac{2}{r-1}+2\ln{(r-1)}\right ]
& & {\rm for}\ \ a = 1\, .
\end{myeqnarray}
There is a minus sign in front of the brackets because the direction
of integration is from infinity to the disc.
For contour graphs of the relative time delay see figures in
Appendix~\ref{atlas} and for plots of maximum and minimum delay at a constant
radius see Fig.~\ref{mm_delay}.
\section{Change of polarization angle}
Various physical effects can influence polarization of light as it propagates
towards an observer. Here we examine only the influence of the gravitational
field represented by the vacuum Kerr space-time.
The change of the polarization angle is defined as the angle by which a vector
parallelly transported along the light geodesic rotates with respect to some
chosen frame. We define it in this
way because in vacuum the polarization vector is parallelly transported along
the light geodesic. This angle depends on the choice of the local frame at
the disc and at infinity. See Fig.~\ref{pol_angle} for an exact definition.
The change of polarization angle is \citep[see][]{
connors1977,connors1980}
\begin{equation}
\label{polar}
\tan{\Psi}=\frac{Y}{X}\, ,
\end{equation}
where
\begin{eqnarray}
X & = & -(\alpha-a\sin{\theta_{\rm o}})\kappa_1-\beta\kappa_2\, ,\\
Y & = & \phantom{-}(\alpha-a\sin{\theta_{\rm o}})\kappa_2-\beta\kappa_1\, ,
\end{eqnarray}
with $\kappa_1$ and $\kappa_2$ being components of the complex constant
of motion $\kappa_{\rm pw}$ (see Walker \& Penrose \citeyear{walker1970})
\begin{eqnarray}
\kappa_1 & = & arp_{\rm e}^\theta f^t-r\,[a\,p_{\rm e}^t-(r^2+a^2)\,
p_{\rm e}^\varphi]f^\theta-r(r^2+a^2)\,p_{\rm e}^\theta f^\varphi\, ,\\
\kappa_2 & = & -r\,p_{\rm e}^rf^t+r\,[p_{\rm e}^t-a\,p_{\rm e}^\varphi]f^r+
arp_{\rm e}^rf^\varphi\, .
\end{eqnarray}
Here the polarization vector $f^\mu$ is a four-vector corresponding to the
three-vector
$\vec{f_1}$ from Fig.~\ref{pol_angle} which is chosen in such a way that it
is a unit vector parallel with $\vec{n'_1}$ (i.e.\ $\Psi_1=0$)
\begin{equation}
f^\mu = \frac{n^\mu-\mu_{\rm e}\left( g\,p_{\rm e}^\mu-U^\mu\right)}
{\sqrt{1-\mu_{\rm e}^2}}\, .
\end{equation}
For contour graphs of the change of the polarization angle see figures in
Appendix~\ref{atlas}.
\section{Azimuthal emission angle}
\label{azimuth_angle}
We define the azimuthal emission angle as the angle between the projection of
the three-momentum of the emitted photon into the equatorial plane (in the local
rest frame co-moving with the disc) and the radial tetrad vector:
\begin{equation}
\label{azim_angle}
\Phi_{\rm e}=\arctan\frac{p_{\rm e}^\alpha\,e_{(\varphi)\,\alpha}}
{p_{\rm e}^\mu\,e_{(r)\,\mu}}=\arctan\left( g\,\frac{\Delta}{r}\,
\frac{-p_{\rm e}^t\,U^\varphi+p_{\rm e}^\varphi\,U^t}{g\,p_{\rm e}^r-U^r}\right)\, .
\end{equation}
The explicit formula for this angle is not necessary for computations of
light curves and spectral profiles, but it appears in discussions of
polarimetry. This is also the reason why we have computed $\Phi_{\rm{}e}$.
Like the quantities discussed previously, the resulting values of the azimuthal
angle depend on the adopted geometry of the source and the rotation law of
the medium. They are determined by mutual interplay of special- and
general-relativistic effects. Therefore, we remind the reader that our
analysis applies to geometrically thin Keplerian discs residing in the
equatorial plane, although generalization to more complicated situations
should be fairly straightforward. For contour graphs of the change of the
azimuthal emission angle see figures in Appendix~\ref{atlas}.
\chapter*{Introduction}
\section*{On black holes and accretion discs}
\addcontentsline{toc}{section}{On black holes and accretion discs}
It is not so long ago that the general theory of relativity was brought
to life by Albert \cite{einstein1916}. Black holes belong to the first of
its offspring, the static ones being described as early as in 1916 by
Karl \citeauthor{schwarzschild1916}. These objects are characterized by
strong gravity and an imaginary horizon that enwraps a region from which nothing
can escape, not even the light. Black holes have their ancestors in the ideas of
John \cite{michell1784} and Pierre Simon Laplace (\citeyear{laplace1796,
laplace1799}), who defined similar objects more than one century before --
stars with such large masses that the escape velocity from their
surface is larger than the speed of light. But this would be a different story,
a Newtonian one. Soon the family of black holes grew bigger, several new types
were classified -- black holes with a non-zero
angular momentum \citep{Kerr1963}, with an electric charge or with a hypothetic
magnetic monopole \citep{reissner1916,nordstrom1918,newman1965}.
However, electrically charged non-rotating black holes are
not of direct astrophysical interest since, if the hole is immersed in a cloud
of ionized or partially ionized plasma, which is a typical situation in cosmic
environments and we assume it hereafter in this work, then any excess
charge becomes rapidly neutralized by selective accretion. Nevertheless, charged
black holes have a role as a model for the realistic black holes formed in a
spinning collapse, see \cite{wald1974,blandford1977,damour1978,damour1980,
price1986,karas1991}. The properties of black holes have been studied by
numerous authors in various contexts;\break see e.g.\ \cite{thorne1986,kormendy1995,
wald1998} for a
survey of the problems and the general discussion of astrophysical connotations
of black holes. As far as the Kerr (rotating) black holes are concerned, the
possibility to extract its
rotational energy either in the ergosphere by Penrose process
\citep{penrose1971} or by Blandford-Znajek process, when
the black hole is embedded in the electromagnetic field \citep{blandford1977},
are really intriguing.
The problem of the origin of black holes is very exciting as well.
Oppenheimer
considered black holes as the last stage in the evolution of massive stars when
after having burnt all their fuel neither
the pressure of electrons \citep{chandrasekhar1931} nor the pressure of
neutrons \citep{oppenheimer1939, rhoades1974} can stop them from collapsing.
There are theories that
the black holes may be nearly as old as the universe itself. These primordial
black holes \citep{hawking1971} may have been the seeds for the formation of
galaxies.
From the astrophysical point of view the motion of matter and the propagation of
light near black holes has a special significance because the matter can emit
light that can be observed and thus we can deduce what is going on out there.
The emission
from the stellar-mass and supermassive black holes themselves is rather low
\citep{hawking1975}. Many authors investigated these issues and calculated
either the time-like and null geodesics in the black-hole space-times
\citep{sharp1979,chandrasekhar1983,bicak1993} or even the more complex
hydrodynamical models for the accreting matter or the outflowing wind. The
spherical accretion onto the black hole was firstly considered as early as in
\citeyear{bondi1944} by Bondi and Hoyle. The disc-like accretion
\citep{lynden-bell1969, pringle1972, novikov1973, shakura1973, shapiro1976}
is potentially much more efficient in converting the gravitational energy into
the thermal energy which can be radiated away. It is also a more realistic
configuration when we consider that the accreting matter has a non-zero angular
momentum.
The family of accretion discs is quite large. Let us just mention
the Shakura-Sunyaev or standard discs (SS), the advection dominated accretion
flow (ADAF), the convection dominated accretion flows (CDAF), the advection
dominated inflows-outflows solution (ADIOS), the Shapiro-Lightman-Eardley discs
(SLE) and the slim discs.
Their behaviour depends mainly on the density, pressure, viscosity and opacity
of the matter that they consist of as well as on the equation of state and their
accretion rate. For a basic review on accretion disc theories see e.g.\
\cite{kato1998} or \cite{frank2002}.
The interaction between an accretion disc and the matter above and below the
disc may also play an important role in the accretion process and therefore the
configuration of the black hole, accretion disc and corona seems to be more
appropriate \citep{paczynski1978,haardt1993}.
As can be seen from the multitude of the disc models, some of the processes
involved in the accretion are quite well understood. Nevertheless, there are
still many open issues, one of them being the origin of viscosity
in accretion discs. Turbulence and
magnetic fields may play an important role here. Another important phenomenon
is the existence of jets emerging from the vicinity of the central black hole,
which may be due to the interaction between the black hole and the accretion
disc. To tackle
these problems much more complex three dimensional numerical models of accretion
discs in curved space-time with magnetic field present have to be investigated
\citep[e.g.][]{hawley1995,nishikawa2001,villiers2003}.
If the disc is thick and/or dense, its own gravity, in particular farther away
from the central black hole, cannot be neglected.
These self-gravitating discs were investigated by various
authors in various approximations, starting with \cite{ostriker1964},
\cite{bardeen1973}, \cite{fishbone1976}, \cite{paczynski1978a} and
\cite{karas1995}. A very useful and pertinent
review of this subject was recently written by Karas, Hur\'{e} \&
Semer\'{a}k \citeyearpar{karas2004}.
\section*{Linking theory with observation}
\addcontentsline{toc}{section}{Linking theory with observation}
Last century was rich in discovering highly energetic and violent objects in
astronomy. In 1918 \citeauthor{curtis1918} discovered a long jet of material
coming out from elliptical galaxy M87. Later, galaxies with particularly
bright and compact nuclei were observed \citep{seyfert1943} and they were
classified as a new type of galaxies, the so-called Seyfert galaxies.
But truly amazing
discoveries came when radio astronomy fully developed. When quasars were
observed for the first time in 1960s, they were called radio stars
\citep{matthews1960, matthews1962} because their emission in the radio part
of the spectrum was enormous and they appeared like point sources. It was three
years later that \cite{schmidt1963} realized that these objects were not hosted
by our Galaxy but that they were rather distant extragalactic objects with
highly
redshifted spectral lines. Since then many strange radio sources with jets and
lobes of various shapes and sizes have been discovered, mostly associated with
optical extragalactic objects -- centres of distant galaxies. The luminosity,
size and variability of these sources indicate that highly energetic
processes were once at work there. Because of their huge activity they are
called active galactic nuclei.
In 1960s it was realized that the only reasonable way of explaining the
production of such large amounts of energy was to be sought in the gravitational
energy conversion by means of an accreting massive black hole
\citep{salpeter1964,zeldovich1964}.
A new branch of astronomy has emerged, in which many novel discoveries were made:
it was the X-ray astronomy and it developed quickly. The first observations were
made by Geiger counters carried by rockets in 1950s and balloons in 1960s and
later by specially designed X-ray astronomy satellites that have been orbiting
the Earth since 1970s.
The first Galactic X-ray source was discovered in 1962 -- a neutron star
Sco X-1 \citep{giacconi1962}. Soon more discoveries followed, X-ray binary
source Cygnus~X-1 \citep{bowyer1965} being one of them. The fact that this
object is a
strong X-ray emitter and that the optical and X-ray emission varies on very
short time scales (as short as one thousandth of a second) suggests that the
companion might be a black hole. Many Galactic objects (most of them binaries
containing neutron stars) and active galactic nuclei emitting X-rays have been
observed since then. It is believed that AGN
sources host a supermassive black hole with
a mass of approximately $10^6-10^8M_{\odot}$ and that some of the Galactic X-ray
binaries contain a black hole with the mass equal to several solar masses.
A breakthrough has come with the two most recent X-ray satellites, {\it
Chandra}\/ and {\it XMM-Newton}, which provide us with the most detailed
images and spectra with an unprecedented resolution.
Some of the observed objects exhibit redshifted, broad or narrow features in
their spectra, thus suggesting that the emission is coming from the very close
vicinity of a black hole.
But still, the data acquired by the instruments on board these satellites
are not as complete and detailed as would be necessary for an unambiguous
interpretation and comparison with the theoretical models. New
X-ray missions are being planned -- {\it Astro-E2}\/ (already being assembled),
{\it Constellation-X}\/ and {\it Xeus}.
Hopefully, with the data measured by the satellites involved in these
observations, it
will be possible not only to decide whether there is a black hole in the
observed system but also to determine its properties, its mass and angular
momentum, as well as the properties of the surrounding accretion disc.
\section*{This thesis}
\addcontentsline{toc}{section}{This thesis}
Nowadays both the theory describing accreting black-hole systems and the
instruments for observing such objects are quite advanced.
Computational tools for comparing acquired data with
theories have been developed to such an advanced state that we are able now
to find out properties of
particular observed systems on the one hand and check if our models are good
enough for describing these systems on the other. One of the tools used for
processing X-ray data, which also implements the possibility to fit the
data within a certain set of models,\footnote{Notice that, in accordance with
traditional terminology, we use
the term `model' in two slightly different connotations: first, it can
mean a physical model of an astronomical system or a process that is under
discussion; other possible meaning of the word is more technical, referring
to a computational representation, or a routine that is employed in order
to compare actual observational data with the theory.} is {\sc{}xspec}.
This X-ray spectral fitting
package contains various models for explaining the measured spectra but
it lacks fully general relativistic models.
One of the aims of this thesis has been to help to fill this gap and provide
new general
relativistic models that would be fast and flexible enough and would be
able to fit parameters describing a black hole, mainly its mass and angular
momentum. Although it is not possible to fit time resolved data within
{\sc{}xspec} and polarimetric data have not been available for X-ray sources yet
(though they may become available when future X-ray missions materialize), we
developed new models able to deal also with these issues.
Our objective is to model X-ray spectra from accretion discs
near compact objects. The observed spectrum depends not only on the local
radiation emitted from the disc but it is also affected by strong
gravitation on its way to the observer. Emission from different parts of the
disc may be either amplified or reduced by gravitational effects. We will focus
our attention on these effects in Chapter~\ref{chapter1}, where we deal with
calculations of six functions
that are needed for the integration of the total spectra of the accretion disc
(including time varying spectra and polarimetric information). These functions
include the Doppler and gravitational shifts, gravitational lensing, relative
time delay, two emission angles and change of the polarization angle.
The basic types of local emission coming from an accretion disc are summarized
in Chapter~\ref{chapter2}, where also the equations for the observed
flux and polarization at infinity are derived.
In Chapter~\ref{chapter3} we describe new {\sc ky} models that we have
developed and that can be used either
inside {\sc{}xspec} or as standalone programs for studying the non-stationary
emission from accretion discs or for studying polarimetry. These are
models for a general relativistic line and Compton reflection continuum,
general convolution models and a model for an orbiting spot.
We also compare these models with some of the models already present in
{\sc{}xspec}.
Some applications of the new models are summarized in
Chapter~\ref{chapter4}. Firstly, we employ the models to high-quality X-ray data
that are currently available for the Seyfert galaxy MCG--6-30-15.
Then we calculate the flux from X-ray illuminated orbiting spot
and polarization from an accretion disc illuminated from a primary source
located above the black hole.
There are several appendixes in this thesis.
There is a summary of the basic equations for Kerr space-time in
Appendix~\ref{kerr}.
Appendix~\ref{fits} describes the layout of data files used in this thesis.
A detailed description of the integration
routines that we have developed and that can be used inside the {\sc xspec}
framework for general
relativistic computations is included in Appendix~\ref{ide}.
In Appendix~\ref{atlas} an atlas of contour figures of all transfer
functions is shown.
\enlargethispage*{\baselineskip}
This thesis describes my original work except where references are given
to the results of other authors. Parts of the research were carried out in
collaboration and published in papers, as indicated in the text. In
particular, various features and usage of the newly developed code have
been described in \cite{dovciak2004} and \cite{dovciak2004b}.
Chapter~\ref{chapter4} is based
on papers \cite{dovciak2004c};\break\cite{dovciak2004} and \cite{dovciak2004a}.
Another application to random emitting spots in the
accretion disc can be found in \cite{czerny2004} but it is not included in
the present thesis.
\section{Kerr metric}
The Kerr metric in the Boyer-Lindquist coordinates $(t,r,\theta,\varphi)$ is
\begin{equation}
g_{\mu\nu} = \mat{-(1-\frac{2r}{\rho^2}) & 0 & 0
& -\frac{2ar\sin^{2}{\! \theta}}{\rho^2} \\[2mm]
0 & \frac{\rho^2}{\Delta} & 0 & 0 \\[2mm] 0 & 0 & \rho^2 & 0 \\[2mm]
-\frac{2ar\sin^{2}{\!\theta}}{\rho^2} & 0 & 0
& \frac{{\cal A}\sin^{2}{\!\theta}}{\rho^2}}\, ,
\end{equation}
where $\rho^2 \equiv r^2+a^2\cos^2\!\theta$, $\Delta \equiv r^2-2r+a^2$ and
${\cal A} \equiv (r^2+a^2)^2-\Delta a^2\sin^2\!\theta$.
Let's define special Kerr ingoing coordinates $(\hat{t},\hat{u},\hat{\mu},\hat{\varphi})$ by
the following tetrad vectors
\begin{equation}
\begin{array}{cclccl}
\fr{\partial}{\partial \hat{t}} & = & \fr{\partial}{\partial t}\ , &
{\rm d} \hat{t} & = & {\rm d} t + \fr{r^{2}+a^{2}}{\Delta}{\rm d} r\ ,
\\[4mm]
\fr{\partial}{\partial \hat{u}} & = & -r^2\fr{\partial}{\partial r}+
r^2\fr{r^{2}+a^{2}}{\Delta}\left(\fr{\partial}{\partial t}+
\Omega^{\rm H}\fr{\partial}{\partial\varphi}\right)\ , \hspace{0.7cm} &
{\rm d} \hat{u} & = & -\fr{1}{r^2}{\rm d} r\ ,
\\[4mm]
\fr{\partial}{\partial \hat{\mu}} & = & -\fr{1}{\sin\theta}\fr{\partial}
{\partial \theta}\ , & {\rm d} \hat{\mu} & = & -\sin\theta {\rm d} \theta\ ,\\[4mm]
\fr{\partial}{\partial \hat{\varphi}} & = & \fr{\partial}{\partial \varphi}\ , &
{\rm d} \hat{\varphi} & = & {\rm d} \varphi + \fr{a}{\Delta} {\rm d} r\ ,
\end{array}
\end{equation}
with $\ \Omega^{\rm H}=\fr{a}{r^{2}+a^{2}}$.
The advantage of these coordinates, where $\hat{u}\equiv r^{-1}$ and
$\hat{\mu}\equiv\cos{\theta}$, is that the Kerr metric is not singular on the
horizon of the black hole and spatial infinity
($r\rightarrow\infty$) is brought to a finite value ($\hat{u}\rightarrow 0$).
Another advantage for numerical computations is that we get rid of the
cosine function.
The relationship between the Boyer-Lindquist coordinate $\varphi$ and the Kerr
ingoing coordinate $\varphi_{\rm K}$, for geodesics coming from infinity
to the equatorial plane, can be expressed in the following way:
\begin{myeqnarray}
\varphi & = & \varphi_{\rm K}+\frac{a}{r_+-r_-}\ln{\frac{r-r_+}{r-r_-}} & &
{\rm for} \ \ a<1\, ,\\
\varphi & = & \varphi_{\rm K}-\frac{1}{r-1} & & {\rm for}\ \ a=1\, ,
\end{myeqnarray}
with $r_\pm=1\pm\sqrt{1-a^2}$ being inner ($-$) and outer ($+$) horizon of the
black hole.
Another useful way to express the transformation between the Boyer-Lindquist
and special Kerr ingoing coordinates is by matrices of transformation
\begin{myeqnarray}
\fr{\partial x^{\mu}}{\partial \hat{x}^{\nu}} & = & \mat{1 & \frac{r^2(r^2+a^2)}
{\Delta} & 0 & 0 \\[2mm]
0 & -r^2 & 0 & 0 \\[2mm]
0 & 0 & -\frac{1}{\sin\theta} & 0 \\[2mm]
0 & \frac{r^2a}{\Delta} & 0 & 1} & = &
\mat{1 & \frac{(1+a^2\hat{u}^2)}{\hat{u}^2\tilde{\Delta}} & 0 & 0 \\[2mm]
0 & -\frac{1}{\hat{u}^2} & 0 & 0 \\[2mm]
0 & 0 & -\frac{1}{\sqrt{1-\hat{\mu}^2}} & 0 \\[2mm]
0 & \frac{a}{\tilde{\Delta}} & 0 & 1}\, ,\\[4mm]
\fr{\partial \hat{x}^{\mu}}{\partial x^{\nu}} & = & \hspace*{2mm} \mat{1 &
\frac{r^{2}+a^{2}}{\Delta} & 0 & 0 \\[2mm]
0 & -\frac{1}{r^2} & 0 & 0 \\[2mm]
0 & 0 & -\sin\theta & 0 \\[2mm]
0 & \frac{a}{\Delta} & 0 & 1} & = &
\mat{1 & \frac{1+a^2\hat{u}^2}{\tilde{\Delta}} & 0 & 0 \\[2mm]
0 & -\hat{u}^2 & 0 & 0 \\[2mm]
0 & 0 & -\sqrt{1-\hat{\mu}^2} & 0 \\[2mm]
0 & \frac{a\hat{u}^2}{\tilde{\Delta}} & 0 & 1}\, .\hspace*{9mm}
\end{myeqnarray}
The Kerr metric in special Kerr ingoing coordinates is
\begin{eqnarray}
g_{\hat{\mu}\hat{\nu}} = \mat{-(1-\frac{2\hat{u}}{\tilde{\rho}^2}) & -\frac{1}{\hat{u}^2} & 0 &
-\frac{2a\hat{u}(1-\hat{\mu}^2)}{\tilde{\rho}^2} \\[2mm]
-\frac{1}{\hat{u}^2} & 0 & 0 & \frac{a(1-\hat{\mu}^2)}{\hat{u}^2} \\[2mm] 0 & 0 &
\frac{\tilde{\rho}^2}{\hat{u}^2(1-\hat{\mu}^2)} & 0 \\[2mm]
-\frac{2a\hat{u}(1-\hat{\mu}^2)}{\tilde{\rho}^2} & \frac{a(1-\hat{\mu}^2)}{\hat{u}^2} & 0 &
\frac{\tilde{\cal A}(1-\hat{\mu}^2)}{\hat{u}^2\tilde{\rho}^2}}\, ,
\end{eqnarray}
with $\tilde{\rho}^2 \equiv \rho^2/r^2=1+a^2\hat{u}^2\hat{\mu}^2$,
$\tilde{\Delta} \equiv \Delta/r^2=1-2\hat{u}+a^2\hat{u}^2$ and
$ \tilde{\cal A} \equiv {\cal A}/r^4=(1+a^2\hat{u}^2)^2-a^2\hat{u}^2\tilde{\Delta}(1-\hat{\mu}^2)$.
\section{Light rays in Kerr space-time}
The four-momentum $p^\mu\equiv\frac{{\rm d}x^{\mu}}{{\rm d}\lambda'}$
of photons travelling in Kerr space-time in the
Boyer-Lindquist coordinates is (see e.g.\ \citealt{carter1968} or
\citealt{misner1973})
\begin{eqnarray}
\label{carter1}
p^t & \equiv & \frac{{\rm d} t}{{\rm d} \lambda'} = [a(l-a\sin^2\!\theta)+(r^2+a^2)
(r^2+a^2-al)/\Delta]/\rho^2\, ,\\
\label{carter2}
p^r & \equiv & \frac{{\rm d} r}{{\rm d} \lambda'} = R_{\rm sgn} \{ (r^2+a^2-al)^2 -
\Delta [(l-a)^2+q^2] \}^{1/2}/\rho^2\, ,\\
\label{carter3}
p^\theta & \equiv & \frac{{\rm d} \theta}{{\rm d} \lambda'} =
\Theta_{\rm sgn}[ q^2-\cot^2\!\theta(l^2-a^2\sin^2\!\theta)]^{1/2}/\rho^2\, ,\\
\label{carter4}
p^\varphi & \equiv & \frac{{\rm d} \varphi}{{\rm d} \lambda'} = [ l/\sin^2\!\theta - a +
a(r^2+a^2-al)/\Delta ]/\rho^2\, ,
\end{eqnarray}
where $l=\alpha(1-\mu_{\rm o}^2)^{1/2}=\alpha\sin\theta_{\rm o}$ and
$q^2=\beta^2+\mu_{\rm o}^2(\alpha^2-a^2)$ are Carter's constants of motion with
$\alpha$ and $\beta$ being impact parameters measured perpendicular and
parallel, respectively, to the spin axis of the
black hole projected onto the observer's sky. Here we define $\alpha$ to
be positive when a photon travels in the direction of the four-vector
$\frac{\partial}{\partial \varphi}$ at infinity, and $\beta$ to be positive if
it travels in the direction of $-\frac{\partial}{\partial\theta}$ at infinity.
The parameter $\theta_{\rm o}$ (and $\mu_{\rm o}=\cos{\theta_{\rm o}}$) defines
a point at infinity through which the light ray passes (we consider only light
rays coming to or from the observer at infinity).
Furthermore, we have denoted the sign of the radial component of the momentum
by ${\rm R}_{\rm sgn}$ and the sign of the $\theta$-component of the momentum
by ${\Theta}_{\rm sgn}$. We have chosen an affine parameter $\lambda'$ along
light geodesics in such a way that the conserved energy is normalized to
$-p_t=1$.
The four-momentum transformed into special Kerr ingoing coordinates is
\begin{eqnarray}
\label{ptk}
p^{\hat{t}} \equiv \frac{{\rm d}\hat{t}}{{\rm d}\lambda'} & = & \frac{1}{\tilde{\rho}^2}
\left[1+a\hat{u}^2(l+a\hat{\mu}^2)-(1+a^2\hat{u}^2)\frac{1+2\hat{u}-(l^2+q^2)\hat{u}^2}
{\hat{u}(2-a\hat{u} l)+U_{\rm sgn}\sqrt{U}}\right]\, , \\
p^{\hat{u}} \equiv \frac{{\rm d}\hat{u}}{{\rm d}\lambda'} & = &
U_{\rm sgn}\frac{\hat{u}^2\sqrt{U}}{\tilde{\rho}^2}\, ,\\
p^{\hat{\mu}} \equiv\frac{{\rm d}\hat{\mu}}{{\rm d}\lambda'} & = &
M_{\rm sgn}\frac{\hat{u}^2\sqrt{M}}{\tilde{\rho}^2}\, , \\
p^{\hat{\varphi}}\equiv\frac{{\rm d}\hat{\varphi}}{{\rm d}\lambda'} & = &
\label{pphik}
\frac{\hat{u}^2}{\tilde{\rho}^2}\left[\frac{l}{1-\hat{\mu}^2}-a\frac{1+2\hat{u}-(l^2+q^2)\hat{u}^2}
{\hat{u}(2-a\hat{u} l)+U_{\rm sgn}\sqrt{U}}\right]\, ,\hspace*{12mm}
\end{eqnarray}
with $U\equiv 1+(a^2-l^2-q^2)\hat{u}^2+2[(a-l)^2+q^2]\hat{u}^3-a^2q^2\hat{u}^4$ and
$M\equiv q^2+(a^2-l^2-q^2)\hat{\mu}^2-a^2\hat{\mu}^4$. The sign of the $\hat{u}$- and
$\hat{\mu}$-components of the four-momentum is denoted by $U_{\rm sgn}$ and
$M_{\rm sgn}$, respectively. Note that in these coordinates the
four-momentum is not singular on the horizon as opposed to the expressions
in the Boyer-Lindquist coordinates.
\section{Keplerian disc in Kerr space-time}
Matter moves along free stable circular orbits
with a rotational velocity
(see e.g.\ Novikov \& Thorne \citeyear{novikov1973})
\begin{equation}
\omega=\frac{{\rm d} \varphi}{{\rm d}t}=\frac{1}{r^{3/2}+a}\, .
\end{equation}
in a Keplerian disc. The disc resides in the equatorial plane of the
black hole.
Individual components of the four-velocity of the Keplerian disc in the
Boyer-Lindquist coordinates are
\begin{eqnarray}
U^t & = & \frac{r^2+a\sqrt{r}}{r\sqrt{r^2-3r+2a\sqrt{r}}}\, ,\\
U^r & = & 0\, ,\\
U^\theta & = & 0\, ,\\
U^\phi & = & \frac{1}{\sqrt{r(r^2-3r+2a\sqrt{r})}}\, .
\end{eqnarray}
There is no free stable circular orbit below the marginally stable orbit defined by
\begin{eqnarray}
r_{\rm ms} & = & 3+Z_2-\sqrt{(3-Z_1)(3+Z_1+2Z_2)]}
\end{eqnarray}
with $Z_1 = 1+(1-a^2)^{1/3}[(1+a)^{1/3} + (1-a)^{1/3}]$ and
$Z_2 = \sqrt{3a^2+Z_1^2}$.
We suppose that below this orbit the matter is in a free fall down to the
horizon. Thus the matter conserves its specific energy $-U_t$ and its specific
angular momentum $U_\varphi$
\begin{myeqnarray}
-U_t(r<r_{\rm ms}) & \equiv & E_{\rm ms} \ \equiv\ -U_t(r_{\rm ms}) & = &
\frac{r^2_{\rm ms}-2r_{\rm ms}+a\sqrt{r_{\rm ms}}}{r_{\rm ms}
\sqrt{r^2_{\rm ms}-3r_{\rm ms}+2a\sqrt{r_{\rm ms}}}}\, , \\
\ \ U_\varphi(r<r_{\rm ms}) & \equiv & L_{\rm ms}\ \equiv\ \ \ U_\phi(r_{\rm ms}) & = &
\frac{r_{\rm ms}^2+a^2-2a\sqrt{r_{\rm ms}}}
{\sqrt{r_{\rm ms}(r^2_{\rm ms}-3r_{\rm ms}+2a\sqrt{r_{\rm ms}})}}\, .
\end{myeqnarray}
When we also consider the normalization condition for the four-velocity,
$U^\mu\,U_\mu=-1$,
then we get the following expressions for its contravariant components
\begin{eqnarray}
U^t(r<r_{\rm ms}) & = & \frac{1}{r\Delta}\{[r(r^2+a^2)+2a^2]E_{\rm ms}-2aL_{\rm ms}\}\, ,\\
U^r(r<r_{\rm ms}) & = & -\frac{1}{r\sqrt{r}}\sqrt{[r(r^2+a^2)+2a^2]E^2_{\rm ms}-4aE_{\rm ms}
L_{\rm ms}-(r-2)L^2_{\rm ms}-r\Delta}\, ,\hspace*{14mm}\\
U^\theta(r<r_{\rm ms}) & = & 0\, ,\\
U^\varphi(r<r_{\rm ms}) & = & \frac{1}{r\Delta}[2aE_{\rm ms}+(r-2)L_{\rm ms}]\, .
\end{eqnarray}
In our calculations we use the following local orthonormal tetrad connected
with the matter in the disc
\begin{eqnarray}
e_{(t)\,\mu} & \equiv & U_\mu\, ,\\
e_{(r)\,\mu} & \equiv & \frac{r}{\sqrt{\Delta(1+U^rU_r)}}\left[\ U^r( U_t,\,
U_r,\,0,\,U_\varphi)+(0,\, 1,\, 0,\, 0)\ \right]\, ,\\
e_{(\theta)\,\mu} & \equiv & (0,\,0,\,r,\,0)\, ,\\
e_{(\varphi)\,\mu} & \equiv & \sqrt{\frac{\Delta}{1+U^rU_r}}\left (-U^\varphi,\,
0,\,0,\,U^t\right )\, .
\end{eqnarray}
\section{Initial conditions of integration}
\label{initial}
In this part of the Appendix we will sum up the initial conditions for
integration of the equation of the geodesic
\begin{equation}
\frac{{\rm D} p^\mu}{{\rm d} \lambda'} = \frac{{\rm d} p^\mu}{{\rm d} \lambda'} +
\Gamma^\mu_{\sigma\tau}p^\sigma p^\tau\,,\quad\mu=\hat{t},\,\hat{u},\,\hat{\mu},\,\hat{\varphi}
\end{equation}
and equation of the geodesic deviation
\begin{equation}
\frac{{\rm d}^2Y_{\rm j}^\mu}{{\rm d}\lambda'^2}+2\Gamma^\mu_{\sigma\gamma}p^\sigma
\frac{{\rm d} Y_{\rm j}^\gamma}{{\rm d}\lambda'}+\Gamma^\mu_{\sigma\tau,\gamma}p^\sigma
p^\tau Y_{\rm j}^\gamma=0\, ,\quad \mu=\hat{t},\,\hat{u},\,\hat{\mu},\,\hat{\varphi}\, ,
\quad{\rm j}=1,\,2\,
\end{equation}
which we solve in order to compute transfer functions over the accretion disc
(see Chapter~\ref{chapter1} for details).
Here $p^{\mu}\equiv\frac{{\rm d} x^{\mu}}{{\rm d}\lambda'}$ is a four-momentum
of light, $\lambda'$ is an affine parameter for which the conserved energy
$-p_t=1$, $Y_{\rm j}^\mu$ are two vectors characterizing the distance between
nearby geodesics and $\Gamma_{\mu\nu}^\sigma$ are Christoffel symbols for the
Kerr metric. We solve these equations in special Kerr ingoing coordinates
$\hat{t},\,\hat{u},\,\hat{\mu},\,\hat{\varphi}$ and we start integrating from the following initial
point:
\begin{equation}
\hat{t}_{\rm i} = 0\, ,\quad \hat{u}_{\rm i} = 10^{-11}\, ,\quad
\hat{\mu}_{\rm i} = \mu_{\rm o} = \cos\theta_{\rm o}\, ,\quad \hat{\varphi}_{\rm i} = 0\, ,
\end{equation}
where $\theta_{{\rm o}}$ is the inclination angle of the observer at infinity.
The initial values of the four-momentum of light were defined by this point and
eqs.~(\ref{ptk})--(\ref{pphik}) with $U_{\rm sgn}=+1$, $M_{\rm sgn}=0$ for
$\beta=0$ and $M_{\rm sgn}={\rm sign}(\beta)$ for $\beta\neq 0$.
Four-vectors $Y_1^{\mu}$ and $Y_2^{\mu}$ have the initial values
\begin{eqnarray}
\label{Y}
\begin{array}{cclccl}
Y_{1{\rm i}}^{\hat{t}} &=& \beta\hat{u}_{\rm i}\, , &
Y_{2{\rm i}}^{\hat{t}} &=& \alpha\hat{u}_{\rm i}\, ,\\[2mm]
Y_{1{\rm i}}^{\hat{u}} &=& -\beta\hat{u}_{\rm i}^3\, , &
Y_{2{\rm i}}^{\hat{u}} &=& -\alpha\hat{u}_{\rm i}^3\, ,\\[2mm]
Y_{1{\rm i}}^{\hat{\mu}} &=& \hat{u}_{\rm i}\sqrt{1-\hat{\mu}_{\rm i}^2}\, , &
Y_{2{\rm i}}^{\hat{\mu}} &=& -\alpha\hat{u}_{\rm i}^2\hat{\mu}_{\rm i}\, ,\\[2mm]
Y_{1{\rm i}}^{\hat{\varphi}} &=& \alpha\hat{u}_{\rm i}^2\hat{\mu}_{\rm i}/(1-\hat{\mu}_{\rm i}^2)\, ,
\hspace{6em} &
Y_{2{\rm i}}^{\hat{\varphi}} &=& \hat{u}_{\rm i}/\sqrt{1-\hat{\mu}_{\rm i}^2}\, ,
\end{array}
\end{eqnarray}
and their derivatives have the initial values (${\rm d} Y_{\rm j}^{\hat{\nu}}/
{\rm d} \lambda'=p^{\hat{\sigma}}\,\partial Y_{\rm j}^{\hat{\nu}}/\partial
x^{\hat{\sigma}}$)
\begin{eqnarray}
\label{dY}
\begin{array}{cclccl}
\displaystyle\frac{{\rm d} Y_{1{\rm i}}^{\hat{t}}}{{\rm d}\lambda'} &=& \beta\hat{u}_{\rm i}^2\, , &
\displaystyle\frac{{\rm d} Y_{2{\rm i}}^{\hat{t}}}{{\rm d}\lambda'} &=& \alpha\hat{u}_{\rm i}^2\, ,
\\[4mm]
\displaystyle\frac{{\rm d} Y_{1{\rm i}}^{\hat{u}}}{{\rm d}\lambda'} &=& -3\beta\hat{u}_{\rm i}^4\, ,
& \displaystyle\frac{{\rm d} Y_{2{\rm i}}^{\hat{u}}}{{\rm d}\lambda'} &=& -3\alpha\hat{u}_{\rm i}^4
\, ,\\[4mm]
\displaystyle\frac{{\rm d} Y_{1{\rm i}}^{\hat{\mu}}}{{\rm d}\lambda'} &=& \hat{u}_{\rm i}^2\sqrt{1-
\hat{\mu}_{\rm i}^2}\, , &
\displaystyle\frac{{\rm d} Y_{2{\rm i}}^{\hat{\mu}}}{{\rm d}\lambda'} &=&
-2\alpha\hat{u}_{\rm i}^3\hat{\mu}_{\rm i}\, ,\\[4mm]
\displaystyle\frac{{\rm d} Y_{1{\rm i}}^{\hat{\varphi}}}{{\rm d}\lambda'} &=& 2\alpha\hat{u}_{\rm i}^3\hat{\mu}_{\rm i}/(1-\hat{\mu}_{\rm i}^2)\, , \hspace{4.7em} &
\displaystyle\frac{{\rm d} Y_{2{\rm i}}^{\hat{\varphi}}}{{\rm d}\lambda'} &=& \hat{u}_{\rm i}^2/
\sqrt{1-\hat{\mu}_{\rm i}^2}\, .\hspace{0.6em}
\end{array}
\end{eqnarray}
The initial vectors $Y_{1{\rm i}}^{\mu}$ and $Y_{2{\rm i}}^{\mu}$ were chosen in such a way that
in the initial point, where we suppose the metric to be a flat-space Minkowski
one, they are perpendicular to each other and to the four-momentum of light,
are space-like and have unit length. We kept only the largest terms in
$\hat{u}_{\rm i}$ in eqs.~(\ref{Y}) and (\ref{dY}).
\section{Motivation}
There is now plausible evidence that the emission in some
active galactic nuclei and some Galactic X-ray binary black-hole
candidates originates, at least in part, from an accretion
disc in a strong gravitational field. A lively debate is aimed at
addressing the question of what the spectral line profiles and the
associated continuum can tell us about the central black hole, and
whether they can be used to constrain parameters of the accretion disc
in a nearby zone, about ten gravitational radii or less from the
centre. For a recent review of AGNs see \cite{fabian2000,
reynolds2003}, and references cited therein. For BHCs see
\cite{miller2002a,mcclintock2003} and references
therein. In several sources there is indication of iron K$\alpha$ line
emission from within the last stable orbit of a Schwarzschild black hole,
e.g.\ in the Seyfert galaxy MCG--6-30-15 \break \citep[see][]{iwasawa1996,
fabian2000,wilms2001,martocchia2002a} or from the region near above
the marginally stable orbit, as in the case of the X-ray transient source
XTE J1650--500 \citep{miniutti2004} that has been identified with a
Galactic black-hole candidate. In other cases the emission
appears to arise farther from
the black hole (e.g.\ in the microquasar GRS 1915+105, see
\citealt{martocchia2002b}; for AGNs see \citealt{yaqoob2004} and references
therein). Often, the results from X-ray line spectroscopy are
inconclusive, especially in the case of low spectral resolution data.
For example, a spinning black hole is allowed but not required by the
line model of the microquasar V4641 Sgr \citep{miller2002b}. The
debate still remains open, but there are good prospects for future
X-ray astronomy missions to be able to use the iron K$\alpha$ line to probe
the space-time in the vicinity of a black hole, and in particular to
measure the angular momentum, or the spin, associated with the metric.
One may also be able to study the `plunge region'
(about which very little is known), between the event horizon and the
last stable orbit, and to determine if any appreciable contribution to
the iron K$\alpha$ line emission originates from there
\citep{reynolds1997,krolik2002}. In addition to the Fe~K
lines, there is some evidence for relativistic soft X-ray emission
lines due to the Ly$\alpha$ transitions of oxygen, nitrogen, and
carbon \citep{mason2003}, although the observational support
for this interpretation is still controversial \citep{lee2001}.
\begin{table}[tbh]
\begin{center}
\begin{footnotesize}
\dummycaption\label{tab:models}
\begin{tabular}{lcccccc}
\hline
\multicolumn{1}{c}{Model}
& \multicolumn{6}{c}{Effects that are taken into account\rule[-1.5ex]{0mm}{4.5ex}} \\
\cline{2-7}
& \rule[-3.5ex]{0mm}{8ex}\parbox{20mm}{\footnotesize Energy shift/ Lensing effect}
& {\footnotesize $a$}
& \parbox{26mm}{\centering\footnotesize Non-axisymmetric emission region}
& \parbox{20.5mm}{\centering\footnotesize Emission from plunge region}
& {\footnotesize Timing}
& {\footnotesize Polarization} \\
\hline
{\sc{}diskline} \rule{0mm}{3ex} & yes/no~ & $0$~
& no & no & no & no \\
{\sc{}laor} & yes/yes & $0.998$~
& no & no & no & no \\
{\sc{}kerrspec} & yes/yes & $\langle0,1\rangle^{\dag}$
& yes~ & no & no & no \\
{\sc{}ky} & yes/yes & $\langle0,1\rangle$
& yes$^{\ddag}$ & yes & yes & yes \\
\hline
\end{tabular}
\end{footnotesize}
\vspace*{2mm}\par{}
{\parbox{0.9\textwidth}{\footnotesize $^{\dag}$~The value of the dimensionless
$a$ parameter is kept frozen.\\
$^{\ddag}$~A one-dimensional version is available for the case of an
axisymmetric disc. In this axisymmetric mode, {\sc{}ky} still allows $a$ and
other relevant parameters to be fitted (in which case the computational speed of
{\sc{}ky} is then comparable to {\sc{}laor}). The results can be more accurate
than those obtained with other routines because of the ability to tune the grid
resolution.}}
\mycaption{Basic features of the new model in comparison with other
black-hole disc-line models. For references see \cite{fabian1989},
\cite{laor1991}, \cite{martocchia2000} and \cite{dovciak2004b},
respectively.}
\end{center}
\end{table}
With the greatly enhanced spectral resolution and throughput of
future X-ray astronomy missions, the need arises
for realistic theoretical models of the disc emission
and computational tools that are powerful enough to
deal with complex models and to
allow actual fitting of theoretical models to observational
data. It is worth noting that some of the current data
have been used to address the issue of distinguishing
between different space-time metrics around a black hole,
however, the current models available for fitting X-ray data are
subject to various restrictions.
In this chapter and the next one we describe a generalized scheme and a code
which can be
used with the standard X-ray spectral fitting package {\sc{}xspec}
\citep{arnaud1996}. We have in mind general relativity models for black-hole
accretion discs. Apart from a better numerical resolution, the principal
innovations compared to the currently available schemes
(see Tab.~\ref{tab:models}) are that the new
model allows one to (i)~fit for the black-hole spin, (ii)~study the
emission from the plunge region, and (iii)~specify a more general form
of emissivity as a function of the polar coordinates in the disc plane
(both for the line and for the continuum). Furthermore, it is also
possible to (iv)~study time variability of the observed signal and
(v)~compute Stokes parameters of a polarized signal. Items
(i)--(iii) are immediately applicable to current data and modelling,
while the last two mentioned features are still mainly of theoretical
interest at the present time. Time-resolved analysis and polarimetry of
accretion discs are directed towards future applications when the necessary
resolution and the ability to do polarimetry are available in X-rays.
Thus our code has the advantage that it can be used with time-resolved
data for reverberation\break studies of relativistic accretion discs
\citep{stella1990,reynolds1999,ruszkowski2000,goyder2004}.
Also polarimetric analysis can be performed, and this will be extremely
useful because it can add very specific information on
strong-gravitational field effects \citep{connors1980,matt1993,bao1997}.
Theoretical spectra with
temporal and polarimetric information can be analysed with the current
version of our code and such analysis should provide tighter constraints
on future models than is currently possible.
\section{Basic spectral components of X-ray sources}
There are several components in the spectra of X-ray sources. Not all of them
are always present and some of them are more prominent in certain objects than
the others.
\begin{figure}[tbh]
\begin{center}
\dummycaption\label{compton_reflection}
\begin{tabular}{cc}
a) \hspace{5.8cm} & b) \hspace{5.8cm} \\[-3mm]
\includegraphics[height=3.5cm]{lamp_a} \hspace{2mm}
& \includegraphics[height=3.5cm]{reflection}
\end{tabular}
\mycaption{Reflection models: a) lamp-post model; b) diffuse corona model.}
\end{center}
\end{figure}
One of the characteristic spectral features that is almost always present is a
power-law component. It is assumed that this feature results from inverse
Compton scattering of thermal photons in a hot corona above the accretion disc.
Usually two different configurations are considered. The first one assumes a
patch of hot corona placed on the rotational axis of the central black hole
at some height above it (lamp-post model). It is usually supposed to be an
isotropic and
point-like source of stationary primary power-law emission. The second
configuration is a diffuse optically thin hot corona near above the accretion
disc. For simple sketches of both configurations see
Fig.~\ref{compton_reflection}.
In both cases the shape of this primary power-law continuum (in the rest
frame of the corona) is not affected by the relativistic effects acting on
photons during their journey to the observer at infinity. These
effects change only the normalization of the spectra.
This component extends up to a cut-off energy of some tenths or a few hundreds
keV.
Often, mainly in the spectra of active galactic nuclei, additional continuum
emission, a ``hump'', is added to the primary component. It is assumed that
this is due to the reflection of the primary emission from the illuminated disc.
The shape of the reflected continuum in the local rest frame co-moving with
the accretion disc depends mainly on photoelectric absorption and Compton
scattering of photons hitting the disc. The local emission is then smeared by
relativistic effects -- this
concerns mainly its sharp features (e.g.\ iron edge). The shape of the observed
spectra also depends on the illumination of the disc. This differs for the
lamp-post model and the diffuse corona model. In the former case the radial
dependence of illumination is determined by the height at which the patch of
corona is placed. In the latter case the illumination of the disc depends on
the emissivity of the diffuse corona near above the disc which may have quite
a complicated radial dependence (but usually is assumed to be decreasing with
radius as a power law).
Spectral lines are an important feature observed in X-ray spectra. We
assume that the origin of lines is the same as in the previous case --
the illumination of the cold
disc by primary (power-law) emission and reflection, in this case, by
fluorescence. Originally
narrow spectral lines are blurred by relativistic effects and thus they become
broad, their width being as large as several keV in several sources. The most
prominent examples are the iron lines K$\alpha$ and K$\beta$.
\begin{figure}[tbh]
\begin{center}
\dummycaption\label{denomination}
\begin{tabular}{lll}
a) \hspace{4.3cm} & b) \hspace{4.3cm} & c) \hspace{4.3cm} \\%[-2mm]
\hspace*{-1.65mm}\includegraphics[height=3cm]{denomination1} \hspace{2mm}
& \includegraphics[height=3cm]{denomination2a} \hspace{2mm}
& \includegraphics[height=3cm]{denomination3}
\end{tabular}
\mycaption{Denomination of various elements of solid angles and areas
defined in the text: a) the light source appears to the observer to
be point-like;
b) the light rays received by the detector are coming from different parts
of the disc (closer view of the disc than in previous figure);
c) area of a light tube changes as the light rays travel close to the black
hole (the disc is edge on).}
\end{center}
\end{figure}
In the present thesis we are concerned with the spectral components described
above but here, we should not omit to mention other two important components --
the thermal emission and the warm absorber.
The thermal emission is more noticeable in the spectra of X-ray binaries with
the black-hole candidate as the central object. This is due to the fact that in
these sources the temperature of the accretion disc is much higher and therefore
the black body emission extends as far as the soft X-ray energy band. This type
of emission manifests itself in the spectra of active galactic nuclei as a
``big blue bump'' in the optical and UV frequencies.
If there is ionized matter, a warm absorber, in the line of sight of the
observer then the absorption spectral features are present in the observed
spectra. These can include edges or resonant absorption lines due to oxygen and
other ions.
\section{Photon flux from an accretion disc}
Properties of radiation are described in terms of photon numbers. The
source appears as a point-like object for a distant observer, so that
the observer measures the flux entering the solid angle ${\rm d}
\Omega_{\rm o}$, which is associated with the detector area
${\rm d}S_{\rm o}{\equiv}D^2\,{\rm d}\Omega_{\rm o}$
(see Fig.~\ref{denomination}a). This relation
defines the distance $D$ between the observer and the source.
We denote the total photon flux received by a detector,
\begin{equation}
\label{flux}
N^{S}_{\rm{}o}(E)\equiv\frac{{\rm d}n(E)}{{\rm d}t\,{\rm d}S_{\rm o}}
={\int}{\rm d}\Omega\,N_{{\rm l}}(E/g)\,g^2\,,
\end{equation}
where
\begin{equation}
N_{{\rm l}}(E_{{\rm l}})\equiv
\frac{{\rm d}n_{{\rm l}}(E_{{\rm l}})}{{\rm d}\tau\,{\rm d}S_{{\rm l}}\,
{\rm d}\Omega_{{\rm l}}}
\end{equation}
is a local photon flux emitted from the surface of the disc, ${\rm
d}n(E)$ is the number of photons with energy in the interval
${\langle}E,E+{\rm d}E\,\rangle$ and $g=E/E_{{\rm l}}$ is the redshift
factor. The local flux, $N_{\rm l}(E_{\rm l})$, may vary over the disc as well
as in time, and it can also depend on the local emission angle. This
dependency is emphasized explicitly only in the final formula
(\ref{emission}), otherwise it is omitted for brevity.
The emission arriving at the detector within the solid angle ${{\rm d}\Omega}$
(see Fig.~\ref{denomination}b) originates from the proper area ${\rm
d}S_{{\rm l}}$ on the disc (as measured in the rest frame co-moving with
the disc). Hence, in our computations we want to integrate the flux
contributions over a fine mesh on the disc surface. To achieve this aim, we
adjust eq.~(\ref{flux}) to the form
\begin{equation}
\label{N_o^S}
N^S_{\rm o}(E)=\frac{1}{D^2}\int {\rm d}S\,\frac{D^2{\rm d}\Omega}{{\rm d}S}\,
N_{{\rm l}}(E/g)\,g^2=\frac{1}{D^2}\int {\rm d}S\,
\frac{{\rm d}S_{\perp}}{{\rm d}S}\,\frac{{\rm d}S_{\rm f}}{{\rm d}S_{\perp}}\,
N_{{\rm l}}(E/g)\,g^2\, .
\end{equation}
Here ${{\rm{}d}S_{\rm f}}$ stands for an element of area perpendicular to light
rays corresponding to the solid angle ${\rm d}\Omega$ at a distance $D$,
${{\rm{}d}S_{\perp}}$ is the proper area measured in the local frame
of the disc and perpendicular to the
rays, and ${\rm d}S$ is the coordinate area for integration.
We integrate in a two-dimensional slice of a four-dimensional space-time, which
is specified by coordinates $\theta=\pi/2$ and $t=t_{\rm o}-\Delta t$ with
$\Delta t$ being a time delay with which photons from different parts of the
disc (that lies in the equatorial plane) arrive to the observer (at the same
coordinate time $t_{\rm o}$). Therefore, let us define the coordinate area by
(we employ coordinates $t',\,\theta,\,r,\,\varphi$
with $t'=t-\Delta t$ and $\Delta t=\Delta t(r,\theta,\varphi)$)
\begin{equation}
\label{dS}
{\rm d}S\equiv|{\rm d}^2{\!S_{t'}}^{\theta}|=\left|\frac{\partial x^\mu}
{\partial t'}{\rm d}^2{\!S_{\mu}}^{\theta}\right|=|{\rm d}^2{\!S_t}^{\theta}|=
|g^{\theta\mu}{\rm d}^2\!S_{t\mu}|\, .
\end{equation}
We define the tensor ${\rm d}^2\!S_{\alpha\beta}$ by two four-vector elements
${\rm d}x_1^\mu\equiv({\rm d}t_1,{\rm d}r,0,0)$ and
${\rm d}x_2^\mu\equiv({\rm d}t_2,0,0,{\rm d}\varphi)$ and by
Levi-Civita tensor
$\varepsilon_{\alpha\beta\gamma\delta}$. The time components of these vectors,
${\rm d}t_1$ and ${\rm d}t_2$, are such that the vectors ${\rm d}x_1^\mu$ and
${\rm d}x_2^\mu$ lie in the tangent
space to the above defined space-time slice. Then we obtain
\begin{equation}
\label{dS2}
{\rm d}S=|g^{\theta\theta}\varepsilon_{t\theta\alpha\beta}\,{\rm d}x_1^{[\alpha}
{\rm d}x_2^{\beta]}|=g^{\theta\theta}\sqrt{-\|g_{\mu\nu}\|}\,{\rm d}r\,{\rm
d}\varphi={\rm d}r\,{\rm d}\varphi\, ,
\end{equation}
where $g_{\mu\nu}$ is the metric tensor and
$\|g_{\mu\nu}\|$
is the determinant of the metric. The proper area, ${\rm d}S_\perp$,
perpendicular to the light ray can be expressed covariantly in the following
way:
\begin{equation}
\label{dSperp}
{\rm d}S_\perp=-\frac{U^{\alpha}\,p^{\beta}\,{\rm d}^2\!S_{\alpha\beta}}
{U^{\mu}\,p_\mu}\, .
\end{equation}
Here, ${\rm d}S_\perp$ is the projection of an element of area,
defined by ${\rm d}^2\!S_{\alpha\beta}$, on a spatial slice of an observer
with velocity $U^\alpha$ and perpendicular to light rays.
$U^\alpha$ is four-velocity of an observer measuring the
area ${\rm d}S_\perp$, and $p^\beta$ is four-momentum of the photon.
The\break proper area ${\rm d}S_\perp$ corresponding to the same flux tube
is identical for all observers\break (see \citealt{schneider1992}). This means that the
last equation holds
true for any four-velocity $U^\alpha$, and we can express it as
\begin{equation}
\label{dS3}
p^{\beta}\,{\rm d}^2\!S_{\alpha\beta}+p_\alpha\,{\rm d}S_\perp = 0\, ,
\quad\alpha = t,\,r,\,\theta,\,\varphi\, .
\end{equation}
For $\alpha=t$ (note that
${\rm d}^2\!S_{tr}={\rm d}^2\!S_{t\varphi}=0$) we get
\begin{equation}
\label{ratio_dS}
\frac{{\rm d}S_\perp}{{\rm d}S}=\left|\frac{1}{g^{\theta\theta}}
\frac{{\rm d}S_\perp}{{\rm d}^2\!S_{t\theta}}\right| =
\left|-\frac{p_\theta}{p_t}\right|=\frac{r\mu_{\rm e}}{g}\, .
\end{equation}
In the last equation we used the formula for the cosine of local emission
angle $\mu_{\rm e}$, see eq.~(\ref{cosine}), and the fact that we have chosen
such an affine parameter of the light geodesic that $p_t=-1$.
From eqs.~(\ref{N_o^S}), (\ref{dS}) and (\ref{ratio_dS}) we get for
the observed flux per unit solid angle
\begin{equation}
\label{emission1}
N^{\Omega}_{\rm o}(E)\equiv\frac{{\rm d}n(E)}{{\rm d}t\,{\rm d}\Omega_{\rm o}}=
{N_0}\int_{r_{\rm in}}^{r_{\rm out}}{\rm d}r\,\int_{\phi}^{\phi+{\Delta\phi}}
{\rm d}\varphi\,N_{{\rm l}}(E/g)\,g\,l\,\mu_{\rm e}\,r,
\end{equation}
where $N_0$ is a normalization constant and
\begin{equation}
l=\frac{{\rm d}S_{\rm f}}{{\rm d}S_\perp}
\end{equation}
is the lensing factor in the limit $D\rightarrow\infty$
(keeping $D^2{\rm d}\Omega$ constant, see Fig.~\ref{denomination}c).
For the line emission, the normalization constant $N_0$ is chosen in
such a way that the total flux from the disc is unity. In the case of a
continuum model, the flux is normalized to unity at a certain value of the
observed energy (typically at $E=1$~keV, as in other {\sc{}xspec}
models).
Finally, the integrated flux per energy bin, $\Delta E$, is
\begin{eqnarray}
\label{emission}
\nonumber
& & \hspace*{-3em} {\Delta}N^{\Omega}_{\rm o}(E,\Delta E,t) =
\int_{E}^{E+\Delta E}{\rm d}\bar{E}\,N^{\Omega}_{\rm o}(\bar{E},t)=\\
& & = N_0\int_{r_{\rm in}}^{r_{\rm out}}{\rm d}r\,
\int_{\phi}^{\phi+{\Delta\phi}}{\rm d}\varphi\,\int_{E/g}^{(E+\Delta E)/g}
{\rm d}E_{{\rm l}}\,N_{{\rm l}}(E_{{\rm l}},r,\varphi,\mu_{\rm e},t-\Delta t)\,g^2\,l\,
\mu_{\rm e}\,r\, ,\hspace*{2em}
\end{eqnarray}
where $\Delta t$ is the relative time delay with which photons arrive to the
observer from different parts of the disc. The transfer functions
$g,\,l,\,\mu_{\rm e}$ and $\Delta t$ are read from the FITS file {\tt
KBHtablesNN.fits} described in Appendix~\ref{appendix3a}. This equation
is numerically integrated for a given local flux
$N_{\rm l}(E_{{\rm l}},r,\varphi,\mu_{\rm e},t-\Delta t)$ in all hereby
described new general relativistic {\it non-axisymmetric models}.
Let us assume that the local emission is stationary and the dependence on the
axial coordinate is only through the prescribed dependence on the local emission
angle $f(\mu_{\rm e})$ (limb darkening/brightening law) together with an
arbitrary radial dependence $R(r)$, i.e.
\begin{equation}
N_{\rm l}(E_{{\rm l}},r,\varphi,\mu_{\rm e},t-\Delta t)\equiv N_{\rm l}(E_{\rm l})\,R(r)\,
f(\mu_{\rm e}).
\end{equation}
The observed flux $N_{\rm o}^{\Omega}(E)$
is in this case given by
\begin{equation}
N_{\rm o}^{\Omega}(E)=\int_{-\infty}^{\infty}{\rm
d}E_{\rm l}\,N_{\rm l}(E_{\rm l})\,G(E,E_{\rm l}),
\end{equation}
where
\begin{equation}
G(E,E_{\rm l})=N_0\int_{r_{\rm in}}^{r_{\rm out}}{\rm
d}r\,R(r)\int_{0}^{2\pi}{\rm d}\varphi\,f(\mu_{\rm e})\,g^2\,l\,\mu_{\rm
e}\,r\,\delta(E-gE_{\rm l}).
\end{equation}
In this case, the integrated flux can be expressed in the following way:
\begin{eqnarray}
\nonumber
{\Delta}N^{\Omega}_{\rm o}(E,\Delta E) & = &
\int_{E}^{E+\Delta E}{\rm d}\bar{E}\,N^{\Omega}_{\rm o}(\bar{E})\quad
=\hspace{19.5em}\\
\nonumber
& & \hspace*{-8.5em} = \int_{E}^{E+\Delta E}{\rm d}\bar{E}\,
{N_0}\int_{r_{\rm in}}^{r_{\rm out}}{\rm d}r\,R(r)\,\int_{0}^{2\pi}
{\rm d}\varphi\,f(\mu_{\rm e})\,N_{{\rm l}}(\bar{E}/g)\,g\,l\,\mu_{\rm e}\,r\,
\int_{-\infty}^{\infty}{\rm d}E_{\rm l}\,\delta(E_{\rm l}-\bar{E}/g)=\\
\label{axisym_emission}
& & \hspace*{-8.5em} = N_0\int_{r_{\rm in}}^{r_{\rm out}}{\rm d}r\,R(r)\,
\int_{-\infty}^{\infty}{\rm d}E_{{\rm l}}\,N_{{\rm l}}(E_{{\rm l}})
\int_{E/E_{{\rm l}}}^{(E+\Delta E)/E_{{\rm l}}}{\rm d}\bar{g}\,F(\bar{g})\, ,
\end{eqnarray}
where we substituted $\bar{g}=\bar{E}/E_{\rm l}$ and
\begin{equation}
\label{conv_function}
F(\bar{g}) = \int_{0}^{2\pi}{\rm d}\varphi\,f(\mu_{\rm e})\,g^2\,l\,
\mu_{\rm e}\,r\,\delta(\bar{g}-g)\, .
\end{equation}
Eq.~(\ref{axisym_emission}) is numerically integrated in all
{\it axially symmetric models}. The function
${\rm d}F(\bar{g})\equiv {\rm d}\bar{g}\,F(\bar{g})$ has been
pre-calculated for several limb darkening/brightening laws
$f(\mu_{\rm e})$ and stored in separate files,
{\tt KBHlineNN.fits} (see Appendix~\ref{appendix3b}).
\section{Stokes parameters in a strong gravity regime}
\label{stokes_param}
For polarization studies, Stokes parameters are used. Let us define specific
Stokes parameters in the following way:
\begin{equation}
i_\nu\equiv \frac{I_{\nu}}{E}\, ,\quad q_\nu\equiv \frac{Q_{\nu}}{E}\, ,\quad
u_\nu\equiv \frac{U_{\nu}}{E}\, ,\quad v_\nu\equiv \frac{V_{\nu}}{E}\, ,
\end{equation}
where $I_{\nu}$, $Q_{\nu}$, $U_{\nu}$ and $V_{\nu}$ are Stokes
parameters for light with frequency $\nu$, $E$ is the energy of a photon at
this frequency. Further on, we drop the index $\nu$ but we will always
consider these quantities for light of a given frequency. We can calculate
the integrated specific Stokes parameters (per energy bin), i.e.\ $\Delta
i_{\rm o}$, $\Delta q_{\rm o}$, $\Delta u_{\rm o}$ and $\Delta v_{\rm
o}$. These are the quantities that the observer determines from the local
specific Stokes parameters $i_{\rm l}$, $q_{\rm l}$, $u_{\rm l}$ and $v_{\rm l}$ on the disc
in the following way:
\begin{eqnarray}
\label{S1}
{\Delta}i_{\rm o}(E,\Delta E) & = & N_0\int{\rm d}S\,\int{\rm d}E_{{\rm l}}\,
i_{{\rm l}}(E_{{\rm l}})\,Fr\, ,\\
\label{S2}
{\Delta}q_{\rm o}(E,\Delta E) & = & N_0\int{\rm d}S\,\int{\rm d}E_{{\rm l}}\,
[q_{{\rm l}}(E_{{\rm l}})\cos{2\Psi}-u_{{\rm l}}(E_{{\rm l}})\sin{2\Psi}]\,Fr\, ,\\
\label{S3}
{\Delta}u_{\rm o}(E,\Delta E) & = & N_0\int{\rm d}S\,\int{\rm d}E_{{\rm l}}\,
[q_{{\rm l}}(E_{{\rm l}})\sin{2\Psi}+u_{{\rm l}}(E_{{\rm l}})\cos{2\Psi}]\,Fr\, ,\\
\label{S4}
{\Delta}v_{\rm o}(E,\Delta E) & = & N_0\int{\rm d}S\,\int{\rm d}E_{{\rm l}}\,
v_{{\rm l}}(E_{{\rm l}})\,Fr\, .
\end{eqnarray}
Here, $F\equiv F(r,\varphi)=g^2\,l\,\mu_{\rm e}$ is a transfer
function, $\Psi$ is the angle by which a vector parallelly transported
along the light geodesic rotates. We refer to this angle also as a
change of the polarization angle, because the polarization vector is parallelly
transported along light geodesics. See Fig.~\ref{pol_angle} for an exact
definition of the angle $\Psi$. The integration boundaries are the
same as in eq.~(\ref{emission}). As can be seen from the
definition, the first specific Stokes parameter is equal to the photon
flux, therefore, eqs.~(\ref{emission}) and (\ref{S1}) are
identical. The local specific Stokes parameters may depend on $r$,
$\varphi$, $\mu_{\rm e}$ and $t-\Delta t$, which we did not state in the
eqs.~(\ref{S1})--(\ref{S4}) explicitly for simplicity.
The specific Stokes parameters that the observer measures may vary in time
in the case when the local parameters also depend on time. In eqs.\
(\ref{S1})--(\ref{S4}) we used a law of transformation of the Stokes
parameters by the rotation of axes (eqs.~(I.185) and (I.186) in
\citealt{chandrasekhar1960}).
An alternative way for expressing polarization of light is by using the
degree of polarization $P_{\rm o}$ and two polarization angles
$\chi_{\rm o}$ and $\xi_{\rm o}$, defined by
\begin{eqnarray}
P_{\rm o} & = & \sqrt{q_{\rm o}^2+u_{\rm o}^2+v_{\rm o}^2}/i_{\rm o}\, , \\
\tan{2\chi_{\rm o}} & = & u_{\rm o}/q_{\rm o}\, ,\\
\sin{2\xi_{\rm o}} & = & v_{\rm o}/\sqrt{q_{\rm o}^2+u_{\rm o}^2
+v_{\rm o}^2}\, .
\end{eqnarray}
\section{Local emission in lamp-post models}
\label{lamp-post}
The local emission from a disc is proportional to the incident illumination from
a power-law primary source placed on the axis at height $h$ above the black hole.
To calculate the incident illumination we need to integrate the geodesics from
the source to the disc.
The four-momentum of the incident photons which were emitted by a primary source
and which are striking the disc at radius $r$ is
(see eqs.~(\ref{carter1})--(\ref{carter4}) with $l=0$ and $\theta=\pi/2$
or also \citealt{carter1968} and \citealt{misner1973})
\begin{eqnarray}
p_{\rm i}^t & = & 1+2/r+4/\Delta\, ,\\
p_{\rm i}^r & = & {\rm R}^{\prime}_{\rm sgn}[(r^2+a^2)^2-\Delta
(a^2+q_{\rm L}^2)]^{1/2}/r^2\, ,\\
p_{\rm i}^\theta & = & q_{\rm L}/r^2\, ,\\
p_{\rm i}^\varphi & = & 2a/(r\Delta)\, ,
\end{eqnarray}
where
$q_{\rm L}^2 = \sin^2{\!\theta_{\rm L}}\,(h^2+a^2)^2/\Delta_{\rm L}-a^2$ is
Carter's constant of motion with $\Delta_{\rm L}=h^2-2h+a^2$, and with the angle
of emission $\theta_{\rm L}$ being the local angle
under which the photon is emitted from a primary source (it is measured in the rest
frame of the source). We define this angle by
$\tan{\theta_{\rm L}}=-{p_{\rm L}^{(\theta)}/p_{\rm L}^{(r)}}$, where
$p_{\rm L}^{(r)}=p_{\rm L}^\mu\,e_{{\rm L}\,\mu}^{(r)}$ and
$p_{\rm L}^{(\theta)}=p_{\rm L}^\mu\,e_{{\rm L}\,\mu}^{(\theta)}$ with
$p_{\rm L}^\mu$ and $e_{{\rm L}\,\mu}^{(a)}$ being the four-momentum of emitted
photons and the local tetrad connected with a primary source, respectively.
The angle is $0^\circ$ when the photon is emitted downwards and $180^\circ$ if
the photon is emitted upwards.
We denoted the sign of the radial component of the momentum by
${\rm R}^{\prime}_{\rm sgn}$. We have chosen such an affine parameter for the light
geodesic that the conserved energy of the light is
$-p_{{\rm i}\,t}=-p_{{\rm L}\,t}=1$. The conserved angular momentum of incident
photons is zero ($l_{\rm L}=0$).
The gravitational and Doppler shift of the photons striking the disc which were
emitted by a primary source is
\begin{equation}
\label{gfac_lamp}
g_{\rm L}= \frac{\nu_{\rm i}}{\nu_{\rm L}}=
\frac{p_{{\rm i}\,\mu} U^\mu}{p_{{\rm L}\,\alpha} U_{\rm L}^\alpha}=
-\frac{p_{{\rm i}\,\mu} U^\mu}{U_{\rm L}^t}\, .
\end{equation}
Here $\nu_{\rm i}$ and $\nu_{\rm L}$ denote the frequency of the incident and
emitted photons, respectively and $U_{\rm L}^\alpha$ is a four-velocity of the
primary source with the only
non-zero component \hbox{$U_{\rm L}^t=\sqrt{\frac{h^2+a^2}{\Delta_{\rm L}}}$}.
Cosine of the local incident angle is
\begin{equation}
\label{cosine_inc}
\mu_{\rm i}=|\cos{\delta_{\rm i}}\,|=
\displaystyle\frac{{p_{{\rm i}\,\alpha}\,n^{\alpha}}}
{{p_{{\rm i}\,\mu}\,U^{\mu}}}\, ,
\end{equation}
where $n^\alpha=-e_{(\theta)}^\alpha$ is normal to the disc with respect to the
observer co-moving with the matter in the disc.
We further define the azimuthal incident angle as the angle between the
projection of the three-momentum of the incident photon into the disc (in the
local rest frame co-moving with the disc) and the radial tetrad vector,
\begin{equation}
\label{azim_angle_inc}
\Phi_{\rm i}=-{\rm R}_{\rm sgn}^{\rm i}\arccos\left(
\frac{-1}{\sqrt{1-\mu_{\rm i}^2}}\frac{\,{p_{{\rm i}\,\alpha}}
\,e_{(\varphi)}^{\alpha}}
{p_{{\rm i}\,\mu}U^\mu}\right)+\frac{\pi}{2}\, ,
\end{equation}
where ${\rm R}_{\rm sgn}^{\rm i}$ is positive if the incident photon travels
outwards ($p_{\rm i}^{(r)}>0$) and negative if it travels inwards
($p_{\rm i}^{(r)}<0$) in the local rest frame of the disc.
In lamp-post models the emission of the disc will be proportional to the
incident radiation $N_{\rm i}^{S}(E_{\rm l})$ which comes from a primary source
\begin{equation}
\label{illumination}
N_{\rm i}^{S}(E_{\rm l})=N_{\rm L}^{\Omega}(E_{\rm L})\frac{{\rm d}
\Omega_{\rm L}}{{\rm d}S_{{\rm l}}}\, .
\end{equation}
Here $N_{\rm L}^{\Omega}(E_{\rm L})=N_{0 {\rm L}}\,E_{\rm L}^{-\Gamma}$ is
an isotropic and stationary power-law emission from a primary source which is
emitted into
a solid angle ${\rm d}\Omega_{\rm L}$ and which illuminates local area
${\rm d}S_{\rm l}$ on the disc. The energy of the photon striking the disc
(measured in the local frame co-moving with the disc) will be redshifted
\begin{equation}
E_{\rm l}=g_{\rm L}\,E_{\rm L}\, .
\end{equation}
The ratio ${\rm d}\Omega_{\rm L}/{\rm d}S_{{\rm l}}$ is
\begin{equation}
\frac{{\rm d}\Omega_{\rm L}}{{\rm d}S_{{\rm l}}}=\frac{{\rm d}\Omega_{\rm L}}
{{\rm d}S} \frac{{\rm d}S}{{\rm d}S_{\rm l}}=\frac{\sin{\theta_{\rm L}{\rm d}
\theta_{\rm L}\,{\rm d}\varphi}}{{\rm d}r\,{\rm d}\varphi}\frac{{\rm d}S}
{{\rm d}S_{\rm l}}\, ,
\end{equation}
where (see eqs.~(\ref{dS2}) and (\ref{dS3}))
\begin{equation}
{\rm d}S={\rm d}r\,{\rm d}\varphi=|{\rm d}^2{\!S_t}^{\theta}|=
-g^{\theta\theta}\,
\frac{p_{{\rm i} t}}{p_{\rm i}^\theta}\,{\rm d}S_\perp=
\,\frac{g^{\theta\theta}}{p_{\rm i}^\theta}\,{\rm d}S_\perp\, .
\end{equation}
Here we used
the same space-time slice as in the discussion above eq.~(\ref{dS}) and thus
the element
${\rm d}^2\!S_{\alpha\beta}$ is defined as before, see eq.~(\ref{dS2}). Note
that here the area ${\rm d}S_\perp$ is defined by the incident flux tube as
opposed to ${\rm d}S_\perp$ in eq.~(\ref{ratio_dS}) where it was defined by
the emitted flux tube.
The coordinate area ${\rm d}S$ corresponds to the proper area ${\rm d}S_\perp$
which is perpendicular to the incident light ray (in the local rest frame
co-moving with the disc). The corresponding proper area (measured in the same local
frame) lying in the equatorial plane will be
\begin{eqnarray}
\nonumber
{\rm d}S_{{\rm l}} & = &
|{\rm d}^2{\!S_{(t)}}^{\!\!(\theta)}|=|e_{(t)}^\mu\,e^{{(\theta)}\,\nu}\,
{\rm d}^2\!S_{\mu\nu}|=|g_{\theta\theta}^{-1/2}\,U^{\mu}\,
{\rm d}^2\!S_{\mu\theta}|=\\
& = & -g_{\theta\theta}^{-1/2}\,
\frac{p_{{\rm i} \mu}\,U^\mu}{p_{\rm i}^\theta}\,{\rm d}S_\perp =
g_{\theta\theta}^{-1/2}\frac{U_{\rm L}^t}{p_{\rm i}^\theta}\,g_{\rm L}\,
{\rm d}S_\perp\, .
\label{dSl}
\end{eqnarray}
Here we have used eq.~(\ref{dS}) for the tetrad components of the element
${\rm d}^2\!S_{\alpha\beta}$, eqs.~(\ref{dSperp}) and (\ref{gfac_lamp}).
It follows from eqs.~(\ref{illumination})--(\ref{dSl}) that the incident
radiation will be again a power law with the same photon index
$\Gamma$ as in primary emission
\begin{equation}
N_{\rm i}^{S}(E_{\rm l})=N_{0 {\rm i}}\,E_{\rm l}^{-\Gamma}\, ,
\end{equation}
with the normalization factor
\begin{equation}
N_{0 {\rm i}}=N_{0 {\rm L}}\,g_{\rm L}^{\Gamma-1}\,
\sqrt{1-\frac{2h}{h^2+a^2}}\,\frac{\sin{\theta_{\rm L}}\,
{\rm d}\theta_{\rm L}}{r\,{\rm d}r}\, .
\end{equation}
The emission of the disc due to illumination will be proportional to
this factor.
\chapter{New models for {\fontfamily{phv}\fontshape{sc}\selectfont xspec}}
\label{chapter3}
\thispagestyle{empty}
We have developed several general relativistic models for line emission
and Compton reflection continuum. The line models are supposed to be
more accurate and versatile than the {\sc laor} model \citep{laor1991}, and
substantially faster than the {\sc kerrspec} model \citep{martocchia2000}.
Several models of intrinsic emissivity were employed, including the lamp-post
model \citep{matt1992}. Among other features,
these models allow various parameters to be fitted such as the black-hole
angular momentum, observer inclination, accretion disc size and some of the
parameters characterizing disc emissivity and primary illumination
properties. They also
allow a change in the grid resolution and, hence, to control accuracy and
computational speed. Furthermore, we developed very general
convolution models. All these models are based on pre-calculated tables
described in Chapter~\ref{transfer_functions} and thus the geodesics
do not need to be calculated each time one integrates the disc emission.
These tables are calculated for the vacuum Kerr space-time and for a Keplerian
co-rotating disc plus matter that is freely falling below the marginally
stable orbit. The falling matter has the energy and angular momentum of
the matter at the marginally stable orbit. It is possible to use
different pre-calculated tables if they are stored in a specific FITS
file (see Appendix~\ref{appendix3a} for its detailed description).
There are two types of new models. The first type of model integrates
the local disc emission in both of the polar coordinates on the disc and thus
enables one to choose non-axisymmetric area of integration (emission
from spots or partially obscured discs). One can also choose the
resolution of integration and thus control the precision and speed of
the computation. The second type of model is axisymmetric -- the
axially dependent part of the emission from rings is pre-calculated and
stored in a FITS file (the function ${\rm d}F(\bar{g})={\rm
d}\bar{g}\,F(\bar{g})$ from eq.~(\ref{conv_function}) is integrated for
different radii with the angular grid having $20\,000$ points). These
models have less
parameters that can be fitted and thus are less flexible even though more
suited to the standard analysis approach. On the other
hand they are fast because the emission is integrated only in one
dimension (in the radial coordinate of the disc). It may be worth emphasizing
that the assumption about axial symmetry concerns only the form of intrinsic
emissivity of the disc, which cannot depend on the polar angle in this case, not
the shape of individual light rays, which is always
complicated near a rotating black hole.
\begin{table}[tbh]
\begin{center}
\dummycaption\label{common_par1}
\begin{tabular}[t]{l|c|c|c|c}
parameter & unit & default value & minimum value & maximum value \\ \hline
\hspace*{0.5em}{\tt a/M} & $GM/c$ & 0.9982 & 0. & 1. \\
\hspace*{0.5em}{\tt theta\_o} & deg & 30. & 0. & 89. \\
\hspace*{0.5em}{\tt rin-rh} & $GM/c^2$ & 0. & 0. & 999. \\
\hspace*{0.5em}{\tt ms} & -- & 1. & 0. & 1. \\
\hspace*{0.5em}{\tt rout-rh} & $GM/c^2$ & 400. & 0. & 999. \\
\hspace*{0.5em}{\tt zshift} & -- & 0. & -0.999. & 10. \\
\hspace*{0.5em}{\tt ntable} & -- & 0. & 0. & 99.
\end{tabular}
\mycaption{Common parameters for all models.}
\end{center}
\end{table}
\begin{table}[tbh]
\begin{center}
\dummycaption\label{common_par2}
\begin{tabular}[h]{l|c|c|c|c}
parameter & unit & default value & minimum value & maximum value \\ \hline
\hspace*{0.5em}{\tt phi} & deg & 0. & -180. & 180. \\
\hspace*{0.5em}{\tt dphi} & deg & 360. & 0. & 360. \\
\hspace*{0.5em}{\tt nrad} & -- & 200. & 1. & 10000. \\
\hspace*{0.5em}{\tt division} & -- & 1. & 0. & 1. \\
\hspace*{0.5em}{\tt nphi} & -- & 180. & 1. & 20000. \\
\hspace*{0.5em}{\tt smooth} & -- & 1. & 0. & 1. \\
\hspace*{0.5em}{\tt Stokes} & -- & 0. & 0. & 6.
\end{tabular}
\mycaption{Common parameters for non-axisymmetric models.}
\end{center}
\end{table}
There are several parameters and switches that are common for all new models
(see Tab.~\ref{common_par1}):
\begin{description} \itemsep -2pt
\item[{\tt a/M}] -- specific angular momentum of the Kerr black hole in units
of $GM/c$ ($M$ is the mass of the central black hole),
\item [{\tt theta\_o}] -- inclination of the observer in degrees,
\item [{\tt rin-rh}] -- inner radius of the disc relative to the black-hole
horizon in units of $GM/c^2$,
\item [{\tt ms}] -- switch for the marginally stable orbit,
\item [{\tt rout-rh}] -- outer radius of the disc relative to the black-hole
horizon in units of $GM/c^2$,
\item [{\tt zshift}] -- overall redshift of the object,
\item [{\tt ntable}] -- number of the FITS file with pre-calculated tables to
be used.
\end{description}
The inner and outer radii are given relative to the black-hole horizon and,
therefore, their minimum value is zero. This becomes handy when one fits the
{\tt a/M} parameter, because the horizon of the black hole as well as the
marginally stable orbit change with {\tt a/M}, and so the lower limit for
inner and outer disc edges
cannot be set to constant values. The {\tt ms} switch determines whether
we also want to integrate emission below the marginally stable orbit.
If its value is set to zero and the inner radius of the disc is below this
orbit then the emission below the marginally stable orbit is taken
into account, otherwise it is not.
The {\tt ntable} switch determines which of the pre-calculated tables
should be used for intrinsic emissivity.
In particular, ${\tt ntable}=0$ for {\tt KBHtables00.fits}
({\tt KBHline00.fits}), ${\tt ntable}=1$ for {\tt KBHtables01.fits}
({\tt KBHline01.fits}), etc., corresponding to non-axisym\-met\-ric
(axisymmetric) models.
\begin{figure}[tbh]
\begin{center}
\dummycaption\label{sector}
\includegraphics[width=5cm]{sector}
\mycaption{Segment of a disc from which emission comes (view from above).}
\end{center}
\end{figure}
\begin{table}[tbh]
\begin{center}
\dummycaption\label{stokes}
\begin{tabular*}{11.5cm}{c|l}
value & photon flux array {\tt photar} contains$^{\dag}{}^{\ddag}$\\ \hline
0 & $i=I/E$, where $I$ is the first Stokes parameter (intensity) \\
1 & $q=Q/E$, where $Q$ is the second Stokes parameter \\
2 & $u=U/E$, where $U$ is the third Stokes parameter \\
3 & $v=V/E$, where $V$ is the fourth Stokes parameter \\
4 & degree of polarization, $P=\sqrt{q^2+u^2+v^2}/i$ \\
5 & angle $\chi$[deg] of polarization, $\tan{2\chi}=u/q$\\
6 & angle $\xi$[deg] of polarization, $\sin{2\xi}=v/\sqrt{q^2+u^2+v^2}$\\ \hline
\multicolumn{2}{l}{\parbox{11.cm}{\vspace*{1mm}\footnotesize
$^{\dag}$~the {\tt photar} array contains the values described in the table
and multiplied by the width of the corresponding energy bin\\
$^{\ddag}~E$ is the energy of the observed photons}}\\
\end{tabular*}
\mycaption{Definition of the {\tt Stokes} parameter.}
\end{center}
\end{table}
The following set of parameters is relevant only for non-axisymmetric models
(see Tab.~\ref{common_par2}):
\begin{description} \itemsep -2pt
\item[{\tt phi}] -- position angle of the axial sector of the disc in degrees,
\item[{\tt dphi}] -- inner angle of the axial sector of the disc in degrees,
\item[{\tt nrad}] -- radial resolution of the grid,
\item[{\tt division}] -- switch for spacing of radial grid
($0$ -- equidistant, $1$ -- exponential),
\item[{\tt nphi}] -- axial resolution of the grid,
\item[{\tt smooth}] -- switch for performing simple smoothing
($0$ -- no, $1$ -- yes),
\item[{\tt Stokes}] -- switch for computing polarization
(see Tab.~\ref{stokes}).
\end{description}
The {\tt phi} and {\tt dphi} parameters determine the axial sector of the disc
from which emission comes (see Fig.~\ref{sector}). The {\tt nrad} and {\tt nphi}
parameters determine the grid for numerical integration.
If the {\tt division} switch is zero, the radial grid is equidistant;
if it is equal to unity then the radial grid is exponential
(i.e.\ more points closer to the black hole).
If the {\tt smooth} switch is set to unity then a simple smoothing
is applied to the final spectrum. Here $N_{\rm o}^{\Omega}(E_{\rm j})=
[N_{\rm o}^{\Omega}(E_{\rm j-1})+2N_{\rm o}^{\Omega}(E_{\rm j})+
N_{\rm o}^{\Omega}(E_{\rm j+1})]/4$.
If the {\tt Stokes} switch is different from zero, then the model also
calculates polarization. Its value determines which of the Stokes parameters
should be computed by {\sc{}xspec}, i.e.\ what will be stored in the output
array for the photon flux {\tt photar}; see Tab.~\ref{stokes}.
(If ${\tt Stokes}\neq0$ then a new {\sf ascii} data file {\tt stokes.dat} is
created in the current directory,
where values of energy $E$ together with all Stokes parameters
$i,\,q,\,u,\,v,\,P,\,\chi$[deg] and $\xi$[deg] are stored, each in one
column.)
A realistic model of polarization has been currently implemented only in the
{\sc kyl1cr} model (see Section~\ref{kyl1cr}
below). In other models, a simple assumption is made -- the local emission
is assumed to be linearly polarized in the direction perpendicular to
the disc (i.e.\ $q_{{\rm l}}=i_{{\rm l}}=N_{\rm l}$ and $u_{\rm l}=v_{\rm l}=0$). In all
models (including {\sc kyl1cr}) there is always no final circular polarization
(i.e.\ $v=\xi=0$), which follows from the fact that the fourth local
Stokes parameter is zero in each model.
\section{Models for a relativistic spectral line}
Three general relativistic line models are included in the new set of
{\sc{}xspec} routines -- non-axisymmetric Gaussian line model
{\sc kyg1line}, axisymmetric Gaussian line model {\sc kygline} and
fluorescent lamp-post line model {\sc kyf1ll}.
\subsection{Non-axisymmetric Gaussian line model
{\fontfamily{phv}\fontshape{sc}\selectfont kyg1line}}
\label{section_kyg1line}
The {\sc kyg1line} model computes the integrated flux from the
disc according to eq.~(\ref{emission}). It assumes that the local
emission from the disc is
\begin{myeqnarray}
\label{line_emiss1a}
N_{{\rm l}}(E_{\rm l}) \hspace*{-0.7em} & = & \hspace*{-0.7em}
\frac{1}{r^{\tt alpha}}\,f(\mu_{\rm e})\,\exp{\left
[-\left (1000\,\frac{E_{\rm l}-{\tt Erest}}{\sqrt{2}\,{\tt sigma}}\right )^2
\right ]} & {\rm for} & \hspace*{-0.5em} r\ge r_{\rm b}\, ,\\
\label{line_emiss1b}
N_{{\rm l}}(E_{\rm l}) \hspace*{-0.7em} & = & \hspace*{-0.7em}
{\tt jump}\ r_{\rm b}^{{\tt beta} - {\tt alpha}}\,
\frac{1}{r^{\tt beta}}\,f(\mu_{\rm e})\,\exp{\left [-\left (1000\,
\frac{E_{\rm l}-{\tt Erest}}{\sqrt{2}\,{\tt sigma}}\right )^2\right ]} &
{\rm for} & \hspace*{-0.5em} r<r_{\rm b}\, .\hspace{12mm}
\end{myeqnarray}
The local emission is assumed to be a Gaussian line with its peak flux
depending on the radius
as a broken power law. The line is defined by nine points equally spaced with
the central point at its maximum.
The normalization constant $N_0$ in eq.~(\ref{emission}) is such that the total
integrated flux of the line is unity.
The parameters defining the Gaussian line are (see Tab.~\ref{kyg1line_par}):
\begin{description} \itemsep -2pt
\item[{\tt Erest}] -- rest energy of the line in keV,
\item[{\tt sigma}] -- width of the line in eV,
\item[{\tt alpha}] -- radial power-law index for the outer region,
\item[{\tt beta}] -- radial power-law index for the inner region,
\item[{\tt rb}] -- parameter defining the border between regions with different
power-law indices,
\item[{\tt jump}] -- ratio between flux in the inner and outer regions at
the border radius,
\item[{\tt limb}] -- switch for different limb darkening/brightening laws.
\end{description}
There are two regions with different power-law dependences with indices
{\tt alpha} and {\tt beta}. The power law changes at the border radius
$r_{\rm b}$ where the local emissivity does not need to be continuous
(for ${\tt jump}\neq 1$). The {\tt rb} parameter defines this radius in the
following way:
\begin{myeqnarray}
\label{rb1}
r_{\rm b} &=& {\tt rb} \times r_{\rm ms} & & {\rm for}\quad {\tt rb}\ge
0\, ,\\
\label{rb2}
r_{\rm b} &=& -{\tt rb}+r_{\rm h} & & {\rm for}\quad {\tt rb}< 0\, ,
\end{myeqnarray}
where $r_{\rm ms}$ is the radius of the marginally stable orbit and $r_{\rm h}$
is the radius of the horizon of the black hole.
\begin{table}[tbh]
\begin{center}
\dummycaption\label{kyg1line_par}
\begin{tabular}[h]{r@{}l|l|c|c|c}
\multicolumn{2}{c|}{parameter} & unit & default value & minimum value &
maximum value \\ \hline
&{\tt a/M} & $GM/c$ & 0.9982 & 0. & 1. \\
&{\tt theta\_o} & deg & 30. & 0. & 89. \\
&{\tt rin-rh} & $GM/c^2$ & 0. & 0. & 999. \\
&{\tt ms} & -- & 1. & 0. & 1. \\
&{\tt rout-rh} & $GM/c^2$ & 400. & 0. & 999. \\
&{\tt phi} & deg & 0. & -180. & 180. \\
&{\tt dphi} & deg & 360. & 0. & 360. \\
&{\tt nrad} & -- & 200. & 1. & 10000. \\
&{\tt division} & -- & 1. & 0. & 1. \\
&{\tt nphi} & -- & 180. & 1. & 20000. \\
&{\tt smooth} & -- & 1. & 0. & 1. \\
&{\tt zshift} & -- & 0. & -0.999 & 10. \\
&{\tt ntable} & -- & 0. & 0. & 99. \\
{*}&{\tt Erest} & keV & 6.4 & 1. & 99. \\
{*}&{\tt sigma} & eV & 2. & 0.01 & 1000. \\
{*}&{\tt alpha} & -- & 3. & -20. & 20. \\
{*}&{\tt beta} & -- & 4. & -20. & 20. \\
{*}&{\tt rb} & $r_{\rm ms}$ & 0. & 0. & 160. \\
{*}&{\tt jump} & -- & 1. & 0. & 1e6 \\
{*}&{\tt limb} & -- & -1. & -10. & 10. \\
&{\tt Stokes} & -- & 0. & 0. & 6. \\
\end{tabular}
\mycaption{Parameters of the non-axisymmetric Gaussian line model {\sc{}kyg1line}.
Model parameters that are not common for all non-axisymmetric models are
denoted by asterisk.}
\end{center}
\end{table}
The function $f(\mu_{\rm e})=f(\cos{\delta_{\rm e}})$ in eqs.~(\ref{line_emiss1a})
and (\ref{line_emiss1b}) describes the limb darkening/ brightening law, i.e.\
the dependence of the local emission on the local emission angle. Several limb
darkening/brightening laws are implemented:
\begin{myeqnarray}
\label{isotropic}
f(\mu_{\rm e}) & = & 1 & & {\rm for} \quad {\tt limb}=0\, , \\
\label{laor}
f(\mu_{\rm e}) & = & 1 + 2.06 \mu_{\rm e} & & {\rm for} \quad {\tt limb}=
-1\, ,\\
\label{haardt}
f(\mu_{\rm e}) & = & \ln{(1+\mu_{\rm e}^{-1})} & & {\rm for} \quad {\tt limb}=
-2\, ,\\
\label{other_limb}
f(\mu_{\rm e}) & = & \mu_{\rm e}^{\tt limb} & & {\rm for} \quad {\tt limb}
\ne 0,-1,-2\, .
\end{myeqnarray}
Eq.~(\ref{isotropic}) corresponds to the isotropic local emission,
eq.~(\ref{laor}) corresponds to
limb\break darkening in an optically thick electron scattering atmosphere
(used by Laor, see\break \citealt{phillips1986,laor1990,laor1991}),
and eq.~(\ref{haardt}) corresponds
to limb brightening predicted by some models of a fluorescent line emitted
by an accretion disc due to X-ray illumination
\citep{haardt1993a,ghisellini1994}.
\begin{figure}
\dummycaption\label{fig:example0}
\includegraphics[width=0.5\textwidth]{rin1}
\includegraphics[width=0.5\textwidth]{rout1}
\mycaption{Comparative examples of simple line profiles,
showing a theoretical line ($E_{\rm rest}=6.4\,$keV) with relativistic
effects originating from a black-hole accretion disc. Different sizes of
the annular region (axially symmetric) have been considered, assuming
that the intrinsic emissivity obeys a power law in the radial direction
($\alpha=3$). Resolution of the line-emitting region was
$n_r{\times}n_\varphi=3000\times1500$ with a non-equidistant layout
of the grid in Kerr ingoing coordinates, as described in the text. Left:
Dependence on the inner edge. Values of $r_{\rm{}in}$ are indicated in the
plot (the outer edge has been fixed at the maximum
radius covered by our tables, $r_{\rm{}out}=10^3$). Right: Dependence
on $r_{\rm{}out}$ (with the inner edge at horizon,
$r_{\rm{}in}=r_{\rm{}h}$). Other key parameters are:
$\theta_{\rm{}o}=45^{\circ}$, $a=1.0$. Locally isotropic emission was
assumed in the disc co-rotating frame.}
\vspace*{2.5\bigskipamount}
\dummycaption\label{fig:example1}
\includegraphics[width=0.5\textwidth]{alpha1}
\includegraphics[width=0.5\textwidth]{phi1}
\mycaption{
More calculated line profiles, as in the previous figure. Left: Line
profiles for different values of $\alpha$. Notice the enhanced red tail
of the line when the intrinsic emission is concentrated to the centre of
the disc. Right: Line emission originating from four different azimuthal
segments of the disc. This plot can serve as a toy model of
non-axisymmetric emissivity or to illustrate the expected effects of
disc obscuration. Obviously, the receding segment of the disc
contributes mainly to the low-energy tail of the line while the
approaching segment constitutes the prominent high-energy peak. These
two plots illustrate a mutual interplay between the effect of changing
$\alpha$ and the impact of obscuration, which complicates interpretation
of time-averaged spectra. The radial range is $r_{\rm{}h}<r<10^3$ in both
panels.}
\end{figure}
There is also a similar model {\sc kyg2line} present among the new {\sc{}xspec}
models, which is useful when fitting two general relativistic lines
simultaneously. The parameters are the same as in the {\sc kyg1line} model
except that there are two sets of those parameters
describing the local Gaussian line emission. There is one more parameter
present, {\tt ratio21}, which is the ratio of the maximum of the second local
line to the maximum of the first local line. Polarization computations are
not included in this model.
\begin{figure}[tb]
\dummycaption\label{fig:example3}
\includegraphics[width=0.5\textwidth]{incl_a0}
\includegraphics[width=0.5\textwidth]{incl_a1}\\
\mycaption{Dependence on the observer inclination, $\theta_{\rm{}o}$, as given
in the plot. Left: Non-rotating black hole, $a=0$. Right: Maximally rotating
black hole, $a=1$. Other parameters as in Fig.~\ref{fig:example0} and
\ref{fig:example1}.}
\end{figure}
Results of an elementary code test are shown in
Figs.~\ref{fig:example0}--\ref{fig:example3}. The intrinsic emissivity
was assumed to be a narrow Gaussian line (width
$\sigma\dot{=}0.42\,\mbox{FWHM}=2$\,eV) with the amplitude decreasing
$\propto{}r^{-\alpha}$ in the local frame co-moving with the disc
medium. Typically, the slope of the low-energy wing is rather sensitive to
the radial dependence of emissivity.
No background continuum is included here, so these lines can be
compared with similar pure disc-line profiles obtained in previous
papers \citep{kojima1991,laor1991} which also imposed the assumption of
axially symmetric and steady emission from an irradiated thin disc.
Again, the intrinsic width of the line is assumed to be much less than
the effects of broadening due to bulk Keplerian motion and the central
gravitational field.
Furthermore,
Figs.~\ref{fig:example4}--\ref{fig:example5} compare model spectra
of widely used {\sc{}xspec} models. In these examples one can see
that the {\sc{}laor} model gives zero contribution at energy below
$0.1E_{\rm{}rest}$ (in the disc local frame). This is because its grid has
only $35$ radial points distributed in the whole range $1.23{\leq}r{\leq}400$.
That is also why, in spite of a very efficient interpolation and smoothing of
the final spectrum, the {\sc{}laor} model does not accurately reproduce the
line originating from a narrow ring. Also, the dependence on the limb
darkening/brightening cannot be examined with this model, because the
form of directionality of the intrinsic emission is hard-wired in the
code, together with the position of the inner edge at
$r{\geq}r_{\rm{}ms}$. This affects especially the spectrum originating
near the black hole, where
radiation is expected to be very anisotropic and flow lines non-circular.
The {\sc{}diskline} model has also been frequently
used in the context
of spectral fitting, assuming a disc around a non-rotating black hole.
This model is analytical, and so it has clear advantages in
{\sc{}xspec}. Notice, however, that photons move along straight lines in this
model and that the lensing effect is neglected.
\begin{figure}[tb]
\dummycaption\label{fig:example4}
\includegraphics[width=0.5\textwidth]{compar1a_30_6-7}
\includegraphics[width=0.5\textwidth]{compar3b_70_100-200}
\mycaption{Comparison of the output from {\sc{}xspec} models
for the disc-line problem: {\sc{}laor}, {\sc{}diskline}, and
{\sc{}kyg1line} (line 1 corresponds to the same limb darkening
law and $a=0.9982$ as in {\sc{}laor}; line 2 assumes locally
isotropic emission and $a=0$ as in {\sc{}diskline}). Left panel:
$\theta_{\rm{}o}=30^{\circ}$, $r_{\rm{}in}=6$, $r_{\rm{}out}=7$.
Right panel: $\theta_{\rm{}o}=70^{\circ}$, $r_{\rm{}in}=100$, $r_{\rm{}out}=200$.
Radial decay of intrinsic emissivity follows $\alpha=3$ power law.}
\vspace*{0.9\bigskipamount}
\dummycaption\label{fig:example5}
\includegraphics[width=0.5\textwidth]{comp5_30_rms-400}
\includegraphics[width=0.5\textwidth]{comp4_70_rms-400}
\mycaption{Similar to previous figure but in logarithmic scale
and for three choices of the darkening law in {\sc{}kyg1line}
-- (1)~$f(\mu_{\rm e})=1+2.06\mu_{\rm e}$; (2)~$f(\mu_{\rm e})=1$;
(3) $f(\mu_{\rm e})=\log(1+1/\mu_{\rm e})$. Left panel: $\theta_{\rm{}o}=
30^{\circ}$; Right panel: $\theta_{\rm{}o}=70^{\circ}$. In both panels,
$r_{\rm{}in}=r_{\rm{}ms}$, $r_{\rm{}out}=400$, $a=0.9982$.}
\end{figure}
\subsection{Axisymmetric Gaussian line model
{\fontfamily{phv}\fontshape{sc}\selectfont kygline}}
This model uses eq.~(\ref{axisym_emission}) for computing the
disc emission with local flux being
\begin{eqnarray}
N_{\rm l}(E_{\rm l}) & = & \delta(E_{\rm l}-{\tt Erest})\, ,\\
R(r) & = & r^{-{\tt alpha}}\, .
\end{eqnarray}
The function ${\rm d}F(\bar{g})\equiv {\rm d}\bar{g}\,F(\bar{g})$ in
eq.~(\ref{conv_function}) was pre-calculated for three different limb
darkening/brightening laws (see eqs.~(\ref{isotropic})--(\ref{haardt})) and stored in
corresponding FITS files {\tt KBHline00.fits} -- {\tt KBHline02.fits}. The local
emission is a delta function with its maximum depending on the radius as a power
law with index {\tt alpha} and also depending on the local emission angle. The
normalization constant $N_0$ in eq.~(\ref{axisym_emission}) is such that the
total integrated flux of the line is unity.
There are less parameters defining the line in this model than in
the previous one (see Tab.~\ref{kygline_par}):
\begin{description} \itemsep -2pt
\item[{\tt Erest}] -- rest energy of the line in keV,
\item[{\tt alpha}] -- radial power-law index.
\end{description}
Note that the limb darkening/brightening law can be chosen by means of
the {\tt ntable} switch.
\begin{table}[tbh]
\begin{center}
\dummycaption\label{kygline_par}
\begin{tabular}[h]{r@{}l|c|c|c|c}
\multicolumn{2}{c|}{parameter} & unit & default value & minimum value &
maximum value \\ \hline
&{\tt a/M} & $GM/c$ & 0.9982 & 0. & 1. \\
&{\tt theta\_o} & deg & 30. & 0. & 89. \\
&{\tt rin-rh} & $GM/c^2$ & 0. & 0. & 999. \\
&{\tt ms} & -- & 1. & 0. & 1. \\
&{\tt rout-rh} & $GM/c^2$ & 400. & 0. & 999. \\
&{\tt zshift} & -- & 0. & -0.999 & 10. \\
&{\tt ntable} & -- & 1. & 0. & 99. \\
{*}&{\tt Erest} & keV & 6.4 & 1. & 99. \\
{*}&{\tt alpha} & -- & 3. & -20. & 20. \\
\end{tabular}
\mycaption{Parameters of the axisymmetric Gaussian line model {\sc{}kygline}.
Model parameters that are not common for all axisymmetric models are denoted
by asterisk.}
\end{center}
\end{table}
This model is much faster than the non-axisymmetric {\sc kyg1line} model.
Although it is not possible to change the resolution grid on the disc,
it is hardly needed because the resolution is set to be very large,
corresponding to ${\tt nrad}=500$,
${\tt division}=1$ and ${\tt nphi}=20\,000$ in the {\sc kyg1line} model,
which is more than sufficient in most cases.
(These values apply if the maximum range of radii is selected,
i.e.\ {\tt rin}=0, {\tt ms}=0 and {\tt rout}=999; in case of
a smaller range the number of points decreases accordingly.)
This means that the resolution of the {\sc kygline} model is much
higher than what can be achieved with the {\tt laor} model, and the
performance is still very good.
\subsection{Non-axisymmetric fluorescent lamp-post line model
{\fontfamily{phv}\fontshape{sc}\selectfont kyf1ll}}
\label{section_kyf1ll}
The line in this model is
induced by the illumination of the disc from the primary power-law source
located on the axis at {\tt height} above the black hole.
This model computes the final spectrum according to eq.~(\ref{emission})
with the local photon flux
\begin{eqnarray}
\nonumber
N_{{\rm l}}(E_{\rm l}) & = & g_{\rm L}^{{\tt PhoIndex}-1}\frac{\sin\theta_{\rm L}
{\rm d}\theta_{\rm L}}{r\,{\rm d}r}\,\sqrt{1-\frac{2\,{\tt height}}
{{\tt height}^2+{\tt (a/M)}^2}}\,f(\mu_{\rm i},\mu_{\rm e})\\
\label{fl_emission}
& & \times\ \exp{\left [-\left
(1000\,\frac{E_{\rm l}-{\tt Erest}}{\sqrt{2}\,{\tt sigma}}\right )^2\right ]}.
\end{eqnarray}
Here, $g_{\rm L}$ is ratio of the energy of a photon received by the
accretion disc to the energy of the same photon when emitted from a source
on the axis, $\theta_{\rm L}$ is an angle under which the photon is emitted
from the source (measured in the local frame of the source) and
$\mu_{\rm i}\equiv\cos{\,\delta_{\rm i}}$ is the cosine of the incident angle
(measured in the local frame of the disc) -- see
Fig.~\ref{compton_reflection}.
All of these functions depend on {\tt height} above the black hole at which
the source is located and on the rotational parameter {\tt a/M} of the black
hole. Values of $g_{\rm L}$, $\theta_{\rm L}$ and $\mu_{\rm i}$ for a given
height and rotation are read from the lamp-post tables {\tt lamp.fits}
(see Appendix~\ref{appendix3c}). At present, only tables for
${\tt a/M}=0.9987492$ (i.e.\ for the horizon of the black hole $r_{\rm h}=1.05$)
and ${\tt height}=$ $2$,$\,3,\,4,\,5,\,6,\,8,\,10,\,12$,
$\,15,\,20,\,30,\,50,\,75$ and $100$ are included in {\tt lamp.fits},
therefore, the {\tt a/M} parameter is used only for the negative values
of {\tt height} (see below).
The factor in front of the function $f(\mu_{\rm i},\mu_{\rm e})$ gives
the radial dependence of the disc emissivity, which is different from the
assumed broken power law in the {\sc kyg1line} model.
For the derivation of this factor, which characterizes the illumination from
a primary source on the axis see Section~\ref{lamp-post}.
\begin{table}[tbh]
\begin{center}
\dummycaption\label{kyf1ll_par}
\begin{tabular}[h]{r@{}l|c|c|c|c}
\multicolumn{2}{c|}{parameter} & unit & default value & minimum value &
maximum value \\ \hline
&{\tt a/M} & $GM/c$ & 0.9982 & 0. & 1. \\
&{\tt theta\_o} & deg & 30. & 0. & 89. \\
&{\tt rin-rh} & $GM/c^2$ & 0. & 0. & 999. \\
&{\tt ms} & -- & 1. & 0. & 1. \\
&{\tt rout-rh} & $GM/c^2$ & 400. & 0. & 999. \\
&{\tt phi} & deg & 0. & -180. & 180. \\
&{\tt dphi} & deg & 360. & 0. & 360. \\
&{\tt nrad} & -- & 200. & 1. & 10000. \\
&{\tt division} & -- & 1. & 0. & 1. \\
&{\tt nphi} & -- & 180. & 1. & 20000. \\
&{\tt smooth} & -- & 1. & 0. & 1. \\
&{\tt zshift} & -- & 0. & -0.999 & 10. \\
&{\tt ntable} & -- & 0. & 0. & 99. \\
{*}&{\tt PhoIndex} & -- & 2. & 1.5 & 3. \\
{*}&{\tt height} & $GM/c^2$ & 3. & -20. & 100. \\
{*}&{\tt Erest} & keV & 6.4 & 1. & 99. \\
{*}&{\tt sigma} & eV & 2. & 0.01 & 1000. \\
&{\tt Stokes} & -- & 0. & 0. & 6. \\
\end{tabular}
\mycaption{Parameters of the fluorescent lamp-post line model {\sc{}kyf1ll}.
Model parameters that are not common for all non-axisymmetric models are
denoted by asterisk.}
\end{center}
\end{table}
The function $f(\mu_{\rm i},\mu_{\rm e})$ is a coefficient of
reflection. It depends on the incident and reflection angles.
Although the normalization of this function also depends on the photon
index of the power-law emission from a primary source,
we do not need to take this into account because the final spectrum
is always normalized to unity. Values of this function are read
from a pre-calculated table which is stored in
{\tt fluorescent\_line.fits} file (see \citealt{matt1991} and
Appendix~\ref{appendix3d}).
The local emission (\ref{fl_emission}) is defined in nine points
of local energy $E_{\rm l}$ that are equally spaced with the
central point at its maximum. The normalization constant
$N_0$ in the formula (\ref{emission}) is such that the total
integrated flux of the line is unity. The parameters defining
local emission in this model are (see Tab.~\ref{kyf1ll_par}):
\begin{description} \itemsep -2pt
\item[{\tt PhoIndex}] -- photon index of primary power-law illumination,
\item[{\tt height}] -- height above the black hole where the primary source
is located for ${\tt height}>0$, and radial power-law index for
${\tt height}\le0$,
\item[{\tt Erest}] -- rest energy of the line in keV,
\item[{\tt sigma}] -- width of the line in eV.
\end{description}
If positive, the {\tt height} parameter works as a switch -- the exact
value present in the tables {\tt lamp.fits} must be chosen.
If the {\tt height} parameter is negative, then this model assumes that
the local emission is the same as in the {\sc kyg1line} model with the
parameters ${\tt alpha}=-{\tt height}$, ${\tt rb=0}$ and ${\tt limb}=-2$
({\tt PhoIndex} parameter is unused in this case).
\begin{figure}[tb]
\begin{center}
\dummycaption\label{kyf}
\begin{tabular}{cc}
\hspace*{-2.5mm}\includegraphics[width=6.75cm]{kyf_height} &
\hspace*{-2.5mm}\includegraphics[width=6.75cm]{kyf_pho}
\end{tabular}
\mycaption{An example of a line profile originating from a disc in
equatorial plane of a Kerr black hole ($a=0.9987$, i.e.\ $r_{\rm h}=1.05$)
due to the illumination
from a primary source on the axis. The {\sc kyf1ll} model was used. Left:
Dependence of the line profile on the
height (in $GM/c^2$) of a primary source with photon index $\Gamma=2$.
Right: Dependence of the line profile on the photon index of the primary
emission with a source at height $3\,GM/c^2$ above the black hole.}
\end{center}
\end{figure}
In Fig.~\ref{kyf} we demonstrate that the broad iron emission lines due to
illumination from the source placed on the axis depend heavily on the height
where the ``lamp'' is located (left), as well as on the photon index of the
primary emission (right). These graphs correspond to the iron K$\alpha$ line
with the rest energy of $6.4$~keV.
\section{Compton reflection models}
We have developed two new relativistic continuum models -- the lamp-post Compton
reflection model {\sc kyl1cr} and the {\sc kyh1refl} model which is a
relativistically blurred {\sc hrefl} model that is already present in
{\sc{}xspec}. Both of these models are non-axisymmetric.
\subsection{Non-axisymmetric lamp-post Compton reflection model
{\fontfamily{phv}\fontshape{sc}\selectfont kyl1cr}}
\label{kyl1cr}
The emission in this model is induced by the illumination of the disc from the
primary power-law source located on the axis at {\tt height} above the
black hole. As in every non-axisymmetric model, the observed spectrum is
computed according to eq.~(\ref{emission}). The local emission is
\begin{myeqnarray}
\label{cr_emission}
N_{{\rm l}}(E_{\rm l}) \hspace*{-0.8em} & = & \hspace*{-0.8em}
g_{\rm L}^{{\tt PhoIndex}-1}\frac{\sin\theta_{\rm L}
{\rm d}\theta_{\rm L}}{r\,{\rm d}r}\,\sqrt{1-\frac{2\,{\tt height}}
{{\tt height}^2+{\tt (a/M)}^2}}\,f(E_{\rm l};\mu_{\rm i},\mu_{\rm e}) & &
\hspace*{-1.3em} {\rm for}\ {\tt height} > 0\, ,\hspace*{10mm}\\
\label{cr_emission_neg}
N_{{\rm l}}(E_{\rm l}) \hspace*{-0.8em} & = & \hspace*{-0.8em}
r^{\tt height}\,\bar{f}(E_{\rm l};\mu_{\rm e}) & &
\hspace*{-1.3em} {\rm for}\ {\tt height \le 0}\, .
\end{myeqnarray}
For the definition of $g_{\rm L}$, $\theta_{\rm L}$ and $\mu_{\rm i}$ see
Section~\ref{section_kyf1ll} and Appendix~\ref{appendix3c}, where pre-calculated
tables of these functions in {\tt lamp.fits} are described.
\begin{table}[tbh]
\begin{center}
\dummycaption\label{kyl1cr_par}
\begin{tabular}[h]{r@{}l|c|c|c|c}
\multicolumn{2}{c|}{parameter} & unit & default value & minimum value &
maximum value \\ \hline
&{\tt a/M} & $GM/c$ & 0.9982 & 0. & 1. \\
&{\tt theta\_o} & deg & 30. & 0. & 89. \\
&{\tt rin-rh} & $GM/c^2$ & 0. & 0. & 999. \\
&{\tt ms} & -- & 1. & 0. & 1. \\
&{\tt rout-rh} & $GM/c^2$ & 400. & 0. & 999. \\
&{\tt phi} & deg & 0. & -180. & 180. \\
&{\tt dphi} & deg & 360. & 0. & 360. \\
&{\tt nrad} & -- & 200. & 1. & 10000. \\
&{\tt division} & -- & 1. & 0. & 1. \\
&{\tt nphi} & -- & 180. & 1. & 20000. \\
&{\tt smooth} & -- & 1. & 0. & 1. \\
&{\tt zshift} & -- & 0. & -0.999 & 10. \\
&{\tt ntable} & -- & 0. & 0. & 99. \\
{*}&{\tt PhoIndex} & -- & 2. & 1.5 & 3. \\
{*}&{\tt height} &$GM/c^2$ & 3. & -20. & 100. \\
{*}&{\tt line} & -- & 0. & 0. & 1. \\
{*}&{\tt E\_cut} & keV & 300. & 1. & 1000. \\
&{\tt Stokes} & -- & 0. & 0. & 6. \\
\end{tabular}
\mycaption{Parameters of the lamp-post Compton reflection model {\sc{}kyl1cr}.
Model parameters that are not common for all non-axisymmetric models are
denoted by asterisk.}
\end{center}
\end{table}
The function $f(E_{\rm l};\mu_{\rm i},\mu_{\rm e})$ gives the dependence of the
locally
emitted spectrum on the angle of incidence and the angle of emission, assuming a
power-law illumination. This function depends on the photon index {\tt PhoIndex}
of the power-law emission from a primary source.
Values of this function for various photon indices of primary emission
are read from the pre-calculated tables stored in {\tt refspectra.fits}
(see Appendix~\ref{appendix3e}). These tables were calculated by the Monte Carlo
simulations of Compton scattering in\break \cite{matt1991}. At present,
tables for ${\tt PhoIndex}=1.5,\ 1.6,\ \dots,\ 2.9,\ 3.0$ and for local
energies in the range from $2\,$keV to $300$~keV are available.
The normalization constant $N_0$ in eq.~(\ref{emission}) is such
that the final photon flux at an energy of $3$~keV is equal to unity,
which is different from what is usual for
continuum models in {\sc xspec} (where the photon flux is unity at $1\,$keV).
The choice adopted is due to the fact that current tables in
{\tt refspectra.fits} do not extend below $2\,$keV.
The function $\bar{f}(E_{\rm l};\mu_{\rm e})$, which is used for negative
{\tt height}, is an averaged function $f(E_{\rm l};\mu_{\rm i},\mu_{\rm e})$ over
$\mu_{\rm i}$
\begin{equation}
\bar{f}(E_{\rm l};\mu_{\rm e})\equiv\int_0^1 {\rm d}\mu_{\rm i}\,f(E_{\rm l};
\mu_{\rm i},\mu_{\rm e})\, .
\end{equation}
The local emission (\ref{cr_emission_neg}) can be interpreted as emission induced
by illumination from clouds localized near above the disc rather than from a
primary source on the axis (see Fig.~\ref{compton_reflection}). In this case
photons strike the disc from all directions.
For positive values of {\tt height} the {\sc kyl1cr} model includes a physical
model of polarization based on Rayleigh scattering in single scattering
approximation. The specific local Stokes parameters describing local
polarization of light are
\begin{eqnarray}
\label{polariz1}
i_{\rm l}(E_{\rm l}) & = & \frac{I_{\rm l}+I_{\rm r}}{\langle I_{\rm l}+
I_{\rm r}\rangle}\, N_{\rm l}(E_{\rm l})\, ,\\[1mm]
q_{\rm l}(E_{\rm l}) & = & \frac{I_{\rm l}-I_{\rm r}}{\langle I_{\rm l}+
I_{\rm r}\rangle}\, N_{\rm l}(E_{\rm l})\, ,\\[1mm]
u_{\rm l}(E_{\rm l}) & = & \frac{U}{\langle I_{\rm l}+
I_{\rm r}\rangle}\,N_{\rm l}(E_{\rm l})\, ,\\[1mm]
\label{polariz4}
v_{\rm l}(E_{\rm l}) & = & 0\, ,
\end{eqnarray}
where the functions $I_{\rm l}$, $I_{\rm r}$ and $U$ determine the angular
dependence of the Stokes parameters in the following way
\begin{eqnarray}
\label{polariz2}
\nonumber
I_{\rm l} & = & \mu_{\rm e}^2(1+\mu_{\rm i}^2)+2(1-\mu_{\rm e}^2)
(1-\mu_{\rm i}^2)-4\mu_{\rm e}\mu_{\rm i}\sqrt{(1-\mu_{\rm e}^2)
(1-\mu_{\rm i}^2)}\,\cos{(\Phi_{\rm e}-\Phi_{\rm i})}\hspace*{3em} \\
& & -\mu_{\rm e}^2(1-\mu_{\rm i}^2)\cos{[2(\Phi_{\rm e}-\Phi_{\rm i})]}\, ,\\
I_{\rm r} & = & 1+\mu_{\rm i}^2+(1-\mu_{\rm i}^2)\cos{[2(\Phi_{\rm e}-
\Phi_{\rm i})]}\, ,\\
U & = & -4\mu_{\rm i}\sqrt{(1-\mu_{\rm e}^2)(1-\mu_{\rm i}^2)}\,
\sin{(\Phi_{\rm e}-\Phi_{\rm i})}-2\mu_{\rm e}(1-\mu_{\rm i}^2)
\sin{[2(\Phi_{\rm e}-\Phi_{\rm i})]}\, .
\end{eqnarray}
Here $\Phi_{\rm e}$ and $\Phi_{\rm i}$ are the azimuthal emission
and the incident angles in the local rest frame co-moving with the accretion
disc (see Sections~\ref{azimuth_angle} and \ref{lamp-post} for their
definition). For the derivation of these formulae see the definitions (I.147)
and eqs.~(X.172) in \cite{chandrasekhar1960}.
We have omitted a common multiplication factor, which
would be cancelled anyway in eqs.~(\ref{polariz1})--(\ref{polariz4}).
The symbol $\langle\ \rangle$ in definitions of the local Stokes parameters
means value averaged over
the difference of the azimuthal angles $\Phi_{\rm e}-\Phi_{\rm i}$. We divide
the parameters by
$\langle I_{\rm l}+I_{\rm r}\rangle$ because the function
$f(E_{\rm l};\mu_{\rm i},\mu_{\rm e})$, and thus
also the local photon flux $N_{\rm l}(E_{\rm l})$, is averaged over the difference of
the azimuthal angles.
The parameters defining local emission in this model are
(see Tab.~\ref{kyl1cr_par}):
\begin{description} \itemsep -2pt
\item[{\tt PhoIndex}] -- photon index of primary power-law illumination,
\item[{\tt height}] -- height above the black hole where the primary source
is located for ${\tt height}>0$, and radial power-law index for
${\tt height}\le0$,
\item[{\tt line}] -- switch whether to include the iron lines
(0 -- no, 1 -- yes),
\item[{\tt E\_cut}] -- exponential cut-off energy of the primary source in keV.
\end{description}
The tables {\tt refspectra.fits} for the function
$f(E_{\rm l};\mu_{\rm i},\mu{\rm e})$ also contain the emission in the iron lines
K$\alpha$ and K$\beta$. The two lines can be excluded from
computations if the {\tt line} switch is set to zero.
The {\tt E\_cut} parameter sets the upper boundary in energies
where the emission from a primary source ceases to follow a power-law
dependence. If the {\tt E\_cut} parameter is lower than both the maximum energy
of the considered dataset and the maximum energy in the tables
for $f(E_{\rm l};\mu_{\rm i},\mu{\rm e})$ in {\tt refspectra.fits} (300~keV),
then this model is not valid.
\begin{figure}[tb]
\begin{center}
\dummycaption\label{kyl_kyl}
\begin{tabular}{cc}
\includegraphics[width=6.5cm]{kyl_kyl_30_2_002} &
\includegraphics[width=6.5cm]{kyl_kyl_30_2_100}
\end{tabular}
\mycaption{General relativistic lamp-post Compton reflection model {\sc kyl1cr}
with (dashed) and without (solid) iron lines K$\alpha$ and K$\beta$.
The emission from the disc is induced by illumination from a primary source
placed $2\,GM/c^2$ (left) and $100\,GM/c^2$ (right) above the black hole.}
\end{center}
\end{figure}
\begin{table}[tbh]
\begin{center}
\dummycaption\label{kyh1refl_par}
\begin{tabular}[h]{r@{}l|c|c|c|c}
\multicolumn{2}{c|}{parameter} & unit & default value & minimum value &
maximum value \\ \hline
&{\tt a/M} & $GM/c$ & 0.9982 & 0. & 1. \\
&{\tt theta\_o} & deg & 30. & 0. & 89. \\
&{\tt rin-rh} & $GM/c^2$ & 0. & 0. & 999. \\
&{\tt ms} & -- & 1. & 0. & 1. \\
&{\tt rout-rh} & $GM/c^2$ & 400. & 0. & 999. \\
&{\tt phi} & deg & 0. & -180. & 180. \\
&{\tt dphi} & deg & 360. & 0. & 360. \\
&{\tt nrad} & -- & 200. & 1. & 10000. \\
&{\tt division} & -- & 1. & 0. & 1. \\
&{\tt nphi} & -- & 180. & 1. & 20000. \\
&{\tt smooth} & -- & 1. & 0. & 1. \\
&{\tt zshift} & -- & 0. & -0.999 & 10. \\
&{\tt ntable} & -- & 0. & 0. & 99. \\
{*}&{\tt PhoIndex} & -- & 1. & 0. & 10. \\
{*}&{\tt alpha} & -- & 3. & -20. & 20. \\
{*}&{\tt beta} & -- & 4. & -20. & 20. \\
{*}&{\tt rb} & $r_{\rm ms}$ & 0. & 0. & 160. \\
{*}&{\tt jump} & -- & 1. & 0. & 1e6 \\
{*}&{\tt Feabun} & -- & 1. & 0. & 200. \\
{*}&{\tt FeKedge} & keV & 7.11 & 7.0 & 10. \\
{*}&{\tt Escfrac} & -- & 1. & 0. & 1000. \\
{*}&{\tt covfac} & -- & 1. & 0. & 1000. \\
&{\tt Stokes} & -- & 0. & 0. & 6. \\
\end{tabular}
\mycaption{Parameters of the reflection {\sc{}kyh1refl} model. Model parameters
that are not common for all non-axisymmetric models are denoted by asterisk.}
\end{center}
\end{table}
Examples of the Compton reflection emission component of the spectra with and
without the fluorescent K$\alpha$ and K$\beta$ lines are shown in
Fig.~\ref{kyl_kyl}. It can be seen that originally narrow lines can
contribute substantially to the continuum component.
\subsection{Non-axisymmetric Compton reflection model
{\fontfamily{phv}\fontshape{sc}\selectfont kyh1refl}}
This model is based on an existing multiplicative {\sc hrefl} model in
combination with the {\sc powerlaw} model, both of which are present in
{\sc xspec}. Local emission in eq.~(\ref{emission}) is the same as the spectrum
given by the model {\sc hrefl*powerlaw} with the parameters ${\tt thetamin}=0$ and
${\tt thetamax}=90$ with a broken power-law radial dependence added:
\begin{myeqnarray}
N_{\rm l}(E_{\rm l}) & = & r^{-{\tt alpha}}\,\textsc{hrefl*powerlaw} & & {\rm for}
\quad r\ge r_{\rm b}\, ,\hspace*{1em}\\
N_{\rm l}(E_{\rm l}) & = & {\tt jump}\ r_{\rm b}^{{\tt beta}-{\tt alpha}}\,
r^{-{\tt beta}}\,\textsc{hrefl*powerlaw} & & {\rm for} \quad r<r_{\rm b}\, .
\end{myeqnarray}
\begin{figure}[tb]
\begin{center}
\dummycaption\label{kyh_href}
\begin{tabular}{cc}
\includegraphics[width=6.5cm]{kyh_href_30_2} &
\includegraphics[width=6.5cm]{kyh_href_30_2_6}
\end{tabular}
\mycaption{Comparison of the general relativistic {\sc kyh1refl} model with the
non-relativistic {\sc hrefl(powerlaw)}. The relativistic blurring of the iron
edge is clearly visible. The power-law index of the primary source is
{\tt PhoIndex}=2 (left) and {\tt PhoIndex}=2.6 (right).}
\end{center}
\bigskip
\begin{center}
\dummycaption\label{kyl_kyh}
\begin{tabular}{cc}
\includegraphics[width=6.5cm]{kyl_kyh_30_2_2_3_4} &
\includegraphics[width=6.5cm]{kyl_kyh_30_2_100_1_5}
\end{tabular}
\mycaption{Comparison of the two new general relativistic Compton reflection
models {\sc kyl1cr} and
{\sc kyh1refl}. The lamp-post {\sc kyl1cr} model is characterized by the height
$h$ above the disc where a primary source of emission is placed, the
reflection {\sc kyh1refl} model is characterized by the radial power-law index
$\alpha$. Left: $h=2\,GM/c^2$, $\alpha=3.4\,$. Right:
$h=100\,GM/c^2$, $\alpha=1.5\,$.}
\end{center}
\end{figure}
For a definition of the boundary radius $r_{\rm b}$ by the {\tt rb} parameter
see eqs.~(\ref{rb1})--(\ref{rb2}), and
for a detailed description of the {\sc hrefl} model see \cite{dovciak2004}
and the {\sc xspec} manual. The {\sc{}kyh1refl} model can be interpreted
as a Compton-reflection model for which the source of primary
irradiation is near above the disc, in contrast to the lamp-post
scheme with the source on the axis (see Fig.~\ref{compton_reflection}).
The approximations for Compton reflection used in {\sc{}hrefl}
(and therefore also in {\sc{}kyh1refl})
are valid below $\sim15$~keV in the disc rest-frame.
The normalization of the final spectrum in this model is the same as
in other continuum models in {\sc xspec}, i.e.\ photon flux is unity at
the energy of $1$~keV.
The parameters defining the local emission in {\sc kyh1refl}
(see Tab.~\ref{kyh1refl_par}) are
\begin{description} \itemsep -2pt
\item[{\tt PhoIndex}] -- photon index of the primary power-law illumination,
\item[{\tt alpha}] -- radial power-law index for the outer region,
\item[{\tt beta}] -- radial power-law index for the inner region,
\item[{\tt rb}] -- parameter defining the border between regions with different
power-law indices,
\item[{\tt jump}] -- ratio between flux in the inner and outer regions at
the border radius,
\item[{\tt Feabun}] -- iron abundance relative to solar,
\item[{\tt FeKedge}] -- iron K-edge energy,
\item[{\tt Escfrac}] -- fraction of the direct flux from the power-law primary
source seen by the observer,
\item[{\tt covfac}] -- normalization of the reflected continuum.
\end{description}
Smearing of sharp features in continuum (iron edge) by relativistic effects
is demonstrated in Fig.~\ref{kyh_href}, where
non-relativistic reflection model {\sc hrefl(powerlaw)} is compared with
our relativistic {\sc kyh1refl} model. Here, we set the radial power-law index
$\alpha=1$ in {\sc kyh1refl}. Other parameters defining these models were set to
their default values.
We compare the two new relativistic reflection models {\sc kyl1cr} and
{\sc kyh1refl} in Fig.~\ref{kyl_kyh}. Note that the {\sc kyl1cr} model is
valid only above approximately $2\,{\rm keV}$ and the {\sc kyh1refl} model only
below approximately $15\,{\rm keV}$.
\section{General relativistic convolution models}
We have also produced two convolution-type
models, {\sc ky1conv} and {\sc kyconv}, which can be applied to any existing
{\sc{}xspec} model for the intrinsic X-ray emission from a disc around a Kerr
black hole. We must stress
that these models are substantially more powerful than the usual
convolution models in {\sc{}xspec} (these are commonly
defined in terms of one-dimensional integration over energy bins).
Despite the fact that our convolution models still use the standard
{\sc{}xspec} syntax in evaluating the observed spectrum
(e.g.\ {\sc kyconv(powerlaw)}), our code
accomplishes a more complex operation. It still performs ray-tracing
across the disc surface so that the intrinsic model contributions are
integrated from different radii and azimuths on the disc.
There are several restrictions that arise from the fact that we use existing
{\sc xspec} models:
\begin{itemize} \itemsep -2pt
\item[--] by local {\sc xspec} models only the energy dependence of the photon
flux can be defined,
\item[--] only a certain type of radial dependence of the local photon flux can
be imposed -- we have chosen to use a broken power-law radial dependence,
\item[--] there is no azimuthal dependence of the local photon flux, except
through limb darkening law,
\item[--] local flux depends on the binning of the data because it is defined
in the centre of each bin, a large number of bins is needed for highly varying
local flux.
\end{itemize}
For emissivities that cannot be defined by existing {\sc xspec} models,
or where the limitations mentioned above are too restrictive, one has
to add a new user-defined model to {\sc{}xspec} (by adding a new
subroutine to {\sc xspec}). This method is more flexible and faster than
convolution models (especially when compared with non-axisymmetric
one), and hence it is recommended even for cases when these
prefabricated models could be used. In any new model for {\sc
xspec} one can use the common ray-tracing driver for relativistic smearing
of the local emission: {\tt ide} for non-axisymmetric
models and {\tt idre} for axisymmetric ones. For a detailed description
see Appendixes~\ref{appendix4a} and \ref{appendix4b}.
\subsection{Non-axisymmetric convolution model
{\fontfamily{phv}\fontshape{sc}\selectfont kyc1onv}}
The local emission in this model is computed according to the
eq.~(\ref{emission}) with the local emissivity equal to
\begin{table}[tbh]
\begin{center}
\dummycaption\label{kyc1onv_par}
\begin{tabular}[h]{r@{}l|c|c|c|c}
\multicolumn{2}{c|}{parameter} & unit & default value & minimum value &
maximum value \\ \hline
&{\tt a/M} & $GM/c$ & 0.9982 & 0. & 1. \\
&{\tt theta\_o} & deg & 30. & 0. & 89. \\
&{\tt rin-rh} & $GM/c^2$ & 0. & 0. & 999. \\
&{\tt ms} & -- & 1. & 0. & 1. \\
&{\tt rout-rh} & $GM/c^2$ & 400. & 0. & 999. \\
&{\tt phi} & deg & 0. & -180. & 180. \\
&{\tt dphi} & deg & 360. & 0. & 360. \\
&{\tt nrad} & -- & 200. & 1. & 10000. \\
&{\tt division} & -- & 1. & 0. & 1. \\
&{\tt nphi} & -- & 180. & 1. & 20000. \\
&{\tt smooth} & -- & 1. & 0. & 1. \\
{*}&{\tt normal} & -- & 1. & -1. & 100. \\
&{\tt zshift} & -- & 0. & -0.999 & 10. \\
&{\tt ntable} & -- & 0. & 0. & 99. \\
{*}&{\tt ne\_loc} & -- & 100. & 3. & 5000. \\
{*}&{\tt alpha} & -- & 3. & -20. & 20. \\
{*}&{\tt beta} & -- & 4. & -20. & 20. \\
{*}&{\tt rb} & $r_{\rm ms}$ & 0. & 0. & 160. \\
{*}&{\tt jump} & -- & 1. & 0. & 1e6 \\
{*}&{\tt limb} & -- & 0. & -10. & 10. \\
&{\tt Stokes} & -- & 0. & 0. & 6. \\
\end{tabular}
\mycaption{Parameters of the non-axisymmetric convolution model {\sc{}kyc1onv}.
Model parameters that are not common for all non-axisymmetric models are
denoted by asterisk.}
\end{center}
\end{table}
\begin{myeqnarray}
N_{\rm l}(E_{\rm l}) & = & r^{-{\tt alpha}}\,f(\mu_{\rm e})\,\textsc{model} & &
{\rm for}\quad r > r_{\rm b} \, ,\\
N_{\rm l}(E_{\rm l}) & = & {\tt jump}\ r_{\rm b}^{{\tt beta}-{\tt alpha}}\,
r^{-{\tt beta}}\,f(\mu_{\rm e})\,\textsc{model} & & {\rm for}\quad r
\le r_{\rm b} \, .\
\end{myeqnarray}
For a definition of the boundary radius $r_{\rm b}$ by the {\tt rb} parameter
see eqs.~(\ref{rb1})--(\ref{rb2}) and for the definition of different limb
darkening laws $f(\mu_{\rm e})$ see eqs.~(\ref{isotropic})--(\ref{other_limb}).
The local emission is given by the
{\sc model} in the centre of energy bins used in {\sc xspec} with the
broken power-law radial dependence and limb darkening law added. Apart from the
parameters of the {\sc model}, the local emission is defined also by the
following parameters (see Tab.~\ref{kyc1onv_par}):
\begin{description} \itemsep -2pt
\item[{\tt normal}] -- switch for the normalization of the final spectrum,\\
$=$ 0 -- total flux is unity (usually used for the line),\\
$>$ 0 -- flux is unity at the energy = {\tt normal} keV (usually used for
the continuum),\\
$<$ 0 -- flux is not normalized,
\item[{\tt ne\_loc}] -- number of points in the energy grid where the local
photon flux is defined,
\item[{\tt alpha}] -- radial power-law index for the outer region,
\item[{\tt beta}] -- radial power-law index for the inner region,
\item[{\tt rb}] -- parameter defining the border between regions with different
power-law indices,
\item[{\tt jump}] -- ratio between the flux in the inner and outer regions at
the border radius,
\item[{\tt limb}] -- switch for different limb darkening/brightening laws.
\end{description}
\begin{table}[tbh]
\begin{center}
\dummycaption\label{kyconv_par}
\begin{tabular}[h]{r@{}l|c|c|c|c}
\multicolumn{2}{c|}{parameter} & unit & default value & minimum value &
maximum value \\ \hline
&{\tt a/M} & $GM/c$ & 0.9982 & 0. & 1. \\
&{\tt theta\_o} & deg & 30. & 0. & 89. \\
&{\tt rin-rh} & $GM/c^2$ & 0. & 0. & 999. \\
&{\tt ms} & -- & 1. & 0. & 1. \\
&{\tt rout-rh} & $GM/c^2$ & 400. & 0. & 999. \\
&{\tt zshift} & -- & 0. & -0.999 & 10. \\
&{\tt ntable} & -- & 0. & 0. & 99. \\
{*}&{\tt alpha} & -- & 3. & -20. & 20. \\
{*}&{\tt ne\_loc} & -- & 100. & 3. & 5000. \\
{*}&{\tt normal} & -- & 1. & -1. & 100. \\
\end{tabular}
\mycaption{Parameters of the axisymmetric convolution model {\sc{}kyconv}.
Model parameters that are not common for all axisymmetric models are denoted
by asterisk.}
\end{center}
\end{table}
The local emission in each {\sc ky} model has to by defined either on
equidistant or exponential (i.e.\ equidistant in logarithmic scale)
energy grid. Because the energy grid used in the convolution model depends on
the binning of the data, which may be arbitrary, the flux has to be
rebinned. It is always rebinned into an exponentially spaced
energy grid in {\sc ky} convolution models.
The {\tt ne\_loc} parameter defines the number of points in which the rebinned
flux will be defined.
\subsection{Axisymmetric convolution model
{\fontfamily{phv}\fontshape{sc}\selectfont kyconv}}
The local emission in this model is computed according to
eq.~(\ref{axisym_emission}) with the local emissivity equal to
\begin{eqnarray}
N_{\rm l}(E_{\rm l}) & = & \textsc{model}\, ,\\
R(r) & = & r^{-{\tt alpha}}\, .
\end{eqnarray}
Except for the parameters of the {\sc model}, the local emission is defined also
by the following parameters (see Tab.~\ref{kyconv_par}):
\begin{description} \itemsep -2pt
\item[{\tt alpha}] -- radial power-law index,
\item[{\tt ne\_loc}] -- number of points in energy grid where local photon
flux is defined,
\item[{\tt normal}] -- switch for the normalization of the final spectrum,\\
$=$ 0 -- total flux is unity (usually used for the line),\\
$>$ 0 -- flux is unity at the energy = {\tt normal} keV (usually used for
the continuum),\\
$<$ 0 -- flux is not normalized.
\end{description}
Note that the limb darkening/brightening law can be chosen through the
{\tt ntable} switch. This model is much faster than the non-axisymmetric
convolution model {\sc kyc1onv}.
\section{Non-stationary model
{\fontfamily{phv}\fontshape{sc}\selectfont kyspot}}
Emission in this model originates from a localized spot on the disc. The spot
may either move along a stable circular orbit or fall from the vicinity of the
marginally stable orbit down to the horizon.
In the former case the spot is orbiting with a Keplerian velocity, in the latter
it falls with energy and angular momentum of the matter on the marginally
stable orbit.
Because the emission changes in time this model cannot be included into
{\sc{}xspec} which
cannot handle non-stationary problems. In spite of this we may include
integrated emission received by the observer in a certain time span
(corresponding to a certain part of the orbit) in future.
In this model it is assumed that the corona above the disc is heated by a flare.
Thus a hot cloud is formed, which illuminates the disc below it by X-rays.
The disc reflects this radiation by Compton scattering and by fluorescence.
In the current model we consider only the fluorescent part of the reflection.
The local emission from the disc is
\begin{eqnarray}
\label{spot1a}
N_{{\rm l}}(E_{\rm l})\hspace*{-0.5em} & = & \hspace*{-0.5em}
f(\mu_{\rm e})\,\exp{\left
[-\left (\frac{E_{\rm l}-{\tt Erest}}{\sqrt{2}\,\sigma}\right )^2
\right ]}\,\exp{\left[-{\tt beta}\,(\Delta r)^2\right]}\ \
{\rm for}\ \ {\tt beta}\,(\Delta r)^2 < 4\, ,\hspace*{10mm}\\
\label{spot1b}
N_{{\rm l}}(E_{\rm l})\hspace*{-0.5em} & = & \hspace*{-0.5em}
0\quad {\rm for}\quad {\tt beta}\,(\Delta r)^2 \ge 4\, .
\end{eqnarray}
Here the function $f(\mu_{\rm e})$ describes the limb darkening/brightening
law (see Section~\ref{section_kyg1line} for more details), {\tt Erest} is
the local energy of the
fluorescent line, $\sigma=2\,$eV is its width, {\tt beta} determines the size of
the spot and $(\Delta r)^2=r^2+r_{\rm spot}^2-2\,r\,r_{\rm spot}\cos{(\varphi-
\varphi_{\rm spot})}$ with $r_{\rm spot}$ and $\varphi_{\rm spot}$ being polar
coordinates of the centre of the spot. The emission is largest at the centre
of the spot and decreases towards its edge as is obvious from eq.~(\ref{spot1a}).
The Gaussian line in energies is defined by nine points equally spaced with
the central point at its maximum.
\begin{figure}[tb]
\dummycaption\label{fig:spot1}
\includegraphics[width=0.5\textwidth]{large_kepler_spot}
\includegraphics[width=0.5\textwidth]{obscured_small_kepler_spot}
\mycaption{Dynamical profile of an iron line ($E_{\rm rest}=6.4\,$keV) produced by
an orbiting spot. Energy is on the ordinate, time on
the abscissa. The horizontal range spans the interval of one
orbital period. Time zero corresponds to the time when the observer receives
photons from the spot at the closest approach to the observer. Left: Large spot
($\beta=0.1$) at the radius $r_{\rm spot}=8GM/c^2$.
Right: Obscured small spot ($\beta=4$) at the radius $r_{\rm spot}=5GM/c^2$.
Obscuration occurs between $315^\circ-540^\circ$ measured from the closest
approach. In both cases the dimensionless angular momentum parameter
of the black hole is $a=1$ and observer inclination is
$\theta_{\rm{}o}=45^{\circ}$. The observed photon flux
is colour-coded (logarithmic scale with arbitrary units).}
\end{figure}
\begin{table}[tbh]
\begin{center}
\dummycaption\label{kyspot_par}
\begin{tabular}[h]{r@{}l|c|c|c|c}
\multicolumn{2}{c|}{parameter} & unit & default value & minimum value &
maximum value \\ \hline
&{\tt a/M} & $GM/c$ & 0.9982 & 0. & 1. \\
&{\tt theta\_o} & deg & 30. & 0. & 89. \\
&{\tt rin-rh} & $GM/c^2$ & 0. & 0. & 999. \\
&{\tt rout-rh} & $GM/c^2$ & 20. & 0. & 999. \\
&{\tt phi} & deg & 0. & -180. & 180. \\
&{\tt dphi} & deg & 360. & 0. & 360. \\
&{\tt nrad} & -- & 300. & 1. & 10000. \\
&{\tt division} & -- & 0. & 0. & 1. \\
&{\tt nphi} & -- & 750. & 1. & 20000. \\
&{\tt smooth} & -- & 1. & 0. & 1. \\
&{\tt zshift} & -- & 0. & -0.999 & 10. \\
&{\tt ntable} & -- & 0. & 0. & 99. \\
{*}&{\tt Erest} & keV & 6.4 & 1. & 99. \\
{*}&{\tt sw} & -- & 1. & 1. & 2. \\
{*}&{\tt beta} & -- & 0.1 & 0.001 & 1000. \\
{*}&{\tt rsp} & $GM/c^2$ & 8. & 0. & 1000. \\
{*}&{\tt psp} & deg & 90. & -360. & 360. \\
{*}&{\tt Norbits} & -- & 1. & 0. & 10 \\
{*}&{\tt nt} & -- & 500. & 1. & 1e6 \\
{*}&{\tt limb} & -- & 0. & -10. & 10. \\
{*}&{\tt polar} & -- & 0. & 0. & 1. \\
\end{tabular}
\mycaption{Parameters of the non-stationary model {\sc{}kyspot}.
Model parameters that are not common for all non-axisymmetric models are denoted
by asterisk.}
\end{center}
\end{table}
The local emission is defined by the following parameters
(see Tab.~\ref{kyspot_par}):
\begin{description} \itemsep -2pt
\item[{\tt Erest}] -- rest energy of the line in keV,
\item[{\tt sw}] -- switch for choosing the type of spot (1 -- orbiting, 2 --
falling),
\item[{\tt beta}] -- parameter defining the size of the spot,
\item[{\tt rsp}] -- radius (in $GM/c^2$) at which the spot is orbiting
or the initial radius in units of the marginally stable orbit
$r_{\rm ms}$ if spot is in-falling,
\item[{\tt psp}] -- azimuthal angle in degrees where the spot is initially
located ($90^\circ$ for the closest approach),
\item[{\tt Norbits}] -- number of orbits for an in-falling spot (not used for
an orbiting spot),
\item[{\tt nt}] -- time resolution of the grid,
\item[{\tt limb}] -- switch for different limb darkening/brightening laws,
\item[{\tt polar}] -- switch for polarization calculations (0 -- not performed,
1 -- performed).
\end{description}
The {\sc kyspot} model creates an {\sf ascii} file {\tt kyspot.dat}, where the
dependence of the observed spectrum (columns) on time (rows) are stored. If the
{\tt polar} switch is set to unity then two other {\sf ascii} files are created
-- {\tt kyspot\_poldeg.dat} and {\tt kyspot\_psi.dat}. Here, the dependence of
the degree of polarization and the angle of polarization on time are stored.
The polarization calculations in this model are based on the same assumptions
as in the {\sc kyg1line} model.
\begin{figure}[tb]
\dummycaption\label{fig:example2}
\hspace*{0.05\textwidth}
\includegraphics[width=0.33\textwidth]{trajectory}
\hspace*{0.05\textwidth}
\includegraphics[width=0.5\textwidth]{falling_spot}
\mycaption{Dynamical profile of an iron line produced by
an in-spiralling small spot ($\beta=100$) in free fall. Left: Trajectory
(in the equatorial plane) of the spot,
defined by a constant energy and angular momentum during the in-fall.
The observer is located to the top of the figure.
Right: Spectrum of the spot. Energy is on the ordinate, time on
the abscissa. The horizontal range spans the interval of $2.4$
orbital periods at the corresponding initial radius,
$r_{\rm spot}=0.93\,r_{\rm{}ms}(a)$. Here, the dimensionless angular
momentum parameter
of the black hole is $a=0.9$, observer inclination is
$\theta_{\rm{}o}=45^{\circ}$. The observed photon flux
is colour-coded (logarithmic scale
with arbitrary units). Gradual decay of the signal and an
increasing centroid redshift can be observed as the spot
completes over two full revolutions and eventually plunges
into the black hole.}
\end{figure}
Examples of dynamical spectra of orbiting and in-falling
spots can be seen in Figs.~\ref{fig:spot1}--\ref{fig:example2}.
The first figure represents an evolving spectral line
in a dynamical diagram for an orbiting spot of two different sizes
one of which is being obscured on part of its orbit around the black hole.
The characteristic radius of the larger spot is $\sim6.3GM/c^2$ and
the radius of the smaller spot is $\sim1GM/c^2$. The second figure represents
the dynamical spectra of the spot falling from the marginally stable orbit down
towards the black-hole horizon. The
characteristic radius of the spot is $\sim0.2GM/c^2$. This figure reflects the
effect of changing the energy shift along the spot trajectory.
The in-spiral motion starts just below the
marginally stable orbit (${\tt rsp}=0.93$) and it proceeds down
to the horizon, maintaining the specific energy and
angular momentum of its initial circular orbit at $r_{\rm ms}(a)$.
One can easily recognize that variations in the redshift could
hardly be recovered from data if only time-averaged spectrum were
available. In particular, the motion above $r_{\rm{}ms}(a)$ near a
rapidly spinning black hole is difficult to distinguish from a
descent below $r_{\rm{}ms}(0)$ in the non-rotating case.
\begin{table}[tbh]
\begin{center}
\dummycaption\label{tab:versions}
\begin{tabular*}{\textwidth}{p{0.1\textwidth}p{0.13\textwidth}p{0.20\textwidth}
p{0.46\textwidth}}
\hline\hline
Name \rule[-1.5ex]{0mm}{4.5ex}& Type$^{\dag}$~ & Symmetry & Usage \\
\hline\hline
{\sf{}KYGline} \break {\sf{}KYG1line} & additive & axisymmetric \break
non-axisymmetric & spectral line from a black-hole disc \\
\hline
{\sf{}KYF1ll} & additive & non-axisymmetric & lamp-post fluorescent line\\
\hline
{\sf{}KYL1cr} & additive & non-axisymmetric & lamp-post Compton reflection
model, physical model of polarization is included \\
\hline
{\sf{}KYH1refl} & additive & non-axisymmetric & Compton reflection with an
incident power-law (or a broken power-law) continuum \\
\hline
{\sf{}KYConv} \break {\sf{}KYC1onv} & convolution & axisymmetric \break
non-axisymmetric & convolution of the relativistic kernel with intrinsic
emissivity across the disc \\
\hline
{\sf{}KYSpot} & \multicolumn{2}{p{0.33\textwidth}}{to be used as an independent
code outside {\sc{}xspec}} & time-dependent spectrum of a pattern co-orbiting
with the disc \\
\hline\hline
\end{tabular*}
\parbox{\textwidth}{\vspace*{1mm}\footnotesize{}$^{\dag}$~Different
model types correspond to the {\sc{}xspec} syntax and are
defined by the way they act in the overall model and form the
final spectrum. According to the usual convention in {\sc{}xspec},
additive models represent individual {\it emission}\/ spectral components,
which may, for instance, originate in different regions of the source.
Additive models are simply superposed in the total signal.
Multiplicative components (e.g.\ {\sc{}hrefl}) multiply the current model by an
energy-dependent
factor. Convolution models modify the model in a more non-trivial
manner. See \cite{arnaud1996} for details.}
\mycaption{Different versions of the {\sc{}ky} model.}
\end{center}
\end{table}
\section{Summary of the new model}
The new {\sc{}ky} model is suited for use with the {\sc{}xspec}
package \citep{arnaud1996}. Several mutations of {\sc{}ky} were developed,
with the aim to specifically provide different applications, and linked
with a common ray-tracing subroutine, which therefore does not have to
be modified when the intrinsic emissivity function in the model is
changed.
When the relativistic line distortions are computed, the new model is
more accurate than the {\sc{}laor} model \citep{laor1991} and faster than
{\sc{}kerrspec} \citep{martocchia2000}. These are the other
two {\sc{}xspec} models with a similar usage (see also Pariev \& Bromley
\citeyear{pariev1998};
\citealt{gierlinski2001,beckwith2004,schnittman2004}). It is also important
to compare the results from fully independent relativistic codes since
the calculations are sufficiently complex for significant differences
to arise. Our code is more general than the currently available
alternatives in {\sc xspec}.
In our {\sc{}ky} model it is possible to choose from various limb
darkening/brightening laws and thus change the angular distribution
of the local emission. The problem of directional distribution of the reflected
radiation is quite complicated and the matter has not been completely
settled yet (e.g.\ \citealt{george1991,zycki1994,magdziarz1995}, and
references cited therein). A specific angular
dependence is often assumed in models, such as limb darkening
$f(\mu_{\rm e})=1+2.06\mu_{\rm e}$ in the {\sc{}laor} model. However,
it has been argued that limb brightening may actually occur in the case
of strong primary irradiation of the disc. This is relevant for
accretion discs near black holes, where the effects of emission
anisotropy are crucial \citep{beckwith2004,czerny2004}. We
find that the spectrum of the inner disc turns out to be very sensitive
to the adopted angular dependence of the emission, and so the
possibility to modify this profile and examine the results using
{\sc{}ky} appears to be rather useful.
Among its useful features, the {\sc{}ky} model allows one to fit various
parameters such as the black-hole angular momentum ($a$), the observer
inclination angle relative to the disc axis ($\theta_{\rm{}o}$),
and the size and shape of the emission area on the disc,
which can be non-axisymmetric. A straightforward modification of a single
subroutine suffices to alter the prescription for the disc emissivity,
which is specified either by an analytical formula or in a tabular form.
Our code allows one to change the mesh spacing and resolution for
the (two-dimensional) polar grid that covers the disc plane, as well
as the energy vector (the output resolution is eventually determined
by the detector in use when the model is folded through the instrument
response). Hence, there is sufficient control of the (improved) accuracy and
computational speed.
Furthermore, {\sc{}ky} can be run as a stand-alone program
(detached from {\sc{}xspec}). In this mode there is an option
for time-variable sources such as orbiting spots,
spiral waves or evolving flares (e.g.\ \citealt{czerny2004},
who applied a similar approach to compute the predicted {\sf{}rms}
variability in a specific flare/spot model).
The improved accuracy of the new model has been achieved in several
ways: (i)~photon rays are integrated in Kerr ingoing coordinates
which follow principal photons, (ii)~simultaneous integration of
the geodesic deviation equations ensures accurate evaluation of the
lensing effect, and (iii)~non-uniform and rather fine grids
have been carefully selected.
Several versions of the routine have been prefabricated for different
types of sources (Tab.~\ref{tab:versions}): (i)~an intrinsically narrow
line produced by a disc, including the lamp-post fluorescent line model
\citep{martocchia2000}, (ii)~the Compton reflection model in two
variants -- a relativistically blurred Compton-reflection continuum
including a primary power-law component and a lamp-post model
\citep{martocchia2000}, (iii) general convolution models,
and (iv) the time-dependent spectrum of an orbiting or a free-falling spot.
Default parameter values for the line model correspond to those in the
{\sc{}laor} model, but numerous options have been added. For example, in
the new model one is able to set the emission inner radius below the
marginally stable orbit, $r_{\rm{}in}<r_{\rm{}ms}(a)$. One can also
allow $a$ to vary independently, in which case the horizon radius,
$r_{\rm{}h}(a)\equiv1+\sqrt{1-a^2}$, has to be, and indeed is, updated
at each step of the fit procedure. We thus define emission radii in
terms of their offset from the horizon. Several arguments have been
advocated in favour of having $r_{\rm{}in}{\neq}r_{\rm{}ms}$ for the disc
emission, but this possibility has never been tested rigorously against
observational data. The set of {\sc{}ky-}routines introduced above
provide the tools to explore black-hole disc models and to actually fit
for their key parameters, namely, $a$, $\theta_{\rm{}o}$, and
$r_{\rm{}in}$.
We have also produced a convolution-type
model, {\sc{}kyconv}, which can be applied to any existing
{\sc{}xspec} model of intrinsic X-ray emission (naturally,
a meaningful combination of the models is the responsibility of
the user). We remind the reader that {\sc{}kyconv} is substantially more
powerful than the usual convolution models in {\sc{}xspec}, which are
defined in terms of a one-dimensional integration over energy bins.
Despite the fact that {\sc{}kyconv}
still uses the standard {\sc{}xspec} syntax in evaluating the observed
spectrum (e.g.\ \hbox{\sc{}kyconv(gaussian)}), our code
performs a more complex operation. It still performs
ray-tracing across the disc surface so that the intrinsic model
contributions are integrated from different radii. Thus the
{\sc{}kyconv(gaussian)} model gives the same results as the {\sc{}kygline}
model if corresponding parameters are set to the same values. The price that
one has to pay for the enhanced functionality is a higher
demand on computational power.
Other user-defined emissivities can be easily adopted. This can be
achieved either by using the above mentioned convolution model
or by adding a new user-defined model to {\sc{}xspec}. The latter method is
more flexible and faster, and hence recommended. In both approaches, the
ray-tracing routine is linked and used for relativistic blurring.
\chapter*{Preface}
\thispagestyle{empty}
\input{preface.tex}
\clearpage
\addcontentsline{toc}{chapter}{Introduction}
\markboth{INTRODUCTION}{INTRODUCTION}
\chapter*{Introduction}
\thispagestyle{empty}
\input{introduction.tex}
\chapter{Transfer functions for a thin disc in Kerr space-time}
\chaptermark{Transfer functions for a thin disc \ldots}
\label{chapter1}
\thispagestyle{empty}
\input{functions.tex}
\chapter{Radiation of accretion discs in strong gravity}
\thispagestyle{empty}
\input{models.tex}
\chapter{Applications}
\label{chapter4}
\thispagestyle{empty}
\input{yaqoob.tex}\clearpage
\input{spot.tex}\clearpage
\input{polar.tex}
\clearpage
\addcontentsline{toc}{chapter}{Summary and future prospects}
\chapter*{Summary and future prospects}
\thispagestyle{empty}
\input{summary.tex}
\section{Polarization signatures of strong gravity in AGN accretion discs}
Accretion discs in central regions of active galactic nuclei are
subject to strong external illumination originating from some kind
of corona and giving rise to specific spectral features in the X-ray
band. In particular, the K-shell lines of iron are found to be prominent
around $6-7$~keV. It has been shown that the shape of the intrinsic
spectra must be further modified by the strong gravitational field of
the central mass, and so X-ray spectroscopy could allow us to explore
the innermost regions of accretion flows near supermassive black holes
\citep[for recent reviews see][]{fabian2000,reynolds2003}.
Similar mechanisms operate also in some Galactic black-hole candidates.
A rather surprising result from recent {\it XMM-Newton}\/ observations is
that relativistic iron lines are not as common as previously believed,
see \cite{bianchi2004} and references therein and also \cite{yaqoob2003}.
This does not necessarily mean that the iron line is not produced in
the innermost regions of accretion discs. The situation is likely to be more
complex than in simple, steady scenarios, and indeed some evidence for
line emission arising from orbiting spots is present in the
time-resolved spectra of a few AGNs \citep{dovciak2004a}. Even
when clearly observed, relativistic lines behave differently than
expected. The best example is the puzzling lack of correlation between
line and continuum emission in MCG--6-30-15 \citep{fabian2002},
unexpected because the very broad line profile clearly indicates that
the line originates in the innermost regions of the accretion disc,
hence very close to the illuminating source. \cite{miniutti2003}
have proposed a solution to this problem in terms of an illuminating
source moving along the black-hole rotation axis or very close to it.
In this section we show that polarimetric studies could provide additional
information about accretion discs in a strong gravity regime, which may be
essential to discriminate between different possible geometries of the
source. The idea of using polarimetry to gain additional information
about accreting compact objects is not a new one. In this context it was
proposed by \cite{rees1975} that polarized X-rays are of high
relevance.\break
\cite{pozdnyakov1979} studied spectral profiles of iron
X-ray lines that result from multiple Compton scattering. Later on,
various influences affecting polarization (due to magnetic fields,
absorption as well as strong gravity) were examined for black-hole
accretion discs \citep{agol1997}. Temporal variations of polarization were
also discussed, in particular the case of orbiting spots near a black
hole (\citealt{connors1980};\break \citealt{bao1996}). With
the promise of new polarimetric detectors \citep{costa2001},
quantitative examination of specific models becomes timely.
\begin{figure}[tbh!]
\vspace*{1.em}
\dummycaption\label{pol}
\includegraphics[width=0.328\textwidth]{pol_angle_0_30}
\hfill
\includegraphics[width=0.328\textwidth]{pol_angle_0_60}
\hfill
\includegraphics[width=0.328\textwidth]{pol_angle_0_80}
\includegraphics[width=0.328\textwidth]{pol_deg_0_30}
\hfill
\includegraphics[width=0.328\textwidth]{pol_deg_0_60}
\hfill
\includegraphics[width=0.328\textwidth]{pol_deg_0_80}
\includegraphics[width=0.328\textwidth]{pola_deg_0_30}
\hfill
\includegraphics[width=0.328\textwidth]{pola_deg_0_60}
\hfill
\includegraphics[width=0.328\textwidth]{pola_deg_0_80}
\mycaption{Energy dependence of polarization angle (top panels) and polarization
degree (middle panels) due to reflected radiation for different observer's
inclination angles ($\theta_{\rm o}=30^\circ,\,60^\circ$ and $80^\circ$) and for
different heights of the primary source ($h=2,\,6,\,15$ and $100$).
Polarization degree for reflected plus direct radiation is also plotted (bottom
panels). The emission comes from a disc within $r_{\rm{}in}=6$ and
$r_{\rm out}=400$. Isotropic primary radiation with photon index $\Gamma=2$ and
angular momentum of the central black hole $a=0.9987$ were assumed.}
\end{figure}
Since the reflecting medium has a disc-like geometry, a substantial
amount of linear polarization is expected in the resulting spectrum
because of Compton scattering. Polarization properties of the disc
emission are modified by the photon propagation in a gravitational field,
providing additional information on its structure. Here we
calculate the observed polarization of the reflected radiation assuming
the lamp-post model for the stationary power-law illuminating source
\citep{martocchia1996, petrucci1997}. We assume a rotating (Kerr) black hole
as the only source of the gravitational field, having a common symmetry
axis with an accretion disc. The disc is
also assumed to be stationary and we restrict ourselves to the
time-averaged analysis. In other words, we examine processes that vary
at a much slower pace than the light-crossing time at the corresponding
radius. Intrinsic polarization of the emerging
light can be computed locally, assuming a plane-parallel scattering
layer which is illuminated by light radiated from the primary source.
This problem was studied extensively in various
approximations \citep[e.g.][]{chandrasekhar1960,sunyaev1985}.
Here we employ the Monte Carlo computations \citep{matt1991,
matt1993b} and thus we find the intrinsic emissivity of an
illuminated disc.
Then we integrate contributions to the total signal
across the disc emitting region using a general relativistic
ray-tracing technique described in previous chapters and we compute the
polarization angle and degree as measured by a distant observer (see
Section~\ref{stokes_param} and equations therein). We show the
polarization properties of scattered light as a function of model
parameters, namely, the height $z=h$ of the primary source on the symmetry
axis, the dimensionless angular momentum $a$ of the black hole, and the
viewing angle $\theta_{\rm{}o}$ of the observer.
In the first set of figures (Figs.~\ref{pol}--\ref{pol1}) we show the energy
dependence of polarization angle and degree due to reflected and reflected
plus direct radiation for different inclination angles and different heights
of the primary source. One can see that the polarization of reflected radiation
can be as high as thirty percent or even more for small inclinations and
small heights. Polarization of the reflected radiation does not depend on energy
very much except for the region close to the iron edge at approximately
$7.2\,$keV, where it either decreases for small inclinations or increases
for large ones.
\begin{figure}[tb]
\vspace*{1em}
\dummycaption\label{pol1}
\includegraphics[width=0.328\textwidth]{pol_angle_1_30}
\hfill
\includegraphics[width=0.328\textwidth]{pol_angle_1_60}
\hfill
\includegraphics[width=0.328\textwidth]{pol_angle_1_80}
\includegraphics[width=0.328\textwidth]{pol_deg_1_30}
\hfill
\includegraphics[width=0.328\textwidth]{pol_deg_1_60}
\hfill
\includegraphics[width=0.328\textwidth]{pol_deg_1_80}
\includegraphics[width=0.328\textwidth]{pola_deg_1_30}
\hfill
\includegraphics[width=0.328\textwidth]{pola_deg_1_60}
\hfill
\includegraphics[width=0.328\textwidth]{pola_deg_1_80}
\mycaption{Same as in the previous figure but for disc starting at
$r_{\rm in}=1.20\,$.}
\vspace*{1em}
\end{figure}
\begin{figure}
\dummycaption\label{poldeg}
\includegraphics[width=0.49\textwidth]{pol_0}
\hfill
\includegraphics[width=0.49\textwidth]{pol_1}
\mycaption{Polarization degree and angle due to reflected radiation integrated
over the whole surface of the disc and propagated to
the point of observation. Dependence on height $h$ is plotted.
Left panel: $r_{\rm{}in}=6$; right panel: $r_{\rm{}in}=1.20$. In both
the panels the energy range was assumed $9-12$~keV, the photon index of
incident radiation $\Gamma=2$, the angular momentum $a=0.9987$.}
\end{figure}
\begin{figure}
\dummycaption\label{polangle1}
\includegraphics[width=0.48\textwidth]{pol0_0}
\hfill
\includegraphics[width=0.48\textwidth]{pol0_1}
\mycaption{Polarization degree and angle as functions of
$\mu_{\rm{}o}$ (cosine of observer inclination, $\mu_{\rm{}o}=0$
corresponds to the edge-on view of the disc). The same model
is shown as in the previous figure.}
\end{figure}
\begin{figure}
\dummycaption\label{poldeg1}
\includegraphics[width=0.48\textwidth]{pol1_0}
\hfill
\includegraphics[width=0.48\textwidth]{pol1_1}
\mycaption{Net polarization degree of the total (primary
plus reflected) signal as a function of $h$.
Left panel: $r_{\rm{}in}=6$; right panel: $r_{\rm{}in}=1.20$.
The curves are parametrized by the corresponding energy range.}
\end{figure}
\begin{figure}
\dummycaption\label{poldeg2}
\includegraphics[width=0.48\textwidth]{pol2_0}
\hfill
\includegraphics[width=0.48\textwidth]{pol2_1}
\mycaption{Net polarization degree of the total (primary
plus reflected) signal as a function of $\mu_{\rm{}o}$.
The same model is shown as in the previous figure.}
\end{figure}
In order to compute observable characteristics one has to
combine the primary power-law continuum with the reflected component.
The polarization
degree of the resulting signal depends on the mutual proportion of the two
components and also on the energy range of an observation. The overall degree of
polarization increases with energy (see bottom panels in
Figs.~\ref{pol}--\ref{pol1}) due to the fact that the intensity of radiation
from the primary source decreases exponentially, the intensity of the reflected
radiation increases with energy (in the energy range $3-15\,$keV) and
the polarization of the reflected light alone is more or less constant.
In our computations we assumed that the irradiating source
emits isotropically and its light is affected only by gravitational redshift
and lensing, according to the source location at $z=h$ on axis. This
results in a dilution of primary light by factor
${\sim}g_{\rm{}h}^2(h,\theta_{\rm{}o})\,l_{\rm{}h}(h,\theta_{\rm{}o})$,
where $g_{\rm{}h}=\sqrt{1-2h/(a^2+h^2)}$ is the redshift of primary
photons reaching directly the
observer, $l_{\rm{}h}$ is the corresponding lensing factor. Here, the redshift
is the dominant relativistic term, while lensing of primary photons is a few
percent at most and it can be safely ignored. Anisotropy of primary radiation
may further attenuate or amplify the polarization degree of the final signal,
while the polarization angle is rather independent of this influence as long
as the primary light is itself unpolarized.
The polarization of scattered light is also shown in Fig.~\ref{poldeg},
where we plot the polarization degree and the change of the polarization
angle as functions of $h$. Notice that in the Newtonian case only
polarization angles of $0^{\circ}$ or $90^{\circ}$ would be expected
for reasons of symmetry. The change in angle is due to gravitation for
which we assumed a rapidly rotating black hole.
The two panels in the figure correspond to different locations of
the inner disc edge: $r_{\rm{}in}=6$ and $r_{\rm{}in}=1.20$,
respectively. The curves are strongly sensitive to $r_{\rm{}in}$ and
$h$, while the dependence on $r_{\rm{}out}$ is weak for a large disc
(here $r_{\rm{}out}=400$). Sensitivity to $r_{\rm{}in}$ is particularly
appealing if one remembers the practical difficulties in estimating
$r_{\rm{}in}$ by fitting spectra. The effect is clearly visible even
for $h\sim20$. Graphs
corresponding to $r_{\rm{}in}=6$ and $a=0.9987$, resemble, in essence quite
closely, the non-rotating case ($a=0$) because dragging effects are most
prominent near the horizon.
Fig.~\ref{polangle1} shows the polarization
degree and angle as functions of the observer's inclination. Again, by
comparing the two cases of different $r_{\rm{}in}$ one can clearly
recognize that the polarization is sensitive to details of the flow near
the inner disc boundary.
The dependence of the polarization degree of overall radiation (primary plus
reflected) on the height of the primary source and the observer inclination
in different energy ranges is shown in Figs.~\ref{poldeg1}--\ref{poldeg2}.
In this section we examined the polarimetric properties of X-ray illuminated
accretion discs in the lamp-post model. From the figures shown it is clear
that observed values of polarization angle and degree are
rather sensitive to the model parameters. The approach adopted
provides additional information with respect to traditional
spectroscopy and so it has great potential for discriminating between
different models. It offers an improved way of measuring rotation of the
black hole because the radiation properties of the inner disc region
most likely reflect the value of the black-hole angular momentum.
While our calculations have been
performed assuming a stationary situation, in reality it is likely that
the height of the illuminating source changes with time, and indeed such
variations have been invoked by \cite{miniutti2003} to explain the
primary and reflected variability patterns of MCG--6-30-15. A complete
time-resolved analysis (including all consequences of the light travel
time in curved space-time) is beyond the scope of this section and we defer
it to future work, assuming that the primary source varies on a
time-scale longer than light-crossing time in the system. This is also a
well-substantiated assumption from a practical point of view,
since feasible techniques will anyway require sufficient integration
time (i.e.\ order of several ksec). Once full temporal resolution is
possible, the analysis described above can be readily extended. Here,
it suffices to note that a variation of $h$ implies a variation of the
observed polarization angle of the reflected radiation. As it is hard to
imagine a physical and/or geometrical effect giving rise to the same
effect, time variability of the polarization angle can be considered
(independently of the details) a very strong signature of strong-field
general relativity effects at work.
New generation photoelectric polarimeters \citep{costa2001} in the
focal plane of large area optics (such as those foreseen for
{\it{}Xeus}) can probe the polarization degree of the order of one percent
in bright AGNs, making polarimetry, along with timing and spectroscopy,
a tool for exploring the properties of the accretion flows in the
vicinity of black holes.
\section[Relativistic spectral features from X-ray illuminated spots]
{Relativistic spectral features from X-ray illuminated \break spots}
Relativistic iron line profiles may provide a powerful tool for
measuring the mass of the black hole in active galactic nuclei
and Galactic black-hole candidates. For this aim,
\cite{stella1990} proposed to use temporal changes in the line
profile following variations of the illuminating primary source
(which at that time was assumed to be located on the disc axis
for simplicity). Along the same line of thought,
\cite{matt1992a} proposed to employ, instead, variations of the
integrated line properties such as equivalent width, centroid energy and
line width. These methods are very similar conceptually to the
classical reverberation mapping method, widely and successfully applied
to optical broad lines in AGNs. Sufficiently long
monitoring of the continuum and of the line emission is required,
as well as large enough signal-to-noise ratio. However, the
above-mentioned methods have not provided many results yet. Even in the
best studied case of the Seyfert galaxy MCG--6-30-15, the mass estimate is
hard to obtain due to the apparent lack of correlation between
the line and continuum emission \citep{fabian2002}. It was also suggested
that these complications are possibly caused by an interplay of
complex general relativistic effects \citep{miniutti2003}.
X-ray spectra from high throughput and high energy resolution
detectors should resolve the problem of interpretation of
observed spectral features. However, before such high quality data are
available it is desirable to examine existing spectra and attempt to
constrain physical parameters of the models.
A simple, direct and potentially robust way to measure the
black-hole mass would be available if the line emission originates
at a given radius and azimuth, as expected if the disc
illumination is provided by a localized flare just above the
disc (possibly due to magnetic reconnection), rather than a central
illuminator or an extended corona. If a resulting `hot spot' co-rotates with the
disc and lives for at least a significant part of an orbit, by fitting
the light curve and centroid energy of the line flux, the inclination
angle $\theta_{\rm{}o}$ and the orbit radius could be derived (radius in units of
the gravitational radius $r_{\rm{}g}$). Further, assuming Keplerian rotation,
the orbital period is linked with radius in a well-known manner. The equation
for the orbital period then contains the black-hole mass $M_{\bullet}$
explicitely, and so this parameter can be determined, as
discussed later.
Hot spots in AGN accretion discs were popular for a while, following the
finding of apparent periodicity in the X-ray emission of the Seyfert~1
galaxy NGC~6814. They, however, were largely abandoned when this periodicity
was demonstrated to be associated with an AM~Herculis system in the field of view
rather than the AGN itself\break \citep{madejski1993}. Periodicities in AGNs
were subsequently reported in a few sources \citep{iwasawa1998,lee2000,
boller2001}. The fact that they were not confirmed in different
observations of the same sources is not
surprising -- quite on the contrary, it would be hard to imagine a hot spot
surviving for several years.
Recently, the discovery of narrow emission features in the X-ray spectra
of several AGNs \citep{turner2002,guainazzi2003,yaqoob2003,turner2004}
has renewed interest in hot spots. There is a tentative
explanation (even if not the only one) for these features,
typically observed in the $5-6$~keV energy range, in terms of iron
emission produced in a small range of radii and distorted by
joint action of Doppler and gravitational shift of photon energy.
Iron lines would be produced by localized flares
which illuminate the underlying disc surface, producing the line by
fluorescence. Indeed, the formation of
magnetic flares on the disc surface is one of the
most promising scenarios for the X-ray emission of AGNs. A particularly
strong flare, or one with a very large anisotropic emission towards the
disc, could give rise to the observed features. Small width of the
observed spectral
features implies that the emitting region must be small, and that
it is seen for only a fraction of the entire orbit
(either because the flare dies out, or
because emission goes below detectability, see next section).
If the flares co-rotate with the disc and if they last
for a significant part of the orbit, it may be possible
by observing their flux and energy
variations with phase to determine the orbital parameters, and thence
$M_{\bullet}$.
The basic properties of line emission from the innermost regions of an
accretion disc around a black hole are well-known \citep[see e.g.]
[for recent reviews]{fabian2000,reynolds2003}. Let us here briefly summarize
several formulae most relevant to our purposes.
If $r$ is the orbital radius and $a$ is the dimensionless black-hole
angular momentum, the orbital period of matter co-rotating along a
circular trajectory $r=\mbox{const}$ around the black hole
is given by \citep{bardeen1972}
\begin{equation}
T_{\rm{}orb} \doteq 310~\left(r^\frac{3}{2}+a\right)
\frac{M_{\bullet}}{10^7M_{\odot}}\quad\mbox{[sec]},
\label{torb}
\end{equation}
as measured by a distant observer. We express lengths
in units of the gravitational radius
$r_{\rm{}g}{\equiv}G{M_{\bullet}}/c^2{\doteq}1.48\times10^{12}M_7$~cm, where
$M_7$ is the mass of the black hole in units of $10^7$ solar masses.
Angular momentum $a$ (per unit mass) is in geometrized units
($0\leq{a}\leq1$). See e.g.\ \cite{misner1973} for useful conversion
formulae between geometrized and physical units.
The innermost stable orbit,
$r_{\rm ms}$, occurs for an equatorial disc at radius
\begin{equation}
r_{\rm ms} = 3+Z_2-\big[\left(3-Z_1)(3+Z_1+2Z_2\right)\big]^{1 \over 2},
\label{velocity1}
\end{equation}
where
$Z_1 = 1+(1-a^2)^{1 \over 3}[(1+a)^{1 \over 3}+(1-a)^{1 \over 3}]$
and $Z_2 = (3a^2+Z_1^2)^{1 \over 2}$;
$r_{\rm ms}$ spans the range of radii from $r=1$ ($a=1$, i.e.\ the case
of a maximally rotating black hole) to $6$
($a=0$, a static black hole). Rotation of a black hole
is believed to be limited by an equilibrium value
$a\dot{=}0.998$ because of the capture of photons from the disc
\citep{thorne1974}.
This would imply $r_{\rm{}ms}\dot{=}1.23$. Different specific models
of accretion can result in somewhat different limiting values
of $a$ and the corresponding $r_{\rm{}ms}$. Notice that in the static case,
the radial dependence $T_{\rm{}orb}(r)_{{\mid}a=0}$ is identical to that
in purely Newtonian gravity.
\begin{figure}
\dummycaption\label{orbits_1}
\includegraphics[width=0.325\textwidth,height=6cm]{mat3b_10_30_00}
\hfill
\includegraphics[width=0.325\textwidth,height=6cm]{mat3b_10_60_00}
\hfill
\includegraphics[width=0.325\textwidth,height=6cm]{mat3b_10_85_00}
\vspace*{3mm}\\
\includegraphics[width=0.325\textwidth,height=6cm]{mat3b_09_30_00}
\hfill
\includegraphics[width=0.325\textwidth,height=6cm]{mat3b_09_60_00}
\hfill
\includegraphics[width=0.325\textwidth,height=6cm]{mat3b_09_85_00}
\vspace*{3mm}\\
\includegraphics[width=0.325\textwidth,height=6cm]{mat3b_00_30_00}
\hfill
\includegraphics[width=0.325\textwidth,height=6cm]{mat3b_00_60_00}
\hfill
\includegraphics[width=0.325\textwidth,height=6cm]{mat3b_00_85_00}
\vspace*{0mm}
\mycaption{Line flux and centroid energy as functions of the
orbital phase of a spot, for three values of angular momentum
($a=1$, $0.9$, and $0$) and three inclination angles
($\theta_{\rm{}o}=30^{\circ}$, $60^{\circ}$, $85^{\circ}$).
The centre of the spot is located at radial distance $r$, which corresponds
to the last stable orbit $r_{\rm{}ms}(a)$ for that angular momentum
plus a small displacement given by the spot radius, ${\rm{}d}r$.
The intrinsic energy of the line emission is assumed to be at
$6.4$~keV (indicated by a dotted line). Prograde rotation is assumed.
Time is expressed in orbital periods.}
\end{figure}
\begin{figure}
\dummycaption\label{orbits_2}
\includegraphics[width=0.325\textwidth,height=6cm]{mat3b_00_30_10}
\hfill
\includegraphics[width=0.325\textwidth,height=6cm]{mat3b_00_60_10}
\hfill
\includegraphics[width=0.325\textwidth,height=6cm]{mat3b_00_85_10}
\vspace*{3mm}\\
\includegraphics[width=0.325\textwidth,height=6cm]{mat3b_00_30_20}
\hfill
\includegraphics[width=0.325\textwidth,height=6cm]{mat3b_00_60_20}
\hfill
\includegraphics[width=0.325\textwidth,height=6cm]{mat3b_00_85_20}
\vspace*{3mm}\\
\includegraphics[width=0.325\textwidth,height=6cm]{mat3b_00_30_50}
\hfill
\includegraphics[width=0.325\textwidth,height=6cm]{mat3b_00_60_50}
\hfill
\includegraphics[width=0.325\textwidth,height=6cm]{mat3b_00_85_50}
\vspace*{0mm}
\mycaption{The same as the previous figure, but with $a=0$ and
$r=10$ (top), $20$ (middle), and $50$ (bottom).}
\end{figure}
\begin{figure}
\dummycaption\label{prof_phi_1}
\includegraphics[width=0.325\textwidth,height=6cm]{mat3ii_10_30_00}
\hfill
\includegraphics[width=0.325\textwidth,height=6cm]{mat3ii_10_60_00}
\hfill
\includegraphics[width=0.325\textwidth,height=6cm]{mat3ii_10_85_00}
\vspace*{5mm}\\
\includegraphics[width=0.325\textwidth,height=6cm]{mat3ii_09_30_00}
\hfill
\includegraphics[width=0.325\textwidth,height=6cm]{mat3ii_09_60_00}
\hfill
\includegraphics[width=0.325\textwidth,height=6cm]{mat3ii_09_85_00}
\vspace*{5mm}\\
\includegraphics[width=0.325\textwidth,height=6cm]{mat3ii_00_30_00}
\hfill
\includegraphics[width=0.325\textwidth,height=6cm]{mat3ii_00_60_00}
\hfill
\includegraphics[width=0.325\textwidth,height=6cm]{mat3ii_00_85_00}
\vspace*{0mm}
\mycaption{Line profiles integrated over twelve consecutive
temporal intervals of equal duration. Each interval covers $1/12$
of the orbital period at corresponding radius. As explained in the text,
top-left frame of each panel corresponds to the spot being observed
at the moment of passing through lower conjunction. Energy is on abscissa
(in keV). Observed photon flux is on ordinate (arbitrary units, scaled
to the maximum flux which is reached during the complete revolution
of the spot). Notice the occurrences of narrow and prominent peaks which
appear for relatively brief fraction of the total period.}
\end{figure}
\begin{figure}
\dummycaption\label{prof_phi_2}
\includegraphics[width=0.325\textwidth,height=6cm]{mat3ii_00_30_10}
\hfill
\includegraphics[width=0.325\textwidth,height=6cm]{mat3ii_00_60_10}
\hfill
\includegraphics[width=0.325\textwidth,height=6cm]{mat3ii_00_85_10}
\vspace*{5mm}\\
\includegraphics[width=0.325\textwidth,height=6cm]{mat3ii_00_30_20}
\hfill
\includegraphics[width=0.325\textwidth,height=6cm]{mat3ii_00_60_20}
\hfill
\includegraphics[width=0.325\textwidth,height=6cm]{mat3ii_00_85_20}
\vspace*{5mm}\\
\includegraphics[width=0.325\textwidth,height=6cm]{mat3ii_00_30_50}
\hfill
\includegraphics[width=0.325\textwidth,height=6cm]{mat3ii_00_60_50}
\hfill
\includegraphics[width=0.325\textwidth,height=6cm]{mat3ii_00_85_50}
\vspace*{0mm}
\mycaption{The same as in previous figure, but with $r=10$ (top), $20$
(middle), and $50$ (bottom). The black hole was assumed non-rotating,
$a=0$ in this figure.}
\end{figure}
In order to compute a synthetic profile of an observed spectral line
one has to link the points of emission in the disc with corresponding
pixels in the detector plane at spatial infinity. This can be achieved by
solving the ray-tracing problem in curved space-time of the black hole.
Appropriate methods were discussed by several authors; see\break
\cite{reynolds2003} for a recent review and for further references.
This way one finds the redshift factor, which determines the
energy shift of photons, the lensing effect (i.e.\ the change of solid angle
due to strong gravity), and the effect of aberration (which influences
the emission direction of photons from the disc; this must be taken
into account if the intrinsic emissivity is non-isotropic). We consider these
effects in our computations, assuming a rotating (Kerr)
black-hole space-time \citep{misner1973}. We also consider the time of arrival
of photons originating at different regions of the disc plane.
Variable travel time results in mutual time delay between different
photons, which can be ignored when analyzing time-averaged data
but it may be important for time-resolved data.
Assuming purely azimuthal Keplerian motion of a spot, one obtains
for its orbital velocity (with respect to a locally non-rotating
observer at corresponding radius $r$):
\begin{equation}
v^{(\phi)} = \frac{r^2-2a\sqrt{r}+{a}^2}{\sqrt{\Delta}\left(r^{3/2}
+{a}\right)}.
\end{equation}
In order to derive time and frequency as measured by a distant observer,
one needs to take into account the Lorentz factor associated with
this orbital motion,
\begin{equation}
\Gamma = \frac{\left(r^{3/2}+{a}\right)\sqrt{\Delta}}{r^{1/4}\;
\sqrt{r^{3/2}-3r^{1/2}+2{a}}\;\sqrt{r^3+{a}^2r+2{a}^2}}\,.
\end{equation}
The corresponding angular velocity of orbital motion is
$\Omega=(r^{3/2}+{a})^{-1}$, which also
determines the orbital period in eq.~(\ref{torb}).
The redshift factor $g$ and the emission angle $\vartheta$ (with respect to
the normal direction to the disc) are then given by
\begin{equation}
g = \frac{{\cal{C}}}{{\cal{B}}-r^{-3/2}\xi},
\quad
\vartheta = \arccos\frac{g\sqrt{\eta}}{r},
\end{equation}
where ${\cal{B}}=1+{a}r^{-3/2}$,
${\cal{C}}=1-3r^{-1}+2{a}r^{-3/2}$; $\xi$ and $\eta$ are constants
of motion connected with the photon ray in an axially symmetric
and stationary space-time.
For practical purposes formula (\ref{torb}) with $a=0$ is also accurate
enough in the case of a spinning black hole,
provided that $r$ is not
very small. For instance, even for $r=6$ (the last stable orbit in
Schwarzschild metric), $T_{\rm orb}(r_{\rm ms})$ calculated for a
static and for a maximally rotating ($a=1$) black hole differ by
about $6.8$\%. The relative difference decreases, roughly linearly, down to
$1.1$\% at $r=20$.
This implies that eq.~(\ref{torb}) can be used
in most cases to estimate the black-hole mass even if the angular momentum
is not known (deviations are relevant only for $r<6$, when the radius itself can be
used to constrain the allowed range of $a$).
Various
pseudo-Newtonian formulae have been devised for accreting black holes
to model their observational properties, which are connected with
the orbital motion of surrounding matter \citep[e.g.][]{abramowicz1996,
artemova1996,semerak1999}.
Although this approach
is often used and found to be practical, we do not employ it here because
error estimates are not possible within the pseudo-Newtonian scheme.
Due to Doppler and gravitational energy shift the line shape changes along
the orbit. Centroid energy is redshifted with respect to the rest
energy of the line emission for most of the orbit.
Furthermore, light aberration and bending cause the flux to
be strongly phase-dependent. These effects are shown in
Figs.~\ref{orbits_1}--\ref{orbits_2}. In these plots,
the arrival time of photons is defined in orbital periods,
i.e.\ scaled with $T_{\rm{}orb}(r;a)$. The
orbital phase of the spot is of course linked with the
azimuthal angle in the disc, but the relation is made complex
by time delays which cannot be neglected, given the large velocities of
the orbiting matter and frame-dragging effects near the black hole.
Here, zero time corresponds to the moment
when the centre of the spot was at the nearest point on its orbit with
respect to the observer (a lower conjunction).
The plots in Fig.~\ref{orbits_1} refer to the case of a spot
circulating at the innermost stable
orbit $r_{\rm{}ms}(a)$ for $a=0$, $0.9$ and $1$.
The effect of black-hole rotation becomes prominent for almost
extreme values of $a$; one can check, for example, that the
difference between cases $a=0$ and $a=0.5$ is very small.
Worth remarking is a large difference in the orbital phase of
maximum emission between the extreme case, $a\rightarrow1$, in
contrast to the non-rotating case, $a\rightarrow0$. The reason is
that for large $a$ the time delay and the effect of frame-dragging
on photons emitted behind the black hole are very substantial. It
is also interesting to note that, for very high inclination angles, most
of the flux comes from the far side of the disc, due to very strong
light bending, as pointed out by \cite{matt1992,matt1993a} and
examined further by many authors who performed detailed ray-tracing,
necessary to determine the expected variations of the line flux and
shape. A relatively simple fitting formula has also been derived
\citep{karas1996} and can be useful for practical computations.
Three more orbits (centred at $r=10$, $20$ and $50$) are shown for
$a=0$ (Fig.~\ref{orbits_2}). As said above, at
these radii differences between spinning and static black holes are
small. Indeed, it can be verified that the dependence on $a$ is only
marginal if $r\gtrsim20$, and so it can be largely neglected for
present-day measurements.
\begin{figure}[tb]
\dummycaption\label{profiles}
\includegraphics[width=0.325\textwidth]{mat2a_00_00}
\hfill
\includegraphics[width=0.325\textwidth]{mat2a_09_00}
\hfill
\includegraphics[width=0.325\textwidth]{mat2a_10_00}
\vspace*{3mm}\\
\includegraphics[width=0.325\textwidth]{mat2a_00_10}
\hfill
\includegraphics[width=0.325\textwidth]{mat2a_00_20}
\hfill
\includegraphics[width=0.325\textwidth]{mat2a_00_50}
\vspace*{0mm}
\mycaption{Time-averaged synthetic spectra in terms of photon
flux (in arbitrary units) versus energy (in keV). These
profiles represent the mean, background-subtracted spectra
of the Fe K$\alpha$ iron-line originating from spots at
different radii. Top panels correspond to $r=r_{\rm{}ms}$ and
$a=0$ (left), $a=0.9$ (middle), and $a=1$ (right). In bottom panels
we fix $a=0$ and choose $r=10$, $20$, and $50$, respectively
(other values of $a$ give very similar profiles).
Three consecutively increasing values of observer
inclination $\theta_{\rm{}o}$ are shown, as indicated inside
the frames.}
\end{figure}
In Figs.~\ref{prof_phi_1}--\ref{prof_phi_2} we show
the actual form of the line profiles for the same sets of parameters as
those explored in Figs.~\ref{orbits_1}--\ref{orbits_2}. The entire
revolution was split into
twelve different phase intervals. The intrinsic flux $I$ is
assumed to decrease exponentially with the distance ${\rm{}d}r$ from the
centre of the spot
(i.e.\ ${\log}I\propto-[\kappa{\rm{}d}r/r]^2$, where $r$ is the
location of the spot centre and $\kappa\sim10$ is a constant).
The illumination is supposed to
cease at ${\rm{}d}r=0.2r$, which also defines the illuminated area in the
disc. Let us remark that we concentrate on a spectral line
which is intrinsically narrow and unresolved in the rest frame of
the emitting medium. Such a line can be produced by a spot which
originates due to sharply
localized illumination by flares, as proposed and discussed
by various authors \citep[e.g.][]{haardt1994,poutanen1999,merloni2001}.
Very recently, \cite{czerny2004} have examined
the induced {\sf{}rms} variability in the flare/spot
model with relativistic effects.
In many cases, and especially for small radii and intermediate to large
inclination angles, the line emission comes from a relatively minor fraction
of the orbit. This implies in practice that for observations with
{\it{}moderate signal-to-noise ratios, only a narrow blue horn can be visible, and
only for a small part of the orbit.}
These large and rapid changes of the line shape get averaged
when integrating over the entire orbit, and so an important
piece of information is missing in the mean spectra.
The line profiles integrated over the whole revolution
are shown in Fig.~\ref{profiles}. Effectively, the mean
profile of a spot is identical
to the profile of an annulus whose radius is equal to the
distance of the spot centre and the width is equal to the spot
size.
We discussed the possibility that the narrow features in the
$5-6$~keV range, recently discovered in a few AGNs and usually interpreted as
redshifted iron lines, could be due to illumination by localized
orbiting spots just above the accretion disc. If this is indeed the
case, these features may provide a powerful and direct way to measure
the black-hole mass in active galactic nuclei. To achieve this aim, it is
necessary to follow the line emission along the orbit.
The orbital radius (in units of $r_{\rm{}g}$) and the disc inclination
can be inferred from the variations of the line flux and centroid energy.
Furthermore, $M_{\bullet}$ can be estimated by comparing the measured orbital
period with the value expected for the derived radius.
We must point out, however, that present-day X-ray
instruments do not have enough collecting area to perform this task
accurately. This capability should be achieved by the planned
high-performance X-ray missions such as
{\it{}Constellation-X}\/ and {\it{}Xeus}.
\section{Non-axisymmetric integration routine
{\fontfamily{pcr}\fontshape{tt}\selectfont ide}}
\label{appendix4a}
This subroutine integrates the local emission and local Stokes parameters
for (partially) polarized emission of the accretion disc near a rotating
(Kerr) black hole (characterized by the angular momentum $a$) for an observer
with an inclination angle $\theta_{\rm o}$.
The subroutine has to be called with ten parameters:
\begin{center}
{\tt ide(ear,ne,nt,far,qar,uar,var,ide\_param,emissivity,ne\_loc)}
\end{center}
\begin{description} \itemsep -2pt
\item[{\tt ear}] -- real array of energy bins (same as {\tt ear} for local
models in {\sc xspec}),
\item[{\tt ne}] -- integer, number of energy bins (same as {\tt ne} for local
models in {\sc xspec}),
\item[{\tt nt}] -- integer, number of grid points in time (${\tt nt}=1$ means
stationary model),
\item[{\tt far(ne,nt)}] -- real array of photon flux per bin
(same as {\tt photar} for local models in {\sc xspec} but with
the time resolution),
\item[{\tt qar(ne,nt)}] -- real array of the Stokes parameter Q divided by the
energy,
\item[{\tt uar(ne,nt)}] -- real array of the Stokes parameter U divided by the
energy,
\item[{\tt var(ne,nt)}] -- real array of the Stokes parameter V divided by the
energy,
\item[{\tt ide\_param}] -- twenty more parameters needed for the integration
(explained below),
\item[{\tt emissivity}] -- name of the external emissivity subroutine, where
the local emission of the disc is defined (explained in detail below),
\item[{\tt ne\_loc}] -- number of points (in energies) where local photon flux
(per keV) in the emissivity subroutine is defined.
\end{description}
The description of the {\tt ide\_param} parameters follows:
\begin{description} \itemsep -2pt
\item[{\tt ide\_param(1)}] -- {\tt a/M} -- the black-hole angular momentum
($0 \le {\tt a/M} \le 1$),
\item[{\tt ide\_param(2)}] -- {\tt theta\_o} -- the observer inclination in
degrees ($0^\circ$ -- pole, $90^\circ$ -- equatorial plane),
\item[{\tt ide\_param(3)}] -- {\tt rin-rh} -- the inner edge of the non-zero
disc emissivity relative to the black-hole horizon (in $GM/c^2$),
\item[{\tt ide\_param(4)}] -- {\tt ms} -- determines whether we also integrate
emission below the marginally stable orbit; if its value is set to zero and
the inner radius of the disc is below the marginally stable orbit then the
emission below this orbit is taken into account, if set to unity it is not,
\item[{\tt ide\_param(5)}] -- {\tt rout-rh} -- the outer edge of the non-zero
disc emissivity relative to the black-hole horizon (in $GM/c^2$),
\item[{\tt ide\_param(6)}] -- {\tt phi} -- the position angle of the axial
sector of the disc with non-zero emissivity in degrees,
\item[{\tt ide\_param(7)}] -- {\tt dphi} -- the inner angle of the axial sector
of the disc with non-zero emissivity in degrees (${\tt dphi} \le 360^\circ$),
\item[{\tt ide\_param(8)}] -- {\tt nrad} -- the radial resolution of the grid,
\item[{\tt ide\_param(9)}] -- {\tt division} -- the switch for the spacing of
the radial grid ($0$ -- equidistant, $1$ -- exponential),
\item[{\tt ide\_param(10)}] -- {\tt nphi} -- the axial resolution of the grid,
\item[{\tt ide\_param(11)}] -- {\tt smooth} -- the switch for performing simple
smoothing ($0$ -- no, $1$ -- yes),
\item[{\tt ide\_param(12)}] -- {\tt normal} -- the switch for normalizing of
the final spectrum,\\
if $=$ 0 -- total flux is unity (usually used for the line),\\
if $>$ 0 -- flux is unity at the energy = {\tt normal} keV (usually used for
the continuum),\\
if $<$ 0 -- final spectrum is not normalized,
\item[{\tt ide\_param(13)}] -- {\tt zshift} -- the overall redshift of the
object,
\item[{\tt ide\_param(14)}] -- {\tt ntable} -- tables to be used, it defines
a double-digit number {\tt NN} in the name of the FITS file
{\tt KBHtablesNN.fits} containing the tables ($0 \le {\tt ntable} \le 99$),
\item[{\tt ide\_param(15)}] -- {\tt edivision} -- the switch for spacing the
grid in local energies (0 -- equidistant, 1 -- exponential),
\item[{\tt ide\_param(16)}] -- {\tt periodic} -- if set to unity then local
emissivity is periodic if set to zero it is not (need not to be set if
${\tt nt}=1$),
\item[{\tt ide\_param(17)}] -- {\tt dt} -- the time step (need not to be set if
${\tt nt}=1$),
\item[{\tt ide\_param(18)}] -- {\tt polar} -- whether the change of the
polarization angle and/or azimuthal emission angle will be read from FITS
tables (0 -- no, 1 -- yes),
\item[{\tt ide\_param(19)}] -- {\tt r0-rh} and
\item[{\tt ide\_param(20)}] -- {\tt phi0} -- in dynamical computations the
initial time will be set to the time when photons emitted from the point
[{\tt r0}, {\tt phi0}] on the disc (in the Boyer-Lindquist coordinates) reach
the observer.
\end{description}
The subroutine {\tt ide} needs an external emissivity subroutine in which the
local emission and local Stokes parameters are defined. This subroutine has
twelve parameters:\\[3mm]
\hspace*{4mm}{\tt emissivity(\parbox[t]{\textwidth}
{ear\_loc,ne\_loc,nt,far\_loc,qar\_loc,uar\_loc,var\_loc,r,phi,cosine,\\
phiphoton,first\_emis)}}
\begin{description} \itemsep -2pt
\item[{\tt ear\_loc(0:ne\_loc)}] -- real array of the local energies where
the local photon flux {\tt far\_loc} is defined, with special meaning of
{\tt ear\_loc(0)}
-- if its value is larger than zero then the local emissivity consists of two
energy regions where the flux is non-zero; the flux between these
regions is zero and {\tt ear\_loc(0)} defines the number of points in local
energies with the zero local flux,
\item[{\tt ne\_loc}] -- integer, the number of points (in energies) where the
local photon flux (per keV) is defined,
\item[{\tt nt}] -- integer, the number of grid points in time (${\tt nt}=1$
means stationary model),
\item[{\tt far\_loc(0:ne\_loc,nt)}] -- real array of the local photon flux
(per keV)
-- if the local emissivity consists of two separate non-zero regions
(i.e.\ ${\tt ear\_loc(0)} > 0$) then {\tt far\_loc(0,it)} is the index of
the last point of the first non-zero local energy region,
\item[{\tt qar\_loc(ne\_loc,nt)}] -- real array of the local Stokes parameter Q
divided by the local energy,
\item[{\tt uar\_loc(ne\_loc,nt)}] -- real array of the local Stokes parameter U
divided by the local energy,
\item[{\tt var\_loc(ne\_loc,nt)}] -- real array of the local Stokes parameter V
divided by the local energy,
\item[{\tt r}] -- the radius in $GM/c^2$ where the local photon flux
{\tt far\_loc} at the local energies {\tt ear\_loc} is wanted
\item[{\tt phi}] -- the azimuth (the Boyer-Lindquist coordinate $\varphi$) where
the local photon flux {\tt far\_loc} at the local energies {\tt ear\_loc} is
wanted,
\item[{\tt cosine}] -- the cosine of the local angle between the emitted ray
and local disc normal,
\item[{\tt phiphoton}] -- the angle between the emitted ray projected onto the
plane of the disc (in the local frame of the moving disc) and the radial
component of the local tetrad (in radians),
\item[{\tt first\_emis}] -- boolean, TRUE if we enter the emissivity subroutine
from the subroutine {\tt ide} for the first time, FALSE if we have already been
in this subroutine (this is convenient if we want to calculate some initial
values when we are in the emissivity subroutine for the first time, e.g.\
trajectory of the falling spot).
\end{description}
\section{Axisymmetric integration routine
{\fontfamily{pcr}\fontshape{tt}\selectfont idre}}
\label{appendix4b}
This subroutine integrates the local axisymmetric emission of an accretion disc
near a rotating (Kerr) black hole (characterized by the angular momentum $a$)
for an observer with an inclination angle~$\theta_{\rm o}$.
The subroutine has to be called with eight parameters:
\begin{center}
{\tt idre(ear,ne,photar,idre\_param,cmodel,ne\_loc,ear\_loc,far\_loc)}
\end{center}
\begin{description} \itemsep -2pt
\item[{\tt ear}] -- real array of energy bins (same as {\tt ear} for local
models in {\sc xspec}),
\item[{\tt ne}] -- integer, the number of energy bins (same as {\tt ne} for local
models in {\sc xspec}),
\item[{\tt photar}] -- real array of the photon flux per bin
(same as {\tt photar} for local models in {\sc xspec}),
\item[{\tt idre\_param}] -- ten more parameters needed for the integration
(explained below),
\item[{\tt cmodel}] -- 32-byte string with a base name of a FITS file with
tables for axisymmetric emission (e.g.\ ``{\tt KBHline}'' for
{\tt KBHlineNN.fits}),
\item[{\tt ne\_loc}] -- the number of points (in energies) where the local
photon flux (per keV) is defined in the emissivity subroutine,
\item[{\tt ear\_loc}] -- array of the local energies where the local
photon flux {\tt far\_loc} is defined,
\item[{\tt far\_loc}] -- array of the local photon flux (per keV).
\end{description}
The description of the {\tt idre\_param} parameters follows:
\begin{description} \itemsep -2pt
\item[{\tt idre\_param(1)}] -- {\tt a/M} -- the black-hole angular momentum
($0 \le {\tt a/M} \le 1$),
\item[{\tt idre\_param(2)}] -- {\tt theta\_o} -- the observer inclination in
degrees ($0^\circ$ -- pole, $90^\circ$ -- equatorial plane),
\item[{\tt idre\_param(3)}] -- {\tt rin-rh} -- the inner edge of the non-zero
disc emissivity relative to the black-hole horizon (in $GM/c^2$),
\item[{\tt idre\_param(4)}] -- {\tt ms} -- determines whether we also integrate
emission below the marginally stable orbit; if its value is set to zero and
the inner radius of the disc is below the marginally stable orbit then the
emission below this orbit is taken into account, if set to unity it is not,
\item[{\tt idre\_param(5)}] -- {\tt rout-rh} -- the outer edge of the non-zero
disc emissivity relative to the black-hole horizon (in $GM/c^2$),
\item[{\tt idre\_param(6)}] -- {\tt smooth} -- the switch for performing simple
smoothing ($0$ -- no, $1$ -- yes),
\item[{\tt idre\_param(7)}] -- {\tt normal} -- the switch for normalizing the
final spectrum,\\
if $=$ 0 -- total flux is unity (usually used for the line),\\
if $>$ 0 -- flux is unity at the energy = {\tt normal} keV (usually used for
the continuum),\\
if $<$ 0 -- final spectrum is not normalized,
\item[{\tt idre\_param(8)}] -- {\tt zshift} -- the overall redshift of the
object,
\item[{\tt idre\_param(9)}] -- {\tt ntable} -- tables to be used, it defines
a double-digit number {\tt NN} in the name of the FITS file
(e.g.\ in {\tt KBHlineNN.fits}) containing the tables
($0 \le {\tt ntable} \le 99$),
\item[{\tt idre\_param(10)}] -- {\tt alpha} -- the radial power-law index.
\end{description}
The subroutine {\tt idre} does not need any external emissivity subroutine.
\section*{Acknowledgements}
Firstly I would like to thank my supervisor, Vladim{\' \i}r Karas, for all of
his helpful advice, useful ideas and pertinent comments on this text and my
research from which it has germinated.
I wish to thank Andrea Martocchia and Giorgio Matt for discussing the lamp-post
model and polarization.
Many thanks go to my colleague, Ladislav \v{S}ubr,
with whom I had many long discussions and who initiated me into the great open
world of Linux. I also want to thank Adela Kawka for reading the manuscript and
filling in all the missing (mostly definite) articles.
I thank everyone whom I worked with at my {\em Alma Mater}, Charles University
in Prague, and at the Institute of Astronomy in Ond\v{r}ejov for their
friendship. I gratefully acknowledge the support from the
Czech Science Foundation grants 202/02/0735 and 205/03/0902, and from the Charles
University grant GAUK 299/2004.
Last but not least, I would like to thank my parents, who supported me during
the whole period of my PhD studies.
\section{Seyfert galaxy MCG--6-30-15}
The Seyfert~1 galaxy MCG--6-30-15 is a unique source in which the evidence
of a broad and skewed {Fe~K$\alpha$} line has led to a wide acceptance of
models with an accreting black hole in the nucleus
\citep{tanaka1995,iwasawa1996,nandra1997,guainazzi1999,fabian2003}.
Being a nearby AGN (the galaxy redshift is $z=0.0078$), this source offers
an unprecedented opportunity to explore directly the pattern of
the accretion flow onto the central hole.
The Fe K line shape and photon redshifts indicate that a large
fraction of the emission originates from $r\lesssim10$ ($GM/c^2$).
The mean line profile derived from {\it{}XMM-Newton}\/ observations
is similar to the one observed previously using {\it ASCA}.
The X-ray continuum shape in the hard spectral band was well
determined from {\it BeppoSAX}\/ data \citep{guainazzi1999}.
\begin{table}[tbh]
\begin{center}
\newlength{\mylen}
\settowidth{\mylen}{{\mbox{$2.1\pm0.2^{\,\dag}$}~}}
\dummycaption\label{tab:best-fit-models}
\begin{tabular}{ccccc@{}c@{}c}
\hline
\multicolumn{1}{c}{{\normalsize{}\#}\rule[-1.5ex]{0mm}{4.5ex}} &
\multicolumn{1}{c}{{\normalsize$a$}} &
\multicolumn{1}{c}{{\normalsize$\theta_{\rm{}o}$}} &
\multicolumn{2}{c}{{\normalsize{}Continuum}} &
\multicolumn{1}{c}{} &
\multicolumn{1}{c}{{$\!$\normalsize$\chi^2$}} \\
\cline{4-5}\cline{7-7}
& & & $\Gamma_{\rm{}c}$ & $\alpha_{\rm{}c}$ & &
~({\sf{}dof})\rule[-1ex]{0mm}{4ex}\\
\hline
1 & $0.35^{+0.57}_{-0.30}$ & $31.8\pm0.3$ & $2.01\pm0.02$ & $1.0^{+9}_{-1}$ &
& {\small$\frac{368.8}{(329)}$}\rule[0ex]{0mm}{4ex}\\%comb73.xcm
2a & $0.99\pm0.01$ & $40.4\pm0.6$ & $2.03\pm0.02$ & $5.5^{+6}_{-2}$ & &
{\small$\frac{308.6}{(330)}$}\rule[0ex]{0mm}{4ex}\\%comb8g.xcm
2b & $0.72^{+0.12}_{-0.30}$ & $28.5\pm0.5$ & $2.01\pm0.01$ & $0.1^{+2}_{-0.1}$ &
& {\small$\frac{313.5}{(330)}$}\rule[0ex]{0mm}{4ex}\\%$comb8g-varianta3.xcm
2c & $0.25\pm0.03$ & $27.6\pm0.6$ & $1.97\pm0.02$ & $3.1^{+0.3}_{-0.1}$ & &
{\small$\frac{313.9}{(330)}$}\rule[0ex]{0mm}{4ex}\rule[-3ex]{0mm}{7ex}\\
\hline
\end{tabular}\\[\bigskipamount]
\begin{tabular}{ccp{\mylen}cccc}
\hline
\multicolumn{1}{c}{{\normalsize{}\#}\rule[-1.5ex]{0mm}{4.5ex}} &
\multicolumn{6}{c}{{\normalsize{}Broad {Fe~K$\alpha$} line}} \\
\cline{2-7}
\rule[-0.5em]{0em}{1.6em} & $r_{\rm{}in}\!-r_{\rm{}h}$ & $r_{\rm{}b}\!-r_{\rm{}h}$ &
$r_{\rm{}out}\!-r_{\rm{}h}$ & $\alpha_{\rm{}in}$ & $\alpha_{\rm{}out}$ & EW\\
\hline
\rule[-0.8em]{0em}{2.4em} 1 & $5.1\pm0.2$ & -- & $11.4\pm0.8$ & -- & $3.9\pm0.6$ &
$258^{+26}_{-13}$\\%comb73.xcm
\rule[-0.9em]{0em}{2.4em} 2a & $0.67\pm0.04$ & $3.35\pm0.05$ & $40^{+960}_{-33}$ &
$6.9^{+0.5}_{-0.4}$ & $9.7^{+0.3}_{-0.8}$ & $268\pm13$ \\%comb8g.xcm
\rule[-1.1em]{0em}{2.4em} 2b & $0.65\pm0.35$ &
\parbox{\mylen}{\mbox{$2.1\pm0.2^{\,\dag}$}\break\mbox{$7.2\pm0.2$}} & $48^{+200}_{-25}$ &
$8.1^{+1.4}_{-0.9}$ & $4.9^{+0.4}_{-0.3}$ & $241^{+13}_{-10}$ \\
\rule[-1.2em]{0em}{2.4em} 2c & $1.23\pm0.06$ & $4\pm0.02$ & $109^{+20}_{-10}$ &
$9.2\pm0.2$ & $3.1\pm0.1$ & $267\pm10$ \\%a025_kyg_3-10.xcm
\hline
\end{tabular}
\vspace*{2mm}\par{}
{\parbox{0.95\textwidth}{\footnotesize{}
Best-fitting values of the important parameters and their
statistical errors for models \#1--2, described in the text.
The models include broad {Fe~K$\alpha$} and {Fe~K$\beta$} emission
lines, a narrow Gaussian line at $\sim 6.9$~keV, and
a Compton-reflection continuum from a relativistic disc.
These models illustrate different assumptions
about intrinsic emissivity of the disc (the radial
emissivity law does not need to be a simple power law, but axial
symmetry has been still imposed here).
The inclination angle $\theta_{\rm{}o}$
is in degrees, relative to the rotation axis; radii are expressed
as an offset from the horizon (in $GM/c^2$);
the equivalent width, EW, is in electron volts.\\
$^{\dag}$~Two values of the transition radius define the interval
$\langle{}r_{\rm{}b-},r_{\rm{}b+}\rangle$ where reflection is
diminished.}}
\mycaption{Spectral fitting results for MCG--6-30-15 using the {\sc ky} model.}
\end{center}
\end{table}
To illustrate the new {\sc ky} model capabilities,
we used our code to analyze the mean EPIC PN spectrum which we
compiled from the long {\it{}XMM-Newton}\/ 2001
campaign (e.g., as described in \citealt{fabian2002}).
The data were cleaned and reduced using standard data reduction
routines, employing
SAS version 5.4.1.\footnote{See {\tt{}\href{http://xmm.vilspa.esa.es/sas/}
{http://xmm.vilspa.esa.es/sas/}.}}
We summed the EPIC PN data from five observations
made in the interval 11 July 2001 to 04 August 2001,
obtaining a total good exposure time of $\sim290$~ks (see
\citealt{fabian2002} for details of the observations).
The energy range was restricted to $3-10$~keV with $339$ energy
bins, unless otherwise stated, and models were fitted by minimizing
the $\chi^{2}$ statistic. Statistical errors quoted correspond to
$90$\% confidence for one interesting parameter
(i.e.\ $\Delta\chi^{2}=2.706$), unless otherwise stated.
No absorber was taken into account; the assumption
here is that any curvature in the spectrum above 3~keV is
not due to absorption, and only due to the Fe K line.
We remind the reader that the Fe~K emission line dominates
around the energy $6-7$~keV, but it has been supposed to stretch
down to $\sim3$~keV or even further.
We emphasize that our aim here is to test the hypothesis that
{\it{}if all of the curvature is entirely due to the broad
Fe~K feature, is it possible to constrain $a$
of the black hole?} Obviously, if the answer to this is `no', it
will also be negative if some of the spectral curvature
between $\sim3-5$~keV is due to processes other than
the Fe~K line emission.
We considered {Fe~K$\alpha$} and {Fe~K$\beta$}
iron lines (with their rest-frame energy fixed at $E_{\rm{}rest}=6.400$
and $7.056$~keV, respectively), a narrow-line feature
(observed at $E_{\rm{}obs}\sim6.9$~keV) modelled as a Gaussian with the central
energy, width, and intensity as free parameters, plus a power-law
continuum. We used a superposition of two {\sc kygline} models
to account for the broad relativistic K$\alpha$ and K$\beta$ lines,
{\sc zgauss} for
the narrow, high-energy line (which we find to be centred at
$6.86$~keV in the source frame, slightly redshifted relative
to $6.966$~keV, the rest-energy of Fe~Ly$\alpha$),
and {\sc kyh1refl} for the Comptonized continuum convolved
with the relativistic kernel. We assumed that the high-energy narrow
line is non-relativistic. Note that an alternative interpretation of the
data (as pointed out by \citealt{fabian2002}) is that there
is not an emission line at $\sim 6.9$~keV, but He-like resonance
absorption at $\sim 6.7$~keV. This would not affect our conclusions.
The ratio of the intensities of the two relativistic lines was fixed at
$I_{\rm{K\beta}}/I_{\rm{K\alpha}}=0.1133$ ($=17/150$), and the iron
abundance for the Compton-reflection continuum was assumed to be three times
the solar value \citep{fabian2002}. We used the angles
$\theta_{\rm{}min}=0^{\circ}$ and $\theta_{\rm{}max}=90^{\circ}$ for
local illumination in {\sc kyh1refl} (these values are equivalent to a central
illumination of an infinite disc).
The normalization of the Compton-reflection continuum relative
to the direct continuum is controlled by the
effective reflection `covering factor', $R_{\rm{}c}$, which is the
ratio of the actual reflection normalization to that expected
from the illumination of an infinite disc.
Also $r_{\rm{}out}$ (outer radius of the disc), $\alpha_{\rm{}c}$
(slope of power-law continuum radial emissivity in the {\sc kyh1refl}
model component), as well as $R_{\rm{}c}$, were included among the
free parameters, but we found them to be only very poorly constrained.
Weak constraints on $\alpha_{\rm{}c}$ and
$R_{\rm{}c}$ are actually expected in this model for two reasons.
Firstly, in these data the continuum is indeed rather featureless. Secondly,
the model continuum was blurred with the relativistic kernel of {\sc kyh1refl},
and so the dependence of the final spectrum on the exact form of
the emissivity distribution over the disc must be quite weak.
\begin{figure}[tb]
\dummycaption\label{fig:confcont1}
\includegraphics[width=0.5\textwidth]{f3a}
\includegraphics[width=0.5\textwidth]{f3b}
\mycaption{Confidence contours around the best-fitting parameter
values (indicated by a cross). Left: the case of a single
line-emitting region (model~\#1 with zero emissivity for
$r<r_{\rm{}ms}$). Right:
the case of a non-monotonic radial emissivity, model~\#2b.
The joint two-parameter contour levels
for $a$ versus $\theta_{\rm{}o}$ correspond to
68\%, 95\% and 99\% confidence.}
\end{figure}
\textit{Model \#1.}
Using a model with a plain power-law radial emissivity
on the disc, we obtained the best-fitting values for the
following set of parameters: $a$ of the black hole, disc
inclination angle $\theta_{\rm{}o}$, radii of the line-emitting region,
$r_{\rm{}in}<r<r_{\rm{}out}$, the corresponding radial emissivity
power-law index $\alpha$ of the line emission, as well as the radial
extent of the continuum-emitting region, its photon index $\Gamma_{\rm{}c}$,
and the corresponding $\alpha_{\rm{}c}$ for the continuum.
Notice that $\Gamma_{\rm{}c}$ and $\alpha_{\rm{}c}$ refer
to the continuum component {\it{}before\/} the relativistic
kernel was applied to deduce the observed spectrum.
As a result of the integration across the disc, the model
weakly constrains these parameters.
There are two ways to interpret this. A model which is
over-parameterized is undesirable from the point of view of deriving
unique model parameters from modelling the data. However, another
interpretation is that a model with a greater number of parameters may
more faithfully reflect the real physics and it is the actual physical
situation which leads to degeneracy in the model parameters. The latter
implies that some model parameters can never be constrained uniquely,
regardless of the quality of the data. In practice one must apply both
interpretations and assess the approach case by case, taking into
account the quality of the data, and which parameters can be constrained
by the data and which cannot. If preliminary fitting shows that large
changes in a parameter do not affect the fit, then that parameter can be
fixed at some value obtained by invoking physically reasonable arguments
pertaining to the situation.
We performed various fits with the inner edge tied
to the marginally stable orbit and also fits where $r_{\rm{}in}$
was allowed to vary independently. Free-fall motion with constant
angular momentum was assumed below $r_{\rm{}ms}$, if the emitting region
extended that far. Tab.~\ref{tab:best-fit-models}
gives best-fitting values of the
key relativistic line and continuum model parameters
for the case in which $r_{\rm{}in}$ and $a$ were independent.
Next, we froze some of the parameters at their best-fit
values and examined the $\chi^2$ space by varying the remaining
free parameters. That way we constructed joint confidence contours
in the plane $a$ versus $\theta_{\rm{}o}$
(see left panel of Fig.~\ref{fig:confcont1}).
These representative plots demonstrate that $\theta_{\rm{}o}$
appears to be tightly constrained, while $a$ is allowed to vary
over a large interval around the best fit, extending down to $a=0$.
\textit{Model \#2.}
We explored the possibility that the broad-line
emission does not conform to a unique power-law radial
emissivity but that, instead,
the line is produced in two concentric rings (a `dual-ring' model).
This case can be considered as a toy model for a more complex
(non-power law) radial dependence of the line emission than the
standard monotonic decline, which we represent here by
allowing for different values of $\alpha$ in the inner and outer
rings: $\alpha_{\rm{}in}$ and $\alpha_{\rm{}out}$.
Effectively, large values of the power-law index
represent two separate rings.
The two regions are matched at the transition radius, $r=r_{\rm{}b}$, and
so this is essentially a broken power law. We explored both cases of
continuous and discontinuous line emissivity at $r=r_{\rm{}b}$. Notice that
the double power-law emissivity arises naturally in the lamp-post model
\citep{martocchia2000} in which the disc irradiation and
the resulting {Fe~K$\alpha$} reflection are substantially anisotropic
due to fast orbital motion in the inner ring. Although the lamp-post
model is very simplified in several respects, namely the way in which the
primary source is set up on the rotation axis, one can expect fairly
similar irradiation to arise from more sophisticated schemes of
coronal flares distributed above the disc plane.
Also, in order to provide a physical picture of the steep emissivity
found in {\it{}XMM-Newton}\/ data of MCG--6-30-15,
\cite{wilms2001} invoked strong magnetic stresses acting
in the innermost part of the system, assuming that they are able to
dissipate a considerable amount of energy in the disc
at very small radii. Intense self-irradiation
of the inner disc may further contribute to the effect.
This more complex model is consistent
with the findings of \cite{fabian2002}. Indeed, the
fit is improved relative to
models with a simple emissivity law because the
enormous red wing and relatively sharp core
of the line are better reproduced thanks to
the contribution from a highly redshifted inner disc
(see Tab.~\ref{tab:best-fit-models}, model~\#2a).
For the same reason that the more
complex model reproduces the line core along
with the red wing well, the model prefers higher values of
$a$ and $\theta_{\rm{}o}$ than what we found for
case \#1. Notice that $a\rightarrow1$
implies that all radiation is produced above $r_{\rm{}ms}$.
Maximum rotation is favoured with
both $a$ and $\theta_{\rm{}o}$
appearing to be tightly constrained near their
best-fit values. Likewise for the continuum radial emissivity
indices. There is a certain freedom
in the parameter values that can be accommodated by this model.
By scanning the remaining parameters,
we checked that the reduced $\chi^2\sim1$ can be achieved
also for $a$ going down to $\sim0.9$ and
$\theta_{\rm{}o}\sim37^{\circ}$. This conclusion is also consistent with
the case for large $a$ in \cite{dabrowski1997}; however, we actually
do not support the claim that the current data {\it{}require}\/
a large value of $a$. As shown below, reasonable
assumptions about the intrinsic emissivity can fit the data with small $a$
equally well.
In model \#2a, small residuals remain near $E\sim4.8$~keV
(at about the $\sim1$\% level),
the origin of which cannot be easily clarified with
the time-averaged data that we employ now. The
excess is reminiscent of a Doppler horn typical of relativistic line
emission from a disc, so it may
also be due to {Fe~K$\alpha$} emission which is locally enhanced
on some part of the disc. We were able to reproduce the peak
by modifying the emissivity at the transition radius,
where the broken power-law emissivity changes its slope (model~\#2b).
We can even allow non-zero emissivity below $r_{\rm{}ms}$ (the inner ring)
with a gap of zero emissivity between the outer edge of the inner
ring and the inner edge of the outer one. The inner ring,
$r_{\rm{}in}{\leq}r{\leq}r_{\rm{}b-}$, contributes to the red tail
of the line while the outer ring, $r_{\rm{}b+}{\leq}r{\leq}r_{\rm{}out}$,
forms the main body of the broad line. The resulting plot of
joint confidence contours of $\theta_{\rm{}o}$ versus $a$ is shown
in Fig.~\ref{fig:confcont1} (right panel).
In order to construct the confidence contours we scanned a broad
interval of parameters, $0<a<1$ and $0<\theta_{\rm{}o}<45^{\circ}$;
here, detail is shown only around the minimum $\chi^2$ region.
Two examples of the spectral profiles are shown in
Fig.~\ref{fig:dualline3-10kev_v3}. It can be seen that the overall
shapes are very similar and the changes concern mainly the red wing of
the profile. For completeness we also fitted several modifications of
the model \#2 and found that with this type of disc emissivity (i.e.\
one which does not decrease monotonically with radius) we can still
achieve comparably good fits which have small values of $a$. The case
\#2c in Tab.~\ref{tab:best-fit-models} gives another example.
This shows that one cannot draw firm conclusions about $a$ (and some of the
other model parameters) based on the redshifted part of the line
using current data. The data used here represent the highest signal-to-noise
relativistically broadened Fe K line profile yet available for any AGN.
\begin{figure}[tb]
\dummycaption\label{fig:dualline3-10kev_v3}
\includegraphics[width=0.5\textwidth]{f4a}
\includegraphics[width=0.5\textwidth]{f4b}
\mycaption{Spectrum and best-fitting model for the {\it XMM-Newton}\/
data for MCG--6-30-15 in which the Fe~K line originates in a dual-ring.
The models 2a (left) and 2b (right) are shown in comparison.
See Tab.~\ref{tab:best-fit-models} for the model parameters.
The continuum component ({\sc kyh1refl}) is also plotted.
The data points are not unfolded: the spectrum in these units was
made by multiplying the ratio of measured counts to the counts
predicted by the best-fitting model and then this ratio was multiplied by
the best-fitting model and then by $E^{2}$.}
\end{figure}
The differences in $\chi^2$ between model \#1 and models
\#2a, 2b, and 2c are very large ($\sim60$) for only one
additional free parameter, so all variations of model 2 shown
are better fits than the case \#1. We note that formally, model \#2a
and \#2c appear to constrain $a$ much more tightly than model \#1.
The issue here is that the values of $a$ can be completely different,
with the statistical errors on $a$ not overlapping (compare, for example,
model \#2a and \#2c in Tab.~\ref{tab:best-fit-models}).
This demonstrates that, at least for the parameter $a$,
it is not simply only the statistical error which determines
whether a small $a$ value is or is not allowed by the data.
We confirmed this conclusion with more complicated models obtained
in {\sc ky} by relaxing the assumption of axial symmetry;
additional degrees of freedom do not change the previous results.
One should bear in mind a well-known technical difficulty
which is frequently encountered while scanning
the parameter space of complex models
and producing confidence contour plots similar to
Fig.~\ref{fig:confcont1}. That is, in a rich parameter space
the procedure may be caught in a local minimum which produces
an acceptable statistical measure of the goodness of fit
and appears to tightly constrain parameters near
the best-fitting values. However, manual searching revealed
equally acceptable results in rather remote
parts of the parameter space. Indeed, as the
results above show, we were able to find
acceptable fits with the central
black hole rotating either slowly or rapidly (in terms of the $a$
parameter). This fact is not in contradiction with previous
results (e.g.\ \citealt{fabian2002}) because we assumed a
different radial profile of intrinsic emissivity, but it indicates
intricacy of unambiguous determination of
model parameters. We therefore need more observational
constraints on realistic physical mechanisms
to be able to fit complicated models to actual data with sufficient confidence
\citep{ballantyne2001,nayakshin2002,rozanska2002}.
We have seen that the models described above are able
to constrain parameters with rather different
degrees of uncertainty.
It turns out that the more complex type \#2 models (i.e.\ those
that have radial emissivity profiles which are
non-monotonic or even have an appreciable contribution from
$r<r_{\rm{}ms}$) provide better fits to the
data but a physical interpretation is not obvious.
\cite{ballantyne2003} also deduce a
dual-reflector model from the same data and propose that the
outer reflection is due to the disc being warped or flared
with increasing radius.
|
1,108,101,564,585 | arxiv | \section{Introduction}
Inspired by Michael Berry and Jonathan Robbins's approach in \cite{BR1997} to understand the spin-statistics theorem quantum mechanically (and, essentially, geometrically!), Michael Atiyah proposed in a series of related papers in 2000-2001 (\cite{Ati-2000}, \cite{Ati-2001}) a geometric construction which associates smoothly to each configuration of $n$ distinct points in $\mathbb{R}^3$, $n$ complex polynomials of degree at most $n-1$, each defined up to a complex scalar factor only. Atiyah conjectured that any set of $n$ polynomials obtained via this construction is linearly independent over $\mathbb{C}$.
Atiyah proved linear independence for the case of $n = 3$ points in various ways (cf \cite{Ati-2000} and \cite{Ati-2001} for example), with the $n = 2$ case being trivial. Moreover, in \cite{Ati-2001}, Atiyah defined a normalized determinant $D$ of the $n$ polynomials in this geometric construction. Numerical calculations performed by Atiyah and Sutcliffe in \cite{Ati-Sut-2002} indicated that $|D| \geq 1$ for any configuration of $n$ distinct points for $n$ up to (at least) $32$. So it was natural for the authors to conjecture that this inequality held for any $n \geq 2$. This was known as Conjecture 2, with Conjecture 1 referring to the linear independence conjecture. Clearly, Conjecture 2 implies Conjecture 1. Atiyah and Sutcliffe also made an even stronger conjecture which they referred to as Conjecture 3 (which implies Conjecture 2), but we shall not discuss it in this article.
Eastwood and Norbury in \cite{EasNor2001} proved the linear independence conjecture (Conjecture 1) for the case of $n = 4$ points. This paper is remarkable because, after expanding $D$ using a computer algebra software (Maple) and obtaining an expression for $D$ involving about $200$ terms, the authors were able to express $D$ in terms of geometric quantities which are obviously nonnegative (triangle inequalities, volumes and so on). They even came close to showing conjecture $2$ for $n = 4$ (they showed that $|D| \geq \frac{15}{16}$).
Building on Eastwood and Norbury's work, Bou Khuzam and Johnson proved in 2014 Conjecture 2 (and even Conjecture 3) for $n = 4$ in \cite{KhuJoh2014} by essentially setting up a linear program and solving it using a computer. At around the same time, Dragutin Svrtan gave his arguments for Conjecture 2 (and Conjecture 3) also for the $n = 4$ case in a talk at the 73-rd ``Seminaire Lotharingien de Combinatoire''. His methods also build on Eastwood and Norbury's formula for the expansion of the normalized determinant for the $n = 4$ case (for which he also provides a human-readable proof).
The methods used to attack the $n = 4$ case rely upon the ``brute force'' expansion of the determinant. One may possibly prove the $n = 5$ case in a similar way after a lot of work (and computers with sufficient processing power), but such methods are obviously limited and do not seem to offer much help with the general case. One may perhaps hope to find a pattern in these formulas for low values of $n$, but so far, such efforts have not proved successful.
It seems clear to the author that one needs to at least supplement Eastwood and Norbury's work by finding a different proof for the $n = 4$ case which avoids the full expansion of Atiyah's determinant function and avoids the use of computer algebra software and the like, before tackling the general case. With this goal in mind, we show in this article that the Gram matrix of the $4$ polynomials associated to a configuration of $4$ distinct points in $\mathbb{R}^3$ is always positive definite, using $2$-spinor calculus and the theory of hermitian positive (semi-)definite matrices. This reproves the linear independence theorem by Eastwood and Norbury for the $n = 4$ case.
\section{Background: the Hopf map and $2$-spinors}
The Hopf map $h: S^3 \to S^2$ is a smooth map from $S^3$ onto $S^2$ with fibers being diffeomorphic to circles. The $3$-sphere $S^3$ can be described as follows.
\[ S^3 = \{ (u, v) \in \mathbb{C}^2 \,; \, |u|^2 + |v|^2 = 1 \}. \]
On the other hand, the $2$-sphere can be described as
\[ S^2 = \{ (\zeta, z) \in \mathbb{C} \times \mathbb{R} \,; \,
|\zeta|^2 + z^2 = 1 \}.\]
The Hopf map is then defined by
\[ h(u, v) = (2 \bar{u} v, |v|^2 - |u|^2). \]
Note that, if $(\zeta, z) \in S^2$ with $z < 1$, then
\[ h^{-1}(\zeta, z) = \frac{e^{i \theta}}{\sqrt{2(1-z)}}\left(1 - z, \,\zeta\right)\]
where $\theta$ is real. On the other hand,
\[ h^{-1}(0, 1) = e^{i\alpha} (0, 1),\]
where $\alpha \in \mathbb{R}$.
A vector in $\mathbb{C}^2$ may be identified with a linear form on $\mathbb{C}^2$ using the complex symplectic form on $\mathbb{C}^2$ (unique up to a nonzero complex scalar factor), which we will denote by $\omega$. With respect to the standard basis of $\mathbb{C}^2$, $\omega$ gets represented by the following matrix
\begin{equation} \omega = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}. \label{omega} \end{equation}
Thus for example, a vector $\mathbf{\psi} = (\psi_0, \psi_1)^T \in \mathbb{C}^2$, where $T$ denotes the transpose, gets identified with
\begin{equation} \omega(\mathbf{\psi}, \, \mathbf{w}) = - \psi_1 u + \psi_0 v, \label{identification} \end{equation}
where $\mathbf{w} = (u, v)^T$. So, in particular, if $z < 1$, we have the identification
\[ h^{-1}(\zeta, z) \sim \frac{1}{\sqrt{2(1-z)}}((1-z)v - \zeta u)\]
(up to a phase factor). If we think of $u$ and $v$ as homogeneous coordinates on $\mathbb{P}^1(\mathbb{C})$, then switching to the inhomogeneous coordinate $v/u$, we see that the root of the previous linear form is $\zeta/(1 - z)$, which is nothing but the stereographic projection of the point $(x, y, z) \in S^2 \subset \mathbb{R}^3$, with $x$ and $y$ being the real and imaginary parts of $\zeta$, respectively.
While the original problem was formulated using stereographic projection and the Hopf map, it will be useful to consider a (very slightly) modified Hopf map instead, which is more naturally associated to the Pauli matrices.
We first remark that any hermitian $2 \times 2$ matrix is a linear combination with real coefficients of the $4$ Pauli matrices:
\begin{align*}
\sigma_0 &= \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \\
\sigma_1 &= \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \\
\sigma_2 &= \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix} \\
\sigma_3 &= \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}
\end{align*}
Note that $\sigma_0$ is the identity $2 \times 2$ matrix, which we also denote by $\mathbf{1}$.
If $\mathbf{w} = (u, v)^T \in S^3 \subset \mathbb{C}^2$, we first form
\[ \mathbf{w} \mathbf{w}^* = \begin{pmatrix} |u|^2 & u\bar{v} \\
v \bar{u} & |v|^2 \end{pmatrix} \]
(with the $*$ denoting the conjugate transpose) which is an hermitian $2 \times 2$ matrix with trace equal to $1$. We can then write, in a unique way
\[ \mathbf{w} \mathbf{w}^* = \frac{1}{2}\left(\mathbf{1} - x \sigma_1 - y \sigma_2 - z\sigma_3 \right), \]
where $x$, $y$ and $z$ are real numbers. Noting that $\mathbf{w} \mathbf{w}^*$ has rank $1$, it thus follows that its determinant vanishes. Hence
\[ 0 = \det(\mathbf{w} \mathbf{w}^*) = \frac{1}{4}\left( 1 - x^2 - y^2 - z^2 \right). \]
In other words, $(x, y, z) \in S^2 \subset \mathbb{R}^3$. We now define the modified Hopf map $\tilde{h}: S^3 \to S^2 \subset \mathbb{R}^3$ as follows.
\[ \tilde{h}(u, v) = (x, y, z). \]
The formula for the inverse map, assuming $z < 1$, now reads
\[ \tilde{h}^{-1}(x, y, z) = \frac{e^{i \theta}}{2(1-z)}\left(1-z, -(x+iy)\right)^T \]
and
\[ \tilde{h}^{-1}(0, 0, 1) = e^{i \alpha}(0, 1)^T. \]
Using the modified Hopf map rather than the usual Hopf map does not affect the absolute value of the normalized determinant $D$ mentioned in the previous section (which will be discussed in section \ref{At-det}), so ultimately, it will be equivalent to using the usual Hopf map, as far as the Atiyah problem on configurations is concerned.
\section{A few useful formulas}
If $\mathbf{w}_i \in S^3 \subset \mathbb{C}^2$ with $\mathbf{w}_i = (u_i, v_i)^T$ for $i = 1, 2$, we then denote by $\langle -,\, -\rangle$ the standard hermitian inner product on $\mathbb{C}^2$, i.e.
\begin{equation} \langle \mathbf{w}_1, \, \mathbf{w}_2 \rangle = u_1 \bar{u}_2 + v_1 \bar{v}_2 = \operatorname{tr}(\mathbf{w}_1 \mathbf{w}_2^*) = \mathbf{w}_2^* \mathbf{w}_1. \label{herm} \end{equation}
We therefore deduce that
\begin{align*} & |\langle \mathbf{w}_1 ,\, \mathbf{w}_2 \rangle|^2 \\
= &\, \mathbf{w}_2^* \mathbf{w}_1 \mathbf{w}_1^* \mathbf{w}_2 \\
= &\, \operatorname{tr}(\mathbf{w}_1\mathbf{w}_1^*
\mathbf{w}_2\mathbf{w}_2^*) \\
= &\, \frac{1}{4}\operatorname{tr}\left[ \left( \mathbf{1} - \mathbf{x}_1.\vec{\sigma} \right) \left( \mathbf{1} - \mathbf{x}_2.\vec{\sigma} \right) \right] \\
= &\, \frac{1}{4}\operatorname{tr}\left[ \left(\mathbf{1} - \mathbf{x}_1.\vec{\sigma} - \mathbf{x}_2.\vec{\sigma} + (\mathbf{x}_1, \mathbf{x}_2) \mathbf{1}\right)\right]
\end{align*}
where $\mathbf{x}_i = (x_i, y_i, z_i)^T \in S^2 \subset \mathbb{R}^3$, which is the (modified) Hopf image of $\mathbf{w}_i$, for $i = 1, 2$, $(-,\, -)$ denotes the standard Euclidean inner product on $\mathbb{R}^3$, $\vec{\sigma}$ is the $3$-vector of Pauli matrices indexed by $1$, $2$ and $3$ and the dot also denotes the Euclidean inner product of two $3$-vectors. Note that in the previous string of equalities, we have made use of the algebra of Pauli matrices. For example,
\[ \sigma_i^2 = \mathbf{1} \quad \text{for $i = 1, \ldots, 3$} \]
and
\[ \sigma_1 \sigma_2 = -\sigma_2 \sigma_1 = i \sigma_3 \]
and cyclically permuted versions of this formula over $\{1, 2, 3\}$.
But
\[ \operatorname{tr}(\mathbf{x}_i.\vec{\sigma}) = 0, \quad \text{for $i = 1, 2$,}\]
so we obtain
\begin{equation} |\langle \mathbf{w}_1 \, \mathbf{w}_2 \rangle|^2 = \frac{1}{2} \left( 1 + (\mathbf{x}_1, \mathbf{x}_2) \right). \label{2-cycle} \end{equation}
\begin{definition} Given $\mathbf{w}_i$ and $\mathbf{x}_i$ as above (for $i = 1, 2$), we define
\[ \rho_{12} = \frac{1}{2} \left( 1 + (\mathbf{x}_1, \mathbf{x}_2) \right).\]
\end{definition}
It follows from the Cauchy-Schwarz inequality that $0 \leq \rho_{12} \leq 1$.
We can rephrase what we have proved, as follows.
\begin{equation} |\langle \mathbf{w}_1 \, \mathbf{w}_2 \rangle|^2 = \rho_{12}. \label{2-cycle-a} \end{equation}
Let $\mathbf{w}_i \in \mathbb{C}^2$ for $i = 1, \ldots, 3$. We have
\begin{align*}
& \langle \mathbf{w}_1 ,\, \mathbf{w}_2 \rangle \langle \mathbf{w}_2 ,\, \mathbf{w}_3 \rangle \langle \mathbf{w}_3 ,\, \mathbf{w}_1 \rangle \\
= & \, \mathbf{w}_2^* \mathbf{w}_1 \mathbf{w}_1^* \mathbf{w}_3 \mathbf{w}_3^* \mathbf{w}_2 \\
= & \, \operatorname{tr}\left( \mathbf{w}_1 \mathbf{w}_1^* \mathbf{w}_3 \mathbf{w}_3^* \mathbf{w}_2 \mathbf{w}_2^* \right) \\
= & \, \frac{1}{8} \operatorname{tr}\left[(\mathbf{1} - \mathbf{x}_1.\vec{\sigma}) (\mathbf{1} - \mathbf{x}_3.\vec{\sigma}) (\mathbf{1} - \mathbf{x}_2.\vec{\sigma})\right] \\
= & \, \frac{1}{8} \operatorname{tr}\left[(1 + (\mathbf{x}_1, \mathbf{x}_2) + (\mathbf{x}_1, \mathbf{x}_3) + (\mathbf{x}_2, \mathbf{x}_3)) \mathbf{1} - (\mathbf{x}_1.\vec{\sigma}) (\mathbf{x}_3.\vec{\sigma}) (\mathbf{x}_2.\vec{\sigma}) \right], \\
\end{align*}
where we have used, in the last line, the fact that $\operatorname{tr}(\mathbf{x}_i.\vec{\sigma}) = 0$ for $i = 1, \ldots, 3$. But
\[ (\mathbf{x}_3.\vec{\sigma}) (\mathbf{x}_2.\vec{\sigma}) = (\mathbf{x}_2, \mathbf{x}_3) \mathbf{1} - i (\mathbf{x}_2 \times \mathbf{x}_3).\vec{\sigma}, \]
so that
\begin{align*}
& \langle \mathbf{w}_1 ,\, \mathbf{w}_2 \rangle \langle \mathbf{w}_2 ,\, \mathbf{w}_3 \rangle \langle \mathbf{w}_3 ,\, \mathbf{w}_1 \rangle \\
= & \, \frac{1}{8} \operatorname{tr}\left[(1 + (\mathbf{x}_1, \mathbf{x}_2) + (\mathbf{x}_1, \mathbf{x}_3) + (\mathbf{x}_2, \mathbf{x}_3) + i \det(\mathbf{x}_1, \mathbf{x}_2, \mathbf{x}_3)) \mathbf{1}\right] \\
= & \, \frac{1}{4} \left(1 + (\mathbf{x}_1, \mathbf{x}_2) + (\mathbf{x}_1, \mathbf{x}_3) + (\mathbf{x}_2, \mathbf{x}_3) + i \det(\mathbf{x}_1, \mathbf{x}_2, \mathbf{x}_3)\right),
\end{align*}
where $\det(\mathbf{x}_1, \mathbf{x}_2, \mathbf{x}_3)$ is the determinant of the real $3 \times 3$ matrix having $\mathbf{x}_1$, $\mathbf{x}_2$ and $\mathbf{x}_3$ as its $3$ columns, in that order.
Using the $\rho$ notation, we have proved
\begin{equation} \begin{split} & \langle \mathbf{w}_1 ,\, \mathbf{w}_2 \rangle \langle \mathbf{w}_2 ,\, \mathbf{w}_3 \rangle \langle \mathbf{w}_3 ,\, \mathbf{w}_1 \rangle \\
= & \, \frac{1}{2} \left(-1 + \rho_{12} + \rho_{13} + \rho_{23}\right) + \frac{i}{4} \det(\mathbf{x}_1, \mathbf{x}_2, \mathbf{x}_3).
\end{split} \label{3-cycle}
\end{equation}
\section{The Atiyah problem on configurations}
Let $C_n(\mathbb{R}^3)$ be the configuration space of $n$ distinct points in $\mathbb{R}^3$. Given $\mathbf{x} = (\mathbf{x}_1, \ldots, \mathbf{x}_n) \in C_n(\mathbb{R}^3)$, for each pair of indices $i$, $j$ with $1 \leq i, j \leq n$ and $i \neq j$, we let
\[ \nu_{ij} = \frac{\mathbf{x}_j - \mathbf{x}_i}{\lVert \mathbf{x}_j - \mathbf{x}_i \rVert} \in S^2. \]
We choose for each such pair of indices $i$, $j$ a Hopf lift $\mathbf{w}_{ij} \in S^3 \subset \mathbb{C}^2$ of $\nu_{ij}$, by which we mean that $h(\mathbf{w}_{ij}) = \nu_{ij}$. Later on, we will be using the modified Hopf map rather than the Hopf map, but for the time being, we will describe the problem first using the usual Hopf map.
Using $\omega$ defined in \eqref{omega}, we identify each $\mathbf{w}_{ij}$ with a corresponding homogeneous linear form $p_{ij}(\mathbf{w})$ on $\mathbb{C}^2$ via
\[ p_{ij}(\mathbf{w}) = \omega(\mathbf{w}_{ij}, \mathbf{w}), \]
where $\mathbf{w} = (u, v)^T$ are the coordinates of a variable point in $\mathbb{C}^2$.
Given $i$, with $1 \leq i \leq n$, we form
\[ p_i(\mathbf{w}) = \prod_{j \neq i} p_{ij}(\mathbf{w}), \]
which is a complex homogeneous polynomial of degree $n - 1$ in two complex variables ($u$ and $v$). Note that each $p_i(\mathbf{w})$ is well defined only up to a phase factor.
\begin{conjecture1} Given $n$, the polynomials $p_i(\mathbf{w})$, for $i = 1, \ldots, n$, associated to any given configuration $\mathbf{x} \in C_n(\mathbb{R}^3)$, are linearly independent (over $\mathbb{C}$).\end{conjecture1}
Conjecture 1 is actually a theorem by M. F. Atiyah for $n = 3$ (cf. \cite{Ati-2000}, \cite{Ati-2001}) and was proved by Eastwood and Norbury for $n = 4$ in \cite{EasNor2001}, though it is open for the general $n > 4$ case. The case $n = 4$ is the first ``hard'' case though, as discussed in the introduction.
\section{The Atiyah determinant} \label{At-det}
Building on the description in the previous section, we may consider ($u$, $v$) as a basis of $(\mathbb{C}^2)^*$ (the complex dual space of $\mathbb{C}^2$). We may similarly consider
\[ (u^{n-1}, u^{n-2}v, \ldots, u^iv^{n-1-i}, \ldots, v^{n-1}) \]
as a basis of $\operatorname{Sym}^{n-1}(\mathbb{C}^2)^*$, i.e. of the $n-1$-st symmetric tensor power of $(\mathbb{C}^2)^*$. With respect to these two bases respectively, we may think of $p_{ij}(\mathbf{w})$, respectively $p_i(\mathbf{w})$, as a vector in $\mathbb{C}^2$, respectively $\mathbb{C}^n$. So it does make sense then to talk about $\det(p_{ij}(\mathbf{w}), p_{ji}(\mathbf{w}))$, which is the determinant of the complex $2 \times 2$ matrix whose column vectors are $p_{ij}(\mathbf{w})$ and $p_{ji}(\mathbf{w})$ in that order.
\begin{definition} Atiyah's normalized determinant function $D: C_n(\mathbb{R}^3) \to \mathbb{C}$ is defined by
\[ D(\mathbf{x}) = \frac{\det(p_1(\mathbf{w}), \ldots, p_n(\mathbf{w}))}{\prod_{1 \leq i < j \leq n} \det(p_{ij}(\mathbf{w}), p_{ji}(\mathbf{w}))}. \]
\end{definition}
It can be checked that $D$ is well defined. Indeed, both the numerator and denominator are homogeneous in the $p_{ij}(\mathbf{w})$, where $1 \leq i, j \leq n$ and $i \neq j$, of degree $1$ in each single $p_{ij}(\mathbf{w})$, so that a choice of different phase factors for the $p_{ij}(\mathbf{w})$ would lead to the same ratio and thus to a well-defined value for $D(\mathbf{x})$.
As a remark, one may define $D(\mathbf{x})$ alternatively as follows and this is the definition we will adopt in this article. At the step where one chooses Hopf lifts, one may assume that once a choice of Hopf lift $\mathbf{w}_{ij}$ of $\nu_{ij}$ was made, where $1 \leq i < j \leq n$, then $\mathbf{w}_{ji}$ is then taken to be as follows.
\[ \begin{pmatrix} u_{ji} \\ v_{ji} \end{pmatrix} =
\begin{pmatrix*}[r]
-\bar{v}_{ij} \\ \bar{u}_{ij}
\end{pmatrix*}. \]
It can be easily checked that $\mathbf{w}_{ji}$ is then a Hopf lift of $\nu_{ji} = -\nu_{ij}$. The experts will recognize that the map sending $\mathbf{w}_{ij}$ to $\mathbf{w}_{ji}$ above is a quaternionic structure on $\mathbb{C}^2$ and is what the antipodal map in $\mathbb{R}^3$ (the so-called parity transformation in physics) corresponds to on the $2$-spinor level.
That the $2$ definitions of $D$ are equivalent is left as an exercise to the reader (it is partly based on the observation that if we follow the above prescription, then $\det(p_{ij}(\mathbf{w}), p_{ji}(\mathbf{w})) = 1$).
\begin{conjecture2} Given $n$, for any given configuration $\mathbf{x} \in C_n(\mathbb{R}^3)$, $|D(\mathbf{x})| \geq 1$.\end{conjecture2}
Conjecture 2 is actually a theorem by M. F. Atiyah for $n = 3$ and was proved for $n = 4$ by Bou Khuzam and Johnson in \cite{KhuJoh2014} (who also proved the stronger Conjecture 3). At around the same time, Dragutin Svrtan gave a talk at the 73-rd ``Seminaire Lotharingien de Combinatoire'' presenting his arguments for the Atiyah-Sutcliffe conjectures 1-3 for the $n = 4$ case, though they do not seem to be published (to the best of the author's knowledge). Conjecture 2 is, at the time of writing, open for the general $n > 4$ case.
As an important note, we will diverge from the standard description of the Atiyah problem on configurations in the following points:
\begin{enumerate}
\item We make use of the modified Hopf map $\tilde{h}$ rather than $h$, so that we have, for example,
\[ \tilde{h}(\mathbf{w}_{ij}) = \nu_{ij} \quad \text{($1 \leq i, j \leq n$, $i \neq j$). }\]
\item We do not ``dualize'' each $\mathbf{w}_{ij}$ using the complex symplectic form $\omega$. We also replace polynomial multiplication with the symmetric tensor product (over $\mathbb{C}$). Thus, for example, instead of considering $p_{ij}(\mathbf{w})$, we consider instead $p_{ij} = \mathbf{w}_{ij}$ (for $1 \leq i, j \leq n$ and $i \neq j$) and we define, for $i = 1, \ldots, n$, the following:
\[ p_i = \operatorname{Sym}_{j \neq i} p_{ij} \in \operatorname{Sym}^{n-1}(\mathbb{C}^2), \]
where $\operatorname{Sym}$ denotes the symmetric tensor product (over $\mathbb{C}$). The vector $p_i$ will play the role of $p_i(\mathbf{w})$.
\end{enumerate}
One may mimic the definition of $D$ above, making sure to use the above modifications. Using the modified Hopf map instead of the Hopf map has the effect of multiplying both numerator and denominator by the same sign factor, namely $(-1)^{\binom{n}{2}}$, so that the value of $D(\mathbf{x})$ remains the same. Similarly, using or not the matrix $\omega$ to dualize $\mathbf{w}_{ij}$ has no effect on the numerator, nor the denominator, in the definition of $D(\mathbf{x})$, since $\omega$ has determinant equal to $1$.
\section{The Gram matrix}
If $\mathbb{C}^2$ is equipped with the standard hermitian inner product \eqref{herm}, then this induces an hermitian inner product on $\operatorname{Sym}^m(\mathbb{C}^2)$, the $m$-th symmetric tensor power of $\mathbb{C}^2$, where $m$ is any given positive integer. More precisely, if $p = \mathbf{w}_1 \odot \cdots \odot \mathbf{w}_m$ and $p' = \mathbf{w}'_1 \odot \cdots \odot \mathbf{w}'_m$, where $\odot$ denotes the symmetric tensor product (which some people may call the symmetric Kronecker product), we then define
\[ \langle p, \, p' \rangle = \sum_{\sigma \in S_m} \langle \mathbf{w}_i, \, \mathbf{w}'_{\sigma(i)} \rangle. \]
Note that we do not include a normalization factor, though many authors may choose to include one.
We fix an integer $n \geq 2$. Given $\mathbf{x} \in C_n(\mathbb{R}^3)$, we let $p_1, \ldots, p_n$ be the corresponding vectors in $\operatorname{Sym}^{n-1}(\mathbb{C}^2)$, each defined up to a phase factor. We form the Gram matrix $H_n(\mathbf{x})$ (which we may sometimes simply write as $H_n$, if $\mathbf{x}$ is understood) of these $n$ vectors, namely
\[ H_n = (\langle p_i,\, p_j \rangle) , \quad \text{$1 \leq i,j \leq n$},\]
where $\langle -,\, - \rangle$ is the hermitian inner product induced on $\operatorname{Sym}^{n-1}(\mathbb{C}^2)$ by the standard hermitian inner product on $\mathbb{C}^2$, as described above.
We note however that, while $H_n$ itself is only defined up to conjugation by an $n \times n$ diagonal unitary matrix due to the phase ambiguity, its eigenvalues (with multiplicity) are well defined and so is its determinant. As a matter of fact, if one keeps track carefully of the norms of the vectors
\[ e_1^{\odot k} \odot e_2^{\odot n-1-k} \quad \text{($0 \leq k \leq n-1$)}\]
(where $e_1 = (1, 0)^T$ and $e_2 = (0, 1)^T$), which form a basis of $\operatorname{Sym}^{n-1}(\mathbb{C}^2)$ used in the definition of $D(\mathbf{x})$, one may show that
\begin{equation} \det(H_n(\mathbf{x})) = \left(\prod_{k=0}^{n-1} \lVert e_1^{\odot k} \odot e_2^{\odot n-1-k} \rVert^2 \right) |D|^2 = c_n |D|^2. \label{det_gram1} \end{equation}
where
\begin{equation} c_n = \prod_{k=0}^{n-1} (k!)^2. \label{det_gram2} \end{equation}
Being a Gram matrix, $H_n(\mathbf{x})$ is therefore always positive semidefinite. Moreover, $H_n(\mathbf{x})$ is positive definite iff its determinant is positive (since we already know that all its eigenvalues are real and nonnegative) iff the $n$ vectors $p_i$, $1 \leq i \leq n$, are linearly independent over $\mathbb{C}$, i.e. iff conjecture 1 holds.
Moreover, one may easily see using \eqref{det_gram1} and \eqref{det_gram2}, that conjecture 2 is equivalent to
\[ \det{H_n} \geq c_n. \]
The following is a reformulation of known results.
\begin{theorem} If $n \leq 4$, then the Gram matrix $H_n(\mathbf{x})$ is positive definite for any $\mathbf{x} \in C_n(\mathbb{R}^3)$. \end{theorem}
We stress that, as a statement, this is only a reformulation of the fact that Atiyah's conjecture 1 is known to be true for $n \leq 4$ (proved by M. F. Atiyah for $n = 3$, cf \cite{Ati-2000}, \cite{Ati-2001}, and by Eastwood and Norbury for $n = 4$ in \cite{EasNor2001}). Our contribution is in providing a new proof for the $n = 4$ case (which is the first ``hard'' case) that does not require the full expansion of the $4 \times 4$ Atiyah determinant, nor the use of computer algebra software such as Maple, unlike the ``tour de force'' kind of proof found in \cite{EasNor2001}. The author hopes that this work will complement Eastwood and Norbury's work and may allow us to proceed further and possibly tackle the $n > 4$ case (ideally, at least).
\section{Proof of the $n = 3$ case}
In this section, we tackle the case of $n = 3$ points in $\mathbf{R}^3$. Given $\mathbf{x} \in C_3(\mathbb{R}^3)$, the corresponding Gram matrix $H_3(\mathbf{x})$ is given by
\begin{equation} H_3(\mathbf{x}) =
\begin{pmatrix} 1 + |\herm{12}{13}|^2 & \herm{12}{23} \,\herm{13}{21} & \herm{12}{31} \,\herm{13}{32} \\
\herm{23}{12} \,\herm{21}{13}& 1 + |\herm{21}{23}|^2 & \herm{21}{32} \,\herm{23}{31} \\
\herm{31}{12} \,\herm{32}{13} & \herm{32}{21} \,\herm{31}{23} & 1 + |\herm{31}{32}|^2 \end{pmatrix}, \label{H3} \end{equation}
where $h_{ij, kl} = \langle \mathbf{w}_{ij},\, \mathbf{w}_{kl} \rangle$.
Let
\begin{align*} \mu_1 &= |\herm{12}{13}|^2 = 1 - |\herm{31}{12}|^2 \\
\mu_2 &= |\herm{21}{23}|^2 = 1 - |\herm{12}{23}|^2 \\
\mu_3 &= |\herm{31}{32}|^2 = 1 - |\herm{23}{31}|^2.
\end{align*}
To understand the first line in the previous set of formulas, note that $|\herm{12}{13}|^2 + |\herm{31}{12}|^2 = 1$ since $\mathbf{w}_{12}$ has unit norm and $(\mathbf{w}_{13}, \mathbf{w}_{31})$ is a unitary basis of $\mathbb{C}^2$. The other two lines are similar.
We now expand $\det(H_3(\mathbf{x}))$, obtaining
\[ \begin{split} & \det(H_3(\mathbf{x})) \\
= \, &(1 + \mu_1)(1 + \mu_2)(1 + \mu_3) + 2(1 - \mu_1)(1 - \mu_2)(1 - \mu_3) - \cdots \\
& \cdots -(1 + \mu_1)(1 - \mu_2)(1 - \mu_3) - (1 - \mu_1)(1 + \mu_2)(1 - \mu_3) - \cdots \\
& \cdots -(1 - \mu_1)(1 - \mu_2)(1 + \mu_3) \end{split}
\]
Expanding and simplifying, we obtain
\begin{equation} \det(H_3(\mathbf{x})) = 4(\mu_1 \mu_2 + \mu_1 \mu_3 + \mu_2 \mu_3) - 4 \mu_1 \mu_2 \mu_3. \label{det-H3} \end{equation}
We consider the Gram matrix $G$ of the vectors $\nu_{23}$, $\nu_{31}$ and $\nu_{12}$ in $\mathbb{R}^3$, whose entries are the pairwise Euclidean inner products of these $3$ vectors. Using \eqref{2-cycle}, we find that
\[ G = \begin{pmatrix} 1 & 1 - 2 \mu_3 & 1 - 2 \mu_2 \\
1 - 2 \mu_3 & 1 & 1 - 2 \mu_1 \\
1 - 2 \mu_2 & 1 - 2 \mu_1 & 1 \end{pmatrix}. \]
But the $3$ vectors $\nu_{23}$, $\nu_{31}$ and $\nu_{12}$ are coplanar, so that $G$ has vanishing determinant. Hence
\[ \begin{split} 0 = \det(G) = \, 1 + 2(1-2\mu_1)(1-2\mu_2)(1-2\mu_3) &- (1-2\mu_1)^2 - \cdots \\
\cdots &-(1-2\mu_2)^2 -(1-2\mu_3)^2, \end{split} \]
which gives, after simplifying
\[ 0 = \det(G) = 8(\mu_1\mu_2 + \mu_1\mu_3 + \mu_2\mu_3) - 4(\mu_1^2 + \mu_2^2 + \mu_3^2) - 16\mu_1 \mu_2 \mu_3, \]
from which we obtain
\begin{equation}-4 \mu_1 \mu_2 \mu_3 = \mu_1^2 + \mu_2^2 + \mu_3^2 - 2(\mu_1 \mu_2 + \mu_1 \mu_3 + \mu_2 \mu_3). \label{identity} \end{equation}
Substituting the previous formula into \eqref{det-H3}, we finally obtain
\begin{equation} \det(H_3(\mathbf{x})) = (\mu_1 + \mu_2 + \mu_3)^2. \end{equation}
Note that the RHS of the previous formula is nonnegative, and cannot vanish. Indeed, if the RHS vanished, then each of $\mu_1$, $\mu_2$ and $\mu_3$ must vanish, which would imply that the triangle with vertices $\mathbf{x}_1$, $\mathbf{x}_2$ and $\mathbf{x}_3$ has all $3$ interior angles equal to $\pi$, which is clearly impossible (since the sum of the interior angles of a Euclidean triangle must be $\pi$).
Hence, we have proved that $\det(H_3(\mathbf{x})) > 0$. Thus $H_3(\mathbf{x})$ is an hermitian positive semidefinite matrix with positive determinant, from which we conclude that $H_3(\mathbf{x})$ is positive definite, for any $\mathbf{x} \in C_3(\mathbb{R}^3)$.
As a note, using \eqref{identification} and the fact that $D$ is real and positive if $n = 3$ (cf. \cite{Ati-Sut-2002}), we deduce that
\[ D = \frac{\mu_1 + \mu_2 + \mu_3}{2}, \]
(which is equivalent to formula (3.16) in \cite{Ati-Sut-2002}) if $n = 3$. Moreover, it is known that in this case, $D$ is minimized at a collinear configuration at which $D = 1$ and is maximized at an equilateral triangle at which $D = 9/8$ (also cf. \cite{Ati-2001}, \cite{Ati-Sut-2002}). We thus have, if $n = 3$, the following.
\begin{equation} 2 \leq \mu_1 + \mu_2 + \mu_3 \leq \frac{9}{4}, \label{bounds} \end{equation}
from which it follows that
\[ \det(H_3(\mathbf{x})) \geq 4. \]
Before leaving this section, we will prove the following lemma.
\begin{lemma} Given any $\mathbf{x} \in C_3(\mathbb{R}^3)$, $H_3(\mathbf{x}) - \mathbf{1}$ (where $\mathbf{1}$ here denotes the $3 \times 3$ identity matrix) is (hermitian) positive semidefinite.
\end{lemma}
\begin{proof}
Consider
\begin{equation} H_3(\mathbf{x}) - \mathbf{1} =
\begin{pmatrix} |\herm{12}{13}|^2 & \herm{12}{23} \,\herm{13}{21} & \herm{12}{31} \,\herm{13}{32} \\
\herm{23}{12} \,\herm{21}{13}& |\herm{21}{23}|^2 & \herm{21}{32} \,\herm{23}{31} \\
\herm{31}{12} \,\herm{32}{13} & \herm{32}{21} \,\herm{31}{23} & |\herm{31}{32}|^2 \end{pmatrix}. \label{H3-minus-1} \end{equation}
It is clear that all the entries on the diagonal are nonnegative. Let us consider the leading principal $2 \times 2$ minor $m_{12}$ of the above matrix, given by
\begin{align*} m_{12} &= |\herm{12}{13}|^2 \,|\herm{21}{23}|^2 - |\herm{12}{23}|^2 \,|\herm{13}{21}|^2. \\
&= \mu_1 \mu_2 - (1 - \mu_2) (1 - \mu_1) \\
&= \mu_1 + \mu_2 - 1 \\
&\geq 1 - \mu_3 \\
&\geq 0,
\end{align*}
where we have used \eqref{bounds}. We can similarly prove that the other principal $2 \times 2$ minors of $H_3(\mathbf{x}) - \mathbf{1}$, namely $m_{13}$ and $m_{23}$, are also nonnegative. It remains only to show that the determinant of $H_3(\mathbf{x}) - \mathbf{1}$ is also nonnegative. Expanding, we have
\[ \begin{split}
& \det(H_3(\mathbf{x}) - \mathbf{1}) \\
= \, & \mu_1 \mu_2 \mu_3 + 2(1 - \mu_2)(1 - \mu_1)(1 - \mu_3) - \mu_1(1 - \mu_2)(1 - \mu_3) - \cdots \\
& \cdots - \mu_2(1 - \mu_1)(1 - \mu_3) - \mu_3(1 - \mu_1)(1 - \mu_2),
\end{split}
\]
where we have used that
\begin{align*} \herm{21}{32} &= \overline{\herm{12}{23}} \\
\herm{13}{21} &= - \overline{\herm{31}{12}} \\
\herm{32}{13} &= - \overline{\herm{23}{31}}.
\end{align*}
To understand the minus signs in the last $2$ equations, note that we are using the convention that if $1 \leq i < j \leq n$, then $w_{ji}$ is the quaternionic structure of $\mathbb{C}^2$ applied to $w_{ij}$ (see the paragraph in section \ref{At-det} preceding the statement of Conjecture 2). In particular, this implies that $w_{ij}$ is minus the quaternionic structure applied to $w_{ji}$. We note also that this quaternionic structure is anti-unitary with respect to the standard hermitian inner product on $\mathbb{C}^2$, which explains the appearance of complex conjugation in the above formulas.
Expanding and simplifying, we obtain
\[ \begin{split} & \det(H_3(\mathbf{x}) - \mathbf{1}) \\
= & \,2 - 3(\mu_1 + \mu_2 + \mu_3) + 4(\mu_1\mu_2 + \mu_1 \mu_3 + \mu_2 \mu_3) - 4 \mu_1 \mu_2 \mu_3.
\end{split}
\]
We now invoke \eqref{identity}, thus obtaining
\[ \det(H_3(\mathbf{x}) - \mathbf{1}) = 2 - 3(\mu_1 + \mu_2 + \mu_3) + (\mu_1 + \mu_2 + \mu_3)^2.\]
Factoring out the previous equation, we get
\begin{equation} \det(H_3(\mathbf{x}) - \mathbf{1}) = (\mu_1 + \mu_2 + \mu_3 - 2) (\mu_1 + \mu_2 + \mu_3 - 1),
\end{equation}
which is nonnegative, since $\mu_1 + \mu_2 + \mu_3 \geq 2$, from \eqref{bounds}. This finishes the proof of the lemma.
\end{proof}
\section{Proof of the $n = 4$ case}
Just as in the previous section, we make use of the notation
\[ \herm{ij}{kl} = \langle \mathbf{w}_{ij},\, \mathbf{w}_{kl} \rangle \]
($1 \leq i, j, k, l \leq 4$).
As a warm-up, we first expand
\[ \begin{split} & \langle p_1,\, p_1 \rangle \\
= \, & 1 + \herm{12}{13}\,\herm{13}{14}\,\herm{14}{12} +
\herm{13}{12}\,\herm{14}{13}\,\herm{12}{14}
- |\herm{12}{13}|^2 - |\herm{12}{14}|^2 - |\herm{13}{14}|^2.
\end{split}
\]
It is well known that this hermitian inner product can be written as the permanent of a $3 \times 3$ matrix. More specifically, consider the Gram matrix $T_{11}$ of $(\mathbf{w}_{12}, \mathbf{w}_{13}, \mathbf{w}_{14})$, defined by
\[T_{11} = \begin{pmatrix} 1 & \herm{12}{13} & \herm{12}{14} \\
\herm{13}{12} & 1 & \herm{13}{14} \\
\herm{14}{12} & \herm{14}{13} & 1 \end{pmatrix}. \]
Then $\langle p_1,\, p_1 \rangle = \operatorname{perm}(T_{11})$. But $\mathbf{w}_{12}$, $\mathbf{w}_{13}$ and $\mathbf{w}_{14}$ are $3$ vectors in $\mathbb{C}^2$, which must thus be linearly dependent (over $\mathbb{C}$). Hence their Gram matrix $T_{11}$ is singular. We therefore have
\[ \det(T_{11}) = 0.\]
Using the previous formula, we deduce the following.
\begin{equation}\langle p_1,\, p_1 \rangle = 2(|\herm{12}{13}|^2 + |\herm{12}{14}|^2 + |\herm{13}{14}|^2). \label{p1-p1}\end{equation}
Using a similar approach, we compute $\langle p_1,\, p_2 \rangle$, which is the permanent of the following $3 \times 3$ matrix
\[T_{12} = \begin{pmatrix} 0 & \herm{12}{23} & \herm{12}{24} \\
\herm{13}{21} & \herm{13}{23} & \herm{13}{24} \\
\herm{14}{21} & \herm{14}{23} & \herm{14}{24} \end{pmatrix}. \]
But $\mathbf{w}_{12}$, $\mathbf{w}_{13}$ and $\mathbf{w}_{14}$ are $3$ vectors in $\mathbb{C}^2$, which must thus be linearly dependent (over $\mathbb{C}$), so that $T_{11}$, which is the ``mixed'' Gram matrix between $(\mathbf{w}_{12}, \mathbf{w}_{13}, \mathbf{w}_{14})$ and $(\mathbf{w}_{21}, \mathbf{w}_{23}, \mathbf{w}_{24})$, must be singular. Hence
\[ \det(T_{12}) = 0. \]
Using the previous formula, we obtain that
\begin{equation} \langle p_1,\, p_2 \rangle = \operatorname{per}(T_{12}) = 2(\herm{12}{23}\, \herm{13}{21}\, \herm{14}{24} + \herm{12}{24}\, \herm{13}{23}\, \herm{14}{21}).\label{p1-p2}\end{equation}
There are formulas similar to \eqref{p1-p1} and \eqref{p1-p2} for any $\langle p_i,\, p_j \rangle$.
Let $H_{123}(\mathbf{x})$ be the Gram matrix of the configuration $\mathbf{x}_1$, $\mathbf{x}_2$ and $\mathbf{x}_3$ (i.e. we delete $\mathbf{x}_4$ from the original configuration). Hence $H_{123}(\mathbf{x}) - \mathbf{1}$, where $\mathbf{1}$ is the $3 \times 3$ identity matrix, is given by the RHS of \eqref{H3-minus-1}.
If $\mathbf{w}_i \in \mathbb{C}^2$, for $i = 1, \ldots, 3$, we define their Gram matrix to be
\[ H(\mathbf{w}_1, \mathbf{w}_2, \mathbf{w}_3) = (\langle \mathbf{w}_i,\, \mathbf{w}_j \rangle) \]
($1 \leq i,j \leq 3$), which is a singular hermitian positive semidefinite matrix.
We now define
\[ A_4(\mathbf{x}) =
\begin{pmatrix} \tilde{A}_4 & \mathbf{0} \\
\mathbf{0}^T & 0 \end{pmatrix}, \]
where
\[ \tilde{A}_4 = 2(H_{123} - \mathbf{1}) * H(\mathbf{w}_{14}, \mathbf{w}_{24}, \mathbf{w}_{34}),\]
with $*$ denoting the Hadamard product of matrices and where we have omitted the dependence on $\mathbf{x}$ from our notation, for brevity. It is known that the Hadamard product of two hermitian positive semidefinite matrices is also hermitian positive semidefinite, from which we deduce that $\tilde{A}_4$, and thus also $A_4$, is hermitian positive semidefinite.
Similarly, if $1 \leq i \leq 3$, we can define $\tilde{A}_i$ in a similar fashion. More precisely, we define
\begin{align*}
\tilde{A}_1 &= 2(H_{234} - \mathbf{1}) * H(\mathbf{w}_{21}, \mathbf{w}_{31}, \mathbf{w}_{41}) \\
\tilde{A}_2 &= 2(H_{134} - \mathbf{1}) * H(\mathbf{w}_{12}, \mathbf{w}_{32}, \mathbf{w}_{42}) \\
\tilde{A}_3 &= 2(H_{124} - \mathbf{1}) * H(\mathbf{w}_{13}, \mathbf{w}_{23}, \mathbf{w}_{43})
\end{align*}
For example $H_{234}$ is the $3 \times 3$ Gram matrix of the configuration $\mathbf{x}$ from which we have deleted the point $\mathbf{x}_1$.
We let $A_i$ ($1 \leq i \leq 3$) be the $4 \times 4$ matrix having $\tilde{A}_i$ as its $3 \times 3$ principal submatrix with row and column indices taken from $\{1, 2, 3, 4\} \setminus \{i\}$ and having zeros everywhere else.
Just as for $\tilde{A}_4$ and $A_4$, it is clear that the $\tilde{A}_i$ and $A_i$ ($1 \leq i \leq 3$) are all hermitian positive semidefinite.
By examining carefully the formulas for the $\langle p_i,\, p_j \rangle$ ($1 \leq i,j \leq 4$), for which \eqref{p1-p1} and \eqref{p1-p2} form a representative sample, one deduces the following fundamental decomposition for $H_4(\mathbf{x})$,
\begin{equation} H_4(\mathbf{x}) = \sum_{i = 1}^4 A_i(\mathbf{x}). \label{decomposition}\end{equation}
We now focus our attention onto $\tilde{A}_4(\mathbf{x})$, as we would like to know for which $\mathbf{x} \in C_4(\mathbb{R}^3)$ it is positive definite, which amounts to checking for which $\mathbf{x} \in C_4(\mathbb{R}^3)$ the determinant of $\tilde{A}_4(\mathbf{x})$ is positive.
We introduce some notation:
\begin{align*}
&\mu_1 = |\herm{12}{13}|^2, &\mu_2 = |\herm{21}{23}|^2, \qquad &\mu_3 = |\herm{31}{32}|^2, \\
&\rho_1 = |\herm{24}{34}|^2, &\rho_2 = |\herm{34}{14}|^2, \qquad &\rho_3 = |\herm{14}{24}|^2.
\end{align*}
We also let
\[ \tilde{\mu}_i = 1 - \mu_i \quad \text{and} \quad \tilde{\rho}_i = 1 - \rho_i \qquad \text{($1 \leq i \leq 3$)}. \]
Note that $\mu_i$, $\rho_i$, $\tilde{\mu}_i$ and $\tilde{\rho}_i$ ($1 \leq i \leq 3$) all lie in $[0, 1]$.
Using \eqref{3-cycle}, we get that
\[ \Re(\herm{14}{24}\,\herm{24}{34}\,\herm{34}{14}) = \frac{1}{2}(-1 + T),\]
where $T = \rho_1 + \rho_2 + \rho_3$.
Expanding, we get
\[\begin{split} & \det(\tilde{A}_4/2) \\
= & \,\mu_1 \mu_2 \mu_3 + \tilde{\mu}_1 \tilde{\mu}_2 \tilde{\mu}_3(-1 + T) - \tilde{\mu}_1 \tilde{\mu}_2 \mu_3 \rho_3 - \tilde{\mu}_1 \mu_2 \tilde{\mu}_3 \rho_2 - \mu_1 \tilde{\mu}_2 \tilde{\mu}_3 \rho_1.
\end{split}
\]
Letting $\widetilde{T} = 3 - T \geq 0$, we can rewrite the previous equation as follows.
\[\begin{split} & \det(\tilde{A}_4/2) \\
= & \,\mu_1 \mu_2 \mu_3 + 2\,\tilde{\mu}_1 \tilde{\mu}_2 \tilde{\mu}_3 - \tilde{\mu}_1 \tilde{\mu}_2 \mu_3 - \tilde{\mu}_1 \mu_2 \tilde{\mu}_3 - \mu_1 \tilde{\mu}_2 \tilde{\mu}_3 - \tilde{\mu}_1 \tilde{\mu}_2 \tilde{\mu}_3 \widetilde{T} + \cdots \\
& \qquad \qquad \cdots + \tilde{\mu}_1 \tilde{\mu}_2 \mu_3 \tilde{\rho}_3 + \tilde{\mu}_1 \mu_2 \tilde{\mu}_3 \tilde{\rho}_2 + \mu_1 \tilde{\mu}_2 \tilde{\mu}_3 \tilde{\rho}_1 \\
= & \, \det(H_{123}(\mathbf{x}) - \mathbf{1}) - \tilde{\mu}_1 \tilde{\mu}_2 \tilde{\mu}_3\widetilde{T} + \tilde{\mu}_1 \tilde{\mu}_2 \mu_3 \tilde{\rho}_3 + \tilde{\mu}_1 \mu_2 \tilde{\mu}_3 \tilde{\rho}_2 + \mu_1 \tilde{\mu}_2 \tilde{\mu}_3 \tilde{\rho}_1
\end{split}
\]
Let $S = \mu_1 + \mu_2 + \mu_3$. It can be shown, after expanding and using \eqref{identity}, that
\begin{equation}
4 \tilde{\mu}_1 \tilde{\mu}_2 \tilde{\mu}_3 = (S - 2)^2.
\label{identity2}
\end{equation}
Using \eqref{H3-minus-1} and \eqref{identity2}, we obtain
\[\begin{split} & \frac{1}{2}\det(\tilde{A}_4) \\
= & \, 4(S - 2)(S - 1) - (S - 2)^2 \widetilde{T} + 4(\tilde{\mu}_1 \tilde{\mu}_2 \mu_3 \tilde{\rho}_3 + \tilde{\mu}_1 \mu_2 \tilde{\mu}_3 \tilde{\rho}_2 + \mu_1 \tilde{\mu}_2 \tilde{\mu}_3 \tilde{\rho}_1).
\end{split}
\]
But $\widetilde{T} = 3 - T$, so we obtain
\begin{equation} \frac{1}{2}\det(\tilde{A}_4) = (S - 2)(S + 2) + R, \label{main-eq} \end{equation}
where
\begin{equation} R = (S - 2)^2 T + 4(\tilde{\mu}_1 \tilde{\mu}_2 \mu_3 \tilde{\rho}_3 + \tilde{\mu}_1 \mu_2 \tilde{\mu}_3 \tilde{\rho}_2 + \mu_1 \tilde{\mu}_2 \tilde{\mu}_3 \tilde{\rho}_1) \geq 0.
\label{main-eq2} \end{equation}
But we have seen that, from the previous section (see \eqref{bounds}), that $S \geq 2$ and that equality is attained if $\mathbf{x}_1$, $\mathbf{x}_2$ and $\mathbf{x}_3$ are collinear. We also claim that equality is attained \emph{only if} $\mathbf{x}_1$, $\mathbf{x}_2$ and $\mathbf{x}_3$ are collinear. This can be seen as follows. If $\alpha$, $\beta$ and $\gamma$ are the interior angles of the triangle with vertices $\mathbf{x}_1$, $\mathbf{x}_2$ and $\mathbf{x}_3$, we then have
\[ S - 2 = \operatorname{cos}^2\left(\frac{\alpha}{2}\right) + \operatorname{cos}^2\left(\frac{\beta}{2}\right) + \operatorname{cos}^2\left(\frac{\gamma}{2}\right) - 2, \]
which can be manipulated using some trigonometric identities and shown to yield
\[ S - 2 = 2 \,\operatorname{sin}\left(\frac{\alpha}{2}\right)\, \operatorname{sin}\left(\frac{\beta}{2}\right)\, \operatorname{cos}\left(\frac{\alpha+\beta}{2}\right) \geq 0, \]
since $0 \leq \frac{\alpha}{2}, \frac{\beta}{2}, \frac{\gamma}{2} \leq \frac{\pi}{2}$ and $\alpha + \beta = \pi - \gamma$. We can also see that $S - 2$ vanishes iff one of the interior angles ($\alpha$, $\beta$ and $\gamma$) vanishes, i.e. iff $\mathbf{x}_1$, $\mathbf{x}_2$ and $\mathbf{x}_3$ are collinear.
Going back to \eqref{main-eq} and \eqref{main-eq2}, we easily see that, if $\mathbf{x}_1$, $\mathbf{x}_2$ and $\mathbf{x}_3$ are collinear, then $S = 2$ and $R = 0$, so that $\det(\tilde{A}_4) = 0$, while if they are not collinear, then $\det(\tilde{A}_4) > 0$.
In other words, $\tilde{A}_4$ is (hermitian) positive definite iff the points $\mathbf{x}_1$, $\mathbf{x}_2$ and $\mathbf{x}_3$ are \emph{not} collinear.
We now go back to our main argument. If the $4$ points $\mathbf{x}_1, \ldots, \mathbf{x}_4$ are collinear, then it can be verified directly that $H_4(\mathbf{x})$ is positive definite.
We then assume that $\mathbf{x}_1, \ldots, \mathbf{x}_4$ are not collinear. It is therefore possible to remove one of the four points and still get a non-collinear configuration, so we assume, WLOG, that $\mathbf{x}_1$, $\mathbf{x}_2$ and $\mathbf{x}_3$ are not collinear. There is also a $3$-subset of the configuration of four points $\mathbf{x}$ containing $\mathbf{x}_4$ which is also not collinear. We thus assume, WLOG, that $\mathbf{x}_2$, $\mathbf{x}_3$ and $\mathbf{x}_4$ are not collinear.
Let $v \in \mathbb{C}^4$ such that
\[ \langle H_4(\mathbf{x}) v,\, v \rangle = 0, \]
where $\langle -,\, - \rangle$ now denotes the standard hermitian inner product on $\mathbb{C}^4$. Using \eqref{decomposition}, we obtain
\[ 0 = \sum_{i=1}^4 \langle A_i(\mathbf{x}) v, \, v \rangle. \]
Since the $A_i$ ($1 \leq i \leq 4$) are hermitian positive semidefinite, we therefore have
\[ \langle A_i(\mathbf{x}) v, \, v \rangle = 0, \quad \text{for $i = 1, \dots, 4$}. \]
Since $\mathbf{x}_1$, $\mathbf{x}_2$ and $\mathbf{x}_3$ are not collinear (using our assumptions above), we thus have that $\tilde{A}_4$ is positive definite. Moreover, since we have, in addition, that
\[ \langle A_4(\mathbf{x}) v, \, v \rangle = 0, \]
we therefore deduce that the first $3$ components of $v$ are $0$. Similarly, since $\mathbf{x}_2$, $\mathbf{x}_3$ and $\mathbf{x}_4$ are also assumed to be non-collinear, we therefore obtain that the last $3$ components of $v$ are $0$. Hence $v$ vanishes. We have therefore finished the proof that $H_4(\mathbf{x})$ is hermitian positive definite for any $\mathbf{x} \in C_4(\mathbb{R}^3)$.
\section{Future work}
In a future work, the author would like to attempt to apply this method to the general $n > 4$ case and see how far he would get.
\section*{Acknowledgements}
The author would like to thank Vivecca for her love and support. He also wishes to thank Dennis Sullivan, Peter Olver and Niky Kamran for their patience, after having received many emails about this problem from him, and for their moral support. Many thanks also go to Stephen Drury for some interesting discussions about the permanent of a matrix and various related inequalities. The author would also like to thank Paul Cernea for listening to his ideas and asking relevant questions.
|
1,108,101,564,586 | arxiv | \section{Introduction}\label{sec:introduction}
A Disc Jockey (DJ) is a musician who plays a sequence of existing music tracks or sound sources seamlessly by manipulating the audio content based on musical elements. The outcomes can be medleys (mix), mash-ups, remixes, or even new tracks, depending on how much DJs edit the substance of the original music tracks. Among them, creating a mix is the most basic role of DJs. This involves curating music tracks and their sections to play, deciding the order, and modifying them to splice one section to another as a continuous stream. In each step, DJs consider various
elements of the tracks
such as tempo, key, beat, chord, rhythm, structure, energy, mood and genre.
These days, DJs create the mix not only for a live audience but also for listeners in music streaming services.
Recently, imitating the tasks of DJ using computational methods has drawn research interests \cite{lin2014bridging,bittner2017automatic,schwarz2018heuristic, veire2018raw,Kim2017automatic,huang2018generating,huang2017djnet}.
On the other hand, efforts have been made to understand the creative process of DJ music making.
In the perspective of reverse engineering, tasks extracting useful information from real-world DJ mixes can be useful in such a pursuit.
In the literature, at least the following tasks have been studied.
(1) \textit{Track identification}~\cite{sonnleitner2016landmark,manzano2016audio,lopez2019analyzing}: identifying which tracks are played in DJ music which can be either a mix or a manipulated track.
(2) \textit{Mix segmentation}~\cite{glazyrin2014towards,scarfe2014segmentation}: finding boundaries between tracks in a DJ mix.
(3) \textit{Mix-to-track alignment}~\cite{werthen2018ground,schwarz2019methods}: aligning the original track to an audio segment in a DJ mix.
(4) \textit{Cue point extraction}~\cite{schwarz2019methods}: finding when a track starts and ends in a DJ mix.
(5) \textit{Transition unmixing}~\cite{werthen2018ground,schwarz2019methods}: explaining how DJs apply audio effects to make a seamless transition from one track to another. However, the previous studies only focused on solving the tasks usually with a small dataset and did not provide further analysis using extracted information from the tasks. For example,
Sonnleitner et al. \cite{sonnleitner2016landmark} used 18 mixes for track identification.
Glazyrin~\cite{glazyrin2014towards} and Scarfe et al.~\cite{scarfe2014segmentation} respectively collected 103 and 339 mixes with boundary timestamps for mix segmentation.
The majority of previous studies concentrated on identification and segmentation and few studies on the other three tasks used artificially generated datasets~\cite{werthen2018ground,schwarz2019methods}.
To address the need of a large-scale study, we collected in a total of 1,557 real-world mixes and original tracks played in the mixes from \emph{1001Tracklists}, a community-based DJ music service.\footnote{\url{https://www.1001tracklists.com}}%
The mixes include 13,728 unique tracks and 20,765 transitions.
However, tracks used in DJ mixes usually include various versions so-called ``extended mix'', ``remix'', or ``edit''.
Also, a few tracks in tracklists of the collected dataset are annotated incorrectly by users.
Therefore, an alignment algorithm is required to ensure that the collected tracks are exactly the same versions as the ones used in the mixes.
More importantly, the alignment will be a foundation for further computational analysis of DJ mixes.
With these two motivations, we set up the mix-to-track subsequence dynamic time warping (DTW)~\cite{muller2015fundamentals} such that the mix can be aligned with the original tracks in presence of possible tempo or key changes.
The warping paths from the DTW provide temporally tight mix-to-track matching from which we can obtain cue points, transition lengths, and key/tempo changes in DJ performances in a quantitative way.
To evaluate the performances of the alignment and the cue point extraction methods simultaneously, we evaluate mix segmentation performances regarding the extracted cue points as boundaries dividing two adjacent tracks in mixes, comparing them to human-annotated boundaries.
Furthermore, by observing the performance changes depending on the three different types of cue points, we analyze the human annotating policy of track boundaries.
Although DJ techniques are complicated and different depending on the characteristics of tracks, there has been common knowledge for making seamless DJ mixes.
However, to the best of our knowledge, the domain knowledge has never been addressed in the literature with statistical evidence obtained by computational analysis.
In this study, we analyze the DJ mixes using the results from the subsequence DTW mentioned above for the following hypotheses:
1) DJs tend not to change tempo and/or key of tracks much to avoid distorting the original essence of the tracks.
2) DJs make seamless transitions from one track to another considering the musical structures of tracks.
3) DJs tend to select cue points at similar positions in a single track.
The analysis is performed based on the results obtained from the subsequence alignment and provides insights statistically for tempo adjustment, key transposition, track-to-track transition lengths, and agreements of the cue points among DJs.
We hope that the proposed analysis and various statistics may elucidate the creative process of DJ music making.
The source code for the mix-to-track subsequence DTW, the cue point analysis and the mix segmentation is available at the link.\footnote{\url{https://github.com/mir-aidj/djmix-analysis/}}
\begin{table}[!t]
\centering
\scalebox{0.82}{
\begin{tabular}{lrr}
\toprule
Summary statistic & All & Matched \\
\midrule
The number of mixes & 1,564 & 1,557 \\
The number of unique tracks & 15,068 & 13,728 \\
The number of played tracks & 26,776 & 24,202 \\
The number of transitions & 24,344 & 20,765 \\
Total length of mixes (in hours) & 1,577 & 1,570 \\
Total length of unique tracks (in hours) & 1,038 & 913 \\
Average length of mixes (in minutes) & 60.5 & 60.5 \\
Average length of unique tracks (in minutes) & 4.1 & 4.0 \\
Average number of played tracks in a mix & 17.1 & 15.5 \\
Average number of transitions in a mix & 14.5 & 12.9 \\
\bottomrule
\end{tabular}
}
\vspace{-3mm}
\caption{Statistics of the \emph{1001Tracklists} dataset.
The original dataset size is denoted as `All' and the size after filtering as `Matched'.
}
\label{tab:dataset}
\end{table}
\vspace{-2mm}
\section{The Dataset}\label{sec:dataset}
Our study is based on DJ music from \emph{1001Tracklists}. We obtained a collection of DJ mix metadata via direct personal communication with \emph{1001Tracklists}.
Each entry of mixes contains a list of track, boundary timestamps and genre.
It also contains web links to the audio files of the mixes and tracks. We downloaded them separately from the linked media service websites on our own. We found a small number of web links to tracks are not correct and so filtered them out by a mix-to-track alignment method automatically (see Section~\ref{sec:matchrate}). The boundary timestamps of tracks in a mix are annotated by the users of \emph{1001Tracklists}.
\tablename~\ref{tab:dataset} summarizes statistics of the dataset. The original size of the dataset is denoted as `All' and the size after filtering as `Matched' in \tablename~\ref{tab:dataset}. Note that the number of played tracks is greater than the number of unique tracks as a track can be played in multiple mixes.
The dataset includes a variety of genres but mostly focuses on House and Trance music.
More detailed statistics of the dataset are available on the companion website.\footnote{\url{https://mir-aidj.github.io/djmix-analysis/}}
\begin{figure*}[!t]
\centering
\hspace{-4mm}
\includegraphics[width=1.01\textwidth]{fig_align}
\vspace{-5mm}
\caption{Visualizations of the result of a DTW-based mix-to-track subsequence alignment between a mix and the original tracks played in that mix.
The colored solid lines show the warping paths of the alignment depending on the input feature, and whether or not applying the transposition-invariant method on the subsequence DTW.
The tagged numbers on warping paths and ground truth boundaries indicate played and timestamped indices in the mix, respectively.
A colored bar at the bottom of the figures is added if the alignment of the method is considered successful according to the match rate.
(Top) A correctly matched example.
(Middle) An unsuccessful example, due to the low sound quality of the mix.
(Bottom) The alignment can be improved using the key-invariant chroma. Best viewed in color.
}
\label{fig:align}
\end{figure*}
\vspace{-2mm}
\section{Mix-To-Track Subsequence Alignment}
The objective of mix-to-track subsequence alignment is to find an optimal alignment path between a subsequence of a mix and a track used in the mix. This alignment result will be the basis of diverse DJ mix analysis concerning the cue point, track boundary, key/tempo changes and transition length. We also use it for removing non-matching tracks. This section describes the detail of computational process.
\subsection{Feature Extraction}
When DJs create a mix, they often adjust tempo and/or key of the tracks in the mix or add audio effects to them. Live mixes contain more changes in timbre and even other sound sources such as the voices from the DJ. In order to address the acoustic and musical variations between the original track and the matched subsequence in the mix, we use beat synchronous chroma and mel-frequency cepstral coefficients (MFCC). The beat synchronous feature representations enable tempo invariance and dramatically reduces the computational cost in the alignment. The aggregation of the features from the frame level to the beat level also smooths out local timbre variations. The chroma feature, on the other hand, facilitates key-invariance as circular shift of the 12-dimensional vector corresponds to key transposition. The MFCC feature captures general timbre characteristics. We used Librosa\footnote{\url{https://librosa.github.io/librosa/}} to extract the chroma and MFCC features with the default options except that the dimentionality of MFCC was set to 12 and the type of chroma was to chroma energy normalized statistics (CENS)~\cite{muller2011chroma}.
\subsection{Key-Invariant Subsequence DTW}
\label{sec:dtw}
We compute the alignment by applying subsequence DTW to the beat synchronous features~\cite{muller2015fundamentals}. We used an implementation from Librosa, adopting the transposition-invariant approach from \cite{muller2007transposition}. Specifically, we calculated 12 versions of chroma features by performing all possible circular shifts on the original track side and select the one with the lowest matching cost in the subsequence DTW.
This result returns not only the optimal alignment path but also the key transposition value of the original track.
\figurename~\ref{fig:align} shows three examples of the alignment results when different combinations of features (MFCC, chroma, and key-invariant chroma) are used.
When the alignment path of the subsequence satisfies a match rate (described in Section \ref{sec:matchrate}), we put a color strip corresponding to each feature in the bottom of the figure.
Since we use beat synchronous representations for them, the warping paths become diagonal with a slope of one if a mix and a track are successfully aligned.
The top panel in the figure shows an successfully aligned example for the most of tracks and features where all warping paths have straight diagonal paths.\footnote{\url{https://1001.tl/14jltnct}}
The middle panel shows a failing example because sounds from crowds are also recorded in the mix.\footnote{\url{https://1001.tl/15fulzc1}}
The bottom panel shows a example where chroma with circular shift distinctively works better others as the DJ frequently uses key transposition on the mix.\footnote{\url{https://1001.tl/bcx2z0t}}
\subsection{Filtering Using Match Rates}
\label{sec:matchrate}
As stated above, we can measure the quality of the alignment from the warping path. Ideally, when every single move on the path is diagonal, that is, one beat at a time for both track and mix axis, we will obtain a perfect straight diagonal line. However, the acoustic and musical changes deform the path. We define the ratio of the diagonal moves in a mix (one move per beat) as the \textit{match rate} and use it for filtering out incorrectly annotated tracks. We experimentally chose 0.4 as a threshold. The size of the dataset after the filtering is denoted as ``Matched'' in
\tablename~\ref{tab:dataset}. We only use the matched tracks for the analysis in this paper.
\begin{figure}[t]
\centering
\hspace{-.4cm}
\includegraphics[width=\columnwidth]{fig_cue}
\vspace{-5mm}
\caption{
A zoomed-in view of a visualization of mix-to-track subsequence alignment explaining the three types of extracted cue points. The two solid lines indicate warping paths representing alignment between the mix and tracks.
The vertical colored dotted lines represent the extracted cue points on the mix and the horizontal dotted lines represent the points on each track.
The vertical black dotted line is a human-annotated ground truth boundary between the two tracks.
The solid lines are the fourth and fifth warping paths from the top of \figurename~\ref{fig:align}. Best viewed in color.
}
\label{fig:cue}
\end{figure}
\vspace{-2mm}
\section{Cue Point Extraction}
Cue points are timestamps in a track that indicate where to start and end the track in a mix.
Determining the cue points of played tracks is an essential task of DJ mixing.
This section describes extracting cue points using the warping paths obtained from the aforementioned mix-to-track subsequence alignment.
\subsection{Term Definitions}
We first define terms related to cue points. In the context of the track-to-track transition, a \textit{cue-out} point is a timestamp that the previous track starts fading out and the next track starts fading in, and a \textit{cue-in} point is when the previous track is fully faded out and only the next track is being played. The \textit{transition} region is defined as the time interval from the cue-out point of the previous track to the cue-in point of the next track. Additionally, we define a \textit{cue-mid} point as the middle of a transition, which can technically be considered as a boundary of the transition.
\subsection{Methods}
The mix-to-track alignment results naturally yield cue points of matched tracks.
\figurename~\ref{fig:cue} shows an example of extracted cue points (a zoomed-in view of the top figure in \figurename~\ref{fig:align}).
The two alignment paths drift from the diagonal lines in the transition region (between 2310 and 2324 in mix beat) because the two tracks cross-fades. Based on this observation, we detect the cue-out point of the previous track by finding the last beat where preceding 32 beats have diagonal moves in the alignment path. Likewise, we detect the cue-in point of the next track by finding the first beat where succeeding 32 beats have diagonal moves in the alignment path.
\begin{figure*}[!t]
\centering
\hspace{-3mm}
\includegraphics[width=0.85\textwidth]{fig_align_beat_diff}
\vspace{-5mm}
\caption{Histograms of distances to ground truth boundaries in the number of beats depending on the type of the cue point. The dotted lines are plotted at every 32 beats which is usually considered as a phrase in the context of dance music.}
\label{fig:align_beatdiff}
\end{figure*}
\begin{table*}[!t]
\begin{center}
\scalebox{0.9}{
\begin{tabular}{lrrrrcrrr}
\toprule
& \multicolumn{4}{c}{Median time difference (in seconds)} & & \multicolumn{3}{c}{Cue-in hit rate} \\
\cmidrule{2-5} \cmidrule{7-9}
Feature & Cue-out & Cue-in & Cue-mid & Cue-best$^\dagger$ & & 15 sec & 30 sec & 60 sec \\
\hline
\midrule
MFCC & 27.92 & 14.27 & 13.55 & 5.340 & & 0.5187 & 0.7591 & 0.9023 \\
\hline
Chroma & 23.85 & 11.80 & 12.33 & \textbf{4.230} & & 0.5837 & 0.7973 & 0.9286 \\
Chroma with key-invariant & 23.87 & 11.77 & 12.37 & 4.240 & & 0.5843 & 0.7968 & 0.9282 \\
\hline
Chroma + MFCC & 23.41 & 11.48 & \textbf{12.16} & 4.380 & & 0.5866 & 0.8035 & 0.9284 \\
Chroma with key-invariant + MFCC & \textbf{23.38} & \textbf{11.40} & \textbf{12.16} & 4.380 & & \textbf{0.5881} & \textbf{0.8040} & \textbf{0.9288} \\
\bottomrule
\end{tabular}
}
\end{center}
\vspace{-4mm}
\caption{
Mix segmentation performances depending on the type of cue point and the input feature used to obtain the warping paths.
Median time differences between cue points and ground truths are shown on the left side and hit rates of cue-in points with thresholds in seconds are shown on the right side.
``Key-invariant" indicates applying the key transposition-invariant method for the DTW.
The best score of each criteria is shown in \textbf{bold}.
$\dagger$ indicates the scores are computed using the best score among the three cue types.
}
\label{tab:hit}
\end{table*}
\begin{table}[t]
\centering
\scalebox{0.9}{
\begin{tabular}{ccc}
\toprule
Cue-out & Cue-in & Cue-mid \\
\midrule
6,151 (30\%) & 10,844 (52\%) & 3,770 (18\%) \\
\bottomrule
\end{tabular}
}
\caption{The number of ground truth boundary timestamps closest to the type of cue point.}
\label{tab:policy}
\end{table}
\section{Mix Segmentation}
The goal of mix segmentation is to divide a continuous DJ mix into individual tracks, which can enhance the listening experience and can be a foundation of further analysis or learning of DJ mixes.
Since DJs make seamless transitions, it is difficult to notice that a track is fading in or out.
To quantitatively measure how difficult it is, a study analyzed how accurate humans are at creating the boundary timestamps and found that the standard deviation of the human disagreement for track boundaries in mixes is about 9 seconds, which implies it is difficult to find the optimal boundaries even for humans~\cite{scarfe2014segmentation}. Furthermore, the ambiguous definition of the boundary and long lengths of transitions makes it difficult to annotate the boundary timestamps~\cite{sonnleitner2016landmark}.
\subsection{Cue Point based Estimation}
Given the extracted cue point so far, we can estimate the track boundaries with three possible choices. The first is the position that the next track fully appears (cue-in point), the second is the position that previous track starts to disappear (cue-out point), and the last is the middle of the transition (cue-mid point). By comparing each of them with human-annotated boundary timestamps, we can measure which type of cue point humans tend to consider as a boundary.
\figurename~\ref{fig:align_beatdiff} shows three histograms where each of them is computed from the differences between human-annotated boundary timestamps and one of the cue point types in beat unit. The overall trend shows that the distribution of cue-in point is mostly skewed towards zero. Interestingly, the distribution of cue-out point has more distinctive peaks around every 32 beat than the distribution of cue-in point. Considering the histogram of the transition length has peaks at every 32 beat as shown in \figurename~\ref{fig:cue_translength}, this reflects that human annotators tend to label cue-in points as a boundary compared to cue-out (note that the transition length is computed by subtracting the cue-in point from the cue-out point). On the other hand, the distribution of cue-mid point has a gradually decreasing curve without peaks. While this distribution looks like having better estimates than the cue-out point, \tablename~\ref{tab:policy} shows an opposite result. That is, in terms of the number of cue points closest to the human annotations, the cue-out point is the second and the cue-mid point is the worst among the three types. These results indicate that the cue-mid point is a safe choice. That is, although the cue-mid point is least likely to be a boundary as shown in \tablename~\ref{tab:policy}, the difference between the estimate and human annotation is relatively small because it is the middle of the transition region.
\tablename~\ref{tab:hit} shows the difference between human-annotated boundary timestamps and one of the cue point types in terms of median time (in seconds) on the left side. The overall trend confirms that the cue-in is the best estimate of track boundary and the cue-mid is a safer choice than the cue-out. The table also shows the result of ``cue-best''. This is computed with the minimum difference among the three cue point types for each of the transition region. The result shows that the median time differences are dramatically decreased to 4-5 seconds. \tablename~\ref{tab:hit} also shows the difference between human-annotated boundary timestamps and the cue-in point in terms of hit rates on the right side. The hit rates are computed the ratio of correct estimates given a tolerance window. If the estimate is within the tolerance window on the human-annotated boundary timestamp, it is regarded as a correct estimate. We set three tolerance windows (15, 30, and 60 seconds) considering that the average tempo of tracks in the dataset is 127 beat per minute (BPM) and then the tolerance windows approximately correspond to 32, 64, 128 beats (multiples of a phrase unit). The result shows that the best hit rate with the 30 second window (about 64 beats) is above 80\%. Given the long transition time as shown in \figurename~\ref{fig:cue_translength}, the cue-in point may be considered as a reasonable choice.
\subsection{Effect of Audio Features}
\tablename~\ref{tab:hit} also compares the median time difference between human-annotated boundary timestamps and one of the cue point types for different audio features used in the subsequence DTW. In general, the chroma features are a better choices than MFCC (p-value of t-test < 0.001 for chroma with or without key-invariant). When both of chroma and MFCC are combined, the median time difference slightly reduces but it is statistically insignificant (p-value of t-test > 0.1). However, we observed that the subsequence DTW does not work well for some genres such as Techno which only contain drum and ambient sounds. This might can be improved by using MFCCs with a large number of bins or using mel-spectrograms. The use of key-invariant chroma generally does not make much difference because key transposition does not performed frequently as discussed in Section~\ref{sec:key_trans}.
\section{Musicological Analysis of DJ Mixes}
\label{sec:tempokey_analysis}
We hypothesize that DJs share common practices in the creative process in terms of tempo change, track-to-track transition, and cue point selection. In this section, we validate them using the results from the mix-to-track subsequence alignment and the cue point extraction.
\subsection{Tempo Adjustment}
We compare the estimated tempo of the original track to the tempo of each audio segment where the track is played in a mix. \figurename~\ref{fig:diff_bpm} shows a histogram of percentage differences of tempo between the original track and the audio segment in the mix. For example, a difference of 5\% indicates the tempo of the original track is increased by 5\% while played in the mix. As shown in the histogram, the adjusted tempo has an double exponential distribution, which means the adjusted tempo values are skewed towards zero. In detail, 86.1\% of the tempo are adjusted less than 5\%, 94.5\% are less than 10\%, and 98.6\% are less than 20\%. If one implements an track identification system for DJ mix that is robust to tempo adjustment, this distribution could be a reference.
\subsection{Key Transposition}
\label{sec:key_trans}
A function so-called ``master tempo'' or ``key lock'' that preserves pitch despite tempo adjustments is activated by default in modern DJ systems such as stand-alone DJ systems, DJ softwares, and even turntables for vinyl records.
Therefore, key transposition is usually performed when a DJ intentionally wants to change the key of a track.
As mentioned in Section~\ref{sec:dtw}, the transposition-invariant DTW can provide the number of transposed semitones as a by-product. We computed the statistics of key transposition using them (using DTW taking both MFCCs and key-invariant chroma). \figurename~\ref{fig:diff_key} shows a histogram of key transposition between the original track and the audio segment in the mix. Only 2.5\% among the total 24,202 tracks are transposed and, among those transposed tracks, 94.3\% of them are only one semitone transposed. This result indicates that DJs generally do not perform key transposition much and leave the ``master tempo'' function turned on in most cases.
\begin{figure}[t]
\centering
\hspace{-1cm}
\includegraphics[width=.8\columnwidth]{fig_diff_bpm}
\vspace{-5mm}
\caption{A histogram of adjusted tempo of tracks in mixes.}
\label{fig:diff_bpm}
\end{figure}
\begin{figure}[!t]
\centering
\hspace{-1.1cm}
\includegraphics[width=.65\columnwidth]{fig_diff_key}
\vspace{-3.8mm}
\caption{The number of tracks depending on the number of semitones in mixes.}
\label{fig:diff_key}
\end{figure}
\begin{figure}[t]
\centering
\hspace{-.3cm}
\includegraphics[width=\columnwidth]{fig_cue_translength}
\vspace{-5mm}
\caption{
A histogram of the transition lengths in number of beats.
The dotted lines are plotted at every 32 beats.
}
\label{fig:cue_translength}
\end{figure}
\begin{figure}[t]
\centering
\hspace{-.5cm}
\includegraphics[width=.81\columnwidth]{fig_cue_diff}
\vspace{-5mm}
\caption{
A histogram of distances between cue points of a single track in the number of beats.
}
\label{fig:cue_diff}
\end{figure}
\subsection{Transition Length}
\label{sec:trans_length}
Once we extract cue-in and cue-out points in the transition region, we can calculate the transition length. This can provide some basic hints on how DJ makes the track-to-track transition in a mix. \figurename~\ref{fig:cue_translength} shows a histogram of transition lengths in the number of beats. We annotated the dotted lines every 32 beat which is often considered as a phrase in the context of dance music. The histogram has peaks at every phrase. This indicates that DJs consider the repetitive structures in the dominant genres of music when they make transitions or set cue points.
\subsection{Cue Point Agreement among DJs}
Deciding cue points of played tracks is a creative choice in DJ mixing. Observing the agreement of cue points on a single track among DJs may elucidate the possibility of finding some common rules. To the end, we collected all extracted cue points for each track and computed the statistics of deviations in cue-in points and cue-out points among DJs. Specifically, we computed all possible pairs and their distances separately for cue-in points and cue-out points. Since the two distributions were almost equal, we combined them into a single distribution in \figurename~\ref{fig:cue_diff}. From the results, 23.6\% of the total cue point pairs have zero deviation. 40.4\% of them were within one measure (4 beats), 73.6\% were within 8 measures and 86.2\% were within 16 measures. This indicates that there are some rules that DJs share in deciding the cue points. It would be interesting to perform detailed pattern analysis to estimate the cue points using this data in future work.
\vspace{-2mm}
\section{Conclusions}
We presented various statistics and analysis of 1,557 real-world DJ mixes from \emph{1001Tracklists}.
Based on the mix-to-track subsequence DTW, we conducted cue point analysis of individual tracks in the mixes and showed the possibility of common rules in the music making that DJs share.
We also investigated mix segmentation by comparing the three types of cue point to human-annotated boundary timestamps and showed that humans tend to recognize cue-in points of the next tracks as boundaries. Finally, we showed the statistics of tempo and key changes of the original tracks in DJ performances. We believe this large-scale statistical analysis of DJ mixes can be beneficial for computer-based research on DJ music. The cue point analysis can be the ground for the precise definition of cue points and the tempo and key analysis can provide a guideline of the musical changes during the DJ mixing.
As a future work, we plan to estimate cue points within a track as a step towards automatically generating a mix~\cite{schwarz2018heuristic,veire2018raw}. The cue point estimation has many application such as DJ software and playlist generation on music streaming services. This will require structure analysis or segmentation of a single music track, which is an important topic in MIR.
Furthermore, we plan to analyze the transition region in a mix to investigate DJ mixing techniques. For example, it is possible to estimate the gain changes in the cross-faded region by comparing the two adjacent original tracks and the mix~\cite{schwarz2019methods,werthen2018ground}.
The methods can be extended to the spectrum domain. Such detailed analysis of mixing techniques will allow us to understand how DJs seamlessly concatenate music tracks and provide a guide to develop automatic DJ systems.
\section{Acknowledgement}
We greatly appreciate \emph{1001Tracklists} for offering us the mix metadata employed in this study.
We note that the metadata used for this analysis was obtained with permission from \emph{1001Tracklists}, and suggest that people who are interested in the data contact \emph{1001Tracklists} directly.
This research was supported by BK21 Plus Postgraduate Organization for Content Science (or BK21 Plus Program), Basic Science Research Program through the National Research Foundation of Korea (NRF-2019R1F1A1062908), and a grant from the Ministry of Science and Technology, Taiwan (MOST107-2221-E-001-013-MY2).
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\begin{document}
\preprint{
HU-EP-21/15-RTG
}
\title{Gravitational Bremsstrahlung and Hidden Supersymmetry of Spinning Bodies}
\author{Gustav Uhre Jakobsen}
\email{[email protected]}
\affiliation{%
Institut f\"ur Physik und IRIS Adlershof, Humboldt-Universit\"at zu Berlin,
Zum Gro{\ss}en Windkanal 2, 12489 Berlin, Germany
}
\affiliation{Max Planck Institute for Gravitational Physics (Albert Einstein Institute), Am M\"uhlenberg 1, 14476 Potsdam, Germany}
\author{Gustav Mogull}
\email{[email protected]}
\affiliation{%
Institut f\"ur Physik und IRIS Adlershof, Humboldt-Universit\"at zu Berlin,
Zum Gro{\ss}en Windkanal 2, 12489 Berlin, Germany
}
\affiliation{Max Planck Institute for Gravitational Physics (Albert Einstein Institute), Am M\"uhlenberg 1, 14476 Potsdam, Germany}
\author{Jan Plefka}
\email{[email protected]}
\affiliation{%
Institut f\"ur Physik und IRIS Adlershof, Humboldt-Universit\"at zu Berlin,
Zum Gro{\ss}en Windkanal 2, 12489 Berlin, Germany
}
\author{Jan Steinhoff}
\email{[email protected]}
\affiliation{Max Planck Institute for Gravitational Physics (Albert Einstein Institute), Am M\"uhlenberg 1, 14476 Potsdam, Germany}
\begin{abstract}
The recently established formalism of a worldline quantum field theory,
which describes the classical scattering of massive bodies
(black holes, neutron stars or stars) in Einstein gravity,
is generalized up to quadratic order in spin,
revealing an alternative $\mathcal{N}=2$ supersymmetric description
of the symmetries inherent in spinning bodies.
The far-field time-domain waveform of the gravitational waves produced in such a
spinning encounter is computed at leading order in the post-Minkowskian
(weak field, but generic velocity) expansion, and exhibits this supersymmetry.
From the waveform we extract the
leading-order total radiated angular momentum in a generic reference frame,
and the total radiated energy in the center-of-mass frame
to leading order in a low-velocity approximation.
\end{abstract}
\maketitle
The rise of gravitational wave (GW) astronomy~\cite{Abbott:2016blz,*LIGOScientific:2018mvr,*Abbott:2020niy} offers new paths to explore our universe, including black hole (BH) population and formation studies~\cite{Abbott:2020gyp}, tests of gravity in the strong-field regime~\cite{Abbott:2020jks}, measurements of the Hubble constant~\cite{Abbott:2019yzh},
and investigations of strongly interacting matter inside neutron stars~\cite{Abbott:2018exr}.
This form of astronomy relies heavily on Bayesian methods to infer
probability distributions for theoretical GW predictions (templates),
depending on a source's parameters, to match the measured strain on detectors.
With the network of GW observatories steadily increasing in sensitivity~\cite{TheLIGOScientific:2014jea,*TheVirgo:2014hva,*Aso:2013eba}, theoretical GW predictions need to keep pace with the accuracy requirements placed on templates~\cite{Purrer:2019jcp}.
For the inspiral and merger phases of a binary
an important strategy is to synergistically combine approximate and numerical relativity predictions~\cite{Buonanno:1998gg,*Ajith:2007qp},
each applicable only to a corner of the parameter space~\cite{vandeMeent:2020xgc}.
In this Letter we calculate gravitational waveforms ---
the primary observables of GW detectors ---
produced in the parameter-space region of highly eccentric
(scattering) \emph{spinning} BHs and neutron stars (NSs),
to leading order in the weak-field, or post-Minkowskian (PM), approximation.
Following the above strategy, this is a valuable input for future eccentric waveform models.
Indeed, the extension of contemporary quasi-circular (non-eccentric) waveform models for spinning binaries to eccentric orbits (including scattering) is under active investigation~\cite{Ramos-Buades:2019uvh,*Chiaramello:2020ehz,*Nagar:2021gss,*Liu:2021pkr,*Khalil:2021txt,*Hinderer:2017jcs,*Islam:2021mha}.
This is motivated, for instance, by the potential insight gained on the formation channels or astrophysical environments of binary BHs (BBHs) through measurements of eccentricity~\cite{Samsing:2017xmd,*Rodriguez:2017pec,*Gondan:2020svr} and spins~\cite{LIGOScientific:2018jsj}, or the search for scattering BHs~\cite{Kocsis:2006hq,*Mukherjee:2020hnm,*Zevin:2018kzq,*Gamba:2021gap} in our universe.
Accurate predictions for GWs from BBHs should crucially also account
for the BHs' spins~\cite{Zackay:2019tzo,*Huang:2020ysn},
and this is an important aspect of the present work.
The gravitational waveforms presented here are valid up to quadratic order
in angular momenta (spins) of the compact stars;
that is, we extend Crowley, Kovacs and Thorne's seminal non-spinning result
\cite{1975ApJ...200..245T,*1977ApJ...215..624C,*Kovacs:1977uw,*Kovacs:1978eu}.
We also improve on our earlier reproduction of the non-spinning result~\cite{Jakobsen:2021smu} by presenting results in a compact Lorentz-covariant form,
using an improved integration strategy.
To obtain these results we generalize the recently introduced
worldline quantum field theory (WQFT) formalism
\cite{Mogull:2020sak,Jakobsen:2021smu}
to spinning particles on the worldline.
This is achieved by including anticommuting worldline fields
carrying the spin degrees of freedom,
building upon Refs.~\cite{Howe:1988ft,Gibbons:1993ap,Bastianelli:2005vk,*Bastianelli:2005uy}.
Our formalism manifests an $\mathcal{N}=2$ extended worldline supersymmetry (SUSY)
which holds up to the desired quadratic order in spin.
The SUSY implies conservation of the covariant spin-supplementary condition (SSC),
and thus represents an alternative formulation of the symmetries inherent to spinning bodies.
It also operates on the spinning waveform.
The spinning WQFT innovates over previous approaches to classical spin based on corotating-frame variables~\cite{Porto:2005ac,Levi:2015msa} in the effective field theory (EFT) of compact objects~\cite{Goldberger:2004jt,*Goldberger:2006bd,*Goldberger:2009qd,Porto:2016pyg,*Levi:2018nxp} ---
see Refs.~\cite{Goldberger:2017ogt,*Goldberger:2016iau,*Shen:2018ebu} for the construction of PM integrands and Refs.~\cite{Liu:2021zxr,Kalin:2020mvi} for worldline and spin deflections
(in agreement with scattering amplitude results~\cite{Bern:2020buy,Kosmopoulos:2021zoq}).
The worldline EFT was applied to radiation also in the weak-field and slow-motion,
i.e.~post-Newtonian (PN),
approximation~\cite{Porto:2010zg,*Porto:2012as,*Maia:2017gxn,*Maia:2017yok,*Cho:2021mqw}--- see Refs.~\cite{Mishra:2016whh,*Buonanno:2012rv} for more traditional methods.
Other approaches to PM spin effects can be found in Refs.~\cite{Vines:2017hyw,*Bini:2017xzy,*Bini:2018ywr,*Guevara:2017csg,*Vines:2018gqi,*Guevara:2018wpp,*Chung:2018kqs,*Guevara:2019fsj,*Chung:2019duq,*Damgaard:2019lfh,*Aoude:2020onz,*Guevara:2020xjx}.
\sec{Spinning Worldline Quantum Field Theory}
It has been known since the 1980s \cite{Howe:1988ft} that the relativistic wave equation
for a massless or massive spin-$\mathcal{N}/2$ field in flat spacetime
(generalizing the Klein-Gordon, Dirac and Maxwell or Proca equations)
may be obtained by quantization of an extended supersymmetric particle model
where one augments the bosonic trajectory $x^{\mu}(\tau)$ by $\mathcal{N}$ anticommuting,
real worldline fields.
Generalizing this to a curved background spacetime
comes with consistency problems beyond $\mathcal{N}=2$.
Yet the situation for spins up to one is well understood
\cite{Bastianelli:2005vk,Bastianelli:2005uy},
and sufficient for our purposes of describing
two-body scattering up to quadratic order in spin.
We therefore augment the worldline trajectories $x_i^{\mu}(\tau_{i})$ ($i=1,2$)
of our two massive bodies
by anticommuting \emph{complex} Grassmann fields $\psi_i^{a}(\tau_{i})$.
These are vectors in the flat tangent Minkowski spacetime connected to
the curved spacetime via the vierbein $e^{a}_{\mu}(x)$.
The worldline action in the massive case for each body
takes the form (suppressing the $i$ subscripts)
\cite{Bastianelli:2005uy,Jakobsen:2021zvh}
\begin{align}\label{N=2act}
S= -m\int\!\mathrm{d}\tau \Bigl [&\sfrac{1}{2}g_{\mu\nu}\dot x^{\mu}\dot x^{\nu}
\!+\! i\bar\psi_a\sfrac{D\psi^a}{D\tau}\!+\!\sfrac{1}{2}
R_{abcd}\bar\psi^{a}\psi^{b}\bar\psi^{c}\psi^{d}\Bigr ]\,,
\end{align}
where $g_{\mu\nu}=e^{a}_{\mu}e^{b}_{\nu}\eta_{ab}$ is the metric in mostly minus signature,
$\sfrac{D\psi^a}{D\tau}=\dot\psi^{a}+\dot x^{\mu}{{\omega_\mu}^a}_b\psi^{b}$
includes the spin connection $\omega_{\mu ab}$
and the Riemann tensor is
$R_{\mu\nu ab}=e^c_\mu e^d_\nu R_{abcd}=
2(\partial_{[\mu}\omega_{\nu]ab}+\omega_{[\mu\,a}{}^{c}\omega_{\nu]cb})$.
This theory enjoys a global $\mathcal{N}=2$ SUSY:
it is invariant under
\begin{equation}
\label{N=2SUSY}
\delta x^{\mu} = i\bar\epsilon \psi^{\mu} + i\epsilon \bar\psi^{\mu} \, ,
\quad
\delta \psi^{a}= -\epsilon e^{a}_{\mu}{\dot x}^{\mu} -\delta x^{\mu}\, \omega_{\mu}{}^{a}{}_{b}\psi^{b}\, ,
\end{equation}
with constant SUSY parameters $\epsilon$ and $\bar \epsilon = \epsilon^{\dagger}$.
The connection to a traditional description of spinning bodies in general relativity,
using the spin field $S^{\mu\nu}$ and the Lorentz body-fixed frame
$\Lambda^{A}_{\mu}$~\cite{Vines:2016unv,Porto:2008jj,Porto:2005ac,Levi:2015msa,Porto:2016pyg,*Levi:2018nxp},
comes about upon identifying the spin field $S^{\mu\nu}(\tau)$ with the Grassmann bilinear:
\begin{equation}
S^{\mu\nu}=-2i e^{\mu}_{a}e^{\nu}_{b}\,\bar{\psi}^{[a}\psi^{b]}\,.
\end{equation}
One can easily show that $S^{ab}$ obeys the Lorentz algebra under Poisson brackets $\{\psi^a,\bar\psi^b\}_{\text{P.B.}}=-i\eta^{ab}$.
In fact, the spin-supplementary condition (SSC) and preservation of spin length
may be related to $\mathcal{N}=2$ SUSY-related constraints~\cite{Jakobsen:2021zvh}.
Finally, by deriving the classical equations of motion from the action
these can be shown to match the Mathisson-Papapetrou equations~\cite{Mathisson:1937zz,*Papapetrou:1951pa,*Dixon:1970zza} at quadratic spin order.
This indicates a hidden $\mathcal{N}=2$ SUSY in the
actions of Refs.~\cite{Porto:2008jj,Vines:2016unv,Levi:2015msa}.
The actions of Refs.~\cite{Porto:2008jj,Vines:2016unv,Levi:2015msa}
also carry a first spin-induced \emph{multipole moment term}
at quadratic order in spins with an undertermined Wilson
coefficient $C_{E}$,
where here $C_E=0$ for a Kerr BH.
Translating it to our formalism this term reads
\begin{equation}
S_{ES^{2}}:= -m\int\!\mathrm{d}\tau\, C_E E_{ab} \bar\psi^a \psi^b\, \bar\psi\cdot\psi\,,
\end{equation}
where $E_{ab}:= R_{a \mu b\nu} \dot x^{\mu} \dot x^{\nu}$ is the ``electric'' part of the Riemann tensor.
The $\mathcal{N}=2$ SUSY is now maintained only in an approximate sense~\cite{Jakobsen:2021zvh}:
it survives in the action for terms up to $\mathcal{O}(\psi^5)$,
i.e.~quadratic order in spin.
In order to describe a scattering scenario we expand the worldline fields
about solutions of the equations of motion along straight-line trajectories:
\begin{align}\label{bgexp}
\begin{aligned}
x^\mu_{i}(\tau_{i})&=b^\mu_{i}+v^\mu_{i}\tau_i +z^\mu_{i}(\tau_{i}) \, ,\\
\psi^{a}_{i}(\tau_{i}) & = \Psi_{i}^{a} +\psi_{i}^{\prime a}(\tau_{i})\,,
\end{aligned}
\end{align}
where $\mathcal{S}_{i}^{\mu\nu}:=-2i\bar\Psi_{i}^{[\mu}\Psi_{i}^{\nu]}$
captures the initial spin of the two massive objects.
The weak gravity expansion of the vierbein reads
\begin{equation}
e^{a}_{\mu} = \eta^{a\nu}\left(\eta_{\mu\nu}+ \frac{\kappa}{2}h_{\mu\nu} -
\frac{\kappa^2}{8}h_{\mu\rho}{h^\rho}_\nu + \mathcal{O}(\kappa^3) \right)\, ,
\end{equation}
introducing the graviton field $h_{\mu\nu}(x)$
and the gravitational coupling $\kappa^{2}= 32\pi G$.
Note that in this perturbative framework the
distinction between curved $\mu,\nu,\ldots$ and tangent
$a,b,\ldots$ indices necessarily drops.
The spinning WQFT has the partition function
\begin{align}\label{ZWQFTdef}
\mathcal{Z}_{\text{WQFT}}
&:= \text{const} \times
\int \!\!D[h_{\mu\nu}]
\, e^{i (S_{\rm EH}+S_{\rm gf})}\\ &\quad
\times\int \prod_{i=1}^{2} D [z_i^\mu] D[{\psi_{i}^{\prime}}^\mu]
\exp\Bigl[i\sum_{i=1}^{2} S^{(i)}+S_{ES^{2}}^{(i)}\Bigr ],\nonumber
\end{align}
where $S_{\rm EH}$ is the Einstein-Hilbert action
and the gauge-fixing term $S_{\rm gf}$ enforces de Donder gauge.
The SUSY variations \eqref{N=2SUSY} leave an
imprint on the free energy (or eikonal)
$F_{\text{WQFT}}(b_{i},v_{i},\mathcal{S}_{i}) := -i\log \mathcal{Z}_{\text{WQFT}}$:
after integrating out the fluctuations $h_{\mu\nu}$,
$z^{\mu}$ and $\psi'^\mu$ in the path integral \eqref{ZWQFTdef},
the SUSY variations of the background trajectories \eqref{bgexp} remain intact
in an asymptotically flat spacetime.
That is, the transformations
\begin{align}\label{susybg}
\begin{aligned}
\delta b^{\mu}_{i}&= i \bar\epsilon \Psi^{\mu}_{i} + i \epsilon\bar{\Psi}^{\mu}_{i}\, ,
\quad \delta v^{\mu}_{i}=0 \, ,
\quad \delta\Psi_{i}^\mu=-\epsilon v_{i}^{\mu}\, \\
\Rightarrow&\quad \delta \mathcal{S}_{i}^{\mu\nu} = v_{i}^{\mu}\, \delta b_{i}^{\nu}
-v_{i}^{\nu}\, \delta b_{i}^{\mu}
\end{aligned}
\end{align}
are a symmetry of $F_{\text{WQFT}}(b_{i},v_{i},\mathcal{S}_{i})$
(only up to quadratic spin order when the Wilson coefficients $C_{E,i}$ are included).
As we shall see, this is also a symmetry of the waveform.
Using a suitable shift of the proper times $\tau_i$ we may choose $b\cdot v_i=0$,
where $b^\mu=b_2^\mu-b_1^\mu$ is the relative impact parameter;
by gauge fixing the SUSY transformations \eqref{susybg}
we impose $v_{i,\mu}\mathcal{S}_i^{\mu\nu}=0$ (the covariant SSC).
\sec{Feynman rules}
As the Feynman rules for the Einstein-Hilbert action are conventional we will not dwell on them;
the only subtlety is our use of a \emph{retarded} graviton propagator:
\begin{align}
\begin{tikzpicture}[baseline={(current bounding box.center)}]
\coordinate (x) at (-.7,0);
\coordinate (y) at (0.5,0);
\draw [photon] (x) -- (y) node [midway, below] {$k$};
\draw [fill] (x) circle (.08) node [above] {$\mu\nu$};
\draw [fill] (y) circle (.08) node [above] {$\rho\sigma$};
\end{tikzpicture}&=i\frac{P_{\mu\nu;\rho\sigma}}{(k^{0}+i\epsilon)^{2}-\mathbf{k}^2}\,,
\end{align}
with $P_{\mu\nu;\rho\sigma}:=\eta_{\mu(\rho}\eta_{\sigma)\nu}-
\sfrac12\eta_{\mu\nu}\eta_{\rho\sigma}$.
On the worldline we work in one-dimensional energy (frequency) space:
the propagators for the fluctuations $z^\mu(\omega)$
and anti-commuting vectors $\psi^{\prime\mu}(\omega)$ are respectively
\begin{subequations}\label{eq:Propagators}
\begin{align}
\begin{tikzpicture}[baseline={(current bounding box.center)}]
\coordinate (in) at (-0.6,0);
\coordinate (out) at (1.4,0);
\coordinate (x) at (-.2,0);
\coordinate (y) at (1.0,0);
\draw [zUndirected] (x) -- (y) node [midway, below] {$\omega$};
\draw [dotted] (in) -- (x);
\draw [dotted] (y) -- (out);
\draw [fill] (x) circle (.08) node [above] {$\mu$};
\draw [fill] (y) circle (.08) node [above] {$\nu$};
\end{tikzpicture}&=-i\frac{\eta^{\mu\nu}}{m\,(\omega+i\epsilon)^2}\,, \\
\begin{tikzpicture}[baseline={(current bounding box.center)}]
\coordinate (in) at (-0.6,0);
\coordinate (out) at (1.4,0);
\coordinate (x) at (-.2,0);
\coordinate (y) at (1.0,0);
\draw [zParticle] (x) -- (y) node [midway, below] {$\omega$};
\draw [dotted] (in) -- (x);
\draw [dotted] (y) -- (out);
\draw [fill] (x) circle (.08) node [above] {$\mu$};
\draw [fill] (y) circle (.08) node [above] {$\nu$};
\end{tikzpicture}&=-i\frac{\eta^{\mu\nu}}{m\,(\omega+i\epsilon)}\,,
\end{align}
\end{subequations}
which also both involve a retarded $i\epsilon$ prescription.
The former was already used in \Rcites{Mogull:2020sak,Jakobsen:2021smu}.
Next we consider the worldline vertices.
The simplest of these is the single-graviton emission vertex:
\begin{align}\label{eq:vertexH}
\begin{tikzpicture}[baseline={(current bounding box.center)}]
\coordinate (in) at (-1,0);
\coordinate (out) at (1,0);
\coordinate (x) at (0,0);
\node (k) at (0,-1.3) {$h_{\mu\nu}(k)$};
\draw [dotted] (in) -- (x);
\draw [dotted] (x) -- (out);
\draw [graviton] (x) -- (k);
\draw [fill] (x) circle (.08);
\end{tikzpicture}&=
-i\frac{m\kappa }{2}e^{ik\cdot b}\delta\!\!\!{}^-\!(k\cdot v)
\bigg(v^\mu v^\nu+ik_\rho\mathcal{S}^{\rho(\mu}v^{\nu)} \nonumber \\[-10pt]
&+\frac12k_\rho k_\sigma\mathcal{S}^{\rho\mu}\mathcal{S}^{\nu\sigma}
+\frac{C_E}{2}v^\mu v^\nu(k\cdot\mathcal{S}\cdot\mathcal{S}\cdot k)\bigg)\,,
\end{align}
where $\delta\!\!\!{}^-\!(\omega):=(2\pi)\delta(\omega)$ and we have used
$\mathcal{S}^{\mu\nu}=-2i\bar{\Psi}^{[\mu}\Psi^{\nu]}$.
The other worldline-based vertices required for the 2PM Bremsstrahlung all appear
in Fig.~\ref{fig:1}:
the two-point interaction between a graviton and a single $z^\mu$ mode in (b),
the two-graviton emission vertex in (c),
and the two-point interaction between a graviton and ${\psi'}^\mu$ in (d).
Full expressions for these vertices are provided in the Supplementary Material.
\sec{Waveform from WQFT}
To describe the Bremsstrahlung at 2PM order including spin effects
we compute the expectation value
$k^2\langle h_{\mu\nu}(k)\rangle_{\rm WQFT}$.
This requires us to compute four kinds of Feynman graphs,
illustrated in Fig.~\ref{fig:1}.
Explicit expressions for the first two graphs (a) and (b)
were given in the non-spinning case \cite{Jakobsen:2021smu};
these are now modified by terms up to $\mathcal{O}({\cal S}^2)$.
Graphs (c) and (d) are unique to the spinning case ---
for the latter we sum over both routings of the fermion line.
\begin{figure}[t!]
\label{fig:1a}
\includegraphics[width=6.0cm]{Fig1.pdf}
\caption{The four diagram topologies contributing to the 2PM Bremsstrahlung up to $\mathcal{O}(\mathcal{S}^2)$,
where $\omega_i=k\cdot v_i$ by energy conservation at the worldline vertices.
For diagrams (b)--(d) we also include the corresponding
flipped topologies with massive bodies 1$\leftrightarrow$2;
for diagram (d) (which includes the propagating fermion $\psi_2^{\prime\mu}$)
we also include the graph with the arrow reversed.}
\label{fig:1}
\end{figure}
From this result we seek to obtain the waveform in spacetime in the \emph{wave zone},
where the distance to the observer $|\mathbf{x}|=r$ is large compared to all other lengths.
Following Ref.~\cite{Jakobsen:2021smu} the gauge-invariant \emph{frequency-domain waveform} $4G\,\epsilon^{\mu\nu}{S}_{\mu\nu}(k^{\mu}=\Omega\,(1,\hat{\bf x}))$
is extracted from the WQFT via
\begin{equation}
{S}_{\mu\nu}(k)= \frac{2}{\kappa} k^{2} \vev{h_{\mu\nu}(k)}_{\rm WQFT}\, ,\end{equation}
where $\Omega$ is the GW frequency and $\mathbf{\hat x}=\mathbf{x}/r$ points towards the observer.
However, it is advantageous to study the \emph{time-domain waveform}
$f(u,\mathbf{\hat{x}})$ which is given by a Fourier transform:
\begin{equation}\label{eq:startingpoint}
\kappa \epsilon^{\mu\nu} h_{\mu\nu}=\frac{f(u,\mathbf{\hat{x}})}{r} = \frac{4G}{r}
\int_\Omega e^{-i k\cdot x} \,\epsilon^{\mu\nu}\,
S_{\mu\nu}(k) \Bigr |_{k^{\mu}=\Omega\, \rho^{\mu}}\, .
\end{equation}
We have contracted with a polarization tensor
$\epsilon^{\mu\nu}=\sfrac12\epsilon^{\mu}\epsilon^{\nu}$,
$\int_\Omega:=\int_{-\infty}^\infty\sfrac{{\rm d}\Omega}{2\pi}$, and
$\rho^{\mu}=(1,\hat{\bf x})$;
in a PM decomposition $f=\sum_nG^nf^{(n)}$
we seek the 2PM component $f^{(2)}$.
Note that $k\cdot x = \Omega (t-r)$ yields the retarded time $u=t-r$,
and $\epsilon\cdot\epsilon=\epsilon\cdot\rho=0$.
\sec{Integration}
Our integration procedure follows closely
that used for the non-spinning calculation in Ref.~\cite{Jakobsen:2021smu},
the main difference being that we maintain four-dimensional Lorentz covariance.
Each diagram contributing to
$k^2\langle h_{\mu\nu}(k)\rangle_{\rm WQFT}$ carries
the overall factor
\begin{equation}\label{intmeasureGold}
{\mu}_{1,2}(k)=
e^{i(q_1\cdot{b}_1+q_2\cdot{b}_2)}
\delta\!\!\!{}^-\!(q_1\cdot{v}_1)\delta\!\!\!{}^-\!(q_2\cdot{v}_2)
\delta\!\!\!{}^-\!(k-q_1-q_2)\,.
\end{equation}
We integrate over $q_i$, the momentum emitted from each worldline
(see Fig.~\ref{fig:1}).
When we also integrate over $\Omega$ ---
as in Eq.~\eqref{eq:startingpoint} ---
the full integration measure becomes
\begin{equation}\label{eq:measure}
\int_{\Omega,q_1,q_2}\mu_{1,2}(k)e^{-ik\cdot x}=
\frac1{\rho\cdot v_2}\int_{q_1}\delta\!\!\!{}^-\!(q_1\cdot v_1)e^{-iq_1\cdot\tilde{b}}\,,
\end{equation}
where $\int_{q_i}:=\int\!\sfrac{\mathrm{d}^4 q_i}{(2\pi)^4}$;
the delta function constraints give
$\Omega=\sfrac{q_1\cdot v_2}{\rho\cdot v_2}$ and $q_2=k-q_1$.
The shifted impact parameter,
\begin{align}\label{eq:shifted}
\tilde{b}^\mu=\tilde{b}_2^\mu-\tilde{b}_1^\mu\,, &&
\tilde{b}_i^\mu=b_i^\mu+u_iv_i^\mu\,,
\end{align}
extends the original impact parameter $b^\mu=b_2^\mu-b_1^\mu$
along the undeflected trajectories of the two bodies.
Finally, $u_i$ is the retarded time in the $i$'th rest frame:
\begin{equation}
u_i=\frac{\rho\cdot(x-b_i)}{\rho\cdot v_i}\,,
\end{equation}
This implies $\rho\cdot\tilde{b}_i=\rho\cdot x=u$, so $\rho\cdot\tilde{b}=0$.
Rewriting the integral measure as in Eq.~\eqref{eq:measure}
is convenient for performing the integrals of diagrams (b)--(d),
in the rest frame of body 1.
The mirrored counterparts to these diagrams are easily recovered after
integration using the $1\leftrightarrow2$ symmetry of the waveform.
To integrate diagram (a) we insert the partial-fraction identity
$q_1^{-2}q_2^{-2}=-q_1^{-2}(2k\cdot q_1)^{-1}-q_2^{-2}(2k\cdot q_2)^{-1}$
(which is valid for $k$ on-shell)
and focus on the first term.
The full 2PM waveform is then written schematically as
(dropping the subscript on $q_1$)
\begin{align}\label{eq:schematic}
\frac{f^{(2)}}{m_1 m_2}
&=4\pi \int_{q}\delta\!\!\!{}^-\!(q\cdot v_1)
\frac{e^{-iq\cdot\tilde{b}}}{q^2}\left(
\frac{\mathcal{N}(q)}{q\cdot v_2+i\epsilon}+
\frac{\mathcal{M}(q)}{(q\cdot v_2)(q\cdot\rho)}
\right)\,,\nonumber\\
&\qquad+(1\leftrightarrow2)\,,
\end{align}
the $\mathcal{N}$- and $\mathcal{M}$-contributions corresponding to diagrams
(b)--(d) and (a) in Fig.~\ref{fig:1} respectively.
The numerators $\mathcal{N}(q)$ and $\mathcal{M}(q)$ have a uniform
power counting in $q$ for each spin order:
\begin{align}
\mathcal{N}(q)&=
\mathcal{N}_\mu q^\mu+\mathcal{N}_{\mu\nu} q^\mu q^\nu+\mathcal{N}_{\mu\nu\rho} q^\mu q^\nu q^\rho\,,\\
\mathcal{M}(q)&=
\mathcal{M}_{\mu\nu} q^\mu q^\nu+\mathcal{M}_{\mu\nu\rho}q^\mu q^\nu q^\rho+
\mathcal{M}_{\mu\nu\rho\sigma}q^\mu q^\nu q^\rho q^\sigma\,,\nonumber
\end{align}
and the non-spinning result involves only $\mathcal{N}_\mu$ and $\mathcal{M}_{\mu\nu}$.
We present full expressions for $\mathcal{N}$ and $\mathcal{M}$ in the
ancillary file attached to the \texttt{arXiv} submission of this Letter.
To lowest order in $q^\mu$, the first integral in \eqn{eq:schematic} is
\begin{align}\label{eq:firstIntegral}
\begin{aligned}
&4\pi \int_{q}\delta\!\!\!{}^-\!(q\cdot v_1)
\frac{e^{-iq\cdot\tilde{b}}}{q^2}
\frac{q^\mu}{q\cdot v_2+i\epsilon}\\
&\qquad=
\frac{P_1^{\mu\nu}v_{2,\nu}}{(\gamma^2-1)|\tilde{\mathbf{b}}|_1}-
\frac{b^\mu}{|b|^2}\!
\left(\frac1{\sqrt{\gamma^2-1}}+\frac{u_2}{|\tilde{\mathbf{b}}|_1}\right)\,,
\end{aligned}
\end{align}
where $P_i^{\mu\nu}:=\eta^{\mu\nu}-v_i^\mu v_i^\nu$ is a projector
into the rest frame of the $i$'th body,
$|b|=-\sqrt{b^\mu b_\mu}$ (the impact parameter is spacelike) and
\begin{align}\label{eq:modBDef}
|\tilde{\mathbf{b}}|_{1,2}&:=\sqrt{-\tilde{b}_\mu P_{1,2}^{\mu\nu}\tilde{b}_\nu}
=\sqrt{|b|^2+(\gamma^2-1)u_{2,1}^2}
\end{align}
are the lengths of the shifted impact parameter $\tilde{b}^\mu$
\eqref{eq:shifted} in the two rest frames.
The second integral in \eqn{eq:schematic} is
\begin{align}\label{eq:secondIntegral}
\begin{aligned}
&
4\pi \int_{q}\delta\!\!\!{}^-\!(q\cdot v_1)
\frac{e^{-iq\cdot\tilde{b}}}{q^2}
\frac{q^\mu q^\nu}{q\cdot v_2\ q\cdot\rho}
\\
&\qquad
=
\frac{
K_1^{\mu\nu}\ v_2 \cdot K_1 \cdot \rho
-
2 (v_{2}\cdot K_1)^{(\mu} (\rho\cdot K_1)^{\nu)}
}{(\gamma^2-1)\ (\rho\cdot v_1)^2\ |b|^2\ |\tilde{b}|^2\ |\tilde{\mathbf{b}}|_1}\,,
\end{aligned}
\end{align}
where we have introduced the symmetric tensor
\begin{equation}\label{eq:kTensor}
K_i^{\mu\nu}:=P_i^{\mu\nu}|\tilde{\mathbf{b}}|_i^2
+ (P_i\cdot{\tilde b})^\mu (P_i\cdot{\tilde b})^\nu\,,
\end{equation}
with the property that $K_i^{\mu\nu}v_{i,\nu}=K_i^{\mu\nu}\tilde{b}_\nu=0$.
Both integrals are derived in the Supplementary Material;
one generalizes to higher powers of $q^\mu$ in the numerators
by taking derivatives with respect to $\tilde{b}^\mu$.
\sec{Results}
The 2PM waveform takes the schematic form
\begin{align}\label{eq:result}
\frac{f^{(2)}}{m_1 m_2} & =
\sum_{s=0}^2\frac1{|\tilde{\mathbf{b}}|_1^{2s+1}}\left[
\alpha_1^{(s)}+
\frac{\beta_1^{(s)}}{|\tilde{b}|^{2s+2}}
\right]+(1\leftrightarrow2)\,,
\end{align}
where the coefficients $\alpha_i^{(s)}$, $\beta_i^{(s)}$,
provided in the ancillary file,
are associated with the $\mathcal{N}$- and $\mathcal{M}$-type contributions
in Eq.~\eqref{eq:schematic} respectively;
they are functions of $u_i$, $b^\mu$, $v_i^\mu$, $\rho^\mu$, and $\mathcal{S}_i^{\mu\nu}$
and bi-linear in $\epsilon^\mu$.
The waveform $f$ is invariant under
the SUSY transformations in Eq.~\eqref{susybg}
to quadratic order in spin regardless of the values of $C_{E,i}$.
To see this we expand the waveform at all PM orders in powers of spin:
\begin{align}
f & = f_{0}
\!+\sum_{i=1}^2\mathcal{S}_{i,\mu\nu}f^{\mu\nu}_{i}
+\sum_{i,j=1}^2\!\!\mathcal{S}_{i,\mu\nu}\mathcal{S}_{j,\rho\sigma} f_{ij}^{\mu\nu;\rho\sigma}
+\mathcal{O}(\mathcal{S}^{3})\,,
\end{align}
where $f_i^{\mu\nu}$ and $f_{ij}^{\mu\nu;\rho\sigma}$ are defined
modulo terms that vanish on support of $v_{i,\mu}\mathcal{S}_i^{\mu\nu}=0$.
The SUSY links higher-spin to lower-spin terms:
\begin{align}\label{eq:susyrel}
\frac{1}{2}\frac{\partial f_{0}}{\partial b_{i,\mu}} &=
v_{i,\nu}\, f_{i}^{[\mu\nu]}\, ,
&
\frac{1}{4} \frac{\partial f_{i}^{\mu\nu}}{\partial b_{j,\rho}} &=
v_{j,\sigma}\, f_{ij}^{\mu\nu;[\rho\sigma]}\, ,
\end{align}
and these identities are satisfied by the waveform \eqref{eq:result}.
To illustrate the waveform we consider the \emph{gravitational wave memory}
$\Delta f(\mathbf{\hat{x}}):=f(+\infty,\mathbf{\hat{x}})-f(-\infty,\mathbf{\hat{x}})$.
The constant spin tensors are decomposed in terms of the
Pauli-Lubanski vectors $a_i^\mu$ as
$\mathcal{S}_i^{\mu\nu}={\epsilon^{\mu\nu}}_{\rho\sigma}v_i^\rho a_i^\sigma$,
the latter satisfying $a_i\cdot v_i=0$.
In the aligned-spin case $a_i\cdot b=a_i\cdot v_j=0$,
i.e.~the spin vectors are orthogonal to the plane of scattering. Writing
$|a_i|=\sqrt{-a_{i}^{2}}$ the wave memory is then
proportional to the non-spinning result:
\begin{align}
&\Delta f^{(2)}=\left(1+\frac{2v|a_3|}{b(1+v^2)}+\frac{|a_3|^2}{|b|^2}-
\sum_{i=1}^2\frac{C_{E,i}|a_i|^2}{|b|^2}\right)\!
\Delta f^{(2)}_{\mathcal{S}=0} ,\nonumber\\
&\frac{\Delta f^{(2)}_{\mathcal{S}=0}}{m_1m_2}=
\frac{4(2\gamma^2-1)\epsilon\cdot v_1(2b\cdot\epsilon\,\rho\cdot v_1-b\cdot\rho\,\epsilon\cdot v_1)}
{|b|^2\sqrt{\gamma^2-1}(\rho\cdot v_1)^2}\nonumber\\
&\qquad\qquad\qquad+(1\leftrightarrow2)\,,
\end{align}
where $a_3^\mu=a_1^\mu+a_2^\mu$.
For two Kerr black holes ($C_{E,i}=0$) with equal-and-opposite spins
($a_1^\mu=-a_2^\mu$) we see that $\Delta f^{(2)}=\Delta f^{(2)}_{\mathcal{S}=0}$,
which we observe also when the spins are mis-aligned to the plane of scattering.
\begin{figure}[t!]
\includegraphics[width=8.5cm]{Fig2.pdf}
\caption{Total radiated angular momenta for the scattering of two Kerr-BHs with $v=0.2$
as a function of the angle between the total initial spins
$\mathbf{a_{3}}=\mathbf{a_{1}} + \mathbf{a_{2}}$ and
$\mathbf{b}$ (with $\mathbf{a_{i}}\cdot\mathbf{v_{i}}=0$) for a range of ratios
$|\mathbf{a_{3}}|/|\mathbf{b}|$. We show the normalized ratio of angular momenta emitted orthogonal to the $\mathbf{b},\mathbf{v}$ plane (left plot) and in the $\mathbf{b}$ direction (right plot), normalization is w.r.t.~angular momentum emitted in the spinless case. }
\label{fig:2}
\end{figure}
There is also a 1PM (non-radiating) contribution to the waveform
consisting of single-graviton emission from either massive body:
\begin{equation}\label{eq:res1PM}
f^{(1)}(\mathbf{\hat{x}})=\frac{2m_1}{\rho\cdot v_1}(\epsilon\cdot v_1)^2+
\frac{2m_2}{\rho\cdot v_2}(\epsilon\cdot v_2)^2\,.
\end{equation}
At 1PM order there is manifestly no dependence on either the spins $\mathcal{S}_i^{\mu\nu}$
or impact parameters $b_i^\mu$,
so the SUSY identities in Eq.~\eqref{eq:susyrel} are trivially satisfied.
Finally, the wave memory and 1PM part of the waveform
contribute to the total radiated angular momentum $J_{ij}^{\rm rad}$.
Using three-dimensional Cartesian basis vectors $\hat{\mathbf e}_i$,
we choose a frame of reference with the initial velocities $v_i^\mu$
restricted to the $t$--$x$ plane;
$\mathbf{b}=|b|\,\hat{\mathbf e}_2$ is orthogonal to these.
Then we find two non-zero components of $J_{ij}^{\rm rad}$:
$J_{xy}^{\rm rad}$ and $J_{zx}^{\rm rad}$,
which are conveniently arranged into
\begin{align}\label{eq:radAngMom}
\begin{aligned}
&\frac{J^{\text{rad}}_{xy}+iJ^{\text{rad}}_{zx}}{\left.J^{\rm init}_{xy}\right|_{\mathcal{S}=0}}=
\frac{4G^2m_1m_2}{|b|^2}\frac{(2\gamma^2-1)}{\sqrt{\gamma^2-1}}{\cal I}(v)\\
&\times\left(1-\frac{2iv\,\mathbf{a}_3\cdot\mathbf{l}}{|b|(1+v^2)}
-\frac{(\mathbf{a}_3\cdot\mathbf{l})^2}{|b|^2}
+\sum_{i=1}^2\frac{C_{E,i}}{|b|^2}(\mathbf{a}_i\cdot\mathbf{l})^2
\right)\\
&\qquad+\mathcal{O}(G^3)\,.
\end{aligned}
\end{align}
We normalize with respect to $\left.J^{\rm init}_{xy}\right|_{\mathcal{S}=0}$,
the initial angular momentum in the non-spinning case.
The spin vectors $\mathbf{a}_1$ and $\mathbf{a}_2$
are taken in the rest frame of each massive body;
$\mathbf{a}_3=\mathbf{a}_1+\mathbf{a}_2$, $\mathbf{l}=\hat{\mathbf e}_2+i\hat{\mathbf e}_3$, and
\begin{align}
{\cal I}(v)=-\frac83+\frac1{v^2}+\frac{(3v^2-1)}{v^3}{\rm arctanh}(v)
\end{align}
is a universal prefactor.
Eq.~\eqref{eq:radAngMom} holds in the
rest frame of either body or the center-of-mass (c.o.m.) frame;
see Fig.~\ref{fig:2} for plots. For a derivation we refer the reader to the Supplementary Material.
There we also compute the total radiated energy in the c.o.m.~frame.
Due to the multi-scale nature of the waveform it is difficult
to perform the necessary time and solid
angle-integrals, so we performed a low velocity expansion.
For terms up to $\mathcal{O}(v^{2})$ we find
\begin{widetext}
\begin{align}\label{eq:radEnergy}
&E^{\text{rad,LO}}_{\text{CoM}}
=
\frac{v G^3 m_1^2 m_2^2 \pi}{|b|^3}
\bigg[
\frac{37}{15}
+\frac{
v
(65 m_1+69 m_2) (\mathbf{a}_1\! \!\cdot \! \mathbf{\hat{e}_{3}})
}{
10 |b| (m_1+m_2)
}
+\frac{
1503 (\mathbf{a}_1\!\! \cdot \!\mathbf{{\hat e}_1})( \mathbf{a}_2\!\! \cdot \!\mathbf{{\hat e}_1})
-
3559 (\mathbf{a}_1\!\! \cdot \!\mathbf{{\hat e}_2})( \mathbf{a}_2\!\! \cdot \!\mathbf{{\hat e}_2})
+
1816 (\mathbf{a}_1\!\! \cdot \!\mathbf{{\hat e}_3})( \mathbf{a}_2\!\! \cdot \!\mathbf{{\hat e}_3})
}{320 |b|^2}
\nonumber\\&\quad
+\frac{9 (185- 176 C_{E,1}) (\mathbf{a}_1\!\! \cdot \!\mathbf{{\hat e}_1})^2-
(3385-3472 C_{E,1})
(\mathbf{a}_1\! \!\cdot \!\mathbf{{\hat e}_2})^2+8(245- 236 C_{E,1}) (\mathbf{a}_1\!\! \cdot \!{\mathbf{{\hat e}_3}})^2}{320 |b|^2}
+ (1\leftrightarrow 2)
+\mathcal{O}\left(v ^2\right)
\bigg] ,
\end{align}
\end{widetext}
where the swap $(1\leftrightarrow 2)$ does not affect the basis vectors $\mathbf{{\hat e}_i}$ or the constant term $\frac{37}{15}$.
It is straighforward to extend this result to higher orders in $v$.
\sec{Conclusions}
In this Letter we extended the WQFT to describe spinning compact bodies to quadratic order in spin, and calculated the leading-PM order waveform for highly eccentric (scattering) orbits.
Our accompanying work~\cite{Jakobsen:2021zvh} presents an application to further observables such as the spin kick and deflection \cite{Liu:2021zxr,Kosmopoulos:2021zoq} at 2PM order
and gives details on the approximate SUSY and its relation to the SSC.
The radiated energy~\eqref{eq:radEnergy} should also be
particularly useful for future studies.
In Refs.~\cite{Herrmann:2021lqe,Herrmann:2021tct} the $\mathcal{O}(G^3)$ energy loss
from a scattering of non-spinning black holes was recently computed to all
orders in velocity using the KMOC formalism \cite{Kosower:2018adc}
(see also Ref.~\cite{Maybee:2019jus});
a similar result could conceivably be obtained at $\mathcal{O}(\mathcal{S}^2)$,
and then checked against Eq.~\eqref{eq:radEnergy} in the low-velocity limit.
Similarly, the remarkably simple result for radiated angular momentum
\eqref{eq:radAngMom} at 2PM order is intriguing; it may be important for understanding the high-energy limit, see \Rcite{DiVecchia:2020ymx,Damour:2020tta} for the non-spinning case.
The application of modern on-shell and integration techniques to compute scattering amplitudes~\cite{Bern:1994zx,*Bern:1994cg,*Britto:2004nc,Bern:2008qj,*Bern:2010ue,*Bern:2012uf,*Bern:2017ucb,*Bern:2018jmv,*Bern:2019prr,Bern:2019crd,Parra-Martinez:2020dzs,Bern:2021dqo,Herrmann:2021lqe} holds great promise for pushing calculations to higher PM orders.
This is demonstrated by the impressive calculation of the 4PM conservative dynamics in the potential region~\cite{Bern:2021dqo,Dlapa:2021npj} --- see also Refs.~\cite{Bern:2019nnu,Bern:2019crd,Cheung:2020gyp,*Kalin:2020fhe,DiVecchia:2020ymx,Damour:2020tta,DiVecchia:2021bdo,*Bjerrum-Bohr:2021din,Bini:2020flp,*Cheung:2020sdj,*Haddad:2020que,*Kalin:2020lmz,*Brandhuber:2019qpg,*Huber:2019ugz,*AccettulliHuber:2020oou,*AccettulliHuber:2020oou,*Bern:2020uwk,*Cheung:2020gbf,*Aoude:2020ygw,Amati:1990xe,*DiVecchia:2019myk,*DiVecchia:2019kta,*Bern:2020gjj,*Huber:2020xny,*DiVecchia:2021ndb,*Bautista:2019tdr,*Laddha:2018rle,*Laddha:2018myi,*Sahoo:2018lxl,*Laddha:2019yaj,*Saha:2019tub,*A:2020lub,*Sahoo:2020ryf,*Sahoo:2018lxl}.
The connection between amplitudes and classical physics was studied in Refs.~\cite{Kosower:2018adc,Maybee:2019jus,Damour:2019lcq,*Bjerrum-Bohr:2019kec,*Bjerrum-Bohr:2021vuf}, and Refs.~\cite{Kalin:2020mvi,*Kalin:2019rwq,*Kalin:2019inp} discussed the connection to bound orbits.
Our WQFT framework~\cite{Mogull:2020sak,Jakobsen:2021smu} provides an efficient,
rather intuitive way to connect amplitude and (classical) worldline EFT calculations.
It may therefore benefit from modern amplitude techniques at higher PM orders in future work, building on the compact Lorentz-covariant master integrals provided here.
\medskip
\sec{Acknowledgments}
We would like to thank F.~Bautista, R.~Bonezzi, A.~Buonanno, P.~Pichini and J.~Vines for very helpful discussions.
We are also grateful for use of G.~K\"alin's C\texttt{++} graph library.
GUJ's and GM's research is funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) Projektnummer 417533893/GRK2575 ``Rethinking Quantum Field Theory''.
|
1,108,101,564,588 | arxiv | \section{Introduction}
The starting point of this paper is a
recent solution of the IKKT-type matrix models with mass term \cite{Sperling:2019xar},
which is naturally interpreted as 3+1-dimensional cosmological FLRW quantum space-time.
It was shown that the fluctuation modes around this background
include spin-2 metric fluctuations, as well as a truncated tower of higher-spin modes which are
organized in a higher-spin gauge theory. The 2 standard Ricci-flat massless
graviton modes were found, as well as some additional
vector-like and scalar modes whose significance was not fully clarified.
The aim of the present paper is to study in more detail the metric perturbations,
and in particular to
see if and how the (linearized) Schwarzschild solution can be obtained.
We will indeed find such a solution, which
is realized in the scalar sector of the linearized perturbation modes exhibited in \cite{Sperling:2019xar}.
This means that the model has a good chance to satisfy the precision solar system tests of gravity.
We will also elaborate and discuss in some detail the extra scalar mode, which is not present in GR.
This seems to provide a natural candidate for apparent dark matter.
Since the notorious problems in attempts to quantize gravity
arise primarily from the Einstein-Hilbert action, is is very desirable to find another
framework for gravity, which is more suitable for quantization.
String theory provides such a framework, but the traditional approach using compactifications
leads to a host of issues, notably lack of predictivity.
This suggests to use matrix models as a starting point, and in particular
the IKKT or IIB model \cite{Ishibashi:1996xs}, which was originally proposed as a constructive definition of string theory. Remarkably,
numerical studies in this non-perturbative formulation provide evidence
\cite{Kim:2011cr,Nishimura:2019qal,Aoki:2019tby} that
3+1-dimensional configurations arise at the non-perturbative level,
tentatively interpreted as expanding universe.
However, this requires a new mechanism for gravity on 3+1-dimensional
non-commutative backgrounds
as in \cite{Sperling:2019xar},
which does not rely on compactification.
The present paper provides further evidence and insights for this mechanism.
The (linearized) Schwarzschild metric is clearly the benchmark for any
viable theory of gravity.
There has been considerable effort to find noncommutative analogs of the Schwarzschild metric from various approaches,
leading to a number of proposals
\cite{Schupp:2009pt,Blaschke:2010ye,Ohl:2009pv,Chaichian:2007dr} and references therein, cf. also \cite{Nicolini:2008aj};
however, none is truly satisfactory.
The proposals are typically obtained by some
ad-hoc modification of the classical solution, without any intrinsic role of noncommutativity,
which is put in by hand.
In contrast,
the quantum structure
(or its semi-classical limit) plays a central role in the present framework.
Our solution is a deformation of the noncommutative background which respects an exact $SO(3)$
rotation symmetry, even though there are only finitely many d.o.f. per unit volume.
The solution has a good asymptotics at large distances,
allowing superpositions corresponding to arbitrary mass distributions.
In fact we obtain generic quasi-static Ricci-flat linearized perturbations,
which complement the Ricci-flat propagating gravitons
found in \cite{Sperling:2019xar}.
This realization of the (linearized) Schwarzschild solution
is remarkable and may seem surprising, because the action is of Yang-Mills type, and
no Einstein-Hilbert-like action is required\footnote{It is well-known that gravity
can be obtained from a Yang-Mills-type action by imposing constraints,
cf. \cite{MacDowell:1977jt,Chamseddine:2002fd,Manolakos:2019fle}.
However this essentially amounts to a reformulation of classical GR,
and the usual problems are expected to arise upon quantization.
In contrast, we do not impose any constraints on the Yang-Mills action. Nevertheless,
quantum effects are expected to induce an
Einstein-Hilbert-like term,
as discussed in \cite{Sperling:2019xar}. This may play an important role here as well,
but we focus on the classical mechanism. Another interesting possibility was proposed in \cite{Hanada:2005vr}, which has some
similarities to the present mechanism but leads to many additional fields, possibly including ghosts.}.
This means that the theory has a good chance to survive upon quantization, which is
naturally defined via integration over the space of matrices.
The IKKT model is indeed well suited for quantization, and quite clearly free of ghosts and
other obvious pathologies.
It is background-independent
in the sense that it has a large class of solutions with different geometries,
and defines a gauge theory for fluctuations on any background.
The price to pay is a considerable complexity of the resulting theory.
As explained in \cite{Steinacker:2016vgf,Sperling:2019xar} the background leads to a higher-spin gauge theory, with a truncated tower of higher
spin modes, and many similarities with (but also distinctions from) Vasiliev theory
\cite{Vasiliev:1990en}.
Since space-time itself is part of the background solution, it is not unreasonable to expect
Ricci-flat deformations, cf. \cite{Rivelles:2002ez,Yang:2006dk,Steinacker:2010rh}.
However, Lorentz invariance is very tricky on noncommutative backgrounds. In the present case
the space-like isometries $SO(3,1)$ of the $k=-1$ FLRW space-time are manifest, but invariance under (local) boosts is not.
Nevertheless, the propagation of all physical modes is governed by the same effective metric.
In particular, the concept of spin has to be used with caution, and would-be spin $s$ modes decompose further
into sectors governed by the space-like $SO(3,1)$ isometry.
The tensor fields are accordingly characterized by the transformation under the local $SO(3)$ stabilizer group,
and the term ``scalar modes'' is understood in this sense throughout the paper.
However this complication is in fact helpful to identify physical degrees of freedom
in the physical sector, and to understand the absence of ghosts.
Let us describe the new results in some details.
We focus on the scalar fluctuation mode which was found in \cite{Sperling:2019xar}, and elaborate the associated metric fluctuations.
The main result is that there is a preferred ``quasi-static'' vacuum solution
which leads to the linearized Schwarzschild metric on the
FLRW background.
This strongly suggests that a near-realistic gravity emerges on the background,
however only the vacuum solution is considered here.
Quasi-static
means that the solution is static on local scales at late times, but slowly decays on cosmic scales, in a specific way.
This is a somewhat unexpected result, whose significance is not entirely clear.
The quasi-static solution is singled out because all other solutions lead to a large diffeo term, which makes the linearized treatment
problematic. Hence the Schwarzschild solution is the ``cleanest'' case, while
the generic dynamical scalar modes require non-linear considerations
somewhat reminiscent of the Vainshtein mechanism \cite{Vainshtein:1972sx}. We offer a heuristic way to
understand them, which points to the intriguing - albeit quite speculative - possibility that
these non-Ricci-flat scalar modes might provide a geometrical explanation for
dark matter at galactic scales.
It may seem strange to start with a curved cosmological background rather than flat Minkowski
space, since the Schwarzschild solution is basically a local structure.
The reason is that no flat counterpart of the underlying
quantum-spacetime with the required structure is known. We will thus largely neglect the
contributions of the Schwarzschild solution at the cosmic scales.
Along the way, we also find the missing $4^{th}$ off-shell scalar fluctuation mode, which was missing in \cite{Sperling:2019xar}.
Thus all 10 off-shell metric fluctuation modes for the most general
metric fluctuations are realized, and
the model is certainly rich enough for a realistic theory of gravity. That theory
would clearly deviate from GR at cosmic scale, since the FLRW background solution is
not Ricci flat, but requires no stabilization by matter (or energy) and no fine-tuning.
Finally, it should be stressed that even though the model is
intrinsically noncommutative, it should be viewed
in the spirit of almost-local and almost-classical field theory.
Space-time arises as a condensation of matrices
rather than some non-local holographic image, with dynamical local fluctuations
described by an effective field theory.
The paper is meant to be as self-contained and compact as possible.
We start with a lightning introduction to the $\cM^{3,1}_n$ space-time under consideration,
and elaborate only the specific modes and aspects needed to obtain the Schwarzschild solution.
For some results we have to refer to \cite{Sperling:2019xar}, but the essential new computations are mostly spelled out.
For the skeptical reader, some of the missing steps may be
uncovered from the file in the arXive.
\section{Quantum FLRW space-time \texorpdfstring{$\cM^{3,1}_n$}{M(3,1)}}
\label{sec:projection-Lorentzian}
The quantum space-time under consideration is based on a particular
representation $\cH_n$ of $SO(4,2)$, which is a lowest weight unitary irrep
in the short discrete series
known as \emph{minireps} or \emph{doubletons} \cite{Mack:1975je,Fernando:2009fq}.
Those are the unique irreps which remain irreducible under
the restriction to $SO(4,1) \subset SO(4,2)$.
We denote the generators in this representation by
$\cM^{ab}$, which are Hermitian operators satisfying
\begin{align}
[\cM_{ab},\cM_{cd}] &=\mathrm{i} \left(\eta_{ac}\cM_{bd} - \eta_{ad}\cM_{bc} -
\eta_{bc}\cM_{ad} + \eta_{bd}\cM_{ac}\right) \
\label{M-M-relations-noncompact}
\end{align}
where $\eta^{ab} = \rm diag(-1,1,1,1,1,-1)$ is the invariant metric of $SO(4,2)$.
We then define
\begin{alignat}{2}
X^\mu &\coloneqq r\cM^{\mu 5} , \qquad X^4 := r\cM^{4 5} \nn\\
T^\mu &\coloneqq R^{-1} \cM^{\mu 4} \qquad \mu,\nu = 0,\ldots,3 \ .
\end{alignat}
Then the $X^a$ transform as vector operators under $SO(4,1)$, while the $T^\mu$ are vector operators
under $SO(3,1) \subset SO(4,1)$.
The $SO(3,1)$-invariant fuzzy or quantum space-time $\cM^{3,1}_n$ is then defined through the algebra of functions $\phi(X^\mu)$
generated by the $X^\mu$ for $\mu=0,1,2,3$.
Here $r$ is a microscopic length scale related to the internal quantum structure, while
$R$ is a macroscopic scale as specified in \refeq{X-T-relations-1}.
The commutation relations \eqref{M-M-relations-noncompact} imply
\begin{subequations}
\label{basic-CR-H4}
\begin{align}
[X^\mu,X^\nu] &= - \mathrm{i}\, r^2\cM^{\mu\nu} \eqqcolon \mathrm{i} \Theta^{\mu\nu} \,,
\label{X-X-CR}\\
[T^\mu,X^\nu] &= \mathrm{i} \frac{1}{R}\eta^{\mu\nu} X^4 \,, \label{T-X-CR}\\
[T^\mu, T^\nu] &= -\frac{\mathrm{i}}{r^2 R^2} \Theta^{\mu \nu} \, , \label{T-T-CR} \\
[T^\mu,X^4] &= - \mathrm{i} \frac{1}{R} X^\mu \,,\\
[X^\mu,X^4] &= - \mathrm{i} r^2 R\, T^\mu \, ,
\end{align}
\end{subequations}
and the irreducibility of $\cH_n$ under $SO(4,1)$
implies the relations \cite{Sperling:2019xar}
\begin{subequations}
\label{basic-constraints}
\begin{align}
X_{\mu} X^{\mu} &= -R^2 - X^4 X^4, \
\qquad R^2 = \frac{r^2}{4}(n^2-4) \label{X-T-relations-1}\\
T_{\mu} T^{\mu} &= \frac 1{r^2} + \frac 1{r^2 R^{2}}\, X^4 X^4, \\
X_\mu T^\mu + T^\mu X_\mu &= 0 \ .
\label{X-T-relations}
\end{align}
\end{subequations}
There are some extra constraints involving $\Theta^{\mu\nu}$,
which will only be given in the semi-classical version below.
Unless otherwise stated, indices will be raised and lowered with $\eta^{ab}$ or $\eta^{\mu\nu}$.
Apart from the extra constraints,
the construction is quite close to that of Snyder
\cite{Snyder:1946qz} and Yang \cite{Yang:1947ud}.
The proper interpretation of this structure is not obvious a priori, due to the
extra generators $T^\mathfrak{u}$ and $\Theta^{\mu\nu}$.
These cannot be dropped, because the full algebra $\mathrm{End}(\cH_n)$ is generated by
the $X^\mu$ alone.
A proper geometrical understanding is obtained by considering all the generators
$\cM_{ab}$ of $\mathfrak{so}(4,2)$.
As explained in \cite{Steinacker:2017bhb,Sperling:2017dts,Medina:2002pc},
these are naturally viewed as quantized embedding
functions of a coadjoint orbit $m^{ab}:\ \C P^{1,2} \hookrightarrow \mathfrak{so}(4,2) \cong{\mathbb R}} \def\C{{\mathbb C}} \def\N{{\mathbb N}^{15}$.
Here $\C P^{1,2}$ is a 6-dimensional noncompact analog of $\C P^3$, which is
singled out by the constraints satisfied by $m^{ab}$.
Hence the full algebra $\mathrm{End}(\cH_n)$ can be interpreted as a quantized
algebra of functions on $\C P^{1,2}$, dubbed fuzzy $\C P^{1,2}_n$.
Furthermore, $\C P^{1,2}$ is naturally a $S^2$ bundle over $H^4$,
which is defined by the $X^a$ satisfying \eqref{X-T-relations-1}.
Hence the space $\cM^{3,1}$ generated by the $X^\mu \sim x^\mu$, $\mu = 0,...,3$
can be viewed as projection of $H^4\subset {\mathbb R}} \def\C{{\mathbb C}} \def\N{{\mathbb N}^{4,1}$ to ${\mathbb R}} \def\C{{\mathbb C}} \def\N{{\mathbb N}^{3,1}$ along $X^4$,
as sketched in figure \ref{fig:projection}.
This is the space-time of interest here, which is
covariant under $SO(3,1)$.
For similar covariant quantum spaces see e.g.
\cite{Grosse:1996mz,Heckman:2014xha,Sperling:2017dts,Ramgoolam:2001zx,Hasebe:2012mz,Buric:2017yes,Steinacker:2017vqw}.
\begin{figure}
\hspace{2cm} \includegraphics[width=0.5\textwidth]{Plot-hyperboloid-crop.pdf}
\caption{Sketch of the projection $\Pi$ from $H^4$ to $\cM^{3,1}$ with
Minkowski signature.}
\label{fig:projection}
\end{figure}
\subsection{Semi-classical structure of \texorpdfstring{$\cM^{3,1}$}{M(3,1)}}
We will mostly restrict ourselves to the semi-classical limit
$n \to \infty$
of the above
space, working with commutative functions of $x^\mu\sim X^\mu$ and $t^\mu \sim T^\mu$, but keeping
the Poisson or symplectic structure $[.,.] \sim i \{.,.\}$ encoded in $\theta^{\mu\nu}$.
The constraints \eqref{basic-constraints} etc. imply the following relations
\begin{subequations}
\label{geometry-H-M}
\begin{align}
x_\mu x^\mu &= -R^2 - x_4^2 = -R^2 \cosh^2(\eta) \, ,
\qquad R \sim \frac{r}{2}n \label{radial-constraint}\\
t_{\mu} t^{\mu} &= r^{-2}\, \cosh^2(\eta) \,, \\
t_\mu x^\mu &= 0, \ \\
t_\mu \theta^{\mu\alpha} \def\da{{\dot\alpha}} \def\b{\beta} &= - \sinh(\eta) x^\alpha} \def\da{{\dot\alpha}} \def\b{\beta , \\
x_\mu \theta^{\mu\alpha} \def\da{{\dot\alpha}} \def\b{\beta} &= - r^2 R^2 \sinh(\eta) t^\alpha} \def\da{{\dot\alpha}} \def\b{\beta , \label{x-theta-contract}\\
\eta_{\mu\nu}\theta^{\mu\alpha} \def\da{{\dot\alpha}} \def\b{\beta} \theta^{\nu\b} &= R^2 r^2 \eta^{\alpha} \def\da{{\dot\alpha}} \def\b{\beta\b} - R^2 r^4
t^\alpha} \def\da{{\dot\alpha}} \def\b{\beta t^\b - r^2 x^\alpha} \def\da{{\dot\alpha}} \def\b{\beta x^\b
\end{align}
\end{subequations}
where $\mu,\alpha} \def\da{{\dot\alpha}} \def\b{\beta = 0,\ldots ,3$.
Here $\eta$ is a global time coordinate defined by
\begin{align}
x^4 = R \sinh(\eta) \ ,
\label{x4-eta-def}
\end{align}
which will be related to the scale parameter of the universe \eqref{a-eta}.
Clearly
the $x^\mu:\, \cM^{3,1}\hookrightarrow {\mathbb R}} \def\C{{\mathbb C}} \def\N{{\mathbb N}^{3,1}$ can be viewed as Cartesian coordinate functions. Similarly,
the $t^\mu$ describe the $S^2$ fiber over $\cM^{3,1}$
as discussed above.
On the other hand, the relation \eqref{T-X-CR} implies that
the derivations
\begin{align}
-i[T^\mu,.] \sim \{t^\mu,.\} \ = \sinh(\eta) \partial_\mu
\end{align}
act as momentum generators on $\cM^{3,1}$,
leading to the useful relation
\begin{align}
\partial_\mu \phi = \b\{t_\mu,\phi\}, \qquad \b = \frac{1}{\sinh(\eta)}
\label{del-t-rel}
\end{align}
for $\phi = \phi(x)$.
In particular, a $SO(3,1)$-invariant matrix d'Alembertian can be defined as
\begin{align}
\Box := [T^\mu,[T_\mu,.]] \ \sim \ -\{t^\mu,\{t_\mu,.\}\} \ .
\label{Box-def}
\end{align}
It acts on any $\phi \in \mathrm{End}(\cH)$, and
will play a central role throughout this paper.
We also define a globally defined time-like vector field
\begin{align}
\t := x^\mu \partial_\mu .
\label{tau-def}
\end{align}
To get some insight into the $\theta^{\mu\nu}$, fix some
reference point $\xi$ on $\cM^{3,1}$, which using $SO(3,1)$ invariance
can be chosen as
\begin{align}
\xi = (x^0,0,0,0) , \qquad x^0 = R\cosh(\eta) \ .
\end{align}
Then \eqref{x-theta-contract}
provides a relation between the $t^{\mu}$ and the $\theta^{\mu\nu}$ generators,
\begin{align}
t^\mu = -\frac{1}{R r^2 x^4}\, x_\nu\theta^{\nu\mu}
\ &\stackrel{\xi}{=} \ -\frac{1}{R r^2}\, \frac{1}{\tanh(\eta)}\theta^{0\mu} \ , \
\, \qquad t^0 \stackrel{\xi}{=} \ 0 \ .
\end{align}
Conversely, the self-duality relation on $H^4_n$ \cite{Sperling:2018xrm}
\begin{align}
\epsilon_{abcde} \theta^{ab} x^{c} &= 2R \theta_{de}
\label{SD-H-class}
\end{align}
relates the space-like and the time-like components of $\theta^{\mu\nu}$
on $\cM^{3,1}$,
and an explicit expression of $\theta^{\mu\nu}$
in terms of $t^\mu$ can be derived \cite{Sperling:2019xar}
\begin{align}
\theta^{\mu\nu} &= \ c(x^\mu t^\nu - x^\nu t^\mu) + b \epsilon^{\mu\nu\alpha} \def\da{{\dot\alpha}} \def\b{\beta\b} x_\alpha} \def\da{{\dot\alpha}} \def\b{\beta
t_\b
\label{theta-P-relation} \\
\text{with} \qquad
c &= r^2 \frac{\sinh(\eta)}{\cosh^2(\eta)}
\qquad \text{and} \qquad
b = \frac{r^2}{\cosh^2(\eta)} \,.
\end{align}
\paragraph{Hyperbolic coordinates.}
Now consider the adapted hyperbolic coordinates
\begin{align}
\begin{pmatrix}
x^0 \\ x^1 \\ x^2 \\x^3
\end{pmatrix}
= R \cosh(\eta)
\begin{pmatrix}
\cosh(\chi) \\
\sinh(\chi)\sin(\theta) \cos(\varphi) \\
\sinh(\chi)\sin(\theta) \sin(\varphi) \\
\sinh(\chi)\cos(\theta)
\end{pmatrix} \ .
\label{hyperbolic-coords}
\end{align}
We will see that $\eta$ measures the cosmic time, cf. \eqref{x4-eta-def}, while
the space-like distance from the origin on each time slice $H^3$ is measured by $\chi$.
Noting that
\begin{align}
\frac{x_\mu x_\nu}{R^2\cosh^2(\eta)}\mathrm{d} x^\mu \mathrm{d} x^\nu &=R^2 \sinh^2(\eta) d\eta^2 \
\label{metric-sphericalcoords-trafo}
\end{align}
which follows from \refeq{radial-constraint}, we obtain
the induced (flat) metric of ${\mathbb R}} \def\C{{\mathbb C}} \def\N{{\mathbb N}^{3,1}$ in these coordinates
\begin{align}
ds^2_g &= \eta_{\mu\nu}\mathrm{d} x^\mu \mathrm{d} x^\nu = R^2\Big(-\sinh^2(\eta)d\eta^2
+ \cosh^2(\eta)d\Sigma^2\Big)
\label{ind-metric-explicit}
\end{align}
where $d\Sigma^2$ is the metric on the
unit hyperboloid $H^3$,
\begin{align}
d\Sigma^2 &= d\chi^2 + \sinh^2(\chi)d\Omega^2, \qquad
d\Omega^2 = d\theta^2 + \sin^2(\theta)d\varphi^2 \ .
\label{ds-induced}
\end{align}
However, the effective metric
is a different one, which is also $SO(3,1)$ invariant but not flat.
\subsection{Effective metric and d'Alembertian}
\label{sec:metric}
In the matrix model framework considered below, the
effective metric on the background $\cM^{3,1}$ under consideration is given by
\cite{Sperling:2019xar}
\begin{align}
G^{\mu\nu} &= \alpha} \def\da{{\dot\alpha}} \def\b{\beta\, \g^{\mu\nu} \ = \ \sinh^{-1}(\eta) \eta^{\mu\nu}
\qquad
\qquad \alpha} \def\da{{\dot\alpha}} \def\b{\beta = \sqrt{\frac 1{\tilde\rho^2|\g^{\mu\nu}|}}
= \sinh^{-3}(\eta) \ \nn\\
\g^{\alpha} \def\da{{\dot\alpha}} \def\b{\beta\b} &= \eta_{\mu\nu}\theta^{\mu\alpha} \def\da{{\dot\alpha}} \def\b{\beta}\theta^{\nu\b}
= \sinh^2(\eta) \eta^{\alpha} \def\da{{\dot\alpha}} \def\b{\beta\b} \ .
\label{eff-metric-G}
\end{align}
This is an $SO(3,1)$-invariant FLRW metric with signature $(-+++)$.
Here $\tilde\rho^2$ is an irrelevant constant which adjusts the dimensions.
There are several ways to obtain this metric. One is by rewriting the
kinetic term in covariant form \cite{Sperling:2019xar,Steinacker:2010rh}
\begin{align}
S[\phi] = \mbox{Tr} [T^\mu,\phi][T_\mu,\phi]
\sim \int d⁴ x\,\sqrt{|G|}G^{\mu\nu}\partial_\mu\phi \partial_\nu \phi \ ,
\label{scalar-action-metric}
\end{align}
and another way is given below by showing \eqref{Box-deldel}.
Using \eqref{ind-metric-explicit}, this metric can be written as
\begin{align}
\mathrm{d} s^2_G = G_{\mu\nu} \mathrm{d} x^\mu \mathrm{d} x^\nu
&= -R^2 \sinh^3(\eta) \mathrm{d} \eta^2 + R^2\sinh(\eta) \cosh^2(\eta)\, \mathrm{d} \Sigma^2 \ \nn\\
&= -\mathrm{d} t^2 + a^2(t)\mathrm{d}\Sigma^2 \,
\label{eff-metric-FRW}
\end{align}
and we can read off the cosmic scale parameter $a(t)$
\begin{align}
a(t)^2 &= R^2\sinh(\eta) \cosh^2(\eta) \ \stackrel {t\to\infty}{\sim} \ R^2\sinh^3(\eta) , \label{a-eta}\\
\mathrm{d} t &= R \sinh(\eta)^{\frac{3}{2}} \mathrm{d}\eta \ .
\end{align}
Hence $a(t) \sim \frac 32 t$ for late times, and the Hubble rate is decreasing as $\frac{\dot a}{a} \sim a^{-5/3}$.
This is related to the time-like vector field $\t$ \eqref{tau-def} via
\begin{align}
\frac{\partial}{\partial\eta} &= \tanh(\eta) \t , \qquad
\frac{\partial}{\partial t}
= \frac 1R \frac{1}{\sqrt{\sinh(\eta)\cosh(\eta)}} \t
\ \stackrel {t\to\infty}{\sim} \ \frac 1R \b \t \ .
\label{tau-eta}
\end{align}
As a consistency check,
it is shown in appendix \ref{sec:diffeo-FRW}
that the covariant d'Alembertian $\Box_G$ of a scalar field
is indeed given by $\Box$ up to a factor \cite{Steinacker:2010rh},
\begin{align}
-\Box \phi &= \eta^{\alpha} \def\da{{\dot\alpha}} \def\b{\beta\b}\{t_\alpha} \def\da{{\dot\alpha}} \def\b{\beta,\{t_\b,\phi\}\}
= \eta^{\alpha} \def\da{{\dot\alpha}} \def\b{\beta\b} \b^{-1} (\partial_\alpha} \def\da{{\dot\alpha}} \def\b{\beta \b^{-1}\partial_\b \phi) \nn\\
&= \b^{-2} \big( \eta^{\alpha} \def\da{{\dot\alpha}} \def\b{\beta\b} \partial_\alpha} \def\da{{\dot\alpha}} \def\b{\beta \partial_\b
- \frac{1}{x_4^2} x^\b \partial_\b \big) \phi \nn\\
&= \b^{-3} \nabla^\alpha} \def\da{{\dot\alpha}} \def\b{\beta \partial_\alpha} \def\da{{\dot\alpha}} \def\b{\beta \phi = \b^{-3} \Box_G
\label{Box-deldel}
\end{align}
where $\nabla$ is the covariant derivative w.r.t. $G_{\mu\nu}$.
In particular, we note the useful formula
\begin{align}
\partial^\alpha} \def\da{{\dot\alpha}} \def\b{\beta\partial_\alpha} \def\da{{\dot\alpha}} \def\b{\beta \phi = \b^2\big(-\Box + \frac 1{R^2}\t\big)\phi \ .
\label{deldel-Box-relation}
\end{align}
We would like to decompose $\Box$ into time derivatives $\t$ and
the space-like Laplacian $\Delta^{(3)}$ on $H^3$
\begin{align}
-\Delta^{(3)} \phi &= \nabla_\mu^{(3)}\nabla^{(3)\mu}\phi =
\partial_\mu( P_\perp^{\mu\nu}\partial_\nu\phi)
\label{Laplacian-H3-def}
\end{align}
using the time-like and space-like projectors
\begin{align}
P_\t^{\mu\nu} &:= \frac{1}{x_\alpha} \def\da{{\dot\alpha}} \def\b{\beta x^\alpha} \def\da{{\dot\alpha}} \def\b{\beta} x^\mu x^\nu, \qquad
P_\perp^{\mu\nu} := \eta^{\mu\nu} - P_\t^{\mu\nu} \ .
\label{H3-projector}
\end{align}
After some calculations using \eqref{del-t-rel} and the formulas in section
\ref{sec:useful-formulas}, one obtains
\begin{align}
\boxed{
\Box\phi = \Big( \b^{-2}\Delta^{(3)} + \frac{1}{R^2} \t
+\frac{\sinh^2(\eta)}{R^2\cosh^2(\eta)} (2 + \t)\t\Big)\phi
}
\label{Box-Laplace-tau}
\end{align}
for scalar fields $\phi(x)$.
This can be checked e.g. for $\phi = x^\alpha} \def\da{{\dot\alpha}} \def\b{\beta$.
On the other hand we can use
the above hyperbolic coordinates \eqref{hyperbolic-coords}, where
\begin{align}
G_{\mu\nu} &= R^2\sinh(\eta)\rm diag \Big(-\sinh^2(\eta),\cosh^2(\eta),
\cosh^2(\eta)\sinh^2(\chi),\cosh^2(\eta)\sinh^2(\chi)\sin^2(\theta)\Big)
\end{align}
so that
\begin{align}
\Box_G &= -\frac{1}{\sqrt{|G_{\mu\nu}|}}\partial_\mu\big(\sqrt{|G_{\mu\nu}|}\, G^{\mu\nu}\partial_\nu\big) \nn\\
&= \frac{1}{R^2\sinh^3(\eta)\cosh^3(\eta)}
\partial_\eta\big(\cosh^3(\eta)\partial_\eta \phi\big)
+ \frac{1}{\sinh(\eta)}\Delta^{(3)} \phi \ .
\label{G-Box-relation}
\end{align}
This reduces indeed to \eqref{Box-Laplace-tau} using $\Box = \b^{-3}\Box_G$
and \eqref{tau-eta}.
The Laplacian $\Delta^{(3)}$ \eqref{Laplacian-H3-def}
on the space-like $H^3$ reduces
for rotationally invariant functions $\phi(\chi)$ to
\begin{align}
\Delta^{(3)} \phi(\chi)
&= - \frac{1}{R^2\cosh^2(\eta)} \frac 1{\sinh^2(\chi)}
\partial_\chi\big(\sinh^2(\chi)\partial_\chi\phi\big) \ .
\label{Delta-3-H}
\end{align}
\subsection{Higher spin sectors and filtration}
\label{sec:higher-spin}
Due to the extra generators $t^\mu$, the full algebra of functions decomposes
into sectors $\cC^s$ which correspond to spin $s$ harmonics on the $S^2$ fiber:
\begin{align}
\mathrm{End}(\cH_n) =\cC = \cC^0 \oplus \cC^1 \oplus \ldots \oplus \cC^n\qquad \text{with}
\quad
{\cal S}} \def\cT{{\cal T}} \def\cU{{\cal U}^2|_{\cC^s} = 2s(s+1) \,
\label{EndH-Cs-decomposition}
\end{align}
Here ${\cal S}} \def\cT{{\cal T}} \def\cU{{\cal U}^2 = \frac 12\sum_{a,b< 5} [\cM_{ab},[\cM^{ab},\cdot]]
+ r^{-2} [X_a,[X^a,\cdot]]$ can be viewed as a spin operator\footnote{Since
local Lorentz invariance is not manifest, the usual notion of spin cannot be used,
and ${\cal S}} \def\cT{{\cal T}} \def\cU{{\cal U}^2$ is a substitute.} on $H^4_n$
\cite{Sperling:2018xrm}, which commutes with $\Box$.
In the semi-classical limit, the $\cC^s$ are modules over $\cC^0$,
and can be realized explicitly in terms of totally symmetric traceless space-like
rank $s$ tensor
fields on $\cM^{3,1}$
\begin{align}
\phi^{(s)} = \phi_{\mu_1 ... \mu_s}(x) t^{\mu_1} ... t^{\mu_s} ,
\qquad \phi_{\mu_1 ... \mu_s} x^{\mu_i} = 0
\label{Cs-explicit}
\end{align}
due to \eqref{geometry-H-M}.
The underlying $\mathfrak{so}(4,2)$ structure provides an $SO(3,1)$ -invariant derivation
\begin{align}
D\phi &:= \{x^4,\phi\} \
= r^2 R^2 \frac{1}{x^4} t^\mu \{t_\mu,\phi\}
= -\frac{1}{x^4}x_\mu\{x^\mu,\phi\} \nn\\
&= r^2 R\, t^{\mu_1}\ldots t^{\mu_s} t^\mu \, \nabla^{(3)}_\mu\phi_{\mu_1\ldots\mu_s}(x)
\label{D-properties}
\end{align}
where $\nabla^{(3)}$ is the covariant derivative along the
space-like $H^3 \subset \cM^{3,1}$.
Hence $D$ relates the different spin sectors in \eqref{EndH-Cs-decomposition}:
\begin{align}
D = D^- + D^+: \ \cC^{s} \ &\to \cC^{s-1} \oplus \cC^{s+1}, \qquad
D^\pm \phi^{(s)} = [D\phi^{(s)}]_{s\pm 1} \
\end{align}
where $[.]_{s}$ denotes the projection to $\cC^{s}$ defined through
\eqref{EndH-Cs-decomposition}.
For example, $D x^\mu = r^2 R\, t^\mu$ and $D t^\mu = R^{-1}\, x^\mu$.
This allows to define a further refinement \cite{Sperling:2019xar}
\begin{align}
\cC^{(s,k)} \coloneqq \cK^{(s,k)} / \cK^{(s,k-1)} , \qquad
\cK^{(s,k)} = \ker (D^-)^{k+1} \subset \cC^{s} \ .
\label{C-sk-def}
\end{align}
Then
\begin{align}
D^\pm: \quad \cC^{(s,k)} &\to \cC^{(s-1,k-1)} \ .
\label{D-refined}
\end{align}
In particular, $\cC^{(s,0)} \subset \cC^{s}$ is the space of divergence-free traceless
space-like rank $s$ tensor fields on $\cM^{3,1}$,
while $D^+D\phi^{(0)} = [t^\mu t^\nu]_2 \nabla^{(3)}_\mu\partial_\nu \phi^{(0)} \in \cC^{(2,2)} \subset \cC^2$ encodes the
traceless second derivatives of the scalar field $\phi^{(0)}$.
These will play an important role below. Finally,
$\t$ is extended to $\cC^s$ via \cite{Sperling:2019xar}
\begin{align}
\sinh(\eta)(\t + s)\phi^{(s)} = x^\mu\{t_\mu,\phi^{(s)} \} \ ,
\label{tau-relns}
\end{align}
which gives \eqref{D-tau-relation}.
\paragraph{Averaging.}
We will need some explicit formulas for the projection $[.]_0$ to $\cC^0$:
\begin{align}
[t^\mu t^\nu]_0 &\eqqcolon \frac{\cosh^2(\eta)}{3r^2} P_\perp^{\mu\nu} \,,
\label{kappa-average}
\end{align}
in terms or the projector $P_\perp$ \eqref{H3-projector}
on the time-slices $H^3$. This can be viewed as an averaging over $S^2$.
Explicitly, one finds \cite{Sperling:2019xar}
\begin{subequations}
\label{averaging-relns}
\begin{align}
\left[t^{\alpha} \def\da{{\dot\alpha}} \def\b{\beta} \theta^{\mu\nu}\right]_{0}
&= \frac{1}{3} \Big(\sinh(\eta) ( \eta^{\alpha} \def\da{{\dot\alpha}} \def\b{\beta\nu} x^\mu - \eta^{\alpha} \def\da{{\dot\alpha}} \def\b{\beta\mu} x^\nu) + x_\b \varepsilon^{\b 4\alpha} \def\da{{\dot\alpha}} \def\b{\beta\mu \nu} \Big)\,, \\
[t^{\mu_1} \ldots t^{\mu_4}]_0 &=
\frac 35 \big([t^{\mu_1}t^{\mu_2}][t^{\mu_3} t^{\mu_4}]_0
+ [t^{\mu_1}t^{\mu_3}][t^{\mu_2} t^{\mu_4}]_0 +
[t^{\mu_1}t^{\mu_4}][t^{\mu_2} t^{\mu_3}]_0\big) \,. \nn\\
[t^{\alpha} \def\da{{\dot\alpha}} \def\b{\beta} t^\b t^\g ]_{1}
&= \frac 35 \Big([t^{\alpha} \def\da{{\dot\alpha}} \def\b{\beta} t^\b]_{0} t^\g + t^{\alpha} \def\da{{\dot\alpha}} \def\b{\beta} [t^\b t^\g]_{0}
+ t^\b[t^{\alpha} \def\da{{\dot\alpha}} \def\b{\beta} t^\g ]_{0} \Big) \ .
\label{average-3}
\end{align}
\end{subequations}
As an application, one can derive the following formula
\begin{align}
\{x^\mu,\{x_\mu,\phi\}\}_0
&= \frac{r^2R^2}3(3 -\cosh^2(\eta))\b^2(-\Box+\frac{1}{R^2}\t)\phi
+ \frac {r^2}3 (2\t + 7)\t \phi\,
\label{dAlembertian-x}
\end{align}
for $\phi\in\cC^0$.
This could be another natural d'Alembertian on $\cM^{3,1}$ which exhibits a transition
from a Euclidean to a Minkowski era, as discussed in \cite{Steinacker:2017bhb}.
However in this paper the effective
d'Alembertian will be $\Box$, which respects the spin sectors $\cC^s$
\eqref{EndH-Cs-decomposition}.
\section{Matrix model and higher-spin gauge theory}
\label{sec:fluctuations}
Now we return to the noncommutative setting, and
define a dynamical model for the fuzzy $\cM^{3,1}$ space-time under consideration.
We consider a Yang-Mills matrix model with mass term,
\begin{align}
S[Y] &= \frac 1{g^2}\mbox{Tr} \Big([Y^\mu,Y^\nu][Y^{\mu'},Y^{\nu'}] \eta_{\mu\mu'} \eta_{\nu\nu'} \,
+\frac{6}{R^2} Y^\mu Y^\nu \eta_{\mu\nu} \Big) \ .
\label{bosonic-action}
\end{align}
This includes in particular the IKKT or IIB matrix model \cite{Ishibashi:1996xs} with mass term,
which is best suited for quantization because maximal supersymmetry protects from UV/IR mixing \cite{Minwalla:1999px}.
As observed in \cite{Sperling:2019xar}, $\cM^{3,1}$ is indeed a solution of this model\footnote{any other
positive mass parameter in \eqref{bosonic-action} would of course just result
in a trivial rescaling. For negative mass parameter, $Y^\mu \sim X^\mu$ would be a solution \cite{Steinacker:2017bhb},
but the fluctuations are more difficult to analyze.}, through
\begin{align}
Y^\mu = T^\mu \ .
\end{align}
Now consider tangential
deformations of the above background solution, i.e.
\begin{align}
Y^\mu = T^\mu + \cA^\mu \ ,
\end{align}
where $\cA^\mu \in \mathrm{End}(\cH_n) \otimes \C^4$ is an arbitrary (Hermitian) fluctuation.
The Yang-Mills action \eqref{bosonic-action} can be expanded as
\begin{align}
S[Y] = S[T] + S_2[\cA] + O(\cA^3) \ ,
\end{align}
and the quadratic fluctuations are governed by
\begin{align}
S_2[\cA] = -\frac{2}{g^2} \,\mbox{Tr} \left( \cA_\mu
\Big(\cD^2 -\frac{3}{R^2}\Big) \cA^\mu + \cG\left(\cA\right)^2 \right) .
\label{eff-S-expand}
\end{align}
Here
\begin{align}
\cD^2 \cA = \left(\Box - 2\cI \right)\cA
\label{vector-Laplacian}
\end{align}
is the vector d'Alembertian, which involves the scalar matrix d'Alembertian
$\Box \sim \alpha} \def\da{{\dot\alpha}} \def\b{\beta^{-1} \Box_G$ on the $\cM^{3,1}$ background
\eqref{Box-def}, \eqref{Box-deldel} as discussed before,
and the intertwiner
\begin{align}
\cI (\cA)^\mu = -\mathrm{i} [[ Y^\mu, Y^\nu],\cA_\nu] = \frac{\mathrm{i}}{r^2 R^2}
[\Theta^{\mu\nu},\cA_\nu]
\eqqcolon -\frac{1}{r^2 R^2}\tilde\cI (\cA)^\mu \
\end{align}
using \eqref{T-T-CR}.
As usual in Yang-Mills theories, $\cA$ transforms under gauge transformations as
\begin{align}
\d_\L\cA = -i[T^\mu + \cA^\mu,\L] \sim \{t^\mu,\L\} + \{\cA^\mu,\L\}
\end{align}
for any $\L\in\cC$,
and the scalar ghost mode
\begin{align}
\cG(\cA) = -\mathrm{i} [T^\mu,\cA_\mu] \sim \{t^\mu,\cA_\mu\} ,
\label{gaugefix-intertwiner}
\end{align}
should be removed to get a meaningful theory.
This can be achieved by adding a gauge-fixing term $-\cG(\cA)^2$ to the action
as well as the corresponding Faddeev-Popov (or BRST) ghost. Then the quadratic
action becomes
\begin{align}
S_2[\cA] + S_{g.f} + S_{ghost} &= -\frac{2}{g^2}\mbox{Tr}\,
\left( \cA_\mu \Big(\cD^2 -\frac{3}{R^2} \Big) \cA^\mu + 2 \overline{c} \Box
c \right) \
\label{eff-S-gaugefixed}
\end{align}
where $c$ denotes the fermionic BRST ghost; see e.g.\ \cite{Blaschke:2011qu}
for more details.
\section{Fluctuation modes}
All indices will be raised and lowered with $\eta^{\mu\nu}$ in this section.
We should expand the vector modes into higher spin modes according to
\eqref{EndH-Cs-decomposition}, \eqref{Cs-explicit}
\begin{align}
\cA^\mu &= A^{\mu}(x) + A^{\mu}_\alpha} \def\da{{\dot\alpha}} \def\b{\beta(x)\, t^\alpha} \def\da{{\dot\alpha}} \def\b{\beta + A^{\mu}_{\alpha} \def\da{{\dot\alpha}} \def\b{\beta\b}(x)\, t^\alpha} \def\da{{\dot\alpha}} \def\b{\beta t^\b + \ldots
\ \in \ \cC^0 \oplus \cC^1 \oplus \cC^2 \oplus \ \ldots
\label{A-M31-spins}
\end{align}
However these are neither irreducible nor eigenmodes of $\cD^2$.
In \cite{Sperling:2019xar}, three series of
spin $s$ eigenmodes $\cA_\mu$ were found of the form
\begin{align}
\label{A2-mode-ansatz}
\boxed{ \
\begin{aligned}
\cA_\mu^{(g)}[\phi^{(s)}] &= \{t_\mu,\phi^{(s)}\} \quad \in \cC^{s}\,,
\\
\cA_\mu^{(+)}[\phi^{(s)}] &= \{x_\mu,\phi^{(s)}\}|_{\cC^{s+1}} \ \equiv
\{x_\mu,\phi^{(s)}\}_+ \quad \in \cC^{s+1} \,, \\
\cA_\mu^{(-)}[\phi^{(s)}] &= \{x_\mu,\phi^{(s)}\}|_{\cC^{s-1}} \ \equiv
\{x_\mu,\phi^{(s)}\}_- \quad \in \cC^{s-1} \,
\end{aligned}
}
\end{align}
for any $\phi^{(s)}\in\cC^s$, which satisfy
\begin{align}
\cD^2 \cA_\mu^{(g)}[\phi] &=
\cA_\mu^{(g)}\left[\left(\Box+\frac{3}{R^2}\right)
\phi\right] \,,
\label{puregauge-D2} \\
\cD^2 \cA_\mu^{(+)}[\phi^{(s)}]
&= \cA_\mu^{(+)}\left[\left(\Box + \frac{2s+5}{R^2}
\right)\phi^{(s)}\right]
\label{D2-A2p-eigenvalues} \,, \\
\cD^2 \cA_\mu^{(-)}[\phi^{(s)}]
&= \cA_\mu^{(-)}\left[\left(\Box +
\frac{-2s+3}{R^2}\right)\phi^{(s)}\right] \, .
\label{D2-A2m-eigenvalues}
\end{align}
We provide in appendix \ref{sec:ladder-ops} a simple new derivation for the
last two relations.
Hence diagonalizing $\cD^2$ is reduced to diagonalizing $\Box$ on $\cC^s$,
and we have the on-shell modes $\big(\cD^2 -\frac{3}{R^2} \big) \cA = 0$ for
\begin{align}
\cA^{(+)}[\phi^{(s)}] \qquad & \text{for } \ \ \left(\Box +
\frac{2s+2}{R^2}\right) \phi^{(s)} = 0 \,,\\
\cA^{(-)}[\phi^{(s)}] \qquad& \text{for } \quad \ \left(\Box + \frac{-2s}{R^2}
\right)\phi^{(s)} = 0 \, , \\
\cA^{(g)}[\phi^{(s)}] \qquad &\text{for } \qquad \qquad \ \Box \phi^{(s)} = 0 \ .
\label{on-shell-B2pm}
\end{align}
Of course $\cA^{(g)}$ is a pure gauge mode and hence unphysical.
Furthermore, the following gauge fixing identities\footnote{As a check,
consider e.g. $\cA^{(-)}[\phi^{(1,0)}]$. It satisfies
$\{t_\mu,\cA^\mu\} = 0 = x_\mu\cA^\mu$ due to \eqref{D-properties}, and \eqref{div-A-full} gives
$\nabla_\mu \cA^\mu = 0$ and $\tilde\cI(\cA^\mu) = r^2 \cA^\mu$,
consistent with (A.33) in \cite{Sperling:2019xar}.} were shown in \cite{Sperling:2019xar}
\begin{align}
\{t^{\mu},\cA_\mu^{(+)}[\phi^{(s)}]\} &= \frac{s+3}{R} D^+\phi^{(s)} \,,\\
\{t^{\mu},\cA_\mu^{(-)}[\phi^{(s)}]\} &= \frac{-s + 2}{R} D^-\phi^{(s)} \, .
\label{A2-gaugefix}
\end{align}
In particular
for $s=2$, $\cA_\mu^{(-)}[\phi^{(2)}]$ is
already gauge fixed\footnote{For $s\neq 2$ some linear combinations of
$\cA_\mu^{(+)}$ and $\cA_\mu^{(-)}$ must be taken to obtain a gauge-fixed physical
solution. However, this is not our concern here.}.
This will lead to the physical spin 2 metric fluctuations.
According to the discussion in section \ref{sec:higher-spin}, they decompose into the
modes $\cA_\mu^{(-)}[\phi^{(2,0)}], \cA_\mu^{(-)}[D\phi^{(1,0)}]$ and
$\cA_\mu^{(-)}[D^+D\phi^{(0)}]$, which we will denote
-- in slight abuse of language -- as
helicity 2, 1 and 0 sectors of the would-be massive spin 2 modes, respectively.
We will focus on the physical helicity 0 or scalar mode, with on-shell condition
\begin{align}
\boxed{ \
\cA_\mu^{(-)}[D^+D\phi], \qquad \big(\Box + \frac{2}{R^2}\big)\phi = 0, \qquad \phi \in \cC^0
\ }
\label{onshell-0}
\end{align}
due to \eqref{Box-x4-relations}.
However, one series of spin $s$ (off-shell) eigenmodes $\cA_\mu$
of $\cD^2$ is still missing, and was not known
up to now. We will find the missing scalar mode in section \ref{sec:time-mode}, in terms of
\begin{align}
\cA_\mu^{(\t)}[\phi^{(s)}] = x_\mu \phi^{(s)} \ .
\label{A4-mode-ansatz}
\end{align}
That ansatz was also considered in \cite{Sperling:2019xar}, where it was shown to satisfy
\begin{align}
\cD^2 \cA_\mu^{(\t)}[\phi^{(s)}] &=
\cA^{(\t)}_\mu\left[\left(\Box+\frac{7}{R^2}\right)\phi^{(s)}\right] + 2\eth_\mu \phi ^{(s)}
\, \label{A4-eom} \\
\{t^\mu,\cA_\mu^{(\t)}[\phi^{(s)}]\} &= \sinh(\eta)\big(4 +s + \t \big) \phi^{(s)} \ .
\label{gaugefix-timelike}
\end{align}
Here $\eth_\mu$ will be defined in \eqref{eth-tbracket-0}.
We will show in the following that $\cA^{(-)}[D^+D\phi]$ provides the on-shell
mode leading to the linearized Schwarzschild metric. Moreover, an ansatz based on
$\cA_\mu^{(\t)}$ will give solutions which are
equivalent on-shell, but not off-shell.
\subsection{Scalar $\cA^{(-)}[D^+D\phi]$ mode}
We need the explicit form of $\cA^{(-)}[D^+D\phi]$. This is quite tedious to work out
and delegated to the appendix \ref{sec:eval-A-DD},
where we provide an exact expression in \eqref{A-DDphi-1}.
This simplifies considerably using the on-shell condition
$(\Box + \frac{2}{R^2})\phi = 0$ \eqref{onshell-0}, leading to
\begin{align}
\boxed{ \
\cA_\mu^{(-)}[D^+D^+\phi]
= \frac {2r^4}5 \Big(\b (t^\mu + x^\mu t^\alpha} \def\da{{\dot\alpha}} \def\b{\beta \partial_\alpha} \def\da{{\dot\alpha}} \def\b{\beta)
- \frac 1{3r^2} \theta^{\m\g}\partial_\g (\t+4 + \b^2)\Big)(\t+2)\phi + \{t_\mu,\L\} \
\ }
\label{A-DDphi-explicit}
\end{align}
with $\L$ given in \eqref{gaugeparam-schwarzschild}.
This is a reasonable perturbation of the background $Y^\mu = t^\mu$,
as long as $\phi$ remains bounded.
Remarkably, \eqref{A-DDphi-explicit} can be rewritten
via $\theta^{\m\g}\partial_\g \phi = \cA^{(+)\mu}[\phi]$ as
\begin{align}
\cA_\mu^{(-)}[D^+D^+\phi] &= \frac 25 \frac{r^2}{R} \Big(D(x^\mu \phi')
- \frac R{3} \cA^{(+)\mu}[(\t+4+\b^2)(\t+2)\phi]\Big)
+ \{t^\mu,\L \} \nn\\
&= \frac 25 \frac{r^2}{R} \cA_\mu^{(S)}[\phi']
+ \{t^\mu, \L' \}
\label{ADDphi-onshell-specialform}
\end{align}
where $\cA_\mu^{(S)}[\phi']$ is the new mode defined in \eqref{A-S-def}, with
\begin{align}
\phi' = \b(\t+2)\phi, \qquad
\L' = \L + \frac 2{15}r^2 R D(\t+4+\b^2)(\t+2)\phi \ .
\label{phi-prime}
\end{align}
To see this, the identities
\begin{align}
\b(t^\mu + x^\mu t^\alpha} \def\da{{\dot\alpha}} \def\b{\beta \partial_\alpha} \def\da{{\dot\alpha}} \def\b{\beta)(\t+2)\phi
&= \frac{1}{r^2R} D(x^\mu \phi') \nn\\
(\t+4+\b^2)(\t+2)\phi &= (\sinh(\eta)(\t+5)+2\b)\phi'
\end{align}
and the on-shell equations
\begin{align}
\big(\Box + \frac{2}{R^2}(3+\t-\b^2)\big)\phi' &= 0 \label{eom-phi'}\\
\Box \L' &= 0 \
\end{align}
are needed, which
can be checked using the results of section \ref{sec:useful-formulas}.
The last form implies that $\cA_\mu^{(-)}[D^+D^+\phi]$
differs from $\cA_\mu^{(S)}[\phi']$ by an on-shell pure gauge mode.
This means that even though these are distinct off-shell modes, they
become degenerate on-shell, so that there is only one physical scalar graviton mode.
This is essential for a ghost-free theory.
Strictly speaking the form \eqref{A-DDphi-explicit}
collapses for $\t=-2$.
However, its expression in terms of $\phi'$ -- or alternatively
the form \eqref{ADDphi-onshell-specialform} -- makes sense also in the limit
$\t\to -2$. This is important, because $\t=-2$ gives precisely the
Ricci-flat quasi-Schwarzschild solution, as discussed in section \ref{sec:A-DD-metric}.
For completeness
we also provide the explicit form of
the pure gauge field $\cA^{(g)}$ corresponding to \eqref{gaugeparam-schwarzschild}
\begin{align}
\cA^{(g)\mu}[\L] &= \{t_\mu,\L\}
= \frac 25 r^2 \Big(\theta^{\mu\alpha} \def\da{{\dot\alpha}} \def\b{\beta} \partial_\alpha} \def\da{{\dot\alpha}} \def\b{\beta\
-R\sinh(\eta) D\partial^\mu\Big)(\t+3)\phi \ .
\label{A-puregauge-L}
\end{align}
\paragraph{Gauge fixing.}
A non-trivial consistency check of \eqref{A-DDphi-explicit} is obtained by
verifying that it satisfies the gauge-fixing constraint.
For the pure gauge contribution, this is
\begin{align}
\{t^\mu,\{t_\mu,\L\}\} &=
\frac 25 r^2 R \Box D(\t+3) \phi
= \frac {4r^2}{5R} D \b^2(2+\t) \phi
\end{align}
using \eqref{box-tau-relation-1}.
Together with the relations \eqref{gauge-fixing-relations-scalar},
one verifies indeed $\{t^\mu, \cA_\mu^{(-)}[D^+D^+\phi]\} =0$.
\subsection{Time-like scalar mode $\tilde\cA_\mu^{(\t)}$}
\label{sec:time-mode}
In this section we will show that a refined ansatz involving $\cA^{(\t)}[\phi]$
provides a further scalar eigenmode of $\cD^2$. This will also
provide the missing $10^{th}$ degree of freedom for
the off-shell metric fluctuations. While this is not essential to understand the Schwarzschild solution,
it provides further insights.
First we recall the relation \eqref{A4-eom}, which involves the derivation
\begin{align}
\eth^\mu \phi &= -\frac{1}{r^2 R^2}\theta^{\mu b}\{x_b,\phi\}
= \{t^\mu,\b\phi\}
+ \frac 1{R^2} x^\mu \Big(-\b^2 + \t \Big)\phi ,
\label{eth-tbracket-0}
\end{align}
for $b=0,...,4$ and $\phi\in\cC^0$.
The second form is obtained noting that
\begin{align}
\eth^\mu \phi &= \partial^\mu\phi + \frac 1{R^2} x^\mu \t\phi \qquad \mbox{for}
\qquad \phi\in\cC^0 \ ,
\end{align}
and rewriting the first term using
$\partial^\mu\phi = \{t^\mu,\b\phi\} - \frac 1{R^2} x^\mu \ \b^2\phi$.
Hence \eqref{A4-eom} can be written as
\begin{align}
\cD^2 \cA_\mu^{(\t)}[\phi]
&= \cA^{(\t)}_\mu\left[
\left(\Box + \frac 1{R^2}\Big(- 2\b^2 + 2\t +7 \Big)\right)\phi\right]
+ 2 \{t^\mu,\b\phi\} \ .
\label{D2-timelike-rel}
\end{align}
Since the last term is a pure gauge mode, this provides a new eigenmode of $\cD^2$:
\paragraph{Scalar time-like $\cC^0$ mode.}
Combining the above with
\eqref{puregauge-D2}, the ansatz
\begin{align}
\boxed{ \ \
\tilde\cA_\mu^{(\t)}[\phi] = \cA_\mu^{(\t)}[\phi] + \{t^\mu,\tilde\phi\} \ \
}
\label{tilde-A-t-0}
\end{align}
leads to new scalar eigenmode of $\cD^2$
\begin{align}
\cD^2 \tilde\cA_\mu^{(\t)}[\phi]
&= \lambda} \def\L{\Lambda} \def\la{\lambda} \def\m{\mu \tilde\cA_\mu^{(\t)}[\phi]
\end{align}
provided
\begin{align}
\Big(\Box + \frac 1{R^2}\big(- 2\b^2 + 2\t +7 \big)\Big)\phi
&= \lambda} \def\L{\Lambda} \def\la{\lambda} \def\m{\mu \phi\nn\\
(\Box+\frac{3}{R^2})\tilde\phi + 2\b\phi &= \lambda} \def\L{\Lambda} \def\la{\lambda} \def\m{\mu \tilde\phi \ .
\label{timelikemode-0}
\end{align}
The first equation can be solved, and
has propagating solutions $\phi$. Then $\tilde\phi$
is determined by the second equation, up to solutions of
$(\Box+\frac{3}{R^2}-\lambda} \def\L{\Lambda} \def\la{\lambda} \def\m{\mu)\tilde\phi = 0$.
This $4^{th}$ eigenmode is needed e.g. for the off-shell propagator.
In particular, $\tilde\cA_\mu^{(\t)}[\phi]$ is on-shell, $(\cD^2 -\frac{3}{R^2} \Big) \tilde\cA^{(\t)} = 0$ for
\begin{align}
\Big(\Box + \frac 2{R^2}\big(2+\t -\b^2 \big)\Big)\phi &= 0 \nn\\
\Box \tilde\phi + 2\b\phi &= 0 \ .
\label{timelikemode-0a}
\end{align}
However the gauge fixing condition for this mode is very restrictive on-shell,
\begin{align}
\{t_\mu,\tilde\cA_\mu^{(\t)}[\phi] \}
&= \{t_\mu,\cA_\mu^{(\t)}[\phi] \} - \Box\tilde\phi
= \big(\sinh(\eta)(4 + \t ) + 2\b\big) \phi
\label{gaugefix-At-0}
\end{align}
or
\begin{align}
\t \phi &= -(4 + 2\b^2) \phi \ ,
\end{align}
which means that $\phi$ is decaying in time with a fixed rate.
Hence these modes are ``frozen''
rather than propagating, which is good because they would otherwise be ghosts.
We will see that these $\cA \in\cC^0$ modes do not contribute to the linearized metric fluctuations.
\paragraph{Scalar time-like $\cC^1$ mode.}
Based on the above mode and using the ladder property \eqref{D2-D+-relation}, we can similarly find a
new eigenmode $\cA \in \cC^1$
with the ansatz
\begin{align}
D\tilde\cA_\mu^{(\t)}[\phi]
&= D\big(\cA_\mu^{(\t)}[\phi] + \{t^\mu,\tilde\phi\} \big) \
= r^2 R t^\mu \phi + x^\mu D\phi
+ \frac{1}{R}\{x^\mu,\tilde\phi\} + \{t^\mu,D\tilde\phi\} \ .
\label{D-tilde-A-t-1}
\end{align}
This is an eigenmode of $\cD^2$ provided $\tilde\cA_\mu^{(\t)}$ is an eigenmode, with shifted eigenvalue
\begin{align}
\cD^2 (D\tilde\cA_\mu^{(\t)}[\phi])
&= D(\cD^2 + \frac{2}{R^2}) \tilde\cA_\mu^{(\t)}[\phi] \ .
\end{align}
In particular, $D^+\tilde\cA_\mu^{(\t)}[\phi]$ is on-shell if
$(\cD^2 -\frac{1}{R^2})\tilde\cA_\mu^{(\t)}[\phi] = 0$,
which means by \eqref{timelikemode-0}
\begin{subequations}
\label{onshell-tildephi}
\begin{align}
\Big(\Box + \frac 2{R^2}(-\b^2 + \t + 3)\Big)\phi
&= 0 \label{onshell-tildephi-1} \\
(\Box + \frac{2}{R^2})\tilde\phi + 2 \b\phi &= 0 \ .
\label{onshell-tildephi-2}
\end{align}
\end{subequations}
This provides the missing $4^{th}$ scalar eigenmode in $\cC^1$.
The gauge-fixing condition is
\begin{align}
\{t^\mu,D\tilde\cA_\mu^{(\t)}[\phi] \}
&= \{t^\mu, r^2 R t_\mu \phi + x_\mu D\phi \} + \{t^\mu,D \{t_\mu,\tilde\phi\}\} \nn\\
&= r^2 R t_\mu \{t^\mu, \phi\} + 4\sinh(\eta) D\phi + x_\mu \{t^\mu,D\phi \}
+ \frac 1R \{t^\mu,\{x_\mu,\tilde\phi\}\} - \Box D\tilde\phi \nn\\
&= \sinh(\eta) D\phi + 4\sinh(\eta) D\phi + \sinh(\eta) (\t+1) D\phi
+ \frac 3{R^2} D\tilde\phi - D(\Box+\frac{2}{R^2})\tilde\phi \nn\\
&= D\Big(\big(\sinh(\eta) (\t+5) + 2\b\big)\phi
+ \frac{3}{R^2}\tilde\phi \Big)
\end{align}
using \eqref{gaugefix-At-0}, \eqref{tau-relns}, \eqref{D-tau-relation}, \eqref{D-properties}
and the on-shell equations \eqref{onshell-tildephi}.
This implies
\begin{align}
\big(\sinh(\eta) (\t+5) + 2 \b\big)\phi
+ \frac{3}{R^2}\tilde\phi = f(x^4) \ .
\label{gaugefix-t-relation}
\end{align}
For now we set $f=0$. Then
\begin{align}
\tilde\phi = -\frac{R^2}3\big(\sinh(\eta) (\t+5) + 2 \b\big)\phi \ ,
\label{tilde-phi-sol}
\end{align}
and one can verify that the equations of motion \eqref{onshell-tildephi-2} for $\tilde\phi$
indeed follow from those of $\phi$,
using the relations in section \ref{sec:useful-formulas}.
This means that the gauge-fixing condition leading to \eqref{tilde-phi-sol}
is consistent with the equations of motion, and we have found a physical propagating mode of the form
\begin{align}
\boxed{ \
\cA_\mu^{(S)}[\phi]
:= D\Big(\cA_\mu^{(\t)}[\phi] -\frac{R^2}3 \{t^\mu, \big(\sinh(\eta) (\t+5) + 2 \b\big)\phi \} \Big)
\ }
\label{A-S-def}
\end{align}
with $\phi$ satisfying \eqref{onshell-tildephi-1}.
On-shell, this coincides precisely with the on-shell eigenmode
$\cA^{(-)}[D^+D\phi]$ \eqref{ADDphi-onshell-specialform}, although off-shell (hence in the propagator) they are distinct modes.
In the quasi-static case $\t=-2$, this will give the linearized Schwarzschild metric.
\section{Scalar metric fluctuation modes}
\label{sec:graviton}
In this section, we elaborate the metric fluctuations arising from the above scalar modes.
The effective metric for functions of $\cM^{3,1}$ on a perturbed background $Y = T + \cA$ can be extracted from the kinetic
term in \eqref{scalar-action-metric}, which defines the
bi-derivation
\begin{align}
\begin{aligned}
\g:\quad \cC\times \cC \ &\to \quad \cC \\
(\phi,\phi') &\mapsto \{Y^\alpha} \def\da{{\dot\alpha}} \def\b{\beta,\phi\}\{Y_\alpha} \def\da{{\dot\alpha}} \def\b{\beta,\phi'\}
\label{metric-full}
\end{aligned}
\end{align}
up to a conformal factor as discussed in section \ref{sec:metric}.
Specializing to $\phi=x^\mu, \phi' = x^\nu$ we obtain the coordinate form
\begin{align}
\g_Y^{\mu\nu} &= \overline\g^{\mu\nu} + \d_\cA \g^{\mu\nu} + [\{\cA^\alpha} \def\da{{\dot\alpha}} \def\b{\beta,x^\mu\}\{\cA_\alpha} \def\da{{\dot\alpha}} \def\b{\beta,x^\nu\}]_0
\label{gamma-nonlinear}
\end{align}
where the linearized contribution is given by
\begin{align}
\begin{aligned}
\d_\cA \g^{\mu\nu} &\coloneqq
[\{t^\alpha} \def\da{{\dot\alpha}} \def\b{\beta,x^\mu\}\{\cA_\alpha} \def\da{{\dot\alpha}} \def\b{\beta,x^\nu\}]_0 + (\mu \leftrightarrow \nu)
= \sinh(\eta) \{\cA_\mu,x^\nu\}_0 + (\mu \leftrightarrow \nu) \ .
\label{gravitons-H1}
\end{aligned}
\end{align}
The projection on $\cC^0$ ensures that this is the metric for functions on $\cM^{3,1}$.
We will focus on the linearized contribution in $\cA$ in the following.
To evaluate this explicitly,
it is convenient to consider the following rescaled graviton mode:
\begin{align}
h^{\mu\nu}[\cA] &\coloneqq \{\cA^\mu,x^\nu\}_0 + (\mu \leftrightarrow \nu) \ ,
\qquad h[\cA] = 2\{\cA^\mu,x_\mu\}_0 \ .
\label{tilde-H-def}
\end{align}
Clearly only $\cA \in \cC^1$ can contribute to $ h^{\mu\nu}[\cA]$.
Taking into account the conformal factor as identified in section \ref{sec:metric},
the effective metric $G^{\mu\nu}$ \eqref{eff-metric-G} is
\begin{align}
G^{\mu\nu} &= \overline G^{\mu\nu} + \d G^{\mu\nu} \nn\\
&= \alpha} \def\da{{\dot\alpha}} \def\b{\beta\left[\g^{\mu\nu} + \d_\cA \g^{\mu\nu}
-\frac{1}{2} \eta^{\mu\nu}\,\left(\eta_{\alpha} \def\da{{\dot\alpha}} \def\b{\beta\b}\ \d_\cA \g^{\alpha} \def\da{{\dot\alpha}} \def\b{\beta\b}\right)\right] \nn\\
\d G^{\mu\nu} &= \b^2\big(h^{\mu\nu} -\frac{1}{2} \eta^{\mu\nu}\,h\big) \ .
\label{eq:def_phys_graviton}
\end{align}
Here $\overline G^{\mu\nu} = \alpha} \def\da{{\dot\alpha}} \def\b{\beta\g^{\mu\nu} = \b\eta^{\mu\nu}$ \eqref{eff-metric-G}
is the effective background metric,
$\alpha} \def\da{{\dot\alpha}} \def\b{\beta = \b^3$ is the conformal factor arising from the fixed symplectic measure on $\C P^{1,2}$, and $\b=\sinh(\eta)^{-1}$ \eqref{del-t-rel}.
Equivalently,
\begin{align}
G_{\mu\nu} &=\overline G_{\mu\nu} - \d G_{\mu\nu} , \nn\\
\d G_{\mu\nu} &= \overline G_{\mu\mu'} \overline G_{\nu\nu'} \d G^{\mu'\nu'}
= h_{\mu\nu} -\frac{1}{2} \eta_{\mu\nu}\,h
\end{align}
where $h_{\alpha} \def\da{{\dot\alpha}} \def\b{\beta\b} = \eta_{\alpha} \def\da{{\dot\alpha}} \def\b{\beta\a'}\eta_{\b\b'}h^{\alpha} \def\da{{\dot\alpha}} \def\b{\beta'\b'}$.
One has to be very careful in rising and lowering indices, because there are different metrics in the game.
The indices of the effective metric $G$ will always be raised and lowered with the effective background metric $\overline G^{\mu\nu}$, while
the indices of $h^{\mu\nu}$ and most other tensorial objects will be raised and lowered with $\eta^{\mu\nu}$.
In case of ambiguity, we will typically spell this out.
With this convention, we can write
the fluctuations of the effective background effective metric \eqref{eff-metric-FRW} as
\begin{align}
(G_{\mu\nu} - \d G_{\mu\nu}) \mathrm{d} x^\mu \mathrm{d} x^\nu
&= -\mathrm{d} t^2 + a^2(t)\mathrm{d}\Sigma^2
- (h_{\mu\nu} -\frac{1}{2} \eta_{\mu\nu}\,h) \mathrm{d} x^\mu \mathrm{d} x^\nu \ .
\label{perturbed-metric-cosm}
\end{align}
\subsection{Linearized Ricci tensor}
\label{sec:lin-curv}
To understand the significance of the metric modes, we consider the
linearized Ricci tensor
\begin{align}
2\d R_{(\rm lin)}^{\mu\nu}[G]
&= -\nabla^\alpha} \def\da{{\dot\alpha}} \def\b{\beta \nabla_\alpha} \def\da{{\dot\alpha}} \def\b{\beta \d G^{\mu \nu}
+ \nabla^{\mu} \nabla_\r \d G^{\nu\r} + \nabla^{\nu} \nabla_\r \d G^{\mu\r}
- \nabla^\mu\nabla^\nu \d G
\label{gmunu-lin}
\end{align}
for a metric fluctuation $\d G^{\mu\nu} = \b^2 \tilde h^{\mu\nu}$ with
\begin{align}
\tilde h^{\mu\nu} &= h^{\mu\nu} - \frac 12 \eta^{\mu\nu} h,
\qquad \tilde h = - h
\end{align}
around the background
$\overline G^{\mu\nu} = \b \eta^{\mu\nu}$.
For simplicity, we will neglect contributions of the order of the cosmic background
curvature. Then we can replace $\nabla$ by $\partial$ in Cartesian coordinates, and
\begin{align}
2R_{(\rm lin)}^{\mu\nu}[G] \
&\stackrel{\eta\to\infty}{\approx} \
\b^2\Big(-\partial^\alpha} \def\da{{\dot\alpha}} \def\b{\beta\partial_\alpha} \def\da{{\dot\alpha}} \def\b{\beta \tilde h^{\mu\nu}
+ \partial^\mu\partial_\r \tilde h^{\r\nu}
+ \partial^\nu\partial_\r \tilde h^{\r\mu}
- \partial^\mu\partial^\nu \tilde h\Big) \nn\\
& \ \ = \quad \b^2\Big(-\partial^\alpha} \def\da{{\dot\alpha}} \def\b{\beta\partial_\alpha} \def\da{{\dot\alpha}} \def\b{\beta (h^{\mu\nu} -\frac 12 \eta^{\mu\nu} h)
+ \partial^\mu\partial_\r h^{\r\nu}
+ \partial^\nu\partial_\r h^{\r\mu} \Big)
\label{Ricci-lin-approx-1}
\end{align}
neglecting the $\partial\b$ terms at late times $\eta\to\infty$, because \eqref{x-t-beta-relations}
\begin{align}
\b^{-1}\partial_\mu\b
= \frac{\b^2}{R} G_{\mu\nu } \frac{x^\nu}{x_4} \ = O(\b^2) .
\end{align}
Now we can use the intertwiner relation (6.25) in \cite{Sperling:2019xar}
\begin{align}
\Big(\Box + \frac{2}{R^2r^2}\tilde\cI\Big) h^{\mu\nu}[\cA]
&= h_{\mu\nu}[\cD^2\cA]
+ \frac{2}{R^2}\Big(3 h^{\mu\nu}[\cA] - \eta^{\mu \nu} h[\cA]\Big) \,
\label{Box-h-relation}
\end{align}
and the on-shell relation $(\cD^2-\frac{3}{R^2})\cA = 0$.
We should also drop the contribution from $\tilde\cI$ in the same approximation, because
\begin{align}
\Box \phi &\sim -\sinh^2(\eta)\partial^\alpha} \def\da{{\dot\alpha}} \def\b{\beta\partial_\alpha} \def\da{{\dot\alpha}} \def\b{\beta \phi \ \gg \ \frac 1{R^2 r^2}\tilde\cI(h^{\mu\nu})
\ \sim \frac{x}{R^2}\partial h^{\mu\nu}
\label{M-estimate}
\end{align}
for $\partial \gg \frac 1{x_4}$,
using \eqref{deldel-Box-relation} and $\{\theta^{\mu\alpha} \def\da{{\dot\alpha}} \def\b{\beta},\phi\} = r^2(x^\mu\partial^\nu - x^\nu \partial^\mu)\phi$ \cite{Sperling:2019xar}.
Therefore \eqref{Box-h-relation} reduces on-shell to
\begin{align}
\partial^\alpha} \def\da{{\dot\alpha}} \def\b{\beta\partial_\alpha} \def\da{{\dot\alpha}} \def\b{\beta h^{\mu\nu}
&\approx - \frac{1}{x_4^2}\left(9 h^{\mu\nu} - 2 \eta^{\mu \nu} h\right)
\end{align}
which is negligible at late times compared to the terms involving second derivatives $\partial\del h^{\mu\nu}$
in \refeq{Ricci-lin-approx-1},
and similarly for the trace.
This means that
the linearized Ricci tensor reduces on-shell to
\begin{align}
2R_{(\rm lin)}^{\mu\nu}[G^{\alpha} \def\da{{\dot\alpha}} \def\b{\beta\b}] \
&= \ \b^2\Big( \partial^\mu\partial_\r h^{\r\nu}
+ \partial^\nu\partial_\r h^{\r\mu} \ + O(\frac{\partial h^{\mu\nu}}{x_4}) \Big)
\label{Ricci-tensor-onshell}
\end{align}
on scales much shorter than the cosmic curvature scale, or
for late times i.e. large $\eta$.
\subsection{Pure gauge modes}
Now consider the metric fluctuation corresponding to the pure gauge fields $\cA^{(g)}[\phi]$, where
$\phi = \phi^{(1)}$ is a spin 1 field. This has the form (cf. \cite{Sperling:2019xar})
\begin{subequations}
\label{eq:prop_pure_gauge}
\begin{align}
h^{\mu\nu}_{(g)}[\phi] &\coloneqq
h^{\mu\nu}[\cA^{(g)}]
= -\{t^\mu,\cA^{(-)\nu}[\phi]\} + (\mu \leftrightarrow \nu)
+ \frac{1}{3} h^{(g)} \eta^{\mu\nu} \,,
\label{pure-gauge-metric} \\
h_{(g)}[\phi] &\coloneqq \eta_{\mu\nu} h_{(g)}^{\mu\nu}[\phi]
= \frac 6R D^-\phi\,
= 6 \{t_\mu,\cA^{(-)\mu}[\phi]\} \, .
\label{pure-gauge-H}
\end{align}
\end{subequations}
It is not hard to show the following formulas
\begin{align}
\{t_\mu, h^{\mu\nu}_{(g)}[\phi] \}
\ &= \ -\{\Box \phi,x^\nu\}_- - \frac 2R\, D^-\{t^\nu,\phi\} \,,
\label{t-h-g-relation} \\
x_\nu x_\mu h^{\mu\nu}_{(g)}[\phi]
\ &= \ 2R \sinh^2(\eta) D^-\t\phi \
\label{xxh-puregauge-id}
\end{align}
using \eqref{D-properties} cf. \cite{Sperling:2019xar},
and in particular
\begin{align}
\{t_\mu, h^{\mu\nu}_{(g)}[\phi^{(1,0)}] \}
\ &= - \frac 2{R^2}\, \cA^{(-)\nu}[\phi] \qquad \mbox{for}\ \ \Box\phi^{(1,0)} = 0 \ \nn\\
x_\mu h^{\mu\nu}_{(g)}[\phi^{(1,0)}]
&= - \sinh(\eta)(\t-1)\cA^{(-)\nu}[\phi] \ .
\label{div-puregauge-spin1}
\end{align}
Taking into account the conformal factor \eqref{eq:def_phys_graviton},
the pure gauge contribution to the effective metric is
\begin{align}
\d G^{\mu\nu}_{(g)} &= \b^2\big(h^{\mu\nu} -\frac{1}{2} \eta^{\mu\nu}\,h\big) \nn\\
&= \b^2\Big(-\{t^\mu,\cA^{\nu}\} -\{t^\nu,\cA^{\mu}\}
- \eta^{\mu\nu}\,\{t_\alpha} \def\da{{\dot\alpha}} \def\b{\beta,\cA^\alpha} \def\da{{\dot\alpha}} \def\b{\beta \} \Big) \nn\\
&= -\partial^\mu\cA^{\nu} -\partial^\nu\cA^{\mu} - G^{\mu\nu}\,(\partial_\alpha} \def\da{{\dot\alpha}} \def\b{\beta\cA^\alpha} \def\da{{\dot\alpha}} \def\b{\beta)
\label{eff-graviton-gauge}
\end{align}
where $\cA^{\alpha} \def\da{{\dot\alpha}} \def\b{\beta} = \cA^{(-)\alpha} \def\da{{\dot\alpha}} \def\b{\beta}[\phi]$ and $\partial^\mu = G^{\mu\nu}\partial_\nu$.
This formula is valid in Cartesian coordinates, and we must be
very careful with using upper indices, e.g.
$\{t^\mu,\phi\} = \sinh(\eta)\eta^{\mu\nu}\partial_\nu \phi
= \sinh^{2}(\eta) G^{\mu\nu}\partial_\nu \phi$.
\paragraph{Relation with diffeomorphisms.}
We can rewrite these pure gauge modes as diffeomorphism modes by
comparing with \eqref{puregauge-grav-covar} on the present FLRW background. This gives
\begin{align}
\d G^{\mu\nu}_{(g)}
&= \partial^\mu\xi^{\nu} + \partial^\nu\xi^{\mu} - \frac{1}{x_4^2} G^{\mu\nu}\, x \cdot \xi \nn\\
&= \nabla^\mu\xi^\nu + \nabla^\nu \xi^\mu, \qquad \xi^\mu = - \cA^\mu
\label{puregauge-diffeo-rel}
\end{align}
using
\begin{align}
x_\alpha} \def\da{{\dot\alpha}} \def\b{\beta \cA^\alpha} \def\da{{\dot\alpha}} \def\b{\beta &= \eta_{\alpha} \def\da{{\dot\alpha}} \def\b{\beta\b} x^\alpha} \def\da{{\dot\alpha}} \def\b{\beta\{x^\b,\phi\}_- = -x^4 D^- \phi \nn\\
\sinh \partial_\alpha} \def\da{{\dot\alpha}} \def\b{\beta \cA^\alpha} \def\da{{\dot\alpha}} \def\b{\beta &= \{t_\alpha} \def\da{{\dot\alpha}} \def\b{\beta,\{x^\alpha} \def\da{{\dot\alpha}} \def\b{\beta,\phi\}_-\} = \frac{1}{R} D^- \phi \nn\\
\partial_\alpha} \def\da{{\dot\alpha}} \def\b{\beta \cA^\alpha} \def\da{{\dot\alpha}} \def\b{\beta &= - \frac{1}{x_4^2} x \cdot \cA \
\end{align}
where $\cA^{\alpha} \def\da{{\dot\alpha}} \def\b{\beta} = \cA^{\alpha} \def\da{{\dot\alpha}} \def\b{\beta(-)}[\phi]$, using the notation $x\cdot\cA \equiv \eta_{\alpha} \def\da{{\dot\alpha}} \def\b{\beta\b}x^\alpha} \def\da{{\dot\alpha}} \def\b{\beta\cA^\b$.
Hence the pure gauge metric modes in the present framework can be identified with
diffeomorphisms generated by $\xi = -\cA$. This also provides a non-trivial consistency check for the
correct identification of $G$.
It is easy to check using \eqref{div-A-full} that these diffeomorphisms satisfy the constraint
\begin{align}
\nabla_\alpha} \def\da{{\dot\alpha}} \def\b{\beta \xi^\alpha} \def\da{{\dot\alpha}} \def\b{\beta = -\frac{3}{x_4^2}\, x \cdot \xi
\label{gauge-diffeo-constraint-1}
\end{align}
or equivalently
\begin{align}
\boxed{ \ \nabla_\alpha} \def\da{{\dot\alpha}} \def\b{\beta (\b^{3} \xi^\alpha} \def\da{{\dot\alpha}} \def\b{\beta) = 0 \ . \ }
\label{gauge-diffeo-constraint}
\end{align}
Hence they are essentially volume-preserving diffeos up to the factor $\b^{3}$,
leaving only 3 rather than 4 diffeomorphism d.o.f., unlike in GR.
This reflects the presence of a dynamical scalar metric degree of freedom, which we will study in detail
below.
\subsection{Generalities for the $\cA^{(-)}$ metric modes}
Among the $\cA^{(-)}[\phi^{(s)}]$ modes, only the ones with spin $s=2$ can
contribute to the metric, and these are in fact physical degrees of freedom as shown in
\eqref{on-shell-B2pm}. The corresponding linearized metric fluctuation is \cite{Sperling:2019xar}
\begin{subequations}
\label{eq:prop_A2-mode}
\begin{align}
h_{(-)}^{\mu\nu}[\phi] &\coloneqq
h^{\mu\nu}[\cA^{(-)}[\phi]]
= -2 \{x^\mu,\{x^\nu,\phi\}_-\}_- = -2 \{x^\nu,\{x^\mu,\phi\}_-\}_- \nn\\
h_{(-)}[\phi] &\coloneqq
\eta_{\mu\nu} h_{(-)}^{\mu\nu}
= -2\{x^\mu,\{x_\mu,\phi\}_-\}_- \
= 2 D_- D_-\phi \,
\label{Hmunu-tr}
\end{align}
\end{subequations}
for $\phi = \phi^{(2)}$.
It is not hard to derive the following formulas
\begin{subequations}
\label{eq:prop_A2-prop}
\begin{align}
\{t_\mu, h_{(-)}^{\mu\nu}\}
&= -\frac{2}{R} \{x^\nu, D^-\phi\}_-
\label{Hmunu-div} \\
\{t_\mu,\{t^\alpha} \def\da{{\dot\alpha}} \def\b{\beta, h^{(-)}_{\alpha} \def\da{{\dot\alpha}} \def\b{\beta\nu}\}\}
+ (\mu\leftrightarrow \nu)
&=\frac{2}{R^2} \Big(h^{(g)}_{\mu\nu} - \frac 13\eta_{\mu\nu}h^{(g)} \Big)[D^-\phi] \\
x_\mu h^{\mu\nu}_{(-)}
&= 2 x_4 \{x^\nu,D^-\phi\}_-
\label{x-h-contract-id}
\end{align}
\end{subequations}
since $\{t^\nu,\phi^{(2)}\}_0 = 0$. Comparing \eqref{x-h-contract-id} and \eqref{Hmunu-div}, we obtain
\begin{align}
\partial_\mu h^{\mu\nu}_{(-)} &= - \frac 1{x_4^2} x_\mu h^{\mu\nu}_{(-)}
\label{del-x-hminus-rel}
\end{align}
or equivalently
\begin{align}
\boxed{ \
\partial_\mu(\b h^{\mu\nu}_{(-)}) = 0 \ .
\ }
\label{A-h-nice-id}
\end{align}
This looks like a gauge-fixing condition. We can write it in covariant form
using the explicit form of the Christoffel symbols \eqref{christoffels}, \eqref{christoffels-2}, which gives
\begin{align}
\nabla_\mu h^{\mu\nu} &= \partial_\mu h^{\mu\nu} - \frac 3{x_4^2} x_\mu h^{\mu\nu} + \frac{1}{2x_4^2}x^\nu h \ .
\end{align}
Since the $\cA^{(-)}[\phi^{(2,0)}]$ and the $\cA^{(-)}[\phi^{(2,1)}]$ modes satisfy $h=0$, this can be written
using \eqref{del-x-hminus-rel} as
\begin{align}
\boxed{
\nabla_\mu (\b^{4} h^{\mu\nu}_{(-)}[\phi^{(2,j)}]) = 0 \qquad\mbox{for} \ \ j=0,1
\label{h-covar-cons-21}
}
\end{align}
Since this condition \eqref{del-x-hminus-rel} is not quite the same as
\eqref{div-puregauge-spin1} for the on-shell pure gauge gravitons,
it follows that the extra 2 on-shell metric fluctuations
$h^{\mu\nu}[\cA^{(-)}[\phi^{(2,1)}]]$ are in fact physical.
\paragraph{Linearized Ricci tensor.}
Using the constraint \eqref{A-h-nice-id} for
$h^{\mu\nu}[\cA^{(-)}[\phi^{(2)}]]$, it follows from \eqref{Ricci-tensor-onshell} that all these
on-shell (would-be massive) spin 2 modes are Ricci-flat up to cosmic scales,
\begin{align}
2R_{(\rm lin)}^{\mu\nu} \
&= \ 0 + \ O(\frac{\partial G^{\mu\nu}}{x_4}) \ .
\label{Ricci-tensor-onshell-massive}
\end{align}
This seems to suggest that these modes are exactly massless with only
2 physical degrees of freedom, but this is not true,
as pointed out above. The point is that the $h^{\mu\nu}$ contributions from the
would-be helicity 1 and 0 modes are typically dominated by diffeos,
which are trivially flat. However, we will see in the next section that the
linearized Schwarzschild solution which arises from $h^{\mu\nu}[\cA^{(-)}[D^+D\phi]]$
is {\em not} dominated by diffeos, but a genuine
non-trivial Ricci-flat metric.
\subsection{Scalar modes $\cA^{(-)}[D^+D\phi]$ and the
Schwarzschild metric}
\label{sec:A-DD-metric}
Now we work out the explicit metric perturbation arising from the
on-shell $\cA^{(-)}[D^+D\phi]$ mode, which is
part of the would-be massless spin 2 multiplet $\cA^{(-)}[\phi^{(2)}]$.
We will see that this includes a quasi-static
Schwarzschild metric, as well as other solutions which might be related to
dark matter.
We will use the on-shell condition
$\Box \phi = - \frac{2}{R^2}\phi$ \eqref{onshell-0} throughout,
and focus on the late-time limit $\eta\to\infty$.
Starting with the explicit form \eqref{A-DDphi-explicit} for $\cA^{(-)}[D^+D^+\phi]$,
dropping the pure gauge contribution $\{t^\mu,\L\}$
and using the results of section \ref{sec:metric-contrib}, we obtain
\begin{align}
\frac{5}{2r^2} h^{\mu\nu}[\cA^{(2)}[D^+D\phi]]
\ &= \ h^{\mu\nu}\big[ \big(r^2 \b t^\mu
+ r^2 \b x^\mu t^\alpha} \def\da{{\dot\alpha}} \def\b{\beta \partial_\alpha} \def\da{{\dot\alpha}} \def\b{\beta
- \frac 1{3} \theta^{\m\g}\partial_\g (\t+4) \big)(\t+2) \phi\big] \nn\\
%
\!\! &\stackrel{\eta\to\infty}{=} \frac 29 r^2 \Big(\eta^{\mu\nu}(2 + \t)(3 + \t^2+4\t)
+ \frac{\b^2}{R^2} x^\mu x^\nu (\t^2 -1)\nn\\
&\qquad - (x^\nu\partial^\mu + x^\mu\partial^\nu)(\t^2+3\t+2)
- R^2\partial^\nu \partial^\mu (\t+4)\Big)(\t+2)\phi \ .
\end{align}
Therefore
\begin{align}
h^{\mu\nu}
&= \frac{4r^4}{45}
\Big((2 + \t)(\t + 3)\eta^{\mu\nu}
+ \frac{\b^2}{R^2} x^\mu x^\nu (\t -1)
- (x^\nu\partial^\mu + x^\mu\partial^\nu) (\t+2) \Big)(\t+1)(\t+2)\phi\nn\\
&\quad -\frac{4r^4}{45} R^2\partial^\nu \partial^\mu (\t+4)(\t+2)\phi
\end{align}
with trace
\begin{align}
h
&\stackrel{\eta\to\infty}{=} \frac{4r^4}{45} (\t+1) (2\t+5)(\t+5)(\t+2)\phi \ .
\end{align}
Then the trace-reversed metric fluctuation $\tilde h^{\mu\nu}$ is
\begin{align}
\tilde h^{\mu\nu} &= h^{\mu\nu} - \frac 12 h \eta^{\mu\nu} \nn\\
&= \frac{4r^4}{45}
\Big(-\frac 12 (5\t + 13) \eta^{\mu\nu}
+ \frac{\b^2}{R^2} x^\mu x^\nu (\t -1)
- (x^\nu\partial^\mu + x^\mu\partial^\nu) (\t+2) \Big) (\t+1)(\t+2)\phi \nn\\
&\quad -\frac{4r^4}{45} R^2\partial^\nu \partial^\mu (\t+4)(\t+2)\phi \ .
\label{tilde-h-nogauge}
\end{align}
Observe that for $\t\neq -2$ the term $(x^\nu\partial^\mu + x^\mu\partial^\nu)\phi$ is dominant at late times, since $x^0\sim R\cosh(\eta)$.
However this is essentially a large diffeomorphism contribution, which
can be removed from the effective metric fluctuation
using \eqref{FRW-puregauge}, with the result
\begin{align}
\tilde h^{\mu\nu}
&\sim \frac{4r^4}{45}
\Big( \frac 12(\t - 1)\eta^{\mu\nu}
+ 3\frac{\b^2}{R^2} x^\mu x^\nu (\t +1)\Big)(\t+1)(\t+2)\phi
\label{eff-metric-after-diffeo}
\end{align}
for large $\eta$. Hence
\begin{align}
\tilde h_{\mu\nu}\, \mathrm{d} x^\mu \mathrm{d} x^\nu
\ \ &= \ \frac{2r^4R^2}{45}
\sinh^2(\eta) \big(d\eta^2(5\t +7) + d\Sigma^2(\t - 1)\big)(\t+1)(\t+2)\phi \nn\\
&\stackrel{\t\to-2}{=} \ \frac{2r^4R^2}{15}
\sinh^2(\eta)(\t+2)\phi\, \big(d\eta^2 + d\Sigma^2\big) \nn\\
&= - 4 \phi' (d t^2 + a(t)^2 d\Sigma^2)
\end{align}
using \eqref{metric-sphericalcoords-trafo}
where $\tilde h_{\mu\nu} = \eta_{\mu\mu'}\eta_{\nu\nu'} \tilde h^{\mu'\nu'}$,
and using the explicit form \eqref{eff-metric-FRW} of the scale parameter $a(t)$ for large $\eta$.
Here we define
\begin{align}
\phi' := -\frac{r^4}{30} \b (\t+2)\phi \
\label{phi-prime-new}
\end{align}
as in \eqref{phi-prime} (up to rescaling), which allows to take $\t\to -2$.
We will see that
this reduces to the linearized Schwarzschild metric for $\t\to -2$, while for $\t\neq -2$ it
is a distinct metric which is not Ricci-flat.
However for $\t \neq-2$ the diffeo contribution in \eqref{tilde-h-nogauge}
grows very large at late times, which may invalidate the
linearized approximation as discussed below.
Therefore we focus on $\t\approx -2$, which is the most interesting and most reliable case.
Then the full perturbed metric can be written
in the form \eqref{perturbed-metric-cosm}
\begin{align}
\label{perturbed-metric-cosm-expl}
\boxed{\
\begin{aligned}
ds^2 = (G_{\mu\nu} - \d G_{\mu\nu}) \mathrm{d} x^\mu \mathrm{d} x^\nu
&= (\sinh(\eta)\eta_{\mu\nu}
- \tilde h_{\mu\nu})\, \mathrm{d} x^\mu \mathrm{d} x^\nu \\
&= - \mathrm{d} t^2 + a(t)^2\mathrm{d}\Sigma^2
\ + 4 \phi' (d t^2 + a(t)^2 d\Sigma^2) \ .
\end{aligned}
\ }
\end{align}
The on-shell condition reduces to
$\Delta^{(3)}\phi = 0$ for $\t=-2$ due to \eqref{Box-Laplace-tau},
and in the spherically symmetric case the Newton potential
on a $k=-1$ geometry is recovered \eqref{harmonic-soln}, with
\begin{align}
\phi \ &= \ \frac{e^{- \chi}}{\sinh(\chi)} \frac{1}{\cosh^2(\eta)}
\ \sim \ \frac{1}{\r} e^{-\chi -2\eta} \ , \qquad \r = \sinh(\chi) \ .
\label{phi-tis-2}
\end{align}
Strictly speaking we should use $\phi'$ rather than $\phi$ in the $(\t+2)\phi = 0$ case. Then
the quasi-static condition becomes $(\t+3+\b^2)\phi' = 0$, and the on-shell condition \eqref{eom-phi'} is
$(\Delta^{(3)} - 4\b^4)\phi' = 0$. However
the $\b^2$ contributions can be dropped in the large $\eta$ limit giving again $\Delta^{(3)}\phi' = 0$, so that
\begin{align}
\phi' &\sim \frac{1}{\r} e^{-\chi-3\eta} \ \sim \frac{e^{-\chi}}{\r} \frac{1}{a(t)^2}
\label{phiprime-solution}
\end{align}
for large $\eta$,
using \eqref{tau-eta} and recalling $a(t) \sim e^{-\frac 32 \eta}$ \eqref{a-eta}.
This metric is very close to the Vittie solution \cite{mcvittie1933mass} for the Schwarzschild metric
for a point mass $M$ in a
FRW spacetime, whose linearization for $k=-1$ is given by
\begin{align}
ds^2
&= -dt^2 + a(t)^2 d\Sigma^2 \ + \ 4\mu(dt^2 + a(t)^2 d\Sigma^2) \ + O(\mu^2) \ .
\label{vittie}
\end{align}
Here
\begin{align}
\mu = \mu(t,\chi) &= \frac{M}{2 \r} \frac{1}{a(t)}
\label{mu-vittie}
\end{align}
is the mass parameter, which is not constant but decays during the cosmic expansion;
this is as it should be, because local gravitational systems do not participate in the
expansion of the universe.
Comparing with \eqref{phiprime-solution} we have
\begin{align}
\phi' &\sim \ \mu(t,\chi)\, \frac{e^{-\chi}}{a(t)} \ .
\end{align}
Since $\mu$ \eqref{mu-vittie}
looks like a constant mass for a comoving observer \cite{mcvittie1933mass},
the effective mass parameter in our solution effectively decreases like
$a(t)^{-1}$ during the cosmic evolution.
This might be interpreted in terms of a time-dependent
Newton constant, although this is a bit premature since
we have not properly investigated the
coupling to matter, and quantum effects may modify the result.
Nevertheless, the result is suggestive.
Also, while both metrics have the characteristic $\frac 1\r$ dependence of the
Newton potential, our solution has an extra $e^{-\chi}$ factor,
which reduces its range at space-like curvature scales.
Both effects are irrelevant at solar system scales,
but they will be important for cosmological considerations, reducing the gravitational attraction at long scales.
For completeness, we also recall the
linearized Schwarzschild solution in isotropic coordinates
\begin{align}
ds^2 &= -\frac{(1- \frac M {2r})^2}{(1+\frac M {2r})^2} \ d t^2
+ (1+ \frac M {2r})^4 (dx^2+dx^2+dz^2) \nn\\
&= \eta_{\mu\nu} dx^\mu dx^\nu + \frac {2M} {r} d x_0^2
+ \frac {2M} {r} (dx^2+dx^2+dz^2) \ + O(\frac{1}{r^2}) \ .
\end{align}
\eqref{vittie} reduces to this metric for a local comoving observer
for large $a(t)$,
while we obtained an extra factor $\frac 1{a(t)}$ in the effective mass.
Let us discuss the consistency and significance of these results.
The most striking point is that even though the metric \eqref{eff-metric-after-diffeo}
is Ricci-flat for $\t=-2$, for other values of $\t$ it is not.
This seems to contradict the
general result \eqref{Ricci-tensor-onshell-massive} for the linearized Ricci tensor, which should always vanish at scales shorter than the background curvature i.e. for large $\eta$.
The resolution of this puzzle lies in the diffeo contributions
$(x\partial+x\partial)$ in \eqref{tilde-h-nogauge}, which were eliminated by a change of
coordinates in \eqref{eff-metric-after-diffeo}.
The point is that for $\t\neq -2$, this term becomes very large for large $\eta$
as $x^0\sim R\cosh(\eta)$, and completely dominates
the other, non-trivial contributions to $\tilde h^{\mu\nu}$.
But the Ricci-tensor for a diffeo contribution vanishes trivially,
leading to \eqref{Ricci-tensor-onshell-massive}.
In other words, if the first terms in \eqref{tilde-h-nogauge} are non-trivial, they are
dominated by the $(x\partial+x\partial)$ term, so that for large $\eta$
the linearized approximation becomes invalid\footnote{Recall that \eqref{A-DDphi-explicit}
also contains large pure gauge contributions $\{t^\mu,\L\}$ which were
discarded. This was justified, because these are exact gauge symmetries of the
matrix model. In contrast, the above $(x\partial+x\partial)$ terms are diffeos
which are presumably not part of the 3 diffeo modes in the present
framework, due to the constraint \eqref{gauge-diffeo-constraint}. Therefore this is an approximation
whose validity needs to be checked carefully.}, unless $\t\approx -2$.
Then the effective metric fluctuation
\eqref{gravitons-H1} must be completed by the non-linear contribution,
which will be discussed briefly below.
In contrast for $\t\approx -2$,
the pure gauge contribution in \eqref{tilde-h-nogauge} vanishes,
hence our Schwarzschild-like solution is fully justified.
A similar issue may arise for the would-be helicity 1 modes
$\cA^{(-)}[\phi^{(2,1)}]$, but not for
the helicity-2 gravitons $\cA^{(-)}[\phi^{(2,0)}]$, because there are no
helicity 2 pure gauge contributions. Therefore these are indeed
Ricci-flat and non-trivial, as stated in \cite{Sperling:2019xar}.
One might worry that the restriction to $\t=-2$ of the Schwarzschild solution
is too rigid for real physical systems such as the solar system.
However, systems with non-uniform motion lead to dynamical metric perturbations
corresponding to physical spin 2 gravitons, which are realized here by the
$\cA^{(-)}[\phi^{(2,0)}]$ modes. Therefore there should not be an obstacle
to obtain dynamical Ricci-flat metric perturbations
as a combination of $\cA^{(-)}[D^+D\phi^{(0)}]$ and $\cA^{(-)}[\phi^{(2,0)}]$ modes.
\paragraph{Interpretation and physical significance.}
We found that the scalar on-shell modes provide a Ricci-flat metric perturbation only for the
specific quasi-static time-dependence $\t\approx -2$.
Indeed it should be expected that a dynamical scalar metric mode, which does not
exist in GR, is not Ricci-flat in general.
From a GR point of view, such non-Ricci-flat perturbations would be
interpreted as dark matter.
Nevertheless, there better be a reason why in typical situations
such as the solar system, such non-Ricci-flat
deformations are suppressed.
Strictly speaking
this question can only be settled once the coupling of matter to the
various modes is properly taken into account.
Quantum effects may also be important here, because they typically
lead to an induced Einstein-Hilbert term\footnote{As explained in \cite{Sperling:2019xar}, there is
a priori no matrix analog of the cosmological constant. However, it remains
to be understood whether or not an analogous term
is induced through quantum effects, and what its consequences would be.}
\cite{Sperling:2019xar},
which would distinguish Ricci-flat and -non-flat solutions.
However heuristically, we can give a classical mechanism which achieves that effect at the non-linear level
as follows.
For scalar modes with large diffeo contribution in \eqref{tilde-h-nogauge}, the
linearized metric \eqref{gravitons-H1} must be replaced by the full non-linear expression \eqref{gamma-nonlinear}.
Then the large would-be diffeo contribution no longer decouples from a conserved $T^{\mu\nu}$,
but strongly couples to matter\footnote{These are not expected to be exact gauge symmetries of the
model, since there are only 3 pure gauge d.o.f.}. But if this large contribution governs the dynamics,
the first two terms in \eqref{tilde-h-nogauge} are effectively suppressed, and this suppression
is stronger for shorter wavelengths due to the derivatives.
On the other hand $\cA$ becomes small sufficiently far from matter, so that the linearized
treatment will suffice. Then the large contribution is indeed a flat diffeo, while
the sub-leading non-Ricci-flat contribution in \eqref{eff-metric-after-diffeo}
is strongly suppressed.
This does not apply to the Ricci-flat $\t=-2$ contribution since the diffeo vanishes,
and we conclude that the non-Ricci-flat contributions are strongly suppressed, as desired.
For very long wavelengths, this suppression mechanism becomes weak,
so that some non-Ricci-flat perturbations with very long - possibly galactic - wavelengths are
expected. This would then be interpreted as dark matter from a GR point of view.
Moreover, the suppression mechanism is weaker in the earlier Universe, which
might explain why dark matter seems to be more abundant in older galaxies
\cite{genzel2017strongly}.
This non-linear effect is somewhat reminiscent of the vDVZ discontinuity in massive gravity
and its resolution through the
Vainshtein mechanism \cite{Vainshtein:1972sx}.
Indeed, the present modes arise precisely from would-be massive
spin 2 modes, albeit the details are different.
A time dependence $\sim a(t)^{-1}$ of the Newton constant seems to be somewhat large
in view of recent estimates \cite{Mould:2014iga,Zhao:2018gwk}.
However, we have not properly taken into account the coupling to matter, and the
underlying FLRW cosmology is non-standard.
Including an induced Einstein-Hilbert action
in the quantum effective action could also affect the result.
These issues need to be understood before solid predictions can be made.
Finally, we note that
the case $\t=-1$ is also special.
After suitable rescaling this leads to $\tilde h^{\mu\nu} \sim \eta^{\mu\nu}$,
hence to a modification of the cosmological evolution $a(t)$ for
$\phi \sim e^{-\eta}$.
A similar modification may arise from $f \neq 0$ in \eqref{gaugefix-t-relation}.
This shows that modifications of the cosmic evolution are possible,
but again this needs to be studied in more detail.
\paragraph{Pure gauge contribution and checks.}
An instructive check can be obtained by computing
the metric fluctuation arising from the pure gauge term \eqref{A-puregauge-L}:
\begin{align}
h^{\mu\nu} &= \frac 25 r^2 h^{\mu\nu}[\theta^{\mu\alpha} \def\da{{\dot\alpha}} \def\b{\beta} \partial_\alpha} \def\da{{\dot\alpha}} \def\b{\beta(\t+3)\phi] \
-\frac 25 r^4 R^2 h^{\mu\nu}[ \sinh(\eta) t^\alpha} \def\da{{\dot\alpha}} \def\b{\beta \partial_\alpha} \def\da{{\dot\alpha}} \def\b{\beta\partial_\mu(\t+3)\phi] \nn\\
&\stackrel{\eta\to\infty}{=}
\frac 4{15} r^4 \Big(-\t\eta^{\mu\nu}
+ \frac{1}{R^2} \b^2 x^\mu x^\nu \
- R^2\sinh^2(\eta)\partial_\mu\partial_\nu
+ (x^\nu \partial^\mu + x^\mu \partial^\nu) \Big)(\t+3) (\t + 2)\phi \nn
\end{align}
using \eqref{h-tdeldel-phi},
with trace
\begin{align}
h
&= -\frac {12}{15} (\t+1)(\t + 2)(\t+3) r^4 \phi
\end{align}
consistent with \eqref{pure-gauge-H}.
Then the trace-reversed pure gauge metric fluctuation is
\begin{align}
\tilde h^{\mu\nu} &= h^{\mu\nu} - \frac 12 h \eta^{\mu\nu} \nn\\
&= \frac{4r^4}{15} \Big(
\frac 12 (\t+3) \eta^{\mu\nu}
+ \frac{1}{R^2} \b^2 x^\mu x^\nu \
- R^2\sinh^2(\eta)\partial_\mu\partial_\nu
+ (x^\nu \partial^\mu + x^\mu \partial^\nu) \Big)(\t+3) (\t + 2)\phi
\label{pure-gauge-contrib-DD}
\end{align}
and one can check using the results in section \ref{sec:diffeo-FRW} that
this is indeed a diffeomorphism on the FLRW background.
Various other non-trivial checks were performed for the combined and the
pure gauge contributions, comparing the trace $h$ and the time component
$x_\mu x_\nu h^{\mu\nu}$ with the general formulas
\eqref{xxh-puregauge-id} and \eqref{Hmunu-tr} and \eqref{h-phi-relation}. All tests
work out, so that we can be very confident that the above expressions for
the metric fluctuations are correct.
\subsection{Unphysical scalar $\cA^{(+)}$ modes.}
Among the $\cA^{(+)}[\phi^{(s)}]$ modes, only the
scalar mode $\cA^{(+)}[\phi^{(0)}]$ contributes to the linearized metric.
Even though it is unphysical because it does not satisfy the gauge-fixing
constraint, we give its metric contribution
for completeness:
\begin{subequations}
\label{eq:prop_A2+mode}
\begin{align}
h_{(+)}^{\mu\nu}[\phi^{(0)}] &= - \{x^\mu,\{x^\nu,\phi^{(0)}\}_+\}_- +
(\mu\leftrightarrow\nu) \ \nn\\
&= -2[\theta^{\mu\alpha} \def\da{{\dot\alpha}} \def\b{\beta}\theta^{\nu\b}]_0\partial_\alpha} \def\da{{\dot\alpha}} \def\b{\beta\partial_\b\phi -
\left(\{x^\mu,\theta^{\nu\b}\}\partial_\b + \{x^\nu,\theta^{\mu\b}\}\partial_\b\right)
\phi^{(0)} \nn\\
&= -\frac{2r^2R^2}3\left(P^{\mu\nu}P^{\alpha} \def\da{{\dot\alpha}} \def\b{\beta\b} -
P^{\mu\b}P^{\nu\alpha} \def\da{{\dot\alpha}} \def\b{\beta}\right)\partial_\alpha} \def\da{{\dot\alpha}} \def\b{\beta\partial_\b\phi^{(0)}
- (\{x^\mu,\theta^{\nu\b}\}\partial_\b + \{x^\nu,\theta^{\mu\b}\}\partial_\b)
\phi^{(0)}\nn\\
&= \frac{2r^2R^2}3\left(\partial^\mu \partial^\nu
- (\eta^{\mu\nu} + R^{-2}x^\mu x^\nu)\b^2(-\Box+\frac{1}{R^2}\t)\right)\phi^{(0)} \nn\\
&\quad
- \frac 23 r^2 \eta^{\mu\nu}(\t + 2)\t\phi^{(0)}
+ \frac 13 r^2(x^\nu \partial^\mu + x^\mu \partial^\nu)(1+ 2\t) \phi^{(0)} \, .
\label{th-rel-A2+}
\end{align}
\end{subequations}
This is part of the $D\tilde\cA^{(\t)}$ \eqref{D-tilde-A-t-1} mode.
As a check, we recover \eqref{dAlembertian-x} by taking the trace.
To summarize, the $\cA^{(-)}[\phi^{(2)}],\cA^{(g)}[\phi^{(1)}], \cA^{(+)}[\phi^{(0)}]$ and $D\tilde\cA_\mu^{(\t)}[\phi]$
modes provide all $5+3+1+1 = 10$ off-shell d.o.f. of the most general metric fluctuation.
They lead to 5 physical on-shell modes
comprising $2$ graviton modes from $\cA^{(-)}[\phi^{(2,0)}]$,
one scalar mode $\cA^{(-)}[D^+D\phi^{(0)}]$, and
presumably 2 helicity 1 modes $\cA^{(-)}[\phi^{(2,1)}]$.
\section{Summary and conclusions}
We have studied in detail the scalar
fluctuations of the FLRW quantum space-time solution $\cM^{3,1}$
of Yang-Mills matrix models, based on the general results in \cite{Sperling:2019xar}.
In particular, we recovered the quasi-static linearized Schwarzschild metric
as a solution, which arises from
the scalar sector of the physical would-be massive spin 2 modes.
Quasi-static indicates that the corresponding effective mass is found to decrease slowly
during the cosmic evolution.
It is very remarkable
that the linearized Schwarzschild solution can be obtained within the framework of
Yang-Mills matrix models, as we have shown.
Along with the propagating spin 2 graviton modes found in \cite{Sperling:2019xar},
this strongly supports the claim that 3+1-dimensional gravity
can emerge from the matrix model framework without compactification,
in particular for the IKKT or IIB model \cite{Ishibashi:1996xs}.
The mechanism is very simple in the spirit of
noncommutative but almost-local field theory,
by considering fluctuations around a background solution.
The present result is tied to the specific structure of the background
solution, which is a twisted $S^2$ bundle over space-time, leading to
a tower of higher-spin modes.
It does not seem to work e.g. on simpler Moyal-Weyl type backgrounds, where
the linearized modes only lead to restricted metric fluctuations,
which includes some Ricci-flat metrics \cite{Rivelles:2002ez,Yang:2006dk,Steinacker:2010rh} but not enough.
An important issue is (local) Lorentz invariance, which is only partially manifest in the present framework.
This leads to a different organization of modes in terms of the
space-like $SO(3,1)$ isometry group. For example, the
5 modes of a generic spin 2 irrep decompose into $2+2+1$ modes
of $\phi^{(2,i)}$ as in \eqref{C-sk-def}. This is best understood in space-like gauge,
somewhat reminiscent of helicity modes.
This structure is indicated by the name ``would-be massive'' modes.
Nevertheless, Lorentz-invariance appears to be largely
respected, presumably due to the large underlying gauge invariance.
In particular, the propagation of all physical modes is governed by the same effective metric.
Aside from the higher spin modes, the present model includes extra on-shell metric modes
beyond those of GR. This is not surprising, since the gauge invariance of the metric sector
is reduced to 3 rather than 4 diffeomorphism d.o.f.
We studied in some detail the extra scalar modes, which arise from the
would-be helicity zero sector of $\cA^{(-)}[\phi^{(2)}]$.
Those are in general not Ricci-flat, but their proper
treatment is quite subtle and require non-linear considerations,
except (!) for the quasi-static Schwarzschild case.
We propose a heuristic argument why the non-Ricci-flat modes should be
suppressed at the non-linear level,
somewhat reminiscent of the Vainshtein mechanism \cite{Vainshtein:1972sx}.
They may however play a role at very long wavelengths, in the guise of dark matter.
Similarly, there are presumably two more physical modes arising from the would-be helicity
1 gravitons, which are not studied here, and may also require
the non-linear theory.
This leads us to the list of open issues and questions which need to be addressed
in future work.
One important step is the inclusion of matter, in order to clarify how matter acts as a source of metric deformations.
This was briefly discussed in \cite{Sperling:2019xar},
but it needs to be studied in detail, and at the non-linear level in order to clarify the above mechanism.
Only then a reliable assessment can be made whether a satisfactory behavior arises at the classical
level, or if quantum effects such as an induced Einstein-Hilbert action are essential.
Another obvious tasks is to extend the present Schwarzschild solution,
and more generally the full higher spin theory,
to the non-linear regime as far as possible. Even though some computations in the
present paper are quite involved, the basic structure of the underlying
$\cA^{(-)}[D^+D\phi^{(0)}]$ solution is very simple and based only on
Lie-algebraic structures. This -- along with black hole
solutions in higher spin theories \cite{Didenko:2009td,Iazeolla:2011cb,Iazeolla:2017vng} -- leads to the hope that an exact
analytic solution can be found, not only at the semi-classical level,
but also at the fully non-commutative level.
These are only some of many open questions which can be studied using the tools provided here and in \cite{Sperling:2019xar}.
\paragraph{Acknowledgements.}
I would like to thank Marcus Sperling for related collaboration, and
T. Damour, Carlo Iazeolla, H. Kawai,
Jan Rosseel and A. Tsuchiya for valuable discussions,
notably during workshops at the IHES in Bures-sur-Yvette and
the ESI Vienna. The hospitality and support of these institutes is gratefully acknowledged.
This work was supported by the Austrian Science Fund (FWF) grant P32086.
The work also profited from the COST network QSPACE through various meetings.
\section{Appendix}
\subsection{Ladder operators and eigenmodes.}
\label{sec:ladder-ops}
We provide a simpler and more conceptual derivation of the
eigenmodes $\cA^{(\pm)}$ \eqref{D2-A2p-eigenvalues}, \eqref{D2-A2m-eigenvalues} found in \cite{Sperling:2019xar}.
Starting from the observation
$[\Theta^{\mu\nu},X^4] \sim i\{\theta^{\mu\nu},x^4\} = 0$ we obtain
\begin{align}
\tilde\cI(D^\pm(\cA_\mu)\} = D^\pm( \tilde\cI(\cA_\mu) \ .
\end{align}
Together with the relations \cite{Sperling:2019xar}
\begin{align}
\Box D^+\phi^{(s)} &= D^+\left(\Box+\frac{2s+2}{R^2}\right)\phi^{(s)} \,,\\
\Box D^-\phi^{(s)} &= D^-\left(\Box-\frac{2s}{R^2}\right)\phi^{(s)} \,
\label{Box-x4-relations}
\end{align}
we obtain
\begin{align}
\cD^2 D^+\cA^{(s)}
&= (\Box + \frac{2}{r^2R^2}\tilde\cI)D^+\cA^{(s)}
= (D^+(\Box + \frac{2s+2}{R^2}) + \frac{2}{r^2R^2}D^+\tilde\cI)\cA^{(s)} \nn\\
&= D^+(\cD^2+\frac{2s+2}{R^2}) \cA^{(s)}, \qquad \cA^{(s)} \in \cC^s
\label{D2-D+-relation}
\end{align}
and similarly for $D^-$.
Therefore $D^\pm$ are intertwiners for $\cD^2$ which rise or lower the eigenvalues.
Now observe
\begin{align}
\cD^2 D^+\cA^{(g)}[\phi^{(s)}] &= D^+(\cD^2+\frac{2s+2}{R^2}) \cA^{(g)}[\phi^{(s)}] = D^+\cA^{(g)}[(\Box +\frac{2s+5}{R^2})\phi^{(s)}] \nn\\
\cD^2 \cA^{(g)}[D^+\phi^{(s)}] &= \cA^{(g)}[ (\Box +\frac{3}{R^2})D^+\phi^{(s)}] = \cA^{(g)}[D^+ (\Box +\frac{2s+5}{R^2})\phi^{(s)}] \ .
\end{align}
But this implies that $\cA^{(+)}[.] = D^+\cA^{(g)}[.] - \cA^{(g)}[D^+[.]]$ has the same intertwiner property,
\begin{align}
\cD^2 \cA^{(+)}[\phi^{(s)}] &=\cA^{(+)}[(\Box +\frac{2s+5}{R^2})\phi^{(s)}]\ ,
\end{align}
and a similar argument based on
$D^-\cA^{(g)}[.] = \cA^{(g)}[D^-[.]] + \cA^{(-)}[.]$ gives
\begin{align}
\cD^2 \cA^{(-)}[\phi^{(s)}] &=\cA^{(-)}[(\Box +\frac{-2s+3}{R^2})\phi^{(s)}]
\end{align}
as desired.
These properties originate from the underlying $\mathfrak{so}(4,2)$ Lie algebra structure,
and they should apply to the fully noncommutative case as well as the
semi-classical Poisson limit.
In particular, the solution $\cA^{(-)}[D^+D\phi]$
underlying the Schwarzschild metric should easily generalize
to the noncommutative setting.
\subsection{Useful relations}
\label{sec:useful-formulas}
From the basic commutation relations \eqref{basic-CR-H4} it is easy to obtain
\begin{align}
\b^{-1}\{x^ \mu, \b\} &= - \b\{x^ \mu, \b^{-1}\}
= r^2\b t^{\mu} \nn\\
\b^{-1}\{t^ \mu, \b\} &= - \b\{t^ \mu, \b^{-1}\}
= \frac 1{R^2} \b x^\mu\label{x-t-beta-relations} \\
\b^{-1}\t\b &= - \b \t \b^{-1}
= -(\b^2+1) \ .
\label{tau-beta-relation}
\end{align}
Furthermore, it is not hard to derive
\begin{align}
\Box x^\alpha} \def\da{{\dot\alpha}} \def\b{\beta &= \frac 1{R^2} x^\alpha} \def\da{{\dot\alpha}} \def\b{\beta \nn\\
\Box x^4 &= \frac 4{R^2} x^4
\label{Box-x}
\end{align}
and
\begin{align}
\Box(\sinh(\eta)\phi)
&= \sinh(\eta)\big(\Box + \frac 2{R^2} (\t + 2)\big) \phi \nn\\
\Box\b \phi &= \b\Box\phi - \frac 2{R^2} (\t + 2)\b \phi \label{Box-b2-id-1} \\
\Box \t\phi &= \t\Box \phi + 2 \b^2(-\Box + \frac{1}{R^2}\t) \phi \
\label{box-tau-relation-1}
\end{align}
for scalar functions
$\phi\in\cC^0$.
Finally, we note that \eqref{tau-relns}
gives
\begin{align}
D(\t+s)\phi &= (\t+s)D\phi, \nn\\
D^+D^- \t &= \t D^+ D^- \ .
\label{D-tau-relation}
\end{align}
\subsection{Metric fluctuations from $\cA$ contributions}
\label{sec:metric-contrib}
In this section we obtain the metric fluctuations $h^{\mu\nu}[\cA]$ \eqref{tilde-H-def}
arising from the various
terms in the tangential perturbations $\cA$.
We will use the averaging formulas \eqref{averaging-relns} and
the on-shell relation $\Box \phi = - \frac{2}{R^2}\phi$ throughout, as well as
\begin{align}
\partial^\alpha} \def\da{{\dot\alpha}} \def\b{\beta\partial_\alpha} \def\da{{\dot\alpha}} \def\b{\beta\phi = \b^2(-\Box+\frac{1}{R^2}\t)\phi = \frac{\b^2}{R^2}(2+\t)\phi
\label{deldel-Box-rel}
\end{align}
using \eqref{deldel-Box-relation}.
Consider first $\cA^\mu = \theta^{\mu\nu}\partial_\nu \phi$.
Then
\begin{align}
h^{\mu\nu} &= -\{x^\mu,\cA^\nu\}_0 + (\mu\leftrightarrow \nu) \
= -\{x^\mu,\theta^{\nu\alpha} \def\da{{\dot\alpha}} \def\b{\beta}\partial_\alpha} \def\da{{\dot\alpha}} \def\b{\beta \phi\} + (\mu\leftrightarrow \nu) \nn\\
&= -[\theta^{\nu\alpha} \def\da{{\dot\alpha}} \def\b{\beta} \theta^{\mu\b}]_0 \partial_\alpha} \def\da{{\dot\alpha}} \def\b{\beta\partial_\b \phi
-\{x^\mu,\theta^{\nu\alpha} \def\da{{\dot\alpha}} \def\b{\beta}\}\partial_\alpha} \def\da{{\dot\alpha}} \def\b{\beta \phi + (\mu\leftrightarrow \nu) \nn\\
&= \frac{r^2}3\Big(- 2(\b^2 + \t)(\t + 2)\eta^{\mu\nu}
- \frac{2\b^2}{R^2} x^\mu x^\nu(2+\t)
+ (x^\nu \partial^\mu + x^\mu \partial^\nu)(2\t + 1)
+ 2R^2 \partial^\nu \partial^\mu\Big) \phi
\label{h-theta-del-phi}
\end{align}
using \eqref{deldel-Box-relation} and the on-shell condition.
Next consider $\cA^\mu = \b t^\mu \phi$. Then
\begin{align}
h^{\mu\nu} &= -\{x^\mu,\cA^\nu\}_0 + (\mu\leftrightarrow \nu)
= - \{x^ \mu,\b t^\nu \phi\}_- + (\mu \leftrightarrow \nu) \nn\\
&= \sinh \eta^{\mu\nu} \b\phi - [t^\nu \theta^{\mu\alpha} \def\da{{\dot\alpha}} \def\b{\beta}]_0\partial_\alpha} \def\da{{\dot\alpha}} \def\b{\beta(\b\phi) + (\mu \leftrightarrow \nu) \nn\\
&= \frac 23\eta^{\mu\nu} (2 + \t - \b^2)\phi - \frac 23 \frac{\b^2}{R^2} x^\mu x^\nu \phi
- \frac 13 (x^\mu \partial^\nu + x^\nu \partial^\mu)\phi \ .
\label{h-t-phi}
\end{align}
For $\cA^\mu = \b x^\mu t^\alpha} \def\da{{\dot\alpha}} \def\b{\beta \partial_\alpha} \def\da{{\dot\alpha}} \def\b{\beta \phi = \frac{\b}{r^2R} x^\mu D\phi$, we obtain
\begin{align}
h^{\mu\nu} &= -\{x^\mu,\cA^\nu\}_0 + (\mu\leftrightarrow \nu)
= - \{x^ \mu,\b x^\nu t^\alpha} \def\da{{\dot\alpha}} \def\b{\beta \partial_\alpha} \def\da{{\dot\alpha}} \def\b{\beta \phi\}_0 + (\mu \leftrightarrow \nu) \nn\\
&= x^\nu \partial_\mu \phi
- x^\nu [t^\alpha} \def\da{{\dot\alpha}} \def\b{\beta \{x^ \mu, \b\}]_0 \partial_\alpha} \def\da{{\dot\alpha}} \def\b{\beta \phi
- \b x^\nu [t^\alpha} \def\da{{\dot\alpha}} \def\b{\beta \theta^{\mu\sigma} \def\S{\Sigma} \def\t{\tau} \def\th{\theta}\partial_\sigma} \def\S{\Sigma} \def\t{\tau} \def\th{\theta\partial_\alpha} \def\da{{\dot\alpha}} \def\b{\beta \phi]_0 + (\mu \leftrightarrow \nu) \nn\\
&= - \frac 43\frac{\b^2}{R^2} x^\nu x^\mu (1 + \t)\phi
+ \frac 13(1+\t-\b^2)(x^\nu \partial^\mu + x^\mu \partial^\nu)
\label{h-xtdel-phi}
\end{align}
using \eqref{x-t-beta-relations}. Again the trace provides some check.
Finally, for $\cA^\mu = \sinh(\eta) t^\alpha} \def\da{{\dot\alpha}} \def\b{\beta \partial_\alpha} \def\da{{\dot\alpha}} \def\b{\beta\partial_\mu\phi$ we obtain
\begin{align}
h^{\mu\nu} &= -\{x^\mu,\cA^\nu\}_0 + (\mu\leftrightarrow \nu)
= - \{x^ \mu,\sinh(\eta) t^\alpha} \def\da{{\dot\alpha}} \def\b{\beta \partial_\alpha} \def\da{{\dot\alpha}} \def\b{\beta\partial_\nu\phi\} + (\mu\leftrightarrow \nu) \nn\\
&= r^2 [t^\mu t^\alpha} \def\da{{\dot\alpha}} \def\b{\beta]_0 \partial_\alpha} \def\da{{\dot\alpha}} \def\b{\beta\partial_\nu\phi
+ \sinh^2(\eta) \partial_\mu\partial_\nu\phi
- \sinh(\eta) [t^\alpha} \def\da{{\dot\alpha}} \def\b{\beta \theta^{ \mu\g}]_0\partial_\g\partial_\alpha} \def\da{{\dot\alpha}} \def\b{\beta\partial_\nu\phi
+ (\mu\leftrightarrow \nu) \nn\\
&= \frac 23 \sinh^2(\eta)\partial_\mu\partial_\nu(2 +\t)\phi
-\frac{1}{R^2} (x^\mu\partial_\nu + x^\nu\partial_\mu )\phi
- \frac 4{3R^4} \b^2 x^\mu x^\mu (2+\t)\phi
\label{h-tdeldel-phi}
\end{align}
usng the on-shell relation.
We also note the relations
\begin{align}
\{t^\mu,\b t_\mu(\t+2)\phi\}
&= t^\mu\partial_\mu(\t+2)\phi \nn\\
\{t^\mu,\b x_\mu t^\alpha} \def\da{{\dot\alpha}} \def\b{\beta \partial_\alpha} \def\da{{\dot\alpha}} \def\b{\beta (\t+2)\phi\}
&= t^\alpha} \def\da{{\dot\alpha}} \def\b{\beta \partial_\alpha} \def\da{{\dot\alpha}} \def\b{\beta (\t+3-\b^2)(\t+2)\phi \nn\\
\{t_\nu,\theta^{\nu\alpha} \def\da{{\dot\alpha}} \def\b{\beta} \partial_\alpha} \def\da{{\dot\alpha}} \def\b{\beta (\t+2)\phi\}
&= 3 r^2 t^\alpha} \def\da{{\dot\alpha}} \def\b{\beta \partial_\alpha} \def\da{{\dot\alpha}} \def\b{\beta (\t+2)\phi
\label{gauge-fixing-relations-scalar}
\end{align}
due to $t^\alpha} \def\da{{\dot\alpha}} \def\b{\beta\partial_\alpha} \def\da{{\dot\alpha}} \def\b{\beta\b = 0$, which are used to check gauge invariance.
\subsection{$DD$ operator on scalar fields}
Let $\phi \in \cC^0$. The explicit formula \eqref{D-properties} for $D$
gives
\begin{align}
D D \phi &= r^4 R^2 t^\mu t^\nu \nabla^{(3)}_\mu \nabla^{(3)}_\nu \phi \nn\\
D D D D \phi &= r^8 R^4 t^\mu t^\nu t^\r t^\sigma} \def\S{\Sigma} \def\t{\tau} \def\th{\theta \nabla^{(3)}_\mu \nabla^{(3)}_\nu \nabla^{(3)}_\r \nabla^{(3)}_\sigma} \def\S{\Sigma} \def\t{\tau} \def\th{\theta \phi
\end{align}
where $\nabla^{(3)}$ is the covariant derivative along the space-like $H^3$.
In particular,
\begin{align}
D^- D^+ \phi &= r^4 R^2 [t^\mu t^\nu]_0 \nabla^{(3)}_\mu \nabla^{(3)}_\nu \phi
= \frac{r^2 R^2}3 \cosh^2(\eta) P_\perp^{\mu\nu} \nabla^{(3)}_\mu \nabla^{(3)}_\nu \phi \nn\\
&= -\frac{r^2 R^2}3 \cosh^2(\eta) \Delta^{(3)}\phi
\label{D-D+-explicit}
\end{align}
where $\Delta^{(3)} = -\nabla^{(3)\mu} \nabla^{(3)}_\mu$ is the covariant Laplacian on $H^3$.
Note that both expressions are $SO(3,1)$-invariant second order differential operators.
The averaging is given
in terms of the projector $P_\perp$ on $H^3$ in \eqref{kappa-average}.
Now we compute
\begin{align}
[D D D D \phi]_0 &= r^8 R^4 [t^\mu t^\nu t^\r t^\sigma} \def\S{\Sigma} \def\t{\tau} \def\th{\theta]_0 \nabla^{(3)}_\mu \nabla^{(3)}_\nu \nabla^{(3)}_\r \nabla^{(3)}_\sigma} \def\S{\Sigma} \def\t{\tau} \def\th{\theta \phi \nn\\
&= \frac 35 r^8 R^4 \big([t^{\mu}t^{\nu}][t^{\r} t^{\sigma} \def\S{\Sigma} \def\t{\tau} \def\th{\theta}]_0
+ [t^{\mu}t^{\r}][t^{\nu} t^{\sigma} \def\S{\Sigma} \def\t{\tau} \def\th{\theta}]_0 + [t^{\mu}t^{\sigma} \def\S{\Sigma} \def\t{\tau} \def\th{\theta}][t^{\nu} t^{\r}]_0\big)
\nabla^{(3)}_\mu \nabla^{(3)}_\nu \nabla^{(3)}_\r \nabla^{(3)}_\sigma} \def\S{\Sigma} \def\t{\tau} \def\th{\theta \phi \nn\\
&= \frac {\cosh^4(\eta)}{15} R^4 r^4 \big(P^{\mu\nu}_H P_\perp^{\r\sigma} \def\S{\Sigma} \def\t{\tau} \def\th{\theta}
+ P_\perp^{\mu\r}P_\perp^{\nu\sigma} \def\S{\Sigma} \def\t{\tau} \def\th{\theta} + P_\perp^{\mu\sigma} \def\S{\Sigma} \def\t{\tau} \def\th{\theta}P_\perp^{\nu\r}\big)
\nabla^{(3)}_\mu \nabla^{(3)}_\nu \nabla^{(3)}_\r \nabla^{(3)}_\sigma} \def\S{\Sigma} \def\t{\tau} \def\th{\theta \phi \nn\\
&= R^4 r^4\frac {\cosh^4(\eta)}{5} \big(\Delta^{(3)} + \frac 43\frac{1}{R^2\cosh^2(\eta)} \big) \Delta^{(3)}\phi
\end{align}
where $P^{\mu\nu}_\perp = g^{\mu\nu}_{(3)}$ is the tangential induced metric on $H^3$ which satisfies
$\nabla^{(3)} P_\perp^{\mu\sigma} \def\S{\Sigma} \def\t{\tau} \def\th{\theta} = 0$.
The individual terms are given by
\begin{align}
P^{\mu\nu}_H P_\perp^{\r\sigma} \def\S{\Sigma} \def\t{\tau} \def\th{\theta} \nabla^{(3)}_\mu \nabla^{(3)}_\nu \nabla^{(3)}_\r \nabla^{(3)}_\sigma} \def\S{\Sigma} \def\t{\tau} \def\th{\theta \phi
&= \Delta^{(3)}\Delta^{(3)} \phi \nn\\
P_\perp^{\mu\r}P_\perp^{\nu\sigma} \def\S{\Sigma} \def\t{\tau} \def\th{\theta}\nabla^{(3)}_\mu \nabla^{(3)}_\nu \nabla^{(3)}_\r \nabla^{(3)}_\sigma} \def\S{\Sigma} \def\t{\tau} \def\th{\theta \phi
&= \Delta^{(3)}\Delta^{(3)} \phi
+ \nabla^{(3)}_\mu ( R^{\mu\alpha} \def\da{{\dot\alpha}} \def\b{\beta}_{(3)}\partial^{(3)}_\alpha} \def\da{{\dot\alpha}} \def\b{\beta \phi) \nn\\
&= P_\perp^{\mu\sigma} \def\S{\Sigma} \def\t{\tau} \def\th{\theta}P_\perp^{\nu\r}\nabla^{(3)}_\mu \nabla^{(3)}_\nu \nabla^{(3)}_\r \nabla^{(3)}_\sigma} \def\S{\Sigma} \def\t{\tau} \def\th{\theta \phi
\end{align}
where
\begin{align}
R_{(3)}^{\mu\alpha} \def\da{{\dot\alpha}} \def\b{\beta} = \frac 13 P_\perp^{\mu\nu} R_{(3)}, \qquad
R_{(3)} = -\frac{6}{R^2\cosh^2(\eta)} \
\end{align}
are the Ricci tensor and scalar on $H^3$.
Combining these, we obtain
\begin{align}
D^- D^- D^+ D^+ \phi &=
[D D D D \phi]_0 - D^- D^+ D^- D^+ \phi \nn\\
&= \frac 4{15} R^4 r^4 \cosh^4(\eta) \Big(\frac 13 \Delta^{(3)}
+\frac{1}{R^2\cosh^2(\eta)} \Big)\Delta^{(3)}\phi \ .
\end{align}
Note that this vanishes for $x^\mu$,
consistent with
$D^- D^- D^+ D^+ x^\mu = 0$.
Now we apply this to on-shell solution with
$\big(\Box + \frac 2{R^2}\big) \phi=0$.
Then \eqref{Box-Laplace-tau} gives
\begin{align}
\cosh^2(\eta)\Delta_g^{(3)} \phi
&= -\frac{1}{R^2}(1+ \t + \b^2)(2 +\t)\phi
\label{Delta3-phi-static}
\end{align}
so that
\begin{align}
\frac 12 h = D^- D^- D^+ D^+ \phi
&\stackrel{\eta\to\infty}{\sim} -\frac 4{45} r^4 (1 - 3\t - \t^2)(\t+1)(\t+2) \phi \ .
\label{h-phi-relation}
\end{align}
This can be used as a consistency check for the computation of the trace $h$ in section \ref{sec:A-DD-metric}.
\subsection{Evaluation of $\cA^{(-)}[D^+D\phi]$}
\label{sec:eval-A-DD}
To find the corresponding metric fluctuation mode, we need to
elaborate the fluctuation mode $\cA^{(-)}_\mu$ explicitly.
For $\phi\in \cC^0$, we have
\begin{align}
DD\phi &= r^4 R^2 t^\alpha} \def\da{{\dot\alpha}} \def\b{\beta t^\b \partial_\alpha} \def\da{{\dot\alpha}} \def\b{\beta \partial_\b \phi
- r^2 R \frac{1}{x_4} t_\alpha} \def\da{{\dot\alpha}} \def\b{\beta\theta^{\alpha} \def\da{{\dot\alpha}} \def\b{\beta\b}\partial_\b\phi \nn\\
&= r^4 R^2 t^\alpha} \def\da{{\dot\alpha}} \def\b{\beta t^\b \partial_\alpha} \def\da{{\dot\alpha}} \def\b{\beta \partial_\b \phi + r^2 \t\phi
\end{align}
hence
\begin{align}
\{x^\mu,DD\phi\}_1 &= r^4 R^2 \{x^\mu,t^\alpha} \def\da{{\dot\alpha}} \def\b{\beta t^\b \partial_\alpha} \def\da{{\dot\alpha}} \def\b{\beta \partial_\b \phi\}_1
+ r^2 \{x^\mu, \t\phi \} \nn\\
&= 2 r^4 R^2 t^\alpha} \def\da{{\dot\alpha}} \def\b{\beta \{x^\mu, t^\b\} \partial_\alpha} \def\da{{\dot\alpha}} \def\b{\beta \partial_\b \phi
+ r^4 R^2 [t^\alpha} \def\da{{\dot\alpha}} \def\b{\beta t^\b \theta^{\m\g}]_1 \partial_\g (\partial_\alpha} \def\da{{\dot\alpha}} \def\b{\beta \partial_\b \phi)
+ r^2 \theta^{\mu\nu}\partial_\nu \t\phi \nn\\
&= - 2 r^4 R x_4 t^\alpha} \def\da{{\dot\alpha}} \def\b{\beta \partial_\alpha} \def\da{{\dot\alpha}} \def\b{\beta \partial_\mu \phi
+ \frac 35 r^4 R^2\Big(2 t^\alpha} \def\da{{\dot\alpha}} \def\b{\beta [t^\b \theta^{\m\g}]_1
+ [t^\alpha} \def\da{{\dot\alpha}} \def\b{\beta t^\b]_0 \theta^{\m\g} \Big)\partial_\g (\partial_\alpha} \def\da{{\dot\alpha}} \def\b{\beta \partial_\b \phi)
+ r^2 \theta^{\mu\nu}\partial_\nu \t\phi \nn\\
&= - 2 r^4 R^2 \{t_\mu, t^\alpha} \def\da{{\dot\alpha}} \def\b{\beta \partial_\alpha} \def\da{{\dot\alpha}} \def\b{\beta \phi\} - 2 r^2 \theta^{\mu\alpha} \def\da{{\dot\alpha}} \def\b{\beta} \partial_\alpha} \def\da{{\dot\alpha}} \def\b{\beta \phi
+ \frac 15 r^2 R^2\cosh^2(\eta) \theta^{\m\g}\partial_\g (\partial^\alpha} \def\da{{\dot\alpha}} \def\b{\beta \partial_\alpha} \def\da{{\dot\alpha}} \def\b{\beta \phi)
+ \frac 15 r^2 \theta^{\m\g} x^\alpha} \def\da{{\dot\alpha}} \def\b{\beta x^\b \partial_\g (\partial_\alpha} \def\da{{\dot\alpha}} \def\b{\beta \partial_\b \phi) \nn\\
&\quad + r^2 \theta^{\mu\nu}\partial_\nu \t\phi
+ \frac 25 r^4 R^2
t^\alpha} \def\da{{\dot\alpha}} \def\b{\beta \sinh(\eta) \Big( x^\mu \partial_\alpha} \def\da{{\dot\alpha}} \def\b{\beta \big(-\b^2\Box + \frac{R^2}{\sinh^2(\eta)}\t\big) \phi - x^\g\partial_\g (\partial_\mu \partial_\alpha} \def\da{{\dot\alpha}} \def\b{\beta \phi) \Big)
\end{align}
using the averaging formulas \eqref{averaging-relns}, \eqref{average-3} and \eqref{deldel-Box-relation}.
The first term is pure gauge, and the last term can be rewritten as
\begin{align}
t^\alpha} \def\da{{\dot\alpha}} \def\b{\beta \sinh(\eta) x^\g\partial_\g \partial_\mu \partial_\alpha} \def\da{{\dot\alpha}} \def\b{\beta \phi
&= \{t_\mu, t^\alpha} \def\da{{\dot\alpha}} \def\b{\beta \partial_\alpha} \def\da{{\dot\alpha}} \def\b{\beta(\t-2) \phi\} +\frac{1}{r^2 R^2} \theta^{\mu\alpha} \def\da{{\dot\alpha}} \def\b{\beta}\partial_\alpha} \def\da{{\dot\alpha}} \def\b{\beta((\t-2) \phi)
\end{align}
using $\t\partial = \partial(\t-1)$.
Further,
\begin{align}
x^\alpha} \def\da{{\dot\alpha}} \def\b{\beta x^\b \partial_\g (\partial_\alpha} \def\da{{\dot\alpha}} \def\b{\beta \partial_\b \phi)
&= \partial_\g((\t-1)(\t-2) \phi) \ .
\end{align}
Therefore
\begin{align}
\{x^\mu,DD\phi\}_1 &= - 2 r^2 \theta^{\mu\alpha} \def\da{{\dot\alpha}} \def\b{\beta} \partial_\alpha} \def\da{{\dot\alpha}} \def\b{\beta \phi
+ \frac 15 r^2 R^2\cosh^2(\eta) \theta^{\m\g}\partial_\g (\partial^\alpha} \def\da{{\dot\alpha}} \def\b{\beta \partial_\alpha} \def\da{{\dot\alpha}} \def\b{\beta \phi)
+ \frac 15 r^2 \theta^{\m\g} x^\alpha} \def\da{{\dot\alpha}} \def\b{\beta x^\b \partial_\g (\partial_\alpha} \def\da{{\dot\alpha}} \def\b{\beta \partial_\b \phi) \nn\\
&\quad + r^2 \theta^{\mu\nu}\partial_\nu \t\phi + \{t_\mu,\L\} \nn\\
&\quad + \frac 25 r^4 R^2 x^\mu \sinh(\eta) t^\alpha} \def\da{{\dot\alpha}} \def\b{\beta \partial_\alpha} \def\da{{\dot\alpha}} \def\b{\beta \big(-\b^2\Box + \frac{1}{R^2\sinh^2(\eta)}\t\big) \phi
- \frac 25 r^4 R^2 \frac{1}{r^2 R^2} \theta^{\mu\alpha} \def\da{{\dot\alpha}} \def\b{\beta}\partial_\alpha} \def\da{{\dot\alpha}} \def\b{\beta((\t-2) \phi) \nn\\
&= - \frac 25 r^4 R^2\b^3 \cosh^2(\eta) t^\mu \big(\Box - \frac 1{R^2}\t\big) \phi
- \frac 15 r^2 R^2\frac{\cosh^2(\eta)}{\sinh^2(\eta)} \theta^{\m\g}\partial_\g \big(\Box - \frac 1{R^2}\t\big) \phi
\nn\\
&\quad - \frac 25 r^4 R^2 x^\mu \frac 1{\sinh(\eta)} t^\alpha} \def\da{{\dot\alpha}} \def\b{\beta \partial_\alpha} \def\da{{\dot\alpha}} \def\b{\beta (\Box - \frac 1{R^2}\t) \phi
+ \frac{r^2}5 \theta^{\m\g} \partial_\g(\t^2-4) \phi
+ \{t_\mu,\L\}
\label{A-DDphi-1}
\end{align}
using
\begin{align}
\theta^{\m\g}\partial_\g \b^2
= 2\b^3 r^2 t^\mu
\end{align}
where
\begin{align}
\L
&= -\frac 25 r^2 R D((\t+3) \phi) \ .
\label{gaugeparam-schwarzschild}
\end{align}
\paragraph{Degenerate case.}
In the special case $\big(\Box - \frac 1{R^2}\t\big)\phi = 0$, we obtain
\begin{align}
\{x^\mu,DD\phi\}_1 &= \frac{r^2}5 \theta^{\m\g} \partial_\g(\t^2-4) \phi
+ \{t_\mu,\L\}
= \frac{r^2}5 \{x^\mu,(\t^2-4) \phi\} + \{t_\mu,\L\}
\label{degeneracy-Apm}
\end{align}
i.e. there is a linear dependence between the $\cA^{(\pm)}$ modes.
Imposing also the on-shell condition would imply
$(2 + \t)\phi = 0$. We will see that then $\cA^{(-)}[D^+D\phi]$ vanishes,
but a non-trivial mode can be extracted by taking a suitable limit,
which corresponds precisely to the Schwarzschild solution.
\paragraph{On-shell condition.}
Now consider on-shell solutions, so that $\Box\phi = -\frac{2}{R^2}\phi$. Then
\eqref{A-DDphi-1} becomes
\begin{align}
\{x^\mu,DD\phi\}_1
&= \frac 25 r^4 \b(1+\b^2) t^\mu \big(2 + \t\big) \phi
+ \frac 15 r^2 (1+\b^2)\theta^{\m\g}\partial_\g \big(2 + \t\big) \phi \nn\\
&\quad + \frac 25 r^4 \b x^\mu t^\alpha} \def\da{{\dot\alpha}} \def\b{\beta \partial_\alpha} \def\da{{\dot\alpha}} \def\b{\beta (2 + \t) \phi
+ \frac{r^2}5 \theta^{\m\g} \partial_\g(\t^2-4) \phi
+ \{t_\mu,\L\} \ .
\end{align}
But in fact we need
\begin{align}
\cA_\mu^{(-)}[D^+D^+\phi] = \{x_\mu,DD\phi\}_1 - \cA_\mu^{(+)}[D^-D\phi]
\end{align}
where
\begin{align}
\cA_\mu^{(+)}[D^-D^+\phi] &= \theta^{\mu\nu}\partial_\nu(D^-D^+ \phi)
= \frac{r^2}{3} \theta^{\mu\nu}\partial_\nu(\b^2+ \t+1)(2 +\t)\phi
\end{align}
on-shell, using \eqref{D-D+-explicit} and \eqref{Delta3-phi-static}.
Combining with the above and using
\begin{align}
- r^2 \theta^{\m\g} (\partial_\g\b^2)(\t+2)\phi
= - 2 r^4 t^{\m} \b^3 (\t+2)\phi
\end{align}
one finds the on-shell form
\begin{align}
\cA_\mu^{(-)}[D^+D^+\phi]
= \frac {2r^4}5 \Big(\b (t^\mu + x^\mu t^\alpha} \def\da{{\dot\alpha}} \def\b{\beta \partial_\alpha} \def\da{{\dot\alpha}} \def\b{\beta)
- \frac 1{3r^2} \theta^{\m\g}\partial_\g (\t+4 + \b^2)\Big)(\t+2)\phi + \{t_\mu,\L\} \ .
\label{A-DDphi-explicit-app}
\end{align}
\subsection{Background FLRW geometry and covariant derivatives}
\label{sec:diffeo-FRW}
The effective FLWR metric \eqref{eff-metric-G} is conformally flat,
\begin{align}
G^{\mu\nu} = \b \eta^{\mu\nu}, \qquad \b = \frac 1{\sinh(\eta)} \ .
\end{align}
Then the Christoffel symbols in the Cartesian coordinates $x^\mu$ are
\begin{align}
\Gamma_{\mu\nu}^\r
&= - \frac{1}{2x_4^2} (\d^\r_{\nu}\eta_{\mu\alpha} \def\da{{\dot\alpha}} \def\b{\beta} x^\alpha} \def\da{{\dot\alpha}} \def\b{\beta + \d^{\r}_{\mu}\eta_{\nu\alpha} \def\da{{\dot\alpha}} \def\b{\beta} x^\alpha} \def\da{{\dot\alpha}} \def\b{\beta
- \eta_{\mu\nu} x^\r ) \nn\\
&= - \frac 1{2R^2}\b^3 (\d^\r_{\nu}G_{\mu\alpha} \def\da{{\dot\alpha}} \def\b{\beta} x^\alpha} \def\da{{\dot\alpha}} \def\b{\beta + \d^{\r}_{\mu}G_{\nu\alpha} \def\da{{\dot\alpha}} \def\b{\beta} x^\alpha} \def\da{{\dot\alpha}} \def\b{\beta
- G_{\mu\nu} x^\r )
\label{christoffels}
\end{align}
so that
\begin{align}
\Gamma^\r &= G^{\mu\nu}\Gamma_{\mu\nu}^\r = \frac{R}{x_4^3} x^\r, \qquad
\Gamma_{\mu\nu}^\mu = - \frac{2}{x_4^2}\eta_{\nu\alpha} \def\da{{\dot\alpha}} \def\b{\beta} x^\alpha} \def\da{{\dot\alpha}} \def\b{\beta
\label{christoffels-2}
\end{align}
using \eqref{tau-beta-relation}.
Note that $\G_{\mu\nu}^\r$ is suppressed by the cosmic
curvature scale.
For example, the pure gauge metric perturbations arising from diffeomorphisms
generated by $\xi^\mu$ are given by
\begin{align}
\d_\xi G^{\mu\nu} &= \nabla^\mu\xi^\nu + \nabla^\nu \xi^\mu
= \partial^\mu\xi^\nu + \partial^\nu \xi^\mu
- \frac{1}{x_4^2}G^{\nu\mu} x\cdot\xi \ .
\label{puregauge-grav-covar}
\end{align}
As an application, the divergence of a vector field can be expressed as follows
\begin{align}
\nabla_\mu \cA^\mu &= \partial_\mu \cA^\mu + \Gamma_{\mu\nu}^\mu \cA^\nu
= \partial_\mu \cA^\mu - \frac{2}{x_4^2}x^\alpha} \def\da{{\dot\alpha}} \def\b{\beta \eta_{\alpha} \def\da{{\dot\alpha}} \def\b{\beta\nu} \cA^\nu \ .
\label{div-A-full}
\end{align}
\paragraph{Diffeomorphisms and standard form on the FRW background.}
The terms $(x^\mu \partial^\nu + x^\nu \partial^\mu)\phi$ and $\partial^\mu\partial^\nu \phi$
in the expression \eqref{tilde-h-nogauge} for $\tilde h^{\mu\nu}$ can be eliminated by a suitable
diffeomorphism. Since $(x^\mu \partial^\nu + x^\nu \partial^\mu)\phi$
becomes large at late times, one must be careful
to use the proper covariant derivatives.
For example, consider the following vector fields on the FRW background
\begin{align}
\xi^\mu = x^\mu \b \phi \ .
\end{align}
Then
\begin{align}
\nabla^\mu\xi^\nu + \nabla^\nu \xi^\mu
&= G^{\mu\mu'}\partial_{\mu'}(x^\nu \b\phi) + (\mu\leftrightarrow\nu)
- \frac{1}{x_4^2}\b G^{\nu\mu} x\cdot x \phi \nn\\
&= (2 + \frac{\cosh^2}{\sinh^2}) \b\phi G^{\mu\nu}
+ (x^\nu G^{\mu\mu'}\partial_{\mu'} \b + ...)\phi + \b (x^\nu G^{\mu\mu'}\partial_{\mu'} + ...)\phi \nn\\
&\stackrel{\eta\to\infty}{=} \b^2\Big(3\phi \eta^{\mu\nu}
+ 2 x^\nu x^\mu \frac{\b^2}{R^2}\phi
+ (x^\nu \eta^{\mu\alpha} \def\da{{\dot\alpha}} \def\b{\beta}\partial_\alpha} \def\da{{\dot\alpha}} \def\b{\beta + x^\mu \eta^{\nu\alpha} \def\da{{\dot\alpha}} \def\b{\beta}\partial_\alpha} \def\da{{\dot\alpha}} \def\b{\beta )\phi\Big) \ .
\label{puregauge-FRT-special-1}
\end{align}
Hence
\begin{align}
\boxed{\
\b^2(x^\nu \eta^{\mu\alpha} \def\da{{\dot\alpha}} \def\b{\beta}\partial_\alpha} \def\da{{\dot\alpha}} \def\b{\beta + x^\mu \eta^{\nu\alpha} \def\da{{\dot\alpha}} \def\b{\beta}\partial_\alpha} \def\da{{\dot\alpha}} \def\b{\beta )\phi
\sim - \b^2\big(3\eta^{\mu\nu} + 2x^\nu x^\mu \frac{\b^2}{R^2}\big) \phi
\ }
\label{FRW-puregauge}
\end{align}
where $\sim$ indicates equivalence up to diffeos.
Next, consider the following vector fields
\begin{align}
\xi^\mu = x^\mu \phi \ .
\end{align}
Then
\begin{align}
\nabla^\mu\xi^\nu + \nabla^\nu \xi^\mu
&= \partial^\mu(x^\nu \phi) + (\mu\leftrightarrow\nu)
- \frac{1}{x_4^2} G^{\nu\mu} x\cdot x \phi \nn\\
&\stackrel{\eta\to\infty}{=} \b\Big(3\phi \eta^{\mu\nu}
+ (x^\nu \eta^{\mu\alpha} \def\da{{\dot\alpha}} \def\b{\beta}\partial_\alpha} \def\da{{\dot\alpha}} \def\b{\beta + x^\mu \eta^{\nu\alpha} \def\da{{\dot\alpha}} \def\b{\beta}\partial_\alpha} \def\da{{\dot\alpha}} \def\b{\beta )\phi\Big)
\label{puregauge-FRT-special}
\end{align}
hence
\begin{align}
\boxed{
\b(x^\mu \partial^\nu + x^\nu \partial^\mu)\phi \sim - 3 \b\eta^{\mu\nu} \phi \ .
}
\label{FRW-puregauge-2}
\end{align}
Finally, consider
\begin{align}
\xi^\mu = \b^{-1}\eta^{\mu\nu}\partial_\nu \phi \ .
\end{align}
Then
\begin{align}
\nabla^\mu\xi^\nu + \nabla^\nu \xi^\mu
&= \partial^\mu(\b^{-1}\eta^{\nu\alpha} \def\da{{\dot\alpha}} \def\b{\beta}\partial_\alpha} \def\da{{\dot\alpha}} \def\b{\beta \phi ) + (\mu\leftrightarrow\nu) -\frac{1}{x_4^2} \b^{-1} G^{\nu\mu}\t\phi \nn\\
&= 2\eta^{\mu\mu'}\eta^{\nu\nu'}\partial_{\mu'}\partial_{\nu'} \phi
- \frac{1}{R^2}\b^2 (x^{\mu}\eta^{\nu\nu'}\partial_{\nu'} + x^{\nu}\eta^{\mu\mu'}\partial_{\mu'}) \phi
-\frac{1}{x_4^2} \eta^{\nu\mu} \t\phi \ .
\end{align}
The second term can be rewritten using \eqref{FRW-puregauge},
and therefore
\begin{align}
\boxed{
R^2\eta^{\mu\mu'}\eta^{\nu\nu'}\partial_{\mu'}\partial_{\nu'} \phi
\sim - \b^2\big(\frac 12(3-\t)\eta^{\mu\nu} + x^\nu x^\mu \frac{\b^2}{R^2}\big) \phi \ .
}
\label{FRW-puregauge-3}
\end{align}
One can check with these results that the pure gauge contribution \eqref{pure-gauge-contrib-DD} is
indeed a diffeomorphism.
\subsection{Massless scalar fields $(\Box+\frac{2}{R^2})\phi = 0$}
\label{sec:Box}
Using \eqref{G-Box-relation}, the on-shell relation can be written
for rotationally invariant $\phi(\eta,\chi)$ in the form
\begin{align}
0 &= (\Box+\frac{2}{R^2})\phi = \sinh(\eta)^3 \Box_G \phi + \frac{2}{R^2}\phi \nn\\
&= \frac{\tanh^2(\eta)}{R^2} \Big(\frac{1}{\sinh^2(\eta)\cosh(\eta)} \partial_\eta\big(\cosh^3(\eta)\partial_\eta\big)
+ 2\frac{\cosh^2(\eta)}{\sinh^2(\eta)}
+ R^2 \cosh^2(\eta)\Delta^{(3)} \Big)\phi \ .
\label{eom-phi-1}
\end{align}
We make a separation ansatz
\begin{align}
\phi(\eta,\chi) = f(\eta) g(\chi) \ .
\end{align}
Then the eom becomes
\begin{align}
\frac{1}{\sinh(\eta)^2\cosh(\eta)}\frac 1f \partial_\eta\big(\cosh^3(\eta)\partial_\eta f\big)
+ 2\frac{\cosh^2(\eta)}{\sinh(\eta)^2}
\ = \ -R^2\cosh^2(\eta)\Delta_g^{(3)} g \ .
\end{align}
The factor $\cosh^2(\eta)$ in front of $\Delta_g^{(3)}$ drops out, see \eqref{Delta-3-H},
which leads to two equations
\begin{align}
-R^2\cosh^2(\eta)\Delta_g^{(3)} g &= c\, g
\label{eom-radial-H3-laplace} \\[1ex]
\frac{1}{\sinh(\eta)^2\cosh(\eta)} \partial_\eta\big(\cosh^3(\eta)\partial_\eta f\big)
+2\frac{\cosh^2(\eta)}{\sinh^2(\eta)} f
&= c\, f
\label{separation-eq-eta}
\end{align}
where $c=const$.
\paragraph{Space-like harmonics.}
Consider first the space-like equation \eqref{eom-radial-H3-laplace}.
For rotationally invariant functions $\phi(\chi)$, this reduces using \eqref{Delta-3-H} to
\begin{align}
\frac{1}{\sinh^2(\chi)}\partial_\chi\big(\sinh^2(\chi)\partial_\chi g \big)
&= c\, g \ .
\label{separation-eq-chi}
\end{align}
The general solution is
\begin{align}
g(\chi) = \Big(c_1 e^{-\sqrt{1+c} \chi}
+c_2 e^{\sqrt{1+c} \chi} \Big)\frac 1{\sinh(\chi)} \ .
\label{g-chi-solution}
\end{align}
For $(1+c)>0$, there is at least one solution
which is decreasing for $\chi\to \infty$.
For $(1+c)<0$, the solutions are oscillating in radial direction.
\paragraph{Time dependence.}
The second equation \eqref{separation-eq-eta} is
\begin{align}
\frac{\cosh^2(\eta)}{\sinh(\eta)^2} f'' + 3 \frac{\cosh(\eta)}{\sinh(\eta)} f'
+ 2\frac{\cosh^2(\eta)}{\sinh(\eta)^2} f- c f &= 0 \ .
\label{eq-f-eta-time}
\end{align}
Asymptotically, this is
\begin{align}
e^{-3\eta}\partial_\eta(e^{3\eta}\partial_\eta f) &= (-2+c) f \nn\\
(\partial_\eta^2 + 3 \partial_\eta + 2-c) f &= 0
\label{lambda-eq-timedep}
\end{align}
which is solved by $f= e^{\lambda} \def\L{\Lambda} \def\la{\lambda} \def\m{\mu\eta}$ with
\begin{align}
\lambda} \def\L{\Lambda} \def\la{\lambda} \def\m{\mu^2 + 3\lambda} \def\L{\Lambda} \def\la{\lambda} \def\m{\mu +2 -c &= 0 \nn\\
\lambda} \def\L{\Lambda} \def\la{\lambda} \def\m{\mu_{1,2} &= \frac 12 (-3 \pm \sqrt{1 + 4 c}) \ .
\label{lambda-eq-timedep-2}
\end{align}
The most interesting quasi-static Schwarzschild solution arises for $\t\phi=-2\phi$,
which corresponds to $\Delta^{(3)}\phi = 0$ via \eqref{Box-Laplace-tau} hence to $c=0$.
Then \eqref{g-chi-solution} and \eqref{eq-f-eta-time} have the exact solutions
\begin{align}
g(\chi) = \frac{e^{- \chi}}{\sinh(\chi)},
\qquad f(\eta) = \frac{1}{\cosh^2(\eta)} \sim e^{-2\eta} \ ,
\label{harmonic-soln}
\end{align}
where $\rho = \sinh(\chi)$ \eqref{ds-induced} is the appropriate distance variable on $H^3$.
Thus $\phi = f(\eta) g(\chi)$ exhibits the typical $\frac 1\r$ behavior of the harmonic Newton potential in 3 dimensions,
with time dependence given by
$f(\eta) \sim e^{-2\eta}$.
Note that $\phi(\eta)$ remains finite for $\eta\to 0$, so that the Schwarzschild solution does not blow up at any time.
For $c<-\frac 14$, this will lead to propagating scalar modes.
\bibliographystyle{JHEP}
|
1,108,101,564,589 | arxiv | \section{Higgs Gluon Fusion \texorpdfstring{$\mu = m_{H}$}{Higgs mass scale} Results}
\begin{table}[H]
\centerline{
\begin{tabular}{c|ccc}
\hline
$\sigma$ order & PDF order & $\sigma + \Delta \sigma_{+} - \Delta \sigma_{-}$ (pb) & $\sigma$ (pb) $+\ \Delta \sigma_{+} - \Delta \sigma_{-}$ (\%) \\
\hline
\multicolumn{4}{c}{PDF uncertainties} \\
\hline
\multirow{4}{*}{N$^{3}$LO}
& aN$^{3}$LO (no theory unc.) & 42.709 + 1.282 - 1.342 & 42.709 + 2.81\% - 3.14\% \\
& aN$^{3}$LO & 42.709 + 1.409 - 1.357 & 42.709 + 3.30\% - 3.17\% \\
& aN$^{3}$LO ($K$ correlated) & 42.709 + 1.448 - 1.317 & 42.709 + 3.39\% - 3.08\% \\
& NNLO & 46.243 + 0.524 - 0.563 & 46.243 + 1.13\% - 1.22\% \\
\hline
NNLO & NNLO & 42.129 + 0.472 - 0.510 & 42.129 + 1.12\% - 1.21\% \\
\hline
\multicolumn{4}{c}{PDF + Scale uncertainties} \\
\hline
\multirow{4}{*}{N$^{3}$LO} & aN$^{3}$LO (no theory unc.) & 42.709 + 2.000 - 2.750 & 42.709 + 4.68\% - 6.44\% \\
& aN$^{3}$LO & 42.709 + 2.088 - 2.757 & 42.709 + 4.89\% - 6.45\% \\
& aN$^{3}$LO ($K$ correlated) & 42.709 + 2.114 - 2.738 & 42.709 + 4.95\% - 6.41\% \\
& NNLO & 46.243 + 1.845 - 3.078 & 46.243 + 3.99\% - 6.66\% \\
\hline
NNLO & NNLO & 42.129 + 4.989 - 5.106 & 42.129 + 11.84\% - 12.12\% \\
\end{tabular}
}
\caption{\label{tab: ggH_results_mh}Higgs production cross section results via gluon fusion using N$^{3}$LO and NNLO hard cross sections combined with NNLO and aN$^{3}$LO PDFs. All PDFs are at the standard choice $\alpha_{s} = 0.118$. These results are found with $\mu = m_{H}$ unless stated otherwise, with the values for $\mu = m_{H}/2$ supplied in Table~\ref{tab: ggH_results_mh2}.}
\end{table}
Provided in Table~\ref{tab: ggH_results_mh} are the results analogous to those in Table~\ref{tab: ggH_results_mh2} but with the central scale set to $\mu = \mu_{f} = \mu_{r} = m_{H}$. These results show a higher level of stability for aN$^{3}$LO PDFs with the chosen central scale. By the renormalisation group arguments, this scale dependence should disappear at all orders in perturbation theory. Therefore the results here suggest that the aN$^{3}$LO PDFs are following this trend.
\section{\texorpdfstring{$\chi^{2}$}{Chi-squared} Results without HERA}\label{app: noHERA}
\subsection{NNLO}
Table \ref{tab: no_HERA_fullNNLO} shows the differences in $\chi^{2}$ found when omitting HERA data from a PDF fit using the MSHT NNLO PDFs. This table is copied here from \cite{Thorne:MSHT20} for the ease of the reader. We see similarities between these results and the $\Delta \chi^{2}$'s seen in the case of N$^{3}$LO PDFs. Specifically the ATLAS $8\ \text{TeV}\ Z\ p_{T}$ displaying a substantial reduction from the global NNLO fit. This therefore provides evidence that the inclusion of the N$^{3}$LO contributions is aiding in reducing tensions between the HERA and non-HERA datasets.
\begin{table}[H]
\centerline{
\begin{tabular}{|P{6.5cm}|P{1cm}|P{2cm}|P{2cm}|}
\hline
Dataset & $N_{\mathrm{pts}}$ & $\chi^{2}$ & $\Delta \chi^{2}$ \\
\hline
BCDMS $\mu p$ $F_{2}$ \cite{BCDMS} & 163 & 174.7 & $-5.5$ \\
BCDMS $\mu d$ $F_{2}$ \cite{BCDMS} & 151 & 143.9 & $-2.1$ \\
NMC $\mu p$ $F_{2}$ \cite{NMC} & 123 & 119.6 & $-4.5$ \\
NMC $\mu d$ $F_{2}$ \cite{NMC} & 123 & 96.6 & $-16.1$ \\
SLAC $ep$ $F_{2}$ \cite{SLAC,SLAC1990} & 37 & 33.0 & $+0.9$ \\
SLAC $ed$ $F_{2}$ \cite{SLAC,SLAC1990} & 38 & 24.1 & $+1.1$ \\
E665 $\mu d$ $F_{2}$ \cite{E665} & 53 & 63.5 & $+3.9$ \\
E665 $\mu p$ $F_{2}$ \cite{E665} & 53 & 68.9 & $+4.3$ \\
NuTeV $\nu N$ $F_{2}$ \cite{NuTev} & 53 & 38.0 & $-0.3$ \\
NuTeV $\nu N$ $xF_{3}$ \cite{NuTev} & 42 & 27.5 & $-3.2$ \\
NMC $\mu n / \mu p$ \cite{NMCn/p} & 148 & 132.7 & $+1.9$ \\
E866 / NuSea $pp$ DY \cite{E866DY} & 184 & 228.0 & $+2.9$ \\
E866 / NuSea $pd/pp$ DY \cite{E866DYrat} & 15 & 9.1 & $-1.3$ \\
CCFR $\nu N \rightarrow \mu\mu X$ \cite{Dimuon} & 86 & 66.2 & $-1.5$ \\
NuTeV $\nu N \rightarrow \mu\mu X$ \cite{Dimuon} & 84 & 49.0 & $-9.5$ \\
CHORUS $\nu N$ $F_{2}$ \cite{CHORUS} & 42 & 29.6 & $-0.6$ \\
CHORUS $\nu N$ $xF_{3}$ \cite{CHORUS} & 28 & 18.2 & $-0.3$ \\
CDF II $p\bar{p}$ incl. jets \cite{CDFjet} & 76 & 60.9 & $+0.5$ \\
D{\O} II $Z$ rap. \cite{D0Zrap} & 28 & 16.6 & $+0.3$ \\
CDF II $Z$ rap. \cite{CDFZrap} & 28 & 38.7 & $+1.5$ \\
D{\O} II $W \rightarrow \nu \mu$ asym. \cite{D0Wnumu} & 10 & 17.4 & $+0.1$ \\
CDF II $W$ asym. \cite{CDF-Wasym} & 13 & 19.0 & $+0.0$ \\
D{\O} II $W \rightarrow \nu e$ asym. \cite{D0Wnue} & 12 & 30.0 & $-3.9$ \\
D{\O} II $p\bar{p}$ incl. jets \cite{D0jet} & 110 & 119.3 & $-0.9$ \\
ATLAS $W^{+},\ W^{-},\ Z$ \cite{ATLASWZ} & 30 & 29.5 & $-0.4$ \\
\hline
\end{tabular}
}
\caption{\label{tab: no_HERA_fullNNLO}The change in $\chi^{2}$ for a NNLO fit(with negative indicating an improvement in the fit quality) when the combined HERA data sets including $F_{L}$ and heavy flavour data are removed, illustrating the tensions of these data sets with several of the other data sets in the global fit. $\Delta \chi^{2}$ represents the change from a full global fit at the same order in $\alpha_{s}$.}
\end{table}
\begin{table}\ContinuedFloat
\centerline{
\begin{tabular}{|P{6.5cm}|P{1cm}|P{2cm}|P{2cm}|}
\hline
Dataset & $N_{\mathrm{pts}}$ & $\chi^{2}$ & $\Delta \chi^{2}$ \\
\hline
CMS W asym. $p_{T} > 35\ \text{GeV}$ \cite{CMS-easym} & 11 & 6.6 & $-1.2$ \\
CMS W asym. $p_{T} > 25, 30\ \text{GeV}$ \cite{CMS-Wasymm} & 24 & 7.5 & $+0.1$ \\
LHCb $Z \rightarrow e^{+}e^{-}$ \cite{LHCb-Zee} & 9 & 24.2 & $+1.5$ \\
LHCb W asym. $p_{T} > 20\ \text{GeV}$ \cite{LHCb-WZ} & 10 & 12.1 & $-0.3$ \\
CMS $Z \rightarrow e^{+}e^{-}$ \cite{CMS-Zee} & 35 & 17.3 & $-0.6$ \\
ATLAS High-mass Drell-Yan \cite{ATLAShighmass} & 13 & 16.9 & $-2.0$ \\
Tevatron, ATLAS, CMS $\sigma_{t\bar{t}}$ \cite{Tevatron-top,ATLAS-top7-1,ATLAS-top7-2,ATLAS-top7-3,ATLAS-top7-4,ATLAS-top7-5,ATLAS-top7-6,CMS-top7-1,CMS-top7-2,CMS-top7-3,CMS-top7-4,CMS-top7-5,CMS-top8} & 17 & 14.2 & $-0.4$ \\
CMS double diff. Drell-Yan \cite{CMS-ddDY} & 132 & 134.2 & $-10.3$ \\
LHCb 2015 $W, Z$ \cite{LHCbZ7,LHCbWZ8} & 67 & 97.4 & $-1.9$ \\
LHCb $8\ \text{TeV}$ $Z \rightarrow ee$ \cite{LHCbZ8} & 17 & 24.4 & $-1.8$ \\
CMS $8\ \text{TeV}\ W$ \cite{CMSW8} & 22 & 13.7 & $+0.9$ \\
ATLAS $7\ \text{TeV}$ jets \cite{ATLAS7jets} & 140 & 228.0 & $+6.5$ \\
CMS $7\ \text{TeV}\ W + c$ \cite{CMS7Wpc} & 10 & 9.2 & $+0.6$ \\
ATLAS $7\ \text{TeV}$ high prec. $W, Z$ \cite{ATLASWZ7f} & 61 & 116.8 & $+0.2$ \\
CMS $7\ \text{TeV}$ jets \cite{CMS7jetsfinal} & 158 & 179.5 & $+3.8$ \\
D{\O} $W$ asym. \cite{D0Wasym} & 14 & 11.3 & $-0.8$ \\
ATLAS $8\ \text{TeV}\ Z\ p_{T}$ \cite{ATLASZpT} & 104 & 149.3 & $-39.2$ \\
CMS $8\ \text{TeV}$ jets \cite{CMS8jets} & 174 & 259.5 & $-1.8$ \\
ATLAS $8\ \text{TeV}$ sing. diff. $t\bar{t}$ \cite{ATLASsdtop} & 25 & 24.5 & $-1.1$ \\
ATLAS $8\ \text{TeV}$ sing. diff. $t\bar{t}$ dilep. \cite{ATLASttbarDilep08_ytt} & 5 & 2.3 & $-1.1$ \\
ATLAS $8\ \text{TeV}$ High-mass DY \cite{ATLASHMDY8} & 48 & 60.9 & $+3.7$ \\
ATLAS $8\ \text{TeV}\ W + \text{jets}$ \cite{ATLASWjet} & 30 & 16.4 & $-1.7$ \\
CMS $8\ \text{TeV}$ double diff. $t\bar{t}$ \cite{CMS8ttDD} & 15 & 23.3 & $+0.8$ \\
ATLAS $8\ \text{TeV}\ W$ \cite{ATLASW8} & 22 & 54.4 & $-3.0$ \\
CMS $2.76\ \text{TeV}$ jet \cite{CMS276jets} & 81 & 102.9 & $+0.0$ \\
CMS $8\ \text{TeV}$ sing. diff. $t\bar{t}$ \cite{CMSttbar08_ytt} & 9 & 10.6 & $-2.6$ \\
ATLAS $8\ \text{TeV}$ double diff. $Z$ \cite{ATLAS8Z3D} & 59 & 108.3 & $+22.7$ \\
\hline
Total & 3042 & 3379.6 & $-61.6$ \\
\hline
\end{tabular}
}
\caption{\textit{(Continued)} The change in $\chi^{2}$ (with negative indicating an improvement in the fit quality) when the combined HERA data sets including $F_{L}$ and heavy flavour data are removed, illustrating the tensions of these data sets with several of the other data sets in the global fit. $\Delta \chi^{2}$ represents the change from a full global fit at the same order in $\alpha_{s}$.}
\end{table}
\subsection{\texorpdfstring{aN$^{3}$LO}{aN3LO}}
Table~\ref{tab: no_HERA_fullN3LO} shows the differences in $\chi^{2}$ found when omitting HERA data from a PDF fit using the MSHT aN$^{3}$LO PDFs. These results show that at aN$^{3}$LO the fit no longer experiences large tensions between HERA and ATLAS $8\ \text{TeV}\ Z\ p_{T}$~\cite{ATLASZpT} datasets. The main tensions at N$^{3}$LO are now concerning the Jets data with HERA (and most likely some non-HERA datasets). This result is not unexpected due to the known issues surrounding jets especially as we move to higher precision~\cite{AbdulKhalek:2020jut}.
\begin{table}[H]
\centerline{
\begin{tabular}{|P{6.5cm}|P{1cm}|P{2cm}|P{2cm}|}
\hline
Dataset & $N_{\mathrm{pts}}$ & $\chi^{2}$ & $\Delta \chi^{2}$ \\
\hline
BCDMS $\mu p$ $F_{2}$ \cite{BCDMS} & 163 & 177.4 & $-2.5$ \\
BCDMS $\mu d$ $F_{2}$ \cite{BCDMS} & 151 & 144.4 & $+1.3$ \\
NMC $\mu p$ $F_{2}$ \cite{NMC} & 123 & 108.1 & $-10.6$ \\
NMC $\mu d$ $F_{2}$ \cite{NMC} & 123 & 85.3 & $-20.9$ \\
SLAC $ep$ $F_{2}$ \cite{SLAC,SLAC1990} & 37 & 32.1 & $+0.1$ \\
SLAC $ed$ $F_{2}$ \cite{SLAC,SLAC1990} & 38 & 22.3 & $+0.4$ \\
E665 $\mu d$ $F_{2}$ \cite{E665} & 53 & 68.2 & $+3.5$ \\
E665 $\mu p$ $F_{2}$ \cite{E665} & 53 & 71.3 & $+3.8$ \\
NuTeV $\nu N$ $F_{2}$ \cite{NuTev} & 53 & 39.7 & $+1.0$ \\
NuTeV $\nu N$ $xF_{3}$ \cite{NuTev} & 42 & 30.8 & $-3.1$ \\
NMC $\mu n / \mu p$ \cite{NMCn/p} & 148 & 129.0 & $+0.5$ \\
E866 / NuSea $pp$ DY \cite{E866DY} & 184 & 212.5 & $+3.3$ \\
E866 / NuSea $pd/pp$ DY \cite{E866DYrat} & 15 & 7.1 & $-0.6$ \\
CCFR $\nu N \rightarrow \mu\mu X$ \cite{Dimuon} & 86 & 71.5 & $+2.3$ \\
NuTeV $\nu N \rightarrow \mu\mu X$ \cite{Dimuon} & 84 & 53.0 & $-2.4$ \\
CHORUS $\nu N$ $F_{2}$ \cite{CHORUS} & 42 & 32.6 & $-0.2$ \\
CHORUS $\nu N$ $xF_{3}$ \cite{CHORUS} & 28 & 20.4 & $+0.6$ \\
CDF II $p\bar{p}$ incl. jets \cite{CDFjet} & 76 & 68.8 & $+0.4$ \\
D{\O} II $Z$ rap. \cite{D0Zrap} & 28 & 17.0 & $+0.2$ \\
CDF II $Z$ rap. \cite{CDFZrap} & 28 & 40.6 & $+1.0$ \\
D{\O} II $W \rightarrow \nu \mu$ asym. \cite{D0Wnumu} & 10 & 19.2 & $+2.4$ \\
CDF II $W$ asym. \cite{CDF-Wasym} & 13 & 19.3 & $-0.6$ \\
D{\O} II $W \rightarrow \nu e$ asym. \cite{D0Wnue} & 12 & 27.9 & $-1.3$ \\
D{\O} II $p\bar{p}$ incl. jets \cite{D0jet} & 110 & 112.1 & $-1.5$ \\
ATLAS $W^{+},\ W^{-},\ Z$ \cite{ATLASWZ} & 30 & 29.8 & $-0.2$ \\
\hline
\end{tabular}
}
\caption{\label{tab: no_HERA_fullN3LO}The change in $\chi^{2}$ for an N$^{3}$LO fit (with negative indicating an improvement in the fit quality) when the combined HERA data sets including $F_{L}$ and heavy flavour data are removed, illustrating the tensions of these data sets with several of the other data sets in the global fit. $\Delta \chi^{2}$ represents the change from a full global fit at the same order in $\alpha_{s}$.}
\end{table}
\begin{table}\ContinuedFloat
\centerline{
\begin{tabular}{|P{6.5cm}|P{1cm}|P{2cm}|P{2cm}|}
\hline
Dataset & $N_{\mathrm{pts}}$ & $\chi^{2}$ & $\Delta \chi^{2}$ \\
\hline
CMS W asym. $p_{T} > 35\ \text{GeV}$ \cite{CMS-easym} & 11 & 7.5 & $+0.5$ \\
CMS W asym. $p_{T} > 25, 30\ \text{GeV}$ \cite{CMS-Wasymm} & 24 & 7.6 & $+0.1$ \\
LHCb $Z \rightarrow e^{+}e^{-}$ \cite{LHCb-Zee} & 9 & 21.5 & $+0.9$ \\
LHCb W asym. $p_{T} > 20\ \text{GeV}$ \cite{LHCb-WZ} & 10 & 13.1 & $+0.1$ \\
CMS $Z \rightarrow e^{+}e^{-}$ \cite{CMS-Zee} & 35 & 16.6 & $-0.7$ \\
ATLAS High-mass Drell-Yan \cite{ATLAShighmass} & 13 & 16.8 & $-1.7$ \\
Tevatron, ATLAS, CMS $\sigma_{t\bar{t}}$ \cite{Tevatron-top,ATLAS-top7-1,ATLAS-top7-2,ATLAS-top7-3,ATLAS-top7-4,ATLAS-top7-5,ATLAS-top7-6,CMS-top7-1,CMS-top7-2,CMS-top7-3,CMS-top7-4,CMS-top7-5,CMS-top8} & 17 & 14.4 & $+0.2$ \\
CMS double diff. Drell-Yan \cite{CMS-ddDY} & 132 & 133.9 & $-3.2$ \\
LHCb 2015 $W, Z$ \cite{LHCbZ7,LHCbWZ8} & 67 & 88.0 & $-9.2$ \\
LHCb $8\ \text{TeV}$ $Z \rightarrow ee$ \cite{LHCbZ8} & 17 & 22.9 & $-4.1$ \\
CMS $8\ \text{TeV}\ W$ \cite{CMSW8} & 22 & 11.1 & $-0.8$ \\
ATLAS $7\ \text{TeV}$ jets \cite{ATLAS7jets} & 140 & 224.3 & $+9.8$ \\
CMS $7\ \text{TeV}\ W + c$ \cite{CMS7Wpc} & 10 & 14.2 & $+2.0$ \\
ATLAS $7\ \text{TeV}$ high prec. $W, Z$ \cite{ATLASWZ7f} & 61 & 111.8 & $+1.4$ \\
CMS $7\ \text{TeV}$ jets \cite{CMS7jetsfinal} & 158 & 187.8 & $-2.0$ \\
D{\O} $W$ asym. \cite{D0Wasym} & 14 & 8.9 & $+0.3$ \\
ATLAS $8\ \text{TeV}\ Z\ p_{T}$ \cite{ATLASZpT} & 104 & 110.2 & $+4.5$ \\
CMS $8\ \text{TeV}$ jets \cite{CMS8jets} & 174 & 256.8 & $-15.8$ \\
ATLAS $8\ \text{TeV}$ sing. diff. $t\bar{t}$ \cite{ATLASsdtop} & 25 & 24.4 & $-0.3$ \\
ATLAS $8\ \text{TeV}$ sing. diff. $t\bar{t}$ dilep. \cite{ATLASttbarDilep08_ytt} & 5 & 2.6 & $+0.5$ \\
ATLAS $8\ \text{TeV}$ High-mass DY \cite{ATLASHMDY8} & 48 & 65.9 & $+2.5$ \\
ATLAS $8\ \text{TeV}\ W + \text{jets}$ \cite{ATLASWjet} & 30 & 17.8 & $-1.3$ \\
CMS $8\ \text{TeV}$ double diff. $t\bar{t}$ \cite{CMS8ttDD} & 15 & 23.3 & $-0.6$ \\
ATLAS $8\ \text{TeV}\ W$ \cite{ATLASW8} & 22 & 49.7 & $-5.5$ \\
CMS $2.76\ \text{TeV}$ jet \cite{CMS276jets} & 81 & 106.8 & $-7.2$ \\
CMS $8\ \text{TeV}$ sing. diff. $t\bar{t}$ \cite{CMSttbar08_ytt} & 9 & 10.9 & $+2.4$ \\
ATLAS $8\ \text{TeV}$ double diff. $Z$ \cite{ATLAS8Z3D} & 59 & 86.6 & $+5.9$ \\
\hline
\end{tabular}
}
\caption{\textit{(Continued)} The change in $\chi^{2}$ for an N$^{3}$LO fit (with negative indicating an improvement in the fit quality) when the combined HERA data sets including $F_{L}$ and heavy flavour data are removed, illustrating the tensions of these data sets with several of the other data sets in the global fit. $\Delta \chi^{2}$ represents the change from a full global fit at the same order in $\alpha_{s}$.}
\end{table}
\begin{table}[t]\ContinuedFloat
\centerline{
\begin{tabular}{|c|c|c|c|}
\hline
Low-$Q^{2}$ Coefficient & & & \\
\hline
$c_{q}^{\mathrm{NLL}}$ $ = -3.612$ & 0.038 & $c_{g}^{\mathrm{NLL}}$ $ = -3.690$ & 0.024 \\
\hline
Transition Matrix Elements & & & \\
\hline
$a_{Hg}$ $ = 15886.000$ & 3.329 & $a_{qq,H}^{\mathrm{NS}}$ $ = -62.947$ & 0.000 \\
$a_{gg,H}$ $ = -1257.100$ & 0.020 & & \\
\hline
Splitting Functions & & & \\
\hline
$\rho_{qq}^{NS}$ $ = 0.006$ & 0.015 & $\rho_{gq}$ $ = -1.763$ & 0.572 \\
$\rho_{qq}^{PS}$ $ = -0.601$ & 0.515 & $\rho_{gg}$ $ = 8.668$ & 0.071 \\
$\rho_{qg}$ $ = -0.563$ & 1.636 & & \\
\hline
K-factors & & & \\
\hline
$\mathrm{DY}_{\mathrm{NLO}}$ $ = -0.173$ & 0.030 & $\mathrm{DY}_{\mathrm{NNLO}}$ $ = -0.040$ & 0.002 \\
$\mathrm{Top}_{\mathrm{NLO}}$ $ = -0.130$ & 0.017 & $\mathrm{Top}_{\mathrm{NNLO}}$ $ = 0.496$ & 0.246 \\
$\mathrm{Jet}_{\mathrm{NLO}}$ $ = -0.337$ & 0.114 & $\mathrm{Jet}_{\mathrm{NNLO}}$ $ = -0.788$ & 0.620 \\
$p_{T}\mathrm{Jets}_{\mathrm{NLO}}$ $ = 0.442$ & 0.195 & $p_{T}\mathrm{Jets}_{\mathrm{NNLO}}$ $ = -0.023$ & 0.001 \\
$\mathrm{Dimuon}_{\mathrm{NLO}}$ $ = -0.862$ & 0.743 & $\mathrm{Dimuon}_{\mathrm{NNLO}}$ $ = 0.220$ & 0.048 \\
\hline \hline
\multicolumn{2}{c|}{} & Total & 3304.1 / 3042\\
\multicolumn{2}{c|}{} & $\Delta \chi^{2}$ from N$^{3}$LO & $-47.8$ \\
\cline{3-4}
\end{tabular}
}
\caption{\textit{(Continued)} The change in $\chi^{2}$ for an N$^{3}$LO fit (with negative indicating an improvement in the fit quality) when the combined HERA data sets including $F_{L}$ and heavy flavour data are removed, illustrating the tensions of these data sets with several of the other data sets in the global fit. $\Delta \chi^{2}$ represents the change from a full global fit at the same order in $\alpha_{s}$.}
\end{table}
\section{List of \texorpdfstring{N$^{3}$LO}{N3LO} Ingredients}\label{app: n3lo_known}
\begin{table}[H]
\centerline{
\begin{tabular}{|c|c|c|c|c|}
\hline
N$^{3}$LO & No. of & Moments & \multirow{2}{*}{Small-$x$} & \multirow{2}{*}{Large-$x$}\\
Function & Moments & (Even only) & & \\
\hline
$P_{qq}^{\mathrm{NS}}$ & 7 & $N=2 - 16$ \cite{4loopNS} & \cite{4loopNS} & \cite{4loopNS} \\
$P_{qq}^{\mathrm{PS}}$ & 4 & $N=2 - 8$ \cite{S4loopMoments,S4loopMomentsNew} & LL \cite{Catani:1994sq} & N/A \\
$P_{qg}$ & 4 & $N=2 - 8$ \cite{S4loopMoments,S4loopMomentsNew} & LL \cite{Catani:1994sq} & N/A \\
$P_{gq}$ & 4 & $N=2 - 8$ \cite{S4loopMoments,S4loopMomentsNew} & LL \cite{Lipatov:1976zz,Kuraev:1977fs,Balitsky:1978ic} & N/A \\
$P_{gg}$ & 4 & $N=2 - 8$ \cite{S4loopMoments,S4loopMomentsNew} & LL \& NLL \cite{Lipatov:1976zz,Kuraev:1977fs,Balitsky:1978ic,Fadin:1998py,Ciafaloni:1998gs} & N/A \\
\hline
$A_{qq,H}^{\mathrm{NS}}$ & 7 & $N=2 - 14$ \cite{bierenbaum:OMEmellin} & N/A & N/A \\
$A_{Hq}^{\mathrm{PS}}$ & 6 & $N=2 - 12$ \cite{bierenbaum:OMEmellin} & \cite{ablinger:3loopPS} & \cite{ablinger:3loopPS} \\
$A_{Hg}$ & 5 & $N=2 - 10$ \cite{bierenbaum:OMEmellin} & LL \cite{Vogt:FFN3LO3} & N/A \\
$A_{gq, H}$ & 7 & $N=2 - 14$ \cite{bierenbaum:OMEmellin} & \cite{ablinger:agq} & \cite{ablinger:agq} \\
$A_{gg, H}$ & 5 & $N=2 - 10$ \cite{bierenbaum:OMEmellin} & N/A & N/A \\
\hline
\end{tabular}
}
\caption{\label{tab: known_splitOME}List of all the N$^{3}$LO ingredients used to construct the approximate N$^{3}$LO splitting functions and transition matrix elements. Where only a citation is provided, extensive knowledge i.e. beyond NLL is used. This table is a non-exhaustive list of the current knowledge about these functions, however information beyond that which is provided here is not currently in a usable format for phenomological studies.}
\end{table}
\begin{table}[H]
\centerline{
\begin{tabular}{|c|c|c|c|}
\hline
GM-VFNS N$^{3}$LO & \multirow{2}{*}{Known N$^{3}$LO Components}\\
Function & \\
\hline
$C_{H,q}$ & $C_{H,q}^{(3),\ \mathrm{FF}}\left(Q^{2} \leq m_{h}^{2}\right)$ LL \cite{Catani:FFN3LO1,Laenen:FFN3LO2,Vogt:FFN3LO3}, $C_{H, q}^{\mathrm{VF},\ (3)}$ \cite{Vermaseren:2005qc} \\
$C_{H,g}$ & $C_{H,g}^{(3),\ \mathrm{FF}}\left(Q^{2} \leq m_{h}^{2}\right)$ LL\cite{Catani:FFN3LO1,Laenen:FFN3LO2,Vogt:FFN3LO3}, $C_{H, q}^{\mathrm{ZM},\ (3)}$ \cite{Vermaseren:2005qc} \\
\hline
$C_{q,q}^{\mathrm{NS}}$ & $C_{q, q,\ \mathrm{NS}}^{\mathrm{ZM},\ (3)}$ \cite{Vermaseren:2005qc} \\
$C_{q,q}^{\mathrm{PS}}$ & $C_{q, q,\ \mathrm{PS}}^{\mathrm{ZM},\ (3)}$ \cite{Vermaseren:2005qc} \\
$C_{q,g}$ & $C_{q, g}^{\mathrm{ZM},\ (3)}$ \cite{Vermaseren:2005qc} \\
\hline
\end{tabular}
}
\caption{\label{tab: known_coeff}List of all N$^{3}$LO ingredients used to construct the approximate N$^{3}$LO GM-VFNS coefficient functions. Note that lower order components that contribute to these functions are also known and are cited in the text. This table only considers contributing 3-loop functions.}
\end{table}
Table's~\ref{tab: known_splitOME} and \ref{tab: known_coeff} summarise the available (at the time of writing) and used information regarding the N$^{3}$LO splitting functions and coefficient functions respectively. The formalism presented in Section~\ref{sec: theo_framework} currently makes use of all this information and is able to be adapted as and when more information becomes available.
\section{Dynamic Tolerances}
In this section we provide an exhaustive breakdown of the $\Delta\chi^{2}_{\mathrm{global}}$ behaviour for all eigenvectors found where N$^{3}$LO $K$-factor parameters are considered completely decorrelated ($H_{ij} + K_{ij}$) or correlated ($H_{ij}^{\prime}$) with all other parameters.
\subsection{Case 1: Decorrelated \texorpdfstring{$K$}{K}-factor Parameters}\label{app: tolerance_decorr}
Fig.~\ref{fig: full_tol_decorr} displays the tolerance landscape for each eigenvector found from the decorrelated ($H_{ij} + K_{ij}$) Hessian described in Section~\ref{sec: theo_framework}. Across all 52 eigenvectors (42 PDF + N$^{3}$LO DIS theory and 10 N$^{3}$LO $K$-factor) we show an overall general agreement with the quadratic assumption similar to that found at NNLO.
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.3\textwidth]{figures/section8/section8-3/N3LO/eigenvector_analysis_1.png}
\includegraphics[width=0.3\textwidth]{figures/section8/section8-3/N3LO/eigenvector_analysis_2.png}
\includegraphics[width=0.3\textwidth]{figures/section8/section8-3/N3LO/eigenvector_analysis_3.png}
\includegraphics[width=0.3\textwidth]{figures/section8/section8-3/N3LO/eigenvector_analysis_4.png}
\includegraphics[width=0.3\textwidth]{figures/section8/section8-3/N3LO/eigenvector_analysis_5.png}
\includegraphics[width=0.3\textwidth]{figures/section8/section8-3/N3LO/eigenvector_analysis_6.png}
\end{center}
\caption{\label{fig: full_tol_decorr}Dynamic tolerances for each eigenvector direction in the case of complete decorrelation between the theory and PDF parameters, and the $K$-factor parameters included in the PDF fit.}
\end{figure}
\begin{figure}[H]\ContinuedFloat
\begin{center}
\includegraphics[width=0.3\textwidth]{figures/section8/section8-3/N3LO/eigenvector_analysis_7.png}
\includegraphics[width=0.3\textwidth]{figures/section8/section8-3/N3LO/eigenvector_analysis_8.png}
\includegraphics[width=0.3\textwidth]{figures/section8/section8-3/N3LO/eigenvector_analysis_9.png}
\includegraphics[width=0.3\textwidth]{figures/section8/section8-3/N3LO/eigenvector_analysis_10.png}
\includegraphics[width=0.3\textwidth]{figures/section8/section8-3/N3LO/eigenvector_analysis_11.png}
\includegraphics[width=0.3\textwidth]{figures/section8/section8-3/N3LO/eigenvector_analysis_12.png}
\includegraphics[width=0.3\textwidth]{figures/section8/section8-3/N3LO/eigenvector_analysis_13.png}
\includegraphics[width=0.3\textwidth]{figures/section8/section8-3/N3LO/eigenvector_analysis_14.png}
\includegraphics[width=0.3\textwidth]{figures/section8/section8-3/N3LO/eigenvector_analysis_15.png}
\end{center}
\caption{\textit{(Continued)} Dynamic tolerances for each eigenvector direction in the case of complete decorrelation between the theory and PDF parameters, and the $K$-factor parameters included in the PDF fit.}
\end{figure}
\begin{figure}\ContinuedFloat
\begin{center}
\includegraphics[width=0.3\textwidth]{figures/section8/section8-3/N3LO/eigenvector_analysis_16.png}
\includegraphics[width=0.3\textwidth]{figures/section8/section8-3/N3LO/eigenvector_analysis_17.png}
\includegraphics[width=0.3\textwidth]{figures/section8/section8-3/N3LO/eigenvector_analysis_18.png}
\includegraphics[width=0.3\textwidth]{figures/section8/section8-3/N3LO/eigenvector_analysis_19.png}
\includegraphics[width=0.3\textwidth]{figures/section8/section8-3/N3LO/eigenvector_analysis_20.png}
\includegraphics[width=0.3\textwidth]{figures/section8/section8-3/N3LO/eigenvector_analysis_21.png}
\includegraphics[width=0.3\textwidth]{figures/section8/section8-3/N3LO/eigenvector_analysis_22.png}
\includegraphics[width=0.3\textwidth]{figures/section8/section8-3/N3LO/eigenvector_analysis_23.png}
\includegraphics[width=0.3\textwidth]{figures/section8/section8-3/N3LO/eigenvector_analysis_24.png}
\end{center}
\caption{\textit{(Continued)} Dynamic tolerances for each eigenvector direction in the case of complete decorrelation between the theory and PDF parameters, and the $K$-factor parameters included in the PDF fit.}
\end{figure}
\begin{figure}\ContinuedFloat
\begin{center}
\includegraphics[width=0.3\textwidth]{figures/section8/section8-3/N3LO/eigenvector_analysis_25.png}
\includegraphics[width=0.3\textwidth]{figures/section8/section8-3/N3LO/eigenvector_analysis_26.png}
\includegraphics[width=0.3\textwidth]{figures/section8/section8-3/N3LO/eigenvector_analysis_27.png}
\includegraphics[width=0.3\textwidth]{figures/section8/section8-3/N3LO/eigenvector_analysis_28.png}
\includegraphics[width=0.3\textwidth]{figures/section8/section8-3/N3LO/eigenvector_analysis_29.png}
\includegraphics[width=0.3\textwidth]{figures/section8/section8-3/N3LO/eigenvector_analysis_30.png}
\includegraphics[width=0.3\textwidth]{figures/section8/section8-3/N3LO/eigenvector_analysis_31.png}
\includegraphics[width=0.3\textwidth]{figures/section8/section8-3/N3LO/eigenvector_analysis_32.png}
\includegraphics[width=0.3\textwidth]{figures/section8/section8-3/N3LO/eigenvector_analysis_33.png}
\end{center}
\caption{\textit{(Continued)} Dynamic tolerances for each eigenvector direction in the case of complete decorrelation between the theory and PDF parameters, and the $K$-factor parameters included in the PDF fit.}
\end{figure}
\begin{figure}\ContinuedFloat
\begin{center}
\includegraphics[width=0.3\textwidth]{figures/section8/section8-3/N3LO/eigenvector_analysis_34.png}
\includegraphics[width=0.3\textwidth]{figures/section8/section8-3/N3LO/eigenvector_analysis_35.png}
\includegraphics[width=0.3\textwidth]{figures/section8/section8-3/N3LO/eigenvector_analysis_36.png}
\includegraphics[width=0.3\textwidth]{figures/section8/section8-3/N3LO/eigenvector_analysis_37.png}
\includegraphics[width=0.3\textwidth]{figures/section8/section8-3/N3LO/eigenvector_analysis_38.png}
\includegraphics[width=0.3\textwidth]{figures/section8/section8-3/N3LO/eigenvector_analysis_39.png}
\includegraphics[width=0.3\textwidth]{figures/section8/section8-3/N3LO/eigenvector_analysis_40.png}
\includegraphics[width=0.3\textwidth]{figures/section8/section8-3/N3LO/eigenvector_analysis_41.png}
\includegraphics[width=0.3\textwidth]{figures/section8/section8-3/N3LO/eigenvector_analysis_42.png}
\end{center}
\caption{\textit{(Continued)} Dynamic tolerances for each eigenvector direction in the case of complete decorrelation between the theory and PDF parameters, and the $K$-factor parameters included in the PDF fit.}
\end{figure}
\begin{figure}\ContinuedFloat
\begin{center}
\includegraphics[width=0.3\textwidth]{figures/section8/section8-3/N3LO/eigenvector_analysis_K1.png}
\includegraphics[width=0.3\textwidth]{figures/section8/section8-3/N3LO/eigenvector_analysis_K2.png}
\includegraphics[width=0.3\textwidth]{figures/section8/section8-3/N3LO/eigenvector_analysis_K3.png}
\includegraphics[width=0.3\textwidth]{figures/section8/section8-3/N3LO/eigenvector_analysis_K4.png}
\includegraphics[width=0.3\textwidth]{figures/section8/section8-3/N3LO/eigenvector_analysis_K5.png}
\includegraphics[width=0.3\textwidth]{figures/section8/section8-3/N3LO/eigenvector_analysis_K6.png}
\end{center}
\caption{\textit{(Continued)} Dynamic tolerances for each eigenvector direction in the case of complete decorrelation between the theory and PDF parameters, and the $K$-factor parameters included in the PDF fit.}
\end{figure}
\begin{figure}\ContinuedFloat
\begin{center}
\includegraphics[width=0.3\textwidth]{figures/section8/section8-3/N3LO/eigenvector_analysis_K7.png}
\includegraphics[width=0.3\textwidth]{figures/section8/section8-3/N3LO/eigenvector_analysis_K8.png}
\\
\includegraphics[width=0.3\textwidth]{figures/section8/section8-3/N3LO/eigenvector_analysis_K9.png}
\includegraphics[width=0.3\textwidth]{figures/section8/section8-3/N3LO/eigenvector_analysis_K10.png}
\end{center}
\caption{\textit{(Continued)} Dynamic tolerances for each eigenvector direction in the case of complete decorrelation between the theory and PDF parameters, and the $K$-factor parameters included in the PDF fit.}
\end{figure}
\subsection{Case 2: Correlated \texorpdfstring{$K$}{K}-factor Parameters}\label{app: tolerance_corr}
Fig.~\ref{fig: full_tol_corr} displays the tolerance landscape for each eigenvector found from the correlated ($H_{ij}^{\prime}$) Hessian described in Section~\ref{sec: theo_framework}. Across all 52 eigenvectors we show an overall general agreement with the quadratic assumption similar to that found at NNLO.
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.3\textwidth]{figures/section8/section8-3/N3LO_Kcorr/eigenvector_analysis_1.png}
\includegraphics[width=0.3\textwidth]{figures/section8/section8-3/N3LO_Kcorr/eigenvector_analysis_2.png}
\includegraphics[width=0.3\textwidth]{figures/section8/section8-3/N3LO_Kcorr/eigenvector_analysis_3.png}
\includegraphics[width=0.3\textwidth]{figures/section8/section8-3/N3LO_Kcorr/eigenvector_analysis_4.png}
\includegraphics[width=0.3\textwidth]{figures/section8/section8-3/N3LO_Kcorr/eigenvector_analysis_5.png}
\includegraphics[width=0.3\textwidth]{figures/section8/section8-3/N3LO_Kcorr/eigenvector_analysis_6.png}
\end{center}
\caption{\label{fig: full_tol_corr}Dynamic tolerances for each eigenvector direction in the case of complete correlation between all theory, PDF and $K$-factor parameters included in the PDF fit.}
\end{figure}
\begin{figure}\ContinuedFloat
\begin{center}
\includegraphics[width=0.3\textwidth]{figures/section8/section8-3/N3LO_Kcorr/eigenvector_analysis_7.png}
\includegraphics[width=0.3\textwidth]{figures/section8/section8-3/N3LO_Kcorr/eigenvector_analysis_8.png}
\includegraphics[width=0.3\textwidth]{figures/section8/section8-3/N3LO_Kcorr/eigenvector_analysis_9.png}
\includegraphics[width=0.3\textwidth]{figures/section8/section8-3/N3LO_Kcorr/eigenvector_analysis_10.png}
\includegraphics[width=0.3\textwidth]{figures/section8/section8-3/N3LO_Kcorr/eigenvector_analysis_11.png}
\includegraphics[width=0.3\textwidth]{figures/section8/section8-3/N3LO_Kcorr/eigenvector_analysis_12.png}
\includegraphics[width=0.3\textwidth]{figures/section8/section8-3/N3LO_Kcorr/eigenvector_analysis_13.png}
\includegraphics[width=0.3\textwidth]{figures/section8/section8-3/N3LO_Kcorr/eigenvector_analysis_14.png}
\includegraphics[width=0.3\textwidth]{figures/section8/section8-3/N3LO_Kcorr/eigenvector_analysis_15.png}
\end{center}
\caption{\textit{(Continued)} Dynamic tolerances for each eigenvector direction in the case of complete correlation between all theory, PDF and $K$-factor parameters included in the PDF fit.}
\end{figure}
\begin{figure}\ContinuedFloat
\begin{center}
\includegraphics[width=0.3\textwidth]{figures/section8/section8-3/N3LO_Kcorr/eigenvector_analysis_16.png}
\includegraphics[width=0.3\textwidth]{figures/section8/section8-3/N3LO_Kcorr/eigenvector_analysis_17.png}
\includegraphics[width=0.3\textwidth]{figures/section8/section8-3/N3LO_Kcorr/eigenvector_analysis_18.png}
\includegraphics[width=0.3\textwidth]{figures/section8/section8-3/N3LO_Kcorr/eigenvector_analysis_19.png}
\includegraphics[width=0.3\textwidth]{figures/section8/section8-3/N3LO_Kcorr/eigenvector_analysis_20.png}
\includegraphics[width=0.3\textwidth]{figures/section8/section8-3/N3LO_Kcorr/eigenvector_analysis_21.png}
\includegraphics[width=0.3\textwidth]{figures/section8/section8-3/N3LO_Kcorr/eigenvector_analysis_22.png}
\includegraphics[width=0.3\textwidth]{figures/section8/section8-3/N3LO_Kcorr/eigenvector_analysis_23.png}
\includegraphics[width=0.3\textwidth]{figures/section8/section8-3/N3LO_Kcorr/eigenvector_analysis_24.png}
\end{center}
\caption{\textit{(Continued)} Dynamic tolerances for each eigenvector direction in the case of complete correlation between all theory, PDF and $K$-factor parameters included in the PDF fit.}
\end{figure}
\begin{figure}\ContinuedFloat
\begin{center}
\includegraphics[width=0.3\textwidth]{figures/section8/section8-3/N3LO_Kcorr/eigenvector_analysis_25.png}
\includegraphics[width=0.3\textwidth]{figures/section8/section8-3/N3LO_Kcorr/eigenvector_analysis_26.png}
\includegraphics[width=0.3\textwidth]{figures/section8/section8-3/N3LO_Kcorr/eigenvector_analysis_27.png}
\includegraphics[width=0.3\textwidth]{figures/section8/section8-3/N3LO_Kcorr/eigenvector_analysis_28.png}
\includegraphics[width=0.3\textwidth]{figures/section8/section8-3/N3LO_Kcorr/eigenvector_analysis_29.png}
\includegraphics[width=0.3\textwidth]{figures/section8/section8-3/N3LO_Kcorr/eigenvector_analysis_30.png}
\includegraphics[width=0.3\textwidth]{figures/section8/section8-3/N3LO_Kcorr/eigenvector_analysis_31.png}
\includegraphics[width=0.3\textwidth]{figures/section8/section8-3/N3LO_Kcorr/eigenvector_analysis_32.png}
\includegraphics[width=0.3\textwidth]{figures/section8/section8-3/N3LO_Kcorr/eigenvector_analysis_33.png}
\end{center}
\caption{\textit{(Continued)} Dynamic tolerances for each eigenvector direction in the case of complete correlation between all theory, PDF and $K$-factor parameters included in the PDF fit.}
\end{figure}
\begin{figure}\ContinuedFloat
\begin{center}
\includegraphics[width=0.3\textwidth]{figures/section8/section8-3/N3LO_Kcorr/eigenvector_analysis_34.png}
\includegraphics[width=0.3\textwidth]{figures/section8/section8-3/N3LO_Kcorr/eigenvector_analysis_35.png}
\includegraphics[width=0.3\textwidth]{figures/section8/section8-3/N3LO_Kcorr/eigenvector_analysis_36.png}
\includegraphics[width=0.3\textwidth]{figures/section8/section8-3/N3LO_Kcorr/eigenvector_analysis_37.png}
\includegraphics[width=0.3\textwidth]{figures/section8/section8-3/N3LO_Kcorr/eigenvector_analysis_38.png}
\includegraphics[width=0.3\textwidth]{figures/section8/section8-3/N3LO_Kcorr/eigenvector_analysis_39.png}
\includegraphics[width=0.3\textwidth]{figures/section8/section8-3/N3LO_Kcorr/eigenvector_analysis_40.png}
\includegraphics[width=0.3\textwidth]{figures/section8/section8-3/N3LO_Kcorr/eigenvector_analysis_41.png}
\includegraphics[width=0.3\textwidth]{figures/section8/section8-3/N3LO_Kcorr/eigenvector_analysis_42.png}
\end{center}
\caption{\textit{(Continued)} Dynamic tolerances for each eigenvector direction in the case of complete correlation between all theory, PDF and $K$-factor parameters included in the PDF fit.}
\end{figure}
\begin{figure}\ContinuedFloat
\begin{center}
\includegraphics[width=0.3\textwidth]{figures/section8/section8-3/N3LO_Kcorr/eigenvector_analysis_43.png}
\includegraphics[width=0.3\textwidth]{figures/section8/section8-3/N3LO_Kcorr/eigenvector_analysis_44.png}
\includegraphics[width=0.3\textwidth]{figures/section8/section8-3/N3LO_Kcorr/eigenvector_analysis_45.png}
\includegraphics[width=0.3\textwidth]{figures/section8/section8-3/N3LO_Kcorr/eigenvector_analysis_46.png}
\includegraphics[width=0.3\textwidth]{figures/section8/section8-3/N3LO_Kcorr/eigenvector_analysis_47.png}
\includegraphics[width=0.3\textwidth]{figures/section8/section8-3/N3LO_Kcorr/eigenvector_analysis_48.png}
\end{center}
\caption{\textit{(Continued)} Dynamic tolerances for each eigenvector direction in the case of complete correlation between all theory, PDF and $K$-factor parameters included in the PDF fit.}
\end{figure}
\begin{figure}\ContinuedFloat
\begin{center}
\includegraphics[width=0.3\textwidth]{figures/section8/section8-3/N3LO_Kcorr/eigenvector_analysis_49.png}
\includegraphics[width=0.3\textwidth]{figures/section8/section8-3/N3LO_Kcorr/eigenvector_analysis_50.png}
\includegraphics[width=0.3\textwidth]{figures/section8/section8-3/N3LO_Kcorr/eigenvector_analysis_51.png}
\includegraphics[width=0.3\textwidth]{figures/section8/section8-3/N3LO_Kcorr/eigenvector_analysis_52.png}
\end{center}
\caption{\textit{(Continued)} Dynamic tolerances for each eigenvector direction in the case of complete correlation between all theory, PDF and $K$-factor parameters included in the PDF fit.}
\end{figure}
\section{Availability and Recommended Usage of MSHT20 \texorpdfstring{aN$^{3}$LO}{aN3LO} PDFs}\label{sec: availability}
We provide the MSHT20 aN$^{3}$LO PDFs in \texttt{LHAPDF} format~\cite{Buckley:2014ana}:
\\
\\
\href{http://lhapdf.hepforge.org/}{\texttt{http://lhapdf.hepforge.org/}}
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\\
\noindent as well as on the repository:
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\href{http://www.hep.ucl.ac.uk/msht/}{\texttt{http://www.hep.ucl.ac.uk/msht/}}
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\noindent The approximate N$^{3}$LO functions (for $P_{ij}(x)$ and $A_{ij}(x)$) are provided as lightweight FORTRAN functions or as part of a Python framework in the repository:
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\href{https://github.com/MSHTPDF/N3LO_additions}{\texttt{https://github.com/MSHTPDF/N3LO\_additions}}
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\noindent We present the aN$^{3}$LO eigenvector sets with and without correlated $K$-factors as discussed in Section~\ref{sec: results}, with the default set being provided with decorrelated $K$-factors.
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\href{http://www.hep.ucl.ac.uk/msht/Grids/MSHT20an3lo_as118.tar.gz}{\texttt{MSHT20an3lo\_as118}}\\
\href{http://www.hep.ucl.ac.uk/msht/Grids/MSHT20an3lo_as118_Kcorr.tar.gz}{\texttt{MSHT20an3lo\_as118\_Kcorr}}
\\
\\
Both these PDF sets contain a central PDF accompanied by 104 eigenvector directions (describing 52 eigenvectors) and can be used in exactly the same way as previous MSHT PDF sets i.e. the MSHT20 NNLO PDFs with 64 eigenvector directions.
As presented in this work, the aN$^{3}$LO PDFs include an estimation for MHOUs (the leading theoretical uncertainty) within their PDF uncertainties. Due to this, we argue and motivate in Section~\ref{sec: predictions} that factorisation scale variations are no longer necessary in calculations involving aN$^{3}$LO PDFs. However the renormalisation scale should continue to be varied to provide estimates of MHOUs in the hard cross section piece of physical calculations.
In the case that the hard cross section for a process is available up to N$^{3}$LO the recommendation is to use the aN$^{3}$LO PDFs, since unmatched ingredients in cross section calculations can ignore important cancellations (between the PDFs and hard cross section).
If a process is included within the global fit and the hard cross section is known only up to NNLO (i.e. those discussed in Section~\ref{sec: n3lo_K}), we recommend the use of the decorrelated version of the aN$^{3}$LO PDF set. Using these PDFs and the details provided in Table~\ref{tab: Kij_limits}, the hard cross section can be transformed from NNLO to approximate N$^{3}$LO. From here the two approximate N$^{3}$LO ingredients can be used together to give a full approximate N$^{3}$LO result.
If a process is not included in the global PDF fit and the hard cross section is known only up to NNLO, the standard NNLO PDF set remains the default choice. However, we recommend the use of these aN$^{3}$LO PDFs as an estimate of potential MHOUs. In this case the aN$^{3}$LO PDF set + NNLO hard cross section prediction should be reflected in any MHOU estimates for the full NNLO prediction. For example, when the hard cross section is known only up to NNLO Equation~(3.13) from \cite{anastasiou2016high} can be adapted to be,
\begin{equation}
\delta(\mathrm{PDF} - \mathrm{TH}) = \frac{1}{2}\abs*{\frac{\sigma_{\mathrm{aN}^{3}\mathrm{LO}}^{(2)} - \sigma_{\mathrm{NNLO}}^{(2)}}{\sigma_{\mathrm{aN}^{3}\mathrm{LO}}^{(2)}}}
\end{equation}
where $\delta(\mathrm{PDF} - \mathrm{TH})$ is the predicted PDF theory uncertainty on the $\sigma$ prediction, $\sigma_{\mathrm{aN}^{3}\mathrm{LO}}^{(2)}$ is the NNLO hard cross section with aN$^{3}$LO PDFs and $\sigma_{\mathrm{NNLO}}^{(2)}$ is the full NNLO result. A caveat to this treatment is that the theory uncertainty is sensitive to unmatched cancellations and should therefore be used with care (and caution), therefore the NNLO set remains the default in evaluating PDF uncertainties.
\section{Conclusions}\label{sec: conclusion}
In this paper we have presented the first approximate N$^{3}$LO global PDF fit. This follows the MSHT20 framework~\cite{Thorne:MSHT20}, where the aN$^{3}$LO PDF set also incorporates estimates for theoretical uncertainties from MHOs. In addition, the framework presented for obtaining these PDFs provides a means of utilising higher order information as and when it is available. In contrast, previously complete information of the next order was required for theoretical calculations in PDF fits. This provides a significant advantage moving forward in precision phenomenology, since as we move to higher orders, this information takes increasingly longer to calculate. We have analysed the resulting set of PDFs, denoted MSHT20aN$^{3}$LO, and made two sets available as described in Section~\ref{sec: availability}. The aN$^{3}$LO PDF fits have been performed to the same set of global hard scattering data and PDF parameterisations included for the MSHT20 NNLO PDF fits.
The NNLO theoretical framework for MSHT20 PDFs has been extended in Section~\ref{sec: theo_framework} to include the addition of general N$^{3}$LO theory parameters into the fit. Subsequently, we have outlined how these N$^{3}$LO theory parameters can be included into the Hessian procedure as controllable nuisance parameters where they are not yet known. Two methods of handling subsets of the N$^{3}$LO theory parameters in the Hessian matrix have then been discussed i.e. including or ignoring correlations with aN$^{3}$LO $K$-factors across distinct processes. Finally in this section we explained in detail the approximation framework which can be employed to provide approximate parameterisations for each N$^{3}$LO function considered.
In Section~\ref{sec: n3lo_split} to Section~\ref{sec: n3lo_K} we have presented the N$^{3}$LO additions to the relevant splitting functions, transition matrix elements, heavy coefficient functions and $K$-factors. We present useable and computationally efficient approximations to N$^{3}$LO based on known information in the small and large-$x$ regimes and the available Mellin moments (and make these available as described in Section~\ref{sec: availability}). In all cases the best fit prediction for each N$^{3}$LO function is in good agreement with the prior expected behaviour. Also in Section~\ref{sec: n3lo_K}, we find good agreement with recent progress towards N$^{3}$LO DY and top production $K$-factors~\cite{Gehrmann:DYN3LO,Kidonakis:tt}. As more information becomes available surrounding each of these functions, the framework we present here can be easily adapted, aiding in the reduction in sources of MHOUs.
Combining together all N$^{3}$LO information, in Section~\ref{sec: results} the results of an approximate N$^{3}$LO global PDF fit are presented. The new MSHT20 approximate N$^{3}$LO PDFs show a significant reduction in $\chi^{2}$ from the MSHT20 NNLO PDF set, with the leading NNLO tensions between HERA and non-HERA datasets heavily reduced at aN$^{3}$LO (most notably with the ATLAS 8 TeV $Z\ p_{T}$ dataset~\cite{ATLASZpT}). With this being said, the aN$^{3}$LO set does fit selected Jets datasets worse in an aN$^3$LO global fit than at NNLO, although these are an exception to the behaviour seen for the other datasets. In performing a fit not including ATLAS 8 TeV $Z\ p_{T}$ data we provide evidence that similar tensions seen at NNLO (see \cite{Thorne:MSHT20}) remain between this dataset and jet production data at aN$^{3}$LO. Further to this, we show that since HERA and ATLAS 8 TeV $Z\ p_{T}$ data are more in agreement in the form of the high-$x$ gluon at aN$^{3}$LO, one can observe that the tension with the jet production data is shared between HERA and ATLAS 8 TeV $Z\ p_{T}$ data. Finally, as discussed, we highlight that in future work it will be interesting to observe if this increased tension may be alleviated when considering these jet datasets instead as dijet cross sections.
Investigating the correlations present within an aN$^{3}$LO PDF fit, a natural separation between process independent and process dependent parameters can be observed. With this motivation, a PDF set with decorrelated aN$^{3}$LO $K$-factor eigenvectors is constructed. The validity of this is then also verified by comparison with a second PDF set which includes correlations between all parameters. Each of these sets exhibits similarly well behaved eigenvectors and levels of dynamical tolerance.
Considering the form of the individual PDFs, the aN$^{3}$LO PDFs include a much harder gluon at small-$x$ due to contributions from the splitting functions as discussed in Section~\ref{subsec: n3lo_contrib}. This enhancement then translates into an increase in the charm and bottom PDFs due to the gluon input into the heavy flavour sector via the transition matrix elements. At very low-$Q^{2}$ the result of the N$^{3}$LO additions is a non-negative charm and gluon PDF at small-$x$. As a consistency check, the fit dependence on $\alpha_{s}$ and $m_{c}$ has been investigated. In both of these cases we show a preference for values which suppress the heavy flavour contributions (slightly lower $\alpha_{s}$ and slightly higher $m_{c}$ than NNLO). Considering the predicted aN$^{3}$LO $\alpha_{s}$, we observe a slightly higher than $1\sigma$ effect when comparing with the NNLO world average. While an extensive analysis of the aN$^{3}$LO $\alpha_{s}$ value is left for further study, since the world average is determined by NNLO results, one could expect a small systematic effect from moving to N$^{3}$LO.
Taking this analysis further and using the approximate N$^{3}$LO PDFs as input to N$^{3}$LO cross section calculations, we consider the cases of gluon and vector boson fusion in Higgs production. We present the first aN$^{3}$LO calculation for these cross sections and show how the aN$^{3}$LO prediction differs from the case with NNLO PDFs including scale variations, highlighting the importance of matching orders in calculations.
In VBF we provide an example where cancellation is not realised between orders. However in this case the quark sector is much more constrained and due to the smaller variation between orders, there is naturally less scope for cancellation.
In summary, we have presented a set of approximate N$^{3}$LO PDFs that are able to more accurately predict physical quantities involving PDFs (given that all ingredients in these calculations are included at N$^{3}$LO or aN$^{3}$LO). In producing these PDFs, we have provided a more controllable method for estimating theoretical uncertainties from MHOs in a PDF fit than scale variations. While some ambiguity remains in this method in how the prior variations are chosen, we argue that the current knowledge and intuition surrounding each source of uncertainty can be utilised as and when available. This is therefore much more in line with what one can expect a theoretical uncertainty to encompass. Another potential shortcoming is the possibility of fitting to sources of uncertainty other than higher orders (or higher order corrections elsewhere in theory calculations included in a PDF fit). Although this is a possibility, the position of the considered sources of uncertainty in the underlying theory combined with the prior variations and penalties should act to minimise this effect. In any case, if a separate source of uncertainty is significantly affecting the fit, this will present itself as a source of tension with the N$^{3}$LO penalties and the $\chi^{2}$ (and PDF uncertainty) will be adapted accordingly.
In future work it will be interesting to investigate the effects in the high-$x$ gluon, which is a region of phenomenological importance and where the interpretation of LHC constraints is not always straightforward. We also note that there are N$^{3}$LO results available from di-lepton rapidity in DY processes~\cite{Gehrmann:DYN3LO}. Considering the results in Section~\ref{sec: n3lo_K} which display an agreement with these recent results, we hope that these approximate N$^{3}$LO PDFs may be of interest in this analysis. Similarly for recent results considering top production~\cite{Kidonakis:tt}. Furthermore, any approximate information from these results could be included in the N$^{3}$LO $K$-factor priors, which was not done for this iteration of the aN$^{3}$LO PDFs. Finally, in order to continually improve the description of aN$^{3}$LO PDFs, the inclusion of more sub-leading sources of MHOUs could be addressed. With the upcoming wealth of experimental data from future colliders such as the HL-LHC and the EIC, it will be of interest to gain a better understanding of the transition matrix elements and also describe better the charged current and longitudinal structure functions, where currently theoretical uncertainties are much smaller than the experimental uncertainties.
\section*{Acknowledgements}
J.M. thanks the Science and Technology Facilities Council (STFC) part of U.K. Research and Innovation for support via Ph.D. funding. T.C. and R.S.T. thank STFC for support via grant awards ST/P000274/1 and ST/T000856/1. L.H.L. thanks STFC for support via grant awards ST/L000377/1 and ST/T000864/1. We would like to thank members of the PDF4LHC working group for numerous discussions on PDFs and theoretical uncertainties. We would also like to thank Alan Martin for long collaboration on the MSHT series of PDFs.
\section{Introduction}\label{sec: intro}
In recent years, the level of precision achieved at the LHC has reached far beyond what was once thought possible. This has initiated a new era of high precision phenomenology that has pushed the need for a robust understanding of theoretical uncertainty to new levels. Due to the perturbative nature of calculations in Quantum Chromodynamics (QCD), with respect to the strong coupling constant $\alpha_{s}$, a leading theoretical uncertainty arises from the truncation of perturbative expansions~\cite{Mojaza:scaleamb,Czakon:dynamicscale}. The current state of the art for parton distribution functions (PDFs) is next-to-next-to leading order (NNLO)~\cite{Thorne:MSHT20,Hou:2019efy,NNPDF3.1,NNPDF:2021njg,NNPDF:2021uiq,PDF4LHC22,ABMP16,ATLAS:2021vod}. However, these PDF sets do not generally include theoretical uncertainties arising from the truncation of perturbative calculations that enter the fit. The consideration of these so-called Missing Higher Order Uncertainties (MHOUs), and how to estimate them, is the topic of much discussion among groups involved in fitting PDFs~\cite{NNPDFscales,consistency,Lucian,Ball:corr2021}.
More recently, a method of utilising a scale variation approach to estimating these uncertainties has been included in an NLO PDF fit~\cite{NNPDFscales}. This approach is based upon the fact that to all orders, a physical calculation must not depend on any unphysical scales introduced into calculations. Therefore varying the factorisation and renormalisation scales is, in principle, a first attempt at estimating the level of theory uncertainty from missing higher orders (MHOs). Motivated by the renormalisation group invariance of physical observables, this method is theoretically grounded to all orders. However, the method of scale variations has been shown to be less than ideal in practice~\cite{consistency,Bonvini:mhous}. An obvious difficulty is the arbitrary nature in the chosen range of the scale variation, as well as the choice of central scale. Expanding on this further, even if a universal treatment of scale variations was agreed upon, these variations are unable to predict the effect of various classes of logarithms (e.g. small-$x$, mass threshold and leading large-$x$ contributions) present at higher orders. Since it is these contributions that are often the most dominant at higher orders, this is an especially concerning pitfall in the use of scale variations to estimate MHOUs. Rather more subtle are the challenges encountered when considering and accounting for correlations between fit and predictions of PDFs~\cite{consistency,Ball:corr2021}.
An alternative method to the above is to parameterise the missing higher orders with a set of nuisance parameters, using the available (albeit incomplete) current knowledge~\cite{Tackmann:2019}.
In this paper we present the first study of an approximate ${\rm N}^3{\rm LO}$ (aN$^{3}$LO) PDF fit. In particular, we first consider approximations to the N$^{3}$LO structure functions and DGLAP evolution of the PDFs, including the relevant heavy flavour transition matrix elements. We make use of all available knowledge to constrain an approximate parameterisation of the N$^{3}$LO theory, including the calculated Mellin moments, low-$x$ logarithmic behaviour and the full results where they exist. Then for the case of hadronic observables (where less N$^{3}$LO information is available), we include approximate N$^{3}$LO $K$-factors which are guided by the size of known NLO and NNLO corrections. Based on the uncertainty in our knowledge of each N$^3$LO function, we obtain a theoretical confidence level (C.L.) constrained by a prior. The corresponding theoretical uncertainties are therefore regulated by our theoretical understanding or lack thereof. Applying the above procedure, we have performed a full global fit at approximate N$^{3}$LO, with a corresponding theoretical uncertainty (from MHOs) included within a nuisance parameter framework. As we will show, adopting this procedure allows the correlations and sources of uncertainties to be easily controlled. The preferred form of the aN$^{3}$LO corrections is determined from the fit quality to data, subject to theoretical constraints from the known information about higher orders.
The outline of this paper is as follows. In Section~\ref{sec: structure} we describe the structure functions and their role in QCD calculations. Section~\ref{sec: theo_framework} presents the theoretical framework, describing the method and conventions used for the rest of the paper. In Sections~\ref{sec: n3lo_split}, \ref{sec: n3lo_OME} and \ref{sec: n3lo_coeff} we present our approximations for the N$^{3}$LO DIS theory functions, while in Section~\ref{sec: n3lo_K} we present the $K$-factors at aN$^{3}$LO. In Section~\ref{sec: results} we present the MSHT aN$^{3}$LO PDFs with theoretical uncertainties and analyse the implications of the approximations in terms of a full MSHT global fit. Section~\ref{sec: predictions} contains examples of using these aN$^{3}$LO PDFs in predictions up to N$^{3}$LO. Finally in Sections~\ref{sec: availability} and \ref{sec: conclusion} we present recommendations for how to best utilise these PDFs and summarise our results.
\section{\texorpdfstring{N$^{3}$LO}{N3LO} Transition Matrix Elements}\label{sec: n3lo_OME}
Heavy flavour transition matrix elements, $A_{ij}$, as described in Section~\ref{sec: structure}, are exact quantities that describe the transition of light quark PDFs into the heavy flavour sector. Due to discontinuous nature of $A_{ij}$ at the heavy flavour mass thresholds, they are also present in the coefficient functions to ensure an exact cancellation of this discontinuity in physical quantities. This combination then preserves the smooth nature of the structure function, as demanded by the renormalisation group flows.
The general expansion of the heavy-quark transition matrix elements in powers of $\alpha_s$ reads,
\begin{equation}
A_{i j}=\delta_{i j}+\sum_{\ell=1}^{\infty} a_{\mathrm{s}}^{\ell} A_{i j}^{(\ell)}=\delta_{i j}+\sum_{\ell=1}^{\infty} a_{\mathrm{s}}^{\ell} \sum_{k=0}^{\ell} L_{\mu}^{k} a_{i j}^{(\ell, k)},
\end{equation}
where at each order the terms proportional to powers of $\mathrm{L}_{\mu} = \ln (m_{h}^{2} / \mu^{2})$ are determined by lower order transition matrix elements and splitting functions. Therefore the focus only needs to be on the $a_{ij}^{(l,0)}$ expressions, as the rest are not only known~\cite{Buza:OMENLO,Buza:OMENNLO}, but are guaranteed not to contribute at mass thresholds due to the presence of $L_{\mu}$. These $\mu$-independent terms can be decomposed in powers of $n_{f}$ as
\begin{equation}
A_{ij}^{(3)} =A_{ij}^{(3)\ 0}+n_{f} A_{ij}^{(3)\ 1},
\end{equation}
where a number of the $n_f$-dependent and independent terms are known exactly. The $n_{f}$ parts are however sub-leading and so as a first approximation, are set to zero in this work. In keeping with the framework set out in Section~\ref{subsec: genframe}, we will make use of the available known information (even-integer Mellin moments~\cite{bierenbaum:OMEmellin} and leading small and large-$x$ behaviour~\cite{ablinger:3loopNS,ablinger:3loopPS,ablinger:agq,Blumlein:AQg,Vogt:FFN3LO3}) about the heavy flavour transition matrix elements to approximate the $\mu$-independent contributions $a_{ij}^{(3,0)}$.
We note that we make the choice to completely ignore any terms that do not contribute at mass threshold since not only are these sub-leading but can also be ignored by explicitly setting $\mu^{2} = m_{h}^{2}$.
\subsection{3-loop Approximations}\label{subsec: 3loop_OME}
\subsection*{$A_{Hg}$}\label{subsec: AHg}
The $A_{Hg}^{(3)}$ function is still under calculation at the time of writing. Currently the first five even-integer moments are known for the $\overline{\mathrm{MS}}$ scheme $A_{Hg}^{(3)}$~\cite{bierenbaum:OMEmellin}, along with the leading small-$x$ terms~\cite{Vogt:FFN3LO3}.
The $n_{f}$-dependent contribution to the 3-loop unrenormalised $A_{Hg}$ transition matrix element has also been approximated in \cite{Vogt:FFN3LO3}, while all other contributions to $A_{Hg}^{(3)}(n_{f} = 0)$ were already known. For this approximation we work in the $\overline{\mathrm{MS}}$ scheme using the framework set out in Section~\ref{subsec: genframe}. We then approximate the function using the set of functions,
\begin{alignat}{5}\label{eq: aQgComb}
f_{1,2}(x) \quad &= \quad \ln^{5}(1-x) \quad &&\text{or} \quad \ln^{4}(1-x) \quad &&\text{or} \quad \ln^{3}(1-x) \quad &&\text{or} \quad \ln^{2}(1-x) \nonumber\\ & &&\text{or} \quad \ln(1-x), \nonumber \\
f_{3,4}(x) \quad &= \quad 2 - x \quad &&\text{or} \quad 1 \quad &&\text{or} \quad x \quad &&\text{or} \quad x^{2}, \nonumber \\
f_{5}(x) \quad &= \quad \ln x \quad &&\text{or} \quad \ln^{2} x,\nonumber \\
f_{e}(x, a_{Hg}) \quad &= \quad \bigg(224\ \zeta_{3} - \frac{41984}{27} - 160\ &&\frac{\pi^{2}}{6} \bigg) \frac{\ln 1/x}{x} + a_{Hg}\ &&\frac{1}{x}
\end{alignat}
where $a_{Hg}$ is varied as $6000 < a_{Hg} < 13000$. This variation is chosen from the criteria outlined in Section~\ref{subsec: genframe} and is comparable to that chosen in \cite{Vogt:FFN3LO3}.
\begin{figure}
\includegraphics[width=0.49\textwidth]{figures/section5/section5-1/AHg0approxComb_13000.png}
\includegraphics[width=0.49\textwidth]{figures/section5/section5-1/AHg0approxComb_6000.png}
\caption{\label{fig: AQg} Combinations of functions with an added variational factor ($a_{Hg}$) controlling the
NLL term. Combinations of functions at the upper (left) and lower (right) bounds of the variation are shown. The solid lines indicate the upper and lower bounds for this function chosen from the relevant criteria.}
\end{figure}
Fig.~\ref{fig: AQg} displays the approximation of the $\overline{\mathrm{MS}}$ $A_{Hg}^{(3)}$ with the variation from different combinations of functions in Equation~\eqref{eq: aQgComb} at the chosen limits of $a_{Hg}$. Comparing with Fig.~3 in \cite{Vogt:FFN3LO3}, we see a slightly larger range of allowed variation. A small proportion of this difference can be accounted for by the difference in renormalisation schemes, with the majority of this change being from the differences in the criteria from Section~\ref{subsec: genframe}. The upper ($A_{Hg}^{(3), A}$) and lower ($A_{Hg}^{(3), B}$) bounds in the small-$x$ region (shown in Fig.~\ref{fig: AQg}) are given by,
\begin{multline}
A_{Hg}^{(3), A} = 44.1703\ \ln^{5}(1-x) + 268.024\ \ln^{4}(1-x) + 45271.0\ x - 68401.4\ x^{2} \\+ 36029.8\ \ln x + \bigg(224\ \zeta_{3} - \frac{41984}{27} - 160\ \frac{\pi^{2}}{6}\bigg)\ \frac{\ln 1/x}{x} + 12000\ \frac{1}{x}
\end{multline}
\begin{multline}
A_{Hg}^{(3), B} = -18.9493\ \ln^{5}(1-x) - 138.763\ \ln^{4}(1-x) - 31692.1\ x + 33282.3\ x^{2} \\- 3088.75\ \ln^{2} x + \bigg(224\ \zeta_{3} - \frac{41984}{27} - 160\ \frac{\pi^{2}}{6}\bigg)\ \frac{\ln 1/x}{x} + 6000\ \frac{1}{x}
\end{multline}
Using this information, we then choose the fixed functional form,
\begin{multline}
A_{Hg}^{(3)} = A_{1}\ \ln^{5}(1-x) + A_{2}\ \ln^{4}(1-x) + A_{3}\ x + A_{4}\ x^{2} + A_{5}\ \ln x \\+ \bigg(224\ \zeta_{3} - \frac{41984}{27} - 160\ \frac{\pi^{2}}{6}\bigg)\ \frac{\ln 1/x}{x} + a_{Hg}\ \frac{1}{x}
\end{multline}
where the variation of $a_{Hg}$ remains unchanged as it already encapsulates the predicted variation to within the $\sim 1\%$ level.
\subsection*{$A_{Hq}^{\mathrm{PS}}$}\label{subsec: Aps}
The $A_{Hq}^{\mathrm{PS}}$ transition matrix element has been calculated exactly in \cite{ablinger:3loopPS}. Here we attempt to qualitatively reproduce this result via an efficient parameterisation to an appropriate precision.
Using the expressions for the small and large-$x$ limits~\cite{ablinger:3loopPS} and the known first six even-integer moments converted into $\overline{\mathrm{MS}}$~\cite{bierenbaum:OMEmellin}, we provide a user-friendly approximation as,
\begin{multline}\label{eq: ApsApprox}
A_{Hq}^{\mathrm{PS}, (3)} =\ (1-x)^{2} \bigg\{-152.523\ \ln^{3}(1-x) -107.241\ \ln^{2}(1-x)\bigg\} \\ - 4986.09\ x + 582.421\ x^{2} - 1393.50\ x\ln^{2}x -4609.79\ x\ln x\\ - 688.396\ \frac{\ln 1/x}{x} + (1-x)\ 3812.90\ \frac{1}{x} + 1.6\ \ln^{5}x - 20.3457\ \ln^{4} x \\ + 165.115\ \ln^{3}x - 604.636\ \ln^{2}x + 3525.00\ \ln x \\ + (1 - x) \bigg\{0.246914 \ln^{4}(1-x) - 4.44444 \ln^{3}(1-x) - 2.28231 \ln^{2}(1-x) \\ - 357.427 \ln(1-x) + 116.478\bigg\}
\end{multline}
where the first two lines have been approximated and the last four lines are the exact leading small and large-$x$ terms. We note here that the approximated part of this parameterisation is in a much less important region of $x$ than the exact parts, therefore any small differences in the approximated part from the exact function are unimportant.
\subsection*{$A_{qq, H}^{\mathrm{NS}}$}\label{subsec: AqqHns}
Moving to the non-singlet $A_{qq, H}^{\mathrm{NS}}$ function, we attempt to parameterise the work from \cite{ablinger:3loopNS}. Specifically, we make use of the known even integer moments up to $N=14$~\cite{bierenbaum:OMEmellin}, converted into the $\overline{\mathrm{MS}}$ scheme, with the even moments corresponding to the ($+$) non-singlet distribution.
As for $A_{Hg}^{(3)}$, the approximation is performed using
the set of functions,
\begin{alignat}{5}\label{eq: ansComb}
f_{1}(x) \quad &= \quad \ln x, \qquad f_{2}(x) \quad = \quad \ln^{2}x, \nonumber \\
f_{3,4}(x) \quad &= \quad 1 \quad \text{or} \quad x \quad \text{or} \quad x^{2} \quad \text{or} \quad \ln(1-x),\nonumber \\
f_{5}(x) \quad &= \quad 1/x, \quad f_{6}(x) \quad = \quad \ln^{3}(1-x), \quad f_{7}(x) \quad = \quad \ln^{2}(1-x), \nonumber \\
f_{e}(x, a_{qq,H}^{\mathrm{NS}}) \quad &= \quad a_{qq,H}^{\mathrm{NS}}\ \ln^{3}x
\end{alignat}
where $a_{qq,H}^{\mathrm{NS}}$ is varied as $-90 < a_{qq,H}^{\mathrm{NS}} < -37$.
To contain this variation in a fixed functional form we employ:
\begin{multline}
A_{qq,H}^{\mathrm{NS},\ (3)\ +} = A_{1}\ \frac{1}{(1-x)_{+}} + A_{2}\ \ln^{3}(1-x) + A_{3}\ \ln^{2}(1-x) + A_{4}\ \ln(1-x) + A_{5} \\+ A_{6}\ x + A_{7}\ \ln^{2} x + a_{qq,H}^{\mathrm{NS}}\ \ln^{3} x
\end{multline}
where the variation of $a_{qq,H}^{\mathrm{NS}}$ is unchanged.
\subsection*{$A_{gq, H}$}\label{subsec: AgqH}
The 3-loop $A_{gq, H}$ function has been calculated exactly in \cite{ablinger:agq}. As with the $A_{Hq}^{\mathrm{PS}}$ function above, we attempt to provide a simple and computationally efficient approximation to this exact form. To do this, we use the known even-integer moments (converted to the $\overline{\mathrm{MS}}$ scheme) and small and large-$x$ information from \cite{ablinger:agq, bierenbaum:OMEmellin}. Gathering a fixed set of functions $f_{i}(x)$ and omitting any variational parameter $a_{gq,H}$, due to the higher amount of information available, the resulting approximation to the $\overline{\mathrm{MS}}$ $A_{gq, H}^{(3)}$ is:
\begin{multline}
A_{gq, H}^{(3)} = -237.172\ \ln^{3}(1-x) - 201.497\ \ln^{2}(1-x) + 7247.70\ \ln(1-x) + 39967.3\ x^{2} \\- 22017.7 - 28459.1\ \ln x - 14511.5\ \ln^{2} x \\+ 341.543\ \frac{\ln 1 / x}{x} + 1814.73\ \frac{1}{x} - \frac{580}{243}\ \ln^{4}(1 - x) - \frac{17624}{729}\ \ln^{3}(1 - x) \\- 135.699\ \ln^{2}(1 - x)
\end{multline}
where the first two lines have been approximated and the last two lines are the exact small and large-$x$ limits.
\subsection*{$A_{gg, H}$}\label{subsec: AggH}
Work is ongoing for the 3-loop contribution to $A_{gg, H}$~\cite{gluonContrib, gluonContrib2}. Due to this, the entire approximation of $A_{gg, H}^{(3)}$ presented here is based on the first 5 even-integer Mellin moments~\cite{bierenbaum:OMEmellin}. To reduce the wild behaviour of this approximation from only using the Mellin moment information (converted into the $\overline{\mathrm{MS}}$ scheme), we introduce a second mild constraint in the form of the relations in Equation~\eqref{eq: Relations}. These relations are closely followed by the gluon-gluon functions up to NNLO, but there is no guarantee that this behaviour will continue at N$^{3}$LO. This constraint is given as,
\begin{equation}\label{eq: agg_smallx}
A_{gg,H}(x\rightarrow0) \simeq \frac{C_{A}}{C_{F}}A_{gq, H}(x\rightarrow 0).
\end{equation}
It can be expected that even though this relation may not be followed exactly, it should not stray too far from this general `rule of thumb'. Due to this a generous contingency of $\pm 50\%$ is allowed when using this rule. Furthermore, to ensure this relation is only used as a guide, we allow the variation to move beyond this rule as long as the criteria in Section~\ref{subsec: genframe} are still satisfied. As a result of this change in prescription and because the allowed variation is now on a much larger scale than that of any functional uncertainty, we choose a fixed functional form from the start and use the criteria described above to guide our choice of variation.
\begin{multline}
A_{gg,H}^{(3)} = A_{1}\ \ln^{2}(1-x) + A_{2}\ \ln(1-x) + A_{3}\ x^{2} + A_{4}\ \ln x + A_{5}\ x + a_{gg,H}\ \frac{\ln x}{x}
\end{multline}
where $-2000 < a_{gg,H} < -700$.
\subsection{Predicted \texorpdfstring{aN$^{3}$LO}{aN3LO} Transition Matrix Elements}\label{subsec: expansion_OME}
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.97\textwidth]{figures/section5/section5-2/Aqqhns.png}
\end{center}
\caption{\label{fig: OME_variation_ns}Perturbative expansions for the transition matrix element $A^{\mathrm{NS}}_{qq,\ H}$ including any corresponding allowed $\pm 1\sigma$ variation (shaded green region). This function is shown at the mass threshold value of $\mu = m_{h}$. The best fit value (blue dashed line) displays the prediction for this function determined from a global PDF fit.}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=0.97\textwidth]{figures/section5/section5-2/AHqPS.png}
\includegraphics[width=0.97\textwidth]{figures/section5/section5-2/Ahg.png}
\end{center}
\caption{\label{fig: OME_variation_H}Perturbative expansions for the transition matrix elements $A^{\mathrm{PS}}_{Hq}$ and $A_{Hg}$ including any corresponding allowed $\pm 1\sigma$ variation (shaded green region). These functions are shown at the mass threshold value of $\mu = m_{h}$. The best fit values (blue dashed line) display the predictions for these functions determined from a global PDF fit.}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=0.97\textwidth]{figures/section5/section5-2/Agqh.png}
\includegraphics[width=0.97\textwidth]{figures/section5/section5-2/Aggh.png}
\end{center}
\caption{\label{fig: OME_variation_gluon}Perturbative expansions for the transition matrix elements $A_{gq,H}$ and $A_{gg,H}$ including any corresponding allowed $\pm 1\sigma$ variation (shaded green region). These functions are shown at the mass threshold value of $\mu = m_{h}$. The best fit values (blue dashed line) display the predictions for these functions determined from a global PDF fit.}
\end{figure}
Fig.'s~\ref{fig: OME_variation_ns}, \ref{fig: OME_variation_H} and \ref{fig: OME_variation_gluon} show the perturbative expansions for each of the $n_{f}$-independent contributions to the transition matrix elements at the mass threshold value of $\mu = m_{h}$. Included with these expansions are the predicted variations ($\pm 1 \sigma$) from Section~\ref{subsec: 3loop_OME} (shown in green) and the approximate N$^{3}$LO best fits (shown in blue - discussed further in Section~\ref{sec: results}).
$A_{qq,H}^{\mathrm{NS}}$ in Fig.~\ref{fig: OME_variation_ns} behaves as expected with little variation from NNLO until the magnitude of this function is very small. The approximations for the more dominant $A^{\mathrm{PS}}_{Hq}$ and $A_{Hg}$ functions in Fig.~\ref{fig: OME_variation_H} exhibit some slight sporadic behaviour towards large-$x$ due to the increased logarithmic influence. However, since this is in a region where the magnitude of these functions become small, any instabilities will have a minimal effect on the overall result. The major feature prevalent across both these functions is the large deviation away from the NNLO behaviour, especially at small-$x$ (and also mid-$x$ for $A_{Hg}$).
Similarly for $A_{gq,H}$ in Fig.~\ref{fig: OME_variation_gluon} (upper), we see some irregular behaviour towards large-$x$. As with $A_{Hq}^{\mathrm{PS}}$ and $A_{Hg}$, this behaviour is in a region where the magnitude of $A_{gq,H}$ is small. As discussed in Section~\ref{subsec: 3loop_OME}, $A_{gq, H}^{(3)}$ is approximated without any variation due to the range of available information being large. Due to this, and the fact that the region of potential instability (large-$x$) is highly suppressed, we can accept this function with negligible effect on any results. As more information becomes available about all these functions, it will be interesting to observe how the behaviour across $x$ changes.
The $A_{gg,H}$ function shown in Fig.~\ref{fig: OME_variation_gluon} (lower) displays the $\pm 50\%$ bounds of violation we allow for the relation Equation~\eqref{eq: Relations}. It follows that the allowed variation is conservative enough to include a generous violation of Equation~\eqref{eq: Relations} at N$^{3}$LO, with the prediction that the function is positive at small-$x$. This is an area where small-$x$ information would clearly be very beneficial. With this information currently in progress, it will be very interesting to compare how well this variation captures the true small-$x$ $A_{gg,H}$ behaviour.
The final best fit values shown in Fig.'s~\ref{fig: OME_variation_ns}, \ref{fig: OME_variation_H} and \ref{fig: OME_variation_gluon} are determined from a global PDF fit with various datasets seen to be constraining these functions within the $\pm\ 2 \sigma$ variations. As observed, we are able to show good agreement between the allowed variations and the best fit predictions. The perturbative expansion predicted for $A_{gg, H}$ is the least well constrained while also violating its expected relation with $A_{gq,H}$ more than one may originally expect. Since the small-$x$ region in all cases changes dramatically at N$^{3}$LO, one potential explanation is that this function is compensating for an inaccuracy in another area of the theory. However, when comparing with the relationship between $A_{Hg}$ and $A_{Hq}^{\mathrm{PS}}$, Equation~\eqref{eq: Relations} also exhibits a significant violation at this order. This could suggest that for the N$^{3}$LO transition matrix elements, this relation may not be the best indicator of precision or consistency. Finally, we remember that the best fit in this case may be feeling a larger effect from higher orders, especially due to these functions only existing from NNLO. For example, in Section~\ref{subsec: expansion_split} we observed a high level of divergence introduced at 4-loops in the splitting functions. The best fit results shown here may therefore be sensitive to a similar level of divergence further along in their corresponding perturbative expansions.
As previously discussed, this lack of knowledge is contained within our choice of the predicted variations of these functions. Therefore this treatment only seeks to add to the predicted level of theoretical uncertainty from MHOs, as one expects.
\subsection{Numerical Results}\label{subsec: num_res_OME}
\begin{figure}
\begin{center}
\includegraphics[width=0.49\textwidth]{figures/section5/section5-3/stepH.png}
\includegraphics[width=0.49\textwidth]{figures/section5/section5-3/stepG.png}
\end{center}
\caption{\label{fig: step_results}Heavy flavour evolution contributions to the heavy quark ($H + \overline{H}$ (left)) and gluon (right) PDFs provided at $\mu \simeq 30\ \mathrm{GeV}^{2}$. These results include the $\mu = m_{h}$ contributions from $A^{\mathrm{PS}}_{Hq}$, $A_{Hg}$, $A_{gq,H}$ and $A_{gg,H}$ transition matrix elements up to aN$^{3}$LO.}
\end{figure}
For these results, the same toy PDFs presented in Section~\ref{subsec: num_res_split} are employed which approximate the general order-independent PDF features at $Q^{2} \simeq 30\ \mathrm{GeV}^{2}$. Note that due to the higher $Q^{2}$, these results are more representative of the b-quark. The left plot in Fig.~\ref{fig: step_results} shows the result of including the N$^{3}$LO transition matrix element approximations we have determined into Equation~\eqref{eq: OME_fh}, which is describing the heavy quark distribution $(H + \overline{H})(x, Q^{2} = m_{h}^{2})$. The right plot in Fig.~\ref{fig: step_results} is describing the heavy flavour contribution to the gluon at $(x, Q^{2} = m_{h}^{2})$ in Equation~\eqref{eq: OME_fg} where the delta function describing the leading order contribution to $A_{gg,H}$ has been subtracted out. The dominant contribution to the heavy quark (left plot) is stemming from the $A_{Hg}$ function. Whereas the dominant contribution to the gluon (right plot) is from the $A_{gg,H}$ function. As one might expect, the predictions at N$^{3}$LO are more divergent at small-$x$, however it is also true that the general trend from NNLO is being followed across most values of $x$.
The best fit functions predicted from a global fit show the preferred aN$^{3}$LO contributions for both scenarios. The predicted behaviour from the global fit follows the results for the perturbative expansions in Section~\ref{subsec: expansion_OME}. For the $(H + \overline{H})(x, Q^{2} = m_{h}^{2})$ result (Fig.~\ref{fig: step_results} left), the aN$^{3}$LO result is positive across a much wider range of $x$. Since this is a perturbatively calculated PDF, this is an encouraging result that could potentially eliminate some of the more unphysical shortcomings at NNLO without demanding positivity of the PDF a priori.
\section{\texorpdfstring{N$^{3}$LO}{N3LO} Heavy Coefficient Functions}\label{sec: n3lo_coeff}
The final set of functions considered are the NC DIS coefficient functions which, when combined with the PDFs, form the structure functions discussed in Section~\ref{sec: structure}.\footnote{Charged current structure function data
is limited to relatively high-$x$ values compared to NC data and is either comparatively low statistics, high-$Q^2$ proton target data from HERA or nuclear target data (again often quite low statistics) on heavy nuclear targets. In both cases the effect of N$^3$LO corrections is small compared with uncertainties, especially when considering those involved with nuclear corrections. Also, heavy flavour contributions are less well known at high orders for CC structure functions. Hence, we do not include
N$^3$LO for these processes, except dimuon data, which is particularly important for the poorly constrained strange quark. An improvement would be necessary for more precise proton data, from the EIC for example.} We approximate the N$^{3}$LO heavy quark coefficient functions which accompany the heavy flavour transition matrix elements from Section~\ref{sec: n3lo_OME} and also the N$^{3}$LO light quark coefficient functions.
\subsection{Low-\texorpdfstring{$Q^{2}$}{Q2} \texorpdfstring{N$^{3}$LO}{N3LO} Heavy Flavour Coefficient Functions}\label{subsec: NLL_coeff}
As previously mentioned in Section~\ref{sec: structure}, the standard MSHT theoretical description of NNLO structure functions includes approximations to the low-$Q^{2}$ FFNS coefficient functions $C_{H,\{q,g\}}^{(3),\mathrm{FF}}$ from~\cite{Catani:FFN3LO1,Laenen:FFN3LO2,Vogt:FFN3LO3}. Within these functions are the precisely known LL small-$x$ terms and mass threshold information, along with an approximate NLL small-$x$ term added into the MSHT fit. In the NNLO fit these approximate NLL parameters play a very small role due to not only being sub-leading, but also only affecting the FFNS scheme below the mass thresholds. At NNLO they are therefore heuristically set to a value that is theoretically justified and suits the NNLO best fit. At N$^{3}$LO these functions begin to directly affect the form of the full GM-VFNS scheme across all $(x, Q^{2})$. For this reason, these NLL parameters need to be considered as an independent source of theoretical uncertainty. In the aN$^{3}$LO fit, the NLL parameters are left free and included into the framework set out in Section~\ref{subsec: hessian_method}.
The standard NNLO MSHT fit contains terms of the form,
\begin{equation}
C_{H,i}^{(3),\ \mathrm{NLL}}(Q^{2} \rightarrow 0) \propto -4\ \frac{1}{x}\ + c_{i}^{\mathrm{LL}}\ \frac{\ln 1/x}{x}, \qquad \quad \left(c_{g}^{\mathrm{LL}} = \frac{C_{F}}{C_{A}}\ c_{q}^{\mathrm{LL}}\right),
\end{equation}
where $i = q, g$ and $c_{i}^{\mathrm{LL}}$ is the precisely known leading small-$x$ log coefficient.
In the aN$^{3}$LO fit, the NLL coefficient is allowed to vary by $\pm 50\%$ ($\pm 1\sigma$ variation). This conservative range is chosen to enable the release of tension with the variational parameters associated with the N$^{3}$LO transition matrix elements. Here we stress that this quantity is heuristically set even at NNLO, therefore our treatment is completely justified with the added benefit of now accounting for an uncertainty for this choice.
\subsection{3-loop Approximations}\label{subsec: 3loop_coeff}
\subsection*{$C_{H, q}$}\label{subsec: Chq}
In this section the $C_{H, q}$ coefficient function is investigated. As discussed in Section~\ref{sec: structure}, $C_{H, q}$ contributes to the heavy flavour structure function $F_{2, H}$. We begin by isolating this function from Equation~\eqref{eq: fullN3LO_H} and relating the FFNS and GM-VFNS schemes at all orders from Equation~\eqref{eq: FFexpanse} and Equation~\eqref{eq: expanse_H},
\begin{multline}\label{eq: general_Cq}
C_{H, q}^{\mathrm{FF}}= \left[C_{H, H}^{\mathrm{VF},\ \mathrm{N S}} +C_{H, H}^{\mathrm{VF},\ \mathrm{PS}}\right] \otimes A_{H q}^{\mathrm{P S}} \ +\ C_{H, q}^{\mathrm{VF}} \otimes \left[A_{q q, H}^{\mathrm{NS}} + A_{qq, H}^{\mathrm{PS}}\right]\ +\ C_{H, g}^{\mathrm{VF}} \otimes A_{g q, H}.
\end{multline}
Expanding this function we obtain:
\begin{align}
\mathcal{O}(\alpha_{s}): \hspace{1cm}
C_{H,q}^{\mathrm{FF},\ (1)} = 0
\label{eq: alpha1}
\end{align}
\begin{align}
\mathcal{O}(\alpha_{s}^{2}): \hspace{1cm} &
\begin{multlined}[t]
C_{H,q}^{\mathrm{FF},\ (2)} = C_{H,H}^{\mathrm{VF},\ (0)} \otimes A_{Hq}^{PS,\ (2)} + C_{H, q}^{\mathrm{VF},\ (2)} \otimes A_{qq, H}^{\mathrm{NS},\ (0)}
\end{multlined}
\label{eq: alpha2}
\end{align}
\begin{align}
\mathcal{O}(\alpha_{s}^{3}): \hspace{1cm} &
\begin{multlined}[t]
C_{H,q}^{\mathrm{FF},\ (3)} = C_{H, H}^{\mathrm{VF},\ (1)}\otimes A_{Hq}^{\mathrm{PS},\ (2)} + C_{H, H}^{\mathrm{VF},\ (0)}\otimes A_{Hq}^{\mathrm{PS},\ (3)} \\+\ C_{H, q}^{\mathrm{VF},\ (3)}\otimes A_{qq,H}^{\mathrm{NS},\ (0)} + C_{H, g}^{\mathrm{VF},\ (1)} \otimes A_{gq, H}^{(2)}
\end{multlined}
\label{eq: alpha3}
\end{align}
where we recall that $A_{qq,H}^{\mathrm{NS},\ (0)} = \delta (1 - x)$.
\subsection*{NNLO}
The first contribution from the heavy quarks appears at the $\mathcal{O}(\alpha_{s}^{2})$ level. Fortunately there is a complete picture of this order~\cite{Thorne:GMVFNNLO} which provides some experience with the behaviour of these functions before moving into unknown territory. Fig.~\ref{fig: NNLO_Cq_toZM} shows the case for $C_{H, q}^{\mathrm{VF},\ (2)}$ converging onto $C_{H, q}^{\mathrm{ZM},\ (2)}$ at high-$Q^2$, as required by the definition of the GM-VFNS scheme outlined in Section \ref{sec: structure}.
\begin{figure}
\centering
\includegraphics[width=\textwidth]{figures/section6/section6-2/chq_vf_nnlo_plots.png}
\caption{\label{fig: NNLO_Cq_toZM} The NNLO GM-VFNS function $C_{H, q}^{\mathrm{VF},\ (2)}$ compared with the NNLO ZM-VFNS function $C_{H, q}^{\mathrm{ZM},\ (2)}$ across a variety of $x$ and $Q^{2}$ values. Mass threshold is set at the charm quark level ($m_{h}^{2} = m_{c}^{2} = 1.4\ \text{GeV}^{2}$).}
\end{figure}
From Fig.~\ref{fig: NNLO_Cq_toZM}, immediately some intuition can be built up surrounding the form of these functions. It can be observed that the GM-VFNS function at low-$Q^{2}$ is consistently more positive than at high-$Q^{2}$. However, the values at low and high-$Q^{2}$ are of the same order of magnitude which provides evidence that the behaviour should not be substantially different across values of $Q^{2}$ when estimating our N$^{3}$LO quantities.
Further to this, as $x \rightarrow 0$ the overall magnitude of $C_{H,q}^{(2)}$ becomes much larger, which is consistent with an inherently pure singlet quantity.
\subsection*{N$^{3}$LO}
At $\mathcal{O}(\alpha_{s}^{3})$ the N$^{3}$LO ZM-VFNS and low-$Q^{2}$ FFNS functions are known~\cite{Vermaseren:2005qc,Catani:FFN3LO1,Laenen:FFN3LO2,Vogt:FFN3LO3} and parameterisations/approximations are available (up to the level of precision discussed in Section~\ref{subsec: NLL_coeff}). Nevertheless, there is no direct information on how the full GM-VFNS function behaves at this order which is required for a full treatment of the heavy flavour coefficients. Using Equation~\eqref{eq: alpha3} to estimate the N$^{3}$LO contribution, we have
\begin{equation}\label{eq: ChqGM-VFNS}
C_{H, q}^{\mathrm{VF},\ (3)} = C_{H,q}^{\mathrm{FF},\ (3)} - C_{H, H}^{\mathrm{VF},\ (1)}\otimes A_{Hq}^{\mathrm{PS},\ (2)} - C_{H, g}^{\mathrm{VF},\ (1)} \otimes A_{gq, H}^{(2)} - A_{Hq}^{\mathrm{PS},\ (3)}.
\end{equation}
where $A_{Hq}^{\mathrm{PS},\ (3)}$ is the N$^{3}$LO transition matrix element approximated in Section~\ref{subsec: 3loop_OME}.
It must be the case that the discontinuities introduced into the heavy flavour PDF from the transition matrix elements (at the threshold value of $Q^{2} = m_{h}^{2}$) are cancelled exactly in the structure function. The cancellation of $A_{Hq}^{\mathrm{PS},\ (3)}$ is therefore guaranteed by its inclusion into the GM-VFNS coefficient function in Equation~\eqref{eq: ChqGM-VFNS}. Since in practice the transition matrix elements are convoluted with the PDFs separately to the coefficient functions, to ensure that this statement remains the case, the parameterisation will be performed in the FFNS number scheme. By doing this, we can explicitly switch to the GM-VFNS number scheme by including the subtraction term in Equation~\eqref{eq: ChqGM-VFNS}. This procedure then ensures that $A_{Hq}^{\mathrm{PS},\ (3)}$ is subtracted off exactly with no unphysical discontinuity.
Following the methodology set out in Section~\ref{subsec: genframe_continuous}, the two regimes we wish to interpolate between are the approximate $C_{H, q}^{\mathrm{FF},\ (3)}(Q^{2} \rightarrow 0)$ limit and
\begin{multline}\label{eq: ChqFF}
C_{H, q}^{\mathrm{FF},\ (3)}(Q^{2} \rightarrow \infty) = C_{H,q}^{\mathrm{ZM},\ (3)} + C_{H, H}^{\mathrm{VF},\ (1)}\otimes A_{Hq}^{\mathrm{PS},\ (2)} + C_{H, g}^{\mathrm{VF},\ (1)} \otimes A_{gq, H}^{(2)} + A_{Hq}^{\mathrm{PS},\ (3)},
\end{multline}
where $C_{H,q}^{\mathrm{VF},\ (3)}$ is replaced with $C_{H,q}^{\mathrm{ZM},\ (3)}$ in the high-$Q^{2}$ limit. Equation~\eqref{eq: ChqFF_param} is then stable across all $(x, Q^{2})$, exactly cancelling any discontinuity that would violate the RG flow, whilst also demanding that the known FFNS approximation (for $Q^{2} < m^{2}_{h}$) is followed\footnote{Since in practice the discontinuities from the transition matrix elements are added to PDFs regardless of what order coefficient function they are convoluted with, discontinuities of even higher order (e.g. $\alpha_{s}^{4}$ and beyond) are also present in calculations. Because the order $\alpha_{s}^{3}$ matrix elements are large these even higher order discontinuities are not insignificant. Therefore we add the same contributions to the unknown FFNS contributions below $m_{h}^{2}$ to impose continuity on structure functions. Such corrections are extremely small, except right at the transition point where they eliminate minor unphysical discontinuities.}.
\begin{figure}
\centering
\includegraphics[width=\textwidth]{figures/section6/section6-2/chq_vf_n3lo_plots.png}
\caption{\label{fig: GM-VFNS_estimation}N$^{3}$LO GM-VFNS function $C_{H, q}^{\mathrm{VF},\ (3)}$ compared with the N$^{3}$LO ZM-VFNS function $C_{H, q}^{\mathrm{ZM},\ (3)}$ across a variety of $x$ and $Q^{2}$ values. $C_{H, q}^{\mathrm{VF},\ (3)}$ is parameterised via Equations~\eqref{eq: ChqGM-VFNS}, \eqref{eq: ChqFF} and \eqref{eq: ChqFF_param}. Mass threshold is set at the charm quark level ($m_{h}^{2} = m_{c}^{2} = 1.4\ \text{GeV}^{2}$).}
\end{figure}
Fig.~\ref{fig: GM-VFNS_estimation} shows the result of estimating $C_{H, q}^{\mathrm{VF},\ (3)}$ using the above approximation for $C_{H, q}^{\mathrm{FF},\ (3)}$ and the relevant subtraction term from Equation~\eqref{eq: ChqGM-VFNS}.
\subsection*{$C_{H, g}$}\label{subsec: Chg}
As with $C_{H,q}$, using Equation~\eqref{eq: FFexpanse} and Equation~\eqref{eq: expanse_H} to isolate $C_{H, g}$ and relate the FFNS and GM-VFNS schemes,
\begin{multline}
C_{H,g}^{\mathrm{FF}}= C_{H,g}^{\mathrm{VF}} \otimes A_{gg, H} + C_{H,q}^{\mathrm{VF},\ \mathrm{PS}} \otimes A_{qg, H} + \left[C_{H,H}^{\mathrm{VF},\ \mathrm{NS}}+C_{H,H}^{\mathrm{VF},\ \mathrm{PS}}\right] \otimes A_{H g}
\end{multline}
\begin{align}
\mathcal{O}(\alpha_{s}): \hspace{1cm}
& C_{H,g}^{\mathrm{FF},\ (1)} = C_{H,g}^{\mathrm{VF},\ (1)} + C_{H, H}^{\mathrm{VF},\ (0)}\otimes A_{Hg}^{(1)}
\label{eq: alpha1g}
\end{align}
\begin{align}
\mathcal{O}(\alpha_{s}^{2}): \hspace{1cm} &
\begin{multlined}[t]
C_{H,g}^{\mathrm{FF},\ (2)} = C_{H,g}^{\mathrm{VF},\ (2)} + C_{H,H}^{\mathrm{VF},\ (0)} \otimes A_{Hg}^{(2)} + C_{H, H}^{\mathrm{VF},\ (1)}\otimes A_{Hg}^{(1)}
\end{multlined}
\label{eq: alpha2g}
\end{align}
\begin{align}
\mathcal{O}(\alpha_{s}^{3}): \hspace{1cm} &
\begin{multlined}[t]
C_{H,g}^{\mathrm{FF},\ (3)} = C_{H,g}^{\mathrm{VF},\ (3)} + C_{H, g}^{\mathrm{VF},\ (1)}\otimes A_{gg, H}^{(2)} + C_{H,H}^{\mathrm{VF},\ \mathrm{NS+PS},\ (2)} \otimes A_{Hg}^{(1)} \\+ C_{H, H}^{\mathrm{VF},\ (1)}\otimes A_{Hg}^{(2)} + C_{H, H}^{\mathrm{VF},\ (0)}\otimes A_{Hg}^{(3)}
\end{multlined}
\label{eq: alpha3g}
\end{align}
we uncover a NLO contribution to the heavy flavour structure function. This lower order contribution is a consequence of the gluon being able to directly probe the heavy flavour quarks, whereas a light quark must interact via a secondary interaction (hence the $C_{H,q}$ coefficient function beginning at NNLO).
\subsection*{NLO \& NNLO}
The NLO and NNLO contributions to $C_{H,g}$ are known exactly~\cite{Thorne:GMVFNNLO}. To build some experience and check our understanding, we can observe how the lower order GM-VFNS functions converge onto their ZM-VFNS counterparts in Fig.~\ref{fig: NLO_Cg_toZM} and Fig.~\ref{fig: NNLO_Cg_toZM}.
\begin{figure}
\centering
\includegraphics[width=\textwidth]{figures/section6/section6-2/chg_vf_nlo_plots.png}
\caption{\label{fig: NLO_Cg_toZM}The NLO GM-VFNS function $C_{H, g}^{\mathrm{VF},\ (1)}$ compared with the NLO ZM-VFNS function $C_{H, g}^{\mathrm{ZM},\ (1)}$ across a variety of $x$ and $Q^{2}$ values. Mass threshold is set at the charm quark level ($m_{h}^{2} = m_{c}^{2} = 1.4\ \text{GeV}^{2}$).}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=\textwidth]{figures/section6/section6-2/chg_vf_nnlo_plots.png}
\caption{\label{fig: NNLO_Cg_toZM}The NNLO GM-VFNS function $C_{H, g}^{\mathrm{VF},\ (2)}$ compared with the NNLO ZM-VFNS function $C_{H, g}^{\mathrm{ZM},\ (2)}$ across a variety of $x$ and $Q^{2}$ values. Mass threshold is set at the charm quark level ($m_{h}^{2} = m_{c}^{2} = 1.4\ \text{GeV}^{2}$).}
\end{figure}
At NLO and NNLO the magnitude of the functions is generally higher in the low-$Q^{2}$ limit than at high-$Q^{2}$. In both cases, the function remains at the same order of magnitude across all $Q^{2}$. However, the relative change across $Q^{2}$ is smaller at NLO, and similar to that seen for $C_{H,q}^{(2)}$ at NNLO. Due to this, we can once again expect that although more of a scaling contribution at N$^{3}$LO may be present, it should not be too substantial across the range of $Q^{2}$.
\subsection*{N$^{3}$LO}
As with the $C_{H, q}^{(3)}$ function at $\mathcal{O}(\alpha_{s}^{3})$, the FFNS result at low-$Q^{2}$ is known (up to the level of precision discussed in Section~\ref{subsec: NLL_coeff}), as well as the exact ZM-VFNS function at high-$Q^{2}$~\cite{Vermaseren:2005qc,Catani:FFN3LO1,Laenen:FFN3LO2,Vogt:FFN3LO3}.
Considering the form of $C^{\mathrm{VF},\ (3)}_{H,g}$, there is an extra complication coming from the transition matrix element $A_{Hg}^{(3)}$. As discussed in Section~\ref{subsec: 3loop_OME}, the $A_{Hg}^{(3)}$ function is not as well known as the $A_{Hq}^{(3)}$ function considered earlier and is accompanied by the variational parameter $a_{Hg}$.
Since it is a requirement for $C_{H, g}^{(3)}$ to exactly cancel the PDF discontinuity introduced by $A_{Hg}^{(3)}$, this variation must be compensated for and included in the description,
\begin{multline}\label{eq: ChgGM-VFNS}
C_{H,g}^{\mathrm{VF},\ (3)} = C_{H,g}^{\mathrm{FF},\ (3)} - C_{H, g}^{\mathrm{VF},\ (1)}\otimes A_{gg, H}^{(2)} - C_{H,H}^{\mathrm{VF},\ \mathrm{NS+PS},\ (2)} \otimes A_{Hg}^{(1)} \\- C_{H, H}^{\mathrm{VF},\ (1)}\otimes A_{Hg}^{(2)} - A_{Hg}^{(3)}.
\end{multline}
As in Section~\ref{subsec: Chq}, transitioning to the FFNS number scheme ensures an exact cancellation via the subtraction term in Equation~\eqref{eq: ChgGM-VFNS}. Using the exact information for $C_{H,g}^{\mathrm{FF},\ (3)}(Q^{2} \rightarrow 0)$ and the known high-$Q^{2}$ limit,
\begin{multline}\label{eq: ChgFF}
C_{H,g}^{\mathrm{FF},\ (3)}(Q^{2} \rightarrow \infty) = C_{H,g}^{\mathrm{ZM},\ (3)} + C_{H, g}^{\mathrm{VF},\ (1)}\otimes A_{gg, H}^{(2)} + C_{H,H}^{\mathrm{VF},\ \mathrm{NS+PS},\ (2)} \otimes A_{Hg}^{(1)} \\+ C_{H, H}^{\mathrm{VF},\ (1)}\otimes A_{Hg}^{(2)} + A_{Hg}^{(3)}
\end{multline}
where $C_{H,g}^{\mathrm{VF},\ (3)}$ is replaced with $C_{H,g}^{\mathrm{ZM},\ (3)}$ in the high-$Q^{2}$ limit. Applying the framework set out in Equation~\eqref{eq: ChqFF_param}, the resulting parameterisation is stable across all $(x,Q^{2})$. As $A_{Hg}^{(3)}$ and its variation is explicitly included in Equation~\eqref{eq: ChgGM-VFNS} this ensures the continuity of the structure function with exact cancellations of discontinuities at mass thresholds.
\begin{figure}
\centering
\includegraphics[width=\textwidth]{figures/section6/section6-2/chg_vf_n3lo_plots.png}
\caption{\label{fig: GM-VFNS_estimation_g}The N$^{3}$LO GM-VFNS function $C_{H, g}^{\mathrm{VF},\ (3)}$ compared with the N$^{3}$LO ZM-VFNS function $C_{H, g}^{\mathrm{ZM},\ (3)}$ across a variety of $x$ and $Q^{2}$ values. $C_{H, g}^{\mathrm{VF},\ (3)}$ is parameterised via Equations~\eqref{eq: ChgGM-VFNS}, \eqref{eq: ChgFF} and \eqref{eq: ChqFF_param}. Mass threshold is set at the charm quark level ($m_{h}^{2} = m_{c}^{2} = 1.4\ \text{GeV}^{2}$).}
\end{figure}
Fig.~\ref{fig: GM-VFNS_estimation_g} displays our approximation for the GM-VFNS coefficient function across a range of $(x, Q^{2})$ via a parameterisation for $C_{H, g}^{\mathrm{FF},\ (3)}$ and the relevant subtraction term in Equation~\eqref{eq: ChgGM-VFNS}. Fig.~\ref{fig: GM-VFNS_estimation_g} also contains the uncertainty in this approximation stemming from $A_{Hg}^{(3)}$ (see Section~\ref{subsec: AHg}). This uncertainty is suppressed as we move to high-$Q^{2}$ owing to the required convergence of the GM-VFNS onto the corresponding ZM-VFNS gluon coefficient function at N$^{3}$LO.
Included in Fig.~\ref{fig: GM-VFNS_estimation_g} is the best fit prediction for $C_{H, g}^{\mathrm{VF},\ (3)}$ (corresponding to the best fit of $A_{Hg}^{(3)}$ approximated in Section \ref{sec: n3lo_OME}).
Overall we see the resultant shape of $C_{H,g}^{(3)}$ is within our predicted range and follows a sensible shape that matches with the known high-$Q^{2}$ FFNS behaviour. Contrasting this with NNLO, the shape across the range of $x$ values shown is less consistent. There is no guarantee that this should be the case, since we do not know how the perturbative nature of QCD will behave. However, we do maintain the relatively consistent order of magnitude across the evolution in $Q^{2}$, therefore the exact form of the shape across $Q^{2}$ will be less important in the resultant structure function picture.
\subsection*{$C_{q,q}^{\mathrm{NS}}$}\label{subsec: Cqqns}
The light quark coefficient functions involve small heavy flavour contributions at higher orders from heavy quarks produced away from the photon vertex. As discussed in Section~\ref{subsec: genframe_continuous} the low-$Q^{2}$ FFNS function in this case is unknown. However, since the heavy flavour contributions to the light quark structure function $F_{2,q}(x,Q^{2})$ are very small, any choice of sensible variation in $Q^{2}$ has a near negligible effect on the overall structure function. Further to this, as is apparent from lower order examples, it can be expected that the light quark coefficient functions remain relatively constant across $Q^{2}$.
Using Equation~\eqref{eq: FFexpanse} and Equation~\eqref{eq: expanse_q}, the non-singlet coefficient function is stated as,
\begin{equation}
C_{q,q}^{\mathrm{FF},\ \mathrm{NS}} = A_{qq,H}^{\mathrm{NS}}\otimes C_{q,q}^{\mathrm{VF}\ \mathrm{NS}},
\end{equation}
\begin{subequations}
\begin{align}
\mathcal{O}(\alpha_{s}^{0}): & \hspace{1cm}
\begin{multlined}
C_{q,q,\ \mathrm{NS}}^{\mathrm{FF},\ (0)}\ =\ C_{q,q,\ \mathrm{NS}}^{\mathrm{VF},\ (0)}
\end{multlined}
\\ \mathcal{O}(\alpha_{s}^{1}): & \hspace{1cm}
\begin{multlined}
C_{q,q,\ \mathrm{NS}}^{\mathrm{FF},\ (1)}\ =\ C_{q,q,\ \mathrm{NS}}^{\mathrm{VF},\ (1)}
\end{multlined}
\\ \mathcal{O}(\alpha_{s}^{2}): & \hspace{1cm}
\begin{multlined}\label{eq: NSlightquark - NNLO}
C_{q,q,\ \mathrm{NS}}^{\mathrm{FF},\ (2)}\ =\ C_{q,q,\ \mathrm{NS}}^{\mathrm{VF},\ (2)} + A_{qq,H}^{\mathrm{NS},\ (2)}
\end{multlined}
\\ \mathcal{O}(\alpha_{s}^{3}): & \hspace{1cm}
\begin{multlined}\label{eq: NSlightquark - N3LO}
C_{q,q,\ \mathrm{NS}}^{\mathrm{FF},\ (3)}\ =\ C_{q,q,\ \mathrm{NS}}^{\mathrm{VF},\ (3)} + A_{qq,H}^{\mathrm{NS},\ (3)} + C_{q,q,\ \mathrm{NS}}^{\mathrm{VF},\ (1)}\otimes A_{qq,H}^{\mathrm{NS},\ (2)}.
\end{multlined}
\end{align}
\label{eq: NSlightquark}
\end{subequations}
From Equation~\eqref{eq: NSlightquark} the FFNS contribution at LO and NLO is identical to the GM-VFNS and ZM-VFNS function at high-$Q^{2}$. Physically for heavy quarks to affect light quarks, a larger number of vertices than are allowed at LO and NLO must be present to enable interactions involving heavy quarks. We therefore begin our discussion at NNLO.
\subsubsection*{NNLO}
At NNLO the functions included in Equation~\eqref{eq: NSlightquark - NNLO} are known exactly~\cite{vogt:NNLOns,Buza:OMENNLO}. Assembling these together, we provide an example of how the GM-VFNS function converges to the familiar ZM-VFNS function for the light quark. By performing this exercise, expectations as to how $C_{q,q}^{\mathrm{NS}}$ will behave at N$^{3}$LO can be constructed.
\begin{figure}
\centering
\includegraphics[width=\textwidth]{figures/section6/section6-2/cqqns_vf_nnlo_plots.png}
\caption{\label{fig: Cqq_ns_NNLO} The NNLO GM-VFNS function $C_{q, q,\ \mathrm{NS}}^{\mathrm{VF},\ (2)}$ compared with the NNLO ZM-VFNS function $C_{q, q,\ \mathrm{NS}}^{\mathrm{ZM},\ (2)}$ across a variety of $x$ and $Q^{2}$ values. Mass threshold is set at the charm quark level ($m_{h}^{2} = m_{c}^{2} = 1.4\ \text{GeV}^{2}$).}
\end{figure}
From Fig.~\ref{fig: Cqq_ns_NNLO} $C_{q,q,\ \mathrm{NS}}^{\mathrm{VF},\ (2)}$ quickly converges onto the ZM-VFNS function with the difference between the low and high-$Q^{2}$ being within $10\%$ at large-$x$ and within $0.01\%$ at small-$x$. This weak scaling with $Q^{2}$ reinforces the statement that it is possible to approximate the N$^{3}$LO function relatively well without extensive low-$Q^{2}$ information.
\subsubsection*{N$^{3}$LO}
Equation~\eqref{eq: NSlightquark - N3LO} involves a mixture of functions known exactly (ZM-VFNS high-$Q^{2}$ limit~\cite{Vermaseren:2005qc}) and functions that are completely unknown ($C_{q,q,\ \mathrm{NS}}^{\mathrm{FF},\ (3)}$). This presents an issue as it is no longer possible to rely on $C_{q,q,\ \mathrm{NS}}^{FF,\ (3)}$ to constrain the low-$Q^{2}$ limit.
Nevertheless, by utilising the experience gained from NNLO, it is feasible to choose any sensible choice for the low-$Q^{2}$ limit. In practice, due to the observed weak scaling in $Q^{2}$, the exact form at low-$Q^{2}$ will not present any noticeable differences.
A naive choice for heuristically placing the $C_{q,q,\ \mathrm{NS}}^{\mathrm{FF},\ (3)}(Q^{2}\rightarrow 0)$ function would be a constant value i.e. no scaling in $Q^{2}$. We propose to use the intuition from NNLO and the overall fit quality to give us potentially a more sensible and viable choice for the GM-VFNS approximation\footnote{The differences in fit quality for sensible choices are $<0.05\%$ compared to the overall $\chi^{2}$ for the light quark NS coefficient function.}.
By inserting the high-$Q^{2}$ limit into the NS part of Equation~\eqref{eq: CqFF_param}, the result is a crude approximation to $C_{q,q,\ \mathrm{NS}}^{\mathrm{FF},\ (3)}(Q^{2}\rightarrow 0)$. Combining this with Equation~\eqref{eq: NSlightquark - N3LO}, we obtain a GM-VFNS parameterisation which is relatively constant across $Q^{2}$ (similar to the NNLO behaviour) with any differences arising from the subtraction terms which are known.
\begin{figure}
\centering
\includegraphics[width=\textwidth]{figures/section6/section6-2/cqqns_vf_n3lo_plots.png}
\caption{\label{fig: Cqq_ns_N3LO}The N$^{3}$LO GM-VFNS function $C_{q, q,\ \mathrm{NS}}^{\mathrm{VF},\ (3)}$ compared with the N$^{3}$LO ZM-VFNS function $C_{q, q,\ \mathrm{NS}}^{\mathrm{ZM},\ (3)}$ across a variety of $x$ and $Q^{2}$ values. $C_{q, q,\ \mathrm{NS}}^{\mathrm{VF},\ (3)}$ is parameterised via Equations~\eqref{eq: NSlightquark - N3LO} and \eqref{eq: CqFF_param}. Mass threshold is set at the charm quark level ($m_{h}^{2} = m_{c}^{2} = 1.4\ \text{GeV}^{2}$).}
\end{figure}
Fig.~\ref{fig: Cqq_ns_N3LO} shows the result of this approximation for the full $C_{q, q,\ \mathrm{NS}}^{\mathrm{VF},\ (3)}$ function. We notice that the behaviour is similar to that of NNLO across all $(x, Q^{2})$ and appropriately larger in magnitude to account for the extra contributions obtained at N$^{3}$LO compared to NNLO. By definition, the parameterisation converges well to the ZM-VFNS scheme with the magnitude at high-$Q^{2}$ (ZM-VFNS regime) remaining similar to that at low-$Q^{2}$ for each specific value of $x$. This final point gives assurances that even if this low-$Q^{2}$ guess is not entirely representative of the actual N$^{3}$LO function, the effects of including this approximation are virtually negligible in a PDF fit.
\subsection*{$C_{q,q}^{\mathrm{PS}}$}\label{subsec: Cqqps}
To complete the light-quark GM-VFNS coefficient function picture the pure-singlet contribution from Equation~\eqref{eq: FFexpanse} and Equation~\eqref{eq: expanse_q} is described by,
\begin{equation}
C_{q,q}^{\mathrm{FF},\ \mathrm{PS}} = C_{q,q}^{\mathrm{VF},\ \mathrm{PS}} \otimes A_{qq,H}^{\mathrm{PS}}\ +\ C_{q,g}^{\mathrm{VF}}\otimes A_{gq,H} +\ C_{q,H}^{\mathrm{VF},\ \mathrm{PS}}\otimes A_{Hq}
\end{equation}
\begin{subequations}
\begin{align}
\mathcal{O}(\alpha_{s}^{0}): & \hspace{1cm}
\begin{multlined}
C_{q,q}^{\mathrm{FF},\ \mathrm{PS},\ (0)}\ =\ 0
\end{multlined}
\\ \mathcal{O}(\alpha_{s}^{1}): & \hspace{1cm}
\begin{multlined}
C_{q,q}^{\mathrm{FF},\ \mathrm{PS},\ (1)}\ =\ 0
\end{multlined}
\\ \mathcal{O}(\alpha_{s}^{2}): & \hspace{1cm}
\begin{multlined}
C_{q,q}^{\mathrm{FF},\ \mathrm{PS},\ (2)}\ =\ C_{q,q,\ \mathrm{PS}}^{\mathrm{VF},\ (2)}
\end{multlined}
\\ \mathcal{O}(\alpha_{s}^{3}): & \hspace{1cm}
\begin{multlined}\label{eq: PSlightquark - N3LO}
C_{q,q}^{\mathrm{FF},\ \mathrm{PS},\ (3)}\ =\ C_{q,q,\ \mathrm{PS}}^{\mathrm{VF},\ (3)} + C_{q,g}^{\mathrm{VF},\ (1)}\otimes A_{gq,H}^{(2)}.
\end{multlined}
\end{align}
\label{eq: PSlightquark}
\end{subequations}
As with the non-singlet analysis the heavy flavour contributions to the pure-singlet appear at higher orders to allow for the possibility of heavy quark contributions. In the pure-singlet case, the heavy flavour contributions are pushed one order higher than the non-singlet
due to the requirement for an extra
intermediary gluon.
\subsubsection*{N$^{3}$LO}
In the pure-singlet case, the FFNS function is non-existent up until N$^{3}$LO. Because of this, we choose to parameterise the pure-singlet with a weak constraint suppressing the FFNS function $C_{q,q}^{\mathrm{FF},\ \mathrm{PS},\ (3)}$ across all $x$ for very low-$Q^{2}$. The reason for this is that the coefficient functions acquire more contributions as they exist through higher orders. If $C_{q,q}^{\mathrm{FF},\ \mathrm{PS},\ (3)}$ is beginning at this order, then one could expect the low-$Q^{2}$ form to be relatively small compared to the known ZM-VFNS function~\cite{Vermaseren:2005qc}. This is somewhat justified by the low-$Q^{2}$ kinematic restrictions for the singlet distribution which broadly manifest into a suppression at low-$Q^{2}$. We reiterate here that the low-$Q^{2}$ form of this function is still essentially around the same magnitude across all $Q^{2}$. Therefore, as with $C_{q,q}^{\mathrm{FF},\ \mathrm{NS},\ (3)}$, it will be virtually negligible in the overall structure function.
After constructing the approximation for $C_{q,q}^{\mathrm{FF},\ \mathrm{PS},\ (3)}$ with Equation~\eqref{eq: CqFF_param}, Equation~\eqref{eq: PSlightquark - N3LO} is used to approximate the GM-VFNS function. The exact form of Equation~\eqref{eq: PSlightquark - N3LO} is chosen based on intuition and where the best fit quality can be achieved\footnote{The differences in fit quality for sensible choices are $<0.1\%$ compared to the overall $\chi^{2}$ for the light quark PS coefficient function.}.
\begin{figure}
\centering
\includegraphics[width=\textwidth]{figures/section6/section6-2/cqqps_vf_n3lo_plots.png}
\caption{\label{fig: Cqq_ps_N3LO}The N$^{3}$LO GM-VFNS function $C_{q, q,\ \mathrm{PS}}^{\mathrm{VF},\ (3)}$ compared with the N$^{3}$LO ZM-VFNS function $C_{q, q,\ \mathrm{PS}}^{\mathrm{ZM},\ (3)}$ across a variety of $x$ and $Q^{2}$ values. $C_{q, q,\ \mathrm{PS}}^{\mathrm{VF},\ (3)}$ is parameterised via Equations~\eqref{eq: PSlightquark - N3LO} and \eqref{eq: CqFF_param}. Mass threshold is set at the charm quark level ($m_{h}^{2} = m_{c}^{2} = 1.4\ \text{GeV}^{2}$).}
\end{figure}
It can be seen from Fig.~\ref{fig: Cqq_ps_N3LO} that the overall magnitude of $C_{q, q,\ \mathrm{PS}}^{\mathrm{VF},\ (3)}$ decreases substantially towards large-$x$ as one would expect from a pure-singlet function. Inspecting the predicted values of $C_{q, q,\ \mathrm{PS}}^{\mathrm{VF},\ (3)}$, we can confirm that the non-singlet function from Fig.~\ref{fig: Cqq_ns_N3LO} begins to dominate at large-$x$. Conversely towards small-$x$, $C_{q, q,\ \mathrm{PS}}^{\mathrm{VF},\ (3)}$ is much larger than $C_{q, q,\ \mathrm{NS}}^{\mathrm{VF},\ (3)}$, thereby preserving the familiar interplay between quark distributions. The suppression of the FFNS parameterisation towards low-$Q^{2}$ is also seen to give sensible results in terms of the expected percentage change in magnitude through the range of $Q^{2}$ values. Specifically we see $< 10\%$ difference in magnitude between low and high-$Q^{2}$. Since scale violating terms become more dominant at higher orders and we are essentially at leading order in terms of heavy flavour contributions, a high level of scaling with $Q^{2}$ is not expected at this order.
\subsection*{$C_{q,g}$}\label{subsec: Cqg}
Finally the gluon-light quark coefficient function is constructed from Equation~\eqref{eq: FFexpanse} and Equation~\eqref{eq: expanse_q} to be,
\begin{equation}
C_{q,g}^{\mathrm{FF}} = C_{q,q}^{\mathrm{VF}} \otimes A_{qg,H}\ +\ C_{q,g}^{\mathrm{VF}}\otimes A_{gg,H} +\ C_{q,H}^{\mathrm{VF},\ \mathrm{PS}}\otimes A_{Hg}
\end{equation}
\begin{subequations}
\begin{align}
\mathcal{O}(\alpha_{s}^{0}): & \hspace{1cm}
\begin{multlined}
C_{q,g}^{\mathrm{FF},\ (0)}\ =\ 0
\end{multlined}
\\ \mathcal{O}(\alpha_{s}^{1}): & \hspace{1cm}
\begin{multlined}
C_{q,g}^{\mathrm{FF},\ (1)}\ =\ C_{q,g}^{\mathrm{VF},\ (1)}
\end{multlined}
\\ \mathcal{O}(\alpha_{s}^{2}): & \hspace{1cm}
\begin{multlined}\label{eq: Glightquark - NNLO}
C_{q,g}^{\mathrm{FF},\ (2)}\ =\ C_{q,g}^{\mathrm{VF},\ (2)} + A_{qg, H}^{(2)}
\end{multlined}
\\ \mathcal{O}(\alpha_{s}^{3}): & \hspace{1cm}
\begin{multlined}\label{eq: Glightquark - N3LO}
C_{q,g}^{\mathrm{FF},\ (3)}\ =\ C_{q,g}^{\mathrm{VF},\ (3)} + A_{qg,H}^{(3)} + C_{qg}^{\mathrm{VF},\ (1)}\otimes A_{gg,H}^{(2)} \\+ C_{q,H,\ \mathrm{PS}}^{\mathrm{VF},\ (2)}\otimes A_{Hg}^{(1)}.
\end{multlined}
\end{align}
\label{eq: Glightquark}
\end{subequations}
For $C_{q,g}$, the FFNS function is non-existent up to NNLO, similar to $C_{q,q,\ \mathrm{NS}}^{\mathrm{FF}, (3)}$. However, the $A_{qg,H}$ contribution at NNLO is sub-leading in $n_{f}$~\cite{Buza:OMENNLO} and is therefore not considered here.
\subsubsection*{N$^{3}$LO}
\begin{figure}
\centering
\includegraphics[width=\textwidth]{figures/section6/section6-2/cqg_vf_n3lo_plots.png}
\caption{\label{fig: Cqg_N3LO}The N$^{3}$LO GM-VFNS function $C_{q, g}^{\mathrm{VF},\ (3)}$ compared with the N$^{3}$LO ZM-VFNS function $C_{q, g}^{\mathrm{ZM},\ (3)}$ across a variety of $x$ and $Q^{2}$ values. $C_{q, g}^{\mathrm{VF},\ (3)}$ is parameterised via Equations~\eqref{eq: Glightquark - N3LO} and \eqref{eq: CqFF_param}. Mass threshold is set at the charm quark level ($m_{h}^{2} = m_{c}^{2} = 1.4\ \text{GeV}^{2}$).}
\end{figure}
At N$^{3}$LO in Equation~\eqref{eq: Glightquark - N3LO}, no information is available for the $C_{q, g}^{\mathrm{FF},\ (3)}$ at low-$Q^{2}$. Whereas at high-$Q^{2}$ the ZM-VFNS function is known~\cite{Vermaseren:2005qc}. To construct the parameterisation, we apply the same method described for $C_{q,q,\ \mathrm{PS}}^{\mathrm{FF},\ (3)}$. Specifically, by applying a suppression to the FFNS parameterisation in the low-$Q^{2}$ limit. After constructing the parameterisation for $C_{q, g}^{\mathrm{FF},\ (3)}$ with Equation~\eqref{eq: CqFF_param}, Equation~\eqref{eq: Glightquark - N3LO} is used to approximate the GM-VFNS function. Since there is no information in the low-$Q^{2}$ limit, the parameterisation in Equation~\eqref{eq: CqFF_param} is chosen roughly based on how the fit prefers the evolution in $Q^{2}$ to behave.
Fig.~\ref{fig: Cqg_N3LO} illustrates the GM-VFNS function in Equation~\eqref{eq: Glightquark - N3LO} with Equation~\eqref{eq: CqFF_param} as $C_{q, g}^{\mathrm{FF},\ (3)}$ across a range of $x$ and $Q^{2}$. $C_{q, g}^{\mathrm{VF},\ (3)}$ increases in magnitude when moving to smaller $x$ and by definition converges onto the ZM-VFNS function. The convergence in this case is chosen to be less steep than for the light quark convergences due to some minor tensions in the fit\footnote{The differences in fit quality for sensible choices of Equation~\eqref{eq: CqFF_param} are $<0.5\%$ compared to the overall $\chi^{2}$ for the light quark gluon coefficient function.}. The magnitude of $C_{q, g}^{\mathrm{VF},\ (3)}$ across the entire range of $Q^{2}$ is still relatively constant, although less flat than the behaviour predicted for $C_{q,q,\ \mathrm{PS/NS}}^{\mathrm{VF},\ (3)}$. However, considering Equation~\eqref{eq: Glightquark - N3LO}, some justification for this behaviour can be offered. When comparing the contributions to the FFNS functions in the NS, PS and gluon cases (Equations~\eqref{eq: NSlightquark - N3LO}, \eqref{eq: PSlightquark - N3LO} and \eqref{eq: Glightquark - N3LO} respectively), the $A_{Hg}$ and $A_{gg,H}$ contributions involved in $C_{q,g}^{\mathrm{FF}}$ are much larger than the contributions from $A_{gq,H}$, $A_{Hq}$ and $A^{\mathrm{NS}}_{qq,H}$. Therefore we can expect a larger difference across $Q^{2}$ for the $C_{q, g}^{\mathrm{VF},\ (3)}$ function. With this being said, the specific form at low-$Q^{2}$ is not very important in current PDF fits, only that the form is continuous and valid.
\section{\texorpdfstring{N$^{3}$LO}{N3LO} \texorpdfstring{$K$}{K}-factors}\label{sec: n3lo_K}
Thus far the primary concern has been the N$^{3}$LO additions to the theoretical form of the DIS cross section. However, to complement these changes it is necessary to extend other cross section data to the same order. With these ingredients it is possible to maintain a consistent approximate N$^{3}$LO treatment across all datasets. At the time of writing, $K$-factors which provide exact transformations for each dataset up to NNLO are available\footnote{An exception to this is the CMS $7\ \text{TeV}\ W + c$ \cite{CMS7Wpc} dataset where $K$-factors are available only up to NLO.}. Although there has been progress in N$^{3}$LO calculations for various processes including Drell-Yan (DY), top production and Higgs processes~\cite{DY_N3LO_Kfac, Gehrmann:DYN3LO, duhr:DY2021, Kidonakis:tt,Ball:2013bra,Bonvini:2014jma,Bonvini:2016frm,Ahmed:2016otz,Bonvini:2018ixe,Bonvini:2018iwt,Bonvini:2013kba,anastasiou2014higgs,anastasiou2016high,Mistlberger:2018}, there is still missing information on how these $K$-factors behave above NNLO. In this section we investigate the effects of the $K$-factors for each dataset when extended to N$^{3}$LO. Five process categories are considered separately: Drell-Yan, Jets, $p_{T}$ jets, $t\bar{t}$ production and Dimuon data. Inside each of these process categories we assume a perfect positive correlation between the behaviour of datasets i.e. all Drell-Yan $K$-factor shifts from NNLO are positively correlated. Clearly this treatment is a simplification, based on the expectation of a high degree of correlation between datasets concerned with the same processes. In practice, the uncertainty introduced from including these $K$-factors is already relatively small compared to the other sources of MHOUs already discussed, therefore any correction to this is guaranteed to be small (this will be shown more clearly in Section~\ref{sec: results}).
\subsection{Extension to \texorpdfstring{aN$^{3}$LO}{aN3LO}}
The extension to aN$^{3}$LO is parameterised with a mixture of the NLO and NNLO $K$-factors. This allows control of the magnitude and shape of the transformation from NNLO to aN$^{3}$LO, using the known shifts from lower orders.
The basic idea is presented as,
\begin{equation}\label{eq: k_fac_param}
K^{\mathrm{N}^{3}\mathrm{LO}/\mathrm{LO}} = a_{\mathrm{NNLO}}\ K^{\mathrm{NNLO}/\mathrm{LO}} + a_{\mathrm{NLO}}\ K^{\mathrm{NLO}/\mathrm{LO}},
\end{equation}
where $K^{\mathrm{N}^{3}\mathrm{LO}/\mathrm{LO}}, K^{\mathrm{NNLO}/\mathrm{LO}}\ \mathrm{and}\ K^{\mathrm{NLO}/\mathrm{LO}}$ are the relevant $K$-factors with respect to the LO cross section, and $a_{\mathrm{N(N)LO}}$ are variational parameters controlling the mixture of NNLO and NLO $K$-factors included in the N$^{3}$LO $K$-factor approximation.
The $K$-factors as a perturbative expansion in $\alpha_{s}$ are,
\begin{equation}\label{eq: K_fac_ex}
K(y) = 1 + \frac{\alpha_{s}}{\pi}D(y) + \left(\frac{\alpha_{s}}{\pi}\right)^{2}E(y) + \left(\frac{\alpha_{s}}{\pi}\right)^{3}F(y) + \mathcal{O}(\alpha_{s}^{4}).
\end{equation}
where $y$ is some data point, $D(y)$ and $E(y)$ are the NLO and NNLO contributions to the $K$-factor, and $F(y)$ is the unknown N$^{3}$LO contribution.
To describe this formalism in terms of physical observables we consider the cross section,
\begin{equation}
\sigma = \sigma_{0} + \sigma_{1} + \sigma_{2} + \dots \equiv \sigma_{\mathrm{NNLO}} + \dots,
\end{equation}
where there is an implicit order of $\alpha_{s}^{p+i}$ absorbed into the definition of $\sigma_{i}$ beginning at the relevant LO for each process, i.e. $p = 0$ for DY.
$K^{\mathrm{NLO/LO}}$ is then the relative shift from $\sigma_{\mathrm{LO}}$ to $\sigma_{\mathrm{NLO}}$,
\begin{equation}\label{eq: KNLO}
K^{\mathrm{NLO/LO}} = \frac{\sigma_{0} + \sigma_{1}}{\sigma_{0}} = 1 + \frac{\sigma_{1}}{\sigma_{0}}.
\end{equation}
Similarly for NNLO we have,
\begin{equation}\label{eq: KNNLO}
K^{\mathrm{NNLO/LO}} = \frac{\sigma_{0} + \sigma_{1} + \sigma_{2}}{\sigma_{0}} = 1 + \frac{\sigma_{1}}{\sigma_{0}} + \frac{\sigma_{2}}{\sigma_{0}}.
\end{equation}
Defining this in terms of $D$ and $E$ contributions it is possible to write,
\begin{equation}\label{eq: ED}
D = \frac{\pi}{\alpha_{s}}\left(\frac{\sigma_{0} + \sigma_{1}}{\sigma_{0}} - 1\right) = \frac{\pi}{\alpha_{s}}\frac{\sigma_{1}}{\sigma_{0}}
\end{equation}
\begin{equation}\label{eq: ED2}
E = \left(\frac{\pi}{\alpha_{s}}\right)^{2}\left(\frac{\sigma_{0} + \sigma_{1} + \sigma_{2}}{\sigma_{0}} - \frac{\sigma_{0} + \sigma_{1}}{\sigma_{0}}\right) = \left(\frac{\pi}{\alpha_{s}}\right)^{2}\frac{\sigma_{2}}{\sigma_{0}}.
\end{equation}
To complete this picture Equations~\eqref{eq: ED} and \eqref{eq: ED2} can be substituted into Equation~\eqref{eq: K_fac_ex} up to NNLO,
\begin{equation}
1 + \frac{\alpha_{s}}{\pi}D + \left(\frac{\alpha_{s}}{\pi}\right)^{2}E = \frac{\sigma_{0} + \sigma_{1} + \sigma_{2}}{\sigma_{0}} = K^{\mathrm{NNLO/LO}},
\end{equation}
which then multiplies the LO cross section $\sigma_{0}$ to produce the expanded NNLO cross section $\sigma_{2}$.
Moving to N$^{3}$LO, we write
\begin{equation}
\sigma = \sigma_{0} + \sigma_{1} + \sigma_{2} + \sigma_{3} + \dots \equiv \sigma_{\mathrm{N^{3}LO}} + \dots,
\end{equation}
where $\sigma_{3} = a_{1} \sigma_{1} + a_{2} \sigma_{2}$ is some superposition of the two lower orders, with $(a_{1}, a_{2}) = (0, 0)$ reproducing the NNLO case.
Pushing forward with this approximation and using the definitions for $\sigma_{1,2}$ in terms of $K$-factors (Equations~\eqref{eq: KNLO} and \eqref{eq: KNNLO}) we have,
\begin{align}
\sigma_{\mathrm{N^3LO}} &= \sigma_{\mathrm{NNLO}} + a_{1}\sigma_{1} + a_{2}\sigma_{2} \\
&= \sigma_{\mathrm{NNLO}} + a_{1}\sigma_{0} (K^{\mathrm{NLO/LO}} - 1) + a_{2} \sigma_{0} (K^{\mathrm{NNLO/LO}} - K^{\mathrm{NLO/LO}})
\end{align}
since,
\begin{align}
\sigma_{1} &= \sigma_{0}\left(K^{\mathrm{NLO/LO}} - 1\right) \\
\sigma_{2} &= \sigma_{0}\left(K^{\mathrm{NNLO/LO}} - \sigma_{1} - \sigma_{0}\right)\nonumber\\
&= \sigma_{0}\left(K^{\mathrm{NNLO/LO}} - K^{\mathrm{NLO/LO}}\right).
\end{align}
From here one can obtain,
\begin{equation}\label{eq: Knn-kn}
K^{\mathrm{NNLO/LO}} - K^{\mathrm{NLO/LO}} = \frac{\sigma_{2}}{\sigma_{0}} = \frac{\sigma_{2} + \sigma_{0}}{\sigma_{0}} - 1 \approx \frac{\sigma_{2} + \sigma_{1} + \sigma_{0}}{\sigma_{1} + \sigma_{0}} - 1 = K^{\mathrm{NNLO/NLO}} - 1,
\end{equation}
assuming $\sigma_{1} \ll \sigma_{0}$, which is in general true for a valid perturbative expansion. Using \eqref{eq: Knn-kn} $\sigma_{\mathrm{N^{3}LO}}$ can be expressed by,
\begin{align}
\sigma_{\mathrm{N^3LO}} &= \sigma_{\mathrm{NNLO}} + a_{1}\sigma_{0} (K^{\mathrm{NLO/LO}} - 1) + a_{2} \sigma_{0} (K^{\mathrm{NNLO/NLO}} - 1)\\
& = \sigma_{\mathrm{NNLO}}\left(1 + a_{1}(K^{\mathrm{NLO/LO}} - 1) + a_{2} (K^{\mathrm{NNLO/NLO}} - 1)\right),
\end{align}
where $\sigma_{2} \ll \sigma_{1} \ll \sigma_{0}$.
This defines the proposed approximated N$^{3}$LO cross section. It is given in terms of extra contributions from lower order shifts, which are controlled by variational parameters $a_{1}$ and $a_{2}$. It is also true that the contributions to N$^{3}$LO are expected to be suppressed by $\alpha_{s} / \pi$ in the NNLO case and $(\alpha_{s}/\pi)^{2}$ in the NLO case to account for the strengths of each contribution. Currently this is taken into account within the variational parameters $a_{1}, a_{2}$. However for the purpose of this description, it is more appropriate to explicitly redefine $a_{1}, a_{2} = a_{s}^{2}\hat{a}_{1}, a_{s}\hat{a}_{2}$ where $a_{s} = \mathcal{N}\alpha_{s}$ and $\mathcal{N}$ is some normalisation factor. This then results in,
\begin{equation}\label{eq: meth2}
K^{\mathrm{N^3LO/LO}} = K^{\mathrm{NNLO/LO}}\left(1 + \hat{a}_{1}\mathcal{N}^{2}\alpha_{s}^{2}(K^{\mathrm{NLO/LO}} - 1) + \hat{a}_{2} \mathcal{N}\alpha_{s} (K^{\mathrm{NNLO/NLO}} - 1)\right).
\end{equation}
where the LO cross section $\sigma_{0}$ is cancelled and Equation~\eqref{eq: meth2} is written in terms of the $K$-factor shifts only.
Retracing our steps and using Equations~\eqref{eq: ED} and \eqref{eq: ED2}, Equation~\eqref{eq: meth2} can be recast into,
\begin{equation}\label{eq: Kfac_expanded}
K^{\mathrm{N^3LO/LO}} = K^{\mathrm{NNLO/LO}}\left(1 + \alpha_{s}^{3}\hat{a}_{1}\frac{\mathcal{N}^{2}}{\pi}D + \alpha_{s}^{3}\hat{a}_{2} \frac{\mathcal{N}}{\pi^{2}}E\right).
\end{equation}
where the correct order $\mathcal{O}(\alpha_{s}^{3})$ is explicitly included in the parameterisation and by choosing $\mathcal{N} \sim 3$ we expect $\hat{a}_{1}, \hat{a}_{2} = \mathcal{O}(1)$. The reason for this is that one can expect the perturbative expansion to converge following a decreasing trend in corrections by up to $\sim 30-40\%$ at each order\footnote{This estimate is chosen based on the average $K$-factor shifts for lower orders across the relevant datasets included in a global fit (examples can be seen in Fig.'s~\ref{fig:Kfac_DY}--\ref{fig:Kfac_dimuon}).}. By applying this heuristic choice, we take our NNLO correction to N$^{3}$LO and divide it by the true NNLO correction,
\begin{equation}
\alpha_{s}^{3}\hat{a}_{2} \frac{\mathcal{N}}{\pi^{2}}E\ /\ \left(\frac{\alpha_{s}}{\pi}\right)^{2}E = \alpha_{s}\hat{a}_{2}\mathcal{N} \sim 30\%,
\end{equation}
with $\alpha_s \sim 0.1$ (around the W/Z mass). Following from this, in order for $\hat{a}_{1} = \mathcal{O}(1)$ to be true, it must be the case that $\mathcal{N} \sim 3$.
For the NLO correction, we expect $30\%$ to carry through to the NNLO and a further $30\%$ to remain from the NNLO to N$^{3}$LO. A quick calculation can reveal,
\begin{equation}
\alpha_{s}^{3}\hat{a}_{1}\frac{\mathcal{N}^{2}}{\pi}D\ /\ \frac{\alpha_{s}}{\pi}D = \alpha_{s}^{2}\hat{a}_{1}\mathcal{N}^{2} \sim 9\%.
\end{equation}
where if the same arguments are taken as before, $\mathcal{N} \sim 3$ is also required. With this formalism manipulated to ensure one can expect $\hat{a}_{1}, \hat{a}_{2} = \mathcal{O}(1)$, an allowed variation of $-1 < \hat{a}_{1}, \hat{a}_{2} < 1$ can be chosen, where the central value is exactly the NNLO prediction.
Reflecting on this, it is worth noting that these fitted $K$-factors will be sensitive to all orders, not just N$^{3}$LO. Considering these $K$-factors as approximating asymptotic behaviour to all orders in perturbation theory when assessing the stability of predictions, we can be less concerned with any somewhat large shifts from NNLO to aN$^{3}$LO, as we will specifically see in the case of Fig.'s~\ref{fig:Kfac_pTjet} and \ref{fig:Kfac_top}. Finally, we remind the reader that at higher orders, new terms with more divergent leading logarithms appear which are missed by the current theoretical description.
Due to this, the all-orders asymptotic description will still remain approximate up to the inclusion of more divergent leading logarithms in $(x,Q^{2})$ limits at even higher orders.
\subsection{Numerical Results}\label{subsec: kfac_res}
Using this formalism for the aN$^{3}$LO $K$-factors, we present the global fit results for each of the five process categories considered.
\subsubsection*{Drell-Yan Processes}
\begin{figure}
\centering
\includegraphics[width=\textwidth]{figures/section7/section7-2/DY_LHCb2015WZ_kfac.png}
\caption{$K$-factor expansion up to aN$^{3}$LO shown for the LHCb 2015 $W,\ Z$ dataset \cite{LHCbZ7,LHCbWZ8}. The $K$-factors shown here are absolute i.e. all with respect to LO ($K^{\mathrm{N^{m}LO/LO}}\ \forall\ m \in \{1, 2, 3\}$).}
\label{fig:Kfac_DY}
\end{figure}
For the Drell-Yan processes (all calculated at $\mu_{r,f}=m_{ll}/2$), a reduction of $\sim 1-2\%$ in the $K$-factor shift is predicted across most of the corresponding datasets at aN$^{3}$LO. This is in agreement with recent work~\cite{Gehrmann:DYN3LO}. An example of this reduction is shown in Fig.~\ref{fig:Kfac_DY}.
\begin{figure}
\centering
\includegraphics[width=\textwidth]{figures/section7/section7-2/DY_ATLAShighprec_kfac.png}
\caption{$K$-factor expansion up to aN$^{3}$LO shown for the ATLAS 7 TeV high precision $W,\ Z$ dataset \cite{ATLASWZ7f}. The $K$-factors shown here are absolute i.e. all with respect to LO ($K^{\mathrm{N^{m}LO/LO}}\ \forall\ m \in \{1, 2, 3\}$).}
\label{fig:Kfac_DY_highprec}
\end{figure}
Conversely, Fig.~\ref{fig:Kfac_DY_highprec} displays an example where the $K$-factor shift has much less of a contribution at N$^{3}$LO. This is a feature of the ATLAS datasets included in the fit due to the impact of chosen $p_{T}$ cuts which reduce the sensitivity to higher orders.
\begin{table}
\centerline{
\begin{tabular}{|c|c|c|c|}
\hline
DY Dataset & $\chi^{2}$ & $\Delta \chi^{2}$ & $\Delta \chi^{2}$ from NNLO\\
& & from NNLO & (NNLO $K$-factors)\\
\hline
E866 / NuSea $pp$ DY \cite{E866DY} & 209.2 / 184 & $-15.9$ & $-11.0$ \\
E866 / NuSea $pd/pp$ DY \cite{E866DYrat} & 7.6 / 15 & $-2.7$ & $-2.6$ \\
D{\O} II $Z$ rap. \cite{D0Zrap} & 16.8 / 28 & $+0.5$ & $+0.3$ \\
CDF II $Z$ rap. \cite{CDFZrap} & 39.6 / 28 & $+2.4$ & $+1.6$ \\
D{\O} II $W \rightarrow \nu \mu$ asym. \cite{D0Wnumu} & 16.8 / 10 & $-0.5$ & $+0.2$ \\
CDF II $W$ asym. \cite{CDF-Wasym} & 19.9 / 13 & $+0.9$ & $+0.6$ \\
D{\O} II $W \rightarrow \nu e$ asym. \cite{D0Wnue} & 29.2 / 12 & $-4.7$ & $-5.1$ \\
ATLAS $W^{+},\ W^{-},\ Z$ \cite{ATLASWZ} & 30.0 / 30 & $+0.1$ & $+0.4$ \\
CMS W asym. $p_{T} > 35\ \text{GeV}$ \cite{CMS-easym} & 7.0 / 11 & $-0.8$ & $-0.5$ \\
CMS W asym. $p_{T} > 25, 30\ \text{GeV}$ \cite{CMS-Wasymm} & 7.5 / 24 & $+0.1$ & $+0.0$ \\
LHCb $Z \rightarrow e^{+}e^{-}$ \cite{LHCb-Zee} & 20.6 / 9 & $-2.1$ & $-1.7$ \\
LHCb W asym. $p_{T} > 20\ \text{GeV}$ \cite{LHCb-WZ} & 12.9 / 10 & $+0.5$ & $+0.9$ \\
CMS $Z \rightarrow e^{+}e^{-}$ \cite{CMS-Zee} & 17.3 / 35 & $-0.6$ & $-0.6$ \\
ATLAS High-mass Drell-Yan \cite{ATLAShighmass} & 18.5 / 13 & $-0.4$ & $-1.2$ \\
CMS double diff. Drell-Yan \cite{CMS-ddDY} & 137.1 / 132 & $-7.4$ & $+12.3$ \\
LHCb 2015 $W, Z$ \cite{LHCbZ7,LHCbWZ8} & 97.2 / 67 & $-2.2$ & $-3.4$ \\
LHCb $8\ \text{TeV}$ $Z \rightarrow ee$ \cite{LHCbZ8} & 27.1 / 17 & $+0.9$ & $-0.1$ \\
CMS $8\ \text{TeV}\ W$ \cite{CMSW8} & 12.0 / 22 & $-0.7$ & $+0.3$ \\
ATLAS $7\ \text{TeV}$ high prec. $W, Z$ \cite{ATLASWZ7f} & 110.5 / 61 & $-6.2$ & $-18.2$ \\
D{\O} $W$ asym. \cite{D0Wasym} & 8.6 / 14 & $-3.4$ & $-2.5$ \\
ATLAS $8\ \text{TeV}$ High-mass DY \cite{ATLASHMDY8} & 63.4 / 48 & $+6.3$ & $+2.8$ \\
ATLAS $8\ \text{TeV}\ W$ \cite{ATLASW8} & 55.1 / 22 & $-2.3$ & $-0.8$ \\
ATLAS $8\ \text{TeV}$ double diff. $Z$ \cite{ATLAS8Z3D} & 80.8 / 59 & $-4.8$ & $-2.4$ \\
\hline
Total & 1044.8 / 864 & $-43.1$ & $-30.5$\\
\hline
\end{tabular}
}
\caption{\label{tab: DY_kfac} Table showing the relevant DY datasets and how the individual $\chi^{2}$ changes from NNLO by including the N$^{3}$LO treatment of $K$-factors, and theoretical N$^{3}$LO additions discussed earlier. The result with purely NNLO $K$-factors included for all data in the fit is also given.}
\end{table}
Table~\ref{tab: DY_kfac} demonstrates that in most cases, the new fitted DY aN$^{3}$LO $K$-factors are producing a slightly better fit with a substantial cumulative effect. We remind the reader that we have included a total of 20 extra parameters into the fit (10 controlling aN$^{3}$LO $K$-factors, 5 controlling aN$^{3}$LO splitting functions and 5 controlling heavy flavour aN$^{3}$LO contributions). These extra 20 parameters are fit across all datasets and multiple processes, whereas the decrease here is for a subset of datasets corresponding to the DY processes included in a global fit. Therefore this decrease in $\chi^{2}$ being much greater than the number of extra parameters, even when only considering the DY sets, is evidence that this extra N$^{3}$LO information is helping the fit perform better in general. This will be discussed further in Section~\ref{sec: results} where a full global $\chi^{2}$ breakdown is provided.
Across these datasets, the $K$-factors act to extend the description of these processes to approximate N$^{3}$LO.
The result of including this procedure is a better fit in the DY regime while also relaxing tensions with other processes included in the fit. Comparing the $\Delta \chi^{2}$ results with and without aN$^{3}$LO $K$-factors, we can see the extent to which the $K$-factors and all other N$^{3}$LO additions are reducing the overall $\chi^{2}$.
In some individual cases, the dataset $\chi^{2}$ becomes slightly worse relative to NNLO (e.g. ATLAS $8\ \text{TeV}$ High-mass DY \cite{ATLASHMDY8}), whilst in a few others the $\chi^{2}$ improvement upon addition of the aN$^{3}$LO splitting functions, transition matrix elements and coefficient function pieces is seen to deteriorate upon addition of the aN$^{3}$LO $K$-factors, e.g. such as for the ATLAS $7\ \text{TeV}$ high prec. $W, Z$ \cite{ATLASWZ7f}, which is observed to prefer N$^{3}$LO theory with NNLO $K$-factors. The addition of the aN$^{3}$LO $K$-factors do nonetheless result in a net reduction in $\chi^{2}$ and for a large number of cases the aN$^{3}$LO $K$-factors allow for a slight reduction in the individual $\chi^{2}$. The CMS double diff. Drell-Yan \cite{CMS-ddDY} shows a particularly large reduction when these are added on top of the aN$^{3}$LO theory, this is a dataset which shows some tension with the DIS N$^{3}$LO additions which is then eased by the addition of the aN$^{3}$LO $K$-factors
\subsubsection*{Jet Production Processes}
\begin{figure}
\centering
\includegraphics[width=\textwidth]{figures/section7/section7-2/jet_CMS7jets_kfac.png}
\caption{$K$-factor expansion up to aN$^{3}$LO shown for the CMS 7 TeV jets dataset ($R=0.7$) \cite{CMS7jetsfinal}. The $K$-factors shown here are absolute i.e. all with respect to LO ($K^{\mathrm{N^{m}LO/LO}}\ \forall\ m \in \{1, 2, 3\}$).}
\label{fig:Kfac_jets}
\end{figure}
The jets processes (all calculated for $\mu_{r,f}=p_{T}^{jet}$) show a general increase in the $K$-factor shifts from NNLO as seen in Fig.~\ref{fig:Kfac_jets}, which displays the $K$-factor expansion up to aN$^{3}$LO for the CMS 7 TeV jets dataset~\cite{CMS7jetsfinal}. It is apparent that there is a mild shift to N$^{3}$LO from the NNLO $K$-factor. This behaviour follows what one might expect for a perturbative expansion considering the forms of the NLO and NNLO functions.
\begin{table}
\centerline{
\begin{tabular}{|c|c|c|c|}
\hline
Jets Dataset & $\chi^{2}$ & $\Delta \chi^{2}$ & $\Delta \chi^{2}$ from NNLO\\
& & from NNLO & (NNLO $K$-factors)\\
\hline
CDF II $p\bar{p}$ incl. jets \cite{CDFjet} & 68.4 / 76 & $+8.0$ & $+0.9$ \\
D{\O} II $p\bar{p}$ incl. jets \cite{D0jet} & 113.6 / 110 & $-6.6$ & $-3.6$ \\
ATLAS $7\ \text{TeV}$ jets \cite{ATLAS7jets} & 214.5 / 140 & $-7.1$ & $+2.5$ \\
CMS $7\ \text{TeV}$ jets \cite{CMS7jetsfinal} & 189.8 / 158 & $+14.1$ & $+13.9$ \\
CMS $8\ \text{TeV}$ jets \cite{CMS8jets} & 272.6 / 174 & $+11.3$ & $+22.9$ \\
CMS $2.76\ \text{TeV}$ jet \cite{CMS276jets} & 113.9 / 81 & $+11.1$ & $+13.3$ \\
\hline
Total & 972.9 / 739 & $+30.8$ & $+49.8$\\
\hline
\end{tabular}
}
\caption{\label{tab: Jets_kfac}Table showing the relevant jet datasets and how the individual $\chi^{2}$ changes from NNLO by including the N$^{3}$LO treatment of $K$-factors. The result with purely NNLO $K$-factors included for all data in the fit is also given.}
\end{table}
A $\chi^{2}$ summary of the Jets datasets is provided in Table~\ref{tab: Jets_kfac}. By combining the N$^{3}$LO structure function and DGLAP additions (Section's \ref{sec: n3lo_split} to \ref{sec: n3lo_coeff}) with NNLO $K$-factors, the fit exhibits a substantial increase in the $\chi^{2}$ from Jets data. Including aN$^{3}$LO $K$-factors acts to reduce some of this tension with over half the initial overall $\chi^{2}$ increase still remaining.
We note that in the case of the ATLAS $7\ \text{TeV}$ jets~\cite{ATLAS7jets}, it is well known that there are issues in achieving a good fit quality across all rapidity bins (see~\cite{MMHTjets} for a detailed study as well as~\cite{Bogdan:decorr} where the 8 TeV data are presented and the same issues observed). In~\cite{MMHTjets,Bogdan:decorr} the possibility of decorrelating some of the systematic error sources where the degree of correlation is less well established, was considered and indeed in our study we follow such a procedure, as described in~\cite{Thorne:MSHT20}. Alternatively, however, it might be that the issues in fit quality could at least part be due to deficiencies in theoretical predictions, such as MHOs. To assess this, we revert to the default ATLAS correlation scenario and repeat the global fit. We find that the $\chi^{2}$ deteriorates by $41.2$ points to 255.7, which is very close to the result found in a pure NNLO fit~\cite{Thorne:MSHT20}. In other words, in our framework the impact of MHOUs does not resolve this issue.
The $\chi^{2}$ results for datasets in Table~\ref{tab: Jets_kfac} show evidence for some tensions with the N$^{3}$LO form of the high-$x$ gluon. It is also apparent that the CMS data is in more tension than ATLAS datasets with N$^{3}$LO structure function and DGLAP theory. Therefore it will be interesting to see how this behaviour changes when considering this data as dijets in the global fit~\cite{AbdulKhalek:2020jut}. We do not consider the dijet data here, though this will be addressed in a future publication.
\subsubsection*{$Z\ p_{T}$ \& Vector Boson $+$ Jets Processes}
\begin{figure}
\centering
\includegraphics[width=\textwidth]{figures/section7/section7-2/pT_ATLAS8ZpT_kfac.png}
\caption{$K$-factor expansion up to aN$^{3}$LO shown for the ATLAS 8 TeV $Z\ p_{T}$ dataset \cite{ATLASZpT}. The $K$-factors shown here are absolute i.e. all with respect to LO ($K^{\mathrm{N^{m}LO/LO}}\ \forall\ m \in \{1, 2, 3\}$).}
\label{fig:Kfac_pTjet}
\end{figure}
In the case of $Z\ p_{T}$ \& vector boson $+$ jet processes (all calculated at $\mu_{r,f}=\sqrt{p_{T,ll}^2 + m_{ll}^2}$), the $K$-factor shift is almost completely dominated by the ATLAS 8 TeV $Z\ p_{T}$ dataset~\cite{ATLASZpT} (due to the larger number of data points included in this dataset) shown in Fig.~\ref{fig:Kfac_pTjet}.
The gluon is less directly constrained than the quarks in a global fit. Therefore it can be expected that the significant modifications at small-$x$ will indirectly affect the high-$x$ gluon, where these processes are most sensitive. Considering the jet production processes in Table~\ref{tab: Jets_kfac}, when performing separate PDF fits not including ATLAS 8 TeV $Z\ p_{T}$ data~\cite{ATLASZpT}, we find a reduction of $\Delta\chi^{2} = - 7.8$ in CMS $8\ \text{TeV}$ jets data~\cite{CMS8jets} eliminating most of the tension for this dataset (similar to MSHT20 NNLO results in Table 16 of \cite{Thorne:MSHT20}). Further to this, when not including HERA and ATLAS 8 TeV $Z\ p_{T}$ data we find a reduction of $\Delta\chi^{2} = - 27.6$ in CMS $8\ \text{TeV}$ jet data~\cite{CMS8jets} and $\Delta\chi^{2} = - 11.8$ in CMS $2.76\ \text{TeV}$ jet data~\cite{CMS276jets}. These results therefore suggest tensions in between the Jet production and $Z\ p_{T}$ \& vector boson $+$ Jets (and HERA) datasets in terms of the effect on the gluon.
Although the overall magnitude of the $K$-factor in Fig.~\ref{fig:Kfac_pTjet} may seem large, this new shift is contained within a $15\%$ increase from NNLO (due to the NLO and NNLO $K$-factors also being significant). Moreover, not only does the size of this shift have some dependence on the central scale, but this shift may be more correctly interpreted as the preferred all-orders cross section rather than simply the pure ${\rm N}^3{\rm LO}$ result.
\begin{table}
\centerline{
\begin{tabular}{|c|c|c|c|}
\hline
$p_{T}$ Jets Dataset & $\chi^{2}$ & $\Delta \chi^{2}$ & $\Delta \chi^{2}$ from NNLO\\
& & from NNLO & (NNLO $K$-factors)\\
\hline
CMS $7\ \text{TeV}\ W + c$ \cite{CMS7Wpc} & 12.3 / 10 & $+3.7$ & $+1.3$ \\
ATLAS $8\ \text{TeV}\ Z\ p_{T}$ \cite{ATLASZpT} & 105.8 / 104 & $-82.7$ & $-53.0$ \\
ATLAS $8\ \text{TeV}\ W + \text{jets}$ \cite{ATLASWjet} & 19.1 / 30 & $+0.9$ & $+0.3$ \\
\hline
Total & 137.1 / 144 & $-78.1$ & $-51.5$\\
\hline
\end{tabular}
}
\caption{\label{tab: pt_jet_kfac}Table showing the relevant $Z\ p_{T}$ \& Vector Boson jet datasets and how the individual $\chi^{2}$ changes from NNLO by including the N$^{3}$LO treatment of $K$-factors, and theoretical N$^{3}$LO additions discussed earlier. The result with purely NNLO $K$-factors included for all data in the fit is also given.}
\end{table}
The extent of the $\chi^{2}$ reduction in the $Z\ p_{T}$ datasets is shown in Table~\ref{tab: pt_jet_kfac}. Note that around $\sim 60\%$ of the improvement to the ATLAS $8\ \text{TeV}\ Z\ p_{T}$~\cite{ATLASZpT} $\chi^{2}$ is due to the extra N$^{3}$LO theory included in the DGLAP and DIS descriptions.
It is also known the ATLAS $8\ \text{TeV}\ Z\ p_{T}$ data~\cite{ATLASZpT} previously exhibited a significant level of tension with many datasets (including HERA data) at NNLO~\cite{Thorne:MSHT20}. This was investigated by performing a global PDF fit with and without HERA data and comparing the individual $\chi^{2}$'s from each dataset. At NNLO it was found that the ATLAS $8\ \text{TeV}\ Z\ p_{T}$ dataset~\cite{ATLASZpT} reduced by $\Delta\chi^{2} = -39.2$ when fitting to all non-HERA data (see Table~\ref{tab: no_HERA_fullNNLO}). At N$^{3}$LO we show that the ATLAS $8\ \text{TeV}\ Z\ p_{T}$ dataset~\cite{ATLASZpT} actually increased by $\Delta\chi^{2} = +3.2$ when fitting to all non-HERA data (see Table~\ref{tab: no_HERA_fullN3LO}). This therefore completely eliminates this tension with the N$^{3}$LO additions and confirming the issue previously observed when attempting to fit the ATLAS $8\ \text{TeV}\ Z\ p_{T}$ dataset~\cite{ATLASZpT} was concerned with MHOs. This is in contrast with the result observed for ATLAS $7\ \text{TeV}$ jets~\cite{ATLAS7jets} where the issues with fit quality were not alleviated by the inclusion of MHO information.
Finally we remind the reader that the CMS $7\ \text{TeV}\ W + c$ dataset~\cite{CMS7Wpc} does not include a $K$-factor at NNLO. To overcome this, we tie the overall N$^{3}$LO $K$-factor shift to the NLO value ($K^{\mathrm{NNLO/NLO}} = 1$ in Equation~\eqref{eq: meth2}), therefore contributing as an overall normalisation effect.
\subsubsection*{Top Quark Processes}
\begin{figure}[t]
\centering
\includegraphics[width=0.7\textwidth]{figures/section7/section7-2/top_CMS8singdiff_kfac.png}
\caption{$K$-factor expansion up to aN$^{3}$LO shown for the CMS 8 TeV single diff. $t\bar{t}$ dataset \cite{CMSttbar08_ytt}. The $K$-factors shown here are absolute i.e. all with respect to LO ($K^{\mathrm{N^{m}LO/LO}}\ \forall\ m \in \{1, 2, 3\}$).}
\label{fig:Kfac_top}
\end{figure}
Moving to top quark processes, for the single differential datasets the scale choice for $\mu_{r,f}$ is $H_T/4$ with the exception of data differential in the average transverse momentum of the top or anti-top, $p_T^t,p_T^{\bar{t}}$, for which $m_T/2$ is used. For the double diff. dataset the scale choice is $H_T/4$ and for the inclusive top $\sigma_{t\bar{t}}$ a scale of $m_{t}$ is chosen. Fig.~\ref{fig:Kfac_top} displays the $K$-factor shifts up to N$^{3}$LO for the CMS 8 TeV single diff. $t\bar{t}$ dataset~\cite{CMSttbar08_ytt}, which shows the greatest reduction in its $\chi^2$. A familiar perturbative pattern can be seen for this process's $K$-factors, with the shift at aN$^{3}$LO increasing by around 3--4\% from NNLO. This is in agreement with a recent $\sim3.5\%$ predicted increase in the N$^3$LO $t\bar{t}$ production $K$-factor at $8\ \text{TeV}$ in~\cite{Kidonakis:tt}, whereby an approximate N$^{3}$LO cross section for $t\bar{t}$ production in proton-proton collisions has been calculated employing a resummation formalism~\cite{Kidonakis:1997gm,Kidonakis:2009ev,Kidonakis:2010dk,Kidonakis:2021yrh}.
\begin{table}
\centerline{
\begin{tabular}{|c|c|c|c|}
\hline
Top Dataset & $\chi^{2}$ & $\Delta \chi^{2}$ & $\Delta \chi^{2}$ from NNLO\\
& & from NNLO & (NNLO $K$-factors)\\
\hline
Tevatron, ATLAS, CMS $\sigma_{t\bar{t}}$ \cite{Tevatron-top,ATLAS-top7-1,ATLAS-top7-2,ATLAS-top7-3,ATLAS-top7-4,ATLAS-top7-5,ATLAS-top7-6,CMS-top7-1,CMS-top7-2,CMS-top7-3,CMS-top7-4,CMS-top7-5,CMS-top8} & 14.2 / 17 & $-0.3$ & $-0.7$ \\
ATLAS $8\ \text{TeV}$ single diff. $t\bar{t}$ \cite{ATLASsdtop} & 24.7 / 25 & $-0.9$ & $-0.1$ \\
ATLAS $8\ \text{TeV}$ single diff. $t\bar{t}$ dilep. \cite{ATLASttbarDilep08_ytt} & 2.1 / 5 & $-1.3$ & $-0.8$ \\
CMS $8\ \text{TeV}$ double diff. $t\bar{t}$ \cite{CMS8ttDD} & 23.9 / 15 & $+1.4$ & $+4.9$ \\
CMS $8\ \text{TeV}$ single diff. $t\bar{t}$ \cite{CMSttbar08_ytt} & 8.4 / 9 & $-4.7$ & $-5.5$ \\
\hline
Total & 73.4 / 71 & $-5.9$ & $-2.2$\\
\hline
\end{tabular}
}
\caption{\label{tab: top_kfac}Table showing the relevant Top Quark datasets and how the individual $\chi^{2}$ changes from NNLO by including the N$^{3}$LO treatment of $K$-factors, and theoretical N$^{3}$LO additions discussed earlier. The result with purely NNLO $K$-factors included for all data in the fit is also given.}
\end{table}
The $\chi^{2}$ results in Table~\ref{tab: top_kfac} display a mildly better fit for top processes, with most datasets not feeling a large overall effect from the N$^{3}$LO additions. Comparing with and without aN$^{3}$LO $K$-factors, we see a slightly better fit overall, with most of the reduction in overall $\chi^{2}$ stemming from CMS $8\ \text{TeV}$ double diff. $t\bar{t}$ data~\cite{CMS8ttDD}.
\subsubsection*{Semi-Inclusive DIS Dimuon Processes}
\begin{figure}
\centering
\includegraphics[width=\textwidth]{figures/section7/section7-2/Dimuon_NuTeV_kfac.png}
\caption{$K$-factor expansion up to aN$^{3}$LO shown for the NuTeV $\nu N \rightarrow \mu \mu X$ dataset \cite{Dimuon}. The $K$-factors shown here are absolute i.e. all with respect to LO ($K^{\mathrm{N^{m}LO/LO}}\ \forall\ m \in \{1, 2, 3\}$).}
\label{fig:Kfac_dimuon}
\end{figure}
The final set of results to consider in this Section are the aN$^{3}$LO $K$-factors associated with semi-inclusive DIS dimuon cross sections (with $\mu_{r,f}^2=Q^{2}$). Although the dimuon is associated with the DIS process described from our approximate N$^{3}$LO structure function picture, it is a semi-inclusive DIS process. Therefore it is sensible to treat this process as entirely separate from DIS. The $K$-factors shown in Fig.~\ref{fig:Kfac_dimuon} (for the NuTeV $\nu N \rightarrow \mu \mu X$ data~\cite{Dimuon}) are largely similar to NNLO. The reason for this is mostly due to these datasets also including a branching ratio (${\rm BR}(D\to \mu)$) which absorbs any overall normalisation shifts.
This behaviour is not a concern since in practice these two work in tandem and when combined together it makes no difference where the normalisation factors are absorbed into.
\begin{table}
\centerline{
\begin{tabular}{|c|c|c|c|c|}
\hline
\multirow{2}{*}{} & \multirow{2}{*}{NLO} & \multirow{2}{*}{NNLO} & \multirow{2}{*}{aN$^{3}$LO} & aN$^{3}$LO\\
& & & & (NNLO $K$-factors)\\
\hline
& \multirow{2}{*}{0.095} & \multirow{2}{*}{0.088} & \multirow{2}{*}{0.080} & \multirow{2}{*}{0.079} \\
${\rm BR}(D\to \mu)$ & & & & \\
\hline
\end{tabular}
}
\caption{\label{tab: dimuon_br}Table displaying dimuon branching ratios (BRs) at NLO, NNLO, aN$^{3}$LO and aN$^{3}$LO with NNLO $K$-factors.}
\end{table}
Investigating the change in the BR's with the addition of N$^{3}$LO contributions in Table~\ref{tab: dimuon_br}, the BR at N$^{3}$LO decreases substantially from NNLO, with little difference from the addition of aN$^{3}$LO $K$-factors. The predicted dimuon BR at aN$^{3}$LO is now outside the allowed $\pm 1 \sigma$ range of $0.092 \pm 0.010$. When performing a fit with the BR fixed at its central value (BR = $0.092$), one is able to observe the effect of manually forcing the normalisation into the $K$-factor variation alone. The result of this is a worse global fit quality $\Delta \chi^{2} = + 17.7$, where $+7.5$ units arise from an increased penalty for the Dimuon $K$-factor description and $+3.8$ units from a slightly worse fit to the Dimuon datasets listed in Table~\ref{tab: dimuon_kfac}. The rest of the observed increase in $\chi^{2}_{\mathrm{global}}$ is dominated by a $+4.0$ increase in the ATLAS $7\ \text{TeV}$ high prec.~$W, Z$ \cite{ATLASWZ7f} data due to a smaller strange quark PDF (compensating the higher BR in dimuon datasets). Returning to consider the case of the $K$-factors and BR together, the predicted effect on dimuon datasets is very similar. However due to the errors accounting for a larger allowed shift in the BR relative to the $K$-factors, the fit favours moving the BR by a larger amount to reduce the penalty $\chi^{2}$ contribution from $K$-factors which explains the results shown in Table~\ref{tab: dimuon_br}.
\begin{table}
\centerline{
\begin{tabular}{|c|c|c|c|}
\hline
Dimuon Dataset & $\chi^{2}$ & $\Delta \chi^{2}$ & $\Delta \chi^{2}$ from NNLO\\
& & from NNLO & (NNLO $K$-factors)\\
\hline
CCFR $\nu N \rightarrow \mu\mu X$ \cite{Dimuon} & 69.2 / 86 & $+1.5$ & $+2.9$ \\
NuTeV $\nu N \rightarrow \mu\mu X$ \cite{Dimuon} & 55.3 / 84 & $-3.1$ & $-3.4$ \\
\hline
Total & 124.6 / 170 & $-1.6$ & $-0.5$\\
\hline
\end{tabular}
}
\caption{\label{tab: dimuon_kfac}Table showing the relevant Dimuon datasets and how the individual $\chi^{2}$ changes from NNLO by including the N$^{3}$LO treatment of $K$-factors, and theoretical N$^{3}$LO additions discussed earlier. The result with purely NNLO $K$-factors included for all data in the fit is also given.}
\end{table}
Table~\ref{tab: dimuon_kfac} further confirms the expectation that the Dimuon datasets are not too sensitive to N$^{3}$LO additions. The results with and without a full treatment of aN$^{3}$LO $K$-factors are also similar in magnitude.
It is therefore clear that the dimuon BR's are compensating for any indirect normalisation effects from the form of the PDFs in the full aN$^{3}$LO fit, as opposed to the aN$^{3}$LO $K$-factors.
\section{\texorpdfstring{N$^{3}$LO}{N3LO} Splitting Functions}\label{sec: n3lo_split}
Splitting functions at N$^{3}$LO allow us to more accurately describe the evolution of the PDFs. These functions are estimated here and the resulting approximations are included within the framework described in Section~\ref{subsec: genframe}. In all singlet cases we set $n_{f} = 4$ before constructing our approximations and ignore any corrections to this from any further change in the number of flavours\footnote{An exception to this are the cases of $P_{qg}$ and $P_{qq}^{\mathrm{PS}}$ where we have already defined $P_{qg} \equiv n_{f}P_{qg}$ and $P_{qq}^{\mathrm{PS}} \equiv n_{f}P_{qq}^{\mathrm{PS}}$. Therefore the leading $n_{f}$ dependence is already taken into account.}. In the non-singlet case, we calculate the approximate parts of $P_{qq}^{NS\ (3)}$ with $n_{f}=4$ however, there is a relatively large amount of information about the $n_{f}$-dependence included from~\cite{4loopNS}. Therefore in the final result we choose to allow the full $n_{f}$-dependence to remain for the non-singlet splitting function.
\subsection{4-loop Approximations}\label{subsec: 4loop_split}
\subsection*{$P_{qg}^{(3)}$}
We begin by considering the four-loop quark-gluon splitting function. Here we provide a more detailed explanation of the method described in Section~\ref{subsec: genframe} which will then be applied to the remaining splitting functions considered in this section. Four even-integer moments are known for $P_{qg}^{(3)}(n_{f} = 4)$ from~\cite{S4loopMoments,S4loopMomentsNew}, along with the LL small-$x$ term from \cite{Catani:1994sq}.
The functions made available for the $P_{qg}$ analysis are,
\begin{alignat}{5}\label{eq: PqgComb}
f_{1}(x) \quad &= \quad \frac{1}{x} \quad &&\text{or} \quad \ln^{4}x \quad &&\text{or} \quad \ln^{3} x \quad &&\text{or} \quad \ln^{2} x,\nonumber \\
f_{2}(x) \quad &= \quad \ln x, \nonumber \\
f_{2}(x) \quad &= \quad 1 \quad &&\text{or} \quad x \quad &&\text{or} \quad x^{2},\nonumber \\
f_{3}(x) \quad &= \quad \ln^{4}(1-x) \quad &&\text{or} \quad \ln^{3}(1-x) \quad &&\text{or} \quad \ln^{2}(1-x) \quad &&\text{or} \quad \ln(1-x),\nonumber \\
f_{e}(x,\rho_{qg}) \quad &= \quad \frac{C_{A}^{3}}{3\pi^{4}}\bigg(\frac{82}{81}\ +&&2\zeta_{3}\bigg)\frac{1}{2}\frac{\ln^{2}1/x}{x}\ +\ &&\rho_{qg}\ \frac{\ln 1/x}{x},
\end{alignat}
where $\rho_{qg}$ is the variational parameter. this is varied between $-2.5 < \rho_{qg} < -0.9$, which has been chosen to satisfy the criteria described in Section~\ref{subsec: genframe}. The set of functions in Equation~\eqref{eq: PqgComb} is chosen from the analysis of lower orders. Specifically, following the pattern of functions from lower orders, it can be shown that at this order we expect the most dominant large-$x$ term to be $\ln^{4}(1-x)$ and $\ln^{4}x$ to be the highest power of $\ln x$ at small-$x$.
\begin{figure}
\centering
\includegraphics[width=0.49\textwidth]{figures/section4/section4-1/PqgapproxComb_m0-9.png}
\includegraphics[width=0.49\textwidth]{figures/section4/section4-1/PqgapproxComb_m2-5.png}
\caption{Combinations of functions with an added variational factor ($\rho_{qg}$) controlling the
NLL term. Combinations of functions at the upper (left) and lower (right) bounds of the variation are shown. The solid lines indicate the upper and lower bounds for this function chosen from the relevant criteria.}
\label{fig: PqgRangeofa}
\end{figure}
Fig.~\ref{fig: PqgRangeofa} displays an example of the variation found from the different choices of functions that encapsulate the chosen range of $\rho_{qg}$. We also show the upper (A) and lower (B) bounds (at small-$x$) for the entire uncertainty (solid line) combining the variation in the functions and in the variation of $\rho_{qg}$. The upper ($P_{qg}^{(3),A}$) and lower ($P_{qg}^{(3), B}$) bounds are given by,
\begin{multline}
P_{qg}^{(3), A} = 5.5219\ \frac{1}{x} + 10.401\ \ln x - 6.5373\ x^{2} + 0.0036299\ \ln^{4}(1-x) \\+ \frac{C_{A}^{3}}{3\pi^{4}}\bigg(\frac{82}{81}\ + 2\zeta_{3}\bigg)\frac{1}{2}\frac{\ln^{2}1/x}{x}\ -1.7\ \frac{\ln 1/x}{x},
\end{multline}
\begin{multline}
P_{qg}^{(3), B} = 12.582\ \ln^{2} x + 5.3065\ \ln x + 1.7957\ x^{2} - 0.0041296\ \ln^{4}(1-x) \\+ \frac{C_{A}^{3}}{3\pi^{4}}\bigg(\frac{82}{81}\ + 2\zeta_{3}\bigg)\frac{1}{2}\frac{\ln^{2}1/x}{x}\ -2.5\ \frac{\ln 1/x}{x}.
\end{multline}
Using this information, a fixed functional form is chosen to be,
\begin{multline}\label{eq: pqg_fff}
P_{qg}^{(3)} = A_{1}\ \ln^{2} x + A_{2}\ \ln x + A_{3}\ x^{2} + A_{4}\ \ln^{4}(1-x) \\+ \frac{C_{A}^{3}}{3\pi^{4}}\bigg(\frac{82}{81}\ + 2\zeta_{3}\bigg)\frac{1}{2}\frac{\ln^{2}1/x}{x}\ + \rho_{qg}\ \frac{\ln 1/x}{x}
\end{multline}
and $\rho_{qg}$ is allowed to vary as $-2.5 < \rho_{qg} < -0.8$. This fixed functional form identically matches with the lower bound $P_{qg}^{(3),B}$ and the expansion of the variation of $\rho_{qg}$ enables (to within $\sim 1\%$) the absorption of the small-$x$ upper bound uncertainty (predicted from $P_{qg}^{(3),A}$) into the variation. In other areas of $x$ there are larger deviations from the upper bound ($\sim 10\%$) when using this convenient fixed functional form. However, in these regions the function is already relatively small, therefore any larger percentage deviations are negligible.
Also since the heuristic choice of variation found earlier is intended as a guide, we are not bound by any solid constraints to precisely reconstruct it with our subsequent choice of fixed functional form. Therefore it is entirely justified to be able to slightly adapt the shape of the variation in less dominant regions.
\subsection*{$P_{qq}^{\mathrm{NS}\ (3)}$}
As discussed in Section~\ref{sec: structure},
the quark-quark splitting function
is comprised of a pure-singlet and non-singlet contribution. We approximate each part independently, although the final quark-quark singlet function will be almost completely dominated by the pure-singlet, except at very high-$x$.
The four-loop non-singlet splitting function has been the subject of relatively extensive research and is known exactly for a number of regimes. For example in~\cite{4loopNS}, some important exact contributions to the four-loop non-singlet splitting functions are presented, along with 8 even-integer moments for each of the $+$ and $-$ distributions~\cite{4loopNS}. In this discussion we are exclusively approximating the non-singlet $+$-distribution, as this is the part that contributes to the full singlet quark-quark splitting function. The other relevant non-singlet distributions $P_{\mathrm{NS}}^{(3),\ -}$ and $P_{\mathrm{NS}}^{(3),\ \mathrm{sea}}$ (described in \cite{Moch:2004pa}), are set to the central values predicted from \cite{4loopNS} since any variation in these functions are negligible. All presently known information is used in this approximation, with results similar to that seen in~\cite{4loopNS} but with our own choice of functions.
\begin{alignat}{5}\label{eq: PnsComb}
f_{1}(x) \ &= \ \frac{1}{(1-x)_{+}}, \qquad \quad f_{2}(x) \ = \ (1-x)\ \ln(1-x), \qquad f_{3}(x)\ =\ (1-x)\ \ln^{2}(1-x), \nonumber \\
f_{4}(x) \ &= \ (1-x)\ \ln^{3}(1-x), \quad f_{5}(x) \ = \ 1, \quad f_{6}(x) \ = \ x, \quad
f_{7}(x) \ = \ x^{2}, \quad f_{8}(x) \ =\ \ln^{2} x, \nonumber
\end{alignat}
\begin{multline}
f_{e}(x, \rho_{qq}^{\mathrm{NS}}) \ = \ C_{F} n_{c}^{3} P_{\mathrm{L}, 0}^{(3)}(x)+C_{F} n_{c}^{2} n_{f} P_{\mathrm{L}, 1}^{(3)}(x)+P_{L n_{f}}^{(3)+}(x) + \rho_{qq}^{\mathrm{NS}}\ \ln^{3} x \\ -55.876\ \ln^{4}x-2.8313\ \ln^{5}x-0.14883\ \ln^{6}x-2601.7-2118.9\ \ln (1-x) \\ + n_{f} \left(4.6584\ \ln^{4}x+0.2798\ \ln^{5}x+312.16+337.93\ \ln(1-x)\right)
\end{multline}
where the functions $C_{F} n_{c}^{3} P_{\mathrm{L}, 0}^{(3)}(x)+C_{F} n_{c}^{2} n_{f} P_{\mathrm{L}, 1}^{(3)}(x)$ and $P_{L n_{f}}^{(3)+}(x)$ can be found in Equation (4.11) and Equation (4.14) respectively within~\cite{4loopNS}, and $\rho_{qq}^{\mathrm{NS}}$ is our variational parameter. Note that the ansatz from Equation~\eqref{eq: splitAnsatz} has been extended to include 8 pairs of functions and coefficients, to accommodate 8 known moments. Within the $f_{e}(x, \rho_{qq}^{\mathrm{NS}})$ part of Equation~\eqref{eq: PnsComb}, we have chosen to vary the coefficient of the most divergent unknown small-$x$ term ($\ln^{3}x$) with the variation across $0 < \rho_{qq}^{\mathrm{NS}} < 0.014$. Due to the high level of information and larger number of functions allowed to be included, we ignore any functional uncertainty and explicitly define each function. Therefore the only variation needed to be considered as an uncertainty stems from the variation of $\rho_{qq}^{\mathrm{NS}}$.
The resulting approximation is then,
\begin{multline}\label{eq: pns_fff}
P_{\mathrm{NS}}^{(3),\ +} = A_{1}\ \frac{1}{(1-x)_{+}} + A_{2}\ (1-x)\ \ln(1-x) + A_{3}\ (1-x)\ \ln^{2}(1-x) \\ + A_{4}\ (1-x)\ \ln^{3}(1-x) + A_{5} + A_{6}\ x + A_{7}\ x^{2} + A_{8}\ \ln^{2}x + f_{e}(x,\rho_{qq}^{\mathrm{NS}}),
\end{multline}
where no alterations are made to the allowed range of $0 < \rho_{qq}^{\mathrm{NS}} < 0.014$.
\subsection*{$P_{qq}^{\mathrm{PS}\ (3)}$}
We now restrict our analysis to focus on approximating the pure-singlet part of $P_{qq}^{(3)}$, thereby providing a more accurate set of functions with a focus on the small-$x$ regime. To ensure the $P_{qq}^{\mathrm{PS}\ (3)}$ function does not interfere with the large-$x$ regime (where the non-singlet description dominates) the ansatz from Equation~\eqref{eq: splitAnsatz} is adapted to be:
\begin{equation}
P_{ij}^{(3)}(x)=\bigg\{A_{1}f_{1}(x)+A_{2}f_{2}(x)+A_{3}f_{3}(x)+A_{4}f_{4}(x)\bigg\}(1-x)+f_{e}(x, \rho_{qq}^{\mathrm{PS}}).
\label{eq: splitAnsatzPqq}
\end{equation}
This modified parameterisation guarantees that any instabilities in the pure singlet approximation will not wash out the non-singlet behaviour at large-$x$.
Using four available even-integer moments for $n_{f} = 4$~\cite{S4loopMoments,S4loopMomentsNew} and the exact small-$x$ information~\cite{Catani:1994sq}, the chosen set of functions for this approximation is,
\begin{alignat}{5}\label{eq: PqqComb}
f_{1}(x) \quad &= \quad \frac{1}{x} \quad &&\text{or} \quad \ln^{4} x,\nonumber \\
f_{2}(x) \quad &= \quad \ln^{3} x \quad &&\text{or} \quad \ln^{2} x \quad &&\text{or} \quad \ln x,\nonumber \\
f_{3}(x) \quad &= \quad 1 \quad &&\text{or} \quad x \quad &&\text{or} \quad x^{2},\nonumber \\
f_{4}(x) \quad &= \quad \ln^{4}(1-x) \quad &&\text{or} \quad \ln^{3}(1-x) \quad &&\text{or} \quad \ln^{2}(1-x) \quad &&\text{or} \quad \ln(1-x),\nonumber \\
f_{e}(x, \rho_{qq}^{\mathrm{PS}}) \quad &= \quad \frac{C_{A}^{2}C_{F}}{3\pi^{4}}\bigg(\frac{82}{81}&&+2\zeta_{3}\bigg)\frac{1}{2}\frac{\ln^{2}1/x}{x}\ +\ &&\rho_{qq}^{\mathrm{PS}}\ \frac{\ln 1/x}{x},
\end{alignat}
where $\rho_{qq}^{\mathrm{PS}}$ is varied as $-0.7 < \rho_{qq}^{\mathrm{PS}} < 0$. For the variation produced from stable combinations of these functions, we coincidentally end up with the same functional form for both the upper $P_{\mathrm{PS}}^{(3),\ A}$ and lower $P_{\mathrm{PS}}^{(3),\ B}$ bounds. Therefore trivially, the fixed functional form is defined as:
\begin{multline}
P_{\mathrm{PS}}^{(3)} = \bigg\{A_{1}\ \frac{1}{x} + A_{2}\ \ln^{2}x + A_{3}\ x^{2} + A_{4}\ \ln^{2}(1-x)\bigg\}(1-x)\ + \\\frac{C_{A}^{2}C_{F}}{3\pi^{4}}\bigg(\frac{82}{81}+2\zeta_{3}\bigg)\frac{1}{2}\frac{\ln^{2}1/x}{x}\ +\ \rho_{qq}^{\mathrm{PS}}\ \frac{\ln 1/x}{x}(1-x)
\end{multline}
where the variation of $\rho_{qq}^{\mathrm{PS}}$ is unchanged and the entire predicted variation is encapsulated in this form.
\subsection*{$P_{gq}^{(3)}$}
As with the previous singlet splitting functions, four even-integer moments for $n_{f} = 4$ are known~\cite{S4loopMoments,S4loopMomentsNew} along with the LL small-$x$ information~\cite{Lipatov:1976zz,Kuraev:1977fs,Balitsky:1978ic}. The set of functions made available for the combinations in our approximation are stated as,
\begin{alignat}{5}\label{eq: PgqComb}
f_{1}(x) \quad &= \quad \frac{\ln 1/x}{x} \quad &&\text{or} \quad \frac{1}{x},\nonumber \\
f_{2}(x) \quad &= \quad \ln^{3}x, \nonumber \\
f_{3}(x) \quad &= \quad x \quad &&\text{or} \quad x^{2},\nonumber \\
f_{4}(x) \quad &= \quad \ln^{4}(1-x) \quad &&\text{or} \quad \ln^{3}(1-x) \quad &&\text{or} \quad \ln^{2}(1-x) \quad &&\text{or} \quad \ln(1-x),\nonumber \\
f_{e}(x, \rho_{gq}) \quad &= \quad \frac{C_{A}^{3}C_{F}}{3\pi^{4}}\zeta_{3}\frac{\ln^{3}1/x}{x}\ +\ &&\rho_{gq}\ \frac{\ln^{2} 1/x}{x},
\end{alignat}
where $\rho_{gq}$ is set as $\rho_{gq} = -1.8$. In this case, the variation from the choice of functions is large enough to satisfy the criteria in Section~\ref{subsec: genframe} and encapsulate a sensible $\pm 1\sigma$ variation without including any further variation in $\rho_{gq}$.
Similarly to previous approximations, for stable variations we estimate this variation with the fixed functional form,
\begin{equation}
P_{gq}^{(3)} = A_{1}\ \frac{\ln 1/x}{x} + A_{2}\ \ln^{3} x + A_{3}\ x + A_{4}\ \ln(1-x) + \frac{C_{A}^{3}C_{F}}{3\pi^{4}}\zeta_{3}\frac{\ln^{3}1/x}{x}\ +\ \rho_{gq}\ \frac{\ln^{2} 1/x}{x}
\end{equation}
where the allowed range of $\rho_{gq}$ is expanded to $-1.8 < \rho_{gq} < -1.5$ to approximate the variation from the choice of functions. As with the $P_{qg}^{(3)}$ fixed functional form, this new range recovers a variation which is within $\sim 1\%$ of the original, in the dominant areas of $x$.
\subsection*{$P_{gg}^{(3)}$}
Finally we move to the approximation of the gluon-gluon splitting function, where four available even-integer moments for $P_{gg}^{(3)}(n_{f} = 4)$ are known from \cite{S4loopMoments, S4loopMomentsNew}. The list of functions (including the known small-$x$ LL and NLL terms from \cite{Lipatov:1976zz,Kuraev:1977fs,Balitsky:1978ic,Fadin:1998py,Ciafaloni:1998gs}) used for the approximation is,
\begin{alignat}{6}\label{eq: PggComb}
f_{1}(x) \quad &= \quad \frac{1}{x} \quad &&\text{or} \quad \ln^{3} x \quad &&\text{or} \quad \ln^{2} x,\nonumber \\
f_{2}(x) \quad & = \quad \ln x,\nonumber \\
f_{3}(x) \quad &= \quad 1 \quad &&\text{or} \quad x \quad &&\text{or} \quad x^{2},\nonumber \\
f_{4}(x) \quad &= \quad \frac{1}{(1-x)}_{+} \quad &&\text{or} \quad \ln^{2}(1-x) \quad &&\text{or} \quad \ln(1-x),\nonumber
\end{alignat}
\begin{multline}
f_{e}(x, \rho_{gg}) \quad = \quad \frac{C_{A}^{4}}{3\pi^{4}}\zeta_{3}\frac{\ln^{3}1/x}{x} +\frac{1}{\pi^{4}}\bigg[C_{A}^{4}\bigg(-\frac{1205}{162}+\frac{67}{36} \zeta_{2}+\frac{1}{4} \zeta_{2}^{2}-\frac{11}{2} \zeta_{3}\bigg) \\+n_{f} C_{A}^{3}\bigg(-\frac{233}{162}+\frac{13}{36} \zeta_{2}-\frac{1}{3}\zeta_{3}\bigg) \\+n_{f} C_{A}^{2} C_{F}\bigg(\frac{617}{243} -\frac{13}{18} \zeta_{2}+\frac{2}{3} \zeta_{3}\bigg)\bigg]\frac{1}{2}\frac{\ln^{2}1/x}{x}\ +\ \rho_{gg}\ \frac{\ln 1/x}{x},
\end{multline}
where $\rho_{gg}$ is varied as $-5 < \rho_{gg} < 15$ and $n_{f} = 4$. The fixed functional form is then chosen to be,
\begin{multline}
P_{gg}^{(3)} = A_{1}\ \ln^{2} x + A_{2}\ \ln x + A_{3}\ x^{2} + A_{4}\ \ln^{2}(1-x) + \frac{C_{A}^{4}}{3\pi^{4}}\zeta_{3}\frac{\ln^{3}1/x}{x} \\ +\frac{1}{\pi^{4}}\bigg[C_{A}^{4}\bigg(-\frac{1205}{162}+\frac{67}{36} \zeta_{2}+\frac{1}{4} \zeta_{2}^{2}-\frac{11}{2} \zeta_{3}\bigg) +n_{f} C_{A}^{3}\bigg(-\frac{233}{162}+\frac{13}{36} \zeta_{2}-\frac{1}{3}\zeta_{3}\bigg) \\+n_{f} C_{A}^{2} C_{F}\bigg(\frac{617}{243} -\frac{13}{18} \zeta_{2}+\frac{2}{3} \zeta_{3}\bigg)\bigg]\frac{1}{2}\frac{\ln^{2}1/x}{x}\ +\ \rho_{gg}\ \frac{\ln 1/x}{x}, \quad (n_{f} = 4)
\end{multline}
where we maintain the variation of $\rho_{gg}$ from above, as the fixed functional form manages to encapsulate the variation predicted, without any extra allowed $\rho_{gg}$ variation.
\subsection{Predicted \texorpdfstring{aN$^{3}$LO}{aN3LO} Splitting Functions}\label{subsec: expansion_split}
\begin{figure}
\begin{center}
\includegraphics[width=0.97\textwidth]{figures/section4/section4-2/Pns.png}
\end{center}
\caption{\label{fig: split_variation_NS}Perturbative expansion up to aN$^{3}$LO for the non-singlet splitting function $P_{qq}^{\mathrm{NS},\ +}$ including any corresponding allowed $\pm 1\sigma$ variation (shaded green region). The best fit value (blue dashed line) displays the prediction for this function determined from a global PDF fit.}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=0.97\textwidth]{figures/section4/section4-2/Pps.png}
\includegraphics[width=0.97\textwidth]{figures/section4/section4-2/Pqg.png}
\end{center}
\caption{\label{fig: split_variation_sing_q}Perturbative expansions up to aN$^{3}$LO for the quark singlet splitting functions $P_{qq}^{\mathrm{PS}}$ (top) and $P_{qg}$ (bottom) including any corresponding allowed $\pm 1\sigma$ variation (shaded green region). The best fit values (blue dashed line) display the predictions for each function determined from a global PDF fit.}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=0.97\textwidth]{figures/section4/section4-2/Pgq.png}
\includegraphics[width=0.97\textwidth]{figures/section4/section4-2/Pgg.png}
\end{center}
\caption{\label{fig: split_variation_sing_g}Perturbative expansions for the gluon splitting functions $P_{gq}$ (top) and $P_{gg}$ (bottom) including any corresponding allowed $\pm 1\sigma$ variation (shaded green region). The best fit value (blue dashed line) displays the prediction for this function determined from a global PDF fit.}
\end{figure}
Fig.'s~\ref{fig: split_variation_NS}, \ref{fig: split_variation_sing_q} and \ref{fig: split_variation_sing_g} show the perturbative expansions for each splitting function up to approximate N$^{3}$LO. Included with these expansions are the predicted variations ($\pm 1 \sigma$) from Section~\ref{subsec: 4loop_split} (shown in green) and the aN$^{3}$LO best fits (shown in blue -- discussed further in Section~\ref{sec: results}). As a general feature, we observe that the singlet N$^3$LO approximations are much more divergent than lower orders due to the presence of higher order logarithms at small-$x$, further highlighting the need for an understanding of MHOUs which is not reliant on the NNLO central value.
Considering the non-singlet case shown in Fig.~\ref{fig: split_variation_NS}, we see a very close agreement at large-$x$ between $P_{qq}^{\mathrm{NS}}$ expanded to NNLO and aN$^{3}$LO. This is a general feature of the non-singlet distribution, since by design, this distribution is largely unaffected by small-$x$ contributions. The ratio plot in Fig.~\ref{fig: split_variation_NS} provides clearer evidence for this, since it is only towards small-$x$ (where the non-singlet distribution tends towards 0) that any noticeable difference between NNLO and aN$^{3}$LO can be seen.
The contributions to $P_{qq}^{\mathrm{PS}}$, $P_{qg}$, $P_{gq}$ and $P_{gg}$ shown in Fig.'s~\ref{fig: split_variation_sing_q} and \ref{fig: split_variation_sing_g} respectively, display a much richer description at aN$^{3}$LO. In all cases, the divergent terms (with $x \rightarrow 0$) present in the approximations have a large effect from intermediate-$x$ ($\sim 10^{-2}$) down to very small-$x$ values. The asymptotic relationships (red line) Equation~\eqref{eq: Relations} defined using the best fit values of the aN$^{3}$LO expansions (i.e. comparable to the blue dashed line) are also shown in Fig.'s~\ref{fig: split_variation_sing_q} and \ref{fig: split_variation_sing_g}. As discussed earlier, these relations are violated by large sub-leading small-$x$ terms and are therefore provided here as a qualitative comparison. Furthermore, we also observe a close resemblance to the N$^{3}$LO asymptotic results in Fig.~4 of \cite{Marzani:smallx}. Specifically for quark evolution, we show that the data prefers a similar form ($P_{qq}^{\mathrm{PS}}$ and $P_{qg}$) to the resummed splitting function results in \cite{Marzani:smallx} whereas for gluon evolution, this agreement is less prominent.
Superimposed onto these variations in Fig.'s \ref{fig: split_variation_sing_q} and \ref{fig: split_variation_sing_g} are the best fit values for the splitting functions, as predicted from a global fit of the full MSHT approximate N$^{3}$LO PDFs. The full fit results will be discussed in more detail in Section~\ref{sec: results}, however we note here that the fit produces relatively good agreement with the prior allowed variations for each of the splitting functions. For all functions except for $P_{gg}$, the best fit results lie within their $\pm 1 \sigma$ variation range. This result implies that constraints from the data included in the global fit are in good agreement with the penalties describing quark evolution (i.e. $P_{\mathrm{PS}}$ and $P_{qg}$ in Fig.~\ref{fig: split_variation_sing_q}). For the gluon evolution in Fig.~\ref{fig: split_variation_sing_g} we observe a small level of tension with the data pushing towards a slightly harder small-$x$ gluon than preferred by the penalty constraints for $P_{gg}$. An important caveat to these best fit results is that the data included in the fit is sensitive to all orders in $\alpha_{s}$. Therefore by proxy, the best fit predictions are also sensitive to corrections at all orders. This will certainly be a driving factor for any violations away from the expected N$^{3}$LO behaviour. However, since the ultimate goal of this investigation is to provide a theoretical uncertainty, the violation from higher orders is manifested into the defined penalties and therefore accounted for in the fit as a source of MHOU.
Finally, an important feature that can be seen across all these splitting function plots are points of zero aN$^{3}$LO uncertainty in the high-$x$ regions. The regions where these points occur are where the moments are constraining the chosen fixed functional forms very tightly. In particular, for $N_{m}$ moments (constraints) in Equation~\eqref{eq: splitAnsatz}, we are left with $N_{m}-1$ points of zero uncertainty predicted from our approximations. As stated, these points are dependent on the choice of our fixed functional form and are therefore regions where the uncertainty has been underestimated when compared to the functional uncertainty which the fixed form approximates. To provide a more complete estimate of the uncertainty in these areas, it would be necessary to smooth the uncertainty band out across these regions (or take into account several fixed functional forms). However, this shortcoming only occurs towards large-$x$, where the uncertainty is naturally smaller across these functions. Therefore if the uncertainty was smoothed, the effect would be negligible for the MHOUs this work aims to include in a PDF global fit. Further to this, these functions are ingredients in the DGLAP convolution where any smaller details are washed out by more dominant features inside convolutions with PDFs. For these reasons, we opt for computational efficiency and leave these points as shown.
\subsubsection{Moment Analysis}
\begin{table}
\centerline{
\begin{tabular}{c|c|cccc}
\hline
& Moment & LO & NLO & NNLO & N$^{3}$LO \\
\hline
\multirow{4}{*}{$P^{\mathrm{PS}}_{qq}$}
& $N=2$ & $-0.056588$ & $-0.06362642$ & $-0.06395712$ & $-0.06412109$\\
& $N=4$ & $-0.11104$ & $-0.1261481$ & $-0.12804822$ & $-0.12835549$\\
& $N=6$ & $-0.14329$ & $-0.16188618$ & $-0.16433013$ & $-0.16470246$\\
& $N=8$ & $-0.166448$ & $-0.18751366$ & $-0.19033329$ & $-0.19074888$\\
\hline
\multirow{4}{*}{$P_{qg}$}
& $N=2$ & $0.042442$ & $0.05008496$ & $0.04991043$ & $0.04983007$\\
& $N=4$ & $0.023342$ & $0.02203438$ & $0.02110201$ & $0.02112623$\\
& $N=6$ & $0.016674$ & $0.01387744$ & $0.01311037$ & $0.01316929$\\
& $N=8$ & $0.013086$ & $0.00979920$ & $0.00919186$ & $0.00927006$\\
\hline
\multirow{4}{*}{$P_{gq}$}
& $N=2$ & $0.056588$ & $0.06362642$ & $0.06395712$ & $0.06412109$\\
& $N=4$ & $0.015562$ & $0.01903295$ & $0.0195455$ & $0.01965547$\\
& $N=6$ & $0.008892$ & $0.0112073$ & $0.01158133$ & $0.0116615$\\
& $N=8$ & $0.006232$ & $0.00801547$ & $0.00831037$ & $0.0083761$\\
\hline
\multirow{4}{*}{$P_{gg}$}
& $N=2$ & $-0.042442$ & $-0.05008496$ & $-0.04991043$ & $-0.04983007$\\
& $N=4$ & $-0.242978$ & $-0.26161441$ & $-0.26280015$ & $-0.26326763$\\
& $N=6$ & $-0.32551$ & $-0.35114066$ & $-0.35335022$ & $-0.35384552$\\
& $N=8$ & $-0.38091$ & $-0.41151668$ & $-0.41447721$ & $-0.41495604$\\
\hline
\end{tabular}
}
\caption{\label{tab: split_moments}Numerical moments of singlet and gluon splitting function moments up to N$^{3}$LO for $\alpha_{s}=0.2$ and $n_{f}=4$.}
\end{table}
\begin{figure}[t]
\begin{center}
\includegraphics[width=\textwidth]{figures/section4/section4-2/split_moment_ratio.png}
\end{center}
\caption{\label{fig: split_moment_ratio}The low-integer numerical Mellin moments of relevant singlet splitting functions (excluding $P_{qq}^{\mathrm{NS},\ +}$) as a ratio between orders. In all cases the expected perturbative convergence is demonstrated.}
\end{figure}
Tracking back to the moments found for the splitting functions~\cite{S4loopMoments, S4loopMomentsNew} (shown in Table~\ref{tab: split_moments} and as a ratio in Fig.~\ref{fig: split_moment_ratio}), we are able to identify the expected convergence in the perturbative expansions up to N$^{3}$LO. Fig.~\ref{fig: split_moment_ratio} illustrates the relative size of the NNLO and N$^{3}$LO contributions to the low even-integer moments.
Until recently (at the time of writing), there were only 3 moments available for the functions $P_{gq}$ and $P_{gg}$ approximated here. However, in \cite{S4loopMomentsNew} an extra moment was published for these two gluon splitting functions.
This extra information led to our predictions at small-$x$ being more in line with the resummation results in \cite{Marzani:smallx} mentioned earlier.
This is an example of how extra information can be added as and when it is available to update any approximations and utilise our full knowledge of the next highest order. By adopting this procedure, we immediately benefit from a slightly increased precision (with a relevant theoretical uncertainty) instead of having to delay the inclusion of higher order theory (for potentially decades) until a complete analytical calculation of the next order in $\alpha_{s}$ is known.
\subsection{Numerical Results}\label{subsec: num_res_split}
We now consider the DGLAP evolution equations for the singlet and gluon shown in Equation~\eqref{eq: DGLAP}. We expand this equation to $\alpha^{4}_{s}$ and investigate the effects of the variation in the N$^{3}$LO contributions.
For the purposes of this analysis, the approximate functions~\eqref{eq: singGluApprox}, taken from \cite{Vogt:2004mw}, are used as sample distributions at an energy scale of $\mu_{f}^{2} \simeq 30\ \mathrm{GeV}^{2}$, a scale chosen due to its relevance to DIS processes included in the MSHT global fit.
\begin{subequations}\label{eq: singGluApprox}
\begin{align}
x\Sigma(x,\mu_{f}^{2}=30\ \mathrm{GeV}^{2})\ =&\quad 0.6\ x^{-0.3}(1-x)^{3.5}(1+5x^{0.8}) \label{eq: singApprox} \\ xg(x,\mu_{f}^{2}=30\ \mathrm{GeV}^{2})\ =& \quad 1.6\ x^{-0.3}(1-x)^{4.5}(1-0.6x^{0.3}) \label{eq: gluApprox}
\end{align}
\end{subequations}
The expressions above are order independent and so provide a robust means to isolate the effects arising from higher orders in the splitting functions. For convenience we also assume
\begin{align}
\alpha_{s}(\mu_{r}^{2}=\mu_{f}^{2}=30\ \mathrm{GeV}^{2})\simeq 0.2.
\end{align}
where $\mu_{r}$ and $\mu_{f}$ are the renormalisation and factorisation scales respectively.
\subsubsection*{Singlet Evolution}
\begin{figure}[t]
\centering
\includegraphics[width=0.49\textwidth]{figures/section4/section4-3/evolution_sing.png}
\includegraphics[width=0.49\textwidth]{figures/section4/section4-3/evolution_shiftsing.png}
\caption{\label{fig: SingEvolutionApprox} The flavour singlet quark distribution evolution equation Equation~\eqref{eq: DGLAP} shown for orders up to the approximate N$^3$LO (left). The relative shift between subsequent orders of the flavour singlet evolution (right) where $\dot{\Sigma} = d \ln \Sigma/d \ln \mu_{f}^{2}$. }
\end{figure}
Fig.~\ref{fig: SingEvolutionApprox} demonstrates the result of including the respective N$^3$LO expansions from Section~\ref{subsec: 4loop_split} in an analysis of the evolution equation. Towards small-$x$ this variation increases due to the larger uncertainty in the $P_{qq}^{\mathrm{PS}}$ and $P_{qg}$ splitting functions at aN$^{3}$LO. On the right of Fig.~\ref{fig: SingEvolutionApprox}, the difference plot displays the respective shifts from the previous order and demonstrates how this shift changes up to N$^3$LO. These results predict a reduction in the evolution of the singlet towards small-$x$ from NNLO. Inspecting Fig.~\ref{fig: split_variation_sing_q}, we can see that this reduction is stemming from the contribution of the gluon with the $P_{qg}$ function at 4-loops, which is the dominant contribution to the evolution. Towards larger $x$ values ($10^{-2} < x < 10^{-1}$) we see a fractional increase in the quark evolution, also following the shape of the $P_{qg}$ function. These results can therefore give some indication as to how we expect our gluon PDF to behave at N$^{3}$LO; since the structure functions are directly related to the quarks (through LO), the singlet evolution should remain fairly constant. Therefore we can expect that the fit will prefer a slightly harder gluon at small-$x$ and a softer gluon between $10^{-2} < x < 10^{-1}$ relative to NNLO.
Fig.~\ref{fig: SingEvolutionApprox} displays a good level of agreement between the allowed N$^{3}$LO shift and the evolution at NLO and NNLO (within $\pm1\sigma$ variation bands from theoretical uncertainties). Also shown in Fig.~\ref{fig: SingEvolutionApprox} is the evolution prediction using the best fit results for $P_{qq}^{(3)}$ and $P_{qg}^{(3)}$ (red dashed). This prediction tends to follow slightly below the center of the $1\sigma$ uncertainty band, where the data has balanced the two variations and is more in line with the NLO evolution than NNLO due to a negative contribution below $10^{-2}$. Considering the magnitude of shifts from each order, the predicted shift from NNLO to aN$^{3}$LO is slightly larger than that from NLO to NNLO, contradicting what may be expected from perturbation theory. However, we remind the reader that these best fit results are, to some degree, sensitive to all orders in perturbation theory through the data constraint. Due to this, the resultant best fit can be thought of as an approximate asymptote to all orders. Interpreting the approximation in this way, restores our faith in perturbation theory and becomes an entirely plausible estimation of the missing higher orders.
Fig.~\ref{fig: SingEvolutionApprox} also exhibits an example of how the points of zero uncertainty (discussed in Section \ref{subsec: 4loop_split}) can affect the evolution predictions. We can see that at most the uncertainty is being underestimated by $<1\%$ and therefore, for the reasons discussed earlier, we do not consider these regions further here.
\subsubsection*{Gluon Evolution}
\begin{figure}[t]
\centering
\includegraphics[width=0.49\textwidth]{figures/section4/section4-3/evolution_gluon.png}
\includegraphics[width=0.49\textwidth]{figures/section4/section4-3/evolution_shiftgluon.png}
\caption{\label{fig: GluonEvolutionApprox} The gluon distribution evolution equation Equation~\eqref{eq: DGLAP} shown for orders up to the approximate N$^3$LO (left). The relative shift between subsequent orders of the gluon evolution (right) where $\dot{g} = d \ln g/d \ln \mu_{f}^{2}$.}
\end{figure}
Fig.~\ref{fig: GluonEvolutionApprox} displays the result of including the aN$^{3}$LO splitting function contributions into the gluon evolution equation. As with the singlet evolution case, this extra contribution is currently inducing a notable variation at N$^3$LO. The general trend at small-$x$ is a reduction in the value of the evolution equation due to the N$^3$LO prediction for $P_{gg}$. On the right hand side of Fig.~\ref{fig: GluonEvolutionApprox} we observe the respective shifts from lower orders and how this shift changes up to N$^3$LO.
In the gluon evolution, there is a large variation coming from the uncertainty in the $P_{gg}^{(3)}$ function. Therefore when $P_{gg}^{(3)}$ is convoluted with the gluon PDF at small-$x$, one could expect a potentially large shift from NNLO. The best fit gluon evolution prediction in Fig.~\ref{fig: GluonEvolutionApprox} is produced by utilising the best fit results for $P_{gq}^{(3)}$ and $P_{gg}^{(3)}$ functions (red dashed). In this prediction we see that the fit prefers a reduction in the evolution from NNLO, which is contained within the $\pm 1 \sigma$ band until around $x \lesssim 10^{-4}$. Since at low-$Q^{2}$, the quark and gluon are comparable at small-$x$, this reduction is likely driven from the form of $P_{gq}$ in Fig.~\ref{fig: split_variation_sing_g}. Combining this with the smaller gluon PDF at low-$Q^{2}$ therefore acts to slow the gluon evolution despite $P_{gg}$ increasing. Furthermore, the best fit is seemingly more in line with the perturbative expectation of the evolution than the chosen variation\footnote{Due to the presence of more divergent higher order logarithms at this level, it is not certain or by any means guaranteed that the shift at N$^{3}$LO will follow the same trend outlined from lower orders.}. Since this variation is chosen from the known information about the perturbative expansions, this is a manifestation of how the framework we present here can capture the relevant sources of theoretical uncertainty (and account for these via a penalty in a PDF fit). This is encouraging, as even with the large amount of freedom for this gluon evolution, it seems that the data is constraining and balancing the two contributions from the splitting functions in a sensible fashion. As discussed in the singlet evolution case, the relative shift from NNLO to N$^{3}$LO is slightly larger than one might hope for when dealing with a perturbative expansion. However, since this best fit is impacted to all orders from the experimental data (up to the leading logarithms at N$^{3}$LO i.e. even higher orders involve more divergent logarithms which are missed in this theoretical description), we can interpret this shift as an approximate all order shift and once again restore its validity in perturbation theory.
As with the singlet case above, negligible points of non-zero uncertainty are displayed in Fig.~\ref{fig: GluonEvolutionApprox}. For the reasons discussed in the singlet case and in Section~\ref{subsec: 4loop_split}, these are not an area of concern at the current level of desired uncertainty and are therefore not considered further.
\section{\texorpdfstring{N$^{3}$LO}{N3LO} Predictions}\label{sec: predictions}
With the increasing number of hard cross section calculations at N$^{3}$LO, there is a growing demand for N$^{3}$LO accuracy in PDFs. In this section we investigate the effect of the MSHT approximate N$^{3}$LO PDFs on Higgs production via gluon fusion and vector boson fusion (VBF). The hard cross sections for these processes have been calculated to N$^{3}$LO accuracy~\cite{Ball:2013bra,Bonvini:2014jma,Bonvini:2016frm,Ahmed:2016otz,Bonvini:2018ixe,Bonvini:2018iwt,Bonvini:2013kba,anastasiou2014higgs,anastasiou2016high,Mistlberger:2018,Cacciari:2015,Dreyer:2016}. We present a full N$^{3}$LO computation for each prediction with our approximate N$^{3}$LO PDFs, including theoretical uncertainties. In future work, the intention will be to expand this analysis to include results for N$^{3}$LO DY~\cite{DY_N3LO_Kfac} and approximate N$^{3}$LO top production~\cite{Kidonakis:tt} cross sections.
Note that in this Section we follow the notation used previously and denote the aN$^{3}$LO results with decorrelated $K$-factors as $(H_{ij} + K_{ij})^{-1}$ and those with correlated $K$-factors with $H_{ij}^{\prime\ -1}$. In all cases, scale variations are found via the 9-point prescription~\cite{NNPDFscales} for results with NNLO PDFs. Whereas for aN$^{3}$LO PDFs, although the extra information introduced is at N$^{3}$LO, the data (and therefore all relevant theory nuisance parameters) which are included in the global fit are sensitive to all orders.
In particular, we include theoretical uncertainties into our aN$^3$LO fit which incorporate MHO effects on the PDFs. Therefore we argue (and in these cases demonstrate) that the factorisation scale variation is contained within the enlarged PDF uncertainties. Therefore it is only
the renormalisation scale which requires variation in predictions involving aN$^{3}$LO PDFs\footnote{This is to quantify the theoretical MHOU in the hard cross section, whereas the aN$^{3}$LO PDFs now come with an estimated MHOU.}.
\subsection{Higgs Production -- Gluon Fusion: \texorpdfstring{$gg \rightarrow H$}{gg to Higgs}}\label{subsec: ggH}
\begin{table}
\centerline{
\begin{tabular}{c|ccc}
\hline
$\sigma$ order & PDF order & $\sigma + \Delta \sigma_{+} - \Delta \sigma_{-}$ (pb) & $\sigma$ (pb) $+\ \Delta \sigma_{+} - \Delta \sigma_{-}$ (\%) \\
\hline
\multicolumn{4}{c}{PDF uncertainties} \\
\hline
\multirow{4}{*}{N$^{3}$LO}
& aN$^{3}$LO (no theory unc.) & 44.164 + 1.339 - 1.382 & 44.164 + 3.03\% - 3.13\% \\
& aN$^{3}$LO ($H_{ij} + K_{ij}$) & 44.164 + 1.473 - 1.395 & 44.164 + 3.34\% - 3.15\% \\
& aN$^{3}$LO ($H_{ij}^{\prime}$) & 44.164 + 1.515 - 1.354 & 44.164 + 3.43\% - 3.07\% \\
& NNLO & 47.817 + 0.558 - 0.581 & 47.817 + 1.17\% - 1.22\% \\
\hline
NNLO & NNLO & 46.206 + 0.541 - 0.564 & 46.206 + 1.17\% - 1.22\% \\
\hline
\multicolumn{4}{c}{PDF + Scale uncertainties} \\
\hline
\multirow{4}{*}{N$^{3}$LO} & aN$^{3}$LO (no theory unc.) & 44.164 + 1.339 - 2.214 & 44.164 + 3.03\% - 5.01\% \\
& aN$^{3}$LO ($H_{ij} + K_{ij}$) & 44.164 + 1.473 - 2.222 & 44.094 + 3.34\% - 5.03\% \\
& aN$^{3}$LO ($H_{ij}^{\prime}$) & 44.164 + 1.515 - 2.196 & 44.164 + 3.43\% - 4.97\% \\
& NNLO & 47.817 + 0.577 - 2.210 & 47.817 + 1.21\% - 4.62\% \\
\hline
NNLO & NNLO & 46.206 + 4.284 - 5.414 & 46.206 + 9.27\% - 11.72\% \\
\end{tabular}
}
\caption{\label{tab: ggH_results_mh2}Higgs production cross section results via gluon fusion using N$^{3}$LO and NNLO hard cross sections combined with NNLO and aN$^{3}$LO PDFs. All PDFs are at the standard choice $\alpha_{s} = 0.118$. These results are found with $\mu = m_{H}/2$ unless stated otherwise, with the values for $\mu = m_{H}$ supplied in Table~\ref{tab: ggH_results_mh}.}
\end{table}
\begin{figure}
\begin{center}
\includegraphics[width=0.49\textwidth]{figures/section9/section9-1/ggH_predictions_mh2.png}
\includegraphics[width=0.49\textwidth]{figures/section9/section9-1/ggH_predictions_mh.png}
\end{center}
\caption{\label{fig: ggH_predictions}Higgs production cross section results via gluon fusion at two central scales: $\mu = m_{H}/2$ (left) and $\mu = m_{H}$ (right). Displayed are the results for aN$^{3}$LO PDFs with decorrelated $K$-factors ($(H_{ij} + K_{ij})^{-1}$), correlated $K$-factors ($H_{ij}^{\prime\ -1} = (H_{ij} + K_{ij})^{-1}$) each with a scale variation band from varying $\mu_{r}$ by a factor of 2. In the NNLO and NLO PDF cases, both scales $\mu_{f}$ and $\mu_{r}$ are varied by a factor of 2 following the 9-point convention~\cite{NNPDFscales}.}
\end{figure}
Table~\ref{tab: ggH_results_mh2} and Fig.~\ref{fig: ggH_predictions} (left) show predictions at a central scale of $\mu = \mu_{f}=\mu_{r}=m_{H}/2$ for the Higgs production cross section via gluon fusion\footnote{Results are obtained with the code \href{https://www.ge.infn.it/~bonvini/higgs/}{\texttt{ggHiggs}}\cite{Ball:2013bra,Bonvini:2014jma,Bonvini:2016frm,Ahmed:2016otz,Bonvini:2018ixe,Bonvini:2018iwt,Bonvini:2013kba,anastasiou2014higgs,anastasiou2016high,Mistlberger:2018,ggHiggs}.} where $m_{H}$ is the Higgs mass. Fig.~\ref{fig: ggH_predictions} (right) displays the same analysis for the gluon fusion cross section with $\mu = \mu_{f}=\mu_{r}=m_{H}$ (numerical results provided in Table~\ref{tab: ggH_results_mh}).
Considering the $\mu = m_{H}/2$ and $\mu = m_{H}$ central value results displayed in Table~\ref{tab: ggH_results_mh2} and Fig.~\ref{fig: ggH_predictions}, it can be observed that aN$^{3}$LO PDFs predict a lower central value than NNLO PDFs across all hard cross section orders. One can also notice an overlap in all cases between predictions from NNLO and aN$^{3}$LO PDFs (minimally for $\mu = m_{H}/2$ at the level of the N$^{3}$LO 1$\sigma$ band). However for both choices of $\mu$, whilst the error bands overlap, their central values are outside each other's respective error bands. Since estimating MHOUs via scale variations is a somewhat ambiguous procedure (and is therefore estimated conservatively to reflect this), these results highlight the benefit of being able to exploit a higher level of control over MHOUs i.e. via nuisance parameters. By predicting a different central value we include a more accurate estimation for higher order predictions which may not be contained within scale variations, especially at unmatched orders in perturbation theory.
Examining the predicted central values further, Fig.~\ref{fig: ggH_predictions} suggests that the increase in the cross section theory at N$^{3}$LO is compensated by the PDF theory at N$^{3}$LO, suggesting a cancellation between terms in the PDF and cross section theory at N$^{3}$LO. This point is important to consider when combining unmatched orders in physical calculations, since we must be open to the possibility that unmatched cancellations in physical calculations can lead to inaccurate predictions, as our results suggest here.
Further to this, the change in the gluon PDF is largely driven by the predicted form of $P_{qg}$ at aN$^{3}$LO and DIS data. Therefore the relevant changes in the gluon at aN$^{3}$LO are most likely due to indirect effects i.e. not directly related to gluon fusion predictions. Due to this, there is no reason to believe that the observed level of convergence should happen at aN$^{3}$LO for both choices of $\mu$.
However, owing to the inclusion of known information at higher orders, one can be confident that the prediction is more accurate than NNLO, whichever way it moves.
Comparing PDF uncertainty values calculated using NNLO and aN$^{3}$LO PDFs, another prominent feature one can notice in Table~\ref{tab: ggH_results_mh2} is the increase in PDF uncertainty. We find that the PDF uncertainty without N$^{3}$LO theory uncertainties included (i.e. using only the eigenvector description from the first 32 eigenvectors and with N$^{3}$LO parameters fixed at the best fit) is over double the PDF uncertainty from NNLO.
Mathematically, the reason for this comes back to the fact that the best fit is inherently different from the NNLO theory, residing in a completely novel $\chi^{2}$ landscape. In turn, this means it is not guaranteed that the PDF uncertainty will remain consistent across the distinct PDF sets\footnote{As we can see from Section~\ref{subsec: pdf_results}, the theory uncertainty is also not guaranteed to add to the total uncertainty (and in fact acts to reduce the uncertainty in some areas of $(x, Q^{2})$).}. In fact, for all aN$^{3}$LO PDF results, eigenvectors most associated with the gluon PDF parameters (eigenvector 11 and 29 in the $(H_{ij} + K_{ij})^{-1}$ N$^{3}$LO case -- see Table~\ref{tab: eigenanalysis}) are the leading contributions to this uncertainty.
Phenomenologically, the further remaining increase from the inclusion of the theoretical uncertainties is a reflection of the estimated PDF MHOUs in this particular cross section and acts to replace factorisation scale variation. As a consistency check, we find that when performing a 9-point scale variation procedure with aN$^{3}$LO PDFs, the values calculated (for both choices of $\mu$) are within the predicted PDF uncertainties. This is therefore a further verification of our MHOUs and that the $\mu_{f}$ variation is intrinsic in the PDF uncertainties.
Finally Fig.~\ref{fig: ggH_predictions} also demonstrates the increased stability of predictions when considering the two different central scales $\mu$ at N$^{3}$LO. As predicted from perturbation theory, the scale dependence is reduced and central values become more in agreement when increasing the order of either the PDFs or hard cross section. Furthermore, the aN$^{3}$LO $\sigma$ central predictions for both choices of $\mu$ are contained within the uncertainty bands of each other. This is true by definition for the NNLO PDFs since the factorisation scale $\mu_{f}$ variation includes both choices of $\mu$, whereas for aN$^{3}$LO PDFs this result is not guaranteed and is therefore intrinsic in the PDF (and renormalisation scale $\mu_{r}$ variation) uncertainty.
\subsection{Higgs Production -- Vector Boson Fusion: \texorpdfstring{$qq \rightarrow H$}{qq to Higgs}}
\begin{figure}
\begin{center}
\includegraphics[width=0.49\textwidth]{figures/section9/section9-2/vbf_predictions.png}
\end{center}
\caption{\label{fig: vbf_predictions}Higgs production cross section results via vector boson fusion at a central scale set to the vector boson momentum. Displayed are the results for aN$^{3}$LO PDFs with decorrelated $K$-factors ($(H_{ij} + K_{ij})^{-1}$), correlated $K$-factors ($H_{ij}^{\prime\ -1} = (H_{ij} + K_{ij})^{-1}$) each with a scale variation band from varying $\mu_{r}$ by a factor of 2. In the NNLO and NLO PDF cases, both scales $\mu_{f}$ and $\mu_{r}$ are varied by a factor of 2 following the 9-point convention~\cite{NNPDFscales}.}
\end{figure}
\begin{table}
\centerline{
\begin{tabular}{c|ccc}
\hline
$\sigma$ order & PDF order & $\sigma + \Delta \sigma_{+} - \Delta \sigma_{-}$ (pb) & $\sigma$ (pb) $+\ \Delta \sigma_{+} - \Delta \sigma_{-}$ (\%) \\
\hline
\multicolumn{4}{c}{PDF uncertainties} \\
\hline
\multirow{4}{*}{N$^{3}$LO} & aN$^{3}$LO (no theory unc.) & 4.1378 + 0.0737 - 0.0926 & 4.1378 + 1.78\% - 2.23\% \\
& aN$^{3}$LO ($H_{ij} + K_{ij}$) & 4.1378 + 0.0867 - 0.0869 & 4.1378 + 2.10\% - 2.10\% \\
& aN$^{3}$LO ($H_{ij}^{\prime}$) & 4.1378 + 0.0829 - 0.0889 & 4.1378 + 2.00\% - 2.15\% \\
& NNLO & 3.9941 + 0.0558 - 0.0631 & 3.9941 + 1.40\% - 1.58\% \\
\hline
NNLO & NNLO & 3.9974 + 0.0557 - 0.0633 & 3.9974 + 1.39\% - 1.58\% \\
\hline
\multicolumn{4}{c}{PDF + Scale uncertainties} \\
\hline
\multirow{4}{*}{N$^{3}$LO} & aN$^{3}$LO (no theory unc.) & 4.1378 + 0.0737 - 0.0926 & 4.1378 + 1.78\% - 2.23\% \\
& aN$^{3}$LO ($H_{ij} + K_{ij}$) & 4.1378 + 0.0867 - 0.0869 & 4.1378 + 2.10\% - 2.10\% \\
& aN$^{3}$LO ($H_{ij}^{\prime}$) & 4.1378 + 0.0829 - 0.0889 & 4.1378 + 2.00\% - 2.15\% \\
& NNLO & 3.9941 + 0.0560 - 0.0631 & 3.9941 + 1.40\% - 1.58\% \\
\hline
NNLO & NNLO & 3.9974 + 0.0576 - 0.0642 & 3.9974 + 1.44\% - 1.61\% \\
\end{tabular}
}
\caption{\label{tab: vbf_results}Higgs production cross section results via the vector boson fusion process using N$^{3}$LO and NNLO hard cross sections combined with NNLO and aN$^{3}$LO PDFs. All PDFs are at the standard choice $\alpha_{s} = 0.118$. These results are found with $\mu = Q^{2}$ where $Q^{2}$ is the vector boson momentum.}
\end{table}
\begin{table}
\centerline{
\begin{tabular}{c|ccc}
\hline
$\sigma$ order & PDF order & $\sigma + \Delta \sigma_{+} - \Delta \sigma_{-}$ (pb) & $\sigma$ (pb) $+\ \Delta \sigma_{+} - \Delta \sigma_{-}$ (\%) \\
\hline
\multirow{3}{*}{N$^{3}$LO} & aN$^{3}$LO $n_{f}=5$ & 4.1378 + 0.0867 - 0.0869 & 4.1378 + 2.10\% - 2.10\% \\
& aN$^{3}$LO $n_{f}=4$ & 4.0510 + 0.0853 - 0.0859 & 4.0510 + 2.11\% - 2.12\% \\
& aN$^{3}$LO $n_{f}=3$ & 2.7066 + 0.0610 - 0.0695 & 2.7066 + 2.26\% - 2.57\% \\
\hline
\multirow{3}{*}{NNLO} & NNLO $n_{f}=5$ & 3.9974 + 0.0557 - 0.0633 & 3.9974 + 1.39\% - 1.58\% \\
& NNLO $n_{f}=4$ & 3.9118 + 0.0561 - 0.0634 & 3.9118 + 1.44\% - 1.62\% \\
& NNLO $n_{f}=3$ & 2.6845 + 0.0539 - 0.0641 & 2.6845 + 2.01\% - 2.39\% \\
\end{tabular}
}
\caption{\label{tab: vbf_results_nf}Higgs production cross section results via the vector boson fusion process using N$^{3}$LO and NNLO hard cross sections combined with NNLO and decorrelated aN$^{3}$LO PDFs whilst varying the number of active flavours $n_{f}$. All PDFs are at the standard choice $\alpha_{s} = 0.118$. These results are found with $\mu = Q^{2}$ where $Q^{2}$ is the vector boson momentum.}
\end{table}
Table~\ref{tab: vbf_results} and Fig.~\ref{fig: vbf_predictions} show the predictions at various orders in $\alpha_{s}$ for Higgs production cross sections via vector boson fusion\footnote{Results are obtained with the inclusive part of the code \href{https://provbfh.hepforge.org/}{\texttt{proVBFH}}~\cite{Cacciari:2015,Dreyer:2016,provbfh}.} up to N$^{3}$LO~\cite{Cacciari:2015,Dreyer:2016}. The predictions shown are calculated with $\mu_{f}^2 = \mu_{r}^2 = Q^{2}$ as the central scale where $Q^{2}$ is the vector boson squared momentum.
For this process one can follow the increase in the cross section as higher order PDFs are used. Contrasting with the case of gluon fusion, Fig.~\ref{fig: vbf_predictions} displays little cancellation between the terms added in the aN$^{3}$LO PDF description and the N$^{3}$LO cross section. However, the cross section for VBF produces around a $\sim 3-4 \%$ change order by order and is therefore fairly constant. Considering this relatively small difference between orders, this lack of cancellation is not a major concern. Further to this, the vector boson fusion process is much more reliant on the quark sector which, compared to the gluon, is relatively constant order by order (see Section~\ref{subsec: pdf_results}). The reason for this stems from the more direct data constraints on the shape of quark PDFs.
Comparing the aN$^{3}$LO VBF cross section (with MHO theoretical uncertainties) with the NNLO cross section result (with NNLO PDFs) including MHOUs via scale variations, we see that the scale variation MHOUs are negligible against the PDF uncertainties at aN$^{3}$LO. This result is in part due to the fact that the scale variation for aN$^{3}$LO is only being included for the renormalisation scale. However at NNLO, the extra MHOU predicted was still only a small contribution. Therefore considering these results further, the effects of higher orders in both cases are expected to be small, which provides some agreement with the argument that there is little scope for cancellation between orders for VBF. As for the gluon fusion prediction in Section~\ref{subsec: ggH}, we confirm that any further factorisation scale variation (i.e. using the 9-point prescription) is contained within the predicted PDF uncertainties; hence further motivating our previous argument that factorisation scale variation is not necessary with aN$^{3}$LO PDFs.
Another feature of the VBF results is that the level of uncertainty at full aN$^{3}$LO is only increased slightly from the calculation involving NNLO PDFs. Comparing this to the gluon fusion results, where the uncertainty was increased by a factor of $\sim 2$, it is evident that these approximate N$^{3}$LO additions are having a minimal effect on the VBF calculation. Once again, the origin of this is due to the nature of the process. VBF involves mostly the quark sector and is therefore much less affected by the extra N$^{3}$LO theory we have introduced (due to direct constraints from data). As we have presented in previous sections, most of the uncertainty in the N$^{3}$LO theory resides in the small-$x$ regime which is more directly probed by the gluon sector than in the quark sector.
Lastly we briefly discuss the $n_{f}$ dependence of the VBF cross section. In VBF the scaling of contributions follows as $n_{f}^{2}$ due to the presence of two input quark flavours in the process. In Table~\ref{tab: vbf_results_nf} we observe that the VBF cross section receives a large contribution when including the charm quark ($n_{f} = 3 \rightarrow 4$) due to this scaling. We also show that at aN$^{3}$LO, this is where most of the difference in the central value and uncertainty from NNLO is accounted for. This is a consequence of the predicted enhancement of the charm PDF at aN$^{3}$LO, discussed in Section~\ref{subsec: pdf_results}. Beyond $n_{f} = 4$ the bottom contribution to VBF in the $W^{\pm}$ channel (the dominant channel) is heavily suppressed, since due to the CKM elements $b$ must transition to $t$ most of the time. Therefore the VBF cross section only receives a small contribution moving from $n_{f}=4$ to $n_{f}=5$.
\section{MSHT20 Approximate \texorpdfstring{N$^{3}$LO}{N3LO} Global Analysis}\label{sec: results}
With the inclusion of all N$^{3}$LO approximations discussed in earlier sections, we now present the results for the first approximate N$^{3}$LO global PDF fit with theoretical uncertainties from MHOs.
\subsection{\texorpdfstring{$\chi^{2}$}{Chi-squared} Breakdown}\label{subsec: chi2}
\begin{table}
\centerline{
\begin{tabular}{|P{6.5cm}|P{1cm}|P{2cm}|P{2cm}|}
\hline
Dataset & $N_{\mathrm{pts}}$ & $\chi^{2}$ & $\Delta \chi^{2}$ from NNLO \\
\hline
BCDMS $\mu p$ $F_{2}$ \cite{BCDMS} & 163 & 179.9 & $-0.3$ \\
BCDMS $\mu d$ $F_{2}$ \cite{BCDMS} & 151 & 143.1 & $-2.9$ \\
NMC $\mu p$ $F_{2}$ \cite{NMC} & 123 & 118.7 & $-5.3$ \\
NMC $\mu d$ $F_{2}$ \cite{NMC} & 123 & 106.2 & $-6.4$ \\
SLAC $ep$ $F_{2}$ \cite{SLAC,SLAC1990} & 37 & 32.0 & $+0.0$ \\
SLAC $ed$ $F_{2}$ \cite{SLAC,SLAC1990} & 38 & 21.9 & $-1.1$ \\
E665 $\mu d$ $F_{2}$ \cite{E665} & 53 & 64.7 & $+5.1$ \\
E665 $\mu p$ $F_{2}$ \cite{E665} & 53 & 67.5 & $+2.8$ \\
NuTeV $\nu N$ $F_{2}$ \cite{NuTev} & 53 & 38.7 & $+0.5$ \\
NuTeV $\nu N$ $xF_{3}$ \cite{NuTev} & 42 & 33.9 & $+3.2$ \\
NMC $\mu n / \mu p$ \cite{NMCn/p} & 148 & 128.5 & $-2.3$ \\
E866 / NuSea $pp$ DY \cite{E866DY} & 184 & 209.2 & $-15.9$ \\
E866 / NuSea $pd/pp$ DY \cite{E866DYrat} & 15 & 7.6 & $-2.7$ \\
HERA $ep$ $F_{2}^{\text{charm}}$ \cite{H1+ZEUScharm} & 79 & 134.7 & $+2.4$ \\
NMC/BCDMS/SLAC/HERA $F_{L}$ \cite{BCDMS,NMC,SLAC1990,H1FL,H1-FL,ZEUS-FL} & 57 & 45.5 & $-23.0$ \\
CCFR $\nu N \rightarrow \mu\mu X$ \cite{Dimuon} & 86 & 69.2 & $+1.5$ \\
NuTeV $\nu N \rightarrow \mu\mu X$ \cite{Dimuon} & 84 & 55.3 & $-3.1$ \\
CHORUS $\nu N$ $F_{2}$ \cite{CHORUS} & 42 & 32.7 & $+2.6$ \\
CHORUS $\nu N$ $xF_{3}$ \cite{CHORUS} & 28 & 19.8 & $+1.3$ \\
HERA $e^{+}p$ CC \cite{H1+ZEUS} & 39 & 51.8 & $-0.1$ \\
HERA $e^{-}p$ CC \cite{H1+ZEUS} & 42 & 66.3 & $-3.8$ \\
HERA $e^{+}p$ NC $820\ \text{GeV}$ \cite{H1+ZEUS} & 75 & 83.8 & $-6.0$ \\
HERA $e^{-}p$ NC $460\ \text{GeV}$ \cite{H1+ZEUS} & 209 & 247.4 & $-0.9$ \\
HERA $e^{+}p$ NC $920\ \text{GeV}$ \cite{H1+ZEUS} & 402 & 476.7 & $-36.0$ \\
HERA $e^{-}p$ NC $575\ \text{GeV}$ \cite{H1+ZEUS} & 259 & 248.0 & $-15.0$ \\
HERA $e^{-}p$ NC $920\ \text{GeV}$ \cite{H1+ZEUS} & 159 & 243.3 & $-1.0$ \\
CDF II $p\bar{p}$ incl. jets \cite{CDFjet} & 76 & 68.4 & $+8.0$ \\
D{\O} II $Z$ rap. \cite{D0Zrap} & 28 & 16.8 & $+0.5$ \\
CDF II $Z$ rap. \cite{CDFZrap} & 28 & 39.6 & $+2.4$ \\
D{\O} II $W \rightarrow \nu \mu$ asym. \cite{D0Wnumu} & 10 & 16.8 & $-0.5$ \\
CDF II $W$ asym. \cite{CDF-Wasym} & 13 & 19.9 & $+0.9$ \\
\hline
\end{tabular}
}
\caption{\label{tab: K-fac_firstresults}Full breakdown of $\chi^{2}$ results for the aN$^{3}$LO PDF fit. The global fit includes the N$^{3}$LO treatment for transition matrix elements, coefficient functions, splitting functions and $K$-factor additions with their variational parameters determined by the fit.}
\end{table}
\begin{table}\ContinuedFloat
\centerline{
\begin{tabular}{|P{6.5cm}|P{1cm}|P{2cm}|P{2cm}|}
\hline
Dataset & $N_{\mathrm{pts}}$ & $\chi^{2}$ & $\Delta \chi^{2}$ from NNLO \\
\hline
D{\O} II $W \rightarrow \nu e$ asym. \cite{D0Wnue} & 12 & 29.2 & $-4.7$ \\
D{\O} II $p\bar{p}$ incl. jets \cite{D0jet} & 110 & 113.6 & $-6.6$ \\
ATLAS $W^{+},\ W^{-},\ Z$ \cite{ATLASWZ} & 30 & 30.0 & $+0.1$ \\
CMS W asym. $p_{T} > 35\ \text{GeV}$ \cite{CMS-easym} & 11 & 7.0 & $-0.8$ \\
CMS W asym. $p_{T} > 25, 30\ \text{GeV}$ \cite{CMS-Wasymm} & 24 & 7.5 & $+0.1$ \\
LHCb $Z \rightarrow e^{+}e^{-}$ \cite{LHCb-Zee} & 9 & 20.6 & $-2.1$ \\
LHCb W asym. $p_{T} > 20\ \text{GeV}$ \cite{LHCb-WZ} & 10 & 12.9 & $+0.5$ \\
CMS $Z \rightarrow e^{+}e^{-}$ \cite{CMS-Zee} & 35 & 17.3 & $-0.6$ \\
ATLAS High-mass Drell-Yan \cite{ATLAShighmass} & 13 & 18.5 & $-0.4$ \\
Tevatron, ATLAS, CMS $\sigma_{t\bar{t}}$ \cite{Tevatron-top,ATLAS-top7-1,ATLAS-top7-2,ATLAS-top7-3,ATLAS-top7-4,ATLAS-top7-5,ATLAS-top7-6,CMS-top7-1,CMS-top7-2,CMS-top7-3,CMS-top7-4,CMS-top7-5,CMS-top8} & 17 & 14.2 & $-0.3$ \\
CMS double diff. Drell-Yan \cite{CMS-ddDY} & 132 & 137.1 & $-7.4$ \\
LHCb 2015 $W, Z$ \cite{LHCbZ7,LHCbWZ8} & 67 & 97.2 & $-2.2$ \\
LHCb $8\ \text{TeV}$ $Z \rightarrow ee$ \cite{LHCbZ8} & 17 & 27.1 & $+0.9$ \\
CMS $8\ \text{TeV}\ W$ \cite{CMSW8} & 22 & 12.0 & $-0.7$ \\
ATLAS $7\ \text{TeV}$ jets \cite{ATLAS7jets} & 140 & 214.5 & $-7.1$ \\
CMS $7\ \text{TeV}\ W + c$ \cite{CMS7Wpc} & 10 & 12.3 & $+3.7$ \\
ATLAS $7\ \text{TeV}$ high prec. $W, Z$ \cite{ATLASWZ7f} & 61 & 110.5 & $-6.2$ \\
CMS $7\ \text{TeV}$ jets \cite{CMS7jetsfinal} & 158 & 189.8 & $+14.1$ \\
D{\O} $W$ asym. \cite{D0Wasym} & 14 & 8.6 & $-3.4$ \\
ATLAS $8\ \text{TeV}\ Z\ p_{T}$ \cite{ATLASZpT} & 104 & 105.8 & $-82.7$ \\
CMS $8\ \text{TeV}$ jets \cite{CMS8jets} & 174 & 272.6 & $+11.3$ \\
ATLAS $8\ \text{TeV}$ sing. diff. $t\bar{t}$ \cite{ATLASsdtop} & 25 & 24.7 & $-0.9$ \\
ATLAS $8\ \text{TeV}$ sing. diff. $t\bar{t}$ dilep. \cite{ATLASttbarDilep08_ytt} & 5 & 2.1 & $-1.3$ \\
ATLAS $8\ \text{TeV}$ High-mass DY \cite{ATLASHMDY8} & 48 & 63.4 & $+6.3$ \\
ATLAS $8\ \text{TeV}\ W + \text{jets}$ \cite{ATLASWjet} & 30 & 19.1 & $+0.9$ \\
CMS $8\ \text{TeV}$ double diff. $t\bar{t}$ \cite{CMS8ttDD} & 15 & 23.9 & $+1.4$ \\
ATLAS $8\ \text{TeV}\ W$ \cite{ATLASW8} & 22 & 55.1 & $-2.3$ \\
CMS $2.76\ \text{TeV}$ jet \cite{CMS276jets} & 81 & 113.9 & $+11.1$ \\
CMS $8\ \text{TeV}$ sing. diff. $t\bar{t}$ \cite{CMSttbar08_ytt} & 9 & 8.4 & $-4.7$ \\
ATLAS $8\ \text{TeV}$ double diff. $Z$ \cite{ATLAS8Z3D} & 59 & 80.8 & $-4.8$ \\
\hline
\end{tabular}
}
\caption{\textit{(Continued)} Full breakdown of $\chi^{2}$ results for the aN$^{3}$LO PDF fit. The global fit includes the N$^{3}$LO treatment for transition matrix elements, coefficient functions, splitting functions and $K$-factor additions with their variational parameters determined by the fit.}
\end{table}
\begin{table}[t]\ContinuedFloat
\centerline{
\begin{tabular}{|c|c|c|c|}
\hline
Low-$Q^{2}$ Coefficient & $\chi^{2}$ & Low-$Q^{2}$ Coefficient & $\chi^{2}$ \\
\hline
$c_{q}^{\mathrm{NLL}}$ $ = -3.976$ & 0.000 & $c_{g}^{\mathrm{NLL}}$ $ = -5.857$ & 0.862 \\
\hline
Transition Matrix Elements & $\chi^{2}$ & Transition Matrix Elements & $\chi^{2}$ \\
\hline
$a_{Hg}$ $ = 12039.000$ & 0.526 & $a_{qq,H}^{\mathrm{NS}}$ $ = -59.596$ & 0.022 \\
$a_{gg,H}$ $ = -2028.800$ & 1.091 & & \\
\hline
Splitting Functions & $\chi^{2}$ & Splitting Functions & $\chi^{2}$\\
\hline
$\rho_{qq}^{NS}$ $ = 0.008$ & 0.007 & $\rho_{gq}$ $ = -1.795$ & 0.935 \\
$\rho_{qq}^{PS}$ $ = -0.527$ & 0.255 & $\rho_{gg}$ $ = 20.345$ & 4.280 \\
$\rho_{qg}$ $ = -1.666$ & 0.000 & & \\
\hline
$K$-factors & $\chi^{2}$ & $K$-factors & $\chi^{2}$\\
\hline
$\mathrm{DY}_{\mathrm{NLO}}$ $ = -0.247$ & 0.061 & $\mathrm{DY}_{\mathrm{NNLO}}$ $ = -0.053$ & 0.003 \\
$\mathrm{Top}_{\mathrm{NLO}}$ $ = 0.324$ & 0.105 & $\mathrm{Top}_{\mathrm{NNLO}}$ $ = 0.812$ & 0.659 \\
$\mathrm{Jet}_{\mathrm{NLO}}$ $ = -0.250$ & 0.063 & $\mathrm{Jet}_{\mathrm{NNLO}}$ $ = -0.719$ & 0.517 \\
$p_{T}\mathrm{Jets}_{\mathrm{NLO}}$ $ = 0.662$ & 0.438 & $p_{T}\mathrm{Jets}_{\mathrm{NNLO}}$ $ = -0.105$ & 0.011 \\
$\mathrm{Dimuon}_{\mathrm{NLO}}$ $ = -0.694$ & 0.481 & $\mathrm{Dimuon}_{\mathrm{NNLO}}$ $ = 0.602$ & 0.363 \\
\hline
N$^{3}$LO Penalty Total & 10.7 / 20 & Average Penalty & 0.534 \\
\hline
\multicolumn{2}{c|}{} & Total & 4948.6 / 4363\\
\multicolumn{2}{c|}{} & $\Delta \chi^{2}$ from NNLO & $-172.5$ \\
\cline{3-4}
\end{tabular}
}
\caption{\textit{(Continued)} Full breakdown of $\chi^{2}$ results for the aN$^{3}$LO PDF fit. The global fit includes the N$^{3}$LO treatment for transition matrix elements, coefficient functions, splitting functions and $K$-factor additions with their variational parameters determined by the fit.}
\end{table}
Table~\ref{tab: K-fac_firstresults} shows the global $\chi^{2}$ results for an aN$^{3}$LO best fit, inclusive of penalties associated with the new theory variational parameters (from Equation~\eqref{eq: penalty_example}). The theory parameters are labelled as: $A_{Hg}(a_{Hg})$, $A_{gg,H}(a_{gg,H})$, $A_{qq,H}^{\mathrm{NS}}(a_{qq,H}^{\mathrm{NS}})$ for the transition matrix elements; $P_{qq}^{\mathrm{NS}}(\rho_{qq}^{\mathrm{NS}})$, $P_{qq}^{\mathrm{PS}}(\rho_{qq}^{\mathrm{PS}})$, $P_{qg}(\rho_{qg})$, $P_{gq}(\rho_{gq})$ and $P_{gg}(\rho_{gg})$ for the splitting functions; and $c_{q}^{\mathrm{NLL}}$ and $c_{g}^{\mathrm{NLL}}$ correspond to the NLL parameters discussed in Section~\ref{subsec: NLL_coeff}. These are supplemented by the 10 additional nuisance parameters for the NLO and NNLO $K$-factors for the five process categories. These 20 additional parameters and their associated penalties are also shown in Table~\ref{tab: K-fac_firstresults}.
The extra N$^{3}$LO theory and level of freedom introduced has allowed the fit to achieve a total $\Delta \chi^{2} = - 172.5$ compared to MSHT20 NNLO total $\chi^{2}$ (Table 7 from \cite{Thorne:MSHT20}).
Comparing with lower order PDF fits, we find a smooth convergence in the fit quality which follows what one may expect from an increase in the accuracy of a perturbative expansion ($\chi^{2} / N_{\mathrm{pts}}$ = LO: 2.57, NLO: 1.33, NNLO: 1.17, N$^{3}$LO: 1.13). In part, this is due to the extra freedom in the $K$-factors, which will almost always act to reduce this $\chi^{2}$ due to the minimisation procedure. However, even with this freedom, in most cases the N$^{3}$LO theory (non $K$-factor) contributions include large divergences from NNLO. With this in mind, we must conclude that the fit is preferring a description different from the current NNLO standard.
At NNLO (Table~\ref{tab: no_HERA_fullNNLO}), the tension between HERA and non-HERA datasets accounted for $\Delta \chi^{2} = -61.6$ reduction in the overall fit quality when the former was removed, with the majority of this tension between HERA and ATLAS $8\ \text{TeV}\ Z\ p_{T}$~\cite{ATLASZpT} data. Whereas comparing fit results with and without HERA data at N$^{3}$LO, we find $\Delta \chi^{2} = -47.8$. Although the overall difference is not too substantial we do report a substantial shift in the leading tensions, where most of the tension with HERA data is now residing with NMC $F_{2}$~\cite{NMC} and CMS $8\ \text{TeV}$ jets~\cite{CMS8jets} data. Tensions with NMC $F_{2}$~\cite{NMC} data are also seen to some extent at NNLO where we show a $\Delta \chi^{2} = - 20.6$ in a fit omitting HERA data (combining the NMC $F_{2}$ datasets shown in Table~\ref{tab: no_HERA_fullNNLO}). However at N$^{3}$LO, Table~\ref{tab: no_HERA_fullN3LO} shows a $\Delta \chi^{2} = - 31.5$ reduction in a fit omitting HERA data. Therefore whilst the N$^{3}$LO additions remove tensions with $Z\ p_{T}$ data, it remains that the HERA data is preferring the high-$x$ quarks to be lower than favoured by NMC data. This is suggestive of higher twist effects for NMC data at low-$Q^{2}$ (as we observe a worse fit to low-$Q^{2}$ data). We also emphasise that when conducting a fit at NNLO with $Z\ p_{T}$ data removed, an improvement of $\Delta \chi^{2} = -41.3$ is observed in the rest of the data, whereas at N$^{3}$LO an improvement of $\Delta \chi^{2} = -89.8$ is observed in all other datasets without removing $Z\ p_{T}$, therefore these results are not purely an effect of removing any $Z\ p_{T}$ tension. Considering tensions with CMS $8\ \text{TeV}$ jets~\cite{CMS8jets} data, as discussed in Section~\ref{sec: n3lo_K}, in general the jets datasets show tensions with the N$^{3}$LO description (especially for CMS $8\ \text{TeV}$ jets~\cite{CMS8jets}), therefore it will be interesting to observe how this picture evolves when considering this data in the form of dijets.
Since a naturally richer description of the small-$x$ regime is being included at N$^{3}$LO, which has a direct effect on the HERA datasets, the reduction of important tensions from NNLO is even further justification for the inclusion of the N$^{3}$LO theory. The extra N$^{3}$LO additions are allowing the large-$x$ behaviour of the PDFs to be less dominated by data at small-$x$, while also producing a better fit quality at small-$x$ (i.e. for HERA data).
Reflecting on the chosen prior distributions for each of the sources of N$^{3}$LO MHOUs, Table~\ref{tab: K-fac_firstresults} confirms that no especially large penalties are being incurred in this new description.
These results therefore demonstrate that the fit is succeeding in leveraging contributions (such as $P_{qq}^{(3)}$ and $P_{qg}^{(3)}$ in the quark evolution part of Equation~\eqref{eq: DGLAP}) to produce a better overall fit.
\subsubsection*{DIS Processes}
\begin{table}[t]
\centerline{
\begin{tabular}{|P{6.5cm}|P{3cm}|P{3cm}|P{3.4cm}|}
\hline
DIS Dataset & $\chi^{2}$ & $\Delta \chi^{2}$ & $\Delta \chi^{2}$ from NNLO\\
& & from NNLO & (NNLO $K$-factors)\\
\hline
BCDMS $\mu p$ $F_{2}$ \cite{BCDMS} & 179.9 / 163 & $-0.3$ & $-0.9$ \\
BCDMS $\mu d$ $F_{2}$ \cite{BCDMS} & 143.1 / 151 & $-2.9$ & $-1.2$ \\
NMC $\mu p$ $F_{2}$ \cite{NMC} & 118.7 / 123 & $-5.3$ & $-7.0$ \\
NMC $\mu d$ $F_{2}$ \cite{NMC} & 106.2 / 123 & $-6.4$ & $-10.0$ \\
SLAC $ep$ $F_{2}$ \cite{SLAC,SLAC1990} & 32.0 / 37 & $+0.0$ & $+0.6$ \\
SLAC $ed$ $F_{2}$ \cite{SLAC,SLAC1990} & 21.9 / 38 & $-1.1$ & $-1.4$ \\
E665 $\mu p$ $F_{2}$ \cite{E665} & 64.7 / 53 & $+5.1$ & $+6.0$ \\
E665 $\mu d$ $F_{2}$ \cite{E665} & 67.5 / 53 & $+2.8$ & $+3.1$ \\
NuTeV $\nu N$ $F_{2}$ \cite{NuTev} & 38.7 / 53 & $+0.5$ & $+2.1$ \\
NuTeV $\nu N$ $xF_{3}$ \cite{NuTev} & 33.9 / 42 & $+3.2$ & $+1.7$ \\
NMC $\mu n / \mu p$ \cite{NMCn/p} & 128.5 / 148 & $-2.3$ & $-2.9$ \\
HERA $ep$ $F_{2}^{\text{charm}}$ \cite{H1+ZEUScharm} & 134.7 / 79 & $+2.4$ & $+6.4$ \\
NMC/BCDMS/SLAC/HERA $F_{L}$ \cite{BCDMS,NMC,SLAC1990,H1FL,H1-FL,ZEUS-FL} & 45.5 / 57 & $-23.0$ & $-23.2$ \\
CHORUS $\nu N$ $F_{2}$ \cite{CHORUS} & 32.7 / 42 & $+2.6$ & $+2.6$ \\
CHORUS $\nu N$ $xF_{3}$ \cite{CHORUS} & 19.8 / 28 & $+1.3$ & $+2.2$ \\
HERA $e^{+}p$ CC \cite{H1+ZEUS} & 51.8 / 39 & $-0.1$ & $+0.3$ \\
HERA $e^{-}p$ CC \cite{H1+ZEUS} & 66.3 / 42 & $-3.8$ & $-2.8$ \\
HERA $e^{+}p$ NC $820\ \text{GeV}$ \cite{H1+ZEUS} & 83.8 / 75 & $-6.0$ & $-5.9$ \\
HERA $e^{-}p$ NC $460\ \text{GeV}$ \cite{H1+ZEUS} & 247.4 / 209 & $-0.9$ & $-0.7$ \\
HERA $e^{+}p$ NC $920\ \text{GeV}$ \cite{H1+ZEUS} & 476.7 / 402 & $-36.0$ & $-32.9$ \\
HERA $e^{-}p$ NC $575\ \text{GeV}$ \cite{H1+ZEUS} & 248.0 / 259 & $-15.0$ & $-14.5$ \\
HERA $e^{-}p$ NC $920\ \text{GeV}$ \cite{H1+ZEUS} & 243.3 / 159 & $-1.0$ & $-0.9$ \\
\hline
Total & 2585.2 / 2375 & $-86.4$ & $-79.3$\\
\hline
\end{tabular}
}
\caption{\label{tab: DIS_results}Table showing the relevant DIS datasets and how the individual $\chi^{2}$ changes from NNLO by including the N$^{3}$LO contributions to the structure function $F_{2}(x, Q^{2})$. The result within purely NNLO $K$-factors included for all data in the fit is also given.}
\end{table}
To complement the discussions in Section~\ref{sec: n3lo_K}, we isolate the $\chi^{2}$ results from DIS data in Table~\ref{tab: DIS_results}. This data is directly affected by the N$^{3}$LO structure functions constructed approximately in Section's~\ref{sec: structure} to \ref{sec: n3lo_coeff}. A substantial decrease in the total $\chi^{2}$ from NNLO is observed across DIS datasets. Considering the results in Table~\ref{tab: DIS_results} in the context of Table's~\ref{tab: DY_kfac} to \ref{tab: dimuon_kfac}, a better fit quality is observed for all DIS and non-DIS datasets than at NNLO with the inclusion of N$^{3}$LO contributions. As the DIS data makes up over half of the total data included in a global fit, it is the dominant force in deciding the overall form of the PDFs, especially at small-$x$ (discussed further in Section~\ref{subsec: pdf_results}). Table~\ref{tab: DIS_results} further reinforces the point that the N$^{3}$LO description is flexible enough to fit to HERA and non-HERA data, without being largely constrained by tensions between the small-$x$ (HERA dominated) and large-$x$ (non-HERA dominated) regions.
\subsection{Correlation Results} \label{subsec: correlations}
\begin{figure}
\begin{center}
\includegraphics[width=0.95\textwidth]{figures/section8/section8-2/cov_theory1.png}
\includegraphics[width=0.95\textwidth]{figures/section8/section8-2/cov_theory2.png}
\end{center}
\caption{\label{fig: corr_theory_full}Correlation matrix for all N$^{3}$LO theory parameters included in the fit against the subset of the MSHT20 parameters (shown in black) used in constructing the Hessian eigenvectors. This is shown for the case where the $K$-factors correlations with the first 42 parameters are included. N$^{3}$LO theory parameters associated with the splitting functions are coloured blue, the parameters affecting the transition matrix elements and coefficient functions are in red and the $K$-factor parameters are in green.}
\end{figure}
The correlation matrix shown in Fig.~\ref{fig: corr_theory_full} illustrates the correlations between extra N$^{3}$LO theory parameters and the subset of the MSHT20 parameters which are included in the construction of Hessian eigenvectors (see Section~\ref{subsec: eigenvector_results} and \cite{Thorne:MSHT20} for details). It is apparent that the majority of $K$-factor parameters for each process (shown in green) have a lower correlation with the other PDF and theory parameters, while also being effectively independent of other processes\footnote{The same pattern can be seen for $c_{i}^{\mathrm{NLL}}$ where $i \in \{q,g\}$.}. Due to this, there is an argument that each process' $K$-factor parameters could be treated separately from all other parameters in the Hessian prescription (see Section~\ref{subsec: genframe}). For all other non $K$-factor N$^{3}$LO parameters there are more associated correlations, therefore treating these separately would sacrifice important information about correlations within the fit. This is a fairly intuitive result, since the correlations are showing a natural separation between the process dependent and process independent physics in the DIS picture. Mathematically, the $K$-factors are directly associated with the hard cross section, whereas other N$^{3}$LO theory parameters ($\rho_{ij}$ and $a_{ij}$) are having a direct effect on the PDFs. Fig.~\ref{fig: corr_theory_full} therefore motivates the inclusion of the `pure' theory (splitting functions and transition matrix elements) parameters within the standard MSHT eigenvector analysis~\cite{Thorne:MSHT20}, with the decorrelation of the $K$-factor parameters, as discussed in Section~\ref{subsec: decorr_params}. We investigate and compare both treatments (complete correlation and $K$-factor decorrelation) throughout the rest of this section. Note that although the $c_{i}^{\mathrm{NLL}}$ parameters also show minimal correlation with other parameters, we include these within the `pure' theory group of parameters (i.e. correlated with $\rho_{ij}$ and $a_{ij}$) as they are essential ingredients in the underlying DIS theory.
\subsection{Eigenvector Results}\label{subsec: eigenvector_results}
In the MSHT fitting procedure (described in \cite{Thorne:MSHT20}) the eigenvectors of a Hessian matrix are found, which encapsulate the sources of uncertainties and corresponding correlations. Combining these with the central PDFs, forms the entire PDF set with uncertainties. In this eigenvector analysis a dynamical rescaling of each eigenvector $e_{i}$ is performed via a tolerance factor $t$ to encapsulate the $68\%$ confidence limit (C.L.).
\begin{equation}
a_{i} = a_{i}^{0} \pm t e_{i},
\end{equation}
where $a_{i}^{0}$ is the best fit parameter.
$t$ is then adjusted to give the desired tolerance $T$ for the required confidence interval defined as $T = \sqrt{\Delta \chi_{\mathrm{global}}^{2}}$ (for $68\%$ C.L.). In a quadratic approximation, for suitably well-behaved eigenvectors, $t = T$ is true. Although for eigenvectors with larger eigenvalues, it is possible to observe significant deviations from $t=T$. The standard MSHT fitting procedure involves allowing all relevant parameters from \cite{Thorne:MSHT20} to vary when finding the best fit, now including all N$^{3}$LO theory parameters ($\rho_{ij}, a_{ij}, c_{i}^{\mathrm{NLL}}, K_{\mathrm{NLO/NNLO}}$) discussed in this work.
After accounting for high degrees of correlation between parameters (described in \cite{Thorne:MSTW09}), the result is a Hessian matrix which in general, depends on a subset of the parameters that were allowed to vary in a best fit and provides a set of suitably well-behaved eigenvectors. The standard MSHT NNLO PDF eigenvectors are based on a set of 32 parameters, reduced from the 52 parameters allowed to vary in the full fit. In the following analysis we are therefore concerned with a smaller number of parameters, specifically the 32 parameters from the standard MSHT fitting procedure plus an extra 20 N$^{3}$LO parameters (shown in Fig.~\ref{fig: corr_theory_full}).
A standard choice of tolerance $T$ is $T = \sqrt{\Delta \chi^{2}_{\mathrm{global}}} = 1$ for a 68\% C.L. limit. However, this assumes all datasets are consistent with Gaussian errors. In practice, due to incomplete theory, tensions between datasets and parameterisation inflexibility, this is known not to be the case in a global PDF fit. To overcome this, a 68\% C.L. region for each dataset is defined. Then for each eigenvector, the value of $\sqrt{\chi^{2}_{\mathrm{global}}}$ for each chosen $t$ is recorded (ideally showing a quadratic behaviour). Finally, a value of $T$ is chosen to ensure that all datasets are described within their 68\% CL in each eigenvector direction. For a fuller mathematical description of the dynamical tolerance procedure used in MSHT PDF fits, the reader is referred to \cite{Thorne:MSTW09}. In this section we present a demonstration of how well the resultant eigenvectors follow the quadratic assumption based on $t=T$, including the specific choices of dynamical tolerances and which dataset/penalty constrains this tolerance in each eigenvector direction.
\subsubsection*{PDF + N$^{3}$LO DIS Theory + N$^{3}$LO $K$-factor (decorrelated) Parameters}
As discussed in Section~\ref{subsec: correlations}, when determining the eigenvectors and therefore PDF uncertainties, we can choose to either include the correlations between the 10 $K$-factor parameters added with the other 42 parameters (encompassing the standard 32 MSHT eigenvector parameters and the 10 new theory parameters from the splitting functions, transition matrix elements and coefficient functions) or to decorrelate the 10 $K$-factor parameters.
In this section we address the scenario where we decorrelate the $K$-factors as
\begin{equation}\label{eq: decorr}
H_{ij}^{-1} + \sum_{p = 1}^{N_{p}}K_{ij,p}^{-1}
\end{equation}
and consider each term individually.
\begin{figure}
\begin{center}
\includegraphics[width=\textwidth]{figures/section8/section8-3/eigenvector42.png}
\end{center}
\caption{\label{fig: eigenmap} Correlation matrix of the first 42 (total 52) eigenvectors found with the N$^{3}$LO parameters added into the analysis in the case where the $K$-factors are decorrelated from these first 42 parameters. Parameters associated with the splitting functions are coloured blue, those affecting the transition matrix elements and coefficient functions are in red.}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=0.7\textwidth]{figures/section8/section8-3/eigenvectorK.png}
\end{center}
\caption{\label{fig: eigenmap_K} Map of the 10 $K$-factor eigenvectors found with the N$^{3}$LO parameters added into the analysis in the case where the $K$-factors are decorrelated from these first 42 parameters.. Combined with the 42 eigenvectors shown in Fig.~\ref{fig: eigenmap}, these form the total 52 eigenvectors in the decorrelated case. Parameters associated with the $K$-factor parameters are in green.}
\end{figure}
Fig.~\ref{fig: eigenmap} shows the map of eigenvectors produced from $H_{ij}$ in Equation~\eqref{eq: decorr}, where we have included the new N$^{3}$LO DIS theory parameters (splitting functions in blue and coefficient functions/transition matrix elements in red) correlated with the PDF parameters. Eigenvectors 35 and 36 are prime examples of where the eigenvectors have specifically encompassed the correlation/anti-correlation between the two NLL FFNS coefficient function parameters $c_{i}^{\mathrm{NLL}}$ ($i \in \{q,g\}$). Whereas the splitting functions naturally give rise to a much more complicated mixing with other PDF parameters as these directly affect the evolution of the PDFs. Due to the direct impact of $\rho_{ij}$'s on the PDFs (via DGLAP evolution), combined with the large contributions to the evolution shown at N$^{3}$LO, this result is as expected.
Another somewhat pleasing aspect is the recovery of a natural separation between eigenvectors associated with the N$^{3}$LO coefficient function/transition matrix elements and our original PDF parameters (incl. N$^{3}$LO splitting functions). This separation is reminiscent of our DIS picture, whereby the splitting functions are much more intertwined with the raw PDFs and the transition matrix elements have a symbiotic relationship with the coefficient functions (see GM-VFNS description in Section~\ref{sec: structure}). Due to this, the form of these eigenvectors has not only some level of physical interpretation inherited from our underlying theory, but also offers a useful way to access the different sources of N$^{3}$LO additions within the PDF set.
In Fig.~\ref{fig: eigenmap_K} the eigenvectors resulting from the $\sum_{p = 1}^{N_{p}}K_{ij,p}^{-1}$ terms in Equation~\eqref{eq: decorr} are shown. These eigenvectors are constructed in pairs, describing the correlation and anti-correlation of the two $K$-factor parameters (controlling the NLO and NNLO contributions to N$^{3}$LO) for each process $p$ contained within the corresponding $K_{ij,p}$ correlation matrix.
Table~\ref{tab: Kij_limits} shows further information regarding the $K$-factor parameter limits from each eigenvector. In most cases the parameter limits are well within the allowed variation ($-1 < a < 1$), which is an indication that the data included in the fit is constraining these parameters rather than the individual penalties for each parameter\footnote{We remind the reader that the Dimuon datasets also include a branching ratio factor which is providing some compensation with these $K$-factor parameters (as discussed in Section \ref{sec: n3lo_K}).}.
To assess whether the eigenvectors are violating the quadratic treatment, four examples displaying this behaviour are shown in Fig.~\ref{fig: tol_decorr}, with a full analysis provided in Appendix~\ref{app: tolerance_decorr}. Additionally, Table~\ref{tab: eigenanalysis} provides a summary of all tolerances found within the eigenvector scans.
\begin{table}[t]
\centering
\begin{tabular}{c|c|c|c|c|c|c|c|c}
\hline
\multirow{2}{*}{Matrix} & \multicolumn{2}{c|}{Central Values} & \multirow{2}{*}{Eigenvector} & \multicolumn{2}{c|}{+ Limit} & \multicolumn{2}{c|}{- Limit} & \multirow{2}{*}{Scale} \\
\cline{2-3} \cline{5-8}
& $a_{\mathrm{NLO}}$ & $a_{\mathrm{NNLO}}$ & & $a_{\mathrm{NLO}}$ & $a_{\mathrm{NNLO}}$ & $a_{\mathrm{NLO}}$ & $a_{\mathrm{NNLO}}$ & \\
\hline
\multirow{2}{*}{$K_{ij}^{\mathrm{DY}}$} & \multirow{2}{*}{-0.247} & \multirow{2}{*}{-0.053} & 43 & -0.386 & 0.039 & -0.146 & -0.119 & \multirow{2}{*}{$m/2$}\\
& & & 44 & -0.173 & 0.467 & -0.319 & -0.559 & \\
\hline
\multirow{2}{*}{$K_{ij}^{\mathrm{Top}}$} & \multirow{2}{*}{0.324} & \multirow{2}{*}{0.812} & 45 & -0.015 & 0.227 & 0.674 & 1.476 & \multirow{2}{*}{Section~\ref{subsec: kfac_res}} \\
& & & 46 & 0.059 & 1.295 & 0.721 & 0.088 & \\
\hline
\multirow{2}{*}{$K_{ij}^{\mathrm{Jets}}$} & \multirow{2}{*}{-0.250} & \multirow{2}{*}{-0.719} & 47 & -0.454 & -0.991 & 0.282 & 0.008 & \multirow{2}{*}{$p_{T}^{\mathrm{jet}}$}\\
& & & 48 & -0.659 & -0.060 & 0.072 & -1.237 & \\
\hline
\multirow{2}{*}{$K_{ij}^{p_{T}\ \mathrm{Jets}}$} & \multirow{2}{*}{0.662} & \multirow{2}{*}{-0.105} & 49 & 0.468 & -0.421 & 0.916 & 0.310 & \multirow{2}{*}{$p_{T}$}\\
& & & 50 & 0.558 & 0.566 & 0.763 & -0.760 & \\
\hline
\multirow{2}{*}{$K_{ij}^{\mathrm{Dimuon}}$} & \multirow{2}{*}{-0.694} & \multirow{2}{*}{0.602} & 51 & -1.219 & -0.054 & -0.272 & 1.258 & \multirow{2}{*}{$Q^{2}$}\\
& & & 52 & -1.268 & 1.013 & 1.099 & -0.680 & \\
\hline
\end{tabular}
\caption{Limiting values for specific $K$-factor parameters for each of the processes considered in the decorrelated case. Parameter values are shown in the positive and negative limits for each eigenvector. The scale choices for top quark processes are described in Section~\ref{subsec: kfac_res} to be $H_T/4$ for the single differential datasets with the exception of data differential in the average transverse momentum of the top or antitop, $p_T^t,p_T^{\bar{t}}$, for which $m_T/2$ is used. For the double diff. dataset the scale choice is $H_T/4$ and for the inclusive top $\sigma_{t\bar{t}}$ a scale of $m_{t}$ is chosen.}
\label{tab: Kij_limits}
\end{table}
\begin{figure}
\begin{center}
\includegraphics[width=0.49\textwidth]{figures/section8/section8-3/N3LO/eigenvector_analysis_32.png}
\includegraphics[width=0.49\textwidth]{figures/section8/section8-3/N3LO/eigenvector_analysis_34.png}
\\
\includegraphics[width=0.49\textwidth]{figures/section8/section8-3/N3LO/eigenvector_analysis_41.png}
\includegraphics[width=0.49\textwidth]{figures/section8/section8-3/N3LO/eigenvector_analysis_42.png}
\caption{\label{fig: tol_decorr}Dynamic tolerance behaviour for 4 selected eigenvectors in the case of decorrelated $K$-factor parameters. The black dots show the fixed tolerance relations found for integer values of $t$, whereas the red triangles show the final chosen dynamical tolerances for each eigenvector direction. For an exhaustive analysis of all eigenvectors see Fig.~\ref{fig: full_tol_decorr}.}
\end{center}
\end{figure}
\begin{table}
\centerline{
\begin{tabular}{|p{0.5cm}|p{0.7cm}|p{0.7cm}|p{4.5cm}|p{0.7cm}|p{0.7cm}|p{4.5cm}|p{2.4cm}|}
\hline
\# & $t+$ & $T+$ & Limiting Factor ($+$) & $t-$ & $T-$ & Limiting Factor ($-$) & Primary \\
& & & & & & & Parameter \\ \hline
\hline
1 & 3.45 & 3.56 & ATLAS $7\ \text{TeV}$ high prec. $W, Z$ \cite{ATLASWZ7f} & 2.80 & 2.93 & ATLAS $8\ \text{TeV}$ double diff. $Z$ \cite{ATLAS8Z3D} & $a_{S,3}$ \\
2 & 3.52 & 3.33 & NMC $\mu d$ $F_{2}$ \cite{NMC} & 5.13 & 5.41 & HERA $e^{+}p$ NC $920\ \text{GeV}$ \cite{H1+ZEUS} & $a_{S,6}$ \\
3 & 4.48 & 4.37 & NMC $\mu d$ $F_{2}$ \cite{NMC} & 6.33 & 6.82 & HERA $e^{+}p$ NC $920\ \text{GeV}$ \cite{H1+ZEUS} & $a_{g,3}$ \\
4 & 5.57 & 5.78 & ATLAS $7\ \text{TeV}$ high prec. $W, Z$ \cite{ATLASWZ7f} & 2.47 & 2.42 & NMC $\mu d$ $F_{2}$ \cite{NMC} & $a_{S,2}$ \\
5 & 6.63 & 6.72 & HERA $e^{-}p$ NC $575\ \text{GeV}$ \cite{H1+ZEUS} & 4.16 & 4.39 & HERA $e^{+}p$ NC $920\ \text{GeV}$ \cite{H1+ZEUS} & $\delta_{g^{\prime}}$ \\
6 & 3.97 & 4.23 & D{\O} II $W \rightarrow \nu e$ asym. \cite{D0Wnue} & 2.15 & 2.14 & D{\O} $W$ asym. \cite{D0Wasym} & $\delta_{u}$ \\
7 & 3.09 & 3.53 & D{\O} $W$ asym. \cite{D0Wasym} & 6.69 & 6.44 & HERA $e^{+}p$ NC $920\ \text{GeV}$ \cite{H1+ZEUS} & $\delta_{S}$ \\
8 & 5.97 & 6.06 & LHCb 2015 $W, Z$ \cite{LHCbZ7,LHCbWZ8} & 2.85 & 3.05 & D{\O} $W$ asym. \cite{D0Wasym} & $a_{u,6}$ \\
9 & 3.15 & 3.02 & BCDMS $\mu p$ $F_{2}$ \cite{BCDMS} & 3.07 & 3.39 & D{\O} $W$ asym. \cite{D0Wasym} & $a_{u,6}$ \\
10 & 2.17 & 2.35 & D{\O} $W$ asym. \cite{D0Wasym} & 5.27 & 5.37 & CMS $8\ \text{TeV}\ W$ \cite{CMSW8} & $\delta_{g}$ \\
11 & 5.54 & 5.89 & NuTeV $\nu N$ $F_{2}$ \cite{NuTev} & 2.55 & 2.63 & NMC $\mu d$ $F_{2}$ \cite{NMC} & $a_{g,2}$ \\
12 & 3.58 & 3.68 & CMS $7\ \text{TeV}\ W + c$ \cite{CMS7Wpc} & 3.22 & 3.28 & NuTeV $\nu N \rightarrow \mu\mu X$ \cite{Dimuon} & $a_{s+,5}$ \\
13 & 2.13 & 2.40 & D{\O} $W$ asym. \cite{D0Wasym} & 2.78 & 3.13 & E866 / NuSea $pd/pp$ DY \cite{E866DYrat} & $A_{s-}$ \\
14 & 4.38 & 4.92 & ATLAS $8\ \text{TeV}$ double diff. $Z$ \cite{ATLAS8Z3D} & 1.82 & 1.86 & E866 / NuSea $pd/pp$ DY \cite{E866DYrat} & $a_{\rho,6}$ \\
15 & 5.08 & 4.39 & NuTeV $\nu N$ $xF_{3}$ \cite{NuTev} & 2.09 & 2.13 & NuTeV $\nu N \rightarrow \mu\mu X$ \cite{Dimuon} & $a_{u,2}$ \\
16 & 2.97 & 3.39 & E866 / NuSea $pd/pp$ DY \cite{E866DYrat} & 0.92 & 1.21 & E866 / NuSea $pd/pp$ DY \cite{E866DYrat} & $A_{\rho}$ \\
17 & 2.23 & 2.61 & D{\O} $W$ asym. \cite{D0Wasym} & 2.81 & 3.01 & E866 / NuSea $pd/pp$ DY \cite{E866DYrat} & $a_{d,6}$ \\
18 & 1.68 & 2.15 & E866 / NuSea $pd/pp$ DY \cite{E866DYrat} & 2.51 & 2.53 & NuTeV $\nu N \rightarrow \mu\mu X$ \cite{Dimuon} & $a_{s+,2}$ \\
19 & 1.45 & 1.45 & $\rho_{qq}^{NS}$ & 1.70 & 1.75 & $\rho_{qq}^{NS}$ & $\rho_{qq}^{NS}$ \\
20 & 1.65 & 1.41 & $\rho_{qq}^{NS}$ & 1.41 & 1.68 & $\rho_{qq}^{NS}$ & $\rho_{qq}^{NS}$ \\
21 & 2.77 & 3.30 & E866 / NuSea $pd/pp$ DY \cite{E866DYrat} & 3.04 & 2.89 & CMS $7\ \text{TeV}\ W + c$ \cite{CMS7Wpc} & $a_{s+,2}$ \\
22 & 2.23 & 2.45 & NuTeV $\nu N \rightarrow \mu\mu X$ \cite{Dimuon} & 1.60 & 1.99 & D{\O} $W$ asym. \cite{D0Wasym} & $a_{d,2}$ \\
\hline
\end{tabular}
}
\caption{\label{tab: eigenanalysis}Tolerances resulting from eigenvector scans with decorrelated $K$-factors for each process. The average tolerance for this set of eigenvectors is $T=3.12$.}
\end{table}
\begin{table}\ContinuedFloat
\centerline{
\begin{tabular}{|p{0.5cm}|p{0.7cm}|p{0.7cm}|p{4.5cm}|p{0.7cm}|p{0.7cm}|p{4.5cm}|p{2.4cm}|}
\hline
\# & $t+$ & $T+$ & Limiting Factor ($+$) & $t-$ & $T-$ & Limiting Factor ($-$) & Primary \\
& & & & & & & Parameter \\ \hline
\hline
23 & 6.08 & 5.26 & BCDMS $\mu p$ $F_{2}$ \cite{BCDMS} & 1.96 & 2.18 & E866 / NuSea $pd/pp$ DY \cite{E866DYrat} & $\eta_{u}$ \\
24 & 5.01 & 5.91 & ATLAS $7\ \text{TeV}$ high prec. $W, Z$ \cite{ATLASWZ7f} & 4.77 & 5.67 & HERA $e^{+}p$ NC $920\ \text{GeV}$ \cite{H1+ZEUS} & $\rho_{qq}^{PS}$ \\
25 & 1.61 & 1.81 & E866 / NuSea $pd/pp$ DY \cite{E866DYrat} & 3.41 & 3.65 & CMS $8\ \text{TeV}\ W$ \cite{CMSW8} & $a_{\rho,3}$ \\
26 & 1.94 & 2.22 & D{\O} $W$ asym. \cite{D0Wasym} & 2.45 & 2.90 & E866 / NuSea $pd/pp$ DY \cite{E866DYrat} & $a_{d,3}$ \\
27 & 1.65 & 1.68 & $\rho_{gq}$ & 3.38 & 4.06 & D{\O} $W$ asym. \cite{D0Wasym} & $\rho_{gq}$ \\
28 & 1.20 & 1.45 & D{\O} $W$ asym. \cite{D0Wasym} & 3.21 & 3.61 & CMS $8\ \text{TeV}\ W$ \cite{CMSW8} & $A_{s+}$ \\
29 & 3.87 & 5.08 & ATLAS $8\ \text{TeV}$ double diff. $Z$ \cite{ATLAS8Z3D} & 2.21 & 2.86 & CMS $8\ \text{TeV}$ sing. diff. $t\bar{t}$ \cite{CMSttbar08_ytt} & $\eta_{g}$ \\
30 & 1.06 & 1.07 & D{\O} $W$ asym. \cite{D0Wasym} & 4.22 & 4.60 & D{\O} $W$ asym. \cite{D0Wasym} & $\eta_{d} - \eta_{u}$ \\
31 & 3.58 & 3.93 & CMS $8\ \text{TeV}\ W$ \cite{CMSW8} & 2.12 & 2.43 & NuTeV $\nu N$ $xF_{3}$ \cite{NuTev} & $\eta_{S}$ \\
32 & 4.86 & 6.78 & HERA $e^{+}p$ NC $920\ \text{GeV}$ \cite{H1+ZEUS} & 4.10 & 4.68 & ATLAS $8\ \text{TeV}\ Z\ p_{T}$ \cite{ATLASZpT} & $\rho_{qg}$ \\
33 & 2.65 & 3.80 & BCDMS $\mu d$ $F_{2}$ \cite{BCDMS} & 3.34 & 3.69 & CMS $8\ \text{TeV}\ W$ \cite{CMSW8} & $\eta_{s+}$ \\
34 & 2.29 & 3.63 & D{\O} $W$ asym. \cite{D0Wasym} & 2.02 & 3.28 & CMS $8\ \text{TeV}\ W$ \cite{CMSW8} & $a_{\rho,1}$ \\
35 & 4.07 & 5.08 & HERA $ep$ $F_{2}^{\text{charm}}$ \cite{H1+ZEUScharm} & 2.40 & 2.92 & HERA $ep$ $F_{2}^{\text{charm}}$ \cite{H1+ZEUScharm} & $c_{g}^{\mathrm{NLL}}$ \\
36 & 1.34 & 1.52 & $c_{q}^{\mathrm{NLL}}$ & 1.24 & 1.22 & $c_{g}^{\mathrm{NLL}}$ & $c_{q}^{\mathrm{NLL}}$ \\
37 & 2.63 & 3.83 & NuTeV $\nu N \rightarrow \mu\mu X$ \cite{Dimuon} & 2.97 & 3.16 & E866 / NuSea $pd/pp$ DY \cite{E866DYrat} & $\eta_{s-}$ \\
38 & 0.69 & 0.71 & $\rho_{gg}$ & 3.76 & 3.90 & $\rho_{gq}$ & $\rho_{gg}$ \\
39 & 1.57 & 5.57 & ATLAS $7\ \text{TeV}$ high prec. $W, Z$ \cite{ATLASWZ7f} & 1.57 & 5.01 & ATLAS $7\ \text{TeV}$ high prec. $W, Z$ \cite{ATLASWZ7f} & $A_{S}$ \\
40 & 0.88 & 1.00 & $a_{qq,H}^{\mathrm{NS}}$ & 1.15 & 1.01 & $a_{qq,H}^{\mathrm{NS}}$ & $a_{qq,H}^{\mathrm{NS}}$ \\
41 & 1.71 & 2.72 & HERA $ep$ $F_{2}^{\text{charm}}$ \cite{H1+ZEUScharm} & 2.16 & 2.81 & $\rho_{gg}$ & $a_{Hg}$ \\
42 & 2.09 & 2.19 & $\rho_{gg}$ & 0.60 & 0.54 & $a_{gg,H}$ & $a_{gg,H}$ \\
43 & 2.25 & 2.77 & ATLAS $7\ \text{TeV}$ high prec. $W, Z$ \cite{ATLASWZ7f} & 1.62 & 1.26 & CMS double diff. Drell-Yan \cite{CMS-ddDY} & $\mathrm{DY}_{\mathrm{NLO}}$ \\
44 & 3.90 & 3.86 & E866 / NuSea $pp$ DY \cite{E866DY} & 3.80 & 4.38 & ATLAS $7\ \text{TeV}$ high prec. $W, Z$ \cite{ATLASWZ7f} & $\mathrm{DY}_{\mathrm{NNLO}}$ \\
45 & 1.28 & 1.22 & ATLAS $8\ \text{TeV}$ sing. diff. $t\bar{t}$ dilep. \cite{ATLASttbarDilep08_ytt} & 1.45 & 1.56 & Tevatron, ATLAS, CMS $\sigma_{t\bar{t}}$ \cite{Tevatron-top,ATLAS-top7-1,ATLAS-top7-2,ATLAS-top7-3,ATLAS-top7-4,ATLAS-top7-5,ATLAS-top7-6,CMS-top7-1,CMS-top7-2,CMS-top7-3,CMS-top7-4,CMS-top7-5,CMS-top8} & $\mathrm{Top}_{\mathrm{NNLO}}$ \\
\hline
\end{tabular}
}
\caption{\textit{(Continued)} Tolerances resulting from eigenvector scans with decorrelated $K$-factors for each process. The average tolerance for this set of eigenvectors is $T=3.12$.}
\end{table}
\begin{table}\ContinuedFloat
\centerline{
\begin{tabular}{|p{0.5cm}|p{0.7cm}|p{0.7cm}|p{4.5cm}|p{0.7cm}|p{0.7cm}|p{4.5cm}|p{2.4cm}|}
\hline
\# & $t+$ & $T+$ & Limiting Factor ($+$) & $t-$ & $T-$ & Limiting Factor ($-$) & Primary \\
& & & & & & & Parameter \\ \hline
\hline
46 & 1.08 & 0.92 & $\mathrm{Top}_{\mathrm{NNLO}}$ & 1.62 & 1.34 & $\mathrm{Top}_{\mathrm{NLO}}$ & $\mathrm{Top}_{\mathrm{NLO}}$ \\
47 & 1.47 & 1.44 & CDF II $p\bar{p}$ incl. jets \cite{CDFjet} & 3.85 & 4.10 & $a_{gg,H}$ & $\mathrm{Jet}_{\mathrm{NLO}}$ \\
48 & 3.25 & 3.29 & CMS $2.76\ \text{TeV}$ jet \cite{CMS276jets} & 2.56 & 2.90 & $\mathrm{Jet}_{\mathrm{NNLO}}$ & $\mathrm{Jet}_{\mathrm{NNLO}}$ \\
49 & 2.10 & 2.35 & ATLAS $8\ \text{TeV}\ Z\ p_{T}$ \cite{ATLASZpT} & 2.75 & 2.71 & ATLAS $8\ \text{TeV}\ W + \text{jets}$ \cite{ATLASWjet} & $p_{T}\ \mathrm{Jet}_{\mathrm{NLO}}$ \\
50 & 2.24 & 2.28 & ATLAS $8\ \text{TeV}\ Z\ p_{T}$ \cite{ATLASZpT} & 2.19 & 2.24 & ATLAS $8\ \text{TeV}\ W + \text{jets}$ \cite{ATLASWjet} & $p_{T}\ \mathrm{Jet}_{\mathrm{NNLO}}$ \\
51 & 1.02 & 1.03 & $\mathrm{Dimuon}_{\mathrm{NLO}}$ & 0.82 & 0.90 & $\mathrm{Dimuon}_{\mathrm{NNLO}}$ & $\mathrm{Dimuon}_{\mathrm{NNLO}}$ \\
52 & 0.75 & 0.77 & $\mathrm{Dimuon}_{\mathrm{NLO}}$ & 2.35 & 2.33 & $\mathrm{Dimuon}_{\mathrm{NLO}}$ & $\mathrm{Dimuon}_{\mathrm{NLO}}$ \\
\hline
\end{tabular}
}
\caption{\textit{(Continued)} Tolerances resulting from eigenvector scans with decorrelated $K$-factors for each process. The average tolerance for this set of eigenvectors is $T=3.12$.}
\end{table}
There is relatively consistent agreement between $t$ and $T$ across all eigenvectors with later eigenvectors (i.e. higher \#) generally becoming less quadratic (a feature which is built into the fit). Eigenvectors 32, 41 and 42 displayed in Fig.~\ref{fig: tol_decorr} are shown in Table~\ref{tab: eigenanalysis} to be either dominated or limited by at least one new N$^{3}$LO parameter. Conversely, eigenvector 34 is much more dominated by the original PDF parameters from MSHT20 NNLO. Comparing these cases, the eigenvectors associated more strongly with the N$^{3}$LO parameters exhibit a similar level of agreement (and occasionally better) with the desired quadratic behaviour as eigenvectors more closely associated with the original PDF parameters.
The last 5 sets of eigenvectors (i.e. the last 10 where a set contains 2 eigenvectors for a particular process) we see in Table~\ref{tab: eigenanalysis} are the decorrelated $K$-factor eigenvectors, where there are correlated/anti-correlated eigenvectors for each process. For all $K$-factor cases, Table~\ref{tab: eigenanalysis} provides sensible results with either the dominant datasets or parameter penalties constraining each eigenvector direction. One interesting feature one can observe here is a sign of tension between the ATLAS $8\ \text{TeV}\ Z\ p_{T}$~\cite{ATLASZpT} and ATLAS $8\ \text{TeV}\ W + \text{jets}$~\cite{ATLASWjet} datasets where the limiting factors in Table~\ref{tab: eigenanalysis} for eigenvectors 49 and 50 show that these datasets are preferring a slightly different $K$-factor.
To provide some extra level of comparison between the eigenvectors shown here and the eigenvectors found in the NNLO case, the average tolerance $T$ for aN$^{3}$LO (decorrelated $K$-factors) set is 3.12, compared to the NNLO average $T$ of 3.37.
\subsubsection*{PDF + N$^{3}$LO DIS Theory + N$^{3}$LO $K$-factor (correlated) Parameters}
In this section we address the scenario,
\begin{equation}
H_{ij}^{\prime} = \left(H_{ij}^{-1} + \sum_{p = 1}^{N_{p}}K_{ij,p}^{-1} \right)^{-1}.
\end{equation}
\begin{figure}
\begin{center}
\includegraphics[width=\textwidth]{figures/section8/section8-3/eigenvector52.png}
\end{center}
\caption{\label{fig: eigenmap52}Map of eigenvectors found with the N$^{3}$LO theory and $K$-factor parameters added into the analysis. Parameters associated with the splitting functions are coloured blue, those affecting the transition matrix elements and coefficient functions are in red and the $K$-factor parameters are in green.}
\end{figure}
Moving to an analysis including aN$^{3}$LO $K$-factors as correlated parameters with PDF and other N$^{3}$LO theory parameters. This provides a comparison to the case of decorrelated $K$-factors and justification for treating the cross section behaviour separately to the PDF theory behaviour.
Fig.~\ref{fig: eigenmap52} shows a map of eigenvectors with the extra 10 N$^{3}$LO $K$-factor parameters (shown in green) included into the correlations considered. As expected, the result of including the correlations between PDF parameters and aN$^{3}$LO $K$-factors results in a slightly more intertwined set of eigenvectors (although a high level of decorrelation remains). Specifically, due to the much higher number of DY datasets included in the global fit, these N$^{3}$LO $K$-factor parameters tend to be included across more of a spread of eigenvectors. On the other hand, the Dimuon $K$-factors are almost entirely isolated within two eigenvectors, similar to the decorrelated case.
\begin{figure}
\begin{center}
\includegraphics[width=0.49\textwidth]{figures/section8/section8-3/N3LO_Kcorr/eigenvector_analysis_27.png}
\includegraphics[width=0.49\textwidth]{figures/section8/section8-3/N3LO_Kcorr/eigenvector_analysis_37.png}
\includegraphics[width=0.49\textwidth]{figures/section8/section8-3/N3LO_Kcorr/eigenvector_analysis_40.png}
\includegraphics[width=0.49\textwidth]{figures/section8/section8-3/N3LO_Kcorr/eigenvector_analysis_52.png}
\caption{\label{fig: tol_corr}Dynamic tolerance behaviour for 4 selected eigenvectors in the case of correlated $K$-factor parameters. The black dots show the fixed tolerance relations found for integer values of $t$, whereas the red triangles show the final chosen dynamical tolerances for each eigenvector direction. For an exhaustive analysis of all eigenvectors see Fig.~\ref{fig: full_tol_corr}.}
\end{center}
\end{figure}
\begin{table}
\centerline{
\begin{tabular}{|p{0.5cm}|p{0.7cm}|p{0.7cm}|p{4.5cm}|p{0.7cm}|p{0.7cm}|p{4.5cm}|p{2.4cm}|}
\hline
\# & $t+$ & $T+$ & Limiting Factor ($+$) & $t-$ & $T-$ & Limiting Factor ($-$) & Primary \\
& & & & & & & Parameter \\ \hline
\hline
1 & 2.87 & 2.90 & ATLAS $8\ \text{TeV}$ double diff. $Z$ \cite{ATLAS8Z3D} & 3.50 & 3.47 & ATLAS $7\ \text{TeV}$ high prec. $W, Z$ \cite{ATLASWZ7f} & $a_{S,3}$ \\
2 & 4.22 & 3.99 & NMC $\mu d$ $F_{2}$ \cite{NMC} & 5.14 & 5.38 & HERA $e^{+}p$ NC $920\ \text{GeV}$ \cite{H1+ZEUS} & $a_{S,6}$ \\
3 & 4.54 & 4.25 & ATLAS $8\ \text{TeV}\ Z\ p_{T}$ \cite{ATLASZpT} & 6.55 & 6.98 & HERA $e^{+}p$ NC $920\ \text{GeV}$ \cite{H1+ZEUS} & $a_{g,3}$ \\
4 & 4.97 & 5.10 & ATLAS $7\ \text{TeV}$ high prec. $W, Z$ \cite{ATLASWZ7f} & 2.82 & 2.74 & NMC $\mu d$ $F_{2}$ \cite{NMC} & $a_{S,2}$ \\
5 & 4.18 & 4.29 & HERA $e^{+}p$ NC $920\ \text{GeV}$ \cite{H1+ZEUS} & 6.63 & 6.59 & HERA $e^{-}p$ NC $575\ \text{GeV}$ \cite{H1+ZEUS} & $\delta_{g^{\prime}}$ \\
6 & 4.00 & 4.10 & D{\O} II $W \rightarrow \nu e$ asym. \cite{D0Wnue} & 2.18 & 2.08 & D{\O} $W$ asym. \cite{D0Wasym} & $\delta_{u}$ \\
7 & 6.69 & 6.31 & HERA $e^{+}p$ NC $920\ \text{GeV}$ \cite{H1+ZEUS} & 3.21 & 3.45 & D{\O} $W$ asym. \cite{D0Wasym} & $\delta_{S}$ \\
8 & 4.65 & 4.45 & CMS W asym. $p_{T} > 25, 30\ \text{GeV}$ \cite{CMS-Wasymm} & 2.24 & 2.32 & D{\O} $W$ asym. \cite{D0Wasym} & $\delta_{d}$ \\
9 & 5.29 & 5.49 & D{\O} II $W \rightarrow \nu \mu$ asym. \cite{D0Wnumu} & 2.45 & 2.25 & BCDMS $\mu p$ $F_{2}$ \cite{BCDMS} & $a_{u,6}$ \\
10 & 5.52 & 5.43 & CMS $8\ \text{TeV}\ W$ \cite{CMSW8} & 2.15 & 2.20 & D{\O} $W$ asym. \cite{D0Wasym} & $\delta_{g}$ \\
11 & 2.64 & 2.45 & NMC $\mu d$ $F_{2}$ \cite{NMC} & 5.65 & 5.90 & NuTeV $\nu N$ $F_{2}$ \cite{NuTev} & $a_{g,2}$ \\
12 & 5.76 & 4.47 & CMS $7\ \text{TeV}\ W + c$ \cite{CMS7Wpc} & 4.44 & 3.40 & NuTeV $\nu N \rightarrow \mu\mu X$ \cite{Dimuon} & $a_{s+,5}$ \\
13 & 2.23 & 2.21 & D{\O} $W$ asym. \cite{D0Wasym} & 2.88 & 3.03 & E866 / NuSea $pd/pp$ DY \cite{E866DYrat} & $A_{s-}$ \\
14 & 4.91 & 4.75 & ATLAS $8\ \text{TeV}$ double diff. $Z$ \cite{ATLAS8Z3D} & 2.53 & 2.28 & D{\O} $W$ asym. \cite{D0Wasym} & $a_{u,2}$ \\
15 & 6.05 & 5.57 & NuTeV $\nu N$ $xF_{3}$ \cite{NuTev} & 1.49 & 1.41 & E866 / NuSea $pd/pp$ DY \cite{E866DYrat} & $a_{\rho,6}$ \\
16 & 0.95 & 0.59 & E866 / NuSea $pd/pp$ DY \cite{E866DYrat} & 2.90 & 3.23 & E866 / NuSea $pd/pp$ DY \cite{E866DYrat} & $A_{\rho}$ \\
17 & 2.86 & 3.05 & NuTeV $\nu N \rightarrow \mu\mu X$ \cite{Dimuon} & 3.27 & 3.23 & CMS $8\ \text{TeV}\ W$ \cite{CMSW8} & $a_{d,6}$ \\
18 & 1.97 & 1.78 & D{\O} $W$ asym. \cite{D0Wasym} & 1.73 & 1.82 & E866 / NuSea $pd/pp$ DY \cite{E866DYrat} & $a_{s+,2}$ \\
19 & 2.36 & 2.48 & CCFR $\nu N \rightarrow \mu\mu X$ \cite{Dimuon} & 2.58 & 2.40 & NuTeV $\nu N \rightarrow \mu\mu X$ \cite{Dimuon} & $a_{s+,3}$ \\
20 & 1.57 & 1.82 & $\rho_{qq}^{NS}$ & 1.84 & 1.43 & $\rho_{qq}^{NS}$ & $\rho_{qq}^{NS}$ \\
21 & 3.03 & 2.61 & $\rho_{qq}^{NS}$ & 2.97 & 3.32 & E866 / NuSea $pd/pp$ DY \cite{E866DYrat} & $a_{s+,2}$ \\
22 & 2.17 & 2.10 & NuTeV $\nu N \rightarrow \mu\mu X$ \cite{Dimuon} & 1.64 & 1.77 & D{\O} $W$ asym. \cite{D0Wasym} & $a_{d,2}$ \\
\hline
\end{tabular}
}
\caption{\label{tab: eigenanalysis_2}Tolerances resulting from eigenvector scans with correlated $K$-factors for each process. The average tolerance for this set of eigenvectors is $T=3.12$.}
\end{table}
\begin{table}\ContinuedFloat
\centerline{
\begin{tabular}{|p{0.5cm}|p{0.7cm}|p{0.7cm}|p{4.5cm}|p{0.7cm}|p{0.7cm}|p{4.5cm}|p{2.4cm}|}
\hline
\# & $t+$ & $T+$ & Limiting Factor ($+$) & $t-$ & $T-$ & Limiting Factor ($-$) & Primary \\
& & & & & & & Parameter \\ \hline
\hline
23 & 4.68 & 4.13 & $\rho_{qq}^{NS}$ & 1.88 & 1.78 & E866 / NuSea $pd/pp$ DY \cite{E866DYrat} & $\eta_{u}$ \\
24 & 4.79 & 5.59 & HERA $e^{+}p$ NC $920\ \text{GeV}$ \cite{H1+ZEUS} & 4.76 & 5.42 & ATLAS $7\ \text{TeV}$ high prec. $W, Z$ \cite{ATLASWZ7f} & $\rho_{qq}^{PS}$ \\
25 & 1.65 & 1.58 & E866 / NuSea $pd/pp$ DY \cite{E866DYrat} & 3.46 & 3.56 & CMS $8\ \text{TeV}\ W$ \cite{CMSW8} & $a_{\rho,3}$ \\
26 & 2.16 & 2.14 & D{\O} $W$ asym. \cite{D0Wasym} & 2.26 & 2.43 & E866 / NuSea $pd/pp$ DY \cite{E866DYrat} & $a_{\rho,3}$ \\
27 & 4.92 & 5.71 & Tevatron, ATLAS, CMS $\sigma_{t\bar{t}}$ \cite{Tevatron-top,ATLAS-top7-1,ATLAS-top7-2,ATLAS-top7-3,ATLAS-top7-4,ATLAS-top7-5,ATLAS-top7-6,CMS-top7-1,CMS-top7-2,CMS-top7-3,CMS-top7-4,CMS-top7-5,CMS-top8} & 1.85 & 1.64 & $\rho_{gq}$ & $\rho_{gq}$ \\
28 & 1.19 & 0.91 & D{\O} $W$ asym. \cite{D0Wasym} & 3.39 & 3.58 & CMS $8\ \text{TeV}\ W$ \cite{CMSW8} & $A_{s+}$ \\
29 & 3.95 & 3.47 & CMS double diff. Drell-Yan \cite{CMS-ddDY} & 2.16 & 2.40 & $\rho_{gq}$ & $\eta_{g}$ \\
30 & 1.79 & 1.56 & CMS double diff. Drell-Yan \cite{CMS-ddDY} & 2.41 & 2.59 & ATLAS $7\ \text{TeV}$ high prec. $W, Z$ \cite{ATLASWZ7f} & $\mathrm{DY}_{\mathrm{NLO}}$ \\
31 & 4.43 & 4.66 & D{\O} $W$ asym. \cite{D0Wasym} & 1.06 & 0.92 & D{\O} $W$ asym. \cite{D0Wasym} & $\eta_{d} - \eta_{u}$ \\
32 & 2.94 & 3.21 & ATLAS $8\ \text{TeV}\ Z\ p_{T}$ \cite{ATLASZpT} & 2.84 & 2.60 & ATLAS $8\ \text{TeV}\ W + \text{jets}$ \cite{ATLASWjet} & $p_{T}$ $\mathrm{Jet}_{\mathrm{NLO}}$ \\
33 & 3.46 & 3.68 & CMS $8\ \text{TeV}\ W$ \cite{CMSW8} & 2.44 & 2.34 & NuTeV $\nu N$ $xF_{3}$ \cite{NuTev} & $\eta_{S}$ \\
34 & 4.61 & 4.55 & D{\O} $W$ asym. \cite{D0Wasym} & 3.89 & 4.03 & NuTeV $\nu N$ $xF_{3}$ \cite{NuTev} & $\mathrm{DY}_{\mathrm{NNLO}}$ \\
35 & 4.99 & 6.80 & HERA $e^{+}p$ NC $920\ \text{GeV}$ \cite{H1+ZEUS} & 3.96 & 4.29 & ATLAS $8\ \text{TeV}\ Z\ p_{T}$ \cite{ATLASZpT} & $\rho_{qg}$ \\
36 & 6.56 & 6.80 & CMS $8\ \text{TeV}\ W$ \cite{CMSW8} & 2.06 & 1.90 & CDF II $p\bar{p}$ incl. jets \cite{CDFjet} & $\mathrm{Jet}_{\mathrm{NLO}}$ \\
37 & 2.14 & 2.95 & D{\O} $W$ asym. \cite{D0Wasym} & 1.86 & 2.91 & CMS $8\ \text{TeV}\ W$ \cite{CMSW8} & $a_{\rho,1}$ \\
38 & 3.77 & 3.92 & CDF II $p\bar{p}$ incl. jets \cite{CDFjet} & 3.10 & 3.92 & BCDMS $\mu d$ $F_{2}$ \cite{BCDMS} & $\eta_{s+}$ \\
39 & 2.62 & 2.91 & $\mathrm{Jet}_{\mathrm{NNLO}}$ & 3.25 & 3.28 & CMS $2.76\ \text{TeV}$ jet \cite{CMS276jets} & $\mathrm{Jet}_{\mathrm{NNLO}}$ \\
40 & 1.85 & 1.77 & ATLAS $8\ \text{TeV}$ sing. diff. $t\bar{t}$ dilep. \cite{ATLASttbarDilep08_ytt} & 2.34 & 2.26 & Tevatron, ATLAS, CMS $\sigma_{t\bar{t}}$ \cite{Tevatron-top,ATLAS-top7-1,ATLAS-top7-2,ATLAS-top7-3,ATLAS-top7-4,ATLAS-top7-5,ATLAS-top7-6,CMS-top7-1,CMS-top7-2,CMS-top7-3,CMS-top7-4,CMS-top7-5,CMS-top8} & $\mathrm{Top}_{\mathrm{NLO}}$ \\
41 & 2.60 & 2.47 & ATLAS $8\ \text{TeV}\ W + \text{jets}$ \cite{ATLASWjet} & 2.90 & 2.97 & ATLAS $8\ \text{TeV}$ sing. diff. $t\bar{t}$ dilep. \cite{ATLASttbarDilep08_ytt} & $p_{T}$ $\mathrm{Jet}_{\mathrm{NNLO}}$ \\
42 & 2.42 & 2.92 & HERA $ep$ $F_{2}^{\text{charm}}$ \cite{H1+ZEUScharm} & 4.08 & 5.04 & HERA $ep$ $F_{2}^{\text{charm}}$ \cite{H1+ZEUScharm} & $c_{g}^{\mathrm{NLL}}$ \\
43 & 2.02 & 1.72 & $\mathrm{Top}_{\mathrm{NLO}}$ & 1.07 & 1.00 & $\mathrm{Top}_{\mathrm{NNLO}}$ & $\mathrm{Top}_{\mathrm{NNLO}}$ \\
44 & 0.82 & 0.78 & $\mathrm{Dimuon}_{\mathrm{NNLO}}$ & 1.00 & 1.03 & $\mathrm{Dimuon}_{\mathrm{NLO}}$ & $\mathrm{Dimuon}_{\mathrm{NNLO}}$ \\
45 & 2.37 & 2.35 & $\mathrm{Dimuon}_{\mathrm{NLO}}$ & 0.76 & 0.76 & $\mathrm{Dimuon}_{\mathrm{NLO}}$ & $\mathrm{Dimuon}_{\mathrm{NLO}}$ \\
\hline
\end{tabular}
}
\caption{\textit{(Continued)} Tolerances resulting from eigenvector scans with correlated $K$-factors for each process. The average tolerance for this set of eigenvectors is $T=3.12$.}
\end{table}
\begin{table}\ContinuedFloat
\centerline{
\begin{tabular}{|p{0.5cm}|p{0.7cm}|p{0.7cm}|p{4.5cm}|p{0.7cm}|p{0.7cm}|p{4.5cm}|p{2.4cm}|}
\hline
\# & $t+$ & $T+$ & Limiting Factor ($+$) & $t-$ & $T-$ & Limiting Factor ($-$) & Primary \\
& & & & & & & Parameter \\ \hline
\hline
46 & 1.34 & 1.52 & $c_{q}^{\mathrm{NLL}}$ & 1.24 & 1.22 & $c_{g}^{\mathrm{NLL}}$ & $c_{q}^{\mathrm{NLL}}$ \\
47 & 2.62 & 3.68 & NuTeV $\nu N \rightarrow \mu\mu X$ \cite{Dimuon} & 3.00 & 2.99 & E866 / NuSea $pd/pp$ DY \cite{E866DYrat} & $\eta_{s-}$ \\
48 & 3.76 & 3.76 & $\rho_{gq}$ & 0.69 & 0.68 & $\rho_{gg}$ & $\rho_{gg}$ \\
49 & 1.54 & 5.44 & ATLAS $7\ \text{TeV}$ high prec. $W, Z$ \cite{ATLASWZ7f} & 1.55 & 4.88 & ATLAS $7\ \text{TeV}$ high prec. $W, Z$ \cite{ATLASWZ7f} & $A_{S}$ \\
50 & 0.88 & 1.00 & $a_{qq,H}^{\mathrm{NS}}$ & 1.15 & 1.01 & $a_{qq,H}^{\mathrm{NS}}$ & $a_{qq,H}^{\mathrm{NS}}$ \\
51 & 2.16 & 2.71 & $\rho_{gg}$ & 1.71 & 2.65 & HERA $ep$ $F_{2}^{\text{charm}}$ \cite{H1+ZEUScharm} & $a_{Hg}$ \\
52 & 2.09 & 2.21 & $\rho_{gg}$ & 0.60 & 0.54 & $a_{gg,H}$ & $a_{gg,H}$ \\
\hline
\end{tabular}
}
\caption{\textit{(Continued)} Tolerances resulting from eigenvector scans with correlated $K$-factors for each process. The average tolerance for this set of eigenvectors is $T=3.12$.}
\end{table}
Once again, to investigate deviations from the quadratic behaviour, Fig.~\ref{fig: tol_corr} illustrates examples of the tolerance behaviours of selected eigenvectors, with a full analysis provided in Appendix~\ref{app: tolerance_corr}. Further to this, Table~\ref{tab: eigenanalysis_2} displays the tolerances and limiting datasets/parameters for the 52 correlated eigenvectors. It is difficult to compare and contrast these results with the decorrelated case, since the eigenvectors are inherently different. However in both cases, the eigenvectors are similarly well behaved, exhibit relatively good consistency between $t$ and $T$ and are therefore providing valid descriptions for a PDF fit.
For most of the 10 eigenvectors with N$^{3}$LO $K$-factors as primary parameters, there is expected behaviour, with the eigenvectors constrained either by their own penalties or by dominant datasets for the associated process. However, due to the extra correlations considered, there are a small number of eigenvector directions which are not as trivial to explain (e.g. eigenvector 34). We therefore recover the lack of correlation between $K$-factor parameters seen within Fig.~\ref{fig: corr_theory_full} in the set of correlated PDF eigenvectors presented here. Further to this, comparing the $t$ and $T$ values found for eigenvectors associated with N$^{3}$LO $K$-factors in Table's~\ref{tab: eigenanalysis} and \ref{tab: eigenanalysis_2}, one can observe clear similarities between eigenvectors. This suggests that even when correlating the $K$-factor parameters, the fit succeeds in decorrelating the individual processes, thereby motivating our original assumption that the correlations with $K$-factors can be ignored. Another similarity one can observe between Table~\ref{tab: eigenanalysis} and Table~\ref{tab: eigenanalysis_2} is the suggestion of some tension between ATLAS $8\ \text{TeV}\ Z\ p_{T}$~\cite{ATLASZpT} and ATLAS $8\ \text{TeV}\ W + \text{jets}$ \cite{ATLASWjet} datasets seen in the limiting factors of eigenvector 32 in the correlated case.
Eigenvectors 27, 40 and 52 displayed in Fig.~\ref{fig: tol_corr} can be seen from Table~\ref{tab: eigenanalysis_2} to be associated with the new N$^{3}$LO theory parameters. Whereas eigenvector 37 is primarily focused on an original PDF parameter. One can observe a similar level of quadratic behaviour across all four of these eigenvector tolerances. Comparing all eigenvectors in the decorrelated/correlated cases, the behaviours are similarly well behaved. The average tolerance $T$ for the aN$^{3}$LO (with correlated $K$-factors) case is 3.12, slightly lower than the NNLO average of 3.37 and exactly the same as the aN$^{3}$LO (with decorrelated $K$-factors) average of 3.12 to the quoted level of precision.
\subsection{PDF Results}\label{subsec: pdf_results}
\begin{figure}
\begin{center}
\includegraphics[width=\textwidth]{figures/section8/section8-4/PDF_set_NNLO.png}
\includegraphics[width=\textwidth]{figures/section8/section8-4/PDF_set_N3LO.png}
\end{center}
\caption{\label{fig: pdfs}General forms of NNLO (top) and aN$^{3}$LO (bottom) PDFs at low (left) and high (right) $Q^{2}$. Several main features can be compared and contrasted such as the marked increase in the gluon and charm at small-$x$ (note the difference in y-axis scale between NNLO (top) and aN$^{3}$LO (bottom)).}
\end{figure}
Fig.~\ref{fig: pdfs} displays the overall shape of the PDFs including the N$^{3}$LO additions compared to the standard NNLO set. We provide this comparison to accompany the results described in earlier sections. At small-$x$ and low-$Q^{2}$ the gluon exhibits a marked enhancement due to the large small-$x$ logarithms inserted at N$^{3}$LO. The changes induced from specific N$^{3}$LO contributions are investigated in Section~\ref{subsec: n3lo_contrib}.
\begin{figure}
\begin{center}
\includegraphics[width=0.49\textwidth]{figures/section8/section8-4/NNLO_ratio/xupv_pdf_ratio_NNLO_qlow.png}
\includegraphics[width=0.49\textwidth]{figures/section8/section8-4/NNLO_ratio/xup_pdf_ratio_NNLO_qlow.png}
\includegraphics[width=0.49\textwidth]{figures/section8/section8-4/NNLO_ratio/xubar_pdf_ratio_NNLO_qlow.png}
\includegraphics[width=0.49\textwidth]{figures/section8/section8-4/NNLO_ratio/xdnv_pdf_ratio_NNLO_qlow.png}
\includegraphics[width=0.49\textwidth]{figures/section8/section8-4/NNLO_ratio/xdn_pdf_ratio_NNLO_qlow.png}
\includegraphics[width=0.49\textwidth]{figures/section8/section8-4/NNLO_ratio/xdbar_pdf_ratio_NNLO_qlow.png}
\end{center}
\caption{\label{fig: pdf_ratios_qlow}Low-$Q^{2}$ ratio plots showing the aN$^{3}$LO 68\% confidence intervals with decorrelated ($H_{ij} + K_{ij}$) and correlated ($H_{ij}^{\prime}$) $K$-factor parameters, compared to NNLO 68\% confidence intervals. Also shown are the central values at NNLO when fit to all non-HERA datasets which show similarities with N$^{3}$LO in the large-$x$ region of selected PDF flavours. All plots are shown for $Q^{2} = 10\ \mathrm{GeV}^{2}$ with the exception of the bottom quark shown for $Q^{2} = 25\ \mathrm{GeV}^{2}$.}
\end{figure}
\begin{figure}[t]\ContinuedFloat
\begin{center}
\includegraphics[width=0.49\textwidth]{figures/section8/section8-4/NNLO_ratio/xsplus_pdf_ratio_NNLO_qlow.png}
\includegraphics[width=0.49\textwidth]{figures/section8/section8-4/NNLO_ratio/xchm_pdf_ratio_NNLO_qlow.png}
\includegraphics[width=0.49\textwidth]{figures/section8/section8-4/NNLO_ratio/xbot_pdf_ratio_NNLO_qmid.png}
\includegraphics[width=0.49\textwidth]{figures/section8/section8-4/NNLO_ratio/xglu_pdf_ratio_NNLO_qlow.png}
\end{center}
\caption{\textit{(Continued)} Low-$Q^{2}$ ratio plots showing the aN$^{3}$LO 68\% confidence intervals with decorrelated ($H_{ij} + K_{ij}$) and correlated ($H_{ij}^{\prime}$) $K$-factor parameters, compared to NNLO 68\% confidence intervals. Also shown are the central values at NNLO when fit to all non-HERA datasets which show similarities with N$^{3}$LO in the large-$x$ region of selected PDF flavours. All plots are shown for $Q^{2} = 10\ \mathrm{GeV}^{2}$ with the exception of the bottom quark shown for $Q^{2} = 25\ \mathrm{GeV}^{2}$.}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=0.49\textwidth]{figures/section8/section8-4/NNLO_ratio/xupv_pdf_ratio_NNLO_qhigh.png}
\includegraphics[width=0.49\textwidth]{figures/section8/section8-4/NNLO_ratio/xup_pdf_ratio_NNLO_qhigh.png}
\includegraphics[width=0.49\textwidth]{figures/section8/section8-4/NNLO_ratio/xubar_pdf_ratio_NNLO_qhigh.png}
\includegraphics[width=0.49\textwidth]{figures/section8/section8-4/NNLO_ratio/xdnv_pdf_ratio_NNLO_qhigh.png}
\includegraphics[width=0.49\textwidth]{figures/section8/section8-4/NNLO_ratio/xdn_pdf_ratio_NNLO_qhigh.png}
\includegraphics[width=0.49\textwidth]{figures/section8/section8-4/NNLO_ratio/xdbar_pdf_ratio_NNLO_qhigh.png}
\end{center}
\caption{\label{fig: pdf_ratios_qhigh}High-$Q^{2}$ ratio plots showing the aN$^{3}$LO 68\% confidence intervals with decorrelated ($H_{ij} + K_{ij}$) and correlated ($H_{ij}^{\prime}$) $K$-factor parameters, compared to NNLO 68\% confidence intervals. Also shown are the central values at NNLO when fit to all non-HERA datasets which show similarities with N$^{3}$LO in the large-$x$ region of selected PDF flavours. All plots are shown for $Q^{2} = 10^{4}\ \mathrm{GeV}^{2}$.}
\end{figure}
\begin{figure}[t]\ContinuedFloat
\begin{center}
\includegraphics[width=0.49\textwidth]{figures/section8/section8-4/NNLO_ratio/xsplus_pdf_ratio_NNLO_qhigh.png}
\includegraphics[width=0.49\textwidth]{figures/section8/section8-4/NNLO_ratio/xchm_pdf_ratio_NNLO_qhigh.png}
\includegraphics[width=0.49\textwidth]{figures/section8/section8-4/NNLO_ratio/xbot_pdf_ratio_NNLO_qhigh.png}
\includegraphics[width=0.49\textwidth]{figures/section8/section8-4/NNLO_ratio/xglu_pdf_ratio_NNLO_qhigh.png}
\end{center}
\caption{\textit{(Continued)} High-$Q^{2}$ ratio plots showing the aN$^{3}$LO 68\% confidence intervals with decorrelated ($H_{ij} + K_{ij}$) and correlated ($H_{ij}^{\prime}$) $K$-factor parameters, compared to NNLO 68\% confidence intervals. Also shown are the central values at NNLO when fit to all non-HERA datasets which show similarities with N$^{3}$LO in the large-$x$ region of selected PDF flavours. All plots are shown for $Q^{2} = 10^{4}\ \mathrm{GeV}^{2}$.}
\end{figure}
Shown in Fig.'s~\ref{fig: pdf_ratios_qlow} and \ref{fig: pdf_ratios_qhigh} are the ratios for each flavour of aN$^{3}$LO PDF compared to the NNLO set with their 68\% confidence intervals at low and high-$Q^{2}$ respectively. The shaded aN$^{3}$LO regions indicate the PDF uncertainty produced with the decorrelated ($(H^{-1}_{ij} + \sum_{p=1}^{N_{p}}K^{-1}_{ij,\ p})^{-1}$) aN$^{3}$LO $K$-factors for each process. As a comparison to these shaded regions, the bounds of uncertainty for the fully correlated ($H_{ij}^{\prime}$) N$^{3}$LO $K$-factor parameters is also provided (red dashed line).
Considering Fig.~\ref{fig: pdf_ratios_qlow} we present the aN$^{3}$LO PDF set at $Q^{2} = 10\ \mathrm{GeV}^{2}$ with the bottom quark PDF at $Q^{2} = 25\ \mathrm{GeV}^{2}$. These PDF ratios better display the substantial increase in the gluon at small-$x$. The predicted harder gluon is then accommodated for in the quark sector by reductions in the PDFs at large and small-$x$ from NNLO. Another prominent feature is the enhanced charm and bottom quark at N$^{3}$LO. Since the heavy flavour quarks are perturbatively calculated in the MSHT framework, this amplification is a feature of the transition matrix element $A_{Hg}^{(3)}$ at high-$x$, combined with the increase in the gluon PDF at small-$x$ (as these two ingredients are convoluted together). Comparing with Fig.~95 in \cite{Thorne:MSHT20}, we observe that the approximate N$^{3}$LO charm quark now follows a much closer trend to the CT18 PDF and is therefore even more significantly different from the NNPDF NNLO fitted charm at large-$x$ than MSHT20 at NNLO. In the high-$Q^{2}$ setting shown in Fig.~\ref{fig: pdf_ratios_qhigh} we observe similar albeit less drastic effects to those described above.
Also contained in Fig.'s~\ref{fig: pdf_ratios_qlow} and \ref{fig: pdf_ratios_qhigh} are the relative forms of NNLO PDFs when fit to all non-HERA data (full $\chi^{2}$ results are provided in Appendix~\ref{app: noHERA}). Comparing the \textit{non}-HERA NNLO PDFs with aN$^{3}$LO PDFs, there are some similarities in the shapes and magnitudes of a handful of PDFs in the intermediate to large-$x$ regime, most noticeably the high-$Q^{2}$ light quarks. At small-$x$ the HERA data heavily constrains the PDF fit and therefore these similarities rapidly break down. However, this analysis displays further evidence that including N$^{3}$LO contributions, even though approximate, reduces tensions between the HERA and non-HERA data (when considering the reduction in tension seen in Table~\ref{tab: no_HERA_fullNNLO}). The aN$^{3}$LO PDFs are seemingly able to fit to HERA and non-HERA datasets with superior flexibility than at NNLO.
While in principle the negativity of quarks is possible in the $\overline{MS}$ scheme, it is unlikely to be correct at very high scales and the behaviour can lead to issues concerning negative cross section predictions~\cite{Candido:2020yat,Collins:2021vke}. In the case of the $\overline{d}$, the form of this PDF has a negative magnitude above $x \sim 0.4$ with a minimum of $\sim -0.01$ at $x \sim 0.55$. As a byproduct of this, the $d$ quark also experiences a similar effect at larger-$x$ ($x \sim 0.7$). However in this case, the $d$ quark reaches a minimum of $\sim -0.001$ (an order of magnitude lower than in the $\overline{d}$ case). These features are not uncommon in PDF analyses and are discussed in detail in \cite{PDF4LHC22}. The proposed smoothing of parameterisations employed in \cite{PDF4LHC22} ensures the definite positive nature of PDFs in the high-$x$ region. Comparing the negativity of these approximate N$^{3}$LO $d$ and $\overline{d}$ PDFs and those in \cite{PDF4LHC22}, the PDFs presented here are less negative. Due to this and the fact that this effect is only apparent in the $\overline{d}$ (and even less so in the $d$ quarks), we present these PDFs as they are. We also note that in the current MSHT20 fit, recent results surrounding the $\overline{d}/\overline{u}$ from the SeaQuest collaboration~\cite{SeaQuest:2021zxb} are not included at the time of writing. It is therefore only the E866 / NuSea $pd/pp$ DY dataset~\cite{E866DYrat} that is constraining this ratio, which is not as precise as the more recent results. However SeaQuest results suggest a preference for a higher $\overline{d}$ at large-$x$, therefore including this data may in fact help constrain the high-$x$ $\overline{d}$ behaviour seen here.
\begin{figure}
\begin{center}
\includegraphics[width=0.49\textwidth]{figures/section8/section8-4/N3LO_ratio/xupv_pdf_ratio_N3LO_qlow.png}
\includegraphics[width=0.49\textwidth]{figures/section8/section8-4/N3LO_ratio/xup_pdf_ratio_N3LO_qlow.png}
\includegraphics[width=0.49\textwidth]{figures/section8/section8-4/N3LO_ratio/xubar_pdf_ratio_N3LO_qlow.png}
\includegraphics[width=0.49\textwidth]{figures/section8/section8-4/N3LO_ratio/xdnv_pdf_ratio_N3LO_qlow.png}
\includegraphics[width=0.49\textwidth]{figures/section8/section8-4/N3LO_ratio/xdn_pdf_ratio_N3LO_qlow.png}
\includegraphics[width=0.49\textwidth]{figures/section8/section8-4/N3LO_ratio/xdbar_pdf_ratio_N3LO_qlow.png}
\end{center}
\caption{\label{fig: pdf_ratios_N3LO_qlow}Low-$Q^{2}$ ratio plots showing the aN$^{3}$LO 68\% confidence intervals with decorrelated and correlated $K$-factor parameters, compared to the aN$^{3}$LO central value. Also shown are the central values at aN$^{3}$LO when fit to all non-HERA datasets and the central values with all $K$-factors set at NNLO. All plots are shown for $Q^{2} = 10\ \mathrm{GeV}^{2}$ with the exception of the bottom quark shown for $Q^{2} = 25\ \mathrm{GeV}^{2}$.}
\end{figure}
\begin{figure}[t]\ContinuedFloat
\begin{center}
\includegraphics[width=0.49\textwidth]{figures/section8/section8-4/N3LO_ratio/xsplus_pdf_ratio_N3LO_qlow.png}
\includegraphics[width=0.49\textwidth]{figures/section8/section8-4/N3LO_ratio/xchm_pdf_ratio_N3LO_qlow.png}
\includegraphics[width=0.49\textwidth]{figures/section8/section8-4/N3LO_ratio/xbot_pdf_ratio_N3LO_qmid.png}
\includegraphics[width=0.49\textwidth]{figures/section8/section8-4/N3LO_ratio/xglu_pdf_ratio_N3LO_qlow.png}
\end{center}
\caption{\textit{(Continued)} Low-$Q^{2}$ ratio plots showing the aN$^{3}$LO 68\% confidence intervals with decorrelated and correlated $K$-factor parameters, compared to the aN$^{3}$LO central value. Also shown are the central values at aN$^{3}$LO when fit to all non-HERA datasets and the central values with all $K$-factors set at NNLO. All plots are shown for $Q^{2} = 10\ \mathrm{GeV}^{2}$ with the exception of the bottom quark shown for $Q^{2} = 25\ \mathrm{GeV}^{2}$.}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=0.49\textwidth]{figures/section8/section8-4/N3LO_ratio/xupv_pdf_ratio_N3LO_qhigh.png}
\includegraphics[width=0.49\textwidth]{figures/section8/section8-4/N3LO_ratio/xup_pdf_ratio_N3LO_qhigh.png}
\includegraphics[width=0.49\textwidth]{figures/section8/section8-4/N3LO_ratio/xubar_pdf_ratio_N3LO_qhigh.png}
\includegraphics[width=0.49\textwidth]{figures/section8/section8-4/N3LO_ratio/xdnv_pdf_ratio_N3LO_qhigh.png}
\includegraphics[width=0.49\textwidth]{figures/section8/section8-4/N3LO_ratio/xdn_pdf_ratio_N3LO_qhigh.png}
\includegraphics[width=0.49\textwidth]{figures/section8/section8-4/N3LO_ratio/xdbar_pdf_ratio_N3LO_qhigh.png}
\end{center}
\caption{\label{fig: pdf_ratios_N3LO_qhigh} High-$Q^{2}$ ratio plots showing the aN$^{3}$LO 68\% confidence intervals with decorrelated and correlated $K$-factor parameters, compared to the aN$^{3}$LO central value. Also shown are the central values at aN$^{3}$LO when fit to all non-HERA datasets and the central values with all $K$-factors set at NNLO. All plots are shown for $Q^{2} = 10^{4}\ \mathrm{GeV}^{2}$.}
\end{figure}
\begin{figure}[t]\ContinuedFloat
\begin{center}
\includegraphics[width=0.49\textwidth]{figures/section8/section8-4/N3LO_ratio/xsplus_pdf_ratio_N3LO_qhigh.png}
\includegraphics[width=0.49\textwidth]{figures/section8/section8-4/N3LO_ratio/xchm_pdf_ratio_N3LO_qhigh.png}
\includegraphics[width=0.49\textwidth]{figures/section8/section8-4/N3LO_ratio/xbot_pdf_ratio_N3LO_qhigh.png}
\includegraphics[width=0.49\textwidth]{figures/section8/section8-4/N3LO_ratio/xglu_pdf_ratio_N3LO_qhigh.png}
\end{center}
\caption{\textit{(Continued)} High-$Q^{2}$ ratio plots showing the aN$^{3}$LO 68\% confidence intervals with decorrelated and correlated $K$-factor parameters, compared to the aN$^{3}$LO central value. Also shown are the central values at aN$^{3}$LO when fit to all non-HERA datasets and the central values with all $K$-factors set at NNLO. All plots are shown for $Q^{2} = 10^{4}\ \mathrm{GeV}^{2}$.}
\end{figure}
Fig.'s \ref{fig: pdf_ratios_N3LO_qlow} and \ref{fig: pdf_ratios_N3LO_qhigh} express the aN$^{3}$LO PDFs with decorrelated (light shaded region) and correlated (red dashed lines) aN$^{3}$LO $K$-factors at low and high-$Q^{2}$ respectively (again with the bottom quark provided at $Q^{2} = 25\ \mathrm{GeV}^{2}$ at low-$Q^{2}$) as a ratio to the N$^{3}$LO central value. For comparison we also include the level of uncertainty predicted with all N$^{3}$LO theory fixed (darker shaded region) i.e. only considering the variation \textit{without} N$^{3}$LO theoretical uncertainty.
Comparing the two different aN$^{3}$LO sets in Fig.'s~\ref{fig: pdf_ratios_N3LO_qlow} and \ref{fig: pdf_ratios_N3LO_qhigh}, in general there is good agreement between the total uncertainties considering the cases with correlated (red dash) and decorrelated (light shaded) aN$^{3}$LO $K$-factors. The differences that are apparent between between the two aN$^{3}$LO cases, are relatively small across all PDFs, with slightly larger effects only where the PDF itself tends towards zero i.e. valence quarks at small-$x$.
A larger distinction is observed when comparing the sets \textit{with} and \textit{without} theoretical uncertainty (where N$^{3}$LO theory is fixed at the best fit value). In general there is an expected substantial increase in the PDF uncertainties when taking into account the MHOUs for the gluon (and therefore the heavy quarks). In particular, the form of the N$^{3}$LO bottom quark uncertainty is reminiscent of the $(H+\overline{H})$ prediction from Fig.~\ref{fig: step_results}. One can therefore directly observe the effect of the $A_{Hg}$ MHOU on the bottom quark directly above its mass threshold. In other areas, the \textit{without} theoretical uncertainty PDF set exhibits a comparable uncertainty to aN$^{3}$LO and is even shown to increase the overall 68\% confidence intervals in certain regions of $(x,Q^{2})$ due to N$^{3}$LO parameters being fixed (i.e. $u_{v}$ and $d_{v}$ PDFs in Fig.~\ref{fig: pdf_ratios_N3LO_qlow} and Fig.~\ref{fig: pdf_ratios_N3LO_qhigh}). As the fit now resides in a different $\chi^{2}$ landscape where a best fit has been achieved through fitting the N$^{3}$LO theory, fixing the aN$^{3}$LO theory parameters is likely to have a substantial effect across all PDFs.
An important point made by Fig.'s~\ref{fig: pdf_ratios_N3LO_qlow} and \ref{fig: pdf_ratios_N3LO_qhigh} is that that the difference between the decorrelated and correlated cases is much smaller than the difference of not including theoretical uncertainties at all (darker shaded region). This analysis therefore provides evidence to support the original assumption of being able to decorrelate the cross section (aN$^{3}$LO $K$-factors) and PDF theory (including other N$^{3}$LO theory).
Along with the separate cases of uncertainty illustrated in Fig.'s~\ref{fig: pdf_ratios_N3LO_qlow} and \ref{fig: pdf_ratios_N3LO_qhigh}, we also display the central values of an aN$^{3}$LO fit to all non-HERA data and an aN$^{3}$LO fit with NNLO $K$-factors. Examining the form of the no HERA aN$^{3}$LO PDFs for $x > 10^{-2}$, we show some agreement with the standard N$^{3}$LO central value across most PDFs (more so at high-$Q^{2}$ than low-$Q^{2}$). Whereas the form at small-$x$ gives some insight into the importance of HERA data in constraining PDFs in this region. In slightly better agreement across all $x$ are the aN$^{3}$LO PDFs with NNLO $K$-factors, which compliment the $\chi^{2}$ results in Section~\ref{sec: n3lo_K} and Section~\ref{subsec: chi2} arguing that the form (and fit results) of aN$^{3}$LO PDFs is mostly determined from the extra PDF $+$ DIS coefficient function N$^{3}$LO additions i.e. not aN$^{3}$LO $K$-factors which prefer a softer high-$x$ gluon (similar to the N$^{3}$LO no HERA case -- also shown in Fig.'s~\ref{fig: pdf_ratios_N3LO_qlow} and \ref{fig: pdf_ratios_N3LO_qhigh}).
\subsection{MSHT20\texorpdfstring{aN$^{3}$LO}{aN3LO} PDFs at \texorpdfstring{$Q^{2} = 2\ \mathrm{GeV}^{2}$}{Q2 = 2 GeV2}}
\begin{figure}
\begin{center}
\includegraphics[width=0.49\textwidth]{figures/section8/section8-5/PDF_set_NNLO_low.png}
\includegraphics[width=0.49\textwidth]{figures/section8/section8-5/PDF_set_N3LO_low.png}
\end{center}
\caption{\label{fig: pdfs_low} General forms of NNLO (left) and aN$^{3}$LO (right) PDFs at $Q^{2} = 2\ \mathrm{GeV}^{2}$. Axis are set to the same scale to highlight the main differences between NNLO and aN$^{3}$LO. Specifically in the gluon and heavy flavour sectors.}
\end{figure}
Fig.~\ref{fig: pdfs_low} compares the MSHT NNLO and aN$^{3}$LO PDF sets at $Q^{2} = 2\ \mathrm{GeV}^{2}$. In this very low-$Q^{2}$ regime, some major differences are evident between NNLO and aN$^{3}$LO sets at $Q^{2} = 2\ \mathrm{GeV}^{2}$, especially towards small-$x$. For example, the gluon PDF is predicted to be much harder across this region, such that it is now positive across all $x$ values considered here. The effect of this can be immediately seen in the sea and heavy quarks.
Since the charm quark is directly coupled to the gluon PDF (through a convolution with $A_{Hg}$), the charm PDF receives a notable enhancement at small-$x$ and also remains positive across all $x$ values considered\footnote{Since this is a convolution, it is the higher small-$x$ gluon, combined with the high-$x$ enhancement of $A_{Hg}$ at N$^{3}$LO which gives rise to this increase in the charm PDF.}. Another interesting feature is the reduction in uncertainty of the strange quark at small-$x$. It may seem counter intuitive to have an uncertainty reduction by adding sources of theoretical uncertainty, however we should recall that the underlying theory has also been altered. Although one can expect an uncertainty increase in PDFs across $(x,Q^{2})$, there are exceptions to this e.g. where tensions are relieved by introducing the N$^{3}$LO theory. The shift in the $\chi^{2}$ landscape then has the potential to result in more precise regions of $(x, Q^{2})$ (in this case manifesting in an uncertainty reduction for the strange quark towards small-$x$).
\begin{figure}
\begin{center}
\includegraphics[width=0.49\textwidth]{figures/section8/section8-5/NNLO_ratio/xupv_pdf_ratio_NNLO_qvlow.png}
\includegraphics[width=0.49\textwidth]{figures/section8/section8-5/NNLO_ratio/xup_pdf_ratio_NNLO_qvlow.png}
\includegraphics[width=0.49\textwidth]{figures/section8/section8-5/NNLO_ratio/xubar_pdf_ratio_NNLO_qvlow.png}
\includegraphics[width=0.49\textwidth]{figures/section8/section8-5/NNLO_ratio/xdnv_pdf_ratio_NNLO_qvlow.png}
\includegraphics[width=0.49\textwidth]{figures/section8/section8-5/NNLO_ratio/xdn_pdf_ratio_NNLO_qvlow.png}
\includegraphics[width=0.49\textwidth]{figures/section8/section8-5/NNLO_ratio/xdbar_pdf_ratio_NNLO_qvlow.png}
\end{center}
\caption{\label{fig: pdf_ratios_qvlow}Very low-$Q^{2}$ ratio plots showing the aN$^{3}$LO 68\% confidence intervals with decorrelated and correlated $K$-factor parameters, compared to NNLO 68\% confidence intervals. All plots are shown for $Q^{2} = 2\ \mathrm{GeV}^{2}$.}
\end{figure}
\begin{figure}[t]\ContinuedFloat
\begin{center}
\includegraphics[width=0.49\textwidth]{figures/section8/section8-5/NNLO_ratio/xsplus_pdf_ratio_NNLO_qvlow.png}
\includegraphics[width=0.49\textwidth]{figures/section8/section8-5/NNLO_ratio/xglu_pdf_ratio_NNLO_qvlow.png}
\end{center}
\caption{\textit{(Continued)} Very low-$Q^{2}$ ratio plots showing the aN$^{3}$LO 68\% confidence intervals with decorrelated and correlated $K$-factor parameters, compared to NNLO 68\% confidence intervals. All plots are shown for $Q^{2} = 2\ \mathrm{GeV}^{2}$.}
\end{figure}
Fig.~\ref{fig: pdf_ratios_qvlow} displays the ratios of the aN$^{3}$LO MSHT PDFs to their NNLO counterparts at $Q^{2} = 2\ \mathrm{GeV}^{2}$. Here the specific shifts of each PDF are displayed more clearly. We note that there are many similar features shown here to those discussed for Fig.'s~\ref{fig: pdf_ratios_qlow} and \ref{fig: pdf_ratios_qhigh}. Even in this very low-$Q^{2}$ regime, the uncertainty difference between correlated and decorrelated aN$^{3}$LO $K$-factor PDF sets is minimal in all relevant regions of $x$.
\subsection{\texorpdfstring{N$^{3}$LO}{N3LO} Contributions}\label{subsec: n3lo_contrib}
In this section all but one N$^{3}$LO contribution will be switched off, in particular only splitting functions, or only heavy or light flavour coefficient functions with their relevant transition matrix elements. In all cases the aN$^{3}$LO $K$-factors are left free to allow the fit some freedom in manipulating the cross sections of other datasets. In practice however, fixing these $K$-factors at the NNLO values has a minimal effect on the shape of the PDFs in all cases (as demonstrated in Fig.~\ref{fig: pdf_ratios_N3LO_qlow} and \ref{fig: pdf_ratios_N3LO_qhigh}).
\begin{figure}
\begin{center}
\includegraphics[width=0.49\textwidth]{figures/section8/section8-6/NNLO_ratio/xupv_pdf_ratio_contributions_NNLO_qlow.png}
\includegraphics[width=0.49\textwidth]{figures/section8/section8-6/NNLO_ratio/xup_pdf_ratio_contributions_NNLO_qlow.png}
\includegraphics[width=0.49\textwidth]{figures/section8/section8-6/NNLO_ratio/xubar_pdf_ratio_contributions_NNLO_qlow.png}
\includegraphics[width=0.49\textwidth]{figures/section8/section8-6/NNLO_ratio/xdnv_pdf_ratio_contributions_NNLO_qlow.png}
\includegraphics[width=0.49\textwidth]{figures/section8/section8-6/NNLO_ratio/xdn_pdf_ratio_contributions_NNLO_qlow.png}
\includegraphics[width=0.49\textwidth]{figures/section8/section8-6/NNLO_ratio/xdbar_pdf_ratio_contributions_NNLO_qlow.png}
\end{center}
\caption{\label{fig: pdf_contributions_ratios_qlow}Low-$Q^{2}$ PDF ratios showing aN$^{3}$LO (with decorrelated $K$-factors) 68\% confidence intervals compared to NNLO 68\% confidence intervals with varying theory contributions. All plots are shown for $Q^{2} = 10\ \mathrm{GeV}^{2}$ with the exception of the bottom quark shown for $Q^{2} = 25\ \mathrm{GeV}^{2}$. The PDFs included are: NNLO (green shaded), All N$^{3}$LO contributions (blue shaded), only splitting functions (green dashed), only heavy flavour coefficient functions and transition matrix elements (dark grey dash-dot) and only light flavour coefficient functions and transition matrix elements (red dotted).}
\end{figure}
\begin{figure}\ContinuedFloat
\begin{center}
\includegraphics[width=0.49\textwidth]{figures/section8/section8-6/NNLO_ratio/xsplus_pdf_ratio_contributions_NNLO_qlow.png}
\includegraphics[width=0.49\textwidth]{figures/section8/section8-6/NNLO_ratio/xchm_pdf_ratio_contributions_NNLO_qlow.png}
\includegraphics[width=0.49\textwidth]{figures/section8/section8-6/NNLO_ratio/xbot_pdf_ratio_contributions_NNLO_qmid.png}
\includegraphics[width=0.49\textwidth]{figures/section8/section8-6/NNLO_ratio/xglu_pdf_ratio_contributions_NNLO_qlow.png}
\end{center}
\caption{\textit{(Continued)} Low-$Q^{2}$ PDF ratios showing aN$^{3}$LO (with decorrelated $K$-factors) 68\% confidence intervals compared to NNLO 68\% confidence intervals with varying theory contributions. All plots are shown for $Q^{2} = 10\ \mathrm{GeV}^{2}$ with the exception of the bottom quark shown for $Q^{2} = 25\ \mathrm{GeV}^{2}$. The PDFs included are: NNLO (green shaded), All N$^{3}$LO contributions (blue shaded), only splitting functions (green dashed), only heavy flavour coefficient functions and transition matrix elements (dark grey dash-dot) and only light flavour coefficient functions and transition matrix elements (red dotted).}
\end{figure}
The deconstructed aN$^{3}$LO PDFs as a ratio to the NNLO MSHT PDFs for various flavours at $Q^{2} = 10\ \mathrm{GeV}^{2}$ (with the bottom quark given at $Q^{2} = 25\ \mathrm{GeV}^{2}$) are shown in Fig.~\ref{fig: pdf_contributions_ratios_qlow}. Across the more tightly constrained light quark PDFs, all contributions lie very close to the aN$^{3}$LO $\pm 1\sigma$ uncertainty bands (blue shaded region and solid line). The additive and compensating nature of these contributions is also clear in a handful of the ratios from Fig.~\ref{fig: pdf_contributions_ratios_qlow}. In other areas the full description is biased towards a single contribution, for example the strange, charm and bottom quarks follow the contribution from heavy flavours as one may expect.
Conversely, to some extent the gluon follows the splitting functions much more closely as these contributions indirectly couple the gluon to the more constraining data\footnote{An exception to this can be seen around $x\sim 10^{-2}$ where the contributions act cumulatively. We make this point as this region of $x$ is of interest for Higgs calculations such as those discussed in Section~\ref{sec: predictions}}.
\subsection{\texorpdfstring{$\alpha_{s}$}{Strong Coupling Constant} Variation}
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.7\textwidth]{figures/section8/section8-7/alphaS_fit.png}
\end{center}
\caption{\label{fig: alphaS_fit}Quadratic fit to the total $\chi^{2}$ results from various $\alpha_{s}$ starting scales. The minimum of the quadratic fit provides a rough estimate of $\alpha_{s} = 0.1167$ at aN$^{3}$LO.}
\end{figure}
As in the standard MSHT20 NNLO PDF fit, we present the best fit aN$^{3}$LO PDFs with $\alpha_{s} = 0.118$, the common value chosen in the PDF4LHC combination~\cite{PDF4LHC22}. However, investigating the true minima in $\alpha_{s}$, the $\chi^{2}$ profiles in Fig.~\ref{fig: alphaS_fit} prefer a value of around $\alpha_{s} = 0.1167$. This result follows the trend from lower orders whereby the best fit values are $\alpha_{s} = 0.1174 \pm 0.0013$ at NNLO and $\alpha_{s} = 0.1203 \pm 0.0015$ at NLO~\cite{Thorne:MSHT20_alphaS}. Following from NNLO, the aN$^{3}$LO $\alpha_{s}$ prediction is also slightly lower than the NNLO world average at around $\alpha_{s} = 0.1179 \pm 0.0010$~\cite{PDG2019}. In any case, the preferred aN$^{3}$LO $\alpha_{s}$ value stated here is in agreement with the MSHT20 NNLO result and with a full analysis (left for a further publication) is expected to overlap with the world average within uncertainties.
\subsection{Charm Mass Dependence}
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.7\textwidth]{figures/section8/section8-8/chm_mass_fit.png}
\end{center}
\caption{\label{fig: chm_mass_fit}Quadratic fit to the total $\chi^{2}$ results from various charm masses ($m_{c}$). The minimum of the quadratic fit provides a rough estimate of $m_{c} = 1.45\ \mathrm{GeV}$ at aN$^{3}$LO.}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=0.49\textwidth]{figures/section8/section8-8/NNLO_ratio/xupv_pdf_ratio_charm_NNLO_qlow.png}
\includegraphics[width=0.49\textwidth]{figures/section8/section8-8/NNLO_ratio/xup_pdf_ratio_charm_NNLO_qlow.png}
\includegraphics[width=0.49\textwidth]{figures/section8/section8-8/NNLO_ratio/xubar_pdf_ratio_charm_NNLO_qlow.png}
\includegraphics[width=0.49\textwidth]{figures/section8/section8-8/NNLO_ratio/xdnv_pdf_ratio_charm_NNLO_qlow.png}
\includegraphics[width=0.49\textwidth]{figures/section8/section8-8/NNLO_ratio/xdn_pdf_ratio_charm_NNLO_qlow.png}
\includegraphics[width=0.49\textwidth]{figures/section8/section8-8/NNLO_ratio/xdbar_pdf_ratio_charm_NNLO_qlow.png}
\end{center}
\caption{\label{fig: pdf_charm_ratios_qlow}Low-$Q^{2}$ PDF ratios showing aN$^{3}$LO (with decorrelated $K$-factors) 68\% confidence intervals compared to NNLO 68\% confidence intervals with varying fixed values for the charm mass. All plots are shown for $Q^{2} = 10\ \mathrm{GeV}^{2}$ with the exception of the bottom quark shown for $Q^{2} = 25\ \mathrm{GeV}^{2}$. The PDFs included are: $m_{c} = 1.40\ \mathrm{GeV}$ (standard MSHT20 choice) (blue solid), $m_{c} = 1.30\ \mathrm{GeV}$ (green dashed), $m_{c} = 1.45\ \mathrm{GeV}$ (grey dotted dashed) $m_{c} = 1.50\ \mathrm{GeV}$ (red dotted).}
\end{figure}
\begin{figure}[t]\ContinuedFloat
\begin{center}
\includegraphics[width=0.49\textwidth]{figures/section8/section8-8/NNLO_ratio/xsplus_pdf_ratio_charm_NNLO_qlow.png}
\includegraphics[width=0.49\textwidth]{figures/section8/section8-8/NNLO_ratio/xchm_pdf_ratio_charm_NNLO_qlow.png}
\includegraphics[width=0.49\textwidth]{figures/section8/section8-8/NNLO_ratio/xbot_pdf_ratio_charm_NNLO_qmid.png}
\includegraphics[width=0.49\textwidth]{figures/section8/section8-8/NNLO_ratio/xglu_pdf_ratio_charm_NNLO_qlow.png}
\end{center}
\caption{\textit{(Continued)} Low-$Q^{2}$ PDF ratios showing aN$^{3}$LO (with decorrelated $K$-factors) 68\% confidence intervals compared to NNLO 68\% confidence intervals with varying fixed values for the charm mass. All plots are shown for $Q^{2} = 10\ \mathrm{GeV}^{2}$ with the exception of the bottom quark shown for $Q^{2} = 25\ \mathrm{GeV}^{2}$. The PDFs included are: $m_{c} = 1.40\ \mathrm{GeV}$ (standard MSHT20 choice) (blue solid), $m_{c} = 1.30\ \mathrm{GeV}$ (green dashed), $m_{c} = 1.45\ \mathrm{GeV}$ (grey dotted dashed) $m_{c} = 1.50\ \mathrm{GeV}$ (red dotted).}
\end{figure}
In a standard MSHT fit~\cite{Thorne:MSHT20}, aN$^{3}$LO PDFs are produced with the charm pole mass $m_{c} = 1.40\ \mathrm{GeV}$. Fig.~\ref{fig: chm_mass_fit} displays the $\chi^{2}$ results when varying this charm mass. The predicted minimum at NNLO (for MSHT20 PDFs) is in the range $m_{c} = 1.35-1.40\ \mathrm{GeV}$~\cite{Thorne:MSHT20_alphaS}, whereas at aN$^{3}$LO we show a minimum in the region of $m_{c} = 1.45-1.47\ \mathrm{GeV}$. This aN$^{3}$LO result therefore shows a slightly better agreement with the world average~\cite{PDG2019}\footnote{There is some ambiguity in this value since the transformation from $\overline{MS}$ to the pole mass definition is not well-defined (see \cite{Thorne:MSHT20_alphaS} for more details).} of $m_{c} = 1.5 \pm 0.2\ \mathrm{GeV}$.
Considering Fig.~\ref{fig: pdf_charm_ratios_qlow}, one is then able to analyse the effect of this slightly higher charm mass on the form of the PDFs. As one can expect, the charm PDF is subject to the largest difference and is suppressed by a higher $m_{c}$. The extra suppression from a higher charm mass allows the fit to suppress the $c + \bar{c}$ sea contribution. This is then compensated by an increase in the $\bar{u}$ and $\bar{d}$ distributions which stabilises the overall sea contribution.
\section{Structure Functions at \texorpdfstring{N$^{3}$LO}{N3LO}}\label{sec: structure}
The general form of a structure function $F(x,Q^{2})$ is a convolution between the PDFs $f_{i}(x, Q^{2})$ and some defined process dependent coefficient function $C(x,\alpha_{s}(Q^{2}))$,
\begin{equation}\label{eq: structure_form}
F(x,Q^{2}) = \sum_{i=q,\bar{q},g}\left[C_{i}(\alpha_{s}(Q^{2}))\otimes f_{i}(Q^{2})\right](x)
\end{equation}
where we have the sum over all partons $i$ and implicitly set the factorisation and renormalisation scales as $\mu_{f}^{2} = \mu_{r}^{2} = Q^{2}$, a choice that will be used throughout this paper for DIS scales. We also note that the relevant charge weightings are implicit in the definition of the coefficient function for each parton.
In Equation~\eqref{eq: structure_form}, the perturbative and non-perturbative regimes are separated out into coefficient functions $C_{i}$ and PDFs $f_{i}$ respectively.
Since these coefficient functions are perturbative quantities, they are an important aspect to consider when transitioning to N$^{3}$LO.
The PDFs $f_{i}(x, Q^{2})$ in Equation~\eqref{eq: structure_form} are non-perturbative quantities. However, their evolution in $Q^{2}$ is perturbatively calculable. In a PDF fit, the PDFs are parameterised at a chosen starting scale $Q^{2}_{0}$, which is in general different to the scale $Q^{2}$ at which an observable (such as $F(x,Q^{2})$) is calculated. It is therefore important that we are able to accurately evolve the PDFs from $Q_{0}^{2}$ to the required $Q^{2}$ to ensure a fully consistent and physical calculation. To facilitate this evolution, we introduce the standard factorisation scale $\mu_{f}$.
The flavour singlet distribution is defined as,
\begin{equation}\label{eq:sing}
\Sigma(x,\mu_{f}^{2})=\sum^{n_{f}}_{i=1}\left[q_{i}(x,\mu_{f}^{2})+\overline{q}_{i}(x,\mu_{f}^{2})\right],
\end{equation}
where $q_{i}(x,\mu_{f}^{2})$ and $\overline{q}_{i}(x,\mu_{f}^{2})$ are the quark and anti-quark distributions respectively, as a function of Bjorken $x$ and the factorisation scale $\mu_{f}^{2}$. The summation in Equation~\eqref{eq:sing} runs over all flavours of (anti-)quarks $i$ up to the number of available flavours $n_{f}$.
This singlet distribution is inherently coupled to the gluon density.
Because of this, we must consider the gluon carefully when describing the evolution of the flavour singlet distribution with the energy scale $\mu_{f}$. The Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP)~\cite{DGLAP} equations that govern this evolution are:
\begin{equation}
\frac{d \boldsymbol{f}}{d \ln \mu_{f}^{2}} \equiv \frac{d}{d \ln \mu_{f}^{2}}\left(\begin{array}{c}{\Sigma} \\ {g}\end{array}\right)=\left(\begin{array}{cc}{P_{q q}} & {\ n_{f}P_{q g}} \\ {P_{g q}} & {P_{g g}}\end{array}\right) \otimes\left(\begin{array}{c}{\Sigma} \\ {g}\end{array}\right) \equiv \boldsymbol{P} \otimes \boldsymbol{f}
\label{eq: DGLAP}
\end{equation}
where $P_{ij}: i,j \in q,g$ are the splitting functions and the factorisation scale $\mu_{f}$ is facilitating the required evolution up to the physical scale $Q^{2}$. The matrix of splitting functions $\boldsymbol{P}$ appropriately couples the singlet and gluon distribution by means of a convolution in the momentum fraction $x$. We note here that $P_{qq}\equiv P_{q \rightarrow gq}$ is decomposed into non-singlet (NS) and a pure-singlet (PS) parts defined by,
\begin{equation}
P_{qq}(x)=P^{+}_{\mathrm{NS}}(x)+P_{\mathrm{PS}}(x),
\label{eq: NStoPS}
\end{equation}
where the $P^{+}_{NS}$ is a non-singlet distribution splitting function which has been calculated approximately to four loops in~\cite{4loopNS}\footnote{In this discussion, we only consider the $P^{+}_{NS}$ non-singlet distribution as this is the distribution which contributes to the singlet evolution. Other non-singlet distributions are briefly discussed in Section~\ref{sec: n3lo_split}.}. The non-singlet part of $P_{qq}$ dominates at large-$x$ but as $x\rightarrow 0$, this contribution is highly suppressed due to the relevant QCD sum rules. On the other hand, due to the involvement of the gluon in the pure-singlet splitting function (as described above), this contribution grows towards small-$x$ and therefore begins to dominate.
Turning to the splitting function matrix, each element can be expanded perturbatively as a function of $\alpha_s$ up to N$^{3}$LO as,
\begin{equation}
\boldsymbol{P}(x,\alpha_{s})=\alpha_{s}\boldsymbol{P}^{(0)}(x)+\alpha_{s}^{2}\boldsymbol{P}^{(1)}(x)+\alpha_{s}^{3}\boldsymbol{P}^{(2)}(x)+\alpha_{s}^{4}\boldsymbol{P}^{(3)}(x)+\dots\ ,
\label{eq: splittingExpand}
\end{equation}
where we have omitted the scale argument of $\alpha_{s}(\mu_{r}^{2}=\mu_{f}^{2})\equiv\alpha_{s}$ for brevity and $\boldsymbol{P}^{(0)}$, $\boldsymbol{P}^{(1)}$, $\boldsymbol{P}^{(2)}$ are known~\cite{DGLAP, NLO1, NLO2, NLO3, NLO4, Moch:2004pa, Vogt:2004mw}. $\boldsymbol{P}^{(3)}$ are the four-loop quantities which we approximate in Section~\ref{sec: n3lo_split} using information from~\cite{Catani:1994sq,Lipatov:1976zz,Kuraev:1977fs,Balitsky:1978ic,Fadin:1998py,Ciafaloni:1998gs,Marzani:smallx, S4loopMoments, S4loopMomentsNew, 4loopNS}.
Considering Equation~\eqref{eq: structure_form}, $\Sigma(Q^{2})$ and $g(Q^{2})$ are the singlet and gluon PDFs respectively, evolved to the required $Q^{2}$ energy of the process via Equation~\eqref{eq: DGLAP}. For more information on the relevant formulae used in this convolution, the reader is referred to~\cite{PinkBook}.
Thus far, we have limited our discussion to only light quark flavours. However, as we move through the full range of $Q^{2}$ values, the number of partons which are kinematically accessible increases. More specifically, as we pass over the charm and bottom mass thresholds (where $Q^{2} = m_{c, b}^{2}$) we must account for the heavy quark PDFs and their corresponding contributions.
To deal with the heavy quark contributions to the total structure function, whilst remaining consistent with the light quark picture described above, we consider
\begin{equation}
f_{\alpha}^{n_{f} + 1}(x, Q^{2}) = \left[A_{\alpha i}(Q^{2}/m_{h}^{2}) \otimes f_{i}^{n_{f}}(Q^{2})\right](x),
\end{equation}
where we have an implied summation over partons $i$ and $A_{\alpha i}$ are the heavy flavour transition matrix elements~\cite{Buza:OMENLO, Buza:OMENNLO} which explicitly depend on the heavy flavour mass threshold $m_{h}$, where these contributions are activated\footnote{The indices here run as $\alpha \in \{H,q,g\}$ and $i \in \{q,g\}$, since $n_{f}$ is the number of light flavours.}. We also denote the PDFs as $f_{i}^{n_{f}}$ and $f_{i}^{n_{f} + 1}$ to indicate whether the PDF has been evolved with only light flavours ($n_{f}$) or also with heavy flavours ($n_{f} + 1$). In this work we only consider contributions at heavy flavour threshold i.e. where $Q^{2} = m_{h}^{2}$. We then define the PDFs:
\begin{subequations}
\begin{equation}\label{eq: OME_fq}
f_{q}^{n_{f} + 1}(x, Q^{2}) = \left[A_{qq,H}(Q^{2}/m_{h}^{2}) \otimes f_{q}^{n_{f}}(Q^{2}) + A_{qg,H}(Q^{2}/m_{h}^{2}) \otimes f_{g}^{n_{f}}(Q^{2})\right](x)
\end{equation}
\begin{equation}\label{eq: OME_fg}
f_{g}^{n_{f} + 1}(x, Q^{2}) = \left[A_{gq,H}(Q^{2}/m_{h}^{2}) \otimes f_{q}^{n_{f}}(Q^{2}) + A_{gg,H}(Q^{2}/m_{h}^{2}) \otimes f_{g}^{n_{f}}(Q^{2})\right](x)
\end{equation}
\begin{equation}\label{eq: OME_fh}
f_{H}^{n_{f} + 1}(x, Q^{2}) = \left[A_{Hq}(Q^{2}/m_{h}^{2}) \otimes f_{q}^{n_{f}}(Q^{2}) + A_{Hg}(Q^{2}/m_{h}^{2}) \otimes f_{g}^{n_{f}}(Q^{2})\right](x)
\end{equation}
\end{subequations}
where we have an implicit summation over light flavours of $q$ and a generalised theoretical description to involve heavy flavour contributions\footnote{Note that the notation $A_{\alpha i, H}$ is exactly equivalent to $A_{\alpha i}$. When $H$ is not present in the final state of matrix element interactions, we opt for the $A_{\alpha i,H}$ notation. This is to remind the reader that these elements are considering only those interactions involving a heavy quark.}. Equation~\eqref{eq: OME_fq} and Equation~\eqref{eq: OME_fg} are the light flavour quark and gluon PDFs defined earlier, modified to include contributions mediated by heavy flavour loops. Whereas in Equation~\eqref{eq: OME_fh} we describe the heavy flavour PDF, perturbatively calculated from the light quark and gluon PDFs.
By considering the number of vertices (and hence orders of $\alpha_{s}$) required for each of these transition matrix elements to contribute to their relevant `output' partons, we are immediately able to show:
\begin{minipage}{0.47\linewidth}
\begin{align}
A_{qq,H} &= \delta(1-x)\ +\ \mathcal{O}(\alpha_{s}^{2}) \nonumber \\
A_{qg,H} &= \mathcal{O}(\alpha_{s}^{2}) \nonumber \\
A_{gq,H} &= \mathcal{O}(\alpha_{s}^{2}) \nonumber
\end{align}
\end{minipage}
\begin{minipage}{0.47\linewidth}
\begin{align}
A_{gg,H} &= \delta(1-x)\ +\ \mathcal{O}(\alpha_{s}^{2}) \nonumber \\
A_{Hq} &= \mathcal{O}(\alpha_{s}^{2}) \nonumber \\
A_{Hg} &= \mathcal{O}(\alpha_{s}) \label{eq: OME_orders}
\end{align}
\end{minipage}
\vspace{0.3cm}
\noindent
where $A_{qq,H}$ and $A_{gg,H}$ include LO $\delta$-functions to ensure this description is consistent with the light quark picture discussed earlier. It is therefore the $A_{Hg}$ transition matrix element which provides our lowest order contribution to the heavy flavour sector (i.e. $g \rightarrow H \overline{H}$).
The insertion of scale independent contributions to $A_{\alpha i}$ introduce unwanted discontinuities at NNLO into the PDF evolution.
In order to ensure the required smoothness and validity of the structure functions across $(x, Q^{2})$, these discontinuities must be accounted for elsewhere in the structure function picture.
Equating the coefficient functions above the mass threshold $m_{H}^{2}$ (describing the total number of flavours including heavy flavour quarks) and those below this threshold, discontinuities are able to be absorbed by a suitable redefinition of the coefficient functions. This procedure provides the foundation for the description of different flavour number schemes.
There are two number schemes which are preferred at different points in the $Q^2$ range. Towards $Q^2 \leq m_{h}^2$ we adopt the Fixed Flavour Number Scheme (FFNS). Towards $\frac{Q^2}{m_{h}^2} \rightarrow \infty$, the heavy contributions can be considered massless and therefore the Zero Mass Variable Flavour Number Scheme (ZM-VFNS) is assumed.
In order to join the FFNS and ZM-VFNS schemes seamlessly together, we ultimately wish to describe the General Mass Variable Number Scheme (GM-VFNS)~\cite{Aivazis:gmvf} (which is valid across all $Q^2$). This scheme can then account for discontinuities from transition matrix elements and re-establish a smooth description of the structure functions.
In~\cite{Thorne:GMVF} an ambiguity in the definition of the GM-VFNS scheme was pointed out (namely the freedom to swap $\mathcal{O}(m_{h}^{2}/Q^{2})$ terms without violating the definition of the GM-VFNS). We note here that since~\cite{Thorne:MSTW09}, MSHT PDFs have employed the TR scheme to define the distribution of $\mathcal{O}(m_{h}^{2}/Q^{2})$ terms, the specific details of which are found in~\cite{Thorne:GMVF,Thorne:GMVFNNLO,thorne2012}. The general method to relate the FFNS and GM-VFNS number schemes is to compare the prediction for a result e.g. the $F_2$ structure function in the FFNS scheme:
\begin{align}\label{eq: FFexpanse}
F_{2}(x,Q^2) &= F_{2,q}(x, Q^{2})\ +\ F_{2, H}(x, Q^{2}) \nonumber\\
&= C_{q,i}^{\mathrm{FF},\ n_{f}}\otimes f_{i}^{n_{f}}(Q^2) + C_{H,i}^{\mathrm{FF},\ n_{f}}\otimes f_{k}^{n_{f}}(Q^2) \nonumber\\
&= C_{q,q}^{\mathrm{FF},\ n_{f}}\otimes f_{q}^{n_{f}}(Q^2) + C_{q,g}^{\mathrm{FF},\ n_{f}}\otimes f_{g}^{n_{f}}(Q^2)\nonumber \\
&+ C_{H,q}^{\mathrm{FF},\ n_{f}}\otimes f_{q}^{n_{f}}(Q^2) + C_{H,g}^{\mathrm{FF},\ n_{f}}\otimes f_{g}^{n_{f}}(Q^2)
\end{align}
and the GM-VFNS scheme,
\begin{multline}\label{eq: GM-VFNS_gen}
F_{2}(x,Q^2) = \mathlarger{\mathlarger{\sum}}_{\alpha \in \{H,q,g\}}\left(C_{q,\alpha}^{\mathrm{VF},\ n_{f}+1}\otimes A_{\alpha i}(Q^2/m_{h}^2)\otimes f_{i}^{n_{f}}(Q^2) \right.\\\left.+ C_{H,\alpha}^{\mathrm{VF},\ n_{f}+1}\otimes A_{\alpha i}(Q^2/m_{h}^2)\otimes f_{i}^{n_{f}}(Q^2)\right),
\end{multline}
where $F_{2,q}$ and $F_{2,H}$ are the light and heavy flavour structure functions respectively\footnote{The extra contribution from $F_{2,H}$ allows for the possibility of final state heavy flavours.}. $C^{\mathrm{FF}, n_{f}}$ and $C^{\mathrm{VF}, n_{f}+1}$ are the FFNS (known up to NLO~\cite{FFNLO1,FFNLO2} with some information at NNLO~\cite{Catani:FFN3LO1,Laenen:FFN3LO2,Vogt:FFN3LO3} including high-$Q^{2}$ transition matrix elements at $\mathcal{O}(\alpha_{s}^{3})$~\cite{bierenbaum:OMEmellin,ablinger:3loopNS,ablinger:3loopPS,ablinger:agq,Blumlein:AQg,Vogt:FFN3LO3}) and GM-VFNS coefficient functions respectively, and $A_{\alpha i}(Q^2/m_{h}^2)$ are the transition matrix elements. We note that the above also applies to other structure functions and for clarity, in the following we consider the light and heavy structure functions separately.
\subsection*{$F_{2,q}$}
Expanding the first term in Equation~\eqref{eq: GM-VFNS_gen} in terms of the transition matrix elements results in,
\begin{align}\label{eq: expanse_q}
F_{2,q}(x,Q^2) &= C_{q, H}^{\mathrm{VF},\ n_{f}+1}\otimes\bigg[A_{Hq}(Q^2/m_{h}^2)\otimes f_{q}^{n_{f}}(Q^2) + A_{Hg}(Q^2/m_{h}^2)\otimes f_{g}^{n_{f}}(Q^2)\bigg] \nonumber \\ & + C_{q, q}^{\mathrm{VF},\ n_{f}+1}\otimes\bigg[A_{qq, H}(Q^2/m_{h}^2)\otimes f_{q}^{n_{f}}(Q^2) + A_{qg, H}(Q^2/m_{h}^2)\otimes f_{g}^{n_{f}}(Q^2)\bigg] \nonumber \\ & + C_{q, g}^{\mathrm{VF},\ n_{f}+1}\otimes\bigg[A_{gq, H}(Q^2/m_{h}^2)\otimes f_{q}^{n_{f}}(Q^2) + A_{gg, H}(Q^2/m_{h}^2)\otimes f_{g}^{n_{f}}(Q^2)\bigg],
\end{align}
which is valid at all orders. The first term in Equation~\eqref{eq: expanse_q} is the contribution to the light quark structure function from heavy quark PDFs (since the term contained within square brackets is exactly our definition in Equation~\eqref{eq: OME_fh}). Due to this, the coefficient function $C_{q,H}$ describes the transition of a heavy quark to a light quark via a gluon and is therefore forbidden to exist below NNLO. The second and third terms here are the purely light quark and gluon contributions, with extra corrections from heavy quark at higher orders.
Using the definitions in Equation~\eqref{eq: OME_orders} we can obtain an equation for $F_{2,q}(x, Q^{2})$ up to $\mathcal{O}(\alpha_{s}^{3})$ as,
\begin{multline}\label{eq: fullN3LO_q}
F_{2,q}(x, Q^{2}) = C_{q,q}^{\mathrm{VF},\ (0)} \otimes f_{q}(Q^{2}) + \frac{\alpha_s}{4\pi}\ \bigg\{C_{q,q,\ n_{f}+1}^{\mathrm{VF},\ (1)}\otimes f_{q}(Q^{2}) + C_{q,g,\ n_{f}+1}^{\mathrm{VF},\ (1)}\otimes f_{g}(Q^{2})\bigg\} \\+ \left(\frac{\alpha_s}{4\pi}\right)^{2}\ \bigg\{\bigg[C_{q,q,\ n_{f}+1}^{\mathrm{VF},\ (2)} + C_{q,q}^{\mathrm{VF},\ (0)} \otimes A_{qq,H}^{(2)}\bigg]\otimes f_{q}(Q^{2}) + \bigg[C_{q,g,\ n_{f}+1}^{\mathrm{VF},\ (2)} \\+ C_{q,q}^{\mathrm{VF},\ (0)} \otimes A_{qg,H}^{(2)}\bigg]\otimes f_{g}(Q^{2})\bigg\} \\+ \left(\frac{\alpha_s}{4\pi}\right)^{3}\ \bigg\{\bigg[C_{q,q,\ n_{f}+1}^{\mathrm{VF},\ (3)} + C_{q,q,\ n_{f}+1}^{\mathrm{VF},\ (1)}\otimes A_{qq, H}^{(2)} + C_{q,g,\ n_{f}+1}^{\mathrm{VF},\ (1)}\otimes A_{gq, H}^{(2)} \\+ C_{q,q}^{\mathrm{VF},\ (0)} \otimes A_{qq,H}^{(3)}\bigg]\otimes f_{q}(Q^{2}) \\+ \bigg[C_{q,g,\ n_{f}+1}^{\mathrm{VF},\ (3)} + C_{q,g,\ n_{f}+1}^{\mathrm{VF},\ (1)}\otimes A_{gg, H}^{(2)} + C_{q,q,\ n_{f}+1}^{\mathrm{VF},\ (1)}\otimes A_{qg, H}^{(2)} \\+ C_{q,q}^{\mathrm{VF},\ (0)} \otimes A_{qg,H}^{(3)}\bigg]\otimes f_{g}(Q^{2}) + C_{q,H}^{\mathrm{VF},\ (2)}\otimes A_{Hg}^{(1)}\otimes f_{g}(Q^{2})\bigg\} + \mathcal{O}(\alpha_{s}^{4})
\end{multline}
where $C_{q,q}^{\mathrm{VF},\ (0)} = \delta(1-x)$ up to charge weighting. Equation~\eqref{eq: fullN3LO_q} defines the light quark structure function to N$^{3}$LO including heavy flavour corrections\footnote{We also note that $\alpha_{s}^{n_{f}+1} \ne \alpha_{s}^{n_{f}}$ and account for this, but omit in expressions such as Equation~\eqref{eq: fullN3LO_q} for simplicity.}.
\subsection*{$F_{2,H}$}
Moving to the heavy quark structure function in Equation~\eqref{eq: FFexpanse}, as above the second term in Equation~\eqref{eq: GM-VFNS_gen} can be expanded in terms of the transition matrix elements to obtain,
\begin{align}\label{eq: expanse_H}
F_{2,H}(x,Q^2) &= C_{H, H}^{\mathrm{VF},\ n_{f}+1}\otimes\bigg[A_{Hq}(Q^2/m_{h}^2)\otimes f_{q}^{n_{f}}(Q^2) + A_{Hg}(Q^2/m_{h}^2)\otimes f_{g}^{n_{f}}(Q^2)\bigg] \nonumber \\ & + C_{H, q}^{\mathrm{VF},\ n_{f}+1}\otimes\bigg[A_{qq, H}(Q^2/m_{h}^2)\otimes f_{q}^{n_{f}}(Q^2) + A_{qg, Q}(Q^2/m_{h}^2)\otimes f_{g}^{n_{f}}(Q^2)\bigg] \nonumber \\ & + C_{H, g}^{\mathrm{VF},\ n_{f}+1}\otimes\bigg[A_{gq, H}(Q^2/m_{h}^2)\otimes f_{q}^{n_{f}}(Q^2) + A_{gg, Q}(Q^2/m_{h}^2)\otimes f_{g}^{n_{f}}(Q^2)\bigg],
\end{align}
which is valid at all orders. Similar to Equation~\eqref{eq: expanse_q}, we have a contribution from the heavy flavour quarks, the light quarks and the gluon respectively. However in this case, due to the required gluon intermediary, the coefficient functions associated with the light quark flavours and gluon are forbidden to exist below NNLO. Considering the $C_{H,H}$ function, we are able to choose this to be identically the ZM-VFNS light quark coefficient function $C_{q,q}$ up to kinematical suppression factors, since at $Q^{2} \rightarrow \infty$ these functions must be equivalent~\cite{Aivazis:gmvf,thorne2012,Aivazis:1993kh}.
The full heavy flavour structure function then reads as,
\begin{multline}
F_{2,H}(x, Q^{2}) = \frac{\alpha_s}{4\pi}\bigg[C_{H,g}^{\mathrm{VF},\ (1)} + C_{H,H}^{\mathrm{VF},\ (0)}\otimes A_{Hg}^{(1)}\bigg]\otimes f_{g}(Q^{2}) \\
+ \bigg(\frac{\alpha_s}{4\pi}\bigg)^{2} \bigg\{\bigg[C_{H,q}^{\mathrm{VF},\ (2)} + C_{H,H}^{\mathrm{VF},\ (0)}\otimes A_{Hq}^{(2)}\bigg]\otimes f_{q}(Q^{2}) \\
+ \bigg[C_{H,g}^{\mathrm{VF},\ (2)} + C_{H,H}^{\mathrm{VF},\ (1)}\otimes A_{Hg}^{(1)} + C_{H,H}^{\mathrm{VF},\ (0)}\otimes A_{Hg}^{(2)}\bigg]\otimes f_{g}(Q^{2})\bigg\} \\
+ \bigg(\frac{\alpha_s}{4\pi}\bigg)^{3} \bigg\{\bigg[C_{H,q}^{\mathrm{VF},\ (3)} + C_{H,g}^{\mathrm{VF},\ (1)}\otimes A_{gq,H}^{(2)} \\
+ C_{H,H}^{\mathrm{VF},\ (1)}\otimes A_{Hq}^{(2)} + C_{H,H}^{\mathrm{VF},\ (0)}\otimes A_{Hq}^{(3)}\bigg]\otimes f_{q}(Q^{2}) \\
+ \bigg[C_{H,g}^{\mathrm{VF},\ (2)} + C_{H,g}^{\mathrm{VF},\ (1)}\otimes A_{gg,H}^{(2)} + C_{H,H}^{\mathrm{VF},\ (2)}\otimes A_{Hg}^{(1)} \\ + C_{H,H}^{\mathrm{VF},\ (1)}\otimes A_{Hg}^{(2)} + C_{H,H}^{\mathrm{VF},\ (0)}\otimes A_{Hg}^{(3)}\bigg]\otimes f_{g}(Q^{2})\bigg\} \\
\label{eq: fullN3LO_H}
\end{multline}
where combining Equation~\eqref{eq: fullN3LO_q} and Equation~\eqref{eq: fullN3LO_H}, one can obtain the full structure function $F_{2}(x, Q^{2})$.
Equating the FFNS expansion from Equation~\eqref{eq: FFexpanse} to the above expressions in the GM-VFNS setting, one can find relationships between the two pictures. In Section~\ref{sec: n3lo_coeff} we use this equivalence to enable the derivation of the GM-VFNS functions at N$^{3}$LO.
To summarise, we have identified the leading theoretical ingredients entering the structure functions and detailed how these affect the PDFs. As we will discuss further, when pushing these equations to N$^{3}$LO, there is already some knowledge available. For example, the N$^{3}$LO ZM-VFNS coefficient functions are known precisely for $n_{f}=3$ from~\cite{Vermaseren:2005qc}, as are a handful of Mellin moments~\cite{S4loopMoments,S4loopMomentsNew,4loopNS,bierenbaum:OMEmellin} and leading small and large-$x$ terms~\cite{Catani:1994sq,Lipatov:1976zz,Kuraev:1977fs,Balitsky:1978ic,Fadin:1998py,Ciafaloni:1998gs,Marzani:smallx,ablinger:3loopNS,ablinger:3loopPS,ablinger:agq,Vogt:FFN3LO3} associated with the splitting functions and transition matrix elements at N$^{3}$LO. Using this information, we approximate these functions to N$^{3}$LO and incorporate the results into the first approximate N$^{3}$LO global PDF fit.
\section{Theoretical Procedures}\label{sec: theo_framework}
In this section we describe the mathematical procedures used to implement N$^{3}$LO approximations into the MSHT PDF framework. These procedures are discussed in terms of the Hessian minimisation method employed by the MSHT fit and extended by theoretically grounded arguments to accommodate theoretical uncertainties.
\subsection{Hessian Method with Nuisance Parameters}\label{subsec: hessian_method}
Following the notation and description from~\cite{Ball:corr2021}, in the Hessian prescription, the Bayesian probability can be written as
\begin{equation}\label{eq: bayes_PTD_orig}
P(T|D) \propto \exp \left( -\frac{1}{2}(T - D)^{T} H_{0} (T - D)\right)
\end{equation}
where $H_{0}$ is the Hessian matrix and $T = \{T_{i}\}$ is the set of theoretical predictions fit to $N$ experimental data points $D = \{D_{i}\}$ with $i=1,\ldots, N$. In this section we explicitly show the adaptation of this equation to accommodate extra theoretical parameters (with penalties) into the total $\chi^{2}$ and Hessian matrices.
To adapt this equation to include a single extra theory parameter, we can make the transformation $T \rightarrow T + t u = T^{\prime}$, where $t$ is the chosen central value of the theory parameter considered and $u$ is some non-zero vector. In defining this new theoretical prescription $T^{\prime}$, we are making the general assumption that the underlying theory is now not necessarily identical to our initial NNLO theory\footnote{For the aN$^{3}$LO prescription defined in this paper this is indeed the case, although for any extra theory parameters that do not inherently change the theory from $T$, this transformation still holds in the case that $t = 0$.} $T$.
We now seek to include a nuisance parameter $\theta$, centered around $t$, to allow the fit to control this extra theory addition. We demand that when $\theta = t$, $T^{\prime}$ remains unaffected with the theory addition unaltered from its central value $t$. This leads us to the expression,
\begin{equation}
T^{\prime} + (\theta - t)u = T + t u + (\theta - t)u.
\end{equation}
Redefining the nuisance parameter as the shift from its central value $t$ ($\theta^{\prime} = \theta - t$) we define $\theta^{\prime}$ centered around 0.
To constrain $\theta^{\prime}$ within the fitting procedure, we must also define a prior probability distribution $P(\theta^{\prime})$ centered around zero and characterised by some standard deviation $\sigma_{\theta^{\prime}}$,
\begin{equation}\label{eq: prior_theta_1}
P(\theta^{\prime}) = \frac{1}{\sqrt{2\pi}\sigma_{\theta^{\prime}}}\exp(-\theta^{\prime\ 2} / 2\sigma_{\theta^{\prime}}^{2}).
\end{equation}
Using this information, we can update Equation~\eqref{eq: bayes_PTD_orig} to be
\begin{align}
P(T|D\theta) &\propto \exp \left( -\frac{1}{2}(T + t u + \frac{(\theta - t)}{\sigma_{\theta^{\prime}}} u - D)^{T} H_{0} (T + t u + \frac{(\theta - t)}{\sigma_{\theta^{\prime}}} u - D)\right) \\
P(T^{\prime}|D\theta^{\prime}) &\propto \exp \left( -\frac{1}{2}(T^{\prime} + \frac{\theta^{\prime}}{\sigma_{\theta^{\prime}}} u - D)^{T} H_{0} (T^{\prime} + \frac{\theta^{\prime}}{\sigma_{\theta^{\prime}}} u - D)\right)\label{eq: bayes_PTDt}
\end{align}
From here, Bayes theorem tells us
\begin{equation}
P(T^{\prime}|D\theta^{\prime})P(\theta^{\prime}|D) = P(\theta^{\prime}|T^{\prime}D)P(T^{\prime}|D)
\end{equation}
where our nuisance parameter $\theta^{\prime}$ is assumed to be independent of the data i.e. $P(\theta^{\prime}|D) = P(\theta^{\prime})$. Integrating over $\theta^{\prime}$ gives
\begin{equation}\label{eq: bayes_PTD}
P(T^{\prime}|D) = \underbrace{\int d\theta^{\prime} P(\theta^{\prime}|T^{\prime}D)}_{=1} P(T^{\prime}|D) = \int d\theta^{\prime} P(T^{\prime}|D\theta^{\prime}) P(\theta^{\prime}).
\end{equation}
Combining Equations~\eqref{eq: prior_theta_1},~\eqref{eq: bayes_PTDt} and~\eqref{eq: bayes_PTD} it is possible to show that,
\begin{equation}\label{eq: bayes_PTD2}
P(T^{\prime}|D) \propto \int d\theta \exp\left(-\frac{1}{2}\left[(T^{\prime} + \frac{\theta^{\prime}}{\sigma_{\theta^{\prime}}} u - D)^{T} H_{0} (T^{\prime} + \frac{\theta^{\prime}}{\sigma_{\theta^{\prime}}} u - D) + \theta^{\prime\ 2}/\sigma_{\theta^{\prime}}^{2}\right]\right).
\end{equation}
To make progress with this equation we consider the exponent and refactor terms in powers of $\theta^{\prime}$,
\begin{equation}
\left(u^{T}H_{0} u + 1\right)\frac{\theta^{\prime\ 2}}{\sigma_{\theta^{\prime}}^{2}} + 2u^{T} H_{0} (T^{\prime}-D) \frac{\theta^{\prime}}{\sigma_{\theta^{\prime}}} + (T^{\prime}-D)^{T}H_{0}(T^{\prime}-D).
\end{equation}
Defining $M^{-1} = \frac{1}{\sigma_{\theta^{\prime}}^{2}}\left(u^{T}H_{0}u + 1\right) $ and completing the square gives,
\begin{multline}\label{eq: square_comp}
M^{-1}\left[\theta^{\prime} + \frac{1}{\sigma_{\theta^{\prime}}}M u^{T} H_{0} (T^{\prime}-D) \right]^{2} - \frac{1}{\sigma_{\theta^{\prime}}^{2}}M\left(u^{T} H_{0} (T^{\prime}-D)\right)^{2} \\+ (T^{\prime}-D)^{T}H_{0}(T^{\prime}-D).
\end{multline}
In Equation~\eqref{eq: square_comp}, we are able to simplify the first term by defining,
\begin{equation}
\overline{\theta}^{\prime}(T,D) = \frac{1}{\sigma_{\theta^{\prime}}} M u^{T} H_{0} (D-T^{\prime}).
\end{equation}
Expanding the second term leaves us with,
\begin{equation}
\left(u^{T} H_{0} (T^{\prime}-D)\right)^{2} = (T^{\prime}-D)^{T} H_{0} u u^{T} H_{0} (T^{\prime}-D)
\end{equation}
The second and third term in Equation~\eqref{eq: square_comp} can then be combined to give,
\begin{equation}
(T^{\prime}-D)^{T}\left(H_{0}-\frac{1}{\sigma_{\theta^{\prime}}^{2}}M H_{0} u u^{T} H_{0}\right)(T^{\prime}-D).
\end{equation}
Further to this we note that the following is true:
\begin{multline}\label{eq: relation_hessian}
(H_{0}^{-1} + u u^{T})\left(H_{0}-\frac{1}{\sigma_{\theta^{\prime}}^{2}}M H_{0} u u^{T} H_{0}\right) = 1 + uu^{T}H_{0} - \frac{1}{\sigma_{\theta^{\prime}}^{2}}Muu^{T}H_{0} - \frac{1}{\sigma_{\theta^{\prime}}^{2}}Mu u^{T}H_{0}u u^{T} H_{0} \\= 1 + uu^{T}H_{0} - \frac{1}{\sigma_{\theta^{\prime}}^{2}}Muu^{T}H_{0} - \frac{1}{\sigma_{\theta^{\prime}}^{2}}Mu (\sigma_{\theta^{\prime}}^{2}M^{-1} - 1) u^{T} H_{0} = 1.
\end{multline}
Using Equation~\eqref{eq: relation_hessian} we are finally able to rewrite Equation~\eqref{eq: bayes_PTD2} as,
\begin{equation}\label{eq: single_nuisance_bayes}
P(T^{\prime}|D) \propto \int d\theta^{\prime} \exp \left(-\frac{1}{2} M^{-1} (\theta^{\prime} - \overline{\theta}^{\prime})^{2} - \frac{1}{2} (T^{\prime}-D)^{T}(H_{0}^{-1} + u u^{T})^{-1} (T^{\prime}-D)\right).
\end{equation}
At this point we can make a choice whether to redefine our Hessian matrix as $H = (H_{0}^{-1} + u u^{T})^{-1}$, or keep the contributions completely separate. By redefining the Hessian we can include correlations between the standard set of MSHT parameters included in $H_{0}$ and the new theoretical parameter $\theta^{\prime}$ contained within $u u^{T}$. However, by doing so we lose information about the specific contributions to the total uncertainty i.e. we cannot then decorrelate the theoretical and standard PDF uncertainties a posteriori. Whereas for the decorrelated choice, although we sacrifice knowledge related to the correlations between the separate sources of uncertainty, we are able to treat the sources completely separably. Interpreting Equation~\eqref{eq: single_nuisance_bayes} as in Equation~\eqref{eq: bayes_PTD_orig} we can write down the two $\chi^{2}$ contributions,
\begin{align}
\chi^{2}_{1} &= (T^{\prime} - D)^{T} (H_{0}^{-1} + u u^{T})^{-1} (T^{\prime}-D) = (T^{\prime} - D)^{T} H (T^{\prime}-D), \\
\chi^{2}_{2} &= M^{-1} (\theta^{\prime} - \overline{\theta}^{\prime})^{2}.
\label{eq: penalty_example}
\end{align}
Where $\chi^{2}_{1}$ is the contribution from the fitting procedure and $\chi^{2}_{2}$ is the new penalty contribution applied when the theory addition strays too far from its original `best guess' central value. This will be discussed further in following sections.
\subsection{Multiple Theory Parameters}\label{subsec: mult_params}
In the case of multiple $N_{\theta^{\prime}}$ theory parameters, Equation~\eqref{eq: bayes_PTDt} becomes
\begin{equation}
P(T^{\prime}|D\theta^{\prime}) \propto \exp \left( -\frac{1}{2}\sum_{i,j}^{N_{\mathrm{pts}}}\bigg(T^{\prime}_{i} + \sum_{\alpha = 1}^{N_{\theta^{\prime}}}\frac{\theta^{\prime}_{\alpha}}{\sigma_{\theta^{\prime}_{\alpha}}} u_{\alpha, i} - D_{i}\bigg) H^{0}_{ij} \bigg(T^{\prime}_{j} + \sum_{\beta = 1}^{N_{\theta^{\prime}}}\frac{\theta^{\prime}_{\beta}}{\sigma_{\theta^{\prime}_{\beta}}} u_{\beta,j} - D_{j}\bigg)\right)
\end{equation}
where we have explicitly included the sum over the number of data points $N_{\mathrm{pts}}$ in the matrix calculation for completeness.
The prior probability for all N$^{3}$LO nuisance parameters also becomes
\begin{equation}\label{eq: prior_theta}
P(\theta^{\prime}) =\prod_{\alpha = 1}^{N_{\theta^{\prime}}} \frac{1}{\sqrt{2\pi}\sigma_{\theta_{\alpha}^{\prime}}}\exp(-\theta_{\alpha}^{\prime\ 2} / 2\sigma_{\theta_{\alpha}^{\prime}}^{2}).
\end{equation}
Constructing $P(T^{\prime}|D)$ using Bayes theorem as before, results in the expression,
\begin{multline}
P(T^{\prime}|D) \propto \int d^{N_{\theta^{\prime}}}\theta^{\prime} \exp \Bigg( -\frac{1}{2}\Bigg[\sum_{i,j}^{N_{\mathrm{pts}}}\bigg(T^{\prime}_{i} + \sum_{\alpha = 1}^{N_{\theta^{\prime}}}\frac{\theta^{\prime}_{\alpha}}{\sigma_{\theta^{\prime}_{\alpha}}} u_{\alpha, i} - D_{i}\bigg) H^{0}_{ij} \times \\ \times \bigg(T^{\prime}_{j} + \sum_{\beta = 1}^{N_{\theta^{\prime}}}\frac{\theta^{\prime}_{\beta}}{\sigma_{\theta^{\prime}_{\beta}}} u_{\beta,j} - D_{j}\bigg) + \sum_{\alpha, \beta}^{N_{\theta^{\prime}}}\frac{\theta^{\prime}_{\alpha}}{\sigma_{\theta^{\prime}_{\alpha}}}\frac{\theta^{\prime}_{\beta}}{\sigma_{\theta^{\prime}_{\beta}}}\delta_{\alpha \beta}\Bigg]\Bigg).
\end{multline}
Following the same procedure as laid out in the previous section, defining $M_{\alpha\beta}^{-1} = (\delta_{\alpha\beta} + u_{\alpha, i}H^{0}_{ij}u_{\beta, j}) / \sigma_{\theta_{\alpha}^{\prime}}\sigma_{\theta_{\beta}^{\prime}}$ and completing the square leaves us with,
\begin{multline}
(T^{\prime}_{i} - D^{\prime}_{i}) H^{0}_{ij} (T^{\prime}_{j} - D^{\prime}_{j}) + \sum_{\alpha,\beta}^{N_{\theta^{\prime}}}M_{\alpha\beta}^{-1}\left[\left(\theta^{\prime}_{\alpha} + \sum_{i,j}^{N_{\mathrm{pts}}}\sum_{\delta=1}^{N_{\theta^{\prime}}}\frac{1}{\sigma_{\theta_{\alpha}^{\prime}}}M_{\alpha \delta}u_{\delta,i}H^{0}_{ij} (T^{\prime}_{j} - D_{j}) \right)^{2} \right.\\ \left.- \left(\sum_{i,j}^{N_{\mathrm{pts}}}\sum_{\delta=1}^{N_{\theta^{\prime}}}\frac{1}{\sigma_{\theta_{\alpha}^{\prime}}}M_{\alpha \delta}u_{\delta,i}H^{0}_{ij} (T^{\prime}_{j} - D_{j})\right)^{2}\right],
\end{multline}
where the summation over the $\beta$ index in $M^{-1}_{\alpha\beta}$ is implicit in the squared terms of the squared bracket expressions.
As in the previous section for a single parameter, we can define,
\begin{align}
\overline{\theta}_{\alpha}^{\prime}(T^{\prime},D) &= \sum_{i,j}^{N_{\mathrm{pts}}}\sum_{\delta = 1}^{N_{\theta^{\prime}}}\frac{1}{\sigma_{\theta_{\alpha}^{\prime}}}M_{\alpha\delta}u_{\delta,i}H^{0}_{ij}(D_{j} - T^{\prime}_{j}) \\
H_{ij} &= \left( \left(H^{0}_{ij}\right)^{-1} + \sum^{N_{\theta^{\prime}}}_{\alpha = 1} u_{\alpha,i}u_{\alpha, j}\right)^{-1}
\label{eq: new_Hessian}
\end{align}
which leads to the final expression for $P(T|D)$,
\begin{multline}
P(T^{\prime}|D) \propto \int d^{N_{\theta^{\prime}}}\theta^{\prime} \exp \left( -\frac{1}{2}\left[\sum_{\alpha, \beta}^{N_{\theta^{\prime}}} \left(\theta^{\prime}_{\alpha} - \overline{\theta}_{\alpha}^{\prime}\right)M_{\alpha\beta}^{-1}\big(\theta^{\prime}_{\beta} - \overline{\theta}_{\beta}^{\prime}\big) \right. \right. \\ \left. \left. + \sum_{i,j}^{N_{\mathrm{pts}}}\left(T^{\prime}_{i} - D_{i}\right) H_{ij} \big(T^{\prime}_{j} - D_{j}\big)\right]\right).
\end{multline}
\subsection{Decorrelated parameters}\label{subsec: decorr_params}
In the treatment above we investigated the case of correlated parameters whereby the Hessian matrix was redefined in Equation~\eqref{eq: new_Hessian}. In performing this redefinition we sacrifice the information contained within $u_{\alpha,i}u_{\alpha, j}$ in order to gain information about the correlations between the original PDF parameters making up $H^{0}_{ij}$ and any new N$^{3}$LO nuisance parameters. As stated earlier, in this case, we can perform a fit to find $H_{ij}$ but one is unable to separate this Hessian matrix into individual contributions.
As will be discussed in later sections, the $K$-factors we include in the N$^{3}$LO additions are somewhat more separate from other N$^{3}$LO parameters considered. The reason for this is that not only are they are concerned with the cross section data directly, they are also included for processes separate from inclusive DIS\footnote{It is true that we may still expect some indirect correlation with the parameters controlling the N$^{3}$LO splitting functions, which are universal across all processes. However, as we will show, these correlations are small and can be ignored.}.
Hence, we have some justification to include the aN$^{3}$LO $K$-factor's nuisance parameters as completely decorrelated from other PDF parameters (including other N$^{3}$LO theory parameters).
To do this we rewrite Equation~\eqref{eq: new_Hessian} as,
\begin{equation}
\left( \left(H^{0}_{ij}\right)^{-1} + \sum^{N_{\theta^{\prime}}}_{\alpha = 1} u_{\alpha,i}u_{\alpha, j} + \sum_{p = 1}^{N_{p}}\sum^{N_{\theta_{K}}}_{\delta = 1} u_{\delta,i}^{p}u_{\delta, j}^{p}\right)^{-1} = \left(H_{ij}^{-1} + \sum_{p = 1}^{N_{p}}K_{ij,p}^{-1} \right)^{-1} = H_{ij}^{\prime}
\end{equation}
where $N_{\theta^{\prime}} \rightarrow N_{\theta^{\prime}} + N_{\theta_{K}}$, $K_{ij,p}$ defines the extra decorrelated contributions from the N$^{3}$LO $K$-factor's parameters, stemming from $N_{p}$ processes; $H_{ij}$ is the Hessian matrix including correlations with parameters associated with N$^{3}$LO structure function theory; and $H_{ij}^{\prime}$ is the fully correlated Hessian matrix. It is therefore possible to construct these matrices separately and perform the normal Hessian eigenvector analysis (described in Section \ref{subsec: eigenvector_results}) on each matrix in turn. In doing this, we maintain a high level of flexibility in our description by assuming the sets of parameters (contained in $H_{ij}^{-1}$ and $K_{ij,p}$) to be suitably orthogonal.
\subsection{Approximation Framework: Discrete Moments}\label{subsec: genframe}
In order to estimate the sources of MHOUs from N$^3$LO splitting functions and transition matrix elements, and ultimately include them into the framework described in Section~\ref{subsec: mult_params}, one must acquire some approximation at N$^{3}$LO. Here we discuss using available sets of discrete Mellin moments for each function, along with any exact leading terms already calculated, to obtain N$^{3}$LO estimations. To perform the parameterisation of the unknown N$^{3}$LO quantities, we follow a similar estimation procedure as in~\cite{vogt:NNLOns,vogt:NNLOs} following the form,
\begin{equation}
F(x)=\sum_{i=1}^{N_{m}}A_{i}f_{i}(x) + f_{e}(x).
\label{eq: splitAnsatz}
\end{equation}
In Equation~\eqref{eq: splitAnsatz}, $N_{m}$ is the number of available moments, $A_{i}$ are calculable coefficients, $f_i(x)$ are functions chosen based on our intuition and theoretical understanding of the full function, and $f_{e}(x)$ encapsulates all the currently known leading exact contributions at either large or small-$x$. To describe this, consider a toy situation where we are given four data points described by some unknown degree 6 polynomial. Along with this information, we are told the dominant term at large-$x$ is described by $2x^{6}$. In this case, one may wish to attempt to approximate this function by means of a set of 4 simultaneous equations formed from Equation~\eqref{eq: splitAnsatz} equated to each of the four data points (or constraints). The result of this is then a unique solution for each chosen set of functions $\{f_{i}(x)\}$. However, a byproduct of this is that for each $\{f_{i}(x)\}$, one lacks any means to control the uncertainty in these approximate solutions. In order to allow a controllable level of uncertainty into this approximation, one must introduce an extra degree of freedom. This degree of freedom will be introduced through an unknown coefficient $a \equiv A_{N_{m} + 1}$, which for convenience, will be absorbed into the definition of $f_{e}(x) \rightarrow f_{e}(x, a)$. In this toy example one is then able to choose to define the functions $f_{i}(x)$ as,
\begin{alignat}{5}\label{eq: example_split_ansatz}
f_{1}(x) \quad &= \quad x^{4},\nonumber \\
f_{2}(x) \quad &= \quad x^{3} \nonumber \\
f_{3}(x) \quad &= \quad x^{2} \nonumber \\
f_{4}(x) \quad &= \quad 1 \quad \text{or} \quad x, \nonumber \\
f_{e}(x, a) \quad &= \quad 2x^{6} + ax^{5},
\end{alignat}
where we have prioritised approximating the large-$x$ behaviour more precisely than the small-$x$ behaviour i.e. the small-$x$ behaviour contains an inherent functional uncertainty from the ambiguity in the choice of functions for $f_{4}(x)$. This could easily be adapted and even reversed depending on which region of $x$ we are most sensitive to. Using these functions, one is then able to assemble a set of potential approximations to the overall polynomial, each uniquely defined by a set of functions and corresponding coefficients $\{A_{i}, a, f_{i}\}$.
As mentioned, for the N$^{3}$LO additions considered in this framework we use the available calculated moments as constraints for the corresponding simultaneous equations.
A summary of all the known and used ingredients for all N$^{3}$LO approximations is provided in Appendix~\ref{app: n3lo_known}. The details of these known quantities will be discussed in detail in Section~\ref{sec: n3lo_split} and Section~\ref{sec: n3lo_OME}. We also mention here that towards the small-$x$ regime, the leading terms present in the splitting functions and transition matrix elements exhibit the relations,
\begin{subequations}
\begin{align}
F_{gg}(x\rightarrow 0) \simeq & \frac{C_{A}}{C_{F}}F_{gq}(x\rightarrow 0), \label{eq: PggPgqRelation}\\
F_{qq}(x\rightarrow 0) \simeq & \frac{C_{F}}{C_{A}}F_{qg}(x\rightarrow 0),
\end{align}
\label{eq: Relations}
\end{subequations}
where $F_{ij} \in \{P_{ij}, A_{ij,H}\}$ and $C_{A}, C_{F}$ are the usual QCD constants. Although Equation~\eqref{eq: Relations} are exact at leading order, it is known that as we expand to higher orders, these will break down due to the effect of large sub-leading logarithms. Due to this, we do not demand this relation as a constraint in our approximations. Instead we discuss the validity of Equation~\eqref{eq: Relations} in comparison with the aN$^{3}$LO functions.
Following from~\cite{vogt:NNLOns,vogt:NNLOs}, we must choose a set of candidate functions for each $f_{i}(x)$. Our convention is to assign these functions such that at large-$x$, $f_{1}(x)$ is dominant, while at small-$x$, $f_{N_{m}}(x)$ is dominant. With $f_{i}(x)\ \forall i \in \{2,\dots, N_{m} - 1\}$, dominating in the region between.
The sets of functions assigned to each $f_{i}(x)$ are determined for each N$^{3}$LO function based on knowledge from lower orders and our intuition about what to expect at N$^{3}$LO.
Analogous to our toy polynomial example, we allow the inclusion of an unknown NLL term (NNLL in the $P_{gg}$ case) into the $f_{e}$ function of our parameterisation. The coefficient of this NLL (NNLL) term is then controlled by a variational parameter $a$. This parameter uniquely defines the solution to the sets of simultaneous equations considered i.e. for each set of functions $f_{i}(x)$ there exists a unique solution for every possible choice of $a$. The final step to consider in this approximation is how to choose the prior allowed variation of $a$ in a sensible way for each N$^{3}$LO approximation. To do this, we consider the criteria outlined below:
\vspace{5mm}
\begin{tabular}{ p{3cm} p{11.5cm} }
\textbf{Criteria 1}: & At very small-$x$ ($x < 10^{-5}$), we require the asymptote of $f_{e}$ to be contained within the uncertainty band of the N$^3$LO approximation i.e. the full function cannot be in a large tension with the small-$x$ description.\\
[0.5cm]
\textbf{Criteria 2}: & At large-$x$ ($x > 10^{-2}$) the N$^3$LO contribution should have relatively little effect. More specifically, we do not expect as large of a divergence as we do at small-$x$. Due to this, we require that the trend of the N$^{3}$LO approximation follow the general trend of the NNLO function at large-$x$.\\
\end{tabular}
\vspace{5mm}
The allowed variation in $a$ gives us an uncertainty which, at its foundations, is chosen via a conservative estimate based on all the available prior knowledge about the function and lower orders being considered. To determine a full predicted uncertainty for the function and allow for a computationally efficient fixed functional form, the variation of $a$ can absorb the uncertainty from the ambiguity in the choice of functions $f_{i}(x)$ (essentially expanding the allowed range of $a$). Since the functions are approximations themselves, increasing the allowed variation of $a$ to encapsulate the total uncertainty predicted by the initial treatment described above is a valid simplification.
A worked example following this procedure is provided for the $P_{qg}^{(3)}$ and $A_{Hg}^{(3)}$ functions in Section's~\ref{subsec: 4loop_split} and~\ref{subsec: 3loop_OME} respectively.
\subsection{Approximation Framework: Continuous Information}\label{subsec: genframe_continuous}
In the previous section we described the approximation framework employed for functions with discrete Mellin moment information, combined with any available exact information. For the N$^{3}$LO coefficient function approximations, we have access to a somewhat richer vein of information than the discrete moments discussed for the framework in Section~\ref{subsec: genframe}. More specifically, approximations of the FFNS coefficient functions at N$^{3}$LO are known for the heavy quark contributions to the heavy flavour structure function $F_{2,H}(x, Q^{2})$ at $Q^{2} < m_{c,b}^{2}$~\cite{Catani:FFN3LO1,Laenen:FFN3LO2,Vogt:FFN3LO3}. These approximations include the exact LL and mass threshold contributions, with an approximated NLL term (the details of this are described in Section~\ref{subsec: NLL_coeff}). Furthermore, the N$^{3}$LO ZM-VFNS coefficient functions are known exactly~\cite{Vermaseren:2005qc}. Both of these contributions can then be combined with the transition matrix element approximations to define the GM-VFNS functions in the $Q^{2} \leq m_{c}^{2}, m_{b}^{2}$ and $Q^{2} \rightarrow \infty$ regimes. Due to this, we base our approximations for the $C_{H,\{q,g\}}^{(3)}$ functions on the known continuous information in the low and high-$Q^{2}$ regimes.
To achieve a reliable approximation for $C_{H,\{q,g\}}^{(3)}$, we first fit a regression model with a large number of functions in $(x,Q^{2})$ space made available to the model (in order to reduce the level of functional bias in the parameterisation). This produces an unstable result at the extremes of the parameterisation (large-$x$ and low-$Q^{2}$). However, it provides a basis for manually choosing a stable parameterisation to move between the two known regimes (low-$Q^{2}$ and high-$Q^{2}$).
Using the regression model predictions as a qualitative guide, we choose a stable and smooth interpolation between the two $Q^{2}$ regimes (low-$Q^{2}$ and high-$Q^{2}$) as given in Equation~\eqref{eq: ChqFF_param}. This interpolation is observed to mirror the expected behaviour observed from lower orders, the regression model qualitative prediction having been calculated independently of lower orders and the best fit quality to data. By definition, we also ensure an exact cancellation between the coefficient functions and the transition matrix elements at the mass threshold energies as demanded by the theoretical description in Section~\ref{sec: structure}.
For the contributions to the heavy flavour structure function $F_{2,H}$ the final interpolations in the FFNS regime are defined as,
\begin{multline}\label{eq: ChqFF_param}
C_{H,\ \{q,g\}}^{\mathrm{FF},\ (3)} =
\begin{cases}
C_{H,\ \{q,g\},\ \text{low-$Q^{2}$}}^{\mathrm{FF},\ (3)}(x, Q^{2} = m_{h}^{2})\ e^{0.3\ (1-Q^{2}/m_{h}^{2})} \\ \qquad\qquad\qquad\qquad +\ C_{H,\ \{q,g\}}^{\mathrm{FF},\ (3)}(x, Q^{2} \rightarrow \infty)\big(1 - e^{0.3\ (1-Q^{2}/m_{h}^{2})}\big) &,\ \text{if}\ Q^{2} \geq m_{h}^{2}, \\ \\
C_{H,\ \{q,g\},\ \text{low-$Q^{2}$}}^{\mathrm{FF},\ (3)}(x, Q^{2})&,\ \text{if}\ Q^{2} < m_{h}^{2}.
\end{cases}
\end{multline}
where $C_{H,\ \{q,g\},\ \text{low-$Q^{2}$}}^{\mathrm{FF},\ (3)}$ are the already calculated approximate heavy flavour FFNS coefficient functions at $Q^{2} \leq m_{h}^{2}$, and $C_{H,\ \{q,g\}}^{\mathrm{FF},\ (3)}(Q^{2} \rightarrow \infty)$ is the limit at high-$Q^{2}$ found from the known ZM-VFNS coefficient functions and relevant subtraction terms, themselves found from Equation~\eqref{eq: fullN3LO_H}. Both of these limits will be discussed in detail on a case-by-case basis in Section~\ref{sec: n3lo_coeff}.
For the heavy flavour contributions to $F_{2,q}$, we have no information about the low-$Q^{2}$ N$^{3}$LO FFNS coefficient functions. In this case, we use intuition from lower orders to provide a soft (lightly weighted) low-$Q^{2}$ target for our regression model in $(x,Q^{2})$.
However, since the overall contribution is very small from these functions, the exact form of these functions is not phenomenologically important at present. Further to this, our understanding from lower orders is that these functions have a weak dependence on $Q^{2}$ and so the form of the low-$Q^{2}$ description is even less important. As with the $C_{H,\ \{q,g\}}^{(3)}$ coefficient functions, the regression results provide an initial qualitative guide which exhibits instabilities in the extremes of $(x,Q^{2})$. We therefore employ a similar technique as before to ensure a smooth extrapolation across all $(x, Q^{2})$ into the unknown behaviour at low-$Q^{2}$. For these functions, the ansatz used is given as,
\begin{multline}\label{eq: CqFF_param}
\qquad C_{q,\ \{q,g\}}^{\mathrm{FF},\ (3)} =
\begin{cases}
C_{q,\ q}^{\mathrm{FF},\ \mathrm{NS},\ (3)}(x, Q^{2} \rightarrow \infty)\big(1 + e^{-0.5\ (Q^{2}/m_{h}^{2}) - 3.5}\big), \qquad \qquad \qquad \\ \\
C_{q,\ q}^{\mathrm{FF},\ \mathrm{PS},\ (3)}(x, Q^{2} \rightarrow \infty)\big(1 - e^{-0.25\ (Q^{2}/m_{h}^{2}) - 0.3}\big),\qquad \qquad \qquad \\ \\
C_{q,\ g}^{\mathrm{FF},\ (3)}(x, Q^{2} \rightarrow \infty)\big(1 - e^{-0.05\ (Q^{2}/m_{h}^{2}) + 0.35}\big), \qquad \qquad \qquad
\end{cases}
\end{multline}
where $C_{q,\{q,g\}}^{\mathrm{FF},\ (3)}(x, Q^{2} \rightarrow \infty)$ is the known limit at high-$Q^{2}$. |
1,108,101,564,590 | arxiv | \section{Introduction}
The Gross-Pitaevskii equation for a Bose-Einstein condensate (BEC) with
symmetric harmonic trap is given by
\begin{equation}
-iu_{t}-\Delta u+(x^{2}+y^{2})u+\left\vert u\right\vert ^{2}u=0. \label{Ec
\end{equation}
Periodic solutions of \eqref{Ec} play an important role in the understanding
of the long term behavior of its solutions. In \cite{PeKe13}, symmetric and
asymmetric vortex solutions are obtained and their stability is established.
Solutions with two rotating vortices of opposite vorticity are constructed in
\cite{57}. In \cite{GKC} the authors prove the existence of periodic and
quasi-periodic trajectories of dipoles in anisotropic condensates. The
literature of the study of vortex dynamics in Bose-Einstein condensates is
vast, both on the mathematical and physical side; we refer the reader to
\cite{17,21,34,38,PeKe11,57} and the references therein for a more detailed account.
In this note we prove the existence of several global branches of solutions to
\eqref{Ec} among which there are vortex solutions and dipole solutions. Let
$X$ be the space of functions in $H^{2}(\mathbb{R}^{2};\mathbb{C)}$ for which
$\left\Vert u\right\Vert _{X}^{2}=\left\Vert u\right\Vert _{H^{2}
^{2}+\left\Vert r^{2}u\right\Vert _{L^{2}}^{2}$ is finite. Our main results are:
\begin{theorem}
\label{teo1} Let $m_{0}\geq1$ and $n_{0}$ be fixed non-negative integers. The
equation \eqref{Ec} has a global bifurcation in
\begin{equation}
Fix(\tilde{O}(2))=\{u\in X:u(r,\theta)=e^{im_{0}\theta}u(r)\text{ with
}u(r)\text{ real valued}\text{ }\}\text{.} \label{deffixteo1
\end{equation}
These are periodic solutions to \eqref{Ec} of the form
\[
e^{-i\omega t}e^{im_{0}\theta}u(r)\text{,
\]
starting from $\omega=2(m_{0}+2n_{0}+1)$, where $u(r)$ is a real-valued function.
\end{theorem}
It will be clear from the proof, and standard bifurcation theory, that for
small amplitudes $a$, we have the local expansion
\[
u(r)=av_{m_{0},n_{0}}(r)+\sum_{n\in\mathbb{N}}u_{m_{0},n}v_{m_{0},n}(r)\text{
and }u_{m_{0},n}(a)=O(a^{2})\text{,
\]
where the $v_{m,n}$'s are the eigenfunctions introduced in \eqref{def.vmn} and
the $u_{m,n}$'s are Fourier coefficients. Thus, the number $m_{0}$ is the
degree of the vortex at the origin and $n_{0}$ is the number of nodes of
$u(r)$ in $(0,\infty)$.
Our second theorem is concerned with the existence of multi-pole solutions.
\begin{theorem}
\label{teo2} Let $m_{0}\geq1$ and $n_{0}<m_{0}$ be two fixed non-negative
integers, then the equation \eqref{Ec} has a global bifurcation in
\begin{equation}
Fix(\mathbb{Z}_{2}\times\tilde{D}_{2m_{0}})=\{u\in X:u(r,\theta)=\bar
{u}(r,\theta)=u(r,-\theta)=-u(r,\theta+\pi/m_{0})\}\text{.} \label{deffixteo2
\end{equation}
These are periodic solutions to \eqref{Ec} of the form
\[
e^{-i\omega t}u(r,\theta)\text{,
\]
starting from $\omega=2(m_{0}+2n_{0}+1)$, where $u(r,\theta)$ is a real
function vanishing at the origin, enjoying the symmetrie
\[
u(r,\theta)=u(r,-\theta)=-u(r,\theta+\pi/m_{0})\text{.
\]
\end{theorem}
The requirement $n_{0}<m_{0}$ in Theorem \ref{teo2} is a non-resonance
condition. The solutions of the previous theorem for $(m_{0},n_{0})=(1,0)$
correspond to dipole solutions. This follows locally from the estimate
\[
u(r,\theta)=a(e^{im_{0}\theta}+e^{-im_{0}\theta})v_{m_{0},n_{0}}(r)+\sum
_{m\in\{m_{0},3m_{0},5m_{0},...\}}\sum_{n\in\mathbb{N}}u_{m,n}(e^{im\theta
}+e^{-im\theta})v_{m,n}(r)
\]
for small amplitude $a$ where $u_{m,n}=O(a^{2}).$
For $n_{0}=0$, since the function $v_{m_{0},0}$ is positive for $r\in
(0,\infty)$ and $u_{m,n}=O(a^{2})$, $u(r,\theta)$ is zero only when
$\theta=(k+1/2)\pi/m_{0}$. Moreover, as $u$ is real, then the lines
$\theta=(k+1/2)\pi/m_{0}$ correspond to zero density regions and the phase has
a discontinuous jump of $\pi$ at those lines.
A difficulty when trying to obtain the dipole solution, that is for
$(m_{0},n_{0})=(1,0)$, is the fact that to carry out a local inversion one has
to deal with a linearized operator with a repeated eigenvalue corresponding to
$(m_{0},n_{0})=(1,0)$ and $(-1,0)$. We overcome this by restricting our
problem to a natural space of symmetries which we identify below. In this
space our linearized operator only encounters a simple bifurcation which
yields the global existence result thanks to a topological degree argument;
see Theorem \ref{Theo}.
\section{Reduction to a bifurcation in a subspace of symmetries}
The group of symmetries of \eqref{Ec} is a three torus, corresponding to
rotations, phase and time invariances. The analysis of the group
representations leads to two kinds of isotropy groups, one corresponding to
vortex solutions and the other to dipole solutions. A fixed point argument on
restricted subspaces and Leray-Schauder degree yield the global existence of
these branches.
In \cite{PeKe13}, the authors study the case $(m_{0},n_{0})=(1,0)$ which is
bifurcation of a vortex of degree one. They also obtain second branch stemming
from this one and analyze its stability. The bifurcation from the case
$(m_{0},n_{0})=(0,0)$ is the ground state (see \cite{PeKe13}). The proof of
existence of dipole-like solutions was left open. In the present work we use
the symmetries of the problem, classifying the spaces of irreducible
representations, to obtain these as global branches provided a non-resonance
condition is satisfied.
\subsection{Setting the problem}
We rewrite \eqref{Ec} it in polar coordinates:
\[
-iu_{t}-\Delta_{(r,\theta)}u+r^{2}u+\left\vert u\right\vert ^{2}u=0\text{,
\]
where $\Delta_{(r,\theta)}=\partial_{r}^{2}+r^{-1}\partial_{r}+r^{-2
\partial_{\theta}^{2}$. Periodic solutions of the form $u(t,r,\theta
)=e^{-i\omega t}u(r,\theta)$ are zeros of the map
\[
f(u,\omega)=-\Delta_{(r,\theta)}u+(r^{2}-\omega+\left\vert u\right\vert
^{2})u.
\]
The eigenvalues and eigenfunctions of the linear Schr\"{o}dinger operator
\[
L=-\Delta_{(r,\theta)}+r^{2}:X\rightarrow L^{2}(\mathbb{R}^{2};\mathbb{C}),
\]
are found in Chapter 6, complement D, pg.727-737, on \cite{QM}. The operator
$L$ has eigenfunctions $v_{m,n}(r)e^{im\theta}$, which form and orthonormal
basis of $L^{2}(\mathbb{R}^{2};\mathbb{C}),$ and eigenvalues
\[
\lambda_{m,n}=2(\left\vert m\right\vert +2n+1)\text{ for }(m,n)\in
\mathbb{Z\times N}\text{,
\]
where $v_{m,n}(r)$ is a solution of $\left( -\Delta_{m}+r^{2}\right)
v_{m,n}(r)=\lambda_{m,n}v_{m,n}(r)$, wher
\begin{equation}
-\Delta_{m}+r^{2}:=-(\partial_{r}^{2}+r^{-1}\partial_{r}-r^{-2}m^{2
)+r^{2}\text{,} \label{def.vmn
\end{equation}
with $v_{m,n}(0)=0$ for $m\neq0$. We have that
\[
u=\sum_{(m,n)\in\mathbb{Z\times N}}u_{m,n}v_{m,n}(r)e^{im\theta},\qquad
Lu=\sum_{(m,n)\in\mathbb{Z\times N}}\lambda_{m,n}u_{m,n}v_{m,n}(r)e^{im\theta
}\text{.
\]
Moreover, we know that $v_{m,n}(r)e^{im\theta}$ are orthogonal functions,
where $n$ is the number of nodes of $v_{m,n}(r)$ in $(0,\infty)$, see section
2.9 in \cite{Mo}.
\begin{remark}
Notice, that this is a slightly different orthonormal system that the one in
\cite{PeKe13}, which is more suited for anisotropic traps: $V(x,y)=\alpha
x^{2}+\beta y^{2}$ with $\alpha\neq\beta$.
\end{remark}
We have that the norm of $u$ in $L^{2}(\mathbb{R}^{2};\mathbb{C})$ is
$\left\Vert u\right\Vert _{L^{2}}^{2}=\sum_{(m,n)\in\mathbb{Z\times N
}\left\vert u_{m,n}\right\vert ^{2}$. Then, the inverse operator
$K=L^{-1}:L^{2}(\mathbb{R}^{2};\mathbb{C})\rightarrow X$ is continuous and
given by
\[
Ku=\sum\lambda_{m,n}^{-1}u_{m,n}v_{m,n}(r)e^{im\theta}.
\]
Moreover, the operator $K:X\rightarrow X$ is compact.
Observe that $H^{2}(\mathbb{R}^{2})$ is a Banach algebra and $H^{2
(\mathbb{R}^{2})\subset C^{0}(\mathbb{R}^{2})$, then $\left\Vert uv\right\Vert
_{X}\leq c\left\Vert v\right\Vert _{X}\left\Vert u\right\Vert _{X}$. We then
see that $g(u):=K(\left\vert u\right\vert ^{2}u)=\mathcal{O}(\left\Vert
u\right\Vert _{X}^{3})$ is a \textit{nonlinear compact map} such that
$g:X\rightarrow X$. Therefore, we obtain an equivalent formulation for the
bifurcation as zeros of the ma
\[
Kf(u,\omega)=u-\omega Ku+g(u):X\times\mathbb{R}\rightarrow X\text{.
\]
This formulation has the advantage that allows us to appeal to the global
Rabinowitz alternative \cite{Ra} (Theorem \ref{Theo} below).
\subsection{Equivariant bifurcation}
Let us define the action of the group generated by $(\psi,\varphi
)\in\mathbb{T}^{2}$, $\kappa\in\mathbb{Z}_{2}$ and $\bar{\kappa}\in
\mathbb{Z}_{2}$ in $L^{2}$ a
\[
\rho(\psi,\varphi)u(r,\theta)=e^{i\varphi}u(r,\theta+\psi);\quad\rho
(\kappa)u(r,\theta)=u(r,-\theta);\quad\rho(\bar{\kappa})u(r,\theta)=\bar
{u}(r,\theta)\text{.
\]
Actually, the group generated by these actions is $\Gamma=O(2)\times O(2),$
and the map $Kf$ is $\Gamma$-equivariant.
Given a pair $(m_{0},n_{0})\in\mathbb{Z\times N},$ the operator $K$ has
multiple eigenvalues $\lambda_{m,n}^{-1}=\lambda_{m_{0},n_{0}}^{-1}$ for each
$(m,n)\in\mathbb{Z\times N}$ such that $\left\vert m\right\vert +2n=\left\vert
m_{0}\right\vert +2n_{0}$. To reduce the multiplicity of the eigenvalue
$\lambda_{m_{0},n_{0}}^{-1}$, we assume for the moment that there is a
subgroup $G$ of $\Gamma$ such that in the fixed point space
\[
Fix(G)=\{u\in X:\rho(g)u=u\text{ for }g\in G\}\text{,
\]
the linear map $K$ has only one eigenvalue $\lambda_{m_{0},n_{0}}^{-1}$. Then,
we can apply the following theorem using the fact that $Kf(u,\omega
):Fix(G)\times\mathbb{R}\rightarrow Fix(G)$ is well defined.
\begin{theorem}
\label{Theo}There is a global bifurcating branch $Kf(u(\omega),\omega)=0,$
starting from $\omega=\lambda_{m_{0},n_{0}}$ in the space $Fix(G)\times
\mathbb{R}$, this branch is a continuum that is unbounded or returns to a
different bifurcation point $(0,\omega_{1})$.
\end{theorem}
For a proof see the simplified approach due to Ize in Theorem 3.4.1 of
\cite{Ni2001}, or a complete exposition in \cite{IzVi03}.
We note that if $u\in H^{2}$ is a zero of $Kf$, then $u=K(\omega u-\left\vert
u\right\vert ^{2}u)\in H^{4}$. Using a bootstrapping argument we obtain that
the zeros of $Kf$ are solutions of the equation (\ref{Ec}) in $C^{\infty}.$
Next, we find the irreducible representations and the maximal isotropy groups.
The fixed point spaces of the maximal isotropy groups will have the property
that $K$ has a simple eigenvalue corresponding to $\lambda_{m_{0},n_{0}
^{-1}.$ Thus, Theorems \ref{teo1} and \ref{teo2} will follow from Theorem
\ref{Theo} applied to $G=\tilde{O}(2)$ and $G=\mathbb{Z}_{2}\times\tilde
{D}_{2m_{0}}$ respectively.
\subsection{Isotropy groups}
The action of the group on the components $u_{m,n}$ is given by $\rho
(\varphi,\psi)u_{m,n}=e^{i\varphi}e^{im\psi}u_{m,n}$ and $\rho(\kappa
)u_{m,n}=u_{-m,n}\text{ and }\rho(\bar{\kappa})u_{m,n}=\bar{u}_{-m,n}$. Then,
the irreducible representations are $(z_{1},z_{2})=(u_{m,n},u_{-m,n
)\in\mathbb{C}^{2}$, and the action of $\Gamma$ in the representation
$(z_{1},z_{2})$ is
\begin{align}
\rho(\varphi,\psi)(z_{1},z_{2}) & =e^{i\varphi}(e^{im\psi}z_{1},e^{-im\psi
}z_{2});\label{Action}\\
\rho(\kappa)(z_{1},z_{2}) & =(z_{2},z_{1});\nonumber\\
\rho(\bar{\kappa})(z_{1},z_{2}) & =(\bar{z}_{2},\bar{z}_{1}).\nonumber
\end{align}
Actually, the irreducible representations $(u_{m_{0},n},u_{-m_{0},n
)\in\mathbb{C}^{2}$ are similar for all $n\in\mathbb{N}.$ The spaces of
similar irreducible representations are of infinite dimension. We analyze only
non-radial bifurcations, that is solutions bifurcating from $\omega
=\lambda_{m_{0},n_{0}}$ with $m_{0}\neq0 ;$ the radial bifurcation with
$m_{0}=0$ may be analyzed directly from the operator associated to the
spectral problem \eqref{def.vmn}.
Let us fix $m_{0}\geq1$ and $(z_{1},z_{2})=(u_{m_{0},n},u_{-m_{0},n})$. Then,
possibly after applying $\kappa$, we may assume $z_{1}\neq0$, unless
$(z_{1},z_{2})=(0,0)$. Moreover, using the action of $S^{1}$, the point
$(z_{1},z_{2})$ is in the orbit of $(a,re^{i\theta})$. It is known that there
are only two maximal isotropy groups, one corresponding to $(a,0)$ and the
other one to $(a,a)$, see for instance \cite{GoSc86}.
From (\ref{Action}), we have that the isotropy group of $(a,0)$ is generated
by $(\varphi,-\varphi/m_{0})$ and $\kappa\bar{\kappa}$, that is
\[
\tilde{O}(2)=\left\langle (\varphi,-\varphi/m_{0}),\kappa\bar{\kappa
}\right\rangle \text{.
\]
While the isotropy group of $(a,a)$ is generated by$\ (\pi,\pi/m_{0})$,
$\kappa$ and $\bar{\kappa}$, that is
\[
\mathbb{Z}_{2}\times\tilde{D}_{2m_{0}}=\left\langle \kappa,(\pi,\pi
/m_{0}),\bar{\kappa}\right\rangle \text{.
\]
These two groups are the only maximal isotropy groups of the representation
$(z_{1},z_{2})\in\mathbb{C}^{2}$, and the fixed point spaces have real
dimension one in $\mathbb{C}^{2}$.
\section{Vortex solutions: Proof of Theorem \ref{teo1}}
The functions fixed by the group $\tilde{O}(2)$ satisfy $u(r,\theta
)=e^{im_{0}\theta}u(r)$ from the element $(\varphi,-\varphi/m_{0})$, and
$u(r)=\bar{u}(r)$ from the element $\kappa\bar{\kappa}$. Thus, functions in
the space $Fix(\tilde{O}(2))$ are of the form
\[
u(r,\theta)=\sum_{n\in\mathbb{N}}u_{m_{0},n}e^{im_{0}\theta}v_{m_{0},n}(r)
\]
with $u_{m_{0},n}\in\mathbb{R}$.
Therefore, the map $Kf(u,\omega)$ has a simple eigenvalue $\lambda
_{m_{0},n_{0}}$ in the space $Fix(\tilde{O}(2))\times\mathbb{R}$ (see
\eqref{deffixteo1}). Therefore, from Theorem \ref{Theo}, there is a global
bifurcation in $Fix(\tilde{O}(2))\times\mathbb{R}$ starting at $\omega
=\lambda_{m_{0},n_{0}}.$
\section{Multi-pole-like solutions: Proof of Theorem \ref{teo2}}
The functions fixed by $\mathbb{Z}_{2}\times\tilde{D}_{2m_{0}}$ satisfy
$u(r,\theta)=\bar{u}(r,\theta)=u(r,-\theta)$. Therefore $u_{m,n}$ is real and
$u_{m,n}=u_{-m,n}$. Moreover, since $u(r,\theta)=-u(r,\theta+\pi/m_{0})$, then
$u_{m,n}=-e^{i\pi(m/m_{0})}u_{m,n}$. This relation gives $u_{m,n}=0$ unless
$e^{i\pi(m/m_{0})}=-1$ or $m/m_{0}$ is odd. Thus, functions in the space
$Fix(\mathbb{Z}_{2}\times\tilde{D}_{2m_{0}})$ are of the form
\[
u(r,\theta)=\sum_{m\in\{m_{0},3m_{0},5m_{0},...\}
{\displaystyle\sum\limits_{n\in\mathbb{N}}}
u_{m,n}(e^{im\theta}+e^{-im\theta})v_{m,n}(r)\text{,
\]
where $u_{m,n}$ is real and $m_{0}\geq1$. Therefore, the map $K$ has a simple
eigenvalue $\lambda_{m_{0},n_{0}}=2(m_{0}+2n_{0}+1)$ in $Fix(\mathbb{Z
_{2}\times\tilde{D}_{2m_{0}})$ if $\lambda_{lm_{0},n}\neq\lambda_{m_{0},n_{0
}$ for $n\in\mathbb{N}$ and $l=3,5,7..$.This condition is equivalent to
$lm_{0}+2n\neq m_{0}+2n_{0}$ or $2n_{0}-(l-1)m_{0}\neq2n$. Then, the
eigenvalue $\lambda_{m_{0},n_{0}}$ is simple if $2n_{0}<(l-1)m_{0}$ for
$l=3,5,...$, or $n_{0}<m_{0}$.
From Theorem (\ref{Theo}), the map $Kf(u,\omega)$ has a global bifurcation in
$Fix(\mathbb{Z}_{2}\times\tilde{D}_{2m_{0}})\times\mathbb{R}$ as $\omega$
crosses the value $\lambda_{m_{0},n_{0}}$ (see \textit{\eqref{deffixteo2}}).
\textbf{Acknowledgements.} C. Garcia learned about this problem from D.
Pelinovsky. A. Contreras is grateful to D.Pelinovsky for useful discussions.
Both authors thank the referees for their useful comments which greatly
improved the presentation of this note.
|
1,108,101,564,591 | arxiv | \section{Introduction and main results}
This paper concerns the asymptotic behavior of analytic functions $f$ whose order of growth $\sigma_M(f)$ and the lower order $\lambda_M(f)$, defined via the maximum modulus function $M(r,f)=\max\{ |f(z)|:|z|=r \}$, are distinct. In particular, we search for asymptotic lower estimates for $\log|f|$ and its $L^p$-mean on the circle of radius $r$ centered at the origin. Lindel\"of proximate order $\rho(r)$ has been widely and effectively used in the frame of such problems~\cite{GO,JK,Levin,Linden1956}. It allows one to majorize $\log M(r,f)$ by the flexible function $V(r)=r^{\rho(r)}$, where $\rho(r)\to\rho_M(f)$, as $r$ approaches either to $\infty$ (the case of entire functions) or to $1$ (the case of functions analytic in the unit disc). Indeed, by Valiron's theorem \cite{GO,JK,Levin} for each entire function of finite order there exists a proximate order $\rho(r)$ such that $\log M(r,f)\le V(r)$ for all $r$, and $\log M(r_n,f)=V(r_n)$ for some sequence $(r_n)$ tending to $\infty$. Such a proximate order is called a proximate order of an entire function $f$. This concept plays an essential role in the theory of functions of completely regular growth \cite{Levin}. However, the defect of this approach is that it completely ignores the value of the lower order. For entire functions $f$ of finite lower order $\lambda_M(f)$ there is a notion of a lower proximate order $\lambda(r)$~\cite{GO,Levin}, which allows to majorize $\log M(r,f)$ by $r^{\lambda_M(f)+o(1)}$ on a sequence of values of $r$ tending to $\infty$. Unfortunately, such a conclusion is in many cases far from being satisfactory. This leads us to the problem of constructing a majorant $V(r)$ for $\log M(r,f)$ such that, on one hand, it keeps the information about both the order $\rho_M(f)$ and the lower order $\lambda_M(f)$ well enough, and, on the other hand, it is sufficiently flexible. In particular we require that it satisfies Caramata's condition $V(2r)=O(V(r))$ as $r\to\infty$. It turns out that this is impossible in the frame of proximate order and its known generalizations. We solve this problem by introducing the notion of a quasi proximate order. It allows us to complement and generalize some results of M.~Cartwright \cite{CartWr33} and C.~N.~Linden~\cite{Li_mp,Li56_conj}.
We proceed towards the statements of our findings via necessary definitions. Let $1\le r_0<\infty$. A function $\sigma:[r_0,\infty)\to(0,\infty)$ is called a {\it quasi proximate order} if
there exist two constants $0\le\lambda<\rho<\infty$ and an associated function $A^*=A^*_\sigma:[r_0,\infty)\to(0,\infty)$ such that
\begin{itemize}
\item [(1)] $\sigma \in C^1[r_0,\infty)$;
\item [(2)] $\displaystyle\limsup_{t\to\infty}\sigma(t)=\rho$;
\item [(3)] $\displaystyle\liminf_{t\to\infty}\sigma(t)=\lambda$;
\item [(4)] $\limsup_{t\to\infty}|\sigma'(t)|t\log t<\infty$;
\item [(5)] $A^*$ is nondecreasing and $A^*(t)\le t^{\sigma(t)}\le(1+o(1))A^*(t)$, as $t\to\infty$.
\end{itemize}
Even though $t^{\sigma(t)}$ is not necessarily monotone, it follows from (2) and (4) that $t^{\sigma(t)}\asymp(2t)^{\sigma(2t)}$, as $t\to\infty$. Namely, (2) and (4) yield $\s(t)\le 2\r$, if $t\ge t_0$, and
$$
\s(t)-\frac{C}{\log t}\le\s(2t)\le\s(t)+\frac{C}{\log t},\quad t\ge t_0,
$$
for some constants $C=C(\sigma)>0$ and $t_0=t_0(\sigma)>1$.
These inequalities imply
$t^{\sigma(t)}\asymp(2t)^{\sigma(2t)}$ which together with (5) yields $A^*(2t)\lesssim A^*(t)$, as $t\to\infty$.
A function satisfying properties (1)--(3) and
\begin{itemize}
\item [(4')] $\limsup_{t\to\infty}|\sigma'(t)|t\log t=0$,
\end{itemize}
is called {\it oscillating} or a {\it generalized proximate order}. If, in addition, $m<\lambda\le\rho<m+1$ for some $m\in\N\cup\{0\}$, then $\s$ is called a {\it Boutroux proximate order}.
Such modifications of a proximate order, their properties and applications can be found in \cite[Chap.~2, \S5]{GO}, \cite{Chernolyas}, \cite{Boichuk}, \cite{MalKozSad}.
The notion of an oscillating proximate order was used to generalize the theory of functions of completely regular growth to classes of functions of non-regular growth~\cite{Boichuk,Chernolyas}. The main disadvantage of this approach is the lack of an existence theorem. All aforementioned generalizations can be used provided that the upper limit (2) or the lower limit (3) is finite. The reference \cite{Drasin74} deals with the case when it is not the case.
Finally, if $\rho=\lambda$ then a generalized proximate order coincides with a proximate order.
Our first main result shows that for each non-decreasing function $A$ satisfying $t^\alpha\lesssim A(t)\lesssim t^\beta$ for some $0\le\alpha<\beta<\infty$, there exists a quasi proximate order $\sigma$ such that the function $t^{\sigma(t)}$ is in a sense a smooth pointwise majorant of $A$ and still reflects the behavior of $A$ in a useful and natural way. The precise statement reads as follows.
\begin{theorem}\label{t:valir}
Let $1\le r_0<\infty$ and let $A$ be a positive continuous non-decreasing function on $[r_0,\infty)$ such that
\begin{equation}\label{Eq:lim-sup-inf-conditions in Thm 1}
\limsup_{t\to\infty}\frac{\log A(t)}{\log t}=\rho,
\quad\liminf_{t\to\infty}\frac{\log A(t)}{\log t}=\lambda,
\quad 0\le\lambda <\rho<\infty.
\end{equation}
Then, for each fixed $\eta\in(0,\rho-\lambda)$, there exists a quasi proximate order $\sigma=\sigma_{\r,\lambda,\eta}:[r_0,\infty)\to(0,\infty)$, with the associated function $A^*=A^*_\sigma:[r_0,\infty)\to(0,\infty)$, such that
\begin{itemize}
\item[\rm(1)] $\displaystyle\limsup_{t\to\infty}\sigma(t)=\rho$;
\item[\rm(2)] $\displaystyle\liminf_{t\to\infty}\sigma(t)=\lambda+\eta$;
\item[\rm(3)] $A(t)\le t^{\sigma(t)}$ for all $r_0\le t<\infty$.
\end{itemize}
\end{theorem}
It is tempting to think that in the statement (2) of Theorem~\ref{t:valir} one may replace $\lambda+\eta$ by $\lambda$. However, this is not possible without violating the condition $\limsup_{t\to\infty}|\sigma'(t)|t\log t<\infty$ of a quasi proximate order. In Section~\ref{Sec:example} we construct a non-decreasing function $A$ which shows this last claim. It also follows from the argument used that no generalized proximate order satisfying the limit conditions (2) and (3) of the definition exists for the constructed $A$. The proof of Theorem~\ref{t:valir} is also given in Section~\ref{Sec:example}. The proof itself is a technical handmade construction which relies on the use of suitable auxiliary functions induced by $A$ and inductive defining of $\sigma$ and $A^*$.
Our primary interest in this paper relies on analytic functions in the unit disc $\D=\{z:|z|<1\}$. For this aim we first pull Theorem~\ref{t:valir} to the setting of unit interval $[0,1)$ by the substitution $t=\frac1{1-r}$, and then apply it to the logarithm of a suitable maximum modulus function. To be precise, recall that for an analytic function $f$ in $\D$, the order and the lower order are defined by
$$
\sigma_M (f) = \limsup_{r\to 1^-} \, \frac{\log^+ \log^+ M(r,f)}{\log{\frac{1}{1-r}}}
$$
and
$$
\lambda_M(f) = \liminf_{r\to 1^-} \, \frac{\log^+ \log^+ M(r,f)}{\log{\frac{1}{1-r}}},
$$
respectively. Let now $f$ be an analytic function in $\D$ such that $0\le\lambda_M(f)<\sigma_M(f)<\infty$. Write $A(t)=\log M(1-\frac 1t, f)$ for all $1\le t<\infty$. Then, for given $\eta\in(0,\sigma_M(f)-\lambda_M(f))$, there exist $\lambda$ and its associated function $A^*=A^*_\lambda$ on $[0,1)$ such that
\begin{itemize}
\item [(1)] $\lambda \in C^{1}[0,1)$;
\item [(2)] $\displaystyle\limsup_{r\to1^-} \lambda(t)=\sigma_M(f)$;
\item [(3)] $\displaystyle\liminf_{r\to1^-} \lambda(t)=\lambda_M(f)+\eta$;
\item [(4)] $\displaystyle\limsup_{r\to1^-}|\lambda'(r)|(1-r)\log\frac1{1-r}<\infty$;
\item [(5)] $A^*(r)\le \frac{1}{(1-r)^{\lambda(r)}} \le (1+o(1))A^*(r)$, as $r\to1^-$;
\item [(6)] $A^*$ is nondecreasing and $A^*\left(\frac{1+r}{2}\right)\lesssim A^*(r)$ for all $0\le r<1$;
\item [(7)] $\log M(r,f)\le\frac{1}{(1-r)^{\lambda(r)}}$ for all $0\le r<1$.
\end{itemize}
To see this, it is enough to set $\lambda(r)=\sigma(\frac{1}{1-r})$, where $\sigma$ is that of Theorem~\ref{t:valir} applied to $A$. Then (1)--(6) are immediate while (7) comes from the statement (3) of the said theorem. If the properties (1)--(7) hold, then $\lambda$ is called a {\it quasi proximate order of the analytic function} $f$ in~$\D$, related to $\eta\in(0,\sigma_M(f)-\lambda_M(f))$. Further, if
\begin{itemize}
\item [(4')] $\displaystyle\limsup_{r\to1^-}|\lambda'(r)|(1-r)\log\frac1{1-r}=0$,
\end{itemize}
then $\lambda$ is a {\it generalized proximate order} of $f$.
We now proceed towards the statement of our second main result. Some more definitions are needed. Let
\begin{gather*}
m_p(r,\log|f|)=\left( \frac 1{2\pi} \int_0^{2\pi} |\log|f(re^{i\theta})||^p\, d\theta\right)^{\frac 1p}, \quad 0<r<1,
\end{gather*}
and, by following~\cite{Li_mp}, define
\begin{gather*}
\rho_p(f)=\limsup _{r\to1^-}\frac{\log^+ m_p(r,\log|f|)}{-\log(1-r)}, \quad
\lambda_p(f)=\liminf _{r\to1^-}\frac{\log^+ m_p(r,\log|f|)}{-\log(1-r)}.
\end{gather*}
It is known \cite{Li_mp} that $\rho_p(f)$ is an increasing function of $p$ and $p\rho_p(f)$ is convex on $(0,\infty)$.
Further, define $\rho_\infty(f)=\lim_{p\to\infty}\rho_p(f)$ as in~\cite{rho_infty}. Linden~\cite{Li_mp} proved the identity $\sigma_M(f)=\rho_\infty(f)$ and showed that
\begin{equation}\label{e:linden_ineq}
\rho_p(f)\le \sigma_M(f)\le \rho_p(f)+\frac 1p, \quad 0\le p<\infty,
\end{equation}
provided $\sigma_M(f)\ge 1$. In general, $\sigma_M(f)\le\rho_\infty(f)$ for all analytic functions in $\D$. Observe that the left-hand inequality in \eqref{e:linden_ineq} is no longer true if $\sigma_M(f)<1$. A Blaschke product~$B$ such that $\rho_p(B)=1-\frac 1p$ while $\sigma_M(B)=0$ gives a counter example~\cite{ChIJM}. Further properties of the order $\rho_\infty(f)$ can be found in \cite{rho_infty}.
Our aim is to establish a counterpart of \eqref{e:linden_ineq} for $\lambda_p(f)$ and $\lambda_M(f)$. This is what we obtain from our next result. Recall that the upper density of a measurable set $E\subset[0,1)$ is defined as
$\DU(E)=\limsup_{r\to1^-}\frac{m(E\cap[r,1))}{1-r}$, where $m(F)$ denotes the Lebesgue measure of the set $F$.
\begin{theorem} \label{t:p_lower_order}
Let $f$ be an analytic function in $\D$ such that it admits a generalized proximate order $\lambda$, and either $1<\lambda_M(f)<\sigma_M(f)<\infty$ or $0\le\lambda_M(f)<\sigma_M(f)\le 1$. Let $1\le p<\infty$ and $\varepsilon>0$. Then
\begin{equation}\label{e:mp_final-countdown}
m_p(r,\log|f|)\lesssim\frac{1}{(1-r)^{\lambda(r)\left(1-\frac1{\sigma_M(f)}\right)^++ 1+\varepsilon}}, \quad r\to 1^-.
\end{equation}
In particular, if $\eta\in(0,\sigma_M(f)-\lambda_M(f))$ then
\begin{equation}\label{e:mp_final-countdown2}
m_p(r,\log|f|)\lesssim\frac{1}{(1-r)^{\left(\lambda_M(f)+\eta\right)\left(1-\frac1{\sigma_M(f)}\right)^++ 1+\varepsilon}},\quad r\in E,
\end{equation}
where $E=E_{\varepsilon,\eta}\subset[0,1)$ is of upper density one.
\end{theorem}
The method we employ in the proof does not allow us to treat the case $\lambda_M(f)\le 1<\sigma_M(f)$ and therefore it remains unsettled.
\begin{corollary} \label{c:m_p}
Let $f$ be an analytic function in $\D$ such that it admits a generalized proximate order $\lambda$, and either $1<\lambda_M(f)<\sigma_M(f)<\infty$ or $0\le\lambda_M(f)<\sigma_M(f)\le 1$. Let $\eta\in(0,\sigma_M(f)-\lambda_M(f))$ and $1\le p<\infty$. Then
\begin{equation}\label{pilili}
\lambda_p(f)\le\lambda_M(f)\left(1-\frac1{\sigma_M(f)}\right)^++1, \quad \lambda_M(f)\le \lambda_p(f)+\frac 1p.
\end{equation}
\end{corollary}
Corollary~\ref{c:m_p} complements \eqref{e:linden_ineq}. Indeed, the left-hand inequality in \eqref{e:linden_ineq} is the limit case $\lambda_M(f)=\sigma_M(f)$ of the first inequality in \eqref{pilili} with $\sigma_M(f)\ge1$. However, the sharpness of Corollary~\ref{c:m_p} itself remains unsettled. We just note here that a similar phenomenon appears in estimates of growth for the central index of analytic functions in the unit disc in terms of orders of their maximum term~\cite{Sons68}.
The proof of Theorem~\ref{t:p_lower_order} occupies most of the body of the remaining part of the paper. It follows the scheme of the proof of \cite[Theorem~1]{Li_mp} to some extent but a substantial amount of different arguments is needed. Some of the auxiliary results obtained on the way to the proof are of independent interest. The first step towards to proof of the theorem is to estimate the number of zeros of an analytic function in polar rectangles in terms of its proximate order. Our result in this direction is Proposition~\ref{p:i_est} which is proved in Section~\ref{sec:polar-rectangle}. The second step is to establish an appropriate generalization of the following lemma due to Cartwright~\cite[Lemma~1]{CartWr33}.
\begin{letterlemma}\label{Lemma:B} Let $0<R<\infty$, $0<\alpha<\infty$ and $\frac{\pi}{2\alpha}<\beta<\infty$. Let $F$ be analytic and non-vanishing in the truncated sector $\Omega=\Omega_{R,\beta}=\{re^{i\theta}:R\le r<\infty,\,|\theta|\le \beta\}$ such that
\begin{equation}\label{e:cart_assum}
\log|F(z)|<B|z|^\alpha,\quad z\in\Omega,
\end{equation}
for some constant $B>0$. Then, for given $\delta>0$, there exists a constant $K=K(\delta)>0$ such that
$$
\log|F(re^{i\theta})|>-KBr^\alpha, \quad |\theta|\le \beta-\delta,\quad R\le r<\infty.
$$
\end{letterlemma}
Lemma~\ref{Lemma:B} plays a crucial role in the proof of \cite[Theorem~2]{CartWr33}. Linden~\cite{Li56_conj} generalized Cartwright's lemma to the case in which the power function $|z|^\alpha$ is replaced by $V(|z|)=|z|^{\rho(|z|)}$, where $\rho(r)$ is a proximate order of $F$, and applied it to analytic functions in the unit disc. In \cite{NikNk74} and \cite{Bor_min} the authors considered the question of when the conclusions of the lemma can be strengthened. They showed that indeed a stronger conclusion can be made if certain extra hypothesis on $V$ is imposed.
In this paper we are interested in the question of {\it to what extent the hypotheses of Lemma \ref{Lemma:B} can be relaxed and still have the same lower estimate?} It appears that it is enough to assume that $\rho(r)$ is a generalized proximate order. It is worth underlining here that this allows $f$ to be of non-regular growth. Moreover, it seems that the conclusion is no longer true for a quasi proximate order. Our generalization of Cartwright's lemma reads as follows.
\begin{proposition}\label{cor:cartright} Let $l:[1,\infty)\to(0,\infty)$ be a generalized proximate order such that $0<\liminf_{t\to\infty}l(t)=l_1<\limsup_{t\to\infty}l(t) =l_2<\infty$. Let $\varepsilon>0$, $0<q<1$ and $0<R<\infty$. Let $G$ be analytic and non-vanishing such that
\begin{equation*
\log|G(re^{i\theta})|<r^{\frac{l(r)}{1+\e}}
\end{equation*}
on the domain $\left\{re^{i\theta}:R<r<\infty,\,|\theta|\le\frac{\pi}{2l(r)q}\right\}$. Then, for each $0<\delta<\pi/2l_2q$, we have
\begin{equation*
\log|G(re^{i\theta})|
>-r^{l(r)}, \quad |\theta|\le\frac{\pi}{2l(r)q}-\delta,\quad r\to \infty.
\end{equation*}
\end{proposition}
The proposition allows us to deduce the following result concerning real parts of analytic functions in the unit disc. It is a counterpart of \cite[Theorem 3]{Li56_conj} and of independent interest.
\begin{theorem}\label{t:cartright}
Let $\sigma$ be a generalized proximate order such that $\lambda(r)=\sigma(1/(1-r))$ satisfies $1<\liminf_{r\to1^-}\lambda(r)<\limsup_{r\to1^-}\lambda(r)<\infty$. Let $\varepsilon>0$ and let $f$ be analytic in $\D$ such that $f(0)=0$ and
\begin{equation}\label{Eg:hypothesis-cartright}
\Re f(re^{i\theta})<\frac{1}{(1-r)^{\lambda(r)}},\quad 0\le \theta<2\pi, \quad r_0<r<1,
\end{equation}
for some $r_0\in(0,1)$. Then
$$
|\Re f(re^{i\theta})|<\frac{1}{(1-r)^{(1+\varepsilon)\lambda(r)}},\quad 0\le \theta<2\pi, \quad r\to1^-.
$$
\end{theorem}
Observe that the hypothesis $1<\liminf_{r\to1^-}\lambda(r)$ in Theorem~\ref{t:cartright} is necessary. Namely, if $\limsup_{r\to 1^-} \lambda(r)<1 $ one can only say that $|\Re f(re^{i\theta})|\lesssim\frac{1}{1-r}$ by~\cite{CartWr33}. This upper bound is the best possible as is seen by considering the function $f(z)=1-\frac{1}{(1-z)^\alpha}$ with $\alpha\in(0,1)$.
Proposition~\ref{cor:cartright} and Theorem~\ref{t:cartright} are established in Section~\ref{sec:cartright}. The proofs do not come free but here we only mention one specific tool used which is the Warschawski mapping theorem, stated as Theorem~\ref{t:war}. Finally, in Section~\ref{sec:final} we pull all the discussed things together, and prove Theorem~\ref{t:p_lower_order} and Corollary~\ref{c:m_p}.
To this end, couple of words about the notation used throughout the paper. The letter $C=C(\cdot)$ will denote an absolute constant whose value depends on the parameters indicated
in the parenthesis, and may change from one occurrence to another.
We will use the notation $a\lesssim b$ if there exists a constant
$C=C(\cdot)>0$ such that $a\le Cb$, and $a\gtrsim b$ is understood
in an analogous manner. In particular, if $a\lesssim b$ and
$a\gtrsim b$, then we write $a\asymp b$ and say that $a$ and $b$ are comparable. Moreover, the notation $a(t)\sim b(t)$ means that the quotient $a(t)/b(t)$ approaches one as $t$ tends to its limit.
\section{Existence of quasi proximate order}\label{Sec:example}
In this section we prove Theorem~\ref{t:valir} and discuss the necessity of its hypotheses. We begin with the proof of the theorem.
\begin{proof}[Proof of Theorem \ref{t:valir}]
Define $d(t)=\frac{\log^+A(t)}{\log t}$ for all $t\in[\max\{r_0, e\},\infty)$. Let $(\e_n)_{n=1}^\infty$ be a strictly decreasing sequence of strictly positive numbers such that $\varepsilon_1<\min\{1,\eta\}$ and $\lim_{n\to\infty}\varepsilon_{n}=0$. The continuity of $A$ together with \eqref{Eq:lim-sup-inf-conditions in Thm 1} allows us to pick up the infinite sequences $(r_n)_{n\in\N}$, $(r_n')_{n\in\N}$ and $(r_n^*)_{n\in\N}$ such that
\begin{itemize}
\item[\rm(i)] $r_n<r_n^*<r_n'<r_{n+1}$ for all $n\in\N$;
\item[\rm(ii)] $d(r_n')=\lambda+\frac\eta2$ for all $n\in\N$;
\item[\rm(iii)] $d(r_1)=\rho -\varepsilon_{1}$ and $r_{n+1}=\min\{ r\ge r_n': d(r)=\rho -\varepsilon_{n+1}\}$ for all $n\in\N$;
\item[\rm(iv)] $(r_n^*)^{\lambda+\eta}=(r_n')^{\lambda+\frac\eta2}$ for all $n\in\N$.
\end{itemize}
The existence of such sequences can be seen, for example, by arguing as follows. First, consider a sequence $(r_n')_{n\in\N}$ consisting of infinitely many points satisfying (ii). Then define $(r_n^*)_{n\in\N}$ by (iv), and by passing to suitable subsequences if necessary we have $r_n'<r_{n+1}^*<r_{n+1}'$. By defining $(r_n)_{n\in\N}$ by (iii) we now have $r_n'<r_{n+1}$, and by passing once more to subsequences we obtain (i). Observe that the value of $d$ at the points $r_n$ and $r_n'$ is known precisely by (ii) and (iii), while (i), (ii) and (iv) together with the monotonicity of $A$ yield the estimate
\begin{equation}\label{Eq:estimate d(r_n^*)}
d(r_n^*)=\frac{\log^+A(r_n^*)}{\log r_n^*}
\le\frac{\lambda+\eta}{\lambda+\frac{\eta}{2}}\frac{\log^+A(r_n')}{\log r_n'}
=\frac{\lambda+\eta}{\lambda+\frac{\eta}{2}}d(r_n')=\lambda+\eta,\quad n\in\N.
\end{equation}
Let now $E_1=\{r\in[r_0, r_1^*]:d(r)\ge\rho-\varepsilon_1\}$ and
$$
E_n=\{r\in[r_n, r_n^*]: d(r)\ge \rho-\varepsilon_n\}, \quad n\in\N\setminus\{1\}.
$$
Then each $E_n$ is non-empty by the property (iii). Write $M_n=\max\{d(r):r\in E_n\}=\max\{d(r):r\in[r_n, r_n^*]\}$, and let $R_n\in E_n$ such that $d(R_n)=M_n$. Then $M_n\to\rho$, as $n\to\infty$, and, by fixing $r_1$ sufficiently large, $\rho-\varepsilon_n\le M_n\le\rho+\varepsilon_1$ for all $n\in\N\setminus\{1\}$ by \eqref{Eq:lim-sup-inf-conditions in Thm 1}. Define
\begin{equation}\label{Eq:(a)}
\s(t)=\left\{
\begin{array}{ll}
M_1,&\quad t\in[r_0,R_1]\\
M_n,& \quad t\in[r_n,R_n],\quad n\in\N\setminus\{1\},
\end{array}\right.
\end{equation}
and set
$$
A^*(t)=t^{\sigma(t)},\quad t\in[r_0,R_1]\cup\Bigl(\bigcup_{n\in\N\setminus\{1\}}[r_n,R_n]\Bigr).
$$
Then $A^*$ is trivially nondecreasing and continuous on each $[r_n,R_n]$ and on $[r_0,R_1]$. Since $A$ is nondecreasing by the hypothesis, the definition of $d$ and the property (iv) imply
$$
A(t)\le A(r_n')=(r_n')^{\lambda+\frac\eta2}=(r_n^*)^{\lambda+\eta}\le t^{\lambda+\eta},\quad t\in [r_n^*, r_n'],\quad n\in\N,
$$
and hence $d(t)\le\lambda+\eta$ for all $t\in [r_n^*,r_n']$ and $n\in\N$. This extends \eqref{Eq:estimate d(r_n^*)}.
Let
$$
D(t)=\max \{ \max\{ d(r): t\le r\le r_n^*\}, \lambda+\eta\},\quad t\in [R_n, r_n^*],\quad n\in\N\setminus\{1\}.
$$
Then $D(t)$ is nonincreasing, and $D(r_n^*)=\max\{d(r_n^*),\lambda+\eta\}=\lambda+\eta$. Define
$$
A^*(t)=t^{D(t)},\quad t\in[R_n,r_n^*],\quad n\in\N\setminus\{1\}.
$$
Then $A^*$ is continuous at each $R_n$, and nondecreasing on each $[R_n,r_n^*]$. Namely, $[R_n, r_n^*]=B\sqcup C$, where $B=\{t\in [R_n,r_n^*]:d(t)=D(t)\}$ is a closed set and $C$ is a union of open intervals $\{\Delta_{k,n}\}_{k}$. By the definition of $D$, we have $A^*(t)=e^{\log^+A(t)}$ on $B$, and $A^*(t)=t^{\lambda+\eta}$ on $C$. The monotonicity of $A^*$ on each $[R_n,r_n^*]$ now follows from that of $A$ and the definition of~$D$.
We next define $\sigma$ on $[R_n,r_n^*]$. In order to do so, denote $u_{0,n}=R_n$ and $u_{0,n}^*=\max\{t\in[R_n,r_n^*]:D(t)=D(R_n)\}$, and define
$$
t_{1,n}=\min\{ t\in \mathbb{N}: t\ge u_{0,n}+1, D(t)<D(u_{0,n})=D(R_n)\}.
$$
Then obviously $t_{1,n}\le u_{0,n}^*+2$. We now define
$$
\sigma(t)=\sigma(u_{0,
n})=M_n,\quad R_n=u_{0,n}\le t\le t_{1,n}.
$$
Let
$$
u_{1,n}=\min\{ t> t_{1,n}:D(t)=y_{1,n}(t)\}
$$
be the abscissa of the point in which the graphs $y=D(t)$ and
$$
y=y_{1,n}(t)=M_n-(\rho+1)\log\log t+(\rho+1)\log\log t_{1,n}
$$
intersect. Note that $t^{y_{1,n}(t)}$ is decreasing on $[t_{1,n},r_n^*]$, because
\begin{equation*}
\begin{split}
(y_{n,1}(t)\log t)'
&=\frac 1t\left(-(\rho+1)+M_n-(\rho+1)(\log\log t-\log\log t_{1,n})\right)\\
&<\frac 1t (-\rho-1+M_n)<0.
\end{split}
\end{equation*}
Since $D(t)\log t=\log A^*(t)$, and $A^*$ is nondecreasing and unbounded, the point $u_{1,n}$ exists and is unique. We set
\begin{equation}\label{Eq:(d)}
\sigma(t)= \sigma (t_{1,n})-(\rho+1) \log\log t+ (\rho+1) \log\log t_{1,n}, \quad t_{1,n} \le t\le u_{1,n}.
\end{equation}
Let $u_{1,n}^*=\max \{t\in[u_{1,n},r_n^*]:D(t)=D(u_{1,n})\}$, and then choose
$$
t_{2,n}=\min\{t\in \mathbb{N}: t\ge u_{1,n}+1, D(t)<D(u_{1,n})\}.
$$
Continue the process as above until the function $\s$ is defined on the whole interval $[R_n, r_n^*]$. Since $u_{k+1, n}\ge t_{k+1,n}\ge u_{k,n}+1$, the process finishes after a finite number of steps.
We will show next that
\begin{equation}\label{e:a_astar_com}
A^*(t)\le t^{\sigma(t)} \le (1+o(1))A^*(t), \quad t\in [R_n, r_n^*],\quad n\to\infty.
\end{equation}
The left-hand inequality follows immediately from the construction. Moreover, $A^*(t)=t^{D(t)}=t^{\sigma(t)}$ for each $t\in[u_{k,n}, u_{k,n}^*]$. Note that $t_{k+1,n}\le u_{k,n}^*+2$ for each $k$ by the definition of $t_{k+1,n}$. Further, for $t\in[u_{k,n}^*,t_{k+1,n}]$ we have
\begin{equation*}
\begin{split}
t^{\sigma(t)}
&=t^{D(u_{k,n}^*)}
\le (u_{k,n}^*+2)^{D(u_{k,n}^*)}
=(u_{k,n}^*)^{D(u_{k,n}^*)}\left(1 +\frac 2 {u_{k,n}^*}\right)^{D(u_{k,n}^*)}\\
\le A^*(t)9^{\frac{D(u_{k,n}^*)}{u_{k,n}^*}}\le A^*(t)9^{\frac{\r+1}{R_n}}
\le A^*(t)\left(1+\frac{9^{\r+1}}{R_n}\right),\quad n\in\N\setminus\{1\}.
\end{split}
\end{equation*}
Since $t^{\sigma(t)}$ is decreasing for $t\in [t_{k+1,n}, u_{k+1,n}]$ by its definition, we obtain
$$
t^{\sigma(t)}
\le t_{k+1,n}^{\sigma(t_{k+1,n})}
\le A^*(t_{k+1,n})\left(1+\frac{9^{\r+1}}{R_n}\right)
\le A^*(t)\left(1+\frac{9^{\r+1}}{R_n}\right).
$$
Therefore \eqref{e:a_astar_com} is proved.
It follows from \eqref{e:a_astar_com} that
$$
(r_n^*)^{\lambda+\eta}=A^*(r_n^*)\le (r_n^*)^{\sigma(r_n^*)} =(1+\delta_n) (r_n^*)^{\lambda+\eta},
$$
where $0\le\delta_n\le\frac{9^{\r+1}}{R_n}$. Hence
\begin{equation}\label{Eq:(b)}
\lambda+\eta
\le\sigma(r_n^*)
\le\lambda+\eta +\frac{9^{\rho+1}}{R_n\log R_n}.
\end{equation}
We then define
\begin{equation}\label{e:sigma_new}
\sigma(t)=\sigma(r_n^*)+ C_n(\log\log t-\log\log r_n^*), \quad r_n^*\le t\le r_n',
\end{equation}
where
$$
C_n=\frac{M_{n+1}-\sigma(r_n^*)}{\log\frac{\lambda+\eta}{\lambda+\frac \eta2}}
\le \frac{\rho+\varepsilon_1-\lambda-\eta}{\log\frac{\lambda+\eta}{\lambda+\frac \eta2}}.
$$
Since $(\lambda+\eta)\log r_n^*=(\lambda+\frac \eta2)\log r_n'$ by (iv), this yields $\sigma(r_n')=M_{n+1}$.
We then define
\begin{equation}\label{Eq:(c)}
\sigma(t)=M_{n+1},\quad r_n'\le t\le r_{n+1},
\end{equation}
and $A^*(t)=t^{\sigma(t)}$ for all $t\in [r_n^*, r_{n+1}]$. Then we
make the next step.
The statement (1) follows from the construction, see, in particular, \eqref{Eq:(a)}, the definition of $M_n$ and the hypothesis \eqref{Eq:lim-sup-inf-conditions in Thm 1}. The statement (2) is a consequence of the construction and \eqref{Eq:(b)}.
Statement (3) is obvious on each $[r_n,R_n]$ by the definition of $M_n$. On $[R_n,r_n^*]$ we have
$$
A(t)\le t^{d(t)}\le t^{D(t)}=A^*(t)\le t^{\s(t)}
$$
by \eqref{e:a_astar_com}. On the remaining interval $[r_n^*,r_{n+1}]$ we have (3) by the construction, see \eqref{e:sigma_new} and \eqref{Eq:(c)}, and the hypothesis on the monotonicity of $A$.
It remains to show that $\s$ is a proximate order and $A^*$ satisfies the properties of an associated function. We have already seen that $\s$ satisfies the property (2) and also (3), with $\lambda+\eta$ in place of $\lambda$. Moreover, (5) is satisfied by \eqref{e:a_astar_com} because $A^*(t)=t^{\s(t)}$ for all $t\in[r_0,\infty)\setminus\bigcup_{n\in\N\setminus\{1\}}[R_n,r_n^*]$. Furthermore, by \eqref{Eq:(d)} and \eqref{Eq:(c)}, we also have $|\sigma'(t)|\le\frac{K}{t\log t}$, where $K=\max\{\sup_n C_n, \rho+1\}$, except for countably many points, where the derivative does not exist. We can slightly modify the continuous function $\sigma$ in small neighborhoods of these points without changing the other properties to get (1) and (4). The associated function $A^*$ is continuous and increasing by the construction.
Theorem~\ref{t:valir} is now proved.
\end{proof}
The following example shows that in the statement of Theorem~\ref{t:valir} the parameter $\eta$ must be strictly positive. Observe also that no generalized proximate order satisfying the limit conditions (2) and (3) of the definition exists for the constructed $A$.
\begin{example}
Let $0<\lambda<\rho<\infty$, $r_1=2$ and $r_{n+1}=r_n ^{\frac{\rho}{\lambda}}$ for all $n\in\mathbb{N}$. Define
$$
A(r)=r_n^\rho,\quad r_n<r\le r_{n+1},\quad n\in\mathbb{N}.
$$
Then
$$
A(r_n^+)=r_n^\rho,\quad A(r_{n+1})=r_n^\r=r_{n+1}^\lambda, \quad n\in\N,
$$
and
\begin{equation}\label{Eq:example}
r^\lambda\le A(r)<r^\rho,\quad r\ge r_1=2.
\end{equation}
By redefining the step function $A$ on small intervals $[r_{n+1}-\e_n,r_{n+1}]\subset(r_n,r_{n+1}]$ such that its graph coincides with the line segment joining the points $(r_{n+1}-\e_n,r_n^\r)$ and $(r_{n+1},r_{n+1}^\r)$, it remains non-decreasing, becomes continuous and satisfies
\begin{equation}\label{Eq:example-modified}
r^\lambda< A(r)\le r^\rho,\quad r\ge r_1=2,
\end{equation}
instead of \eqref{Eq:example}, and $A(r_n)=r_n^\rho\le A(r)\le r_{n+1}^\rho$ for all $r\in[r_n,r_{n+1}]$ and $n\in\N$.
Suppose that $\sigma\colon [2,\infty)\to (0,\infty)$ is a quasi proximate order of $A$ such that
$$
\limsup_{r\to \infty} \sigma(r)=\rho, \quad \liminf_{r\to \infty} \sigma(r)=\lambda.
$$
Then there exist infinite sequences $(n_k)_{k\in\N}$ and $(r_{k}')_{k\in\N}$, and a decreasing function $\eta\colon[2,\infty)\to (0,1)$ such that $\lim_{r\to\infty}\eta(r)=0$, $r_{k}'\in(r_{n_k},r_{n_k+1})$ for all $k\in\N$, and
\begin{equation}\label{Ex:2}
\left(r_k'\right)^{\sigma(r_k')}\le\left(r_k'\right)^{\lambda+\eta(r_k')},\quad k\in\N.
\end{equation}
Denote $$C_k=\sup_{[r_{k}', r_{n_k+1}]}|\sigma'(r)|r\log r$$ for all $k\in\N$. By the property (3) of Theorem~\ref{t:valir} we deduce $r_{n}^\r=A(r_{n})\le r_{n}^{\sigma(r_{n})}$, and thus $\rho\le\sigma(r_n)$ for all $n\in\N$. By (3) we also deduce $r_{n_k}^\r\le A(r_k')\le(r_k')^{\s(r_k')}$ for all $k\in\N$. These observations together with \eqref{Ex:2} and the definition of the sequence $(r_n)_{n\in\N}$ yield
\begin{equation*}
\begin{split}
\rho-(\lambda+\eta(r_{k}'))
&\le \sigma(r_{n_k+1})-\sigma (r_k')
\le \int_{r_k'}^{r_{n_k+1}} \frac{C_k}{r\log r} \, dr\\
=C_k \log \frac{ \log r^\lambda _{n_k+1}}{\lambda \log r_k'}
=C_k \log \frac{ \log r^\r _{n_k}}{\lambda \log r_k'}\\
&\le C_k \log \frac{ \log (r' _{k})^{\lambda+\eta(r_k')}}{\lambda \log r_k'}=C_k \log \frac{\lambda+\eta(r_k')}{\lambda}.
\end{split}
\end{equation*}
It follows that
$$
C_k\ge\frac{\rho-\lambda-\eta(r_k')}{\log \frac{\lambda+\eta(r_k')}{\lambda}},\quad k\in\N,
$$
and, consequently, $\limsup_{r\to\infty}|\sigma'(r)|r\log r=\infty$.
\end{example}
\section{Estimates for the number of zeros in polar rectangles}\label{sec:polar-rectangle}
Let $\{a_k\}$ denote the sequence of zeros of $f$, listed according to their multiplicities and ordered by the increasing moduli. Denote
$$
n_1(r,f)=\max_\varphi \# \left\{a_k:r\le\left|a_k\right|\le\frac{1+r}{2},\,\left|\arg a_k-\varphi\right|\le\frac{\pi}{4}(1-r)\right\},\quad 0\le r<1.
$$
The aim of this section is to establish the following sharp estimate for $n_1(r,f)$ in terms of a quasi proximate order of $f$.
\begin{proposition}\label{p:i_est
Let $f$ be an analytic function in $\D$ such that $0\le \lambda_M(f)<\sigma_M(f)<\infty$. Let $\lambda$ be a quasi proximate order of $f$, related to $\eta\in(0,\sigma_M(f)-\lambda_M(f))$, and $\varepsilon>0$. Then
\begin{equation}\label{e:n_est0}
n_1(R, f) \lesssim \frac{1}{(1-R)^{1+\lambda(R)\left(1-\frac{1}{\sigma_M(f)}\right)^+ +\varepsilon}},\quad R\to1^-.
\end{equation}
In particular,
\begin{equation}\label{e:n_est}
n_1(R,f)\lesssim \frac{1}{(1-R)^{1+\left(\lambda_M(f)+\eta\right)\left(1-\frac{1}{\sigma_M(f)}\right)^++\varepsilon}},\quad R\in E,
\end{equation}
where $E\subset[0,1)$ satisfies $\DU(E)=1$.
\end{proposition}
The first step towards Proposition~\ref{p:i_est} is to apply \cite[Theorem~2]{Linden1956}. Namely, if $n(\zeta,h,f)$ denotes the number of zeros of an analytic function $f$ in the closed disc $\overline{D}(\zeta,h)=\{w:|\zeta-w|\le h\}$, then by replacing $z$ by $Rz$ in the said theorem we obtain the following result.
\begin{lettertheorem}\label{TheoremLinden2}
Let $f$ be an analytic function in $\D$ such that $f(0)=1$. Then, for $\a\in[1/2,1)$ and $\widetilde\eta\in(0,1/6)$ there exist constants $R_0=R_0(\a)\in(0,1)$ and $C=C(\a,\widetilde\eta)$ such that
\begin{equation}\label{e:n}
n(\zeta,h,f)\le\frac{C}{(R-r)^\frac1\a}\left(\int_0^R\log^+M(t,f)(R-t)^{\frac1\a-1}\,dt+\log^+M(R_0,f)\right),
\end{equation}
where $|\zeta|=r<R<1$ and $h=\widetilde\eta(R-r)/R$.
\end{lettertheorem}
In view of this theorem the proof of Proposition~\ref{p:i_est} boils down to estimating the integral appearing on the right-hand side of \eqref{e:n}. This is done in the following lemma.
\begin{lemma}\label{l:i_est}
Let $f$ be an analytic function in $\D$ such that $0\le \lambda_M(f)<\sigma_M(f)<\infty $.
For $1/2\le\alpha<1$ and $0\le R_0<1$, define $I_\alpha:[R_0,1)\to[0,\infty)$ by
\begin{equation}\label{e:ir_def}
I_\alpha(R)=\frac{1}{(1-R)^{\frac1\a}}\left(\int_0^R\log^+M(t,f)(R-t)^{\frac1\a-1}\,dt+\log^+M(R_0,f)\right).
\end{equation}
Let $\lambda$ be a quasi proximate order of $f$ related to $\eta\in(0,\sigma_M(f)-\lambda_M(f))$. Then
\begin{equation}\label{e:ir_est0}
I_\alpha(R)\lesssim\frac{1}{(1-R)^{1+\lambda(R)\left(1-\frac{1}{\sigma_M(f)}\right)^+ +\varepsilon}},\quad R\to1^-,
\end{equation}
provided one of the following two conditions is satisfied:
\begin{itemize}
\item[\rm(i)] $0\le\sigma_M(f)<1$ and $\e>0$;
\item[\rm(ii)] $1\le\sigma_M(f)<\infty$ and $1/\a<1+\frac{\varepsilon}2$.
\end{itemize}
In particular,
\begin{equation}\label{e:ir_est}
I_\alpha(R)\lesssim\frac{1}{(1-R)^{1+(\lambda_M(f)+\eta)\left(1-\frac{1}{\sigma_M(f)}\right)^+ +\varepsilon}},\quad R\in E,
\end{equation}
where $E=E_{\varepsilon,\eta}\subset[0,1)$ satisfies $\DU(E)=1$.
\end{lemma}
\begin{proof
By Theorem~\ref{t:valir} there exists a quasi proximate order $\lambda$ of $f$, related to $\eta\in(0,\sigma_M(f)-\lambda_M(f))$, satisfying the properties (1)--(7).
In view of (7) the problem boils down to estimating the quantity
\begin{equation}\label{e:j_def}
J(R)=\frac{1}{(1-R)^{\frac1\a}}\left(\int_0^R \frac{(R-t)^{\frac1\a-1}}{(1-t)^{\lambda(t)}}\,dt+1\right).
\end{equation}
(i) Assume that $0\le\sigma_M(f)<1$ and, without loss of generality, pick up $\e>0$ such that $\sigma_M(f)+\varepsilon<1/\a$. Fix $R_0\in(0,1)$ such that $\lambda(r)\le \sigma_M(f)+\varepsilon$ for all $r\in[R_0,1)$, and let $R\in(R_0,1)$. Then
\begin{equation}\label{pervo1}
J(R)\lesssim\frac 1{(1-R)^{\frac1\a}} \int_0^R \frac{dt}{(1-t)^{\sigma_M(f)+\varepsilon+1-\frac 1\alpha}}
\lesssim \frac 1{(1-R)^{\frac1\a}} \lesssim \frac 1{(1-R)^{1+\varepsilon}},
\end{equation}
and thus \eqref{e:ir_est0} is proved.
(ii) Assume next that $1\le\sigma_M(f)<\infty$ and $\varepsilon<1/\a<1+\frac{\varepsilon}2$. Fix $R_0\in(0,1)$ such that $\lambda(r)\le \sigma_M(f)+\frac\varepsilon2$ for all $r\in[R_0,1)$, and let $R\in(R_0,1)$. If $\lambda(R)\ge\sigma_M(f)$ the required estimate follows from standard arguments applied directly to \eqref{e:ir_def}. Assume that $\lambda(R)<\sigma_M(f)$, and define $R^*\in(0,R)$ by the condition $(1-R^*)^{\sigma_M(f)}=(1-R)^{\lambda(R)}$. Then (5), (6), the inequality $1/\a<1+\frac{\varepsilon}2$, and the definition of $R^*$ yield
\begin{equation*}
\begin{split}
J(R)&=(1-R)^{-\frac1{\alpha}} \left(\int_0^{R^*}+\int_{R^*}^R\right)\frac{(R-t)^{\frac1\a-1}}{(1-t)^{\lambda(t)}}\,dt\\
&\lesssim(1-R)^{-\frac1{\alpha}} \int_0^{R^*}\frac{(R-t)^{\frac1\alpha-1}}{(1-t)^{\sigma_M(f)+\varepsilon/2}}\,dt
+(1-R)^{-\frac1{\alpha}} \int_{R^*}^R A^*(t) (R-t)^{\frac1\alpha-1}\,dt\\
&\le(1-R)^{-\frac1{\alpha}} \int_0^{R^*} \frac1{(1-t)^{\sigma_M(f)+\varepsilon/2-\frac 1\alpha +1}}\, dt+ \alpha (1-R)^{-\frac1{\alpha}} {A^*(R) (R-R^*)^{\frac1\alpha}}\\
&\lesssim \frac{{(1-R)^{-\frac1{\alpha}}}}{(1-R^*)^{\sigma_M(f)+\varepsilon/2 -\frac 1\alpha}}
+\frac{(1-R^*)^{\frac1\alpha}}{(1-R)^{\lambda(R)+\frac 1\alpha }}\\
&=\frac1{(1-R)^{\frac1{\alpha}+\frac{\lambda(R)}{\sigma_M(f)}({\sigma_M(f)+\varepsilon/2 -\frac 1\alpha}) }}
+\frac{1}{(1-R)^{\lambda(R)+\frac 1 \alpha -\frac{\lambda(R)}{\sigma\alpha} }}\\
&\lesssim\frac1{(1-R)^{\frac1{\alpha}+\lambda(R)+ \varepsilon/2 -\frac{\lambda(R)}{\sigma_M(f)\alpha}}}
\lesssim\frac1{(1-R)^{1+\lambda(R)-\frac{\lambda(R)}{\sigma_M(f)}+\varepsilon}}.
\end{split}
\end{equation*}
Therefore \eqref{e:ir_est0} holds in this case also.
It remains to prove \eqref{e:ir_est}. In the case $\sigma_M(f)\le1$ this follows with $E=[R_0,1)$ from \eqref{pervo1}. So assume that $\sigma_M(f)>1$. Pick up an increasing sequence $(R_n)_{n\in\N}$ such that $\lambda(R_n)=\lambda_M(f)+\eta+\frac\varepsilon2$ for all $n\in\N$ and $\lim_{n\to\infty}R_n=1$. Then define $(R^*_n)$ by $(1-R^*_n)^{\lambda_M(f)+\eta+2\varepsilon}=(1-R_n)^{\lambda_M(f)+\eta +\frac {3\varepsilon}2}$ for all $n\in\N$, and set $E=\bigcup_{n=1}^\infty[R_n^*,R_n]$. The properties (5) and (6) imply
$$
\frac1{(1-R)^{\lambda(R)}}\lesssim A^*(R)\le A^*(R_n)\le\frac1{(1-R_n)^{\lambda(R_n)}},\quad R\in [R_n^*,R_n],\quad n\in\N,
$$
and hence \eqref{e:ir_est0}, the definitions of $(R_n)$ and $(R^*_n)$ yield
\begin{equation*}
\begin{split}
J(R)&\lesssim\frac1{(1-R)^{1+\lambda(R)\left(1-\frac{1}{\sigma_M(f)}\right)+\varepsilon}}
=\frac1{(1-R)^{1+\frac \varepsilon{\sigma_M(f)}}}\cdot\frac1{(1-R)^{(\lambda(R)+\varepsilon)\left(1-\frac{1}{\sigma_M(f)}\right)}}\\
&\lesssim\frac1{(1-R)^{1+\frac\varepsilon{\sigma_M(f)}}}
\cdot\frac1{(1-R_n)^{(\lambda(R_n)+\varepsilon)\left(1-\frac{1}{\sigma_M(f)}\right)}}\\
&=\frac1{(1-R)^{1+\frac\varepsilon{\sigma_M(f)}}}\cdot\frac1{(1-R_n)^{(\lambda_M(f)+\eta+\frac{3\varepsilon}2)\left(1-\frac{1}{\sigma_M(f)}\right)}}\\
&=\frac1{(1-R)^{1+\frac\varepsilon{\sigma_M(f)}}}\cdot\frac1{(1-R_n^*)^{(\lambda_M(f)+\eta+2\varepsilon)\left(1-\frac{1}{\sigma_M(f)}\right)}}\\
&\le
\frac1{(1-R)^{1+ (\lambda_M(f)+\eta)\left(1-\frac{1}{\sigma_M(f)}\right)+2\varepsilon-\frac{\varepsilon}{\sigma_M(f)}}},\quad R\in [R_n^*,R_n],\quad n\in\N.
\end{split}
\end{equation*}
By re-defining $\e$, this yields \eqref{e:ir_est} on $E$. Finally, since $R_n^*\to1^-$ as $n\to\infty$, we have
$$
1\ge\DU(E)\ge\lim_{n\to\infty}\frac{R_n-R_n^*}{1-R_n^*}=1-\lim_{n\to\infty}(1-R_n^*)^{\frac{\varepsilon}{2(\lambda+\eta+\frac{3\varepsilon}{2})}}=1.
$$
This finishes the proof of the lemma.
\end{proof}
With these preparations we can prove the proposition we are after in this section.
\begin{proof}[Proof of Proposition~\ref{p:i_est}] For simplicity, we assume that $f(0)=1$. The general case is an easy modification.
Let $\varepsilon>0$ be given, and fix $\alpha\in[1/2,1)$ and $\eta\in(0,\sigma_M(f)-\lambda_M(f))$ as in Lemma~\ref{l:i_est}. Further, let $(R_n)_{n\in\N}$ be the sequence appearing in the proof of the said lemma. Let $r_\nu=1-2^{-\nu}$ for all $\nu\in\N$, and let $\nu$ be sufficiently large such that $r_\nu\ge R_0$, where $R_0=R_0(\alpha)\in(0,1)$ is the constant from Theorem~\ref{TheoremLinden2}.
Let $r\in[r_\nu,r_{\nu+1})$ and $R=\frac{2r}{1+r}\in(r_{\nu},r_{\nu+2})$. Then the estimate \eqref{e:n} yields
\begin{equation}\label{e:nglob}
n(\zeta,\tilde\eta(1-r)/2,f)
\lesssim I_\alpha\left(\frac{2r}{1+r}\right),
\end{equation}
where $|\zeta|=r$ and $0<\widetilde\eta<1/6$. An application of this estimate together with Lemma~\ref{l:i_est} gives the assertions.
\end{proof}
\section{Generalization of Cartwright's lemma and its application}\label{sec:cartright}
In this section we prove Proposition~\ref{cor:cartright} and establish Theorem~\ref{t:cartright} as its consequence. Both results are stated in the introduction. The proposition generalizes the Cartwright's lemma, see Lemma~\ref{Lemma:B}, while the theorem concerns estimates of real parts of analytic functions. To establish our results we will need the following Warschawski mapping theorem. This theorem was efficiently used in the earlier work~\cite{Bor_min} on the topic.
\begin{lettertheorem}[{\cite[Sec.~7]{War41}}]\label{t:war}
Let $\omega:[0,\infty)\to(0,\infty)$ be a continuously differentiable function such that $\int_0^\infty\frac{(\omega'(t))^2}{\omega(t)}\,dt<\infty$ and $\lim_{t\to\infty}\omega'(t)=0$. Let $\zeta$ be a conformal map of the curvilinear strip $\{u+iv:|v|\le\omega(u)\}$ onto the strip ${\mathcal K}=\{z:|\Im z|\le\pi/2\}$ such that $\zeta(u+iv)\to\pm \infty$ as $u\to\pm \infty$. Then there exists a constant $k=k(\omega)\in\mathbb{R}$ such that
$$
{\zeta}(u+iv)=k+\frac{\pi}2\int_0^u\frac{dt}{\omega(t)}+
i\frac{\pi v}
{2\omega(u)}+o(1), \quad u\to\infty.
$$
\end{lettertheorem}
Proposition~\ref{cor:cartright} is a consequence of the following generalization of Cartwright's lemma.
\begin{lemma}\label{l:cartwr}
Let $l:[1,\infty)\to(0,\infty)$ be a quasi proximate order such that $$0<\liminf_{t\to\infty}l(t)=l_1<\limsup_{t\to\infty}l(t) =l_2<\infty.$$ Define $L(r)=\int_1^r \frac{l(s)}{s}\,ds /\log r$ for all $1\le r<\infty$.
Let $\varepsilon>0$, $0<q<1$ and $0<R<\infty$. Let $G$ be analytic and non-vanishing such that
\begin{equation}\label{hypothesis1}
\log|G(re^{i\theta})|<r^{{L(r)}}
\end{equation}
on the domain $\left\{re^{i\theta}:R\le r<\infty,\,|\theta|\le\frac{\pi}{2l(r)q}\right\}$. Then, for each fixed $\delta>0$ there exists $K=K(\delta, l, q)>0$ such that we have
\begin{equation}\label{e:lower_cart_ets}
\log|G(re^{i\theta})|
>-Kr^{L(r)}, \quad |\theta|\le\frac{\pi}{2l(r)q}-\delta,\quad r\to \infty.
\end{equation}
\end{lemma}
\begin{proof}
Let $0<q<1$ and define $\omega(t)=\frac{\pi}{2l(e^t)q}$ for all $0\le t<\infty$. Since
\begin{equation} \label{e:prox_ord_prop}
\limsup_{t\to\infty}|l'(t)|t\log t<\infty
\end{equation}
and $0<\liminf_{t\to\infty}l(t)=l_1<\limsup_{t\to\infty}l(t) =l_2<\infty$ by the hypotheses, we have
$$
|\omega'(t)|=\frac{\pi|l'(e^t)|e^t}{2l^2(e^t)}\lesssim\frac1t, \quad t\to \infty.
$$
Therefore, for sufficiently large $R>0$ we have
\begin{equation*}
\int_R^\infty\frac{(\omega'(t))^2}{\omega(t)}\,dt
\lesssim\int_R^\infty\frac{dt}{t^2}<\infty.
\end{equation*}
By re-defining $\om$ on $[0,R)$ in a suitable way if necessary we deduce that it satisfies the hypotheses of Theorem~\ref{t:war}. Since the statement of the lemma concerns the behavior of $G$ far a way from the origin, this re-definition does not affect to the proof.
For $\alpha>0$, define
$$
S_q(\alpha)=\left\{w=\rho e^{i\varphi}: \rho>0,\,|\varphi| \le \frac{\pi } {2\alpha q}\right\},$$
$$ \widetilde S_q=\left\{z=re^{i\theta}:r>0,\,|\theta|\le\frac{\pi} {2l(r)q}\right\},
$$
and
$$
\log\widetilde S_q=\Bigl\{u+iv: -\infty< u<+\infty,\, |v|\le\om(u)=\frac{\pi}{2l(e^u)q}\Bigr\}.
$$
Let now $\zeta$ be a conformal map from $\log\widetilde S_q$ onto $\mathcal{K}$ such that $\zeta(z)\to\pm \infty$ as $z\to\pm \infty$. Consider the functions
$$
\psi\colon S_q(\alpha)\to \widetilde S_q,
\quad z=\psi(w)=\exp \left( \zeta^{-1}(\alpha q\log w)\right),
$$
and
$$
\psi^{-1}\colon\widetilde S_q \to S_q(\alpha),
\quad w=\psi^{-1}(z)=\exp\left(\frac1{\alpha q}\zeta(\log z)\right),
$$
where $\log$ denotes the principal branch of the logarithm. Define $F=G\circ\psi$ on $S_q(\alpha)$. By the hypothesis \eqref{hypothesis1} we have
\begin{equation}\label{e:Gupp}
\log|F(w)|=\log|G(\psi(w))|<|\psi(w)|^{{L(|\psi(w|)}},\quad|\psi(w)|\ge R.
\end{equation}
In order to apply Lemma~\ref{Lemma:B} we have to verify \eqref{e:cart_assum} for $F$. In view of \eqref{e:Gupp} it is enough to show $|\psi(w)|^{{L(|\psi(w)|)}}<B|w|^\alpha$, or equivalently,
\begin{equation}\label{e:dag}
r^{{L(r)}}\le B|\psi^{-1}(re^{i \theta})|^\alpha.
\end{equation}
By Theorem~\ref{t:war} there exists $k\in \mathbb{R}$ such that
\begin{equation}\label{vehe}
\begin{split}
{\zeta}(\log r +i\theta)
&=k+\frac{\pi}2\int_0^{\log r}\frac{dt}{\omega(t)}+i\frac{\pi \theta}{2\omega(\log r)}+o(1),\quad r\to\infty,\\
&=k+q \int_0^{\log r} l(e^t)\,dt + i q\theta l(r) +o(1),\quad r\to\infty,\\
&=k+q\int_1^r \frac{l(s)}s\,ds +iq\theta l(r) +o(1),\quad r\to\infty,
\end{split}
\end{equation}
provided $|\theta|\le\om(\log r)=\frac{\pi}{2l(r)q}$, and consequently,
\begin{equation*}
\begin{split}
|\psi^{-1}(re^{i\theta})|^\alpha
&=\left| e^{\frac 1q \zeta(\log r+i\theta)}\right|
=\exp\left(\frac kq +\int_1^r\frac{l(s)}{s}\,ds+o(1)\right)\\
&=r^{L(r)} e^{\frac kq +o(1)},\quad r\to\infty,
\end{split}
\end{equation*}
whenever $|\theta|\le\frac{\pi}{2l(r)q}$. It follows that $|\psi^{-1}(re^{i\theta})|^\alpha=r^{L(r)} d(r)$, where $d(r)=(1+o(1))e^{\frac kq}$ as $r\to\infty$, and hence \eqref{e:dag} is valid with $B=\sup_{r>R} d(r)$. Therefore Lemma~\ref{Lemma:B} implies that, for a given $\delta_1>0$, there exists a constant $K=K(\delta_1)>0$ such that
$$
\log|F(w)|=\log|G(\psi(w))|>-KB |w|^\alpha, \quad |\arg w|\le \frac{\pi}{2\alpha q}-\delta_1.
$$
Now \eqref{vehe} yields
$$
\arg \psi^{-1} (re^{i \theta})
=\Im\left(\frac1{\alpha q}\zeta(\log r+ i\theta))\right)
\sim\frac{\theta }{\alpha} l(r), \quad r\to\infty,
$$
and therefore, by choosing $\delta_1=\delta_1(\delta)>0$ sufficiently small we deduce
$$
\log|G(re^{i\theta})|>-K|\psi^{-1}(re^{i\theta})|^\alpha
=-(1+o(1))K r^{L(r)} e^{\frac kq} \ge -K_1 r^{L(r)},
$$
where $ |\theta|\le \frac{\pi}{2ql(r)}-\delta$.
The lemma is now proved.
\end{proof}
What remains to be done in this section is to deduce the results we are after from Lemma~\ref{l:cartwr}.
\begin{proof}[Proof of Proposition~\ref{cor:cartright}]
The proposition is an immediate consequence of Lemma~\ref{l:cartwr} because for a generalized proximate order we have $L(r)\sim l(r)$ as $r\to \infty$. In fact, (4') implies that for each $\e>0$ there exists $R=R(\e,l_1)>1$ such that $|l'(s)s\log s|<\frac{\e l_1}3$ for all $s\ge R$. Therefore, for some constant $C=C(R,l)>0$ we have
\begin{equation*}
\begin{split}
\left|L(r)-l(r)\right
&=\frac1{\log r}\left|\int_1^rl'(s)\log s\,ds\right|
\le\frac1{\log r}\left(\int_1^R|l'(s)|\log s\,ds+\frac{\e l_1}3\int_R^r\frac{ds}{s}\right)\\
&\le\frac{C}{\log r} +\frac{\e l_1}3
\le\frac{\e l_1}2 , \quad r\to\infty,
\end{split}
\end{equation*}
and consequently, $\left|\frac{L(r)}{l(r)}-1\right|<\e$ as $r\to\infty$.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{t:cartright}] As in \cite{CartWr33}, the statement of the theorem follows from Proposition~\ref{cor:cartright}. Let $\e>0$ and $0\le\theta<2\pi$ fixed. Choose $\e_1>0$, $0<q<1$ and $0<R<\infty$ such that
\begin{equation}\label{pillukarva}
\sup_{r\ge R}\e_1\left(1+\left(1+\e_1\right)\left(\sigma(r)\right)^{-1}\right)<\e
\quad\textrm{and}\quad
\inf_{r\ge R}\sigma(r)(1+\varepsilon_1)q>1.
\end{equation}
Then the function
$$
G(w)=\exp\left(f\left(e^{i\theta}\left(1-\frac 1w\right)\right)\right)
$$
satisfies the hypotheses of Proposition~\ref{cor:cartright} with $l(t)=(1+\varepsilon_1)(\sigma(t)+\e_1)$ and sufficiently large $R$. In fact, by writing $w=1/(1-z)$ we have $|\arg w|\le\frac{\pi}{2l(|w|)q}$ if and only if $|\arg \frac1{1-z}|\le\pi\left(2l\left(\frac1{\left|1-z\right|}\right)q\right)^{-1}$, and hence $|1-z|\asymp1-|z|$ by the right hand inequality in \eqref{pillukarva}. This together with the hypothesis \eqref{Eg:hypothesis-cartright} shows that
\begin{equation*}
\begin{split}
\log|G(w)|
&=\Re f\left(e^{i\theta}\left(1-\frac 1w\right)\right)
=\Re f(e^{i\theta}z)
\le\frac{1}{(1-|z|)^{\lambda(|z|)}}
\le\frac{C_1}{|1-z|^{\lambda(|z|)}}\\
&=C_1|w|^{\sigma\left(\frac1{1-\left|1-\frac1w\right|}\right)}
\lesssim |w|^{\sigma\left(\left|w\right|\right)},\quad |w|\to\infty,
\end{split}
\end{equation*}
for some constant $C_1=C_1(\sigma,\e_1,q)>0$. The last equality is valid because $\sigma$ is a generalized proximate order by the hypothesis, and hence there exist constants $C_2=C_2(w,\sigma,\e_1,q)\ge1$ and $C_3=C_3(\sigma,\e_1,q)\ge1$ such that $\sup_wC_2<\infty$ and
\begin{equation*}
\begin{split}
\left|\sigma\left(\frac1{1-\left|1-\frac1w\right|}\right)-\sigma\left(\left|w\right|\right)\right|
&=|\sigma'(C_2|w|)|\left(\frac1{1-\left|1-\frac1w\right|}-|w|\right)\\
&\le|\sigma'(C_2|w|)|C_3|w|=O\Bigl(\frac 1{\log|w|}\Bigr),\quad |w|\to\infty,
\end{split}
\end{equation*}
provided $|\arg w|\le\frac{\pi}{2l(|w|)q}$. Therefore, by choosing $\delta=\frac{\pi}{4l_2q}$ and using the left hand inequality in \eqref{pillukarva} we deduce
\begin{equation*}
\begin{split}
\Re f(re^{i\theta})
&=\log\left|G\left(\frac1{1-r}\right)\right|>-\left(\frac1{1-r}\right)^{(1+\varepsilon_1)\left(\sigma\left(\frac1{1-r}\right)+\e_1\right)+o(1)}\\
&>-\frac{1}{(1-r)^{(1+\varepsilon)\lambda(r)}},\quad r\to1^-.
\end{split}
\end{equation*}
Since $0\le\theta<2\pi$ was arbitrary, the assertion is proved.
\end{proof}
\section{Integral mean estimates and their consequences}\label{sec:final}
In this last section we prove the integral mean estimates stated in Theorem~\ref{t:p_lower_order} and deduce Corollary~\ref{c:m_p} as its consequence. Several auxiliary results are needed. We begin with two known ones concerning the canonical product
\begin{equation}\label{e:canon_prod}
\mathcal{P}(z)
=\mathcal{P}\left(z,(a_k)_{k=1}^\infty,s\right)
=\prod_{k=1}^\infty E(A(z, a_k), s),\quad \sum_{k=1}^\infty(1-|a_k|)^{s+1}<\infty,
\end{equation}
where
$$
E(w,s)=(1-w)\exp\left(w+\frac{w^2}{2} +\dots+ \frac{w^s}{s}\right),\quad s\in \N,
$$
is the Weierstrass primary factor and $A(z,\zeta)=\dfrac{1-|\zeta|^2}{1-z\overline\zeta}$ for all $z\in \mathbb{D}$ and $\zeta\in\overline{\mathbb{D}}$. Both of them concern sharp estimates of $\log|\mathcal{P}|$ in terms of sums of $|A(z,a_k)|$'s.
\begin{lettertheorem}[{\cite[p.224]{Tsuji}}]\label{t:tsuji} For each canonical product $\mathcal{P}$ and $\varepsilon>0$ there exists a constant $K=K(\mathcal{P},\e)>0$ such that
\begin{equation}\label{2.4l}
\log|\mathcal{P}(z)|\le K \sum_{k=1}^\infty|A(z,a_k)|^{\mu+1+\varepsilon},\quad\frac12\le|z|<1,
\end{equation}
where $\mu$ is the exponent of convergence of the zero sequence $(a_k)_{k=1}^\infty$.
Further, if $D_k=D(a_k,(1-|a_k|^2)^{\mu+4})$ for all $k\in\N$, then
\begin{equation}\label{2.5l}
\log|\mathcal{P}(z)|\ge K\log(1-|z|)\sum_{k=1}^\infty|A(z,a_k)|^{\mu+1+\varepsilon},
\quad\frac12\le|z|<1,\quad z\not\in\bigcup_{k=1}^\infty D_k.
\end{equation}
\end{lettertheorem}
It is known~\cite{DrSh} that P\'olya's order $\rho^*(\psi)$ for a non-decreasing function $\psi:[r_0,\infty)\to(0,\infty)$ can be determined by the formula
\begin{equation}\label{e:psi_con}
\rho^*(\psi)
=\sup\left\{\rho:\limsup_{x,C \to\infty}\frac{\psi(Cx)}{C^\rho\psi(x)}=\infty\right\}.
\end{equation}
It is finite if and only if $\psi$ satisfies Caramata's condition $\psi(2t)\lesssim\psi(t)$ as $t\to\infty$. The P\'olya's order $\rho^*(\psi)$ is not smaller than the order of growth of $\psi$, that is, $\psi(x)\lesssim x^\rho$ as $x\to\infty$ for all $\rho>\rho^*(\psi)$, but not vice versa.
\begin{letterlemma}[{\cite[Lemma 9]{ChyShep}}]\label{l:lin_can} Let $f$ be an analytic function in $\D$ with a zero sequence $Z=(a_k)_{k=1}^\infty$ such that
$$
n_1\left(r,f\right)\le\psi\left(\frac1{1-r}\right),\quad 0\le r<1,
$$
where the function $\psi$ satisfies Caramata's condition $\psi(2t)\lesssim\psi(t)$ as $t\to\infty$. Then, for $s>\rho^*(\psi)$, there exists a constant $C>0$ such that the canonical product $\mathcal{P}(z)=\mathcal{P}\left(z,\left(a_k\right)_{k=1}^\infty,s\right)$ admits the estimate
\begin{equation}\label{e:lem1}
\log|\mathcal{P}(z)|\le2^{s+2}\sum_{k=1}^\infty \left|A(z,a_k)\right|^{s+1}
\le C\widetilde\psi\left(\frac1{1-|z|}\right), \quad z\in\D,
\end{equation}
where $\widetilde{\psi}(t)=\int_{1}^t \frac{\psi(x)}{x}\,dx$.
\end{letterlemma}
The main step towards Theorem~\ref{t:p_lower_order} and Corollary~\ref{c:m_p} is the following lemma which is a counterpart of \cite[Lemma 1]{Li_mp} to the language used in this paper. Its proof relies on Proposition~\ref{p:i_est} and Lemma~\ref{l:i_est} as well as on a number of auxiliary results from the existing literature of which two are stated above.
\begin{lemma}\label{l:lin1}
Let $f$ be analytic in $\D$ such that $0\le\lambda_M(f)<\sigma_M(f)<\infty$, and $1\le p<\infty$. Let $\varepsilon>0$ and let $\lambda$ be a quasi proximate order of $f$, related to $\eta\in (0,\sigma_M(f)-\lambda_M(f))$. Then
$$
m_p\left(r,\log\left|\mathcal{P}\left(\cdot,(a_k)_{k=1}^\infty,s\right)\right|\right)
\lesssim\frac{1}{(1-r)^{\lambda(r)\left(1-\frac1{\sigma_M(f)}\right)^++1+\varepsilon}},\quad 0\le r<1,
$$
where $(a_k)_{k=1}^\infty$ is the zero sequence of $f$, $s=[\rho^*(A^*)]+1$ and $A^*$ is the function from the definition of the quasi proximate order $\lambda$.
\end{lemma}
\begin{proof}
If $\sigma_M(f)\le1$, then
the assertion is an immediate consequence of \cite[Lemma~1]{Li_mp}. Assume now that $\sigma_M(f)>1$.
Define
\begin{equation}\label{Eq:def-psi}
\psi(t)=\left(A^*\left(1-\frac 1t\right)\right)^{1-\frac1{\sigma_M(f)}}t^{1+\varepsilon},\quad 1\le t<\infty,
\end{equation}
where $A^*$ is the function from the definition of $\lambda$. By Proposition~\ref{p:i_est} and property (5) we have
\begin{equation}\label{e:zero_est}
n_1(r,f)
\lesssim\left(\frac{1}{(1-r)^{\lambda(r)}}\right)^{1-\frac 1{\sigma_M(f)}}\frac{1}{(1-r)^{1+\varepsilon}}
\lesssim\frac{(A^*(r))^{1-\frac1{\sigma_M(f)}}}{(1-r)^{1+\varepsilon}}
=\psi\left(\frac1{1-r}\right).
\end{equation}
By the property (6) of a quasi proximate order, $\psi$ satisfies Caramata's condition, and consequently, $\psi$ has finite P\'olya order
$\rho^*=\rho^*(\psi)$. Therefore we may apply Lemma~\ref{l:lin_can} with $s=[\rho^*(A^*)]+1$. Moreover, as $A^*$ is nondecreasing by (6), we have
\begin{equation}\label{e:zero_tild_est}
\begin{split}
\widetilde{\psi}(t)
&=\int_1^t \frac{\psi(x)}{x} \, dx=
\int_0^t\left(A^*\left(1-\frac1x\right)\right)^{1-\frac 1{\sigma_M(f)}}x^{\varepsilon}\,dx\\
&\le\frac1{1+\varepsilon}\left(A^*\left(1-\frac1t\right)\right)^{1-\frac 1{\sigma_M(f)}}t^{1+\varepsilon}
\le\psi(t),\quad 1\le t<\infty.
\end{split}
\end{equation}
To establish the statement we follow the scheme of the proof of \cite[Lemma~1]{Li_mp}. Without loss of generality, we may assume that $1/2\le r<1$ and $\frac 34 \le |a_k|<1$ for all $k\in\N$. We cover the set of integration by the intervals $[\tau+r-1,\tau+1-r]$, where $\tau=2k(1-r)$ and $k=0,1,\dots, [\frac{\pi}{1-r}]$, and show that
\begin{equation}\label{2.9l}
\int_{\tau+r-1}^{\tau+1-r}|\log|\mathcal{P}(re^{i\theta})||^p\,d\theta
\lesssim\frac{(1-r)\left(\log\frac1{1-r}\right)^p}{(1-r)^{\left(\lambda(r)\left(1-\frac 1{\sigma_M(f)}\right)+1+\varepsilon\right)p}}
\end{equation}
for each $\tau$. Then the statement of the lemma follows by summing over $\tau$.
Without loss of generality we may suppose that $\tau=0$. For given $r$, let $F$ denote the set of integers $m$ for which the exceptional disc $D_m$ of Theorem \ref{t:tsuji} intersects $\gamma_r=\{ re^{i\theta}: r-1\le \theta \le 1-r\}$.
Application of \eqref{e:nglob} and Lemma~\ref{l:i_est} show that
\begin{equation}\label{2.11l}
\# F\lesssim \psi\left(\frac 1{1-r}\right).
\end{equation}
Consider the factorization $\mathcal{P}=B_1B_2B_3$, where
$$
B_1(z)=\prod_{k\not\in F} E(A(z,a_k),s), \quad B_2(z)=\prod_{k\in F}\exp\left(\sum_{j=1}^s\frac1j(A(z,a_k))^j\right),
$$
and
$$
B_3(z)=\prod_{k\in F}(1-A(z,a_k))=\prod_{k\in F}\frac{\overline a_k(a_k-z)}{1-z\overline a_k}.
$$
It follows from \eqref{e:lem1} that the series $\sum_{k=1}^\infty \Bigl( \frac{1-|a_k|^2}{|1-z\bar a_k|}\Bigr)^{s+1} $ is convergent for every $z\in \D$ provided that $s>\rho^*(\psi)$. Choosing $z=0$ we deduce that the convergence exponent $\mu$ of the sequence $(a_k)_{k=1}^\infty $ satisfies $\mu \le \rho^*(\psi)$. Thus, using Theorem~\ref{t:tsuji} with $\varepsilon=s-\mu$ we obtain
\begin{equation*}
\int_{r-1}^{1-r} |\log|{B_1}(re^{i\theta})||^pd\theta
\lesssim\left(\log\frac1{1-r}\right)^p\int_{r-1}^{1-r}\left(\sum_{k\not\in F}|A(re^{i\theta},a_k)|^{s+1}\right)^p\,d\theta.
\end{equation*}
Then Lemma~\ref{l:lin_can}, \eqref{e:zero_tild_est} and property (5) yield
\begin{equation}\label{e:appl_lema9}
\begin{split}
\int_{r-1}^{1-r}|\log |{B_1}(re^{i\theta})||^pd\theta
&\lesssim\left(\log\frac1{1-r}\right)^p\widetilde\psi\left(\frac1{1-r}\right)^p(1-r)\\
&\lesssim\frac{(1-r)\left(\log\frac1{1-r}\right)^p}{(1-r)^{\left(\lambda(r)\left(1-\frac1{\sigma_M(f)}\right)+1+\varepsilon\right)p}}.
\end{split}
\end{equation}
On the way to \cite[Lemma~1]{Li_mp} it is proved that
\begin{equation*
\int_{r-1}^{1-r} |\log |{B_2}(re^{i\theta})||^pd\theta +\int_{r-1}^{1-r} |\log |{B_3}(re^{i\theta})||^pd\theta \lesssim (1-r) (\# F)^p.
\end{equation*}
Taking into account \eqref{2.11l} we deduce
\begin{equation}\label{e:b2_b3}
\int_{r-1}^{1-r}|\log |{B_2}(re^{i\theta})||^pd\theta
+\int_{r-1}^{1-r}|\log |{B_3}(re^{i\theta})||^pd\theta
\lesssim (1-r)\psi\left(\frac 1{1-r}\right)^p.
\end{equation}
Estimates \eqref{e:appl_lema9} and \eqref{e:b2_b3} now yield \eqref{2.9l}.
\end{proof}
We can now finish this section and the paper by proving Theorem~\ref{t:p_lower_order} and Corollary~\ref{c:m_p}.
\begin{proof}[Proof of Theorem~\ref{t:p_lower_order}]
The statement \eqref{e:mp_final-countdown} is valid by standard estimates if $\sigma_M(f)\le1$. Let now $f$ be an analytic function in $\D$ such that $1<\lambda_M(f)<\sigma_M(f)<\infty$, and let $\lambda$ be its generalized proximate order. Each generalized proximate order is a quasi proximate order. As an associated function one can take $A^*(t)=(1-t)^{-\lambda(t)}$ which is increasing for $r$ sufficiently close to $1$ because of (4'). Thus, the properties (1)--(7) of a quasi proximate order of the analytic function $f$ are satisfied. Further, let $s=[\rho^*(A^*)]+1$, where $\rho^*(A^*)$ is P\'olya's order of $A^*$. Consider the canonical product \eqref{e:canon_prod}, convergent by \cite[Lemma~9]{ChyShep}. It leads us to the factorization
\begin{equation}\label{e:fact}
f(z)=z^m \mathcal{P}(z)g(z),\quad z\in\D,
\end{equation}
where $g$ is a nonvanishing analytic function in $\D$ and $m=m(f)\in\N\cup\{0\}$. By combining Theorem~\ref{t:tsuji} with Lemma~\ref{l:lin_can} we deduce that there exists $n_0\in\N$ such that, for $n\ge n_0$, the interval $[r_n,r_{n+1})$, where $r_n=1-2^{-n}$, contains a point $T_n$ for which the circle $\{ z:|z|=T_n\}$ does not intersect any of exceptional discs of Theorem~\ref{t:tsuji}, and
\begin{equation}\label{e:prod_unif_est}
|\log|\mathcal{P}(z)||
\lesssim\psi\left(\frac1{1-|z|}\right)\log\frac1{1-|z|},\quad|z|=T_n,
\end{equation}
where $\psi$ is defined by \eqref{Eq:def-psi}. By using factorization \eqref{e:fact} and the properties (5) and (7), we obtain
$$
\log|g(z)|
\le\log|f(z)|+|\log|\mathcal{P}(z)||-m\log|z|
\lesssim\psi\left(\frac1{1-|z|}\right)\log\frac1{1-|z|},\quad|z|=T_n.
$$
The maximum modulus principle together with \eqref{Eq:def-psi} and the properties (5) and (6) then implies
\begin{equation*}
\begin{split}
\log M(r,g)
&\le\log M (T_{n+1}, f)
\le\psi\left(\frac1{1-T_{n+1}}\right)\log\frac1{1-T_{n+1}}\\
&\lesssim\frac{1}{(1-r)^{\lambda(r)\left(1-\frac1{\sigma_M(f)}\right)+ 1+2\varepsilon}},\quad r\in [r_n, r_{n+1}),\quad n\to\infty,
\end{split}
\end{equation*}
from which Theorem~\ref{t:cartright} yields
\begin{equation*}
|\log|g(z)||
\lesssim\frac{1}{(1-r)^{(1+\e)\left(\lambda(r)\left(1-\frac1{\sigma_M(f)}\right)+ 1+2\varepsilon\right)}}, \quad r_{n_0}\le |z|<1.
\end{equation*}
Therefore
$$
m_p(r,\log|g|)
\lesssim\frac{1}{(1-r)^{(1+\e)\left(\lambda(r)\left(1-\frac1{\sigma_M(f)}\right)+ 1+2\varepsilon\right)}},\quad r\to1^-,
$$
and $m_p(r,\log|f|)$ obeys the same upper bound by Minkowski's inequality and Lemma~\ref{l:lin1}. The statement \eqref{e:mp_final-countdown} of Theorem~\ref{t:p_lower_order} in the case $\sigma_M(f)>1$ follows by re-defining $\e$.
By arguing as in the last part of the proof of Lemma~\ref{l:i_est} we deduce \eqref{e:mp_final-countdown2} from \eqref{e:mp_final-countdown}. Details are omitted.
\end{proof}
\begin{proof}[Proof of Corollary~\ref{c:m_p}]
It follows from \eqref{e:mp_final-countdown2} that
$$
\lambda_p(f)\le\left(\lambda_M(f)+\eta\right)\left(1-\frac1{\sigma_M(f)}\right)^++1+\varepsilon.
$$
Since $\eta$ and $\varepsilon$ can be chosen arbitrarily small, the first inequality follows. The second inequality is a consequence of the estimate
$$
\log|f(re^{i\theta})|\lesssim \frac{m_p(\frac{1+r}{2}, \log|f|)}{(1-r)^{\frac 1p}},\quad r\to1^-,
$$
which follows from the Poisson-Jensen formula and H\"older's inequality~\cite{Li_mp}.
\end{proof}
|
1,108,101,564,592 | arxiv | \section{The Argument So Far}
In Parts I[1] and II[2], the basic facts concerning zero-divisors
(ZDs) as they arise in the hypercomplex context were presented and
proved. ``Basic,'' in the context of this monograph, means seven
things. First, they emerged as a side-effect of applying CDP a
minimum of 4 times to the Real Number Line, doubling dimension to
the Complex Plane, Quaternion 4-Space, Octonion 8-Space, and 16-D
Sedenions. With each such doubling, new properties were found: as
the price of sacrificing counting order, the Imaginaries made a
general theory of equations and solution-spaces possible; the
non-commutative nature of Quaternions mapped onto the realities of
the manner in which forces deploy in the real world, and led to
vector calculus; the non-associative nature of Octonions, meanwhile,
has only come into its own with the need for necessarily
unobservable quantities (because of conformal field-theoretical
constraints)in String Theory. In the Sedenions, however, the most
basic assumptions of all -- well-defined notions of field and
algebraic norm (and, therefore, measurement) -- break down, as the
phenomena correlated with their absence, zero-divisors, appear
onstage (never to leave it for all higher CDP dimension-doublings).
Second thing: ZDs require at least two differently-indexed
imaginary units to be defined, the index being an integer larger
than 0 (the CDP index of the Real Unit) and less than $2^{N}$ for a
given CDP-generated collection of $2^{N}$-ions. In ``pure CDP,''
the enormous number of alternative labeling schemes possible in any
given $2^{N}$-ion level are drastically reduced by assuming that
units with such indices interact by XOR-ing: the index of the
product of any two is the XOR of their indices. Signing is more
tricky; but, when CDP is reduced to a 2-rule construction kit, it
becomes easy: for index $u < {\bf G}$, ${\bf G}$ the Generator of
the $2^{N}$-ions (i.e., the power of 2 immediately larger than the
highest index of the predecessor $2^{N-1}$-ions), Rule 1 says $i_{u}
\cdot i_{\bf G} = + i_{(u + {\bf G})}$. Rule 2 says take an
associative triplet $(a, b, c)$, assumed written in CPO (short for
``cyclically positive order'': to wit, $a \cdot b = +c$, $b \cdot c
= +a$, and $c \cdot a = + b$). Consider, for instance, any $(u,
{\bf G}, {\bf G} + u)$ index set. Then three more such associative
triplets (henceforth, \textit{trips}) can be generated by adding
{\bf G} to two of the three, then switching their resultants' places
in the CPO scheme. Hence, starting with the Quaternions' $(1, 2, 3)$
(which we'll call a \textit{Rule 0} trip, as it's inherited from a
prior level of CDP induction), Rule 1 gives us the trips $(1, 4,
5)$, $(2, 4, 6)$, and $(3, 4, 7)$, while Rule 2 yields up the other
4 trips defining the Octonions: $(1, 7, 6)$, $(2, 5, 7)$, and $(3,
6, 5)$. Any ZD in a given level of $2^{N}$-ions will then have
units with one index $< {\bf G}$, written in lowercase, and the
other index $> {\bf G}$, written in uppercase. Such pairs,
alternately called ``dyads'' or ``Assessors,'' saturate the diagonal
lines of their planes, which diagonals never mutually zero-divide
each other (or make \textit{DMZs}, for "divisors (or dyads) making
zero"), but only make DMZs with other such diagonals, in other such
Assessors. (This is, of course, the opposite situation from the
projection operators of quantum mechanics, which are diagonals in
the planes formed by Reals and dimensions spanned by Pauli spin
operators contained within the 4-space created by the Cartesian
product of two standard imaginaries.)
Third thing: Such ZDs are not the only possible in CDP spaces; but
they define the ``primitive'' variety from which ZD spaces
saturating more than 1-D regions can be articulated. A not quite
complete catalog of these can be found in our first monograph on the
theme [3]; a critical kind which was overlooked there, involving the
Reals (and hence, providing the backdrop from which to see the
projection-operator kind as a degenerate type), were first discussed
more recently [4]. (Ironically, these latter are the easiest sorts
of composites to derive of any: place the two diagonals of a DMZ
pairing with differing internal signing on axes of the same plane,
and consider the diagonals \textit{they} make with each other!) All
the primitive ZDs in the Sedenions can be collected on the vertices
of one of 7 copies of an Octahedron in the \textit{Box-Kite}
representation, each of whose 12 edges indicates a two-way ``DMZ
pathway,'' evenly divided between 2 varieties. For any vertex V,
and $k$ any real scalar, indicate the diagonals this way:
$\verb|(V,/)| = k \cdot (i_{v} + i_{V})$, while $\verb|(V, \)| = k
\cdot (i_{v} - i_{V})$. 6 edges on a Box-Kite will always have
\textit{negative edge-sign} (with \textit{unmarked} ET cell entries:
see the ``sixth thing''). For vertices M and N, exactly two
DMZs run along the edge joining them, written thus:
\begin{center}
$\verb|(M,/)| \cdot \verb|(N, \)|$ $= \verb|(M, \)|$ $\cdot$
$\verb|(N,/)|$ $ = 0$
\end{center}
The other 6 all have \textit{positive edge-sign}, the diagonals of
their two DMZs having same slope (and \textit{marked} -- with
leading dashes -- ET cell entries):
\begin{center}
$\verb|(Z,/)| \cdot \verb|(V, /)| = $ $\verb|(Z, \)| \cdot
\verb|(V,\)| = 0$
\end{center}
Fourth thing: The edges always cluster similarly, with two opposite
faces among the 8 triangles on the Box-Kite being spanned by 3
negative edges (conventionally painted red in color renderings),
with all other edges being positive (painted blue). One of the red
triangles has its vertices' 3 low-index units forming a trip;
writing their vertex labels conventionally as A, B, C, we find there
are in fact always 4 such trips cycling among them: $(a, b, c)$,
the \textit{L-trip}; and the three \textit{U-trips} obtained by
replacing all but one of the lowercase labels in the L-trip with
uppercase: $(a, B, C)$; $(A, b, C)$; $(A, B, c)$. Such a 4-trip
structure is called a \textit{Sail}, and a Box-Kite has 4 of them:
the \textit{Zigzag}, with all negative edges, and the 3
\textit{Trefoils}, each containing two positive edges extending from
one of the Zigzag vertices to the two vertices opposite its
\textit{Sailing partners}. These opposite vertices are always
joined by one of the 3 negative edges comprising the Vent which is
the Zigzag's opposite face. Again by convention, the vertices
opposite A, B, C are written F, E, D in that order; hence, the
Trefoil Sails are written $(A, D, E)$; $(F, D, B)$, and $(F, C, E)$,
ordered so that their lowercase renderings are equivalent to their
CPO L-trips. The graphical convention is to show the Sails as
filled in, while the other 4 faces, like the Vent, are left empty:
they show ``where the wind blows'' that keeps the Box-Kite aloft. A
real-world Box-Kite, meanwhile, would be held together by 3 dowels
(of wood or plastic, say) spanning the joins between the only
vertices left unconnected in our Octahedral rendering: the
\textit{Struts} linking the \textit{strut-opposite} vertices (A, F);
(B, E); (C, D).
Fifth thing: In the Sedenions, the 7 isomorphic Box-Kites are
differentiated by which Octonion index is missing from the vertices,
and this index is designated by the letter {\bf S}, for
``signature,'' ``suppressed index,'' or \textit{strut constant}.
This last designation derives from the invariant relationship
obtaining in a given Box-Kite between {\bf S} and the indices in the
Vent and Zigzag termini (V and Z respectively) of any of the 3
Struts, which we call the ``First Vizier'' or VZ1. This is one of 3
rules, involving the three Sedenion indices always missing from a
Box-Kite's vertices: ${\bf G}$, ${\bf S}$, and their simple sum
${\bf X}$ (which is also their XOR product, since ${\bf G}$ is
always to the left of the left-most bit in ${\bf S}$). The Second
Vizier tells us that the L-index of either terminus with the U-index
of the other always form a trip with ${\bf G}$, and it true as
written for all $2^{N}$-ions. The Third shows the relationship
between the L- and U- indices of a given Assessor, which always form
a trip with ${\bf X}$. Like the First, it is true as written only
in the Sedenions, but as an unsigned statement about indices only,
it is true universally. (For that reason, references to VZ1 and VZ3
hereinout will be assumed to refer to the \textit{unsigned}
versions.) First derived in the last section of Part I, reprised in
the intro of Part II, we write them out now for the third and final
time in this monograph:
\begin{center}
VZ1: $v \cdot z = V \cdot Z = {\bf S}$
\smallskip
VZ2: $Z \cdot v = V \cdot z = {\bf G}$
\smallskip
VZ3: $V \cdot v = z \cdot Z = {\bf X}$.
\end{center}
Rules 1 and 2, the Three Viziers, plus the standard Octonion
labeling scheme derived from the simplest finite projective group,
usually written as PSL(2,7), provide the basis of our toolkit. This
last becomes powerful due to its capacity for recursive re-use at
all levels of CDP generation, not just the Octonions. The simplest
way to see this comes from placing the unique Rule 0 trip provided
by the Quaternions on the circle joining the 3 sides' midpoints,
with the Octonion Generator's index, 4, being placed in the center.
Then the 3 lines leading from the Rule 0 trip's (1, 2, 3) midpoints
to their opposite angles -- placed conventionally in clockwise order
in the midpoints of the left, right, and bottom sides of a triangle
whose apex is at 12 o'clock -- are CPO trips forming the Struts,
while the 3 sides themselves are the Rule 2 trips. These 3 form the
L-index sets of the Trefoil Sails, while the Rule 0 trip provides
the same service for the Zigzag. By a process analogized to tugging
on a slipcover (Part I) and pushing things into the central zone of
hot oil while wok-cooking (Part II), all 7 possible values of {\bf
S} in the Sedenions, not just the 4, can be moved into the center
while keeping orientations along all 7 lines of the Triangle
unchanged. Part II's critical Roundabout Theorem tells us, moreover,
that all $2^{N}$-ion ZDs, for all $N
> 3$, are contained in Box-Kites as their minimal ensemble size.
Hence, by placing the appropriate ${\bf G}$, ${\bf S}$, or ${\bf X}$
in the center of a PSL(2,7) triangle, with a suitable Rule 0 trip's
indices populating the circle, any and all \textit{candidate}
primitive ZDs can be discovered and situated.
Sixth thing: The word ``candidate'' in the above is critical; its
exploration was the focus of Part II. For, starting with $N = 5$ and
hence ${\bf G = 16}$ (which is to say, in the 32-D Pathions), whole
Box-Kites can be suppressed (meaning, all 12 edges, and not just the
Struts, no longer serve as DMZ pathways). But for all $N$, the full
set of candidate Box-Kites are viable when ${\bf S} \leq 8$ or equal
to some higher power of 2. For all other ${\bf S}$ values, though,
the phenomenon of \textit{carrybit overflow} intervenes -- leading,
ultimately, to the ``meta-fractal'' behavior claimed in our
abstract. To see this, we need another mode of representation, less
tied to 3-D visualizing, than the Box-Kite can provide. The answer
is a matrix-like method of tabulating the products of candidate ZDs
with each other, called \textit{Emanation Tables} or \textit{ETs}.
The L-indices only of all candidate ZDs are all we need indicate
(the U-indices being forced once ${\bf G}$ is specified); these will
saturate the list of allowed indices $< {\bf G}$, save for the value
of ${\bf S}$ whose choice, along with that of ${\bf G}$, fixes an
ET. Hence, the unique ET for given ${\bf G}$ and ${\bf S}$ will
fill a square spreadsheet whose edge has length $2^{N-1}- 2$.
Moreover, a cell entry (r,c) is only filled when row and column
labels R and C form a DMZ, which can never be the case along an ET's
long diagonals: for the diagonal starting in the upper left corner,
R xor R = 0, and the two diagonals within the same Assessor, can
never zero-divide each other; for the righthand diagonal, the
convention for ordering the labels (ascending counting order from
the left and top, with any such label's strut-opposite index
immediately being entered in the mirror-opposite positions on the
right and bottom) makes R and C strut-opposites, hence also unable
to form DMZs.
For the Sedenions, we get a 6 x 6 table, 12 of whose cells (those on
long diagonals) are empty: the 24 filled cells, then, correspond to
the two-way traffic of ``edge-currents'' one imagines flowing
between vertices on a Box-Kite's 12 edges. A computational
corollary to the Roundabout Theorem, dubbed the \textit{Trip-Count
Two-Step}, is of seminal importance. It connects this most basic
theorem of ETs to the most basic fact of associative triplets,
indicated in the opening pages of Part I, namely: for any N, the
number $Trip_{N}$ of associative triplets is found, by simple
combinatorics, to be $(2^{N} - 1)(2^{N} - 2)/3!$ -- 35 for the
Sedenions, 155 for the Pathions, and so on. But, by Trip-Count
Two-Step, we also know that \textit{the maximum number of Box-Kites
that can fill a $2^{N}$-ion ET = $Trip_{N-2}$.} For ${\bf S}$ a
power of 2, beginning in the Pathions (for ${\bf S} = 2^{5 - 2} =
8$), the Number Hub Theorem says the upper left quadrant of the ET
is an unsigned multiplication table of the $2^{N-2}$-ions in
question, with the 0's of the long diagonal (indicated Real negative
units) replaced by blanks -- a result effectively synonymous with
the Trip-Count Two-Step.
Seventh thing: We found, as Part II's argument wound down, that the
2 classes of ETs found in the Pathions -- the ``normal'' for ${\bf
S} \leq 8$, filled with indices for all 7 possible Box-Kites, and
the ``sparse'' so-called Sand Mandalas, showing only 3 Box-Kites
when $8 < {\bf S} < 16$, were just the beginning of the story. A
simple formula involving just the bit-string of ${\bf s}$ and ${\bf
g}$, where the lowercase indicates the values of ${\bf S}$ and ${\bf
G}$ modulo ${\bf G}/2$, gave the prototype of our first
\textit{recipe}: all and only cells with labels R or C, or content P
( = R xor C ), are filled in the ET. The 4 ``missing Box-Kites''
were those whose L-index trip would have been that of a Sail in the
$2^{N-1}$ realm with ${\bf S} = {\bf s}$ and ${\bf G} = {\bf g}$.
The sequence of 7 ETs, viewed in ${\bf S}$-increasing succession,
had an obvious visual logic leading to their being dubbed a
\textit{flip-book}. These 7 were obviously indistinguishable from
many vantages, hence formed a \textit{spectrographic band}. There
were 3 distinct such bands, though, each typified by a Box-Kite
count common to all band-members, demonstrable in the ETs for the
64-D Chingons. Each band contained ${\bf S}$ values bracketed by
multiples of 8 (either less than or equal to the higher, depending
upon whether the latter was or wasn't a power of 2). These were
claimed to underwrite behaviors in all higher $2^{N}$-ion ETs,
according to 3 rough patterns in need of algorithmic refining in
this Part III. Corresponding to the first unfilled band, with ETs
always missing $4^{N-4}$ of their candidate Box-Kites for $N > 4$,
we spoke of \textit{recursivity}, meaning the ETs for constant ${\bf
S}$ and increasing $N$ would all obey the same recipe, properly
abstracted from that just cited above, empirically found among the
Pathions for ${\bf S} > 8$. The second and third behaviors, dubbed,
for ${\bf S}$ ascending, \textit{(s,g)-modularity} and
\textit{hide/fill involution} respectively, make their first
showings in the Chingons, in the bands where $16 < {\bf S} \leq 24$,
and then where $24 < {\bf S} < 32$. In all such cases, we are
concerned with seeing the ``period-doubling'' inherent in CDP and
Chaotic attractors both become manifest in a repeated doubling of ET
edge-size, leading to the fixed-${\bf S}$, $N$ increasing analog of
the fixed-$N, {\bf S}$ increasing flip-books first observed in the
Pathions, which we call \textit{balloon-rides}. Specifying and
proving their workings, and combining all 3 of the above-designated
behaviors into the ``fundamental theorem of zero-division algebra,''
will be our goals in this final Part III. Anyone who has read this
far is encouraged to bring up the graphical complement to this
monograph, the 78-slide Powerpoint show presented at NKS 2006 [5],
in another window. (Slides will be referenced by number in what
follows.)
\section{$8 <{\bf S} < 16, N \rightarrow \infty$ : Recursive Balloon Rides in the Whorfian Sky}
We know that any ET for the $2^{N}$-ions is a square whose edge is
$2^{N-1} - 2$ cells. How, then, can any simply recursive rule
govern exporting the structure of one such box to analogous boxes
for progressively higher $N$? The answer: \textit{include the label
lines} -- not just the column and row headers running across the top
and left margins, but their strut-opposite values, placed along the
bottom and right margins, which are mirror-reversed copies of the
label-lines \textit{(LLs)} proper to which they are parallel. This
increases the edge-size of the ET box to $2^{N-1}$.
\medskip
\noindent {\small Theorem 11.} For any fixed ${\bf S} > 8$ and not
a power of $2$, the row and column indices comprising the Label
Lines (LLs) run along the left and top borders of the $2^{N}$-ion ET
"spreadsheet" for that ${\bf S}$. Treat them as included in the
spreadsheet, \textit{as labels}, by adding a row and column to the
given square of cells, of edge $2^{N-1} - 2$, which comprises the ET
proper. Then add another row and column to include the
strut-opposite values of these labels' indices in ``mirror LLs,''
running along the opposite edges of a now $2^{N-1}$-edge-length box,
whose four corner cells, like the long diagonals they extend, are
empty. When, for such a fixed ${\bf S}$, the ET for the
$2^{N+1}$-ions is produced, the values of the 4 sets of LL indices,
bounding the contained $2^{N}$-ion ET, correspond, \textit{as cell
values}, to actual DMZ P-values in the bigger ET, residing in the
rows and columns labeled by the contained ET's ${\bf G}$ and ${\bf
X}$ (the containing ET's $g$ and $g + {\bf S}$). Moreover, all
cells contained in the box they bound in the containing ET have
P-values (else blanks) exactly corresponding to -- and
\textit{including edge-sign markings} of -- the positionally
identical cells in the $2^{N}$-ion ET: those, that is, for which
the LLs act as labels.
\medskip
\noindent \textit{Proof.} For all strut constants of interest,
${\bf S} < g ( = {\bf G/2})$; hence, all labels up to and including
that immediately adjoining its own strut constant (that is, the
first half of them) will have indices monotonically increasing, up
to and at least including the midline bound, from $1$ to $g - 1$.
When $N$ is incremented by $1$, the row and column midlines
separating adjoining strut-opposites will be cut and pulled apart,
making room for the labels for the $2^{N+1}$-ion ET for same ${\bf
S}$, which middle range of label indices will also monotonically
increase, this time from the current $2^N$-ion generation's $g$ (and
prior generation's ${\bf G}$), up to and at least including its own
midline bound, which will be $g$ plus the number of cells in the LL
inherited from the prior generation, or $g/2 - 1$. The LLs are
therefore contained in the rows and columns headed by $g$ and its
strut opposite, $g + {\bf S}$. To say that the immediately prior
CDP generation's ET labels are converted to the current generation's
P-values in the just-specified rows and columns is equivalent to
asserting the truth of the following calculation:
\smallskip
\begin{center}
$(g + u) + (sg) \cdot ({\bf G} + g + u_{opp})$
\underline{$\;\;\;\;\;\;g \;\;\;\;\; + \;\;\;\;\; ({\bf G} + g +
{\bf S})\;\;\;\;\;\;$}
\smallskip
$- (vz)\cdot ({\bf G} + u_{opp}) \;\;\; + (vz) \cdot (sg) \cdot u$
\underline{$ + u \;\;\;\;\; - \;\;\;\;\; (sg) \cdot ({\bf G} +
u_{opp})$}
\smallskip
$0$ only if $vz = (-sg)$
\end{center}
\smallskip
Here, we use two binary variables, the inner-sign-setting $sg$, and
the Vent-or-Zigzag test, based on the First Vizier. Using the two
in tandem lets us handle the normal and ``Type II'' box-kites in the
same proof. Recall (and see Appendix B of Part II for a quick
refresher) that while the ``Type I'' is the only type we find in the
Sedenions, we find that a second variety emerges in the Pathions,
indistinguishable from Type I in most contexts of interest to us
here: the orientation of 2 of the 3 struts will be reversed (which
is why VZ1 and VZ3 are only true generally when unsigned). For a
Type I, since ${\bf S} < g$, we know by Rule 1 that we have the trip
$({\bf S}, g, g + {\bf S})$; hence, $g$ -- for all $2^{N}$-ions
beyond the Pathions, where the Sand Mandalas' $g = 8$ is the L-index
of the Zigzag B Assessor -- must be a Vent (and its strut-opposite,
$g + {\bf S}$, a Zigzag). For a Type II, however, this is
necessarily so only for 1 of the 3 struts -- which means, per the
equation above, that sg must be reversed to obtain the same result.
Said another way, we are free to assume either signing of $vz$ means
+1, so the ``only if'' qualifying the zero result is informative.
It is $u$ and its relationship to $g + u$ that is of interest
here, and this formulation makes it easier to see that the products
hold for arbitrary LL indices $u$ \textit{or} their strut-opposites.
But for this, the term-by-term computations should seem routine: the
left bottom is the Rule 1 outcome of $(u, g, g+u)$: obviously, any
$u$ index must be less than $g$. To its right, we use the trip
$(u_{opp}, g, g + u_{opp}) \rightarrow ({\bf G} + g + u_{opp}, g,
{\bf G} + u_{opp})$, whose CPO order is opposite that of the
multiplication. For the top left, we use $(u, {\bf S}, u_{opp})$ as
limned above, then augment by $g$, then ${\bf G}$, leaving $u_{opp}$
unaffected in the first augmenting, and $g + u$ in the second.
Finally, the top right (ignoring $sg$ and $vz$ momentarily) is
obtained this way: $(u,{\bf S}, u_{opp}) \rightarrow (u, g +
u_{opp}, g + {\bf S}) \rightarrow (u, {\bf G} + g + s, {\bf G} + g +
u_{opp})$; ergo, $+u$.
Note that we cannot eke out any information about edge-sign marks
from this setup: since labels, as such, have no marks, we have
nothing to go on -- unlike all other cells which our recursive
operations will work on. Indeed, the exact algorithmic
determination of edge-sign marks for labels is not so trivial: as
one iterates through higher $N$ values, some segments of LL indexing
will display reversals of marks found in the ascending or descending
left midline column, while other segments will show them unchanged
-- with key values at the beginnings and ends of such octaves
(multiples of $8$, and sums of such multiples with ${\bf S}\; mod \;
8$) sometimes being reversed or kept the same irrespective of the
behavior of the terms they bound. Fortunately, such behaviors are
of no real concern here -- but they are, nevertheless, worth
pointing out, given the easy predictability of other edge-sign marks
in our recursion operations.
Now for the ET box within the labels: if all values (including
edge-sign marks) remain unchanged as we move from the $2^{N}$-ion ET
to that for the $2^{N+1}$-ions, then one of 3 situations must
obtain: the inner-box cells have labels $u, v$ which belong to some
Zigzag L-trip $(u, v, w)$; or, on the contrary, they correspond to
Vent L-indices -- the first two terms in the CPO triplet $(w_{opp},
v_{opp}, u)$, for instance; else, finally, one term is a Vent, the
other a Zigzag (so that inner-signs of their multiplied dyads are
both positive): we will write them, in CPO order, $v_{opp}$ and
$u$, with third trip member $w_{opp}$. Clearly, we want all the
products in the containing ET to indicate DMZs only if the inner
ET's cells do similarly. This is easily arranged: for the
containing ET's cells have indices identical to those of the
contained ET's, save for the appending of $g$ to both (and ditto for
the U-indices).
\medskip
\textit{Case 1:} If $(u, v, w)$ form a Zigzag L-index set, then so
do $(g + v, g + u, w)$, so markings remain unchanged; and if the
$(u,v)$ cell entry is blank in the contained, so will be that for
$(g + u, g + v)$ in its container. In other words, the following
holds:
\smallskip
\begin{center}
$(g + v) + (sg) \cdot ({\bf G} + g + v_{opp})$
\underline{$\;\;\;(g + u) \;\;\;\; + \;\;\;({\bf G} + g +
u_{opp})\;\;\;$}
\smallskip
$- ({\bf G} + w_{opp}) \;\;\; - (sg) \cdot w$
\underline{$ - w \;\;\;\;\; - (sg) \cdot ({\bf G} + w_{opp})$}
\smallskip
$0$ only if $sg = (-1)$
\end{center}
\pagebreak
$(g + u) \cdot (g + v) = P:$ $(u,v,w) \rightarrow (g + v, g + u,
w)$; hence, $(- w)$.
$(g + u) \cdot (sg) \cdot ({\bf G} + g + v_{opp}) = P:$ $(u,
w_{opp}, v_{opp})$ $\rightarrow (g + v_{opp}, w_{opp}, g + u)$
$\rightarrow ({\bf G} + w_{opp}, {\bf G} + g + v_{opp}, g + u)$;
hence, $(sg) \cdot ( - ({\bf G} + w_{opp}))$.
$({\bf G} + g + u_{opp}) \cdot (g + v) = P:$ $(u_{opp}, w_{opp},
v)$ $\rightarrow (g + v, w_{opp}, g + u_{opp})$ $\rightarrow (g + v,
{\bf G} + g + u_{opp}, {\bf G} + w_{opp})$; hence, $(- ({\bf G} +
w_{opp}))$.
$({\bf G} + g + u_{opp}) \cdot ({\bf G} + g + v_{opp}) = P:$ Rule 2
twice to the same two terms yields the same result as the terms in
the raw, hence $(- w)$.
Clearly, cycling through $(u,v,w)$ to consider $(g + v) \cdot (g +
w)$ will give the exactly analogous result, forcing two (hence
three) negative inner-signs in the candidate Sail; hence, if we have
DMZs at all, we have a Zigzag Sail.
\medskip
\textit{Case 2:} The product of two Vents must have negative
edge-sign, and there's no cycling through same-inner-signed products
as with the Zigzag, so we'll just write our setup as a one-off, with
upper inner-sign explicitly negative, and claim its outcome true.
\smallskip
\begin{center}
$(g + v_{opp}) - ({\bf G} + g + v)$
\underline{$\;(g + w_{opp}) \; + \;({\bf G} + g + w)\;$}
\smallskip
$+ ({\bf G} + u_{opp}) \;\;\;\;\; + u$
\underline{$ - u \;\;\;\;\; - ({\bf G} + u_{opp})$}
\smallskip
$0$
\end{center}
\smallskip
$(g + w_{opp}) \cdot (g + v_{opp}) = P:$ $(w_{opp}, v_{opp}, u)$
$\rightarrow (g + v_{opp}, g + w_{opp}, u)$; hence, $(- u)$.
$(g + w_{opp}) \cdot ({\bf G} + g + v) = P:$ $(w_{opp}, v,
u_{opp})$ $\rightarrow (g + v, g + w_{opp}, u_{opp})$ $\rightarrow
({\bf G} + u_{opp}, g + w_{opp}, {\bf G} + g + v)$; but inner sign
of upper dyad is negative, so $(- ({\bf G} + u_{opp}))$.
$({\bf G} + g + w) \cdot (g + v_{opp}) = P:$ $(v_{opp}, u_{opp},
w)$ $\rightarrow (g + w, u_{opp}, g + v_{opp})$ $\rightarrow ({\bf
G} + u_{opp}, {\bf G} + g + w, g + v_{opp})$; hence, $(+ ({\bf G} +
u_{opp}))$.
$({\bf G} + g + w) \cdot ({\bf G} + g + v) = P:$ Rule 2 twice to the
same two terms yields the same result as the terms in the raw; but
inner sign of upper dyad is negative, so $(+ u)$.
\medskip
\textit{Case 3:} The product of Vent and Zigzag displays same inner
sign in both dyads; hence the following arithmetic holds:
\pagebreak
\begin{center}
$(g + u) + ({\bf G} + g + u_{opp})$
\underline{$\; (g + v_{opp}) \; + \;({\bf G} + g + v)\;$}
\smallskip
$- ({\bf G} + w) \;\;\;\;\; + w_{opp}$
\underline{$ - w_{opp} \;\;\;\;\; + ({\bf G} + w)$}
\smallskip
$0$
\end{center}
\smallskip
The calculations are sufficiently similar to the two prior cases as
to make their writing out tedious. It is clear that, in each of our
three cases, content and marking of each cell in the contained ET
and the overlapping portion of the container ET are identical. $\;\;
\blacksquare$
\medskip
To highlight the rather magical label/content involution that occurs
when $N$ is in- or de- cremented, graphical realizations of such
nested patterns, as in Slides 60-61, paint LLs (and labels proper) a
sky-blue color. The bottom-most ET being overlaid in the central
box has $g =$ the maximum high-bit in ${\bf S}$, and is dubbed the
\textit{inner skybox}. The degree of nesting is strictly measured
by counting the number of bits $B$ that a given skybox's $g$ is to
the left of this strut-constant high-bit. If we partition the inner
skybox into quadrants defined by the midlines, and count the number
$Q$ of quadrant-sized boxes along one or the other long diagonal, it
is obvious that the inner skybox itself has $B = 0$ and $Q = 1$; the
nested skyboxes containing it have $Q = 2^{B}$. If recursion of
skybox nesting be continued indefinitely -- to the fractal limit,
which terminology we will clarify shortly -- the indices contained
in filled cells of any skybox can be interpreted in $B$ distinct
ways, $B \rightarrow \infty$, as representations of distinct ZDs
with differing ${\bf G}$ and, therefore, differing U-indices. By
obvious analogy to the theory of Riemann surfaces in complex
analysis, each such skybox is a separate ``sheet''; as with even
such simple functions as the logarithmic, the number of such sheets
is infinite. We could then think of the infinite sequence of
skyboxes as so many cross-sections, at constant distances, of a
flashlight beam whose intensity (one over the ET's cell count)
follows Kepler's inverse square law. Alternatively, we could ignore
the sheeting and see things another way.
Where we called fixed-$N$, ${\bf S}$ varying sequences of ETs
flip-books, we refer to fixed-${\bf S}$, $N$ varying sequences as
balloon rides: the image is suggested by David Niven's role as
Phineas Fogg in the movie made of Jules Vernes' \textit{Around the
World in 80 Days}: to ascend higher, David would drop a sandbag
over the side of his hot-air balloon's basket; if coming down, he
would pull a cord that released some of the balloon's steam. Each
such navigational tactic is easy to envision as a bit-shift, pushing
${\bf G}$ further to the left to cross LLs into a higher skybox,
else moving it rightward to descend. Using ${\bf S = 15}$ as the
basis of a 3-stage balloon-ride, we see how increasing $N$ from $5$
to $6$ to $7$ approaches the white-space complement of one of the
simplest (and least efficient) plane-filling fractals, the Ces\`aro
double sweep [6, p. 65].
\begin{figure}
\includegraphics[width = 1.\textwidth] {PHSS3_arxiv_1.eps}
\caption{ETs for S=15, N=5,6,7 (nested skyboxes in blue)$\cdots$ and
``fractal limit.''}
\end{figure}
The graphics were programmatically generated prior to the proving of
the theorems we're elaborating: their empirical evidence was what
informed (indeed, demanded) the theoretical apparatus. And we are
not quite finished with the current task the apparatus requires of
us. We need two more theorems to finish the discussion of skybox
recursion. For both, suppose some skybox with $B = k$, $k$ any
non-negative integer, is nested in one with $B = k + 1$. Divide the
former along midlines to frame its four quadrants, then block out
the latter skybox into a $4 \times 4$ grid of same-sized window
panes, partitioned by the one-cell-thick borders of its own midlines
into quadrants, each of which is further subdivided by the outside
edges of the 4 one-cell-thick label lines and their extensions to
the window's frame. These extended LLs are themselves NSLs, and
have $R, C$ values of $g$ and $g + {\bf S}$; for ${\bf S} = 15$,
they also adjoin NSLs along their outer edges whose $R, C$ values
are multiples of $8$ plus ${\bf S} \; mod \; 8$. These pane-framing
pairs of NSLs we will henceforth refer to (as a windowmaker would)
as \textit{muntins}. It is easy to calculate that while the inner
skybox has but one muntin each among its rows and columns, each
further nesting has $2^{B+1} - 1$. But we are getting ahead of
ourselves, as we still have two proofs to finish. Let's begin with
Four Corners, or
\medskip
\noindent {\small Theorem 12.} The 4 panes in the corners of the
16-paned $B = k + 1$ window are identical in contents and marks to
the analogously placed quadrants of the $B = k$ skybox.
\pagebreak
\noindent \textit{Proof.} Invoke the Zero-Padding Lemma with regard
to the U-indices, as the labels of the boxes in the corners of the
$B = k + 1$ ET are identical to those of the same-sized quadrants in
the $B = k$ ET, all labels $\geq$ the latter's $g$ only occurring in
the newly inserted region. $\;\; \blacksquare$
\medskip
\noindent \textit{Remarks.} For $N = 6$, all filled Four Corners
cells indicate edges belonging to $3$ Box-Kites, whose edges they in
fact exhaust. These $3$, not surprisingly, are the zero-padded
versions of the identically L-indexed trio which span the entirety
of the $N = 5$ ET. By calculations we'll see shortly, however, the
inner skybox, when considered as part of the $N = 6$ ET, has filled
cells belonging to all the other 16 Box-Kites, even though the
contents of these cells are identical to those in the $N = 5$ ET. As
$B$ increases, then, the ``sheets'' covering this same central
region must draw upon progressively more extensive networks of
interconnected Box-Kites. As we approach the fractal limit -- and
``the Sky is the limit'' -- these networks hence become scale-free.
(Corollarily, for $N = 7$, the Four Corners' cells exhaust all the
edges of the $N = 6$ ET's 19 Box-Kites, and so on.)
Unlike a standard fractal, however, such a Sky merits the prefix
``meta'': for each empty ET cell corresponds to a point in the
usual fractal variety; and each pair of filled ET cells, having
(r,c) of one = (c,r) of the other), correspond to diagonal-pairs in
Assessor planes, orthogonal to all other such diagonal-pairs
belonging to the other cells. Each empty ET cell, in other words,
not only corresponds to a point in the usual plane-confined fractal,
but belongs to the complement of the filled cells' infinite number
of \textit{dimensions} framing the Sky's \textit{meta-}fractal.
\medskip
We've one last thing to prove here. The French Windows Theorem
shows us the way the cell contents of the pairs of panes contained
between the $B = k + 1$ skybox's corners are generated from those of
the analogous pairings of quadrants in the $B = k$ skybox, by adding
$g$ to L-indices.
\medskip
\noindent {\small Theorem 13}. For each half-square array of cells
created by one or the other midline (the French windows), each cell
in the half-square parallel to that adjoining the midline (one of
the two shutters), but itself adjacent to the label-line delimiting
the former's bounds, has content equal to $g$ plus that of the cell
on the same line orthogonal to the midline, and at the same distance
from it, as \textit{it} is from the label-line. All the empty
long-diagonal cells then map to $g$ (and are marked), or $g + {\bf
S}$ (and are unmarked). Filled cells in extensions of the
label-lines bounding each shutter are calculated similarly, but with
reversed markings; all other cells in a shutter have the same marks
as their French-window counterparts.
\medskip
\noindent \textit{Preamble.} Note that there can be (as we shall
see when we speak of \textit{hide/fill involution}) cells left empty
for rule-based reasons other than $P \; = \; R \veebar C \; = \; 0
\;|
\; {\bf S}$. The shutter-based counterparts of such French-window
cells, unlike those of long-diagonal cells, remain empty.
\medskip
\noindent \textit{Proof.} The top and left (bottom and right)
shutters are equivalent: one merely switches row for column labels.
Top/left and bottom/right shutter-sets are likewise equivalent by
the symmetry of strut-opposites. We hence make the case for the
left shutter only. But for the novelties posed by the initially
blank cells and the label lines (with the only real subtleties
involving markings), the proof proceeds in a manner very similar to
Theorem 11: split into 3 cases, based on whether (1) the L-index
trip implied by the $R, C, P$ values is a Zigzag; (2) $u, v$ are
both Vents; or, (3) the edge signified by the cell content is the
emanation of same-inner-signed dyads (that is, one is a Vent, the
other a Zigzag).
\textit{Case 1:} Assume $(u, v, w)$ a Zigzag L-trip in the French
window's contained skybox; the general product in its shutter is
\begin{center}
$v \;\; - \;\; ({\bf G} + v_{opp})$
\underline{$\;\;\;(g + u) \;\; + \;\;({\bf G} + g +
u_{opp})\;\;\;$}
\smallskip
$- ({\bf G} + g + w_{opp}) \;\;\; + (g + w)$
\underline{$ - (g + w) \;\;\; + ({\bf G} + g + w_{opp})$}
\smallskip
$0$
\end{center}
\smallskip
$(g + u) \cdot v = P:$ $(u,v,w) \rightarrow (g + w, v, g + u)$;
hence, $(- (g + w))$.
$(g + u) \cdot ({\bf G} + v_{opp}) = P:$ $(u, w_{opp}, v_{opp})$
$\rightarrow (g + w_{opp}, g + u, v_{opp})$ $\rightarrow ({\bf G} +
v_{opp}, g + u, {\bf G} + g + w_{opp})$; dyads' opposite inner signs
make $({\bf G} + g + w_{opp})$ positive.
$({\bf G} + g + u_{opp}) \cdot v = P:$ $(u_{opp}, w_{opp}, v)$
$\rightarrow (g + w_{opp}, g + u_{opp}, v)$ $\rightarrow ({\bf G} +
g + u_{opp}, {\bf G} + g + w_{opp}, v)$; hence, $(- ({\bf G} + g +
w_{opp}))$.
$({\bf G} + g + u_{opp}) \cdot ({\bf G} + v_{opp}) = P:$ $(v_{opp},
u_{opp}, w)$ $\rightarrow$ $(v_{opp}, g + w, g + u_{opp})$
$\rightarrow$ $({\bf G} + g + u_{opp}, g + w, {\bf G} + v_{opp})$;
dyads' opposite inner signs make $(g + w)$ positive.
\medskip
\textit{Case 2:} The product of two Vents must have negative
edge-sign, hence negative inner sign in top dyad to lower dyad's
positive. The shutter product thus looks like this:
\pagebreak
\begin{center}
$(u_{opp}) - ({\bf G} + u)$
\underline{$\;(g + v_{opp}) \; + \;({\bf G} + g + v)\;$}
\smallskip
$+ ({\bf G} + g + w_{opp}) \;\;\;\;\; + (g + w)$
\underline{$ - (g + w) \;\;\;\;\; - ({\bf G} + g + w_{opp})$}
\smallskip
$0$
\end{center}
\smallskip
$(g + v_{opp}) \cdot u_{opp} = P:$ $(v_{opp}, u_{opp}, w)$
$\rightarrow (g + w, u_{opp}, g + v_{opp})$; hence, $(- (g + w))$.
$(g + v_{opp}) \cdot ({\bf G} + u) = P:$ $(v_{opp}, u, w_{opp})$
$\rightarrow (g + w_{opp}, u, g + v_{opp})$ $\rightarrow ({\bf G} +
u, {\bf G} + g + w_{opp}, g + v_{opp})$; but dyads' inner signs are
opposite, so $(- ({\bf G} + g + w_{opp}))$.
$({\bf G} + g + v) \cdot u_{opp} = P:$ $(u_{opp}, w_{opp}, v)$
$\rightarrow (u_{opp}, g + v, g + w_{opp})$ $\rightarrow (u_{opp},
{\bf G} + g + w_{opp}, {\bf G} + g + v)$; hence, $(+ ({\bf G} + g +
w_{opp}))$.
$({\bf G} + g + v) \cdot ({\bf G} + u) = P:$ $(u, v, w)$
$\rightarrow$ $(u, g + w, g + v)$ $\rightarrow$ $({\bf G} + g + v, g
+ w, {\bf G} + u)$; but dyads' inner signs are opposite, so $(+ (g +
w))$.
\medskip
\textit{Case 3:} The product of Vent and Zigzag displays same inner
sign in both dyads; hence the following arithmetic holds:
\smallskip
\begin{center}
$(u_{opp}) + ({\bf G} + u)$
\underline{$\; (g + v) \; + \;({\bf G} + g + v_{opp})\;$}
\smallskip
$+({\bf G} + g + w) \;\;\;\;\; +( g + w_{opp})$
\underline{$ - (g + w_{opp}) \;\;\;\;\; - ({\bf G} + g + w)$}
\smallskip
$0$
\end{center}
As with the last case in Theorem 11, we omit the term-by-term
calculations for this last case, as they should seem ``much of a
muchness'' by this point. What is clear in all three cases is that
index values of shutter cells have same markings as their
French-window counterparts, at least for all cells which
\textit{have} markings in the contained skybox; but, in all cases,
indices are augmented by $g$.
The assignment of marks to the shutter-cells linked to blank cells
in French windows is straightforward for Type I box-kites: since
any containing skybox must have $g > {\bf S}$, and since $g + s$ has
$g$ as its strut opposite, then the First Vizier tells us that any
$g$ must be a Vent. But then the $R, C$ indices of the cell
containing $g$ must belong to a Trefoil in such a box-kite; hence,
one is a Vent, the other a Zigzag, and $g$ must be marked. Only if
the $R, C, P$ entry in the ET is necessarily confined to a Type II
box-kites will this not necessarily be so. But Part II's Appendix B
made clear that Type II's are generated by \textit{excluding} $g$
from their L-indices: recall that, in the Pathions, for all ${\bf
S}$ < 8, all and only Type II box-kites are created by placing one
of the Sedenion Zigzag L-trips on the ``Rule 0'' circle of the
PSL(2,7) triangle with $8$ in the middle (and hence excluded). This
is a box-kite in its own right (one of the 7 ``Atlas'' box-kites
with ${\bf S} = 8$); its 3 sides are ``Rule 2'' triplets, and
generate Type II box-kites when made into zigzag L-index sets.
Conversely, all Pathion box-kites containing an '8' in an L-index
(dubbed ''strongboxes'' in Appendix B) are Type I. Whether something
peculiar might occur for large $N$ (where there might be multiple
powers of 2 playing roles in the same box-kite) is a matter of
marginal interest to present concerns, and will be left as an open
question for the present. We merely note that, by a similar
argument, and with the same restrictions assumed, $g + {\bf S}$ must
be a Zigzag L-index, and $R, C$ either both be likewise (hence, $g +
{\bf S}$ is unmarked); or, both are Vents in a Trefoil (so $g + {\bf
S}$ must be unmarked here too).
The last detail -- reversal of label-line markings in their
$g$-augmented shutter-cell extensions -- is demonstrated as follows,
with the same caveat concerning Type II box-kites assumed to apply.
Such cells house DMZs (just swap $u$ for $g + u$ in Theorem 11's
first setup -- they form a Rule 1 trip -- and compute). The LL
extension on top has row-label $g$; that along the bottom, the
strut-opposite $g + {\bf S}$. Given trip $(u,v,w)$, the shutter-cell
index for $R, C = (g, u)$ corresponds to French-window index for $R,
C = (g, g + u)$. But $(u, g, g+u)$ is a Trefoil, since $g$ is a
Vent. So if $u$ is one too, $g + u$ isn't; hence marks are reversed
as claimed. $\;\;\blacksquare$
\section{Maximal High-Bit Singletons: (s,g)-Modularity for ${\bf 16 < S \leq 24}$}
The Whorfian Sky, having but one high bit in its strut constant, is
the simplest possible meta-fractal -- the first of an infinite
number of such infinite-dimensional zero-divisor-spanned spaces. We
can consider the general case of such singleton high-bit
recursiveness in two different, complementary ways. First, we can
supplement the just-concluded series of theorems and proofs with a
calculational interlude, where we consider the iterative embeddings
of the Pathion Sand Mandalas in the infinite cascade of
boxes-within-boxes that a Sky oversees. Then, we can generalize
what we saw in the Pathions to consider the phenomenology of strut
constants with singleton high-bits, which we take to be any bits
representing a power of $2 \geq 3$ if ${\bf S}$ contains low bits
(is not a multiple of $8$), else a power of $2$ strictly greater
than $3$ otherwise. Per our earlier notation, $g = {\bf G/2}$ is the
highest such singleton bit possible. We can think of its
exponential increments -- equivalent to left-shifts in bit-string
terms -- as the side-effects of conjoint zero-padding of $N$ and
${\bf S}$. This will be our second topic in this section.
Maintaining our use of ${\bf S = 15}$ as exemplary, we have already
seen that NSLs come in quartets: a row and column are each headed
by ${\bf S} \; mod \; g$ (henceforth, $s$) and $g$, hence $7$ and
$8$ in the Sand Mandalas. But each recursive embedding of the
current skybox in the next creates further quartets. Division down
the midlines to insert the indices new to the next CDP generation
induces the Sand Mandala's adjoining strut-opposite sets of $s$ and
$g$ lines (the pane-framing muntins) to be displaced to the borders
of the four corners and shutters, with the new skybox's $g$ and $g +
s$ now adjoining the old $s$ and $g$ to form new muntins, on the
right and left respectively, while $g + g/2$ (the old ${\bf G} + g)$
and its strut opposite form a third muntin along the new midlines.
Continuing this recursive nesting of skyboxes generates 1, 3, 7,
$\cdots$, $2^{B+1}-1$ row-and-column muntin pairs involving
multiples of $8$ and their supplementings by $s$, where (recalling
earlier notation) $B = 0$ for the inner skybox, and increments by
$1$ with each further nesting. Put another way, we then have a
muntin number $\mu = (2^{N-4} - 1)$, or $4\mu$ NSL's in all.
The ET for given $N$ has $(2^{N-1}-2)$ cells in each row and column.
But NSLs divvy them up into boxes, so that each line is crossed by
$2 \mu$ others, with the 0, 2 or 4 cells in their overlap also
belonging to diagonals. The number of cells in the overlap-free
segments of the lines, or $\omega$, is then just $4 \mu \cdot
(2^{N-1} - 2 - 2 \mu ) = 24 \mu ( \mu + 1 )$: an integer number of
Box-Kites. For our ${\bf S = 15}$ case, the minimized line shuffling
makes this obvious: all boxes are 6 x 6, with 2-cell-thick
boundaries (the muntins separating the panes), with $\mu$
boundaries, and $( \mu + 1)$ overlap-free cells per each row or
column, per each quartet of lines.
The contribution from diagonals, or $\delta$, is a little more
difficult, but straightforward in our case of interest: 4 sets of
$1, 2, 3, \cdots, \mu$ boxes are spanned by moving along
\textit{one} empty long diagonal before encountering the
\textit{other}, with each box contributing 6, and each overlap zone
between adjacent boxes adding 2. Hence, $\delta = 24 \cdot (2^{N-3}
- 1) (2^{N-3} - 2)/6$ -- a formula familiar from associative-triplet
counting: it also contributes an integer number of Box-Kites. The
one-liner we want, then, is this:
\smallskip
\begin{center}
{\small $BK_{N,\; 8 < {\bf S} < 16} = \omega + \delta =
(2^{N-4})(2^{N-4} - 1) \; + \; (2^{N-3} - 1)(2^{N-3} - 2)/6$}
\end{center}
\smallskip
For $N = 4, 5, 6, 7, 8, 9, 10$, this formula gives $0, 3, 19, 91,
395$, $1643, 6699$. Add $4^{N-4}$ to each -- the immediate
side-effect of the offing of all four Rule 0 candidate trips of the
Sedenion Box-Kite exploded into the Sand Mandala that begins the
recursion -- and one gets ``d\'ej\`a vu all over again'': $1$, $7$,
$35$, $155$, $651$, $2667$, $10795$ -- the full set of Box-Kites for
${\bf S \leq 8}$.
It would be nice if such numbers showed up in unsuspected places,
having nothing to do with ZDs. Such a candidate context does, in
fact, present itself, in Ed Pegg's regular MAA column on ``Math
Games'' focusing on ``Tournament Dice.'' [7] He asks us, ``What
dice make a non-transitive four player game, so that if three dice
are chosen, a fourth die in the set beats all three? How many dice
are needed for a five player non-transitive game, or more?'' The
low solution of 3 explicitly involves PSL(2,7); the next solution of
19 entails calculations that look a lot like those involved in
computing row and column headers in ETs. No solutions to the
dice-selecting game beyond 19 are known. The above formulae,
though, suggest the next should be 91. Here, ZDs have no apparent
role save as dummies, like the infinity of complex dimensions in a
Fourier-series convergence problem, tossed out the window once the
solution is in hand. Can a number-theory fractal, with
intrinsically structured cell content (something other, non-meta,
fractals lack) be of service in this case -- and, if not in this
particular problem, in others like it?
Now let's consider the more general situation, where the singleton
high-bit can be progressively left-shifted. Reverting to the use of
the simplest case as exemplary, use ${\bf S = g + 1 = 9}$ in the
Pathions, then do tandem left-shifts to produce this sequence: $N =
6, \; {\bf S = g + 1}$ ${\bf = 17}$; $N = 7$, ${\bf S = }$ ${\bf g +
1 = 33};\; \cdots;\; N = K$, ${\bf S = g + 1} = 2^{K-2} + 1$. A
simple rule governs these ratchetings: in all cases, the number of
filled cells = $6 \cdot (2^{N-1} - 4)$, since there are two sets of
parallel sides which are filled but for long-diagonal intersections,
and two sets of $g$ and $1$ entries distributed one per row along
orthogonals to the empty long diagonals. Hence, for the series just
given, we have cell counts of $72,\; 168,\; \cdots,\; 6 \cdot (2^{N
- 1} - 4)$ for $BK_{N,\;S} = 3,\;7,\; \cdots,\;2^{N - 3} - 1$, for
$g < {\bf S} < g + 8 = {\bf G}$ in the Pathions, and all $g < {\bf
S} \leq g + 8$ in the Chingons, $2^{7}$-ions, and general
$2^{N}$-ions, in that order.
Algorithmically, the situation is just as easy to see: the
splitting of dyads, sending U- and L- indices to strut-opposite
Assessors, while incorporating the ${\bf S}$ and ${\bf G}$ of the
current CDP generation as strut-opposites in the next, continues.
For ${\bf S = 17}$ in the Chingons, there are now $2^{N-3}-1 = 7$,
not $3$, Box-Kites sharing the new $g = 16$ (at B) and ${\bf S}
\;mod \;g = 1$ (at E) in our running example. The U- indices of the
Sand Mandala Assessors for ${\bf S = g + 1 = 9}$ are now L-indices,
and so on: every integer $< G$ and $\neq {\bf S}$ gets to be an
L-index of one of the $30 (= 2^{N-1} - 2)$ Assessors, as $16$ and
${\bf S} \;mod \; g = 1$ appear in each of the $7$ Box-Kites, with
each other eligible integer appearing once only in one of the $7
\cdot 4 = 28$ available L-index slots.
As an aside, in all 7 cases, writing the smallest Zigzag L-index at
$a$ mandates all the Trefoil trips be ``precessed'' -- a phenomenon
also observed in the ${\bf S = 8}$ Pathion case, as tabulated on p.
14 of [8]. For Zigzag L-index set $(2, 16, 18)$, for instance,
$(a,d,e)$ $=$ $(2,3,1)$ instead of $(1,2,3)$; $(f,c,e)$ $=$
$(19,18,1)$ not $(1,19,18)$; and $(f,d,b)$ $=$ $(19,3,16)$. But
otherwise, there are no surprises: for $N=7$, there are $(2^{7 - 3}
- 1) = 15$ Box-Kites, with all $62 ( = 2^{N-1} - 2)$ available cells
in the rows and columns linked to labels $g$ and ${\bf S}\; mod \;
g$ being filled, and so on.
Note that this formulation obtains for any and all ${\bf S > 8}$
where the maximum high-bit (that is, $g$) is included in its
bitstring: for, with $g$ at B and ${\bf S} \;mod \; g$ at E,
whichever ${\bf R, C}$ label is not one of these suffices to
completely determine the remaining Assessor L-indices, so that no
other bits in ${\bf S}$ play a role in determining any of them.
Meanwhile, cell \textit{contents} ${\bf P}$ containing either $g$ or
${\bf S} \; mod \; g$, but created by XORing of row and column
labels equal to neither, are arrayed in off-diagonal pairs, forming
disjoint sets parallel or perpendicular to the two empty ones. If
we write ${\bf S} \;mod \; g$ with a lower-case $s$, then we could
call the rule in play here \textit{(s,g)-modularity}. Using the
vertical pipe for logical or, and recalling the special handling
required by the 8-bit when ${\bf S}$ is a multiple of 8 (which we
signify with the asterisk suffixed to ``mod''), we can shorthand its
workings this way:
\medskip
\noindent {\small Theorem 14}. For a $2^{N}$-ion inner skybox whose
strut constant ${\bf S}$ has a singleton high-bit which is maximal
(that is, equal to $g = {\bf G/2} = 2^{N-2}$), the recipe for its
filled cells can be condensed thus:
\smallskip
\begin{center}
${\bf R\; |\; C |\; P} = g\; |\; {\bf
S} \; mod^{*} \; g$
\end{center}
Under recursion, the recipe needs to be modified so as to include
not just the inner-skybox $g$ and ${\bf S} \; mod^{*} \; g$
(henceforth, simply lowercase $s$), but all integer multiples $k$ of
$g$ less than the ${\bf G}$ of the outermost skybox, plus their
strut opposites $k \cdot g + s$.
\medskip
\noindent \textit{Proof}. The theorem merely boils down the
computational arguments of prior paragraphs in this section, then
applies the last section's recursive procedures to them. The first
claim of the proof is identical to what we've already seen for Sand
Mandalas, with zero-padding injected into the argument. The second
claim merely assumes the area quadrupling based on midline
splitting, with the side-effects already discussed. No formal
proof, then, is called for beyond these points. $\;\;\blacksquare$
\medskip
\noindent \textit{Remarks}. Using the computations from two
paragraphs prior to the theorem's statement, we can readily
calculate the box-kite count for any skybox, no matter how deeply
nested: recall the formula $6 \cdot (2^{N - 1} - 4)$ for
$BK_{N,\;S} = 2^{N - 3} - 1$. It then becomes a straightforward
matter to calculate, as well, the limiting ratio of this count to
the maximal full count possible for the ET as $N \rightarrow
\infty$, with each cell approaching a point in a standard 2-D
fractal. Hence, for any ${\bf S}$ with a singleton high-bit in
evidence, there exists a Sky containing all recursive redoublings of
its inner skybox, and computations like those just considered can
further be used to specify fractal dimensions and the like. (Such
computations, however, will not concern us.) Finally, recall that,
by spectrographic equivalence, all such computations will lead to
the same results for each ${\bf S}$ value in the same spectral band
or octave.
\section{Hide/Fill Involution: Further-Right High-Bits with ${\bf 24 < S < 32}$.}
Recall that, in the Sand Mandala flip-book, each increment of ${\bf
S}$ moved the two sets of orthogonal parallel lines one cell closer
toward their opposite numbers: while ${\bf S = 9}$ had two
filled-in rows and columns forming a square missing its corners, the
progression culminating in ${\bf S = 15}$ showed a cross-hairs
configuration: the parallel lines of cells now abutted each other
in 2-ply horizontal and vertical arrays. The same basic progression
is on display in the Chingons, starting with ${\bf S = 17}$. But
now the number of strut-opposite cell pairs in each row and column
is 15, not 7, so the cross-hairs pattern can't arise until ${\bf S =
31}$. Yet it never arises in quite the manner expected, as
something quite singular transpires just after flipping past the ET
in the middle, for ${\bf S = 24}$. Here, rows and columns labeled
$8$ and $16$ constrain a square of empty cells in the center
$\cdots$ quickly followed by an ET which seems to continue the
expected trajectory -- except that almost all the non-long-diagonal
cells left empty in its predecessor ETs are now inexplicably filled.
More, there is a method to the ``almost all'' as well: for we now
see not 2, but 4 rows and columns, all being blanked out while those
labeled with $g$ and ${\bf S} \; mod \; g$ are being filled in.
This is an inevitable side effect of a second high-bit in ${\bf S}$:
we call this phenomenon, first appearing in the Chingons,
\textit{hide/fill involution}. There are 4, not 2, line-pairs,
because ${\bf S}$ and ${\bf G}$, modulo a lower power of 2 (because
devolving upon a prior CDP generation's $g$), offer twice the
possibilities: for ${\bf S = 25}$, ${\bf S} \; mod \; 16$ is now
$9$, but ${\bf S} \; mod\; 8$ can result in either $1$ or $17$ as
well -- with correlated \textit{multiples} of $8$ ($8$ proper, and
$24$) defining the other two pairings. All cells with ${\bf R \; |
C \; | \; P}$ equal to one of these 4 values, but for the handful
already set to ``on'' by the first high-bit, will now be set to
``off,'' while all other non-long-diagonal cells set to ``off'' in
the Pathion Sand Mandalas are suddenly ``on.'' What results for
each Chingon ET with $24 < {\bf S} < 32$ is an ensemble comprised of
$23$ Box-Kites. (For the flip-book, see Slides 40 -- 54.) Why does
this happen? The logic is as straightforward as the effect can seem
mysterious, and is akin, for good reason, to the involutory effect
on trip orientation induced by Rule 2 addings of ${\bf G}$ to 2 of
the trip's 3 indices.
In order to grasp it, we need only to consider another pair of
abstract calculation setups, of the sort we've seen already many
times. The first is the core of the Two-Bit Theorem, which we state
and prove as follows:
\medskip
\noindent {\small Theorem 15}. $2^{N}$-ion dyads making DMZs before
augmenting ${\bf S}$ with a new high-bit no longer do so after the
fact.
\medskip
\noindent \textit{Proof}. Suppose the high-bit in the bitstring
representation of ${\bf S}$ is $2^{K},\;K < (N-1)$. Suppose further
that, for some L-index trip $(u,v,w)$, the Assessors $U$ and $V$ are
DMZ's, with their dyads having same inner signs. (This last
assumption is strictly to ease calculations, and not substantive: we
could, as earlier, use one or more binary variables of the $sg$ type
to cover all cases explicitly, including Type I vs. Type II
box-kites. To keep things simple, we assume Type I in what
follows.) We then have $(u + u \cdot X)(v + v \cdot X) = (u + U)(v +
V) = 0$. But now suppose, without changing $N$, we add a bit
somewhere further to the left to ${\bf S}$, so that ${\bf S} <
(2^{K} = L) < {\bf G}$. The augmented strut constant now equals
${\bf S_{L}} = {\bf S + L}$. One of our L-indices, say $v$, belongs
to a Vent Assessor thanks to the assumed inner signing; hence, by
Rule 2 and the Third Vizier, $(V,v,X) \rightarrow (X + L, v, V +
L)$. Its DMZ partner $u$, meanwhile, must thereby be a Zigzag
L-index, which means $(u,U,X) \rightarrow (u, X + L, U + L)$. We
claim the truth of the following arithmetic:
\smallskip
\begin{center}
$v \; + \; (V + L)$ \\
\underline{$ \;\;\; u \; + \; (U + L)\;\;\;$} \\
$+(W \;+ \; L) \; + w$ \\
\underline{$+ \; w \;\; - (W + L)$} \\
NOT ZERO (+w's don't cancel) \\
\end{center}
\smallskip
The left bottom product is given. The product to its right is
derived as follows: since $u$ is a Zigzag L-index, the Trefoil
U-trip $(u,V,W)$ has the same orientation as $(u,v,w)$, so that Rule
2 $\rightarrow (u, W+L, V+L)$, implying the negative result shown.
The left product on the top line, though, has terms derived from a
Trefoil U-trip lacking a Zigzag L-index, so that only after Rule 2
reversal are the letters arrayed in Zigzag L-trip order: $(U + L,
v, W + L)$. Ergo, $+ (W+L)$. Similarly for the top right: Rule 2
reversal ``straightens out'' the Trefoil U-trip, to give $(U + L, V
+ L, w)$; therefore, $(+ w)$ results. If we explicitly covered
further cases by using an $sg$ variable, we would be faced with a
Theorem 2 situation: one or the other product pair cancels, but not
both. $\;\;\blacksquare$
\medskip
\noindent \textit{Remark.} The prototype for the phenomenon this
theorem covers is the ``explosion'' of a Sedenion box-kite into a
trio of interconnected ones in a Pathion sand mandala, with the
${\bf S}$ of the latter = the ${\bf X}$ of the former. As part of
this process, 4 of the expected 7 are ``hidden'' box-kites (HBKs),
with no DMZs along their edges. These have zigzag L-trips which are
precisely the L-trips of the 4 Sedenion Sails. Here, an empirical
observation which will spur more formal investigations in a sequel
study: for the 3 HBKs based on trefoil L-trips, exactly 1 strut has
reversed orientation (a different one in each of them), with the
orientation of the triangular side whose midpoint it ends in also
being reversed. For the HBK based on the zigzag L-trip, all 3
struts are reversed, so that the flow along the sides is exactly the
reverse of that shown in the ``Rule 0'' circle. (Hence, all
possible flow patterns along struts are covered, with only those
entailing 0 or 2 reversals corresponding to functional box-kites:
our Type I and Type II designations.) It is not hard to show that
this zigzag-based HBK has another surprising property: the 8 units
defined by its own zigzag's Assessors plus ${\bf X}$ and the real
unit form a ZD-free copy of the Octonions. This is also true when
the analogous Type II situation is explored, albeit for a slightly
different reason: in the former case, all 3 Catamaran ``twistings''
take the zigzag edges to other HBKs; in the latter, though, the pair
of Assessors in some other Type II box-kite reached by ``twisting''
-- $(a,B)$ and $(A,b)$, say, if the edge be that joining Assessors A
and B, with strut-constant $c_{opp} = d$ -- are \textit{strut
opposites}, and hence also bereft of ZDs. The general picture seems
to mirror this concrete case, and will be studied in ``Voyage by
Catamaran'' with this expectation: the bit-twiddling logic that
generates meta-fractal ``Skies'' also underwrites a means for
jumping between ZD-free Octonion clones in an infinite number of
HBKs housed in a Sky. Given recent interest in pure ``E8'' models
giving a privileged place to the basis of zero-divisor theory,
namely ``G2'' projections (viz., A. Garrett Lisi's ``An
Exceptionally Simple Theory of Everything''); a parallel vogue for
many-worlds approaches; and, the well-known correspondence between
8-D closest-packing patterns, the loop of the 240 unit Octonions
which Coxeter discovered, and E8 algebras -- given all this,
tracking the logic of the links across such Octonionic ``brambles''
might prove of great interest to many researchers.
\medskip
Now, we still haven't explained the flipside of this off-switch
effect, to which prior CDP generation Box-Kites -- appropriately
zero-padded to become Box-Kites in the current generation until the
new high-bit is added to the strut-constant -- are subjected. How
is it that previously empty cells \textit{not} associated with the
second high-bit's blanked-out R, C, P values are now \textit{full}?
The answer is simple, and is framed in the Hat-Trick Theorem this
way.
\medskip
\noindent {\small Theorem 16}. Cells in an ET which represent DMZ
edges of some $2^{N}$-ion Box-Kites for some fixed ${\bf S}$, and
which are offed in turn upon augmenting of ${\bf S}$ by a new
leftmost bit, are turned on once more if ${\bf S}$ is augmented by
yet another new leftmost bit.
\medskip
\noindent \textit{Proof}. We begin an induction based upon the
simplest case (which the Chingons are the first $2^{N}$-ions to
provide): consider Box-Kites with ${\bf S \leq 8}$. If a high-bit be
appended to ${\bf S}$, then the associated Box-Kites are offed.
However, if \textit{another} high-bit be affixed, these dormant
Box-Kites are re-awakened -- the second half of \textit{hide/fill
involution}. We simply assume an L-index set $(u,v,w)$ underwriting
a Sail in the ET for the pre-augmented ${\bf S}$, with Assessors
$(u, U)$ and $(v, V)$. Then, we introduce a more leftified bit
$2^{Q} = M$, where pre-augmented ${\bf S} < L < M < {\bf G}$, then
compute the term-by term products of $(u + (U + L + M))$ and $(v +
sg \cdot (V + L + M))$, using the usual methods. And as these
methods tell us that two applications of Rule 2 have the same effect
as none in such a setup, we have no more to prove.
$\;\;\blacksquare$
\medskip
\noindent \textit{Corollary}. The induction just invoked makes it
clear that strut constants equal to multiples of $8$ not powers of
$2$ are included in the same spectral band as all other integers
larger than the prior multiple. The promissory note issued in the
second paragraph of Part II's concluding section, on 64-D
Spectrography, can now be deemed redeemed.
\medskip
In the Chingons, high-bits $L$ and $M$ are necessarily adjacent in
the bitstring for ${\bf S < G = 32}$; but in the general $2^{N}$-ion
case, $N$ large, zero-padding guarantees that things will work in
just the same manner, with only one difference: the recursive
creation of ``harmonics'' of relatively small-$g$ $(s,g)$-modular
${\bf R, C, P}$ values will propagate to further levels, thereby
effecting overall Box-Kite counts.
In general terms, we have echoes of the formula given for
$(s,g)$-modular calculations, but with this signal difference: there
will be \textit{one} such rule for \textit{each} high-bit $2^{H}$ in
${\bf S}$, where residues of ${\bf S}$ modulo $2^{H}$ will generate
their own near-solid lines of rows and columns, be they hidden or
filled. Likewise for multiples of $2^{H} < {\bf G}$ which are not
covered by prior rules, and multiples of $2^H$ supplemented by the
bit-specific residue (regardless of whether $2^{H}$ itself is
available for treatment by this bit-specific rule). In the simplest,
no-zero-padding instances, all even multiples are excluded, as they
will have occurred already in prior rules for higher bits, and fills
or hides, once fixed by a higher bit's rule, cannot be overridden.
Cases with some zero-padding are not so simple. Consider this
two-bit instance, ${\bf S = 73}, N = 8$: the fill-bit is 64, the
hide-bit is just 8, so that only 9 and 64 generate NSLs of filled
values; all other multiples of 8, and their supplementing by 1
(including 65) are NSLs of hidden values. Now look at a variation on
this example, with the single high-bit of zero-padding removed --
i.~e., ${\bf S = 41}, N = 8$. Here, the fill-bit is 32, and its
multiples 64 and 96, as well as their supplements by ${\bf S} \;
modulo \; 32 \; = \; 9$, or 9 and 73 and 105, label NSLs of filled
values; but all other multiples of 8, plus all multiples of 8
supplemented by 1 not equal to 9 or 73 or 105, label NSLs of hidden
values. Cases with multiple fill and hide bits, with or without
additional zero-padding, are obviously even more complicated to
handle explicitly on a case-by-case basis, but the logic framing the
rules remain simple; hence, even such messy cases are
programmatically easy to handle.
Hide/fill involution means, then, that the first, third, and any
further odd-numbered high-bits (counting from the left) will
generate ``fill'' rules, whereas all the even-numbered high-bits
generate ``hide'' rules -- with all cells not touched by a rule
being either hidden (if the total number of high-bits $B$ is odd) or
filled ($B$ is even).
Two further examples should make the workings of this protocol more
clear. First, the Chingon test case of ${\bf S = 25}$: for $({\bf R
\; | \; C \; | \; P} \; = \; 9 \;| \; 16)$, all the ET cells are
filled; however, for $({\bf R \; | \; C \; | \; P} \; = 1 \;|\; 8
\;|\; 17 \;|\; 24)$, ET cells not already filled by the first rule
(and, as visual inspection of Slide 48 indicates, there are only 8
cells in the entire 840-cell ET already filled by the prior rule
which the current rule would like to operate on) are hidden from
view. Because the 16- and 8- bits are the only high-bits, the count
of same is even, meaning all remaining ET cells not covered by these
2 rules are filled.
We get 23 for Box-Kite count as follows. First, the 16-bit rule
gives us 7 Box-Kites, per earlier arguments; the 8-bit rule, which
gives 3 filled Box-Kites in the Pathions, recursively propagates to
cover 19 hidden Box-Kites in the Chingons, according to the formula
produced last section. But hide/fill involution says that, of the 35
maximum possible Box-Kites in a Chingon ET, $35 - 19 = 16$ are now
made visible. As none of these have the Pathion ${\bf G = 16}$ as
an L-index, and all the 7 Box-Kites from the 16-bit rule
\textit{do}, we therefore have a grand total of $7 + 16 = 23$
Box-Kites in the ${\bf S =25}$ ET, as claimed (and as cell-counting
on the cited Slide will corroborate).
The concluding Slides 76--78 present a trio of color-coded
``histological slices'' of the hiding and filling sequence
(beginning with the blanking of the long diagonals) for the simplest
3-high-bit case, $N = 7, {\bf S = 57}$. Here, the first fill rule
works on 25 and 32; the first hide rule, on 9, 16, 41, and 48; the
second fill rule, on 1, 8, 17, 24, 33, 40, 49, and 56; and the rest
of the cells, since the count of high-bits is odd, are left blank.
We do not give an explicit algorithmic method here, however, for
computing the number of Box-Kites contained in this 3,720-cell ET.
Such recursiveness is best handled programmatically, rather than by
cranking out an explicit (hence, long and tedious) formula, meant
for working out by a time-consuming hand calculation. What we can
do, instead, is conclude with a brief finale, embodying all our
results in the simple ``recipe theory'' promised originally, and
offer some reflections on future directions.
\section{Fundamental Theorem of Zero-Divisor Algebra}
All of the prior arguments constitute steps sufficient to
demonstrate the Fundamental Theorem of Zero-Divisor Algebra. Like
the role played by its Gaussian predecessor in the legitimizing of
another ``new kind of [complex] number theory,'' its simultaneous
simplicity and generality open out on extensive new vistas at once
alien and inviting. The Theorem proper can be subdivided into a
Proposition concerning all integers, and a ``Recipe Theory''
pragmatics for preparing and ``cooking'' the meta-fractal entities
whose existence the proposition asserts, but cannot tell us how to
construct.
\medskip
\noindent \textit{Proposition:} Any integer $K > 8$ not a power of
$2$ can uniquely be associated with a Strut Constant ${\bf S}$ of ZD
ensembles, whose inner skybox resides in the $2^{N}$-ions with
$2^{N-2} < K < 2^{N-1}$. The bitstring representation of ${\bf S}$
completely determines an infinite-dimensional analog of a standard
plane-confined fractal, with each of the latter's points associated
with an empty cell in the infinite Emanation Table, with all
non-empty cells comprised wholly of mutually orthogonal primitive
zero-divisors, one line of same per cell.
\medskip
\noindent \textit{Preparation:} Prepare each suitable ${\bf S}$ by
producing its bitstring representation, then determining the number
of high-bits it contains: if ${\bf S}$ is a multiple of 8,
right-shift 4 times; otherwise, right-shift 3 times. Then count the
number $B$ of 1's in the shortened bitstring that results. For this
set \verb|{B}| of $B$ elements, construct two same-sized arrays,
whose indices range from $1$ to $B$: the array \verb|{i}| which
indexes the left-to-right counting order of the elements of
\verb|{B}|; and, the array \verb|{P}| which indexes the powers of
$2$ of the same element in the same left-to-right order. (Example:
if $K = 613$, the inner skybox is contained in the $2^{11}$-ions; as
the number is not a multiple of $8$, the bistring representation
$1001100101$ is right-shifted thrice to yield the substring of
high-bits $1001100$; $B = 3$, and for $1 \; \leq \; i \; \leq \; 3$,
$P_{1} = 9, \; P_{2} = 6; P_{3} = 5$.)
\medskip
\noindent \textit{Cookbook Instructions:} \begin{description}
\item \verb|[0]| ~ For a given strut-constant ${\bf S}$, compute
the high-bit count $B$ and bitstring arrays \verb|{i}| and
\verb|{P}|, per preparation instructions.
\item \verb|[1]|~ Create a square spreadsheet-cell array, of edge-length $2^{I}$,
where $I \geq {\bf G/2} = g$ of the inner skybox for ${\bf S}$, with
the Sky as the limit when $I \rightarrow \infty$.
\item \verb|[2]| ~ Fill in the labels along all four edges, with those running
along the right (bottom) borders identical to those running along
the left (top), except in reversed left-right (top-bottom) order.
Refer to those along the top as column numbers $C$, and those along
the left edge, as row numbers $R$, setting candidate contents of any
cell (r,c) to $R \veebar C = P$.
\item \verb|[3]| ~ Paint all cells along the long diagonals of the
spreadsheet just constructed a color indicating BLANK, so that all
cells with $R = C$ (running down from upper left corner) else $R
\veebar C = {\bf S}$ (running down from upper right) have their
$P$-values hidden.
\item \verb|[4]| ~ For $1 \; \leq \; i \; \leq \; B$, consider for painting only
those cells in the spreadsheet created in \verb|[1]| with $R \; | \;
C \; | \; P \; = \; m \cdot 2^{\gamma} \; | \; m \cdot 2^{\gamma} \;
+ \; \sigma$, where $\gamma = P_{i}, \sigma = {\bf S} \; mod* \;
2^{\gamma}$, and $m$ is any integer $\geq 0$ (with $m = 0$ only
producing a legitimate candidate for the right-hand's second option,
as an XOR of $0$ indicates a long-diagonal cell).
\item \verb|[5]| ~ If a candidate cell has already been painted
by a prior application of these instructions to a prior value of
$i$, leave it as is. Otherwise, paint it with $R \veebar C$ if $i =
$ odd, else paint it BLANK.
\item \verb|[6]| ~ Loop to \verb|[4]| after incrementing $i$. If $i
< B$, proceed until this step, then reloop, reincrement, and retest
for $i = B$. When this last condition is met, proceed to the next
step.
\item \verb|[7]| ~ If $B$ is odd, paint all cells not already
painted, BLANK; for $B$ even, paint them with $R \veebar C$.
\end{description}
\medskip
In these pseudocode instructions, no attention is given to edge-mark
generation, performance optimization, or other embellishments.
Recursive expansion beyond the chosen limits of the $2^{N}$-ion
starting point is also not addressed. (Just keep all painted cells
as is, then redouble until the expanded size desired is attained;
compute appropriate insertions to the label lines, then paint all
new cells according to the same recipe.) What should be clear,
though, is any optimization cannot fail to be qualitatively more
efficient than the code in the appendix to [9], which computes on a
cell-by-cell basis. For ${\bf S} > 8, \; N > 4$, we've reached the
onramp to the Metafractal Superhighway: new kinds of efficiency,
synergy, connectedness, and so on, would seem to more than
compensate for the increase in dimension.
It is well-known that Chaotic attractors are built up from fractals;
hence, our results make it quite thinkable to consider Chaos Theory
from the vantage of pure Number $\cdots$ and hence the switch from
one mode of Chaos to another as a bitstring-driven -- or, put
differently, a cellular automaton-type -- process, of Wolfram's
Class 4 complexity. Such switching is of the utmost importance in
coming to terms with the most complex finite systems known: human
brains. The late Francisco Varela, both a leading visionary in
neurological research and its computer modeling, and a long-time
follower of Madhyamika Buddhism who'd collaborated with the Dalai
Lama in his ``Tibetan Buddhists talk with brain scientists''
dialogues [10], pointed to just the sorts of problems being
addressed here as the next frontier. In a review essay he
co-authored in 2001 just before his death [11, p. 237], we read
these concluding thoughts on the theme of what lies ``Beyond
Synchrony'' in the brain's workings:
\begin{quote}
The transient nature of coherence is central to the entire idea of
large-scale synchrony, as it underscores the fact that the system
does not behave dynamically as having stable attractors [e.g.,
Chaos], but rather metastable patterns -- a succession of
self-limiting recurrent patterns. In the brain, there is no
``settling down'' but an ongoing change marked only by transient
coordination among populations, as the attractor itself changes
owing to activity-dependent changes and modulations of synaptic
connections.
\end{quote}
Varela and Jean Petitot (whose work was the focus of the intermezzo
concluding Part I, in which semiotically inspired context the Three
Viziers were introduced) were long-time collaborators, as evidenced
in the last volume on \textit{Naturalizing Phenomenology} [12] which
they co-edited. It is only natural then to re-inscribe the theme of
mathematizing semiotics into the current context: Petitot offers
separate studies, at the ``atomic'' level where Greimas' ``Semiotic
Square'' resides; and at the large-scale and architectural, where
one must place L\'evi-Strauss's ``Canonical Law of Myths.'' But the
pressing problem is finding a smooth approach that lets one slide
the same modeling methodology from the one scale to the other: a
fractal-based ``scale-free network'' approach, in other words. What
makes this distinct from the problem we just saw Varela consider is
the focus on the structure, rather than dynamics, of transient
coherence -- a focus, then, in the last analysis, on a
characterization of \textit{database architecture} that can at once
accommodate meta-chaotic transiency and structural linguists'
cascades of ``double articulations.''
Starting at least with C. S. Peirce over a century ago, and
receiving more recent elaboration in the hands of J. M. Dunn and the
research into the ``Semantic Web'' devolving from his work, data
structures which include metadata at the same level as the data
proper have led to a focus on ``triadic logic,'' as perhaps best
exemplified in the recent work of Edward L. Robertson. [13] His
exploration of a natural triadic-to-triadic query language deriving
from Datalog, which he calls Trilog, is not (unlike our Skies)
intrinsically recursive. But his analysis depends upon recursive
arguments built atop it, and his key constructs are strongly
resonant with our own (explicitly recursive) ones. We focus on just
a few to make the point, with the aim of provoking interest in
fusing approaches, rather than in proving any particular results.
The still-standard technology of relational databases based on SQL
statements (most broadly marketed under the Oracle label) was itself
derived from Peirce's triadic thinking: the creator of the
relational formalism, Edgar F. ``Ted'' Codd, was a PhD student of
Peirce editor and scholar Arthur W. Burks. Codd's triadic
``relations,'' as Robertson notes (and as Peirce first recognized,
he tells us, in 1885), are ``the minimal, and thus most uniform''
representations ``where metadata, that is data about data, is
treated uniformly with regular data.'' In Codd's hands (and in
those of his market-oriented imitators in the SQL arena), metadata
was ``relegated to an essentially syntactic role'' [13, p. 1] -- a
role quite appropriate to the applications and technological
limitations of the 1970's, but inadequate for the huge and/or highly
dynamic schemata that are increasingly proving critical in
bioinformatics, satellite data interpretation, Google server-farm
harvesting, and so on. As Robertson sums up the situation motivating
his own work,
\begin{quote}
Heterogeneous situations, where diverse schemata represent
semantically similar data, illustrate the problems which arise when
one person's semantics is another's syntax -- the physical ``data
dependence'' that relational technology was designed to avoid has
been replaced by a structural data dependence. Hence we see the
need to [use] a simple, uniform relational representation where the
data/metadata distinction is not frozen in syntax. [13, pp. 1-2]
\end{quote}
As in relational database theory and practice, the
forming and exploiting of inner and outer \textit{joins} between
variously keyed tables of data is seminal to Robertson's approach as
well as Codd's. And while the RDF formalism of the Semantic Web (the
representational mechanism for describing structures as well as
contents of web artifacts on the World Wide Web) is likewise
explicitly triadic, there has, to date, been no formal mechanism put
in place for manipulating information in RDF format. Hence, ``there
is no natural way to restrict output of these mechanisms to triples,
except by fiat'' [13, p. 4], much less any sophisticated rule-based
apparatus like Codd's ``normal forms'' for querying and tabulating
such data. It is no surprise, then, that Robertson's ``fundamental
operation on triadic relations is a particular three-way join which
takes explicit advantage of the triadic structure of its operands.''
This \textit{triadic join}, meanwhile, ``results in another triadic
relation, thus providing the closure required of an algebra.'' [13,
p. 6]
Parsing Robertson's compact symbolic expressions into something
close to standard English, the trijoin of three triadic relations R,
S, T is defined as some $(a, b, c)$ selected from the universe of
possibilities $(x, y, z)$, such that $(a, x, z) \in R$, $(x, b, y)
\in S$, and $(z, y, c) \in T$. This relation, he argues, is the
most fundamental of all the operators he defines. When supplemented
with a few constant relations (analogs of Tarski's ``infinite
constants'' embodied in the four binary relations of universality of
all pairs, identity of all equal pairs, diversity of all unequal
pairs, and the empty set), it can express all the standard monotonic
operators (thereby excluding, among his primitives, only the
relative complement).
How does this compare with our ZD setup, and the workings of Skies?
For one thing, Infinite constants, of a type akin to Tarski's, are
embodied in the fact that any full meta-fractal requires the use of
an infinite ${\bf G}$, which sits atop an endless cascade of
singleton leftmost bits, determining for any given ${\bf S}$ an
indefinite tower of ZDs. One of the core operators massaging
Robertson's triads is the \textit{flip}, which fixes one component
of a relation while interchanging the other two $\cdots$ but our
Rule 2 is just the recursive analog of this, allowing one to move up
and down towers of values with great flexibilty (allowing, as well,
on and off switching effecting whole ensembles). The integer triads
upon which our entire apparatus depends are a gift of nature, not
dictated ``by fiat,'' and give us a natural basis for generating and
tracking unique IDs with which to ``tag'' and ``unpack'' data (with
``storage'' provided free of charge by the empty spaces of our
meta-fractals: the ``atoms'' of Semiotic Squares have four
long-diagonal slots each, one per each of the ``controls'' Petitot's
Catastrophe Theory reading calls for, and so on.)
Finally, consider two dual constructions that are the core of our
own triadic number theory: if the $(a, b, c)$ of last paragraph, for
instance, be taken as a Zigzag's L-index set, then the other trio of
triples correlates quite exactly with the Zigzag U-trips. And this
3-to-1 relation, recall, exactly parallels that between the 3
Trefoil, and 1 Zigzag, Sails defining a Box-Kite, with this very
parallel forming the support for the recursion that ultimately lifts
us up into a Sky. We can indeed make this comparison to Robinson's
formalism exceedingly explicit: if his X, Y, Z be considered the
angular nodes of PSL(2,7) situated at the 12 o'clock apex and the
right and left corners respectively, then his $(a, b, c)$ correspond
exactly to our own Rule 0 trip's same-lettered indices!
Here, we would point out that these two threads of reflection -- on
underwriting Chaos with cellular-automaton-tied Number Theory, and
designing new kinds of database architectures -- are hardly
unrelated. It should be recalled that two years prior to his
revolutionary 1970 paper on relational databases [14], Codd
published a pioneering book on cellular automata [15]. It is also
worth noting that one of the earliest technologies to be spawned by
fractals arose in the arena of data compression of images, as
epitomized in the work of Michael Barnsley and his Iterative Systems
company. The immediate focus of the author's own commercial efforts
is on fusing meta-fractal mathematics with the context-sensitive
adaptive-parsing ``Meta-S'' technology of business associate Quinn
Tyler Jackson. [16] And as that focus, tautologically, is not
mathematical \textit{per se}, we pass it by and leave it, like so
many other themes just touched on here, for later work.
\pagebreak
\section*{References}
\begin{description}
\item \verb|[1]| Robert P. C. de Marrais, ``Placeholder
Substructures I: The Road From NKS to Scale-Free Networks is Paved
with Zero Divisors,'' \textit{Complex Systems}, 17 (2007), 125-142;
arXiv:math.RA/0703745
\item \verb|[2]| Robert P. C. de Marrais, ``Placeholder
Substructures II: Meta-Fractals, Made of Box-Kites, Fill
Infinite-Dimensional Skies,'' arXiv:0704.0026 [math.RA]
\item \verb|[3]| Robert P. C. de Marrais, ``The 42 Assessors and
the Box-Kites They Fly,'' arXiv:math.GM/0011260
\item \verb|[4]| ~ Robert P. C. de Marrais, ``The Marriage of Nothing
and All: Zero-Divisor Box-Kites in a `TOE' Sky,'' in Proceedings of
the $26^{\textrm{th}}$ International Colloquium on Group Theoretical
Methods in Physics, The Graduate Center of the City University of
New York, June 26-30, 2006, forthcoming from Springer--Verlag.
\item \verb|[5]| Robert P. C. de Marrais, ``Placeholder
Substructures: The Road from NKS to Small-World, Scale-Free
Networks Is Paved with Zero-Divisors,'' http:// \newline
wolframscience.com/conference/2006/
presentations/materials/demarrais.ppt (Note: the author's surname
is listed under ``M,'' not ``D.'')
\item \verb|[6]|~ Benoit Mandelbrot, \textit{The Fractal Geometry of Nature} (W.
H. Freeman and Company, San Francisco, 1983)
\item \verb|[7]| Ed Pegg, Jr., ``Tournament Dice,'' \textit{Math Games} column for July 11, 2005, on the MAA
website at http://www.maa.org/editorial/ mathgames/mathgames
\verb|_07_11_05|.html
\item \verb|[8]| Robert P. C. de Marrais, ``The `Something From
Nothing' Insertion Point,''
http://www.wolframscience.com/conference/2004/ presentations/
\newline
materials/rdemarrais.pdf
\item \verb|[9]| Robert P. C. de Marrais, ``Presto! Digitization,'' arXiv:math.RA/0603281
\item \verb|[10]| Francisco Varela, editor, \textit{Sleeping,
Dreaming, and Dying: An Exploration of Consciousness with the Dalai
Lama} (Wisdom Publications: Boston, 1997).
\item \verb|[11]| F. J. Varela, J.-P. Lachauz, E. Rodrigues and J.
Martinerie, ``The brainweb: phase synchronization and large-scale
integration,'' \textit{Nature Reviews Neuroscience}, 2 (2001), pp.
229-239.
\item \verb|[12]| Jean Petitot, Francisco J. Varela, Bernard Pachoud
and Jean-Michel Roy, \textit{Naturalizing Phenomenology: Issues in
Contemporary Phenomenology and Cognitive Science} (Stanford
University Press: Stanford, 1999)
\item \verb|[13]| Edward L. Robertson, ``An Algebra for Triadic
Relations,'' Technical Report No. 606, Computer Science Department,
Indiana University, Bloomington IN 47404-4101, January 2005; online
at http://www.cs.indiana.edu/ \newline
pub/techreports/TR606.pdf
\item \verb|[14]| E. F. Codd, \textit{The Relational Model for
Database Management: Version 2} (Addison-Wesley: Reading MA, 1990)
is the great visionary's most recent and comprehensive statement.
\item \verb|[15]| E. F. Codd, \textit{Cellular Automata} (Academic
Press: New York, 1968)
\item \verb|[16]| Quinn Tyler Jackson, \textit{Adapting to Babel --
Adaptivity and Context-Sensiti- vity in Parsing: From
$a^{n}b^{n}c^{n}$ to RNA} (Ibis Publishing: P.O. Box3083, Plymouth
MA 02361, 2006; for purchasing information, contact Thothic
Technology Partners, LLC, at their website, www.thothic.com).
\end{description}
\end{document}
|
1,108,101,564,593 | arxiv | \section{introduction
For $m \ge 3$, the {\it $m$-gonal number} is described as the total number of dots to constitute a regular $m$-gon.
One may easily induce a formula
\begin{equation} \label{m number}
P_m(x)=\frac{m-2}{2}x^2-\frac{m-4}{2}x
\end{equation}
for the total number of dots to constitute a regular $m$-gon with $x$ dots for each side.
We especially call the $m$-gonal number of (\ref{m number}) as {\it x-th $m$-gonal number}.
For a long time, the representation of positive integers by a sum of $m$-gonal numbers has been one of popular subjects in the field of number theory.
In 17-th Century, Fermat conjectured that every positive integer may be written as at most $m$ $m$-gonal numbers.
And Lagrange and Gauss resolved his conjecture for $m=4$ and $m-3$ in 1770 and 1796, respectively.
Finally Cauchy presented a proof for his conjecture for all $m \ge 3$ in 1813.
By definition of $m$-gonal number, only positive integer would be admitted to $x$ in (\ref{m number}).
On the other hand, by considering $P_m(x)$ in (\ref{m number}) with $x \in \z_{\le 0}$ as $m$-gonal number too, we may generalize the $m$-gonal number.
As a general version of Fermat's Conjecture, very recently, the second author \cite{det} completed the minimal $\ell_m$ for which every positive integer may be written as $\ell_m$ (generalized) $m$-gonal numbers for all $m \ge 3$ as
$$ \ell_m=
\begin{cases} m-4 & \text{ if } m \ge 9\\
3& \text{ if } m \in \{3,5,6\}\\
4& \text{ if } m \in \{4,7,8\}. \\
\end{cases}$$
To consider the representation of positive integers by $m$-gonal numbers more generally, we may think about the weighted sum of $m$-gonal numbers
\begin{equation}\label{m form}
F_m(\mathbf x)=a_1P_m(x_1)+\cdots+a_nP_m(x_n)
\end{equation}
where $a_i \in \mathbb N$ admitting $x_i \in \z$.
We call $F_m(\mathbf x)$ in (\ref{m form}) as {\it $m$-gonal form}.
In this paper, without loss of generality, we assume that $a_1 \le \cdots \le a_n$.
If the diophantine equation $$F_m(\mathbf x)=N$$ has an integer solution $\mathbf x \in \z^n$ for $N \in \mathbb N$, then we say that the $m$-gonal form $F_m(\mathbf x)$ {\it represents $N$}.
Naturally, an $m$-gonal form which represents every positive integer may be paid attention.
We call an $m$-gonal form which represents every positive integer as {\it universal}.
Every universal $3$-gonal forms was classified by Liouville in $19$-th Century.
And Ramanujam showd all universal $4$-gonal forms, in fact, one of them is not actually universal.
In general, to determine whether a form is universal is not easy.
On the other hand, Conway and Schneeberger announced amazingly simple criterion to determine universality of a quadratic form.
The result which is well known as {\it $15$-Theorem} states that the representability of positive integers up to only $15$ by a quadratic form characterize the universality of the quadratic form.
Kane and Liu \cite{BJ} claimed that such a finiteness theorem holds for any $m$-gonal form too.
On the other words, there is (unique and minimal) $\gamma_m$ for which if an $m$-gonal form represents every positive integer up to $\gamma_m$, then the $m$-gonal form is universal.
They \cite{BJ} also questioned about the growth of $\gamma_m$ (which is asymptotically increasing) and showed that
$$m-4 \le \gamma_m \ll m^{7+\epsilon}.$$
The authors \cite{KP'} obtained the optimal growth of $\gamma_m$ which is exactly linear on $m$ by showing Theorem \ref{C}.
\begin{thm} \label{C}
For $m \ge 3$, there exists an absolute constant $C$ such that $\gamma_m \le C(m-2)$.
\end{thm}
\begin{proof}
See \cite{KP'}.
\end{proof}
On the other hand, Bhargava \cite{CS} suggested a simple proof for $15$-Theorem by introducing {\it escalator tree} of quadratic form.
Following the Bhargava's escalating method, we may consider an escalator tree of $m$-gonal form.
For a non universal form, we call the minimal integer which is not represented by the form as the {\it truant} of the form.
We call a super form of a non-universal form which represents the truant of the non-universal form as {\it escalator} of the form.
The escalator tree is a rooted tree consisting of $m$-gonal forms having the root $\emptyset$.
If a node $\sum_{i=1}^ka_iP_m(x_i)$ of the tree is not universal, then the node spreads branches by taking its children all of its escalotors $\sum_{i=1}^{k+1}a_iP_m(x_i)$ with $a_k \le a_{k+1}$.
And if a node does not have truant (i.e., the node is universal), then the node would be a leaf of the tree.
In other words, an universal $m$-gonal form would only appear on the leaves of the escalating tree, all leaves of the tree are universal and all of the universal $m$-gonal forms on the leaves of the tree are kind of proper universal $m$-gonal forms in the sense that without its last component the universality is broken (i.e., its parents are not universal).
The $\gamma_m$ would be exactly the maximal truant of node of escalator tree of $m$-gonal form.
In effect, in \cite{CS}, Bhargava found the maximal truant of node of escalator tree of quadratic form.
Meanwhile, one may catch that an $m$-gonal form would contain at least one of leaves of escalator tree as its subform.
In this paper, we treat the escalator tree of $m$-gonal form for sufficiently large $m$.
Especially, we determine the minimal rank $r_m$ and maximal rank $R_m$ of leaf of the escalator tree and show there is a leaf of rank $n$ for any $n \in [r_m,R_m]$ for all sufficiently large $m$.
The $r_m$ would be indeed the minimal rank of universal $m$-gonal form.
So we may also obtain the minimal rank of universal $m$-gonal form for sufficiently large $m$.
And any $m$-gonal form would contain a universal $m$-gonal form of rank less than or equal to $R_m$.
Overall, the results would provide the answers of most part on the rank of universal $m$-gonal forms.
In this paper, we basically use the arithmetic theory of quadratic form.
Any unexplained notation and terminology can be found in \cite{O1} and \cite{O}.
\vskip 0.5em
The paper is organized as follows.
In section 2, we introduce escalator tree more concretely and our results.
In Section 3, we determine $r_m$ and see a leaf of rank $r_m$.
In Section 4, we determine $R_m$ and see a leaf of rank $R_m$.
\section{preliminary
Following the Guy's argument \cite{G}, since the smallest (generalized) $m$-gonal number is $m-3$ except $0$ and $1$, we may yield that every node $\sum _{i=1}^ka_iP_m(x_i)$ of the escalator tree must satisfy the following conditions
\begin{equation}\label{coe}\begin{cases}a_1=1 & \\ a_{i+1}\le a_1+\cdots +a_i+1 & \text{if } a_1+\cdots +a_i <m-4\end{cases}\end{equation}
because the truant of a node $\sum_{i=1}^ka_iP_m(x_i)$ with $a_1+\cdots +a_k <m-4$ would be $$a_1+\cdots + a_k +1.$$
From (\ref{coe}), we may have that
\begin{equation}\label{2^i}a_{i+1} \le 2^i \quad \text{ when }a_1+\cdots +a_i <m-4.\end{equation}
By the Guy's argument \cite{G} again, every leaf $\sum _{i=1}^n a_iP_m(x_i)$ which is universal must have
\begin{equation}\label{uni}a_1+\cdots +a_n \ge m-4\end{equation}
since otherwise the integers from $a_1+\cdots+a_n+1$ to $m-4$ cannot be represented by the (universal) leaf, which is a contradiction.
From (\ref{2^i}) and (\ref{uni}), we may obtain that every leaf in the escalator tree would have the rank greater than or equal to $\ceil{\log_2(m-3)}$, i.e., $$\ceil{\log_2(m-3)} \le r_m.$$
And following the Theorem \ref{C}, a truant could not exceed $C(m-2)$.
So we may clearly obtain that $$R_m \le C(m-2).$$
Throughtout this paper, we exactly determine the $r_m$ and $R_m$ for all $m$ sufficiently large.
In Chapter 3, we prove the following theorem.
\begin{thm} \label{min}For $m>2\left(\left(2C+\frac{1}{4}\right)^{\frac{1}{4}}+\sqrt{2}\right)^2$,
\begin{equation}\label{thmrm}r_m=\begin{cases}\ceil{\log_2(m-3)}+1& \text{ when } -3 \le 2^{\ceil{\log_2(m-3)}}-m\le 1\\
\ceil{\log_2(m-3)} & \text{ when } \ \quad 2 \le 2^{\ceil{\log_2(m-3)}}-m.
\end{cases}\end{equation}
\end{thm}
\vskip 0.5em
\begin{rmk}
Furthermore, in the proof of Therem \ref{min}, we claim that for $m>2\left(\left(2C+\frac{1}{4}\right)^{\frac{1}{4}}+\sqrt{2}\right)^2$,
$$P_m(x_1)+2P_m(x_2)+\cdots +2^{r_m-1}P_m(x_{r_m})$$ is universal $m$-gonal form of the minimal $\rank r_m$.
Actually, the authors guess that (\ref{thmrm}) in Theorem \ref{min} holds for much smaller $m$'s too.
But we would not remove the restriction $m>2\left(\left(2C+\frac{1}{4}\right)^{\frac{1}{4}}+\sqrt{2}\right)^2$ in Theorem \ref{min} with the arguments in Chapter 3 even though one could take slightly smller lower bound instead of $2\left(\left(2C+\frac{1}{4}\right)^{\frac{1}{4}}+\sqrt{2}\right)^2$ through more careful care.
\end{rmk}
\vskip 0.5em
In Chapter 4, we determine $R_m$ by claming following theorem.
\begin{thm}\label{main}
For $m>6C^2(C+1)$,
$$R_m=\begin{cases}m-2 & \text{ when } m\nequiv 2 \pmod{3}\\m-3 & \text{ when } m\equiv 2 \pmod{3}.\end{cases}$$
\end{thm}
\vskip 0.5em
\begin{rmk} \label{rmk R}
For $m>6C^2(C+1)$, there would be exactly $3m-12$ and $3m-14$ leaves of the $\rank \ R_m$ when $m\equiv0$ and $m\not\equiv0 \pmod{3}$, respectively.
In particular, we would characterize all of the leaves of $\rank R_m$ as follows.
\begin{itemize}
\item[1.]When $m\equiv 0 \pmod{3}$, all of the leaves of $\rank \ R_m$ are $$P_m(x_1)+P_m(x_2)+\sum \limits_{k=3}^{m-3}3P_m(x_k)+a_{m-2}P_m(x_{m-2})$$ where $3 \le a_{m-2} \le 3m-10.$\\
\item[2.]When $m\equiv 1 \pmod{3}$, all of the leaves of $\rank \ R_m$ are $$P_m(x_1)+P_m(x_2)+\sum \limits_{k=3}^{m-3}3P_m(x_k)+a_{m-2}P_m(x_{m-2})$$ where $3 \le a_{m-2} \le 3m-12.$\\
\item[3.]When $m\equiv 2 \pmod{3}$, all of the leaves of $\rank \ R_m$ are $$P_m(x_1)+2P_m(x_2)+\sum \limits_{k=3}^{m-4}3P_m(x_k)+a_{m-3}P_m(x_{m-3})$$ where $3 \le a_{m-3} \le 3m-12.$
\end{itemize}
\end{rmk}
\vskip 1em
\section{A universal $m$-gonal form of the minimal rank : The most fastly escalated universal $m$-gonal form
We may see that a node $\sum \limits_{i=1}^ka_iP_m(x_i)$ with $a_1+\cdots +a_k <m-4$ of the esacalator tree have the truant $a_1+\cdots+a_k+1$.
So a leaf $\sum \limits_{i=1}^n a_iP_m(x_i)$(which is universal) having no truant of the tree should satisfy the followings
\begin{equation}\label{3.1}
\begin{cases}a_1=1 & \\
a_{i+1}\le a_1+\cdots +a_i+1 & \text{if } a_1+\cdots +a_i <m-4 \\
a_1+\cdots +a_n \ge m-4. & \end{cases}\end{equation}
From the first and second conditions in (\ref{3.1}), we may obtain that for any leaf $\sum \limits_{i=1}^n a_iP_m(x_i)$,
\begin{equation}\label{3.2}a_{k+1}\le 2^k\end{equation}
holds whenever $a_1+\cdots +a_k<m-4$.
And then with the third condition in (\ref{3.1}) and (\ref{3.2}), we may induce that every leaf $\sum \limits_{i=1}^n a_iP_m(x_i)$ has the rank $n$ greater than or equal to $\ceil{\log_2(m-3)}$.
Note that there is a node
\begin{equation}\label{mf}\sum \limits_{i=1}^k2^{i-1}P_m(x_i)\end{equation}
where $k=\ceil{\log_2(m-3)}$ of the tree.
The $m$-gonal form (\ref{mf}) would be one of the most fastly escalated $m$-gonal forms to represent up to $m-4$.
For sufficiently large $m>2\left(\left(2C+\frac{1}{4}\right)^{\frac{1}{4}}+\sqrt{2}\right)^2$ with $2 \le 2^{\ceil{\log_2(m-3)}}-m$, by showing that the $m$-gonal form (\ref{mf}) is universal, we claim that the $m$-gonal form is a leaf of the minimal $\rank r_m$, yielding the $m$-gonal form is indeed a universal $m$-gonal form of the minimal rank.
\begin{lem} \label{rm1}
For $m>2\left(\left(2C+\frac{1}{4}\right)^{\frac{1}{4}}+\sqrt{2}\right)^2$ with $2 \le 2^{\ceil{\log_2(m-3)}}-m$, the $m$-gonal form
\begin{equation}\label{minm}
F_m(\mathbf x)=P_m(x_1)+2P_m(x_2)+\cdots+2^{n-1}P_m(x_n)
\end{equation}
where $n=\ceil{\log_2(m-3)}$ is universal.
\end{lem}
\begin{proof}
In virtue of the Theorem \ref{C}, it may be enough to show that $F_m(\mathbf x)$ represents every positive integer up to only $C(m-2).$
Throughout this proof, we write the integers in $[1,C(m-2)]$ as $$A(m-2)+B$$ where $0\le A \le C$ and $0 \le B \le m-3$.
Note that
\begin{align*}
F_m(\mathbf x) = & (m-2)\{ P_3(x_1-1)+2P_3(x_2-1)+4P_3(x_3-1)+8P_3(x_4-1)\} \\
& + (x_1+2x_2+4x_3+8x_4)+16P_m(x_5)+\cdots+2^{n-1}P_m(x_n).\end{align*}
For a non-negative integer $A$, let $x(A)$ be the largest positive integer satisfying $$P_3(x(A)-1) \le A ,$$
i.e., the integer in the interval $(\sqrt{2A+\frac{1}{4}}-\frac{1}{2}, \sqrt{2A+\frac{1}{4}}+\frac{1}{2}]$.
On the other hand, there would be exactly one $$y_1(A,B) \in \{x(A), x(A)-1, x(A)-2, x(A)-3\}$$ satisfying
$$\begin{cases}
P_3(y_1(A,B)-1) \equiv A \pmod{2} \\
y_1(A,B) \equiv B \pmod{2}.
\end{cases}$$
Since the ternary triangular form $$P_3(x)+2P_3(y)+4P_3(z)$$ is universal, the even integer $A-P_3(y_1(A,B)-1)$ may be written as
$$2P_3(y_1(A,B)-1)+4P_3(y_2(A,B)-1)+8P_3(y_3(A,B)-1)$$
for some $y_i(A,B) \in \z$, i.e., $$A=P_3(y_1(A,B)-1)+2P_3(y_1(A,B)-1)+4P_3(y_2(A,B)-1)+8P_3(y_3(A,B)-1).$$
Beside that since $P_3(x)=P_3(-x-1)$, if it is necessary, by changing $y_i(A,B)-1$ to $-y_i(A,B)$, we may assume that
\begin{equation}\label{y}
\begin{cases} A=P_3(y_1(A,B)-1)+\cdots +8P_3(y_4(A,B)-1) \\B \equiv y_1(A,B)+2y_2(A,B)+4y_3(A,B)+8y_4(A,B) \pmod{16}. \end{cases}
\end{equation}
Since the integer $y_1(A,B)$ in (\ref{y}) is in $[\sqrt{2A+\frac{1}{4}}-\frac{7}{2}, \sqrt{2A+\frac{1}{4}}+\frac{1}{2}]$ we may get that
\begin{align*}
0 & \le A-P_3(y_1(A,B)-1) \\
& =2P_3(y_2(A,B)-1)+4P_3(y_3(A,B)-1)+8P_3(y_4(A,B)-1)<4\sqrt{2A+\frac{1}{4}}-8.\\
\end{align*}
By arranging the above inequality, we may obtain
\begin{equation}\label{sqrt}\left(y_2(A,B)-\frac{1}{2}\right)^2+2\left(y_3(A,B)-\frac{1}{2} \right)^2+4\left(y_4(A,B)-\frac{1}{2} \right)^2<4\sqrt{2A+\frac{1}{4}}-\frac{25}{4}\end{equation}
and through a basic calculation, we may get that $$|2y_2(A,B)+4y_3(A,B)+8y_4(A,B)|<14\sqrt{\frac{4\sqrt{2A+\frac{1}{4}}-\frac{25}{4}}{73}}.$$
So for $y_1(A,B) \in [\sqrt{2A+\frac{1}{4}}-\frac{7}{2}, \sqrt{2A+\frac{1}{4}}+\frac{1}{2}]$, we may get that such the above $y_i(A,B)$ where $160 \le A \le C$ satisfy
\begin{align*}
0 & < \sqrt{2A+\frac{1}{4}}-14\sqrt{\frac{4\sqrt{2A+\frac{1}{4}}-\frac{25}{4}}{73}}-\frac{7}{2}\\
& < y_1(A,B)+2y_2(A,B)+4y_3(A,B)+8y_4(A,B) \\ & <\sqrt{2A+\frac{1}{4}}+14\sqrt{\frac{4\sqrt{2A+\frac{1}{4}}-\frac{25}{4}}{73}}+\frac{1}{2} < \frac{m-2}{2}.
\end{align*}
Through similar processings with the above we may obtain
$$\begin{cases} A=P_3(z_1(A,B)-1)+\cdots+8P_3(z_4(A,B)-1) \\ B \equiv z_1(A,B)+2z_2(A,B)+4z_3(A,B)+8z_4(A,B) \pmod{16}\end{cases}$$
hold for some $z_1(A,B) \in \{-x(A)+1, -x(A)+2,-x(A)+3,-x(A)+4\}$ and $z_i(A,B) \in \z$ for $i=2,3,4$.
And in this case, we may get that such the above $z_i(A,B)$ with $160 \le A \le C$ satisfy
\begin{align*}-\frac{m-2}{2} & < -\sqrt{2A-\frac{1}{4}}-14\sqrt{\frac{4\sqrt{2A+\frac{1}{4}}-\frac{25}{4}}{73}}+\frac{1}{2} \\ & \le z_1(A,B)+2z_2(A,B)+4z_3(A,B)+8z_4(A,B) \\
& <-\sqrt{2A+\frac{1}{4}}+14\sqrt{\frac{4\sqrt{2A+\frac{1}{4}}-\frac{25}{4}}{73}}+\frac{9}{2}<0.\end{align*}
And then one may easily see that for each integer $A(m-2)+B \in [160(m-2), C(m-2)]$, $$\text{either } \ (x_i(A,B))=(y_i(A,B)) \ \text{ or } \ (x_i(A,B))=(z_i(A,B))$$ satisfies
$$0 \le A(m-2)+B-\{P_m(x_1(A,B))+\cdot
+8P_m(x_4(A,B))\} \le m-11$$
with $$A(m-2)+B-\{P_m(x_1(A,B))+\cdot
+8P_m(x_4(A,B))\}\equiv 0 \pmod{16}.$$
On the other hand, remaining $16P_m(x_5)+\cdots +2^{n-1}P_m(x_n)$ may represent all the multiples of $16$ up to $m-11(\le 2^n-16)$
by taking $P_m(x_i) \in \{0,1\}$ for all $5\le i \le n$ which yields that $A(m-2)+B$ may be represented by $F_m(\mathbf x)$ as follows
$$P_m(x_1(A,B))+\cdots+8P_m(x_4(A,B))+16P_m(x_5)+\cdots +2^{n-1}P_m(x_n)$$
for some $(x_5,\cdots, x_n) \in \{0,1\}^{n-4}$.
Until now, we showed that $$F_m(\mathbf x)=P_m(x_1)+2P_m(x_2)+\cdots+2^{n-1}P_m(x_n)$$ represents every positive integer in $[160(m-2),C(m-2)].$
In the remaining of this proof, we show that $F_m(\mathbf x)$ represents every positive integer in $[1, 160(m-2)]$.
Through direct calcuations (the authors used python), we may obtain that for each $(A,r_B) \in \z \times \z/8\z$ with $0 \le A \le 160$, there are integer solutions $(x_1,x_2,x_3)\in \z^3$ for both of
\begin{equation} \label{pos}
\begin{cases}P_3(x_1-1)+2P_3(x_2-1)+4P_3(x_3-1)=A \\
x_1+2x_2+4x_3 \equiv r_B \pmod{8} \\
0 \le x_1+2x_2+4x_3 <100 \ll \frac{m-2}{2}
\end{cases}
\end{equation}
and
\begin{equation} \label{neg}
\begin{cases}
P_3(x_1-1)+2P_3(x_2-1)+4P_3(x_3-1)=A \\
x_1+2x_2+4x_3 \equiv r_B \pmod{8} \\
-\frac{m-2}{2}\ll -100<x_1+2x_2+4x_3 \le 0
\end{cases}
\end{equation}
respectively except the pairs $(A,r_B)$ in $S^+ \cup S^- (\subset \z \times \z/8\z)$ where
\begin{equation}\label{(a,r)pos}
\begin{array} {lllllll}
S^+:= &\{(0,0), \!&(0,1), \!&(0,2),\!& (0,3), \!& (0,4), \!& (0,5), \\
\!\!\!\!&\ (0,6), \!&(0,7), \!& (1,0), \!&(1,1), \!&(1,2),\!& (1,3), \\
\!\!\!\!&\ (1,4), \!& (1,5), \!& (1,6), \!&(2,0), \!&(2,1),\!& (2,2),\\
\!\!\!\!&\ (2,3), \!& (2,4),\!& (2,5), \!&(3,1), \!&(3,2),\!& (3,3), \\
\!\!\!\!&\ (3,4), \!& (3,7), \!& (4,0), \!&(4,1), \!&(4,2),\!& (4,3), \\
\!\!\!\!&\ (5,1), \!& (5,2),\!& (5,7), \!&(6,0), \!&(6,1),\!& (6,6), \\
\!\!\!\!&\ (7,0), \!& (7,5), \!& (8,0), \!&(8,1), \!&(8,2),\!& (8,4), \\
\!\!\!\!&\ (8,5), \!& (8,6), \!& (9,1), \!&(9,3), \!&(9,4),\!& (10,2), \\
\!\!\!\!&\ (10,5), \!& (10,7),\!& (11,0), \!&(11,3), \!&(11,4),\!& (11,5), \\
\!\!\!\!&\ (11,6), \!& (12,4), \!& (13,0), \!&(14,1), \!&(14,3),\!& (15,1),\\
\!\!\!\!&\ (15,2), \!& (16,1),\!& (16,2), \!&(16.3), \!&(17,3),\!& (17,6), \\
\!\!\!\!&\ (18,0), \!& (18,1),\!& (18,2), \!&(19,0), \!&(19,2),\!& (20,7), \\
\!\!\!\!&\ (21,0), \!& (21,1), \!& (22,0), \!&(22,1), \!&(23,1),\!& (23,5), \\
\!\!\!\!&\ (25,2), \!& (26,7),\!& (28,4), \!&(18,6), \!&(29,2),\!& (31,3), \\
\!\!\!\!&\ (32,2), \!& (32,3), \!& (35,2), \!&(36,1), \!&(37,1),\!& (37,2), \\
\!\!\!\!&\ (37,7), \!& (38,0),\!& (43,1), \!&(43,4), \!&(44,0),\!& (44,1), \\
\!\!\!\!&\ (50,2), \!& (51,6),\!& (53,0), \!&(53,3), \!&(53,5),\!& (54,7), \\
\!\!\!\!&\ (58,2), \!&(64,2),\!& (64,3), \!& (65,1), \!& (65,4),\!& (72,2), \\
\!\!\!\!&\ (74,2), \!&(75,0),\!&(81,1), \!& (85,5), \!& (92,4),\!& (106,2), \\
\!\!\!\!&\ (106,3), \!&(110,2),\!&(116,3), \!&(123,1), \!&(128,2),\!& (128,3)\} \\\end{array}
\end{equation}
and
\begin{equation}\label{(a,r)neg}
\begin{array} {lllllll}
S^-:=&\{(1,7), \!&(2,6), \!&(2,7),\!& (4,4), \!& (4,5), \!& (4,6), \\
\!\!\!\!&\ (4,7), \!&(7,7), \!&(8,3),\!& (8,7), \!& (9,6), \!& (10,0), \\
\!\!\!\!&\ (11,2), \!&(11,7), \!&(15,5),\!& (18,6), \!& (18,7), \!& (20,0), \\
\!\!\!\!&\ (22,7), \!&(23,6), \!&(26,0),\!& (28,1), \!& (31,4), \!& (36,6), \\
\!\!\!\!&\ (37,0), \!&(37,5), \!&(44,6),\!& (44,7), \!& (53,2), \!& (53,4), \\
\!\!\!\!&\ (53,7), \!&(54,0), \!& (106,4), \!& (110,5),\!&(116,4), \!&(128,4), \\
\!\!\!\!&\ (128,5)\}. \!& \!& \!& \!& \!& \\
\end{array}
\end{equation}
Each pair $(A,r_B) \in S^+$ has integer solutions for only (\ref{pos}) and not for (\ref{neg}) and each pair $(A,r_B) \in S^-$ has integer solutions for only (\ref{neg}) and not for (\ref{pos}).
And then, we may easily see that for each
$$1 \le A(m-2)+B \le 160(m-2)$$
with $(A,r_B) \notin S^+ \cup S^-$ where $r_B$ is the residue of $B$ modulo $8$ one of the above integer solutions $(x_1,x_2,x_3) \in \z^3$ for (\ref{pos}) or (\ref{neg}) satisfies
\begin{equation}\label{124;ab}\begin{cases} A(m-2)+B\equiv P_m(x_1)+2P_m(x_2)+4P_m(x_3) \pmod{8} \\
0 \le A(m-2)+B-\{ P_m(x_1)+2P_m(x_2)+4P_m(x_3)\} < m-2.
\end{cases}\end{equation}
Denote an integer solution for (\ref{124;ab}) by $(x_1(A,B),x_2(A,B),x_3(A,B)) \in \z^3$.
Since the remaining $$8P_m(x_4)+\cdots +2^{n-1}P_m(x_n)$$ would represents $$A(m-2)+B-\{ P_m(x_1(A,B))+2P_m(x_2(A,B))+4P_m(x_3(A,B))\}$$(which is a multiple of $8$ in $[0,m-3]$ from (\ref{124;ab})) by taking $P_m(x_i) \in \{0,1\}$ for all $4 \le i \le n$, we may obtain that every integer $A(m-2)+B$ in $[1, 160(m-2)]$ with $(A,r_B) \notin S^+ \cup S^-$ is represented by $$F_m(\mathbf x)=P_m(x_1)+2P_m(x_2)+\cdots+2^{n-1}P_m(x_n)$$
To complete the proof, we finally consider the integers $$A(m-2)+B \in [1,160(m-2)]$$ where $(A,r_B) \in S^+ \cup S^-$.
Among the pairs $(A,r_B)$ in $S^+ \cup S^-$, consider $(116,3) \in S^+$.
From
\begin{equation}\label{116,3}P_m(1)+2P_m(-7)+4P_m(6)=116(m-2)+11,\end{equation}
we may yield that every positive integer $A(m-2)+B \in [1,160(m-2)]$ with $(A,r_B)=(116,3)$ may be represented by $F_m(\mathbf x)$ other than $116(m-2)+3$.
On the other hand, $116(m-2)+3$ may be equivalent with one $115(m-2)+B'$ of the
$$\begin{cases}
115(m-2)+32=P_m(-2)+2P_m(1)+4P_m(8)\\
115(m-2)+41=P_m(3)+2P_m(9)+4P_m(5)\\
115(m-2)+42=P_m(2)+2P_m(6)+4P_m(7)\\
115(m-2)+35=P_m(3)+2P_m(0)+4P_m(8)\\
115(m-2)+36=P_m(-2)+2P_m(9)+4P_m(5)\\
115(m-2)+37=P_m(11)+2P_m(1)+4P_m(6)\\
115(m-2)+38=P_m(2)+2P_m(10)+4P_m(4)\\
115(m-2)+39=P_m(-1)+2P_m(6)+4P_m(7)\\
\end{cases}$$
modulo $8$ because $\{32, 41,42,35,36, 37, 38, 39\}$ form a complete set of residues modulo $8$.
By agian using the fact that the remaining $8P_m(x_4)+\cdots +2^{n-1}P_m(x_n)$ represents all the multiples of $8$ up to $m-3$, we may obtaint that $116(m-2)+3$ is also represented by $F_m(\mathbf x)$ because $$(116(m-2)+3)-(115(m-2)+B') $$ would be a multiple of $8$ in $[0,m-3]$.
Similarly with the above case $(A,r_B)=(116,3) \in S^+\cup S^-$, through surveys based on the result of integer solutions for (\ref{pos}) and (\ref{neg}) with the fact that the remaining $m$-gonal subform $8P_m(x_4)+\cdots+2^{n-1}P_m(x_n)$ of $F_m(\mathbf x)$ represents all the multiples of $8$ up to $m-3(\le 2^n-8)
, we may get that every integer $A(m-2)+B$ in $[1,160(m-2)]$ with $(A,r_B) \in S^+ \cup S^-$ other than
\begin{equation}\label{TT}\begin{array} {lll}
\!\!\!\! 37(m-2)+7, \!&37(m-2)+15, \!&37(m-2)+m-18, \\
\!\!\!\!37(m-2)+m-16, \!&37(m-2)+m-11, \!&37(m-2)+m-10 \\
\end{array}\end{equation}
may be represented by $F_m(\mathbf x)$ by taking $(x_4,\cdots,x_n) \in \{0,1\}^{n-3}$.
In the meanwhile, we may directly check that the representability of integers in (\ref{TT}) by $F_m(\mathbf x)$ as follows
$$\begin{cases}
37(m-2)+7= P_m(-1)+2P_m(4)+4P_m(0)+8P_m(-2)+16P_m(1)\\
37(m-2)+15= P_m(-1)+2P_m(0)+4P_m(-2)+8P_m(3)\\
37(m-2)+m-18= P_m(4)+2P_m(0)+4P_m(-3)+8P_m(-1)\\
37(m-2)+m-16= P_m(4)+2P_m(1)+4P_m(-3)+8P_m(-1)\\
37(m-2)+m-11= P_m(-3)+2P_m(-3)+4P_m(-2)+8P_m(-1)+16P_m(1)\\
37(m-2)+m-10= P_m(-4)+2P_m(0)+4P_m(-1)+8P_m(-2)+16P_m(1).\\
\end{cases}$$
Consequently, we may conclude that $$F_m(\mathbf x)=P_m(x_1)+2P_m(x_2)+\cdots+2^{n-1}P_m(x_n)$$ represents every positive integer up to $C(m-2)$, yielding the $F_m(\mathbf x)$ is universal by the Theorem \ref{C}.
\end{proof}
\vskip 0.5em
\begin{rmk} \label{rmk(rm)1}
In Lemma \ref{rm1}, we proved that for sufficiently large $m$ with $2 \le 2^{\ceil{\log_2(m-3)}}-m$, $$r_m=\ceil{\log_2(m-3)}$$
by showing that $$P_m(x_1)+2P_m(x_2)+4P_m(x_3)+\cdots +2^{n-1}P_m(x_n)$$ where $n=\ceil{\log_2(m-3)}$ is universal.
Since $-3 \le 2^{\ceil{\log_2(m-3)}}-m$, in order to complete the proof of Theorem \ref{min} now we need to consider $r_m$ for $-3 \le 2^{\ceil{\log_2(m-3)}}-m\le 4$.
\begin{itemize}
\item[(1)]
When $-3 = 2^{\ceil{\log_2(m-3)}}-m$, there is only one $m$-gonal form which represents every positive integer up to $m-4$ with the $\rank n= \ceil{\log_2(m-3)}$ that is $$P_m(x_1)+2P_m(x_2)+\cdots+2^{n-1}P_m(x_n)$$
which does not represent $m-2$, i.e., there is no $m$-gonal form which represents every positive integer up to $m-2$ of the $\rank$ less than or equal to $\ceil{\log_2(m-3)}$.
Clearly we may have that there is no universal $m$-gonal form of the $\rank$ less than or equal to ${\ceil{\log_2(m-3)}}$ when $-3 = 2^{\ceil{\log_2(m-3)}}-m$ which implies that $${\ceil{\log_2(m-3)}}+1 \le r_m.$$
\item[(2)]
When $-2 = 2^{\ceil{\log_2(m-3)}}-m$, all of the $m$-gonal forms which represent every positive integer up to $m-4$ with the $\rank n= \ceil{\log_2(m-3)}$ are
$$\begin{cases}
P_m(x_1)+2P_m(x_2)+\cdots+2^{n-1}P_m(x_n) \text{ and }\\
$$P_m(x_1)+2P_m(x_2)+\cdots+(2^{n-1}-1)P_m(x_n)\\
\end{cases}$$
which do not represent $m-2$.
So in this case too, we may have that $${\ceil{\log_2(m-3)}}+1 \le r_m.$$
\item[(3)]
When $-1 = 2^{\ceil{\log_2(m-3)}}-m$, all of the $m$-gonal forms which represent every positive integer up to $m-4$ with the $\rank n= \ceil{\log_2(m-3)}$ are
$$\begin{cases}
P_m(x_1)+2P_m(x_2)+\cdots+2^{n-1}P_m(x_n)\\
P_m(x_1)+2P_m(x_2)+\cdots+(2^{n-1}-1)P_m(x_n)\\
P_m(x_1)+2P_m(x_2)+\cdots+(2^{n-1}-2)P_m(x_n)\\
\end{cases}$$
which do not represent $2m-4$, $m-2$, and $m-2$, respectively which yields that $${\ceil{\log_2(m-3)}}+1 \le r_m.$$
\item[(4)]
When $0 = 2^{\ceil{\log_2(m-3)}}-m$, the all of $m$-gonal forms which represent every positive integer up to $m-4$ with the $\rank n= \ceil{\log_2(m-3)}$ are
$$\begin{cases}
P_m(x_1)+2P_m(x_2)+\cdots+2^{n-2}P_m(x_{n-1})+2^{n-1}P_m(x_n)\\
P_m(x_1)+2P_m(x_2)+\cdots+2^{n-2}P_m(x_{n-1})+(2^{n-1}-1)P_m(x_n)\\
P_m(x_1)+2P_m(x_2)+\cdots+2^{n-2}P_m(x_{n-1})+(2^{n-1}-2)P_m(x_n)\\
P_m(x_1)+2P_m(x_2)+\cdots+2^{n-2}P_m(x_{n-1})+(2^{n-1}-3)P_m(x_n)\\
P_m(x_1)+2P_m(x_2)+\cdots+(2^{n-2}-1)P_m(x_{n-1})+(2^{n-1}-2)P_m(x_n)
\end{cases}$$
which do not represent $2m-3$, $2m-4$, $m-2$, $m-2$, and $m-2$, respectively which yields that $${\ceil{\log_2(m-3)}}+1 \le r_m.$$
\item[(5)]
When $1 = 2^{\ceil{\log_2(m-3)}}-m$, all of the $m$-gonal forms which represent every positive integer up to $m-4$ with the $\rank n= \ceil{\log_2(m-3)}$ are
$$\begin{cases}
P_m(x_1)+2P_m(x_2)+\cdots+2^{n-2}P_m(x_{n-1})+2^{n-1}P_m(x_n)\\
P_m(x_1)+2P_m(x_2)+\cdots+2^{n-2}P_m(x_{n-1})+(2^{n-1}-1)P_m(x_n)\\
P_m(x_1)+2P_m(x_2)+\cdots+2^{n-2}P_m(x_{n-1})+(2^{n-1}-2)P_m(x_n) \\
$$ P_m(x_1)+2P_m(x_2)+\cdots+2^{n-2}P_m(x_{n-1})+(2^{n-1}-3)P_m(x_n)\\
$$ P_m(x_1)+2P_m(x_2)+\cdots+2^{n-2}P_m(x_{n-1})+(2^{n-1}-4)P_m(x_n)\\
$$ P_m(x_1)+2P_m(x_2)+\cdots+(2^{n-2}-1)P_m(x_{n-1})+(2^{n-1}-2)P_m(x_n)\\
$$ P_m(x_1)+2P_m(x_2)+\cdots+(2^{n-2}-1)P_m(x_{n-1})+(2^{n-1}-3)P_m(x_n)\\
\end{cases}$$
which do not represent $5(m-2)-1$, $2m-3$, $2m-4$, $m-2$, $m-2$, $m-2$, and $m-2$, respectively which yields that $${\ceil{\log_2(m-3)}}+1 \le r_m.$$
\item[(6)]
When $2\le 2^{\ceil{\log_2(m-3)}}-m \le 4$, one may see that $$P_m(x_1)+2P_m(x_2)+\cdots +2^{n-1}P_m(x_n)$$ where $n=\ceil{\log_2(m-3)}$ is universal by showing that the $m$-gonal form represents every positive integer up to $C(m-2)$ through similar processings with the Lemma \ref{rm1}.
But it would require more delicate care to examine the representability of small integers in $[1,160(m-2)]$ than the Lemma \ref{rm1} because of the tighter condition of integers represented by $8P_m(x_4)+\cdots +2^{n-1}P_m(x_n)$.
We omit the proof in this paper.
\end{itemize}
\end{rmk}
\vskip 0.5em
\begin{lem} \label{rm2}
For $m>2\left(\left(2C+\frac{1}{4}\right)^{\frac{1}{4}}+\sqrt{2}\right)^2$, the $m$-gonal form
\begin{equation}\label{minm'}
F_m(\mathbf x)=P_m(x_1)+2P_m(x_2)+\cdots+2^{n}P_m(x_{n+1})
\end{equation}
where $n=\ceil{\log_2(m-3)}$ is universal.
\end{lem}
\begin{proof}
One may prove this lemma through almost same arguments with the proof of Lemma \ref{rm1}.
Actually, under the assumption of this lemma, the fact that the subform $$8P_m(\mathbf x_4)+\cdots +2^nP_m(\mathbf x_{n+1})$$ of $F_m(\mathbf x)$ represents all the multiples of $8$ up to $2m-14$ make to show this lemma easier than to show Lemma \ref{rm1}.
\end{proof}
\vskip 0.5em
\begin{rmk} \label{rmk(rm)2}
The Lemma \ref{rm2} says that $F_m(\mathbf x)$ is a universal $m$-gonal form of the $\rank \ceil{\log_2(m-3)}+1$.
On the other hand, we observed that there is no universal $m$-gonal form of the $\rank$ less than or equal to $\ceil{\log_2(m-3)}$ when $-3 \le 2^{\ceil{\log_2(m-3)}}-m\le 1$ in Remark \ref{rmk(rm)1}.
Therefore we may obtain that the $F_m(\mathbf x)$ in the Lemma \ref{rm2} is a universal $m$-gonal form of the minimal $\rank$ when $-3 \le 2^{\ceil{\log_2(m-3)}}-m\le 1$ which yields that $r_m=\ceil{\log_2(m-3)}+1$.
With Lemma \ref{rm1}, Remark \ref{rmk(rm)1}, and Lemma \ref{rm2}, we may claim Theorem \ref{min}.
\end{rmk}
\section{The maximal rank for a leaf of the escalator tree
Throughout this section, we prove Theorem \ref{main}.
\begin{prop}\label{node}
A node $F_m(\mathbf x)=\sum_{i=1}^ka_iP_m(x_i)$ of the escalator tree represents every positive integer up to $\sum _{i=1}^ka_i$.
\end{prop}
\begin{proof}
The proof proceeds by induction on the rank $k$ of node.
When $k=1$, it is clear because $a_1=1$.
And now assume that the Proposition is true for all nodes of $\rank \ k-1$, that is, any node $\sum_{i=1}^{k-1}a_iP_m(x_i)$ of $\rank \ k-1$ represents every positive integer up to $\sum _{i=1}^{k-1}a_i$.
For a node $\sum _{i=1}^ka_iP_m(x_i)$ of the $\rank \ k$ to obtain a contradiction, assume that there is an integer $\alpha \le \sum _{i=1}^ka_i$ which is not represented by the node $\sum _{i=1}^ka_iP_m(x_i)$.
Since the truant of $\sum_{i=1}^{k-1}a_iP_m(x_i)$ is less than the truant of $\sum_{i=1}^{k}a_iP_m(x_i)$ (which is less than or equal to $\alpha$), we may get that $a_k \le \alpha$ because $a_k$ must be less than or equal to the truant of $\sum_{i=1}^{k-1}a_iP_m(x_i)$.
Since $0 \le \alpha-a_k \le \sum_{i=1}^{k-1}a_i$, by the induction hypothesis, $\alpha - a_k$ may be represented by $\sum _{i=1}^{k-1}a_iP_m(x_i)$.
Therefore $$\alpha=(\alpha - a_k)+a_k$$ may be represented by $\sum _{i=1}^ka_iP_m(x_i)$ by taking $x_k=1$, which is a contradiction.
This completes the proof.
\end{proof}
\vskip 0.5em
\begin{rmk} \label{key}
We may obtain that a node $$\sum \limits_{i=1}^ka_iP_m(x_i)$$ with $C(m-2) \le a_1+\cdots +a_k$ would be a leaf, i.e., a proper universal $m$-gonal form by Theorem \ref{C} and Proposition \ref{node}
\end{rmk}
\vskip 0.5em
\begin{lem} \label{2}
For $m>6C^2(C+1)$, a leaf $F_m(\mathbf x)=\sum \limits_{i=1}^na_iP_m(x_i)$ with $a_{l_m} \ge C+1$ where $l_m:=\floor {\frac{m-2}{C+1}}$ has the $\rank \ n \le \left(1-\frac{1}{(C+1)^2} \right)(m-2)+\frac{C+2}{C+1}$.
\end{lem}
\begin{proof}
To obtain a contradiction assume that $n >(1-\frac{1}{(C+1)^2})(m-2)+\frac{C+2}{C+1}$.
Then we may get
\begin{align*}\sum \limits_{i=1}^{n-1}a_i & =\sum \limits_{i=1}^{l_m}a_i+\sum \limits_{i={l_m}+1}^{n-1}a_i \\
&>\left (\frac{m-2}{C+1}-1\right)+(C+1)\left(n-1-\frac{m-2}{C+1}\right)>C(m-2)\end{align*}
which yields that the parent $\sum \limits_{i=1}^{n-1}a_iP_m(x_i)$ of $F_m(\mathbf x)$ is already universal by Remark \ref{key}, which is a contradiction.
Consequently, we may conclude that $n \le (1-\frac{1}{(C+1)^2})(m-2)+\frac{C+2}{C+1}$.
\end{proof}
\vskip 0.5em
\begin{rmk}
Now we consider a node $\sum \limits_{i=1}^na_iP_m(x_i)$ with $0 \neq a_{l_m} < C+1$ where $l_m:=\floor {\frac{m-2}{C+1}}$.
One may easily see that there appear 5 consecutively same coefficients between $C$-th coefficient $a_C$ and $5C$-th coefficient $a_{5C}$, i.e., there is $C\le t \le 5C-4$ for which $$a_t=a_{t+1}=\cdots =a_{t+4}$$
because there are $4C+1$ components between $a_CP_m(x_C)$ and $a_{5C}P_m(x_{5C})$ and $1\le a_C \le a_{C+1} \le \cdots \le a_{5C} \le a_{l_m}<C+1$.
\end{rmk}
\vskip 0.5em
\begin{lem} \label{A(m-2)}
For $A\in \mathbb N$, the $m$-gonal form $$\sum \limits_{i=1}^5A\cdot P_m(x_i)$$ represents all the multiples of $A(m-2)$.
\end{lem}
\begin{proof}
See Lemma 2.2 in \cite{KP}.
\end{proof}
\vskip 0.5em
\begin{prop}\label{b}
For $m \ge 6C^2(C+1)$, let $F_m(\mathbf x)=\sum \limits_{i=1}^na_iP_m(x_i)$ be a leaf with $0 \not= a_{l_m} < C+1$ where $l_m:=\floor {\frac{m-2}{C+1}}$.
If there is $C \le t \le 5C-4$ for which $a_t=a_{t+1}=\cdots=a_{t+4}$, then we rearrange the coefficients of $F_m(\mathbf x)$ except the 5 consecutive coefficients $a_t,\cdots,a_{t+4}$ as follows
$$b_i:=\begin{cases}a_i & \text{ when } i<t\\ a_{i+5} & \text{ when } i\ge t.
\end{cases}$$
And then we have the followings.
\begin{itemize}
\item[(1)] For $i \le l_m$, the inequality $b_i\le b_1+\cdots +b_{i-1}+1$ always holds.
\item[(2)] If there is $l_m < i \le C\cdot l_m$ such that $b_i> b_1+\cdots +b_{i-1}+1$, then $n < m-4$.
\end{itemize}
\end{prop}
\begin{proof}
(1) For $i<t$, if $C \le b_1+\cdots +b_{i-1}$, then since $b_i=a_i \le a_{l_m} \le C$ from the assumption, we may get that $b_{i}\le b_1+\cdots +b_{i-1}+1$. If $a_1+\cdots +a_{i-1}=b_1+\cdots +b_{i-1} <C<m-4$, then the truant of the node $a_1P_m(x_1)+\cdots +a_{i-1}P_m(x_{i-1})$ would be $a_1+\cdots +a_{i-1}+1(<m-3)$.
So we obtain that $b_i=a_i \le a_1+\cdots +a_{i-1}+1=b_1+\cdots +b_{i-1}+1$.
If there is $t\le i \le l_m$ such that $b_{i}>b_1+\cdots +b_{i-1}+1$,
then we may get that $$C \le t \le i=(i-1)+1 \le b_1+\cdots+b_{i-1}+1 < b_i \le a_{l_m}.$$
This is a contradiction to $a_{l_m} \le C$.
This yields the claim.
(2) To obtain a contradiction, assume that $n \ge m-4$.
If there is $l_m\left(=\floor{\frac{m-2}{C+1}}\right)<i\le C\cdot l_m$ such that $b_{i}>b_1+\cdots +b_{i-1}+1$, then we may have that $$a_{i+5}>a_1+\cdots +a_{i+4}+1-5A \ge i +5-5A > \frac{m-2}{C+1}-5A \ge (C+1)^2.$$
And then from
\begin{align*}\sum \limits_{i=1}^{n-1}a_i & =\sum \limits_{i=1}^{C\cdot l_m+4}a_i+\sum \limits_{i=C\cdot l_m+5}^{n-1}a_i\\ &\ge (C\cdot l_m +4)+(C+1)^2(n-1-C\cdot l_m-4)\\
&\ge (C\cdot l_m +4)+(C+1)^2(m-9-C\cdot l_m)>C(m-2),\end{align*}
we may obtain a contradiction that the parent $\sum_{i=1}^{n-1}a_iP_m(x_i)$ of $F_m(\mathbf x)$ is already universal by Remark \ref{key}.
Consequently, we may conclude that $n < m-4$.
\end{proof}
\vskip 0.5em
\begin{rmk}
In Lemma \ref{2} and Proposition \ref{b}, we showed that the $\rank$ of leaf under some conditions does not exceed $m-4$.
Actually, what the conditions have in common is that its escalating is not happening that slowly.
Under such the conditions, we reach to leaf node, i.e., a universal $m$-gonal form before $m-4$ escalating steps.
\end{rmk}
\begin{lem} \label{13}
For $m>6C^2(C+1)$, let $F_m(\mathbf x)=\sum \limits_{i=1}^na_iP_m(x_i)$ be a leaf with $0 \not= a_{l_m} < C+1$ where $l_m:=\floor {\frac{m-2}{C+1}}$.
Then $$(a_1,a_2)=(1,1)\text{ or } (a_1,a_2,a_3) \in \{(1,2,2),(1,2,3),(1,2,4)\}$$
and there is $C \le t \le 5C-4$ such that $$a_t=a_{t+1}=\cdots =a_{t+4}=:A.$$
\begin{itemize}
\item[(1)] When $(a_1,a_2)=(1,1)$, if there is above $C \le t \le 5C-4$ for which $a_t=\cdots=a_{t+4}=A>6$, then $n < m-4$.
\item[(2)] When $(a_1,a_2,a_3)=(1,2,2)$, if there is above $C \le t \le 5C-4$ for which $a_t=\cdots=a_{t+4}=A>12$, then $n < m-4$.
\item[(3)] When $(a_1,a_2,a_3)=(1,2,3)$, if there is above $C \le t \le 5C-4$ for which $a_t=\cdots=a_{t+4}=A>12$, then $n < m-4$.
\item[(4)] When $(a_1,a_2,a_3)=(1,2,4)$, $n <m-4$.
\end{itemize}
\end{lem}
\begin{proof}
Since the coefficients of $F_m(\mathbf x)$ follows the conditions (\ref{coe}), we may have that $$(a_1,a_2)=(1,1),\text{ or } (a_1,a_2,a_3) \in \{(1,2,2),(1,2,3),(1,2,4)\}.$$
We prove this lemma only for the case $(a_1,a_2)=(1,1)$ and omit the proof of (2), (3), and (4) in this paper.
One may prove those through similar arguments with the proof of (1).
Under same notation as in Proposition \ref{b}, in virtue of Proposition \ref{b}, we could assume that
\begin{equation}\label{bas}b_i \le b_1+\cdots +b_{i-1}+1\end{equation} for all $1\le i \le \min \{n-5, C\cdot l_m \}$.
Under the assumption (\ref{bas}), we prove that $n \le C\cdot l_m+5 \left( \approx \frac{C}{C+1}m\right)<m-4$.
To obtain a contradiction, assume that $n >C\cdot l_m+5$.
From \begin{align*}\sum \limits_{i=1}^{C\cdot l_m}b_i=\sum \limits_{i=1}^{5C}b_i+\sum \limits_{i=5C+1}^{C\cdot l_m}b_i &\ge 5C+A(C\cdot l_m-5C) \\
& \ge 5C+AC(\frac{m-2}{C+1}-1)-5AC \\
& > (A-1)(m-2)+4
\end{align*}
with (\ref{bas}), we may have that $$\sum _{i=1}^{C\cdot l_m}b_iP_m(x_i)$$ represents every positive integer up to $(A-1)(m-2)+4$ by taking $x_i \in \{0,1\}$ for all $i$.
For integer $N \in [(A-1)(m-2)+5,A(m-2)]$, since $$0 <N-\{(6(m-2)+4\} < (A-1)(m-2)$$ holds, $N-\{(6(m-2)+4\} $ may be written as
\begin{equation}\label{11'}N-\{6(m-2)+4\}=\sum \limits_{i=1}^{C\cdot l_m}b_iP_m(N(x_i))\end{equation} for some $(N(x_i))_i \in \{0,1\} ^{C\cdot l_m}$ by the above argument.
Up to reordering, we may assume that $(N(x_1),N(x_2))=(0,0), (1,0),$ or $(1,1)$.
Note that
\begin{equation}\label{11}\begin{cases} P_m(4)+P_m(0)=6(m-2)+4 \\ P_m(4)+P_m(1)=6(m-2)+5 \\ P_m(3)+P_m(3)=6(m-2)+6. \end{cases}\end{equation}
And then with (\ref{11'}) and (\ref{11}), we may see that the integer $N \in [(A-1)(m-2)+5,A(m-2)]$ is written as follows
$$\begin{cases}
N=P_m(4)+P_m(0)+\sum \limits_{i=3}^{C\cdot l_m}b_iP_m(N(x_i)) & \text{ when }(N(x_1),N(x_2))=(0,0) \\
N=P_m(4)+P_m(1)+\sum \limits_{i=3}^{C\cdot l_m}b_iP_m(N(x_i)) & \text{ when }(N(x_1),N(x_2))=(1,0) \\
N=P_m(3)+P_m(3)+\sum \limits_{i=3}^{C\cdot l_m}b_iP_m(N(x_i)) & \text{ when }(N(x_1),N(x_2))=(1,1). \\
\end{cases}$$
As a result, we may obtain that $\sum_{i=1}^{C\cdot l_m}b_iP_m(x_i)$ represents every positive integer up to $A(m-2)$.
On the other hand, by Lemma \ref{A(m-2)}, all the multiples of $A(m-2)$ may be represented by $$\sum_{i=t}^{t+4}a_iP_m(x_i).$$
Consequently, we may conclude that $$\sum_{i=1}^{C\cdot l_m+5}a_iP_m(x_i)=\sum_{i=1}^{C\cdot l_m}b_iP_m(x_i)+\sum_{i=t}^{t+4}a_iP_m(x_i)$$
represents every positive integer, i.e., the parent of $F_m(\mathbf x)$ is already universal, yielding a contradiction to $F_m(\mathbf x)$ is a leaf.
This completes the proof of (1).
By using the below equations instead of (\ref{11}), one may show (2), (3), and (4) through similar arguments with the above.
(2) When $(a_1,a_2,a_3)=(1,2,2),$ one may use following equations\\
\begin{equation}\label{122}
\begin{cases}
P_m(0)+2P_m(-2)+2P_m(-2)=12m-32\\
P_m(-3)+2P_m(-2)+2P_m(0)=12m-31\\
P_m(-4)+2P_m(-1)+2P_m(0)=12m-30\\
P_m(-3)+2P_m(-2)+2P_m(1)=12m-29\\
P_m(0)+2P_m(-3)+2P_m(1)=12m-28\\
P_m(1)+2P_m(-3)+2P_m(1)=12m-27.\\
\end{cases}
\end{equation}
(3) When $(a_1,a_2,a_3)=(1,2,3), $ one may use following equations\\
\begin{equation}\label{123}
\begin{cases}
P_m(-2)+2P_m(-2)+3P_m(-1)=12m-33\\
P_m(-2)+2P_m(0)+3P_m(-2)=12m-32\\
P_m(-3)+2P_m(-2)+3P_m(0)=12m-31\\
P_m(0)+2P_m(-3)+3P_m(0)=12m-30\\
P_m(1)+2P_m(-3)+3P_m(0)=12m-29\\
P_m(3)+2P_m(-2)+3P_m(-1)=12m-28\\
P_m(3)+2P_m(0)+3P_m(-2)=12m-27.\\
\end{cases}
\end{equation}
(4) When $(a_1,a_2,a_3)=(1,2,4), $ one may use following equations (note that when $a_3=4$, $A\ge 4$)\\
\begin{equation}\label{124}
\begin{cases}
P_m(2)+2P_m(-1)+4P_m(0)=3m-6\\
P_m(-1)+2P_m(-1)+4P_m(1)=3m-5\\
P_m(-2)+2P_m(0)+4P_m(1)=3m-4\\
P_m(3)+2P_m(0)+4P_m(0)=3m-3\\
P_m(-2)+2P_m(1)+4P_m(1)=3m-2\\
P_m(3)+2P_m(1)+4P_m(0)=3m-1\\
P_m(2)+2P_m(2)+4P_m(0)=3m\\
P_m(3)+2P_m(0)+4P_m(1)=3m+1.\\
\end{cases}
\end{equation}
\end{proof}
\vskip 0.5em
\begin{rmk}
In Lemma \ref{13}, we showed that if a leaf $F_m(\mathbf x)=\sum \limits_{i=1}^na_iP_m(x_i)$ with $0 \neq a_{l_m} < C+1$ has the 5 consecutively same coefficients $A$ greater than $12$ between $C$-th component and $5C$-th component, then its $\rank$ could not exceed $m-4$.
Especially under the condition (\ref{bas}), the $\rank$ would be less than or equal to $C\cdot l_m+5 (\approx \frac{C}{C+1}m)$ (more stirict calculations would much reduce the upper bound for $\rank \ n$).
On the other hand, next lemma may help to consider the upper bound for $\rank$ of leaves $F_m(\mathbf x)=\sum \limits_{i=1}^na_iP_m(x_i)$ with $0 \neq a_{l_m} < C+1$ for which every 5 consecutively same coefficients appearing between $C$-th component and $5C$-th component is less than or equal to $12$.
\end{rmk}
\begin{lem} \label{k(A)+1}
For $m>6C^2(C+1)$, let $F_m(\mathbf x)=\sum \limits_{i=1}^na_iP_m(x_i)$ be a leaf with $0 \neq a_{l_m} < C+1$ where $l_m:=\floor {\frac{m-2}{C+1}}$.
Then there is $C \le t \le 5C-4$ such that $$a_t=a_{t+1}=\cdots =a_{t+4}=:A.$$
For $A \ge 2$, let $i(A)$ be the smallest index satisfying $A-1 \le a_1+\cdots+a_{i(A)}$.
\begin{itemize}
\item[(1)]
If $a_{i(A)+1} \le A-1$, then $n < m-4$.
\item[(2)]
If $A+1 \le a_{(C-A-1)l_m+5}$, then $n < m-4$.
\end{itemize}
\end{lem}
\begin{proof}
(1) Similarly with Proposition \ref{b}, we rearrange the coefficients of $F_m(\mathbf x)$ except 6 coefficients $a_{i(A)+1},a_t,\cdots,a_{t+4}$ as follows $$c_i:=\begin{cases}a_i & \text{ when } i \le i(A)\\ a_{i+1} & \text{ when } i(A)+1 \le i < t-1 \\ a_{i+6} & \text{ when } t-1 \le i.
\end{cases}$$
Through similar arguments with the proof of Proposition \ref{b}, one may induce that if $c_i> c_1+\cdots +c_{i-1}+1$ for some $ i \le C\cdot l_m$, then $n < m-4$.
From now on, we prove the lemma under the assumption \begin{equation}\label{cas}\text{$c_i\le c_1+\cdots +c_{i-1}+1$ for any $i \le C\cdot l_m$}.\end{equation}
To obtain a contradiction assume that $m-4\le n$.
Through similar arguments with the proof of Lemma \ref{13}, one may obtain that $\sum_{i=1}^{C\cdot l_m}c_iP_m(x_i)$ represents every positive integer up to $(A-1)(m-2)$ by taking $x_i \in \{0,1\}$ for all $i$.
For an integer $N \in [(A-1)(m-2),A(m-2)]$, we may see that \begin{equation} \label{eq2}0<N-a_{i(A)+1}P_m(x_{i(A)+1})<(A-1)(m-2)\end{equation} holds for some $x_{i(A)+1} \in \{-1,2\}$.
From the above argument, since the $N-a_{i(A)+1}P_m(x_{i(A)+1})$ of (\ref{eq2}) is represented by $\sum _{i=1}^{C\cdot l_m}c_iP_m(x_i)$, we may yield that
$N$ is represented by $\sum _{i=1}^{C\cdot l_m}c_iP_m(x_i)+a_{i(A)+1}P_m(x_{i(A)+1})$ by taking $x_{i(A)+1} \in \{-1,2\}$.
So we may obtain that $$\sum \limits _{i=1}^{C\cdot l_m}c_iP_m(x_i)+a_{i(A)+1}P_m(x_{i(A)+1})$$ represents every positive integer up to $A(m-2)$.
On the other hand, by Lemma \ref{A(m-2)}, all the multiples of $A(m-2)$ may be represented by $\sum_{i=t}^{t+4}a_iP_m(x_i)$.
Finally, we may conclude that
$$\sum \limits_{i=1}^{C\cdot l_m +6}a_iP_m(x_i)=a_{i(A)+1}P_m(x_{i(A)+1})+\sum \limits _{i=1}^{C\cdot l_m}c_iP_m(x_i)+\sum \limits_{i=t}^{t+4}a_iP_m(x_i)$$ $\left(C\cdot l_m +6 \approx \frac{C}{C+1}m\right)$ is already universal, yielding a contradiction to the fact that $F_m(\mathbf x)$ is a leaf.
This completes the proof of (1).
(2) In virtue of Proposition \ref{b}, under same notation as in the Proposition \ref{b}, we could assume that
\begin{equation}\label{bas'}b_i \le b_1+\cdots +b_{i-1}+1\end{equation} for all $1\le i \le \min \{n-5, C\cdot l_m\}$.
Under the assumption (\ref{bas'}), we prove that $n \le C\cdot l_m+5<m-4$.
To obtain a contradiction, assume that $n >C\cdot l_m+5$.
With (\ref{bas'}), from
\begin{align*}
\sum_{i=1}^{C\cdot l_m}b_i & =\sum_{i=1}^{5C}b_i+\sum_{i=5C+1}^{(C-A-1)\cdot l_m-1}b_i+\sum_{i=(C-A-1)l_m}^{C\cdot l_m}b_i\\
&\ge 5C+A((C-A-1)l_m-1-5C)+(A+1)(A+1)l_m\\
&=5C+A(C\cdot l_m-1)+(A+1)l_m\\
&>5C+A\left(C\cdot\frac{m-2}{C+1}-2\right)+(A+1)\left(\frac{m-2}{C+1}-1\right)\\
&=5C+A(m-2)-A\left(\frac{m-2}{C+1}+2\right)+(A+1)\left(\frac{m-2}{C+1}-1\right) \\
&>A(m-2),
\end{align*}
we may induce that $\sum_{i=1}^{C\cdot l_m}b_iP_m(x_i)$ represents every positive integer up to $A(m-2)$.
So we may conclude that $$\sum_{i=1}^{C\cdot l_m+5}a_iP_m(x_i)=\sum_{i=1}^{C\cdot l_m}b_iP_m(x_i)+\sum_{i=t}^{t+4}a_iP_m(x_i)$$
is already universal, yielding a contradiction since $\sum_{i=t}^{t+4}a_iP_m(x_i)$ represents all the multiples of $A(m-2)$ by Lemma \ref{A(m-2)}.
This completes the proof.
\end{proof}
\begin{table}
\caption{} \label{t2}
\begin{tabular}{|c|c|}
\hline \rule[-2.4mm]{0mm}{7mm}
A & $(a_1,\cdots,a_{i(A)})$ \\
\hline \hline
$1$ & \\
\hline
$2$ & $(1)$ \\
\hline
$3$ & $(1,1),(1,2)$ \\
\hline
$4$ & $(1,1,1),(1,1,2),(1,1,3),(1,2)$ \\
\hline
$5$ & $(1,1,1,1), (1,1,1,2), (1,1,1,3), (1,1,1,4), (1,2,2), (1,2,3), (1,2,4)$ \\
\hline
& $(1,1,1,1,1), (1,1,1,1,2), (1,1,1,1,3), (1,1,1,1,4),(1,1,1,1,5),$ \\
$6$ & $(1,1,1,2), (1,1,1,3), (1,1,1,4),(1,1,2,2), (1,1,2,3), (1,1,2,4), (1,1,2,5),$\\
& $ (1,2,2), (1,2,3), (1,2,4)$ \\
\hline
$7$ & $(1,2,2,2), (1,2,2,3), (1,2,2,4), (1,2,2,5), (1,2,2,6),(1,2,3)$ \\
\hline
$8$ & $(1,2,2,2),(1,2,2,3), (1,2,2,4), (1,2,2,5), (1,2,2,6), (1,2,3,3), (1,2,3,4), $\\
& $ (1,2,3,5), (1,2,3,6), (1,2,3,7)$\\
\hline
& $(1,2,2,2,2),(1,2,2,2,3),(1,2,2,2,4),(1,2,2,2,5),(1,2,2,2,6),(1,2,2,2,7), $\\
$9$&$(1,2,2,2,8),(1,2,2,3), (1,2,2,4), (1,2,2,5), (1,2,2,6), (1,2,3,3), (1,2,3,4), $\\
& $ (1,2,3,5), (1,2,3,6), (1,2,3,7)$\\
\hline
& $(1,2,2,2,2),(1,2,2,2,3),(1,2,2,2,4),(1,2,2,2,5),(1,2,2,2,6),(1,2,2,2,7), $\\
$10$&$(1,2,2,2,8),(1,2,2,3,3),(1,2,2,3,4), (1,2,2,3,5), (1,2,2,3,6), (1,2,2,3,7), $\\
&$(1,2,2,3,8),(1,2,2,3,9),(1,2,2,4), (1,2,2,5), (1,2,2,6), (1,2,3,3), (1,2,3,4), $\\
& $ (1,2,3,5), (1,2,3,6), (1,2,3,7)$\\
\hline
& $(1,2,2,2,2,2),(1,2,2,2,2,3),(1,2,2,2,2,4),(1,2,2,2,2,5),(1,2,2,2,2,6),$\\
&$(1,2,2,2,2,7),(1,2,2,2,2,8),(1,2,2,2,2,9),(1,2,2,2,2,10),$\\
&$(1,2,2,2,3),(1,2,2,2,4),(1,2,2,2,5),(1,2,2,2,6),(1,2,2,2,7), $\\
$11$&$(1,2,2,2,8),(1,2,2,3,3),(1,2,2,3,4), (1,2,2,3,5), (1,2,2,3,6), (1,2,2,3,7), $\\
&$(1,2,2,3,8),(1,2,2,3,9),(1,2,2,4,4),(1,2,2,4,5),(1,2,2,4,6),(1,2,2,4,7),$\\
&$(1,2,2,4,8),(1,2,2,4,9),(1,2,2,4,10),(1,2,2,5), (1,2,2,6),$\\
&$ (1,2,3,3,3),(1,2,3,3,4),(1,2,3,3,5),(1,2,3,3,6),(1,2,3,3,7),$\\
&$(1,2,3,3,8),(1,2,3,3,9),(1,2,3,3,10), (1,2,3,4),(1,2,3,5), (1,2,3,6), $\\
& $ (1,2,3,7)$\\
\hline
& $(1,2,2,2,2,2),(1,2,2,2,2,3),(1,2,2,2,2,4),(1,2,2,2,2,5),(1,2,2,2,2,6),$\\
&$(1,2,2,2,2,7),(1,2,2,2,2,8),(1,2,2,2,2,9),(1,2,2,2,2,10)$\\
&$(1,2,2,2,3,3),(1,2,2,2,3,4),(1,2,2,2,3,5),(1,2,2,2,3,6),(1,2,2,2,3,7),$\\
&$(1,2,2,2,3,8),(1,2,2,2,3,9),(1,2,2,2,3,10),(1,2,2,2,3,11),(1,2,2,2,4),$\\
&$(1,2,2,2,5),(1,2,2,2,6),(1,2,2,2,7), (1,2,2,2,8),(1,2,2,3,3),(1,2,2,3,4),$\\
$12$&$(1,2,2,3,5), (1,2,2,3,6), (1,2,2,3,7),(1,2,2,3,8),(1,2,2,3,9),(1,2,2,4,4), $\\
&$(1,2,2,4,5),(1,2,2,4,6),(1,2,2,4,7),(1,2,2,4,8),(1,2,2,4,9),(1,2,2,4,10),$\\
&$(1,2,2,5,5),(1,2,2,5,6),(1,2,2,5,7),(1,2,2,5,8),(1,2,2,5,9),(1,2,2,5,10), $\\
&$ (1,2,2,5,11),(1,2,2,6), (1,2,3,3,3),(1,2,3,3,4),(1,2,3,3,5),(1,2,3,3,6),$\\
&$(1,2,3,3,7),(1,2,3,3,8),(1,2,3,3,9),(1,2,3,3,10), (1,2,3,4,4), $\\
&$(1,2,3,4,5),(1,2,3,4,6),(1,2,3,4,7),(1,2,3,4,8),(1,2,3,4,9),$\\
& $(1,2,3,4,10),(1,2,3,4,11), (1,2,3,5), (1,2,3,6),(1,2,3,7)$\\
\hline
\end{tabular}
\end{table}
\vskip 0.5em
\begin{rmk}
In Section 4, until now, we showed that a great part of leaf has $\rank$ smaller than $m-4$,
more precisely, every leaf other than the leaves $F_m(\mathbf x)=\sum \limits_{i=1}^na_iP_m(x_i)$
with
\begin{equation}\label{fin}\begin{cases}(A;a_1,\cdots,a_{i(A)}) \text{ in Table \ref{t2} and} \\ a_{i}=A \ \text{ for all }i(A)<i \le \min \{n,(C-A-1)l_m\} \end{cases}\end{equation}
has the $\rank$ less than $m-4$.
We lastly consider the $\rank$ of leaves $F_m(\mathbf x)$ with (\ref{fin}).
One may notice that the leaves $F_m(\mathbf x)$ with (\ref{fin}) are very slowly escalated proper universal $m$-gonal form.
\end{rmk}
\vskip 0.5em
\begin{lem} \label{AAAAA}
The $m$-gonal forms $$a_1P_m(x_1)+\cdots +a_{i(A)}P_m(x_{i(A)})+AP_m(x_{i(A)+1})+\cdots+AP_m(x_n)$$
where $n=\ceil{\left(1-\frac{1}{2A}\right)(C+1)l_m+\left(1-\frac{1}{2A}\right)(C+1)}+i(A)+5$ with $(A;a_1,\cdots,a_{i(A)})$ in Table \ref{t2} other than $ (1;\ ),(3;1,1),$ and $(3;1,2)$ are universal.
\end{lem}
\begin{proof}
Similarly with Proposition \ref{b}, we rearrange the coefficients except the $5$ coefficients $a_t,\cdots, a_{t+4}$ as
$$b_i:=\begin{cases}a_i & \text{ when } i\le i(A)\\ a_{i+5} & \text{ when } i\ge i(A)+1.
\end{cases}$$
From
\begin{align*}
\sum_{i=1}^{n}b_i&=\sum_{i=1}^{i(A)}b_i+\sum_{i=i(A)+1}^{n-5}b_i\\
& =\sum_{i=1}^{i(A)}b_i+\sum_{i=i(A)+1}^{n-5}A\\
& \ge \sum_{i=1}^{i(A)}b_i+\left(A-\frac{1}{2}\right)(m-2),
\end{align*}
we may see that the $m$-gonal form $\sum_{i=1}^{n-5}b_iP_m(x_i)$ represents every positive integer up to $\floor{\left(A-\frac{1}{2}\right)(m-2)}$.
On the other hand, one may directly check that for each $(A;a_1,\cdots,a_{i(A)})=(A;b_1,\cdots,b_{i(A)})$ in Table \ref{t2} other than $ (1;\ ),(3;1,1),$ and $(3;1,2)$, $$a_1P_m(x_1)+\cdots+a_{i(A)}P_m(x_{i(A)})=b_1P_m(x_1)+\cdots+b_{i(A)}P_m(x_{i(A)})$$ represents complete residues modulo $A$ in $\left[m-3,\floor{\left(A-\frac{1}{2}\right)(m-2)}\right]$.
For example, for $(A;a_1,\cdots, a_{i(A)})=(2;1)$, we have a complete system of residues
$$P_m(-1)=m-3, \quad P_m(2)=m$$ modulo $A=2$
and for $(A;a_1,\cdots, a_{i(A)})=(4;1,1,1)$, we have a complete system of residues
\begin{align*} P_m(-1)+P_m(0)+P_m(0)=m-3, \quad & P_m(-1)+P_m(1)+P_m(0)=m-2,\\
P_m(-1)+P_m(1)+P_m(1)=m-1, \quad & P_m(2)+P_m(0)+P_m(0)=m\\\end{align*}
modulo $A=4$.
Without difficulty, one may check that for the other cases too by hand.
We omit the lengthy calcuations in this paper.
And then an integer $N \in \left[\floor{\left(A-\frac{1}{2}\right)(m-2)},A(m-2)\right]$ may be written as
$$N=b_1P_m(N_1)+\cdots +b_{i(A)}P_m(N_{i(A)})+AP_m(N_{i(A)+1})+\cdots +AP_m(N_{n-5})$$
where $N':=b_1P_m(N_1)+\cdots +b_{i(A)}P_m(N_{i(A)})$ is an integer in $[m-3,(A-1)(m-2)]$ which is equivalent with $N$ modulo $A$ and $(N_{i(A)+1},\cdots , N_{n-5}) \in \{0,1\}^{n-i(A)-5}$ since $$0\le N-N'\le (A-1)(m-2)+1$$ is a multiple of $A$.
So we may get that $\sum_{i=1}^{n-5}b_iP_m(x_i)$ represents every positive integer up to $A(m-2)$.
By Lemma \ref{A(m-2)}, since $$AP_m(x_{i(A)+1})+\cdots +AP_m(x_{i(A)+5})$$ represents all the multiples of $A(m-2)$, we may conclude that $$\sum_{i=1}^na_iP_m(x_i)=AP_m(x_{i(A)+1})+\cdots +AP_m(x_{i(A)+5})+\sum_{i=1}^{n-5}b_iP_m(x_i)$$ is universal.
\end{proof}
\begin{lem}
Let $F_m(\mathbf x)=\sum \limits_{i=1}^na_iP_m(x_i)$ be a leaf with $0 \neq a_{l_m} < C+1$ where $l_m:=\floor {\frac{m-2}{C+1}}$.
If there is $ t \le 5C-4$ for which $$a_t=a_{t+1}=\cdots =a_{t+4}=:A.$$
with $A\neq 1,3$, then we have $n <m-4$.
\end{lem}
\begin{proof}
From Lemma \ref{b}, Lemma \ref{k(A)+1}, and Lemma \ref{AAAAA}, one may yield this.
\end{proof}
\vskip 0.5em
\begin{rmk}
More delicate care may reduce the upper bound for the $\rank$ of $F_m(\mathbf x):=a_1P_m(x_1)+\cdots +a_{i(A)}P_m(x_{i(A)})+AP_m(x_{i(A)+1})+\cdots+AP_m(x_n)$ $$n\approx \left(1-\frac{1}{2A}\right)(m-2)$$ which makes $$a_1P_m(x_1)+\cdots +a_{i(A)}P_m(x_{i(A)})+AP_m(x_{i(A)+1})+\cdots+AP_m(x_n)$$ universal in the above lemma.
Actually, following \cite{KP}, especially for $(A;a_1,\cdots, a_{i(A)})$ of the form $(A;1,\cdots, 1)$ with $A \neq 1,3$, we may take the below $n$
\begin{equation}\label{n}
n=\begin{cases}
\floor{\frac{m}{2}} & \text{ when }A=2 \\
\ceil{\frac{m-2}{4}}+2 & \text{ when }A=4 \\
\ceil{\frac{m-3}{A}}+(A-2) & \text{ when } 5\le A \le 12 \\
\end{cases}
\end{equation}
instead of $n=\ceil{\left(1-\frac{1}{2A}\right)(C+1)l_m+\left(1-\frac{1}{2A}\right)(C+1)}+i(A)+5\approx \left(1-\frac{1}{2A}\right)(m-2)$ in Lemma \ref{AAAAA}
and the $n$ of (\ref{n}) would be optimal.
The authors guess that for the most $(A;a_1,\cdots, a_{i(A)})$ in Table \ref{t2} other than $(3;1,1),$ and $(3;1,2)$ the optimal $n$
which makes $$a_1P_m(x_1)+\cdots +a_{i(A)}P_m(x_{i(A)})+AP_m(x_{i(A)+1})+\cdots+AP_m(x_n)$$ universal would be close to $\frac{m}{A}$ but not all.
Lastly, we consider the $\rank$ of leaves $F_m(\mathbf x)=\sum \limits_{i=1}^na_iP_m(x_i)$ with
$$(a_1,\cdots,a_{(C-2)l_m+5})=(1,\cdots,1)$$ or $$(a_1,\cdots, a_{\min \{n,(C-4)l_m+5\}})=(1,1,3.\cdots,3) \text{ or }(1,2,3,\cdots, 3).$$
Actually among such the above leaves, the maximal $\rank R_m$ of leaves of $m$-gonal form's escalator tree would appear.
\end{rmk}
\begin{lem} \label{A=1}
Let $F_m(\mathbf x)=\sum \limits_{i=1}^na_iP_m(x_i)$ be a node of the escalator tree with
\begin{equation}\label{A=1,cond'}a_1=a_2=\cdots =a_{(C-2)l_m+5}=1.\end{equation}
If \begin{equation}\label{A=1,cond} m-4 \le a_1+a_2+\cdots+a_n,\end{equation} then the node would be universal, i.e., the node would become a leaf of the tree.
\end{lem}
\begin{proof}
Note that the $m$-gonal form
$$f_m(\mathbf x):=a_6P_m(x_6)+\cdots +a_nP_m(x_n)$$ represents every positive integer up to $m-3$ except at most $5$ integers by taking $P_m(x_6) \in \{0,1,m-3\}$ and $P_m(x_i) \in \{0,1\}$ for all $7 \le i \le n$.
And the integers not represented by $f_m(\mathbf x)$ would be consecutive.
On the specific,
if there is an integer not represented by $f_m(\mathbf x)$, then the integer always would be in $[(C-2)l_m+1,m-4]$ and such the situation would happen only when
\begin{equation}\label{A=1case}\begin{cases}a_1+\cdots+a_n-5<m-4 & \text{ or } \\ a_n>a_1+\cdots +a_{n-1}+1-5. & \end{cases}\end{equation}
When $a_1+\cdots+a_n-5<m-4$, the consecutive integers not represented by $f_m(\mathbf x)$ in $[(C-2)l_m+1,m-4]$ would be $$a_1+\cdots + a_n-4,a_1+\cdots + a_n-3,\cdots, m-4$$ and when $a_n>a_1+\cdots +a_{n-1}+1-5$, the consecutive integers not represented by $f_m(\mathbf x)$ in $[(C-2)l_m+1,m-4]$ would be $$a_1+\cdots+a_{n-1}-4,a_1+\cdots+a_{n-1}-3,\cdots,a_n-1.$$
In the cases that $f_m(\mathbf x)$ represents every positive integer up to $m-3$, we may conclude that $F_m(\mathbf x)$ is universal by using the fact that $$a_1P_m(x_1)+\cdots+a_5P_m(x_5)=P_m(x_1)+\cdots+P_m(x_5)$$ represents all the multiples of $m-2$ from Lemma \ref{A(m-2)}.
For the other cases, let $$E_1< E_2(=E_1+1) < \cdots < E_s(=E_1+(s-1))$$ where $s \le 5$ be all of the positive integers which are not represented by $f_m(\mathbf x)$ in $[(C-2)l_m+1,m-4]$.
By using Lemma \ref{A(m-2)} again, we may yield that $F_m(\mathbf x)$ represents every positive integer which is not congruent to $E_1,E_2,\cdots, E_s$ modulo $m-2$.
On the other hand, one may observe that
$E_1-1$ is written as $f_m(\mathbf x)=a_6P_m(x_6)+\cdots +a_nP_m(x_n)$ with $x_i=1$ for all $6 \le i \le n-1$ so from the representation of $E_1-1$ by changing $P_m(x_6)=1$ to $P_m(2)=m$, we may obtain $$E_1+(m-2)=P_m(2)+P_m(0)+P_m(0)+\cdots+P_m(x_n),$$
by changing both of $P_m(x_6)=P_m(x_7)=1$ to $P_m(2)=m$, we may obtain
$$E_{2}+2(m-2)=P_m(2)+P_m(2)+P_m(0)+\cdots+P_m(x_n),$$ and by changing all of $P_m(x_6)=P_m(x_7)=P_m(x_8)=1$ to $P_m(2)=m$, we may obtain
$$E_{3}+3(m-2)=P_m(2)+P_m(2)+P_m(2)+\cdots+P_m(x_n),$$
i.e., $E_1+(m-2), E_2+2(m-2)$ and $E_3+3(m-2)$ may be represented by $f_m(\mathbf x)$.
And $E_s+1$ may be written as $f_m(\mathbf x)=a_6P_m(x_6)+\cdots +a_nP_m(x_n)$ with $x_i=0$ for all $7 \le i \le n-1$ so from the representation of $E_s+1$ by changing $P_m(x_7)=0$ to $P_m(-1)=m-3$, we may obtain $$E_s+(m-2)=P_m(x_6)+P_m(-1)+P_m(0)+P_m(0)+\cdots+P_m(x_n)$$
and by changing both of $P_m(x_7)=P_m(x_8)=0$ to $P_m(-1)=m-3$, we may obtain
$$E_{s-1}+2(m-2)=P_m(x_6)+P_m(-1)+P_m(-1)+P_m(0)+\cdots+P_m(x_n),$$
i.e., $E_s+(m-2)$ and $E_{s-1}+2(m-2)$ may be represented by $f_m(\mathbf x)$.
So from Lemma \ref{A(m-2)}, we may conclude that every positive integer except the below at most $9$ positive integers
\begin{equation}\label{1,exc}
\begin{array}{lllll}
E_1 \quad \quad & \quad \quad E_2 & \quad \quad \ \cdots & \quad \quad E_{s-1} & \quad \quad E_s \\
& E_2+(m-2) & \quad \quad \ \cdots & E_{s-1}+(m-2) & \\
& & E_3+2(m-2) & & \\
\end{array}
\end{equation} where $s \le 5$ is represented by $F_m(\mathbf x)$.
On the other hand, the representability of the above integers in (\ref{1,exc}) by $F_m(\mathbf x)$ may be directly confirmed.
Since $a_1+\cdots +a_n \ge m-4$, by Proposition \ref{node}, the integers $$E_1,E_2\cdots, E_s$$ (smaller than $m-4$) are represented by the node $F_m(\mathbf x)$ by taking $x_i \in \{0,1\}$ for all $i$ and moreover, in this situation, we may additionally assume that $P_m(x_1)=P_m(x_2)=1$.
And then from a representation of $E_1,\cdots ,E_{s-2}$ by $F_m(\mathbf x)$ with $P_m(x_1)=1$ by changing $P_m(x_1)=1$ to $P_m(2)=m$, we may see that $$E_2+(m-2),\cdots ,E_{s-1}+(m-2)$$ are represented by $F_m(\mathbf x)$ and from a representation of $E_1$ by $F_m(\mathbf x)$ with $P_m(x_1)=P_m(x_2)=1$ by changing both of $P_m(x_1)=P_m(x_1)=1$ to $P_m(2)=m$, we may see that $$E_3+2(m-2)$$ is represented by $F_m(\mathbf x)$.
This completes the proof.
\end{proof}
\vskip 0.5em
\begin{rmk}\label{A=1rmk}
Following the Guy's argument \cite{G}, the total sum of all coefficients of a leaf $\sum \limits_{i=1}^na_iP_m(x_i)$ must exceed $m-4$, i.e., $a_1+\cdots +a_n \ge m-4$ because otherwise, the integers in $[1,m-4]$ could not all be represented by the (universal) leaf.
So the coefficient condition (\ref{A=1,cond}) in Lemma \ref{A=1} on the total sum of all coefficients
$$a_1+\cdots +a_n \ge m-4$$is essential for any leaf $\sum _{i=1}^n a_iP_m(x_i)$.
By Lemma \ref{A=1}, we may easily induce that the $\rank$ of a leaf $\sum _{i=1}^na_iP_m(\mathbf x)$ with (\ref{A=1,cond'}) does not exceed $m-4$ since $a_1+\cdots +a_n \ge n$ and (\ref{A=1,cond}) holds for $n=m-4$ only if $a_1= \cdots = a_{m-5}=1$.
Since the truant of the node $$P_m(x_1)+\cdots +P_m(x_{m-5})$$ is $m-4$, it would have its childeren
\begin{equation}\label{A=1,leaf}P_m(x_1)+\cdots +P_m(x_{m-5})+a_{m-4}P_m(x_{m-4})\end{equation} where $1\le a_{m-4}\le m-4$ and that's all.
Conesequently, we may yield that a leaf $$a_1P_m(x_1)+\cdots +a_nP_m(x_n)$$ with (\ref{A=1,cond'}) has $n \le m-4$ and there are excatly $m-4$ leaves with (\ref{A=1,cond'}) of the $\rank n= m-4$ of (\ref{A=1,leaf}).
Until now, we showed that the $\rank$ of a leaf $\sum \limits_{i=1}^na_iP_m(x_i)$ otherwise
\begin{equation}\label{3cond} a_3=a_4=\cdots =a_{(C-4)l_m+5}=3 \end{equation}
does not exceed $m-4$.
In Lemma \ref{A=3}, we treat the leaves $\sum \limits_{i=1}^na_iP_m(x_i)$ with (\ref{3cond}).
\end{rmk}
\vskip 0.5em
\begin{lem} \label{A=3}
Let $F_m(\mathbf x)=\sum \limits_{i=1}^na_iP_m(x_i)$ be a node of the escalator tree with
$a_3=a_4=\cdots =a_{(C-4)l_m+5}=3$.
\begin{itemize}
\item[(1) ] If $3(m-3)-1 \le a_1+a_2+\cdots+a_n$ with $a_i \equiv 0 \pmod{3} \text{ for all $3\le i \le n$}$, then the node would be universal, i.e., the node would become a leaf of the tree.
\item[(2) ] If $a_{n} \nequiv 0 \pmod{3}$, then the node would be universal, i.e., the node would become a leaf of the tree.
\end{itemize}
\end{lem}
\begin{proof}
(1) For $$b_i:=\begin{cases}a_i & \text{ for }i=1,2\\ a_{i+5} & \text{ for } i \ge 3,\end{cases}$$ similarly with the proof of Lemma \ref{A=1}, we may get that the $m$-gonal form $f_m(\mathbf x):=\sum_{i=1}^{n-5}b_iP_m(x_i)$ may represent every positive integer up to $3(m-2)$ except at most 5 positive integers in $[3(C-2)l_m+3,3(m-3)-1]$ and these integers would have the same residue modulo $3$ and the congruent integers would be consecutive.
On the specific, the situation that there is an integer in $[3(C-2)l_m+3,3(m-3)-1]$ not represented by $f_m(\mathbf x)$ happens only when
\begin{equation}\label{A=3case}\begin{cases}a_1+\cdots+a_n-15=b_1+\cdots+b_{n-5}<3(m-3)-1 & \text{ or } \\ a_n>a_1+\cdots +a_{n-1}+1-15. & \end{cases}\end{equation}
If $f_m(\mathbf x)$ represents every positive integer up to $3(m-2)$, then we may conclude that $F_m(\mathbf x)$ is universal from Lemma \ref{A(m-2)}.
Now assume that there is an integer in $[3(C-2)l_m+3,3(m-3)-1]$ not represented by $f_m(\mathbf x)$ and let $$E_1 < E_2(=E_1+3) < \cdots < E_s(=E_1+3(s-1))$$ where $1\le s\le 5$ be all of the positive integers not represented by $f_m(\mathbf x)$ in $[1,3(m-3)-1]$.
Through similar arguments with the proof of Lemma \ref{A=1}, we may obtain that $F_m(\mathbf x)$ represents every positive integer except the below at most $9$ integers
\begin{equation}\label{3,exc}
\begin{array}{lllll}
E_1 \quad \quad & \quad \quad E_2 & \quad \quad \ \cdots & \quad \quad E_{s-1} & \quad \quad E_s \\
& E_2+(m-2) & \quad \quad \ \cdots & E_{s-1}+(m-2) & \\
& & E_3+2(m-2) & & \\
\end{array}
\end{equation} by using the fact that $a_3P_m(x_3)+\cdots +a_7P_m(x_7)$ represents all the multiples of $3(m-2)$ from Lemma \ref{A(m-2)}.
On the other hand, the representability of the above exception integers in (\ref{3,exc}) by $F_m(\mathbf x)$ may also be directly confirmed similarly with the argument of the proof of Lemma \ref{A=1}.
(2) In virtue of (1), we may have that $a_1+\cdots +a_{n-1}<3(m-3)-1$.
And then we may see that the truant of $\sum_{i=1}^{n-1}a_iP_m(x_i)$ would be one of
$$a_1+\cdots +a_{n-1}+1, \ a_1+\cdots +a_{n-1}+2, \text{ or }a_1+\cdots +a_{n-1}+3,$$
which implies that $$a_n\le a_1+\cdots +a_{n-1}+3<3(m-3)+2.$$
For $$b_i:=\begin{cases}a_i & \text{ for }i=1,2\\ a_{i+5} & \text{ for } i \ge 3,\end{cases}$$ similarly with the proof of Lemma \ref{A=1}, we may obtain that $f_m(\mathbf x):=\sum_{i=1}^{n-5}b_iP_m(x_i)$ represents every positive integer up to $3(m-2)$ except at most 5 positive integers in $[3(C-2)l_m+3,3(m-3)-1]$ and these integers would have the same residue modulo $3$ and the congruent integers would be consecutive.
On the specific, the situation that there is an integer in $[3(C-2)l_m+3,3(m-3)-1]$ not represented by $f_m(\mathbf x)$ happen only when
\begin{equation}\label{A=3case'}a_n>a_1+\cdots +a_{n-1}+1-15.\end{equation}
If $f_m(\mathbf x)$ represents every positive integer up to $3(m-2)$, then we may conclude that the $F_m(\mathbf x)$ is universal from Lemma \ref{A(m-2)}.
Now assume that there is an integer in $[3(C-2)l_m+3,3(m-3)-1]$ not represented by $f_m(\mathbf x)$ and let
$$E_1 < E_2(=E_1+3) < \cdots < E_s(=E_1+3(s-1))$$
where $1\le s \le 5$ be all of the positive integers not represented by $f_m(\mathbf x)$ in $[3(C-2)l_m+3,3(m-3)-1]$.
Through similar arguments with the proof of Lemma \ref{A=1}, we may obtain that $F_m(\mathbf x)$ represents every positive integer except the below at most $9$ integers
\begin{equation}\label{3,exc'}
\begin{array}{lllll}
E_1 \quad \quad & \quad \quad E_2 & \quad \quad \ \cdots & \quad \quad E_{s-1} & \quad \quad E_s \\
& E_2+(m-2) & \quad \quad \ \cdots & E_{s-1}+(m-2) & \\
& & E_3+2(m-2) & & \\
\end{array}
\end{equation} in virtue of the fact that $a_3P_m(x_3)+\cdots +a_7P_m(x_7)$ represents all the multiples of $3(m-2)$ from Lemma \ref{A(m-2)}.
On the other hand, the representability of the above exception integers in (\ref{3,exc'}) by $F_m(\mathbf x)$ may also be directly confirmed similarly with the argument of the proof of Lemma \ref{A=1}.
\end{proof}
\begin{proof}[Proof of Theorem \ref{main}]
From Lemma \ref{A=3}, we may yield that
the $\rank$ of a leaf $\sum \limits_{i=1}^na_iP_m(x_i)$ with
\begin{equation} \label{3cond'}a_3=a_4=\cdots =a_{(C-4)l_m+5}=3\end{equation}
would not exceed $m-2$ because $a_1+a_2+a_3+\cdots+a_n\ge 2+3(n-2)$ under (\ref{3cond'}).
Overall, with Remark \ref{A=1rmk}, we may get an upper bound $m-2$ for the $\rank$ of leaf of the $m$-gonal form's escalating tree.
On the other hand, we may have that the below nodes
\begin{equation}\label{A=3,last}
\begin{cases}
P_m(x_1)+P_m(x_2)+\sum \limits_{i=3}^{m-3}3P_m(x_i) & \text{ when } m\equiv 0 \pmod{3}\\
P_m(x_1)+P_m(x_2)+\sum \limits_{i=3}^{m-3}3P_m(x_i) & \text{ when } m\equiv 1 \pmod{3}\\
P_m(x_1)+2P_m(x_2)+\sum \limits_{i=3}^{m-4}3P_m(x_i) & \text{ when } m\equiv 2 \pmod{3}
\end{cases}
\end{equation}
are not a leaf (i.e., not a universal form) with the truant $3m-10,3m-12$, and $3m-12,$ respectively and we may see that from Lemma \ref{A=3}, that's all of the nodes which are not leaves of the $\rank$ greater than or equal to $m-3, m-3,$ and $m-4$, respectively because
\begin{equation}\label{A=3,last'}
\begin{cases}P_m(x_1)+2P_m(x_2)+\sum \limits _{i=3}^{\frac{2m-3}{3}}3P_m(x_i) & \text{ when } m\equiv 0 \pmod{3} \\
P_m(x_1)+2P_m(x_2)+\sum \limits _{i=3}^{\frac{2m-5}{3}}3P_m(x_i) & \text{ when } m\equiv 1 \pmod{3} \\
P_m(x_1)+P_m(x_2)+\sum\limits _{i=3}^{\frac{2m-4}{3}}3P_m(x_i) & \text{ when } m\equiv 2 \pmod{3}
\end{cases}
\end{equation}
are universal.
This completes the proof of Theorem and also bring out Remark \ref{rmk R}.
\end{proof}
\vskip 0.5em
\begin{rmk}
In this paper, we proved for $m>2\left(\left(2C+\frac{1}{4}\right)^{\frac{1}{4}}+\sqrt{2}\right)^2$,
$$r_m=\begin{cases}\ceil{\log_2(m-3)}+1& \text{ when } -3 \le 2^{\ceil{\log_2(m-3)}}-m\le 1\\
\ceil{\log_2(m-3)} & \text{ when } \ \quad 2 \le 2^{\ceil{\log_2(m-3)}}-m
\end{cases}
$$
and for $m>6C^2(C+1)$,
$$R_m=\begin{cases}m-2 & \text{ when } m\nequiv 2 \pmod{3}\\m-3 & \text{ when } m\equiv 2 \pmod{3}.\end{cases}$$
Especially, we showed that $$F_m^{(0)}(\mathbf x):=P_m(x_1)+2P_m(x_2)+\cdots +2^{r_m-1}P_m(x_{r_m})$$ is a leaf of the minimal $\rank r_m$, i.e., an universal $m$-gonal form of the minimal $\rank r_m$.
Now for $m>$ with $2 \le 2^{\ceil{\log_2(m-3)}}-m$, consider a sequence $\{F_m^{(j)}(\mathbf x)\}_{j=0}^{m-4-r_m}$ of leaves (proper universal $m$-gonal forms) which is inductively constructed from $F_m^{(0)}(\mathbf x)$ by splitting the first component $a_iP_m(x_i)$ with non-one coefficient $a_i(>1)$ into two components $P_m(\cdot)$ and $(a_i-1)P_m(\cdot)$, namely,
from the $(j+1)$-th leaf of $\{F_m^{(j)}(\mathbf x)\}_{j=0}^{m-4-r_m}$ with $a_i > 1$
$$F_m^{(j)}(\mathbf x):=P_m(x_1)+\cdots +P_m(x_{i-1})+a_iP_m(x_i)+\cdots +a_{n}P_m(x_{n})$$
the next $(j+2)$-th leaf is constructed as
$$F_m^{(j+1)}(\mathbf x):=P_m(x_1)+\cdots +P_m(x_{i-1})+P_m(x_i)+(a_i-1)P_m(x_{i+1})+\cdots +a_{n}P_m(x_{n+1})$$
by rearranging variables.
By using the notation $\mathcal C_j:=[a_1,\cdots, a_n]$ for $F_m^{(j)}(\mathbf x)=\sum \limits_{i=1}^na_iP_m(x_i)$, we may describe the stream of coefficients of the sequence $\{F_m^{(j)}(\mathbf x)\}_{j=0}^{m-4-r_m}$ as follows
$$\begin{array}{lllllllll}
\mathcal C_0=&[1, & \ 2, & \ \quad 4, & \ \quad 8, & 16, & \cdots, & 2^{r_m-2},& 2^{r_m-1}] \\
\mathcal C_1=&[1, &1,1, & \ \quad 4, & \ \quad 8, & 16, & \cdots , & 2^{r_m-2},& 2^{r_m-1}]\\
\mathcal C_2=&[1, &1,1, & \ \ \ 1,3, & \ \quad 8, & 16, & \cdots , & 2^{r_m-2},& 2^{r_m-1}]\\
\mathcal C_3=&[1, &1,1, & \ 1,1,2, & \ \quad 8, & 16, & \cdots , & 2^{r_m-2},& 2^{r_m-1}]\\
\mathcal C_4=&[1, &1,1, & 1,1,1,1, & \ \quad8, & 16, & \cdots , & 2^{r_m-2},& 2^{r_m-1}]\\
\mathcal C_5=&[1, &1,1, & 1,1,1,1, & \ \ \ 1,7, & 16, & \cdots , & 2^{r_m-2},& 2^{r_m-1}]\\
\mathcal C_6=&[1, &1,1, & 1,1,1,1, & \ 1,1,6, & 16, & \cdots , & 2^{r_m-2},& 2^{r_m-1}]\\
\mathcal C_7=&[1, &1,1, & 1,1,1,1, & 1,1,1,5, & 16, & \cdots , & 2^{r_m-2},& 2^{r_m-1}]\\
\quad \vdots & & & & \ \ \quad \vdots & & & & \ \quad \quad \quad .\\
\end{array}$$
Then we may see $\rank F_m^{(j)}(\mathbf x)=r_m+j$ ultimately there appear leaves of $\rank$ from $r_m$ and to $m-4$ in the sequence $\{F_m^{(j)}(\mathbf x)\}_{j=0}^{m-4-r_m}$.
On the other hand for $m>$ with $-3 \le 2^{\ceil{\log_2(m-3)}}-m\le 1$, even though every $m$-gonal form of the inductively constructed sequence $\{ F_m^{(j)}(\mathbf x)\}_{j=0}^{m-4+r_m}$ as the same manner with the above is universal, but not all of them are proper universal, i.e., for some $F_m^{(j)}(\mathbf x)=\sum \limits_{i=1}^na_iP_m(x_i)$, its proper subform $$F_m^{(j)}(\mathbf x)-2^{r_m-1}P_m(x_n)$$ would also be universal and the subform would be indeed a leaf(i.e., proper univeral $m$-gonal form).
For the smallest $J$ for which $F_m^{(J)}(\mathbf x)$ is not a leaf, by replacing the $j$-th $m$-gonal form $F_m^{(j)}(\mathbf x)=\sum \limits_{i=1}^na_iP_m(x_i)$ for all where $j \ge J$ of the above sequence $\{F_m^{(j)}(\mathbf x)\}_{j=0}^{m-4-r_m}$ as its subform $$F_m^{(j)}(\mathbf x)-2^{r_m-1}P_m(x_n)$$
which is a leaf, we may get a sequence of leaf $\{F_m^{(j)}(\mathbf x)\}_{j=0}^{m-4-r_m}$.
And the $$\rank F_m^{(j)}(\mathbf x)=\begin{cases}r_m+j & \text{when } j<J \\ r_m+j-1 & \text{when } j \ge J. \end{cases}$$
So in this case, there appear leaves of $\rank$ from $r_m$ and to $m-5$ in the sequence $\{F_m^{(j)}(\mathbf x)\}_{j=0}^{m-4-r_m}$.
And $P_m(x_1)+\cdots +P_m(x_{m-4})$ is a leaf of the $\rank m-4$.
When $m \nequiv 2 \pmod{3}$, we may show that $$P_m(x_1)+P_m(x_2)+3P_m(x_3)+\cdots +3P_m(x_{r_m-2})+6P_m(x_{r_m-1})$$ is a leaf of $\rank r_m-1=m-3$ by using Lemma \ref{A=3}.
From the above arguments, we may see that there is a leaf of $\rank n$ for each $r_m \le n \le R_m$.
\end{rmk}
|
1,108,101,564,594 | arxiv | \section{Introduction}
\IEEEPARstart{B}{iometrics} offers a secure and convenient way for access control. Face biometrics is one of the most convenient
modalities for biometric authentication due to its non-intrusive nature. Even though face recognition systems are reaching human performance in identifying persons in many challenging datasets \cite{learned2016labeled}, most face recognition systems are still vulnerable to presentation attacks (PA), also known as spoofing \footnote{The term spoofing should be deprecated in favour of presentation attacks to comply with the ISO standards.} attacks \cite{marcel2014handbook}, \cite{ISO1}. Merely presenting a printed photo to an unprotected face recognition system could be enough to fool it \cite{anjos2011counter}. Vulnerability to presentation attacks limits the reliable deployment of such systems for applications in unsupervised conditions.
As per the ISO standard \cite{ISO1}, presentation attack is defined as ``a presentation to the biometric data capture subsystem with the goal of interfering with the operation of the biometric system''. Presentation attacks include both `impersonation' as well as `obfuscation' of identity. The `impersonation' refers to attacks in which the attacker wants to be recognized as a different person, whereas in `obfuscation' attacks, the objective is to hide the identity of the attacker. The biometric characteristic or object used in a presentation attack is known as presentation attack instrument (PAI).
\begin{figure}[t]
\centering
\includegraphics[width=0.95\linewidth]{pics/All_attacks.png}
\caption{Figure showing \textit{bonafide}, print and replay attacks from different PAD databases, Replay-Attack \cite{chingovska2012effectiveness} (first row), Replay-Mobile \cite{costa2016replay} (second row), and MSU-MFSD \cite{wen2015face} (third row).}\label{fig:example_attacks}
\end{figure}
Different kinds of PAIs can be used to attack face recognition systems.
The presentation of a printed photo, or replaying a video of a subject, are common examples of 2D PAI which have been extensively explored in the available literature. Examples of \textit{bonafide} and 2D PAs
from publicly available databases are shown in Fig. \ref{fig:example_attacks}. More sophisticated attacks could involve manufacturing custom 3D masks which correspond to a target identity for impersonation or to evade identification. For reliable usage of face recognition technology, it is necessary to develop presentation attack detection (PAD) systems to detect such PAs automatically.
The majority of available research deals with the detection of print and replay attacks using visible spectral data. Most of the methods relies on the limitations of PAIs and quality degradation of the recaptured sample. Features such as color, texture \cite{boulkenafet2015face}, \cite{maatta2011face}, motion \cite{anjos2011counter}, and physiological cues \cite{ramachandra2017presentation}, \cite{heusch2019remote} are often leveraged for PAD in images from visible spectrum.
While it is useful to have visible spectrum image based PAD algorithms for legacy face recognition systems, we argue that using only visual spectral information may not be enough for the detection of sophisticated attacks and generalization to new unseen PAIs. The quality of presentation attack instruments (PAI) evolves together with advances in cameras, display devices, and manufacturing methods. Tricking a multi-channel system is harder than a visual spectral one. An attacker would have to mimic real facial features across different representations. The PAD approaches which work in existing PAD databases may not work in real-world conditions when encountered with realistic attacks. Complementary information from multiple channels could improve the accuracy of PAD systems.
The objective of this work is to develop a PAD framework which can detect a variety of 2D and 3D attacks in obfuscation or impersonation settings. To this end, we propose the new Multi-Channel Convolutional Neural Network (MC-CNN) architecture, efficiently combining multi-channel information for robust detection of presentation attacks. The proposed network uses a pre-trained LightCNN model as the base network, which obviates the requirement to train the framework from scratch. In the proposed MC-CNN only low-level LightCNN features across multiple channels are re-trained, while high-level layers of pre-trained LightCNN remain unchanged.
Databases containing a wide variety of challenging PAIs are essential for developing and benchmarking PAD algorithms. In this context, we introduce a Wide Multi-Channel presentation Attack (WMCA) dataset, which contains a broad variety of 2D and 3D attacks. The data split and evaluation protocols are predefined and publicly available. The algorithms, baselines, and the results are reproducible. The software and models to reproduce the results are available publicly \footnote{Source code available at: \url{https://gitlab.idiap.ch/bob/bob.paper.mccnn.tifs2018}}.
The main contributions from this paper are listed below.
\begin{itemize}
\item We propose a novel framework for face presentation attack detection based on multi-channel CNN (MC-CNN). MC-CNN uses a face recognition sub-network, namely LightCNN, making the framework reusable for both PAD and face recognition. The source codes for the network and instructions to train the model are made publicly available allowing to reproduce the findings. We benchmark the proposed method against selected \textbf{reproducible} baseline available in recent publications on the topic~\cite{nikisins2018effectiveness}, as well as reimplementations of recent literature in multi-channel PAD \cite{lucena2017transfer}. We demonstrate that the multi-channel approach is beneficial in both proposed and baseline systems.
\item The new WMCA database is introduced: the subjects in the database are captured using multiple capturing devices/channels, and the MC data is spatially and temporally aligned. The channels present are color, depth, thermal and infrared. The database contains a wide variety of 2D and 3D presentation attacks, specifically, 2D print and replay attacks, mannequins, paper masks, silicone masks, rigid masks, transparent masks, and non-medical eyeglasses.
\end{itemize}
The rest of the paper is organized as follows. Section 2 revisits available literature related to face presentation attack detection. Section 3 presents the proposed approach. The details about the sensors and the dataset are described in section 4. Experimental procedure followed, and the baseline systems are described in Section 5. Extensive testing and evaluations of the proposed approach, along with comparisons with the baselines, discussions, and limitations of the proposed approach are presented in Section 6. Conclusions and future directions are described in Section 7.
\section{Related work}
Most of the work related to face presentation attack detection addresses detection of 2D attacks, specifically print and 2D replay attacks. A brief review of recent PAD methods is given in this section.
\begin{table*}[t]
\caption{Recent multi-channel face PAD datasets}
\label{tab:multichannel_datasets}
\begin{tabular}{l|p{1.cm}|p{2cm}|p{3cm}|p{3cm}|p{2cm}}
\toprule
Database & Year & Samples & Attacks & Channels & \multicolumn{1}{c}{\begin{tabular}[c]{@{}c@{}}Synchronous\\ Capture\end{tabular}} \\ \midrule
3DMAD \cite{3dmad} & 2013 & 17 subjects & 3D: Mask attacks & Color and depth &\cmark \\ \hline
I\textsuperscript{2}BVSD \cite{dhamecha2014recognizing} & 2013 & 75 subjects & 3D: Facial disguises & Color and thermal &---\\ \hline
GUC-LiFFAD \cite{raghavendra2015presentation} & 2015 & 80 subjects & 2D: Print and replay & Light-field imagery &\cmark \\ \hline
MS-Spoof \cite{chingovska2016face} &2016 & 21 subjects & 2D: Print & Color and NIR (800nm) &\cmark \\ \hline
BRSU \cite{steiner2016design} &2016 & 50+ subjects & 3D: Masks, facial disguise & Color \& 4 SWIR bands &\cmark \\ \hline
EMSPAD \cite{raghavendra2017vulnerability} &2017 & 50 subjects & 2D: Print(laser \&Inkjet) & 7-band multi-spectral data & \cmark \\ \hline
MLFP \cite{agarwal2017face} &2017 & 10 subjects & 3D: Obfuscation with latex masks & Visible, NIR and thermal bands & \xmark \\ \bottomrule
\end{tabular}
\end{table*}
\subsection{Feature based approaches for face PAD}
For PAD using visible spectrum images, several methods such as detecting motion patterns \cite{anjos2011counter}, color texture, and histogram based methods in different color spaces, and variants of Local Binary Patterns (LBP) in grayscale \cite{boulkenafet2015face} and color images \cite{chingovska2012effectiveness}, \cite{maatta2011face}, have shown good performance. Image quality based features \cite{galbally2014image} is one of the successful feature based methods available in prevailing literature. Methods identifying moir{\'e} patterns \cite{patel2015live}, and image distortion analysis \cite{wen2015face}, use the alteration of the images due to the replay artifacts. Most of these methods treat PAD as a binary classification problem which may not generalize well for unseen attacks~\cite{nikisins2018effectiveness}.
Chingovska \textit{et al}. \cite{chingovska2015use} studied the amount of client-specific information present in features used for PAD. They used this information to build client-specific PAD methods. Their method showed a 50\% relative improvement and better performance in unseen attack scenarios.
Arashloo \textit{et al}. \cite{arashloo2017anomaly} proposed a new evaluation scheme for unseen attacks. Authors have tested several combinations of binary classifiers and one class classifiers. The performance of one class classifiers was better than binary classifiers in the unseen attack scenario. BSIF-TOP was found successful in both one class and two class scenarios. However, in cross-dataset evaluations, image quality features were more useful. Nikisins \textit{et al}. \cite{nikisins2018effectiveness} proposed a similar one class classification framework using one class Gaussian Mixture Models (GMM). In the feature extraction stage, they used a combination of Image Quality Measures (IQM). The experimental part involved an aggregated database consisting of replay attack \cite{chingovska2012effectiveness}, replay mobile \cite{costa2016replay}, and MSU-MFSD \cite{wen2015face} datasets.
Heusch and Marcel \cite{Heuch2018}, \cite{heusch2019remote} recently proposed a method for using features derived from remote photoplethysmography (rPPG).
They used the long term spectral statistics (LTSS) of pulse signals obtained from available methods for rPPG extraction. The LTSS features were combined with SVM for PA detection.
Their approach obtained better performance than state of the art methods using rPPG in four publicly available databases.
\subsection{CNN based approaches for face PAD}
Recently, several authors have reported good performance in PAD using convolutional neural networks (CNN). Gan \textit{et al}. \cite{gan20173d} proposed a 3D CNN based approach, which utilized the spatial and temporal features of the video. The proposed approach achieved good results in the case of 2D attacks, prints, and videos. Yang \textit{et al}. \cite{yang2014learn} proposed a deep CNN architecture for PAD. A preprocessing stage including face detection and face landmark detection is used before feeding the images to the CNN. Once the CNN is trained, the feature representation obtained from CNN is used to train an SVM classifier and used for final PAD task. Boulkenafet \textit{et al}. \cite{boulkenafet2017competition} summarized the performance of the competition on mobile face PAD. The objective was to evaluate the performance of the algorithms under real-world conditions such as unseen sensors, different illumination, and presentation attack instruments. In most of the cases, texture features extracted from color channels performed the best. Li \textit{et al}. \cite{li2018learning} proposed a 3D CNN architecture, which utilizes both spatial and temporal nature of videos. The network was first trained after data augmentation with a cross-entropy loss, and then with a specially designed generalization loss, which acts as a regularization factor. The Maximum Mean Discrepancy (MMD) distance among different domains is minimized to improve the generalization property.
There are several works involving various auxiliary information in the CNN training process, mostly focusing on the detection of 2D attacks. Authors use either 2D or 3D CNN. The main problem of CNN based approaches mentioned above is the lack of training data, which is usually required to train a network from scratch. One commonly used solution is fine-tuning, rather than a complete training, of the networks trained for face-recognition, or image classification tasks. Another issue is the poor generalization in cross-database and unseen attacks tests. To circumvent these issues, some researchers have proposed methods to train CNN using auxiliary tasks, which is shown to improve generalization properties. These approaches are discussed below.
Liu \textit{et al}. \cite{liu2018learning} presented a novel method for PAD with auxiliary supervision. Instead of training a network end-to-end directly for PAD task, they used CNN-RNN model to estimate the depth with pixel-wise supervision and estimate remote photoplethysmography (rPPG) with sequence-wise supervision. The estimated rPPG and depth were used for PAD task. The addition of the auxiliary task improved the generalization capability.
Atoum \textit{et al}. \cite{atoum2017face} proposed a two-stream CNN for 2D presentation attack detection by combining a patch-based model and holistic depth maps.
For the patch-based model, an end-to-end CNN was trained. In the depth estimation, a fully convolutional network was trained using the entire face image. The generated depth map was converted to a feature vector by finding the mean values in the $N \times N$ grid. The final PAD score was obtained by fusing the scores from the patch and depth CNNs.
Shao \textit{et al}. \cite{shao2017deep} proposed a deep convolutional network-based architecture for 3D mask PAD. They tried to capture the subtle differences in facial dynamics using the CNN. Feature maps obtained from the convolutional layer of a pre-trained VGG \cite{Simonyan15} network was used to extract features in each channel. Optical flow was estimated using the motion constraint equation in each channel. Further, the dynamic texture was learned using the data from different channels. The proposed approach achieved an AUC (Area Under Curve) score of 99.99\% in the 3DMAD dataset.
Lucena \textit{et al}. \cite{lucena2017transfer} presented an approach for face PAD using transfer learning from pre-trained models (FASNet). The VGG16 \cite{Simonyan15} architecture which was pre-trained on ImageNet \cite{ILSVRC15} dataset was used as the base network as an extractor, and they modified the final fully connected layers. The newly added fully connected layers in the network were fine-tuned for PAD task. They obtained HTERs of 0\% and 1.20\% in 3DMAD and Replay-Attack dataset respectively.
\subsection{Multi-channel based approaches and datasets for face PAD}
In general, most of the visible spectrum based PAD methods try to detect the subtle differences in image quality when it is recaptured. However, this method could fail as the quality of capturing devices and printers improves. For 3D attacks, the problem is even more severe. As the technology to make detailed masks is available, it becomes very hard to distinguish between \textit{bonafide} and presentation attacks by just using visible spectrum imaging. Several researchers have suggested using multi-spectral and extended range imaging to solve this issue \cite{raghavendra2017extended}, \cite{steiner2016reliable}.
Akhtar \textit{et al}. \cite{akhtar2015biometric} outlines the major challenges and open issues in biometrics concerning presentation attacks. Specifically, in case of face PAD, they discuss a wide variety of possible attacks and possible solutions. They pointed out that sensor-based solutions which are robust against spoofing attempts and which works even in `in the wild' conditions require specific attention.
Hadid \textit{et al}. \cite{hadid2015biometrics} presented the results from a large scale study on the effect of spoofing on different biometrics traits. They have shown that most of the biometrics systems are vulnerable to spoofing. One class classifiers were suggested as a possible way to deal with unseen attacks. Interestingly, countermeasures combining both hardware (new sensors) and software were recommended as a robust PAD method which could work against a wide variety of attacks.
Raghavendra \textit{et al}. \cite{raghavendra2017extended} presented an approach using multiple spectral bands for face PAD. The main idea is to use complementary information from different bands. To combine multiple bands, they observed a wavelet-based feature level fusion and a score fusion methodology. They experimented with detecting print attacks prepared using different kinds of printers. They obtained better performance with score level fusion as compared to the feature fusion strategy.
Erdogmus and Marcel \cite{erdogmus2014spoofing} evaluated the performance of a number of face PAD approaches against 3D masks using 3DMAD dataset. This work demonstrated that 3D masks could fool PAD systems easily. They achieved HTER of 0.95\% and 1.27\% using simple LBP features extracted from color and depth images captured with Kinect.
Steiner \textit{et al}. \cite{steiner2016reliable} presented an approach using multi-spectral SWIR imaging for face PAD. They considered four wavelengths - 935\textit{nm}, 1060\textit{nm}, 1300\textit{nm} and 1550\textit{nm}. In their approach, they trained an SVM for classifying each pixel as a skin pixel or not. They defined a Region Of Interest (ROI) where the skin is likely to be present, and skin classification results in the ROI is used for classifying PAs. The approach obtained 99.28 \% accuracy in per-pixel skin classification.
Dhamecha \textit{et al}. \cite{dhamecha2013disguise} proposed an approach for PAD by combining the visible and thermal image patches for spoofing detection.
They classified each patch as either \textit{bonafide} or attack and used the \textit{bonafide} patches for subsequent face recognition pipeline.
Agarwal \textit{et al}. \cite{agarwal2017face} proposed a framework for the detection of latex mask attacks from multi-channel data, which comprised of visible, thermal and infrared data. The dataset was collected independently for different channels and hence lacks temporal and spatial alignment between the channels. They have performed experiments using handcrafted features independently in the multiple channels. For PAD, the best performing system was based on redundant discrete wavelet transform (RDWT) and Haralick \cite{haralick1979statistical} features. They computed the features from RDWT decompositions of each patch in a $4 \times 4$ grid and concatenated them to obtain the final feature vector. The computed feature vectors were used with SVM for the PAD task. From the experiments, it was shown that the thermal channel was more informative as compared to other channels obtaining 15.4\% EER in the frame-based evaluation. However, experiments with fusion could not be performed since the channels were recorded independently.
In \cite{Bhattacharjee:256262} Bhattacharjee \textit{et al}. showed that it is possible to spoof commercial face recognition systems with
custom silicone masks. They also proposed to use the mean temperature of the face region for PAD.
Bhattacharjee \textit{et al}. \cite{bhattacharjee2017you} presented a preliminary study of using multi-channel information for PAD. In addition to visible spectrum images, they considered thermal, near infrared, and depth channels. They showed that detecting 3D masks and 2D attacks are simple in thermal and depth channels respectively. Most of the attacks can be detected with a similar approach with combinations of different channels, where the features and combinations of channels to use are found using a learning-based approach.
Several multi-channel datasets have been introduced in the past few years for face PAD.
Some of the recent ones are shown in Table \ref{tab:multichannel_datasets}. From the
table it can be seen that the variety of PAIs is limited in most of the available datasets.
\subsection{Discussions}
In general, presentation attack detection in a real-world scenario is challenging. Most of the PAD methods available in prevailing literature try to solve the problem for a limited number of presentation attack instruments. Though some success has been achieved in addressing 2D presentation attacks, the performance of the algorithms in realistic 3D masks and other kinds of attacks is poor.
As the quality of attack instruments evolves, it becomes increasingly difficult to discriminate between \textit{bonafide} and PAs in the visible spectrum alone. In addition, more sophisticated attacks, like 3D silicone masks, make PAD in visual spectra challenging. These issues motivate the use of multiple channels, making PAD systems harder to by-pass.
We argue that the accuracy of the PAD methods can get better with a multi-channel acquisition system. Multi-channel acquisition from consumer-grade devices can improve the performance significantly. Hybrid methods, combining both extended hardware and software could help in achieving good PAD performance in real-world scenarios. We extend the idea of a hybrid PAD framework and develop a multi-channel framework for presentation attack detection.
\section{Proposed method}
A Multi-Channel Convolutional Neural Network (MC-CNN) based approach is proposed for PAD. The main idea is to use the joint representation from multiple channels for PAD, using transfer learning from a pre-trained face recognition network. Different stages of the framework are described in the following.
\subsection{Preprocessing}
\label{subsec:preprocess}
Face detection is performed in the color channel using the MTCNN algorithm \cite{zhang2016joint}. Once the face bounding box is obtained, face landmark detection is performed in the detected face bounding box using Supervised Descent Method (SDM) \cite{xiong2013supervised}. Alignment is accomplished by transforming image, such that the eye centers and mouth center are aligned to predefined coordinates. The aligned face images are converted to grayscale, and resized, to the resolution of $128 \times 128$ pixels.
The preprocessing stage for non-RGB channels requires the images from different channels to be aligned both spatially and temporally with the color channel. For these channels, the facial landmarks detected in the color channel are reused, and a similar alignment procedure is performed. A normalization using Mean Absolute Deviation (MAD) \cite{leys2013detecting} is performed to convert the range of non-RGB facial images to 8-bit format.
\begin{figure*}[t]
\centering
\includegraphics[width=0.99\textwidth]{pics/Diagg10.jpg}
\caption{Block diagram of the proposed approach. The gray color blocks in the CNN part represent layers which are not retrained, and other colored blocks represent re-trained/adapted layers. }
\label{fig:mcnn_general_block}
\end{figure*}
\subsection{Network architecture}
Many of previous work in face presentation attack detection utilize transfer learning from pre-trained networks. This is required since the data available for PAD task is often of a very limited size, being insufficient to train a deep architecture from scratch. This problem becomes more aggravated when multiple channels of data are involved. We propose a simpler way to leverage a pretrained face recognition model for multi-channel PAD task, adapting a minimal number of parameters.
The features learned in the low level of CNN networks are usually similar to Gabor filter masks, edges and blobs \cite{yosinski2014transferable}.
Deep CNNs compute more discriminant features as the depth increases \cite{mallat2016understanding}. It has been observed in different studies~\cite{yosinski2014transferable}, \cite{li2015lcnn}, that features, which are closer to the input are more general, while features in the higher levels contain task specific information. Hence, most of the literature in the transfer learning attempts to adapt the higher level features for the new tasks.
Recently, Freitas Pereira \textit{et al}. \cite{freitas2018heterogeneous} showed that the high-level features in deep convolutional
neural networks, trained in visual spectra, are domain independent, and they can be used to encode face images collected from different
image sensing domains. Their idea was to use the shared high-level features for heterogeneous face recognition task, retraining
only the lower layers. In their method they split the parameters of the CNN architecture into two, the higher level features are shared among the different channels, and the lower level features (known as Domain Specific Units (DSU)) are adapted separately for different modalities. The objective was to learn the same face encoding for different channels, by adapting the DSUs only. The network was trained using contrastive loss (with Siamese architecture) or triplet loss. Re-training of only low-level features has the advantage of modifying a minimal set of parameters.
We extend the idea of domain-specific units (DSU) for multi-channel PAD task. Instead of forcing the representation from different channels to be the same, we leverage the complementary information from a joint representation obtained from multiple channels. We hypothesize that the joint representation contains discriminatory information for PAD task. By concatenating the representation from different channels, and using fully connected layers, a decision boundary for the appearance of \textit{bonafide} and attack presentations can be learned via back-propagation. The lower layer features, as well as the higher level fully connected layers, are adapted in the training phase.
The main idea used from \cite{freitas2018heterogeneous} is the adaptation of lower layers of CNN, instead of adapting the whole network when limited amount of target data is available. The network in \cite{freitas2018heterogeneous} only has one forward path, whereas in MC-CNN the network architecture itself is extended to accommodate multi-channel data. The main advantage of the proposed framework is the adaptation of a minimal amount of network weights when the training data is limited, which is usually the case with available PAD datasets. The proposed framework introduces a new way to deal with multi-channel PAD problem, reusing a large amount of face recognition data available when a limited amount of data is available for training PAD systems.
In this work, we utilize a LightCNN model \cite{wu2018light}, which was pre-trained on a large number of face images for face recognition. The LightCNN network is especially interesting as the number of parameters is much smaller than in other networks used for face recognition. LightCNN achieves a reduced set of parameters using a Max-Feature Map (MFM) operation as an alternative to Rectified Linear Units (ReLU), which suppresses low activation neurons in each layer.
The block diagram of the proposed framework is shown in Fig. \ref{fig:mcnn_general_block}. The pre-trained LightCNN model produces a 256-dimensional embedding, which can be used as face representation. The LightCNN model is extended to accept four channels. The 256-dimensional representation from all channels are concatenated, and two fully connected layers are added at the end for PAD task. The first fully connected layer has ten nodes, and the second one has one node. Sigmoidal activation functions are used in each fully connected layer. The higher level features are more related to the task to be solved. Hence, the fully connected layers added on top of the concatenated representations are tuned exclusively for PAD task. Reusing the weights from a network pre-trained for face recognition on a large set of data, we avoid plausible over-fitting, which can occur due to a limited amount of training data.
Binary Cross Entropy (BCE) is used as the loss function to train the model using the ground truth information for PAD task.
The equation for BCE is shown below.
\begin{equation}
\mathcal{L}=-{(y\log(p) + (1 - y)\log(1 - p))}
\end{equation}
where $y$ is the ground truth, ($y=0$ for attack and $y=1$ for \textit{bonafide}) and $p$ is predicted probability.
Several experiments were carried out by adapting the different groups of layers, starting from the low-level features. The final fully connected layers are adapted for PAD task in all the experiments.
While doing the adaptation, the weights are always initialized from the weights of the pre-trained layers. Apart from the layers adapted, the parameters for the rest of the network remain shared.
The layers corresponding to the color channel are not adapted since the representation from the color channel can be reused for face recognition, hence making the framework suitable for simultaneous face recognition and presentation attack detection.
\section{The Wide Multi-Channel Presentation Attack Database}
The Wide Multi-Channel presentation Attack (WMCA) database consists of short video recordings for both \textit{bonafide} and presentation attacks from 72 different identities. In this section, we provide the details on the data collection process, and statistics of the database.
\subsection{Camera set up for data collection}
For acquisition of face data, different sensors were selected to provide a sufficient range of high-quality information in both visual and infrared spectra. In addition, 3D information was provided by one sensor adding a depth map channel to the video stream. Overall, the data stream is composed of a standard RGB video stream, a depth stream (called RGB-D when considered together with the color video), a Near-Infrared (NIR) stream, and a thermal stream. While the RGB-D and NIR data are provided by an Intel RealSense SR300 sensor, the thermal data is provided by a Seek Thermal Compact PRO camera, both being relatively cheap devices aimed at the consumer market. The hardware specifications of these devices are described below.
\subsubsection{Intel RealSense SR300 sensor}
The Intel RealSense SR300 camera is a consumer grade RGB-D sensor aimed at gesture recognition and 3D scanning (Fig. \ref{fig:idiap_setup}(c)). It features a full-HD RGB camera, capable of capturing resolution of $1920\times1080$ pixels in full-HD mode at 30 frames-per-second (fps) or $1260\times720$ pixels in HD mode at 60 fps.
\subsubsection{Seek Thermal Compact PRO sensor}
The Seek Thermal Compact PRO sensor is a commercial thermal camera aimed at the consumer market (Fig. \ref{fig:idiap_setup}(b)). It provides a QVGA resolution of $320\times240$. This camera range is primarily intended to be mounted on smart-phone devices. The sensor is capable of capturing at approximately 15 fps, with a non-continuous operation due to an electro-mechanical shutter used to calibrate the sensor at regular intervals (approx. $2s$).
\subsection{Camera integration and calibration}
\begin{figure}[h]
\begin{center}
\includegraphics[width=0.45\textwidth]{pics/Setup2.png}
\caption{The integrated setup used for WMCA data collection; a) rendering of the integrated system, b) Seek Thermal Compact PRO sensor , c) Intel RealSense SR300 sensor. }\label{fig:idiap_setup}
\end{center}
\end{figure}
For multi-sensor capture, it is essential that all sensors are firmly attached to a single mounting frame to maintain alignment and minimize vibrations. The setup was built using standard optical mounting posts, giving an excellent strong and modular mounting frame with the ability to precisely control the orientation of the devices, Fig.~\ref{fig:idiap_setup}.
To calibrate the cameras and to provide relative alignment of the sensors to the software architecture, we used a checkerboard pattern made from materials with different thermal characteristics. The data from this checkerboard was captured simultaneously from all the channels. For the pattern to be visible on the thermal channel, the target was illuminated by high power halogen lamps. Custom software was then implemented to automatically extract marker points allowing precise alignment of the different video streams. Sample images from all the four channels after alignment is shown in Fig. \ref{fig:db_sample}.
\begin{figure}[ht]
\centering
\includegraphics[width=1\linewidth]{pics/db_sample.png}
\caption{Sample images of a) Bonafide and b) Silicone mask attack from the database for all channels after alignment. The images from all channels are aligned with the calibration parameters and normalized to eight bit
for better visualization.}
\label{fig:db_sample}
\end{figure}
\subsection{Data collection procedure}
The data was acquired during \textbf{seven} sessions over an interval of five months. The sessions were different (Fig. \ref{fig:bf_rgb}) in their environmental conditions such as background (uniform and complex) and illumination (ceiling office light, side LED lamps, and day-light illumination) (Table \ref{tab:session-info}).
At each session, 10 seconds of data both for \textit{bonafide} and at least two presentation attacks performed by the study participant was captured. Session four was dedicated to presentation attacks only.
\begin{table}[ht]
\centering
\caption{Session description for WMCA data collection}
\label{tab:session-info}
\begin{tabular}{lcr}
\specialrule{.1em}{.05em}{.05em}
\textbf{Session} & \textbf{Background} & \textbf{Illumination} \\
\specialrule{.1em}{.05em}{.05em}
1 & uniform & ceiling office light \\ \hline
2 & uniform & day-light illumination \\ \hline
3 & complex & day-light illumination \\ \hline
4 & uniform & ceiling office light \\ \hline
5 & uniform & ceiling office light \\ \hline
6 & uniform & side illumination with LED lamps \\ \hline
7 & complex & ceiling office light \\
\specialrule{.1em}{.05em}{.05em}
\end{tabular}
\end{table}
Participants were asked to sit in front of the custom acquisition system and look towards the sensors with a neutral facial expression. If the subjects wore prescription glasses, their \textit{bonafide} data was captured twice, with and without the medical glasses. The masks and mannequins were heated using a blower prior to capture to make the attack more challenging.
The distance between the subject and the cameras was approximately 40\textit{cm} for both \textit{bonafide} and presentation attacks.
The acquisition operator adjusted the cameras so that the subject's face was frontal and located within the field of view of all the sensors at the desired distance. Then they launched the capturing program which recorded data from the sensors for 10 seconds.
\begin{figure}[ht]
\centering
\includegraphics[width=.7\linewidth]{pics/sessions_rgb.jpg}
\caption{Examples of \textit{bonafide} data in 6 different sessions. Top left is session one and bottom right is session seven. There is no \textit{bonafide} data for session four.}
\label{fig:bf_rgb}
\end{figure}
\subsection{Presentation attacks}
The presentation attacks were captured under the same conditions as \textit{bonafide}. More than \textbf{eighty} different presentation attack instruments (PAIs) were used for the attacks most of which were presented both by a study participant and on a fixed support.
The PAIs presented in this database can be grouped into seven categories. Some examples can be seen in Fig. \ref{fig:pais}.
\begin{itemize}
\item \textbf{glasses:} Different models of disguise glasses with fake eyes (funny eyes glasses) and paper glasses. These attacks constitute partial attacks.
\item \textbf{fake head:} Several models of mannequin heads were used, some of the mannequins were heated with a blower prior to capture.
\item \textbf{print:} Printed face images on A4 matte and glossy papers using professional quality Ink-Jet printer (Epson\_XP-860) and typical office laser printer (CX c224e). The images were captured by the rear camera of an ``iPhone S6'' and re-sized so that the size of the printed face is human like.
\item \textbf{replay:} Electronic photos and videos. An ``iPad pro 12.9in'' was used for the presentations. The videos were captured in HD at 30 fps by the front camera of an ``iPhone S6'' and in full-HD at 30 fps by the rear camera of the ``iPad pro 12.9in''. Some of the videos were re-sized so that the size of the face presented on the display is human like.
\item \textbf{rigid mask:} Custom made realistic rigid masks and several designs of decorative plastic masks.
\item \textbf{flexible mask:} Custom made realistic soft silicone masks.
\item \textbf{paper mask:} Custom made paper masks based on real identities. The masks were printed on the matte paper using both printers mentioned in the print category.
\end{itemize}
\begin{figure}[ht]
\centering
\includegraphics[width=1\linewidth]{pics/pais.png}
\caption{Examples of presentation attacks with different PAIs. (a): glasses (paper glasses), (b): glasses (funny eyes glasses), (c): print, (d): replay, (e): fake head, (f): rigid mask (Obama plastic Halloween mask), (g): rigid mask (transparent plastic mask), (h): rigid mask (custom made realistic), (i): flexible mask (custom made realistic), and (j): paper mask.}
\label{fig:pais}
\end{figure}
The total number of presentations in the database is 1679, which contains 347 \textit{bonafide} and 1332 attacks. More detailed information can be found in Table \ref{tab:BATL-data} \footnote{ For downloading the dataset, visit \url{https://www.idiap.ch/dataset/wmca} }.
Each file in the dataset contains data recorded at 30 fps for 10 seconds amounting to 300 frames per channel, except for thermal channel, which contains approximately 150 frames captured at 15 fps. All the channels are recorded in uncompressed format, and the total size of the database is 5.1 TB.
\begin{table}[ht]
\centering
\caption{Statistics for WMCA database.}
\label{tab:BATL-data}
\begin{tabular}{lr}
\specialrule{.1em}{.05em}{.05em}
\textbf{Type} & \#Presentations \\
\specialrule{.1em}{.05em}{.05em}
\textit{bonafide} & 347 \\ \hline
glasses & 75 \\ \hline
fake head & 122 \\ \hline
print & 200 \\ \hline
replay & 348 \\ \hline
rigid mask & 137 \\ \hline
flexible mask & 379 \\ \hline
paper mask & 71 \\
\specialrule{.1em}{.05em}{.05em}
\textbf{TOTAL} & \textbf{1679} \\
\specialrule{.1em}{.05em}{.05em}
\end{tabular}
\end{table}
\section{Experiments}
This section describes the experiments performed on the WMCA dataset. All four channels of data obtained from Intel RealSense SR300 and Seek Thermal Compact PRO are used in the experiments. Various experiments were done to evaluate the performance of the system in ``seen'' and ``unseen'' attack scenarios. In the ``seen'' attack protocol, all types of PAIs are present in the train, development and testing subsets (with disjoint client ids in each fold). This protocol is intended to test the performance of the algorithm in the cases where the attack categories are known a priori. The ``unseen'' attack protocols try to evaluate the performance of the system on PAIs which were not present in the training and development subsets. The ``unseen'' attack protocols thus emulate the realistic scenario of encountering an attack which was not present in the training set. The evaluation protocols are described below.
\subsection{Protocols}
As the consecutive frames are correlated, we select only 50 frames from each video which are uniformly sampled in the temporal domain. Individual frames from a video are considered as independent samples producing one score per frame. A biometric sample consists of frames from all four channels which are spatially and temporally aligned.
Two different sets of protocols were created for the WMCA dataset.
\begin{itemize}
\item \textbf{grandtest} protocol : The WMCA dataset is divided into three partitions: $train$, $dev$, and $eval$. The data split is done ensuring almost equal distribution of PA categories and disjoint set of client identifiers in each set. Each of the PAIs had different client id. The split is done in such a way that a specific PA instrument will appear in only one set. The ``grandtest'' protocol emulates the ``seen'' attack scenario as the PA categories are distributed uniformly in the three splits.
\item \textbf{unseen attack} protocols: The unseen attack protocols defined in the WMCA dataset contains three splits similar to the grandest protocol. Seven unseen attack sub-protocols were created for conducting unseen attack experiments using leave one out (LOO) technique. In each of the unseen attack protocols, one attack is left out in the $train$ and $dev$ sets. The $eval$ set contains the \textit{bonafide} and the samples from the attack which was left out in training. For example, in ``LOO\_fakehead'' protocol, the fake head attacks are not present in both $train$ and $dev$ sets. In the $test$ set, only \textit{bonafide} and fake head attacks were present. The training and tuning are done on data which doesn't contain the attack of interest.
\end{itemize}
\subsection{Evaluation metrics}
We train the framework using the data in the $train$ set, and the decision threshold is found from the $dev$ set by minimizing a given criteria (here, we used a BPCER = 1\% for obtaining the thresholds). We also report the standardized ISO/IEC 30107-3 metrics \cite{ISO}, Attack Presentation Classification Error Rate (APCER), and Bonafide Presentation Classification Error Rate (BPCER) in the $test$ set at the previously optimized threshold.
To summarize the performance in a single number, the Average Classification Error Rate (ACER) is used, which is an average of APCER and BPCER. The ACER is reported for both $dev$ and $test$ sets.
Apart from the error metrics, ROC curves are also shown for the baseline and MCCNN method.
\subsection{Baseline experiment setup}
Since a new database is proposed in this paper, baseline experiments are performed for each channel first. The selected baselines are reproducible and have either open-source implementation~\cite{nikisins2018effectiveness} or re-implementation. Three sets of baselines are described in this section.
\subsubsection{IQM-LBP-LR baseline}
This baseline consists of Image Quality Measures (IQM)~\cite{galbally2014image} for the RGB channel, and different variants of Local Binary Patterns (LBP) for non-RGB channels. The detailed description of the baseline systems is given below.
An individual PAD algorithm is implemented for every data stream from the camera. The structure of all PAD algorithms can be split into three blocks: preprocessor, a feature extractor, and classifier. The final unit of the PAD system is a fusion block, combining the outputs of channel-specific PAD algorithms, and producing a final decision.
The preprocessing part is exactly similar to the description in Section \ref{subsec:preprocess}, except for color channel. For the color channel, all three RGB channels are retained in the baseline experiments.
The feature extraction step aims to build a discriminative feature representation. For the color channel, the feature vector is composed of 139 IQMs \cite{nikisins2018effectiveness}. Spatially enhanced histograms of LBPs are selected as features for infrared, depth, and thermal channels \cite{nikisins2018effectiveness}. Optimal LBP parameters have been selected experimentally using grid search for each channel independently.
For classification, Logistic Regression (LR) is used as a classifier for color, infrared, depth, and thermal channels \footnote{ We are pointing out that we investigated other classifiers such as SVM but as no performance improvement was noticed we decided to keep a simple method.}. The features are normalized to zero mean and unity standard deviation before the training. The normalization parameters are computed using samples of \textit{bonafide} class only. In the prediction stage, a probability of a sample being a \textit{bonafide} class is computed given trained LR model.
Scores from all PAD algorithms are normalized to $[0, 1]$ range, and a mean fusion is performed to obtain the final PA score.
\subsubsection{RDWT-Haralick-SVM baseline}
In this baseline we used the re-implementation of the algorithm in \cite{agarwal2017face} for individual channels. We applied similar preprocessing strategy as discussed in the previous section in all channels. Haralick \cite{haralick1979statistical} features computed from the RDWT decompositions in a $4 \times 4$ grid are concatenated and fed to a Linear SVM for obtaining the final scores. Apart from implementing the pipeline independently for each channel, we additionally performed a mean fusion of all channels.
\subsubsection{FASNet baseline}
We also compare our system to a deep learning based FASNet \cite{lucena2017transfer} baseline. The FASNet
uses the aligned color images as input for PAD task. We reimplemented the approach in PyTorch \cite{paszke2017automatic} which is made available publicly.
\subsection{Experiment setup with the proposed MC-CNN approach}
The architecture shown in Fig. \ref{fig:mcnn_general_block} was used for the experiments. The base network used is LightCNN with 29 layers. Further, to accommodate four channels of information, the same network is extended. The embedding layers from different channels are concatenated, and two fully connected layers are added at the end. We performed different experiments by re-training different sets of low-level layers. The layers which are not re-trained are shared across the channels.
The dataset contained an unequal number of samples for \textit{bonafide} and attacks in the training set. The effect of this class imbalance is handled by using a weighted BCE loss function. The weights for the loss function is computed in every mini-batch dynamically based on the number of occurrences of classes in the mini-batch. To compensate for the small amount of training data, we used data augmentation by randomly flipping the images horizontally. All channels are flipped simultaneously to preserve any cross-channel dependencies. A probability of 0.5 was used in this data augmentation. The network was trained using Binary Cross Entropy (BCE) loss using Adam Optimizer \cite{kingma2014adam} with a learning rate of $1\times10^{-4}$. The network was trained for 25 epochs on a GPU grid with a mini-batch size of 32. Implementation was done in PyTorch \cite{paszke2017automatic}.
\section{Results and Discussion}
\subsection{Experiments with \textbf{grandtest} protocol}
\subsubsection{Baseline results}
The performance of the baselines in different individual channels and results from fusion are shown in Table \ref{tab:baseline_results}. From Table \ref{tab:baseline_results}, it can be seen that for individual channels, thermal and infrared provides
more discriminative information. RDWT-Haralick-SVM in infrared channel provides the best accuracy among individual channels.
It is observed that the fusion of multiple channels improves accuracy. Score fusion improves the accuracy of both feature-based baselines. Deep learning-based baseline FASNet achieves better accuracy as compared to IQM and RDWT-Haralick features in color channel. The FASNet architecture was designed exclusively for three channel color images since it uses normalization parameters and weights from a pre-trained model trained on ImageNet \cite{ILSVRC15} dataset. The training stage in FASNet is performed by fine-tuning the last fully connected layers. The usage of three channel images and finetuning of only last fully connected layers limits the straight forward extension of this architecture to other channels. In the baseline experiments, score fusion of individual channels achieved the best performance and is used as the baselines in the subsequent experiments. From this set of experiments, it is clear that the addition of multiple channels helps in boosting the performance of PAD systems. However, the performance achieved with the best baseline systems is not adequate for deployment in critical scenarios. The lower accuracy in the fusion baselines points to the necessity to have methods which utilize multi-channel information more efficiently.
\begin{table*}[t]
\centering
\caption{Performance of the baseline systems and the components in \textbf{grandtest} protocol of WMCA dataset. The values reported are obtained with a threshold computed for BPCER 1\% in $dev$ set.}
\label{tab:baseline_results}
\begin{tabular}{@{}ccc|ccc@{}}
\toprule
\multirow{2}{*}{Method} & \multicolumn{2}{c|}{dev (\%)} & \multicolumn{3}{c}{test (\%)} \\ \cmidrule(l){2-6}
& \multicolumn{1}{c|}{APCER} & ACER & \multicolumn{1}{c|}{APCER} & \multicolumn{1}{c|}{BPCER} & ACER \\ \midrule
Color (IQM-LR) &76.58 &38.79 &87.49 &0 &43.74\\
Depth (LBP-LR) &57.71 &29.35 &65.45 &0.03 &32.74\\
Infrared (LBP-LR) &32.79 &16.9 &29.39 &1.18 &15.28\\
Thermal (LBP-LR) &11.79 &6.4 &16.43 &0.5 &8.47 \\
Score fusion (IQM-LBP-LR Mean fusion) &10.52 &5.76 &13.92 &1.17 &7.54 \\ \hline
Color (RDWT-Haralick-SVM) &36.02 &18.51 &35.34 &1.67 &18.5\\
Depth (RDWT-Haralick-SVM) &34.71 &17.85 &43.07 &0.57 &21.82\\
Infrared (RDWT-Haralick-SVM)&14.03 &7.51 &12.47 &0.05 &6.26 \\
Thermal (RDWT-Haralick-SVM) &21.51 &11.26 &24.11 &0.85 &12.48 \\
Score fusion (RDWT-Haralick-SVM Mean fusion) &6.2 &3.6 &6.39 &0.49 &3.44 \\\hline
FASNet &18.89 &9.94 &17.22 &5.65 &11.44 \\ \bottomrule
\end{tabular}
\end{table*}
\subsubsection{Results with MC-CNN}
The results with the proposed approach in the grandest protocol are shown in Table \ref{tab:perf_mccn}. The corresponding ROCs are shown in Fig. \ref{fig:roc_idiap_mccnn} \footnote{\label{note1} The score distributions from the CNN is bimodal (with low variance in each mode) with most of the values concentrated near zero and one, which explains the lack of points in the lower APCER values in the ROC plots.}.
From Table \ref{tab:perf_mccn}, it can be seen that the proposed MC-CNN approach outperforms the selected baselines by a big margin. From the baseline results, it can be seen that having multiple channels alone doesn't solve the PAD problem. Efficient utilization of information from the various channels is required for achieving good PAD performance. The proposed framework utilizes complementary information from multiple channels with the joint representation. Transfer learning from the pretrained face recognition model proves to be effective for learning deep models for multi-channel PAD task while avoiding overfitting by adapting only a minimal set of DSU's. Overall, the proposed MC-CNN framework uses the information from multiple channels effectively boosting the performance of the system.
The performance breakdown per PAI for BPCER threshold of 1\% is shown in Table \ref{tab:perfbreakdown}. From Table \ref{tab:perfbreakdown} it can be seen that the system achieves perfect accuracy in classifying attacks except for ``glasses''. A discussion about the performance degradation in the ``glasses'' attack is presented in Subsection \ref{subsec:unseen}.
\begin{figure}[t]
\centering
\includegraphics[width=0.95\linewidth,page=2]{pics/mccnn_comparison.pdf}
\caption{ROC for MC-CNN and the baseline methods in WMCA \textbf{grandtest} protocol $eval$ set \textsuperscript{\ref{note1}}.}
\label{fig:roc_idiap_mccnn}
\end{figure}
\begin{table}[t]
\centering
\caption{Performance breakdown across different PAIs. Accuracy for each type of attack at at BPCER 1\% is reported.}
\label{tab:perfbreakdown}
\begin{tabular}{@{}cl@{}}
\toprule
ATTACK\_TYPE & MC-CNN @ BPCER : 1\% \\ \midrule
glasses & 90.82 \\
fake head & 100.0 \\
print & 100.0 \\
replay & 100.0 \\
rigid mask & 100.0 \\
flexible mask & 100.0 \\
paper mask & 100.0 \\
\bottomrule
\end{tabular}
\end{table}
\begin{table*}[t]
\centering
\caption{Performance of the proposed system as compared to the best baseline method on the $dev$ and $test$ set of the \textbf{grandtest} protocol of WMCA dataset.}
\label{tab:perf_mccn}
\begin{tabular}{@{}ccc|ccc@{}}
\toprule
\multirow{2}{*}{Method} & \multicolumn{2}{c|}{dev (\%)} & \multicolumn{3}{c}{test (\%)} \\ \cmidrule(l){2-6}
& \multicolumn{1}{c|}{APCER} & ACER & \multicolumn{1}{c|}{APCER} & \multicolumn{1}{c|}{BPCER} & ACER \\ \midrule
\textbf{MC-CNN} &0.68 &0.84 & 0.6 & 0 &\textbf{0.3}\\
\multicolumn{1}{c}{\begin{tabular}[c]{@{}c@{}}RDWT+Haralick\\ Score fusion\end{tabular}} &6.2 &3.6 &6.39 &0.49 &3.44 \\
\multicolumn{1}{c}{\begin{tabular}[c]{@{}c@{}}IQM+LBP\\ Score fusion\end{tabular}}
&10.52 &5.76 &13.92 &1.17 &7.54 \\ \bottomrule
\end{tabular}
\end{table*}
\subsection{Generalization to unseen attacks}
\label{subsec:unseen}
In this section, we evaluate the performance of the system under unseen attacks. Experiments are done with
the different sub-protocols, which exclude one attack systematically in training. The algorithms are trained with samples from each protocol and evaluated on the $test$ set which contains only the \textit{bonafide} and the attack which was left out in training.
The performance of MC-CNN, as well as the baseline system, are tabulated in Table \ref{tab:unseen_baseline_mccnn}.
\begin{table*}[t]
\centering
\caption{Performance of the baseline and the MC-CNN system with \textbf{unseen attack} protocols. The values reported are obtained with a threshold computed for BPCER 1\% in $dev$ set.}
\label{tab:unseen_baseline_mccnn}
\begin{tabular}{lccc|ccc|lll}
\hline
\multicolumn{1}{c}{\multirow{3}{*}{Protocol}} & \multicolumn{3}{c|}{\begin{tabular}[c]{@{}c@{}}RDWT+Haralick \\ Score fusion\end{tabular}} & \multicolumn{3}{c|}{\begin{tabular}[c]{@{}c@{}}IQM+LBP\\ Score fusion\end{tabular}} & \multicolumn{3}{c}{MC-CNN} \\ \cline{2-10}
\multicolumn{1}{c}{} & \multicolumn{3}{c|}{test (\%)} & \multicolumn{3}{c|}{test (\%)} & \multicolumn{3}{c}{test (\%)} \\ \cline{2-10}
\multicolumn{1}{c}{} & \multicolumn{1}{l|}{APCER} & \multicolumn{1}{l|}{BPCER} & \multicolumn{1}{l|}{ACER} & \multicolumn{1}{l|}{APCER} & \multicolumn{1}{l|}{BPCER} & \multicolumn{1}{l|}{ACER} & \multicolumn{1}{l|}{APCER} & \multicolumn{1}{l|}{BPCER} & \multicolumn{1}{l}{ACER} \\ \hline
LOO\_fakehead & 4.82 & 1.5 & 3.16 & 4.12 & 0.64 & 2.38 & 0 & 0 & 0 \\
LOO\_flexiblemask & 28.06 & 0.03 & 14.05 & 56.36 & 0.8 & 28.58 & 5.04 & 0 & 2.52 \\
LOO\_glasses & 97.09 & 0.61 & 48.85 & 100 & 1.72 & 50.86 & 84.27 & 0 & 42.14 \\
LOO\_papermask & 4.01 & 0.49 & 2.25 & 31.51 & 1.17 & 16.34 & 0 & 0.7 & 0.35 \\
LOO\_prints & 0 & 0 & 0 & 3.52 & 1.08 & 2.3 & 0 & 0 & 0 \\
LOO\_replay & 10.03 & 1.51 & 5.77 & 0.15 & 1.53 & 0.84 & 0 & 0.24 & 0.12 \\
LOO\_rigidmask & 15.3 & 0 & 7.65 & 27.47 & 1.08 & 14.27 & 0.63 & 0.87 & 0.75 \\ \hline
\end{tabular}
\end{table*}
From this table, it can be seen that the MC-CNN algorithm performs well in most of the unseen attacks. The baseline methods also achieve reasonable performance in this protocol. However, MC-CNN achieves much better performance as compared to the fusion baselines, indicating the effectiveness of the approach. The performance in the case of ``glasses'' is very poor for both the baseline and the MC-CNN approach. From Figure \ref{fig:bonafide_funnyeyes}, it can be seen that the appearance of the glass attacks are very similar to \textit{bonafide} wearing medical glasses in most of the channels. Since the ``glasses'' attacks were not present in the training set, they get classified as bonafide and reduce the performance of the system.
The issue mentioned above is especially crucial for partial attacks in face regions with more variability. For example, partial attacks in eye regions would be harder to detect as there is a lot of variabilities introduced by \textit{bonafide} samples wearing prescription glasses. Similarly, attacks in lower chin could be harder to detect due to variability introduced by \textit{bonafide} samples with facial hair and so on.
\begin{figure}[t]
\centering
\includegraphics[width=0.85\linewidth,page=2]{pics/bonafide_funnyeye.jpg}
\caption{Preprocessed data from four channels for \textit{bonafide} with glasses (first row) and funny eyes glasses attack (second row).}
\label{fig:bonafide_funnyeyes}
\end{figure}
\subsection{Analysis of MC-CNN framework}
Here we study the performance of the proposed MC-CNN framework with two different sets of experiments---one with re-training different sets of layers and another with different combinations of channels.
\subsubsection{Experiments with adapting different layers}
Here we try to estimate how adapting a different number of low-level layers affect the performance. The features for the grayscale channel were frozen, and different experiments were done by adapting different groups of low-level layers of the MC-CNN from first layer onwards. In all the experiments, the weights were initialized from the ones trained in the grayscale channel.
Different groups of parameters were re-trained and are notated as follows. The grouping follows the implementation of LightCNN from the authors in the opensource implementation \cite{LightCNN}. The name of the layers is the same as given in 29 layer network described in \cite{wu2018light}.
The notations used for the combination of layers are listed below.
\begin{itemize}
\item \textbf{FFC} : Only two final fully connected (FFC) layers are adapted.
\item \textbf{C1-FFC} (1+FFC) : First convolutional layer including MFM, and FFC are adapted
\item \textbf{C1-B1-FFC} (1-2+FFC) : Adapting \textit{ResNet} blocks along with the layers adapted in the previous set.
\item \textbf{C1-B1-G1-FFC} (1-3+FFC) : Adapts \textit{group1} along with the layers adapted in the previous set.
\item \textbf{1-N+FFC} : Adapts layers from 1 to N along with FFC.
\item \textbf{ALL} (1-10 +FFC) : All layers are adapted.
\end{itemize}
Here $FFC$ denotes the two final fully connected layers, and the rest of the names are for different blocks
corresponding to the opensource implementation of LightCNN from \cite{wu2018light}.
The results obtained with re-training different layers are shown in Table \ref{tab:performance_cnn_layers}.
It can be seen that the performance improves greatly when we adapt the lower layers; however, as we adapt more layers, the performance starts to degrade. The performance becomes worse when all layers are adapted. This can be attributed to over-fitting as the number of parameters to learn is very large. The number of layers to be adapted is selected empirically. The criteria used in this selection is good performance while adapting a minimal number of parameters. For instance, the ACER obtained is 0.3\% for \textit{1-2+FFC} and \textit{1-4+FFC}, the optimal number of layers to be adapted is selected as ``2'' (C1-B1-FFC) since it achieved the best performance adapting a minimal set of parameters. This combination selected as the best system and is used in all the other experiments.
\begin{table*}[t]
\centering
\caption{Performance of the MC-CNN when different combinations of layers were adapted.}
\label{tab:performance_cnn_layers}
\begin{tabular}{@{}ccc|ccc@{}}
\toprule
\multirow{2}{*}{Method} & \multicolumn{2}{c|}{dev (\%)} & \multicolumn{3}{c}{test (\%)} \\ \cmidrule(l){2-6}
& \multicolumn{1}{c|}{APCER} & ACER & \multicolumn{1}{c|}{APCER} & \multicolumn{1}{c|}{BPCER} & ACER \\ \midrule
FFC (FFC) &1.51 &1.26 & 2.88 & 0 &1.44 \\
C1-FFC (1+FFC) &1.77 &1.38 & 2.44 & 0 &1.22 \\
\textbf{C1-B1-FFC(1-2+FFC)} &0.68 &0.84 & 0.6 & 0 &\textbf{0.3} \\
C1-B1-G1-FFC(1-3+FFC) &1.1 &1.05 & 1.11 & 0.05 &0.58 \\
C1-B1-G1-B2-FFC(1-4+FFC) &0.23 &0.61 & 0.58 & 0.02 &0.3 \\
(1-5+FFC) &1.14 &0.57 & 0.99 & 0.56 &0.77 \\
(1-6+FFC) &100 &50 & 100 & 0 &50 \\
(1-7+FFC) &97.56 &48.78 & 96.88& 0 &48.44 \\
(1-8+FFC) &99.99 &49.99 & 100 & 0 &50 \\
(1-9+FFC) &100 &50 &100 & 0 &50 \\
ALL(1-10+FFC) &100 &50 &100 & 0 &50 \\ \bottomrule
\end{tabular}
\end{table*}
\subsubsection{Experiments with different combinations of channels}
Here the objective is to evaluate the performance of the algorithm with different combinations of channels.
This analysis could be useful in selecting promising channels which are useful for the PAD task.
Additionally, the performance of individual channels is also tested in this set of experiments to identify the contribution from individual channels.
It is to be noted that color, depth, and infrared channels are available from the Intel RealSense SR300 device, and the thermal channel is obtained from the Seek Thermal Compact PRO camera. It could be useful to find performance when data from only one sensor is available. We have done experiments
with six different combinations for this task. The combinations used are listed below.
\begin{enumerate}
\item \textbf{G+D+I+T} : All channels, i.e., Grayscale, Depth, Infrared, and Thermal are used.
\item \textbf{G+D+I} : Grayscale, Depth and Infrared channels are used (All channels from Intel RealSense).
\item \textbf{G} : Only Grayscale channel is used.
\item \textbf{D} : Only Depth channel is used.
\item \textbf{I} : Only Infrared channel is used.
\item \textbf{T} : Only Thermal channel is used.
\end{enumerate}
The architecture of the network remains similar to the one shown in Fig. \ref{fig:mcnn_general_block}, where only the layers corresponding to the selected channels are present. Experiments were done using the different combinations of channels with the proposed framework. While training the model, the embeddings from the channels used in a specific experiment are used in the final fully connected layers. The training and testing are performed similarly as compared to experiments conducted in the grandtest protocol.
The results with different combinations of channels are compiled in Table \ref{tab:ablation_channels}. It can be seen that the system with all four channels performs the best with respect to ACER (0.3\%). The combination ``CDI'' achieves an ACER of 1.04\% which is also interesting as all the three channels used is coming from the same device (Intel RealSense). This analysis can be helpful in cases where all the channels are not available for deployment. The performance of the system with missing or with a subset of channels can be computed apriori, and models trained on the available channels can be deployed quickly. Among the individual channels, thermal channel achieves the best performance with an ACER of 1.85\%. However, it is to be noted that the analysis with individual channels is an ablation study of the framework, and the network is not optimized for individual channels. While doing the experiments with individual channels, the architecture is not MC-CNN anymore. The performance boost in the proposed framework is achieved with the use of multiple channels.
\begin{table}[t]
\centering
\caption{Performance of the MC-CNN with various combinations of channels.}
\label{tab:ablation_channels}
\begin{tabular}{@{}ccc|ccc@{}}
\toprule
\multirow{2}{*}{System} & \multicolumn{2}{c|}{dev (\%)} & \multicolumn{3}{c}{test (\%)} \\ \cmidrule(l){2-6}
& \multicolumn{1}{c|}{APCER} & ACER & \multicolumn{1}{c|}{APCER} & \multicolumn{1}{c|}{BPCER} & ACER \\ \midrule
\textbf{G+D+I+T} &0.68 &0.84 & 0.6 & 0 &\textbf{0.3} \\
G+D+I &0.78 &0.89 &2.07 & 0 &1.04\\
G &41.14 & 21.07&65.65 & 0 &32.82\\
D &10.3 &5.65 &11.77 & 0.31 & 6.04\\
I & 3.5 &2.25 &5.03 & 0 &2.51\\
T &4.19 &2.59 &3.14 &0.56 &1.85\\
\bottomrule
\end{tabular}
\end{table}
\subsection{Discussions}
\subsubsection{Performance}
From the experiments in the previous subsections, it can be seen that the performance of the proposed algorithm surpasses the selected feature-based baselines. Transfer learning from face recognition network proves to be effective in training deep multi-channel CNN's with a limited amount of training data.
From the experiments with different channels, it can be seen that the performance of the system with all four channels was the best. We have also tested the same system in a cross-database setting. The data used in this testing was part of the Government Controlled Test (GCT) in the IARPA ODIN \cite{ODIN} project. In the GCT data, it was observed that the system which uses all four channels was performing the best. The addition of complementary information makes the classifier more accurate. The combination of channels makes the framework more robust in general.
The experiments in the unseen attack scenario show some interesting results. Even though the framework is trained as a binary classifier, it is able to generalize well for most of the attacks when the properties of the unseen attacks can be learned from other types of presentations. This can be explained as follows, the 2D PAIs prints and replays can be characterized from depth channel alone. Having one of them in the training set is enough for the correct classification of the other class. The same idea can be extended to other attacks which need information from multiple channels for PAD. For example, if we have silicone masks in the training set; then classifying mannequins as an attack is rather easy.
A PAI is relatively easy to detect when it is distinctive from \textit{bonafide} in at least one of the channels. PAD becomes harder as the representations across channels become similar to that of \textit{bonafide}. This makes the detection of partial attacks such as glasses which occlude a small portion of the face more complex. From the above discussion, it can be seen that, if we have a wide variety of sophisticated attacks in the training set, then the accuracy in detecting simpler unseen attacks seems to be better. This observation is interesting as this could help to tackle the unseen attack scenario, i.e., if we train the system using sufficient varieties of complex PAIs, then the resulting model can perform reasonably well on simpler ``unseen'' attacks. Further, the representation obtained from the penultimate layer of MC-CNN can be used to train one class classifiers/anomaly detectors which could be used to detect unseen attacks.
\subsubsection{Limitations}
One of the main limitations of the proposed framework is the requirement of spatially and temporally aligned channels. Spatial alignment can be achieved by careful calibration of the cameras. Achieving temporal alignment requires the sensors to be triggered in a supervised fashion. However, the proposed framework can handle small temporal misalignments and does not have very stringent requirement on absolute synchronization between channels, as long as there is no significant movement between the frames from different channels. Data from different channels recorded in multiple sessions, as in \cite{agarwal2017face} cannot be used in the proposed framework. In deployment scenarios, the time spent for data capture should be small from the usability point of view; synchronized capture between multiple sensors is suitable for this scenario since it reduces the overall time for data capture. Further, if the multiple channels are not synchronized, face detection in the additional channels is not trivial. Having spatial and temporal alignment obviates the requirement of face detection for all channels since the face location can be shared among different channels. Data capture can be done synchronously as long as the illumination requirements for one sensor is not interfering another sensor and there are no cross sensor interferences. More stringent timing control will be required if there are cross sensor incompatibilities.
From Table \ref{tab:ablation_channels}, it is clear that having multiple channels improves performance significantly. However, it may not be feasible to deploy all the sensors in deployment scenarios. In the absence of certain channels, the proposed framework can be retrained to work with available channels (but with reduced performance). Further, it is possible to extend the proposed framework to work with a different set of additional channels by adding more channels to the proposed framework.
\section{Conclusions}
As the quality of PAIs gets better and better, identifying presentation attacks using visible spectra alone is becoming harder. Secure use of biometrics requires more reliable ways to detect spoofing attempts. Presentation attack detection is especially challenging while presented with realistic 3D attacks and partial attacks. Using multiple channels of information for PAD makes the systems much harder to spoof. In this work, a Multi-channel CNN framework is proposed, which achieves superior performance as compared to baseline methods.
We also introduce a new multi-channel dataset containing various 2D and 3D attacks tackling identity concealment and impersonation. The proposed database includes a variety of attacks including 2D prints, video and photo replays, mannequin heads, paper, silicone, and rigid masks among others. From the experiments, it becomes clear that the performance of algorithms is poor when only the color channel is used. Addition of multiple channels improves the results greatly. Furthermore, the unseen attack protocols and evaluations indicate the performance of the system in the real-world scenarios, where the system encounters attacks which were not present in the training set.
\section*{Acknowledgment}
Part of this research is based upon work supported by the Office of the
Director of National Intelligence (ODNI), Intelligence Advanced Research
Projects Activity (IARPA), via IARPA R\&D Contract No. 2017-17020200005.
The views and conclusions contained herein are those of the authors and
should not be interpreted as necessarily representing the official
policies or endorsements, either expressed or implied, of the ODNI,
IARPA, or the U.S. Government. The U.S. Government is authorized to
reproduce and distribute reprints for Governmental purposes
notwithstanding any copyright annotation thereon.
\ifCLASSOPTIONcaptionsoff
\newpage
\fi
\bibliographystyle{IEEEtran}
|
1,108,101,564,595 | arxiv | \section{Introduction}
The weak mixing angle is defined as a ratio of the gauge
couplings
\begin{equation}
\hat{s}^2(Q) = \frac{\hat{g}^{\prime 2}(Q)}{\hat{g}^2(Q)+\hat{g}^{\prime 2}(Q)} \;,
\label{eq:1}
\end{equation}
where $\hat{g}$ and $\hat{g}^\prime$ are the $SU(2)$ and $U(1)_Y$
gauge couplings respectively. These couplings are running in the
sense that they depend on the energy scale Q, and in supersymmetry
they are evaluated in the $\overline{DR}$ scheme. The weak mixing angle
$\hat{s}^2$ can be easily predicted from a GUT model, {\it i.e.} SU(5) or
from a string derived model like the flipped SU(5). There are many
sources of the determination of the weak mixing angle in terms
of experimentally measurable quantities. From muon
decay for instance and knowing the most accurate parameters,
$M_Z=91.1867 \pm 0.0020 GeV$, $\alpha_{EM}=1/137.036$ and
$G_F=1.16639(1)\times10^{-5}\, GeV^{-2}$, we can define the $\hat{s}^2$
at $M_Z$ by the expression
\begin{equation}
\hat{s}^2 \hat{c}^2=\frac{\pi\, \alpha_{EM}\,
}{\sqrt{2}\,M_Z^2\,G_F\,
(1-\Delta \hat{\alpha})\,\hat{\rho}\,(1-\Delta \hat{r}_W)}\; ,
\label{eq:2}
\end{equation}
where
\begin{eqnarray}
\hat{\rho}^{-1}&=&1-\Delta \hat{\rho}=
1-\frac{\Pi_{ZZ}(M_Z^2)}{M_Z^2}+\frac{\Pi_{WW}(M_W^2)}
{M_W^2}\label{eq:3}\;,\\[3mm]
\Delta \hat {r}_W&=&\frac{\Pi_{WW}(0)-\Pi_{WW}(M_W^2)}{M_W^2}+
\hat{\delta}_{VB}\label{eq:4}\;,\\[3mm]
\hat{\alpha}&=&\frac{\alpha_{EM}}{1-\Delta{\hat{\alpha}}}
\label{eq:5}\;.
\end{eqnarray}
$\Pi$'s are the loop contributions to the transverse Z and W
gauge bosons self energies and they contain both logarithmic and
finite parts. The quantity $\hat{\delta}_{VB}$ contains vertex+box
corrections and the term $\Delta \hat{\alpha}$ contributions from
light quarks and leptons as well contributions from heavy particle
thresholds {\it i.e.} top quark, Higgs bosons, superpartner masses.
The contributions (\ref{eq:3}),(\ref{eq:4}),(\ref{eq:5})
include logarithmic corrections
of the form
\begin{eqnarray}
\log \biggl ( \frac{M_{1/2}^2}{M_Z^2} \biggr ) \;,
\nonumber
\end{eqnarray}
and if the soft gaugino mass $M_{1/2}$ takes on
large values we refer to this quantity as a ``large log" quantity.
We have proven in \cite{Dedes5} that the weak mixing angle behaves like
\begin{equation}
\hat{s}^2 \sim \log \biggl ( \frac{M_{1/2}^2}{M_Z^2} \biggr ) \;.
\label{eq:6}
\end{equation}
This is clear from Figure 1, where if we increase the value of
$M_{1/2}$ then $\hat{s}^2$ takes on large values which means that large
logs have not been decoupled from its value. Through this note we
always assume the unification of gauge couplings up to two loops
and the constraint
from the radiative symmetry breaking \cite{Tamvakis}
where the full one loop minimization
conditions have been taken into account.
\begin{figure}
\centerline{\psfig{figure=sinhat.eps,height=2.2in}}
\caption{The extracted value of $\hat{s}^2$ as a function of $M_{1/2}$.
The logarithmic behavior resulting from the soft gaugino masses is obvious.}
\end{figure}
\twocolumn[\hsize\textwidth\columnwidth\hsize\csname @twocolumnfalse\endcsname
\widetext
\begin{table}
\caption{Partial and total contributions to ${\Delta k}_f$,
($f=lepton,charm,bottom$), for two sets of inputs shown at the top.
Also shown are the predictions for the effective weak mixing angles and the
asymmetries. In the first five rows we display the universal contributions
to $10^3 \; \times \;{\Delta k}$ of squarks ($\tilde q$),
sleptons($\tilde l$), Neutralinos and Charginos (${\tilde Z},{\tilde C}$),
ordinary fermions and Higgses
(The number shown in the middle below the
"charm" column refers to "lepton" and "bottom" as well). In the next five rows
we display the contributions of gauge bosons as well as the supersymmetric
$Electroweak \; (EW)$ and $SQCD$ vertex
and external fermion wave function renormalization corrections
to $10^3 \; \times \;{\Delta k}$. }
\begin{tabular}{ccccccccc}
&$M_0=100$&$M_{1/2}=100$&$A_0=100$& &
&$M_0=600$&$M_{1/2}=600$&$A_0=600$ \\
&$m_t=175$&$tanb=4$&$\mu > 0$& &
&$m_t=175$&$tanb=4$&$\mu > 0$ \\
\tableline
\\
&$lepton$ &$charm$ &$bottom$& &
&$lepton$ &$charm$ &$bottom$ \\ \\
\tableline
\\
${\tilde q}$& &-4.3481 & && & &-10.0677 & \\
${\tilde l}$& &-0.2166 & && & &-0.4042 & \\
${\tilde Z},{\tilde C}$& &7.087 & && & &-12.2366 & \\
$Fermions$& &4.8177 & && & &4.4290 & \\
$Higgs$& &-0.4390 & && & &-1.1523 & \\ \\
$Gauge$&-3.2491 &-3.7027 &2.2999 &&
&-3.0766 &-3.5184 &2.2771 \\ \\
$Vertex(EW)$&-1.9399&2.1535 &6.9837 &&
&2.8229 &5.6099 &21.2526 \\
$Wave(EW)$&-0.1487&-2.210 &-8.3524 &&
&-2.8280 &-5.6037 &-21.0202\\ \\
$Vertex(SQCD)$& - &0.1217 &-1.0491 &&
& - &0.2416 &-1.0202 \\
$Wave(SQCD)$&- &-0.1204 &1.0057 &&
& - &-0.2416 &1.0190\\
\tableline
\\
${\Delta k}\;(\times \; 10^2 ) $ &0.1557 &0.3137 &0.7782 &&
&-2.2513 &-2.2944 &-1.6424 \\
${sin^2} {\theta}_f$& 0.23019 & 0.23055 &0.23162 && &0.23145
& 0.23135 & 0.23277\\
${\cal A}_{LR}^f$& 0.1575 & 0.6708 &0.9355 && &0.1476 &0.6681 & 0.9347\\
${\cal A}_{FB}^f$& 0.0186 & 0.0792 &0.111 && &0.0163 & 0.0740 & 0.1035\\
\end{tabular}
\end{table}
]
\narrowtext
Although the weak mixing angle $\hat{s}^2$ provides a convenient means to
test unification in unified extensions of the Standard Model (SM)
or the MSSM it is {\it not} an experimental quantity.
There are several studies in the literature evaluating the weak mixing angle
both in the SM \cite{Sirlin,gambino,all1,all2,djoua}
and in the MSSM \cite{polon}.
\section{The Effective Weak Mixing Angle}
LEP collaborations employ the effective weak mixing angle
$s_f^2$ first introduced by Degrassi and Sirlin \cite{Sirlin}.
Electroweak corrections to the $Zf\bar{f}$ vertex yield the
effective Lagrangian which defines the effective weak mixing angle as
\begin{equation}
s_f^2 \equiv \hat{s}^2 \hat{k}_f = \hat{s}^2 ( 1 + \Delta \hat{k}_f )\;,
\label{eq:7}
\end{equation}
or in analogous way as we define the eq.(\ref{eq:2})
\begin{equation}
s^2_{f}c^2_{f}=\frac{\pi\, \alpha_{EM}\, (1+\Delta \hat{k}_f)\,
(1-\frac {\hat{s}^2}{\hat{c}^2}\,\Delta \hat{k}_f)}{\sqrt{2}\,M_Z^2\,G_F\,
(1-\Delta \hat{\alpha})\,\hat{\rho}\,(1-\Delta \hat{r}_W)}\;,
\label{eq:8}
\end{equation}
where
\begin{eqnarray}
\Delta \hat{k}_f&=&\frac{\hat{c}}{\hat{s}}\,\frac{\Pi_{Z\gamma}(M_Z^2)-
\Pi_{Z\gamma}(0)}{M_Z^2}+
\delta k_f^{SUSY}+\cdots \;.
\label{eq:9}
\end{eqnarray}
$\Pi_{Z\gamma}$ denotes the Z-$\gamma$ propagator corrections and
the term $\delta k_f^{SUSY}$ the non-universal corrections resulting
from the vertex and wave function renormalization corrections.
The non-universal corrections $\delta k_f^{SUSY}$ contain both diagrams with
electroweak and supersymmetric QCD corrections. There are also other
SM contributions which although included in our numerical analysis,
they are
irrelevant to the discussion here.
The effective weak mixing angle is an experimental measured
quantity and its average LEP+SLD value is 0.23152 $\pm$ 0.00023
\cite{Altarelli}. The difference between the two angles in the SM
is
\begin{eqnarray}
{\rm SM :}\;\; s_l^2-\hat{s}^2 \lesssim O(10^{-4})\;,
\nonumber
\end{eqnarray}
which is less than the error quoted by the experimental groups.
This fact is due to the small total contribution from fermions and
bosons occurring at the one loop level, in the
$\overline{MS}$ scheme and the only dominant contributions are
obtained from the two-loop heavy top contributions and three-loop
QCD effects.
In the MSSM this picture is changed dramatically. The difference
between the two angles can be at one loop,
\begin{eqnarray}
{\rm MSSM :}\;\; s_l^2-\hat{s}^2 \lesssim O(10^{-3})\;.
\nonumber
\end{eqnarray}
So, in the case of the MSSM the two angles may have completely
different values and the difference between them can be much
greater than the experimental error.
As we have shown in ref.\cite{Dedes5} although $\hat{s}^2$ suffers from
large logs in the MSSM, this is not the case for the $s^2_f$.
So far the situation is clear : GUT analysis is able to predict
the weak mixing angle $\hat{s}^2$ and then eq.(\ref{eq:7}) can translate the
result into the experimentally measured quantity, the effective weak
mixing angle $s_f^2$.
As we have previously mentioned, among the interesting features
of the effective weak mixing angle is that large logs get decoupled
from its value. In ref.\cite{Dedes5} we have shown this fact both
analytically and numerically. Here we present only the results of
the analytical proof. Math formulae can be found in ref.\cite{Dedes5}.
\subsection{Analytical Proof of the Decoupling}
There are three sources of large logs which affect the value of the
weak mixing angle $s_f^2$:\\
1. Gauge boson self energies which feed large logs to the quantities
$\Delta \hat r_W$, $\hat\rho$ and $\Delta \hat k_f$. These corrections
are canceled against large logs stemming from the electromagnetic coupling
$\hat{\alpha}(M_Z)$.\\
2. Vertex, external wave function renormalizations and box corrections
to muon decay which affect $\Delta \hat r_W$ through $\delta_{VB}^{SUSY}$.
These corrections are cancelled against each other out.\\
3. Non-universal vertex and external fermion corrections to
$Z \overline{f} f$ coupling which affects $\Delta \hat k_f$.
These corrections are also cancelled against each other out.\\
We conclude that $s_f^2$ behaves like $\sim O(\frac{M_Z}{M_{1/2}})$ which
is the SUSY contribution from the finite parts. It seems that SUSY follows
the well known Appelquist-Carazzone decoupling theorem \cite{Carazzone}.
\subsection{Numerical Proof of the Decoupling}
In order to prove the decoupling of large logs, we have included
the full supersymmetric one loop corrections for the universal and
non-universal part of eq.(\ref{eq:8}).
In all figures that follows we assume the constraint
from the radiative EW symmetry breaking \cite{Tamvakis}
and the universality for soft
breaking masses at the unification scale $M_{GUT}$.
In Figure 2 we plot the leptonic effective weak mixing angle $s_l^2$ as
a function of the soft gaugino mass parameter $M_{1/2}$ for a low
($M_0=100$ GeV) and high ($M_0=600$ GeV) value of the soft scalar
parameter, $M_0$.
\begin{figure}
\centerline{\psfig{figure=mom12.eps,height=3.5in}}
\caption{The effective weak mixing angle $s_l^2$ versus $M_{1/2}$
for two characteristic values of $M_0$. The top quark mass is fixed
at $m_t=175$ GeV.}
\end{figure}
We obtain that increasing the value of $M_{1/2}$, $s_l^2$ takes
on constant values in the limit $M_{1/2}\rightarrow 900$ GeV independently
on the soft squark mass $M_0$. Finite corrections of the form
$O(\frac{M_Z}{M_{1/2}})$ are significant in the region $M_{1/2}\sim M_0
\lesssim 200$ GeV with the tendency to decrease (increase) the $s_l^2$ when
$M_{1/2}$ is greater(smaller) than $M_Z$. In addition lowering $M_0$ we
obtain smaller values for the $s_l^2$. It is worth noting that
MSSM seems to agree with the average LEP+SLD value while in the
vicinity of $M_0 \sim M_{1/2} \simeq 100$ GeV, $s_l^2$ agrees with the
SLD experimental value\footnote{The LEP average $s_{eff}^2=0.23199\pm 0.00028$
differs by 2.9$\sigma$ from SLD value $s_{eff}^2=0.23055\pm 0.00041$
obtained from the single measurement of left-right asymmetry.}.
The parameters $A_0$ and $\tan\beta$\footnote{The
independent parameter $\tan\beta$ plays an important role
in the case of $s_b^2$.} do not affect significantly the
extracted value of $s_l^2$.
Going deeper, we display in Table I the partial and total
contributions to $\Delta k_f$. Also shown are the predictions
for the effective weak mixing angle $s_f^2$ and the asymmetries.
We would like to stress five interesting points on this Table :
$\bullet$ The bulk of the SUSY corrections to $\Delta \hat{k}_f$ is
carried by the universal corrections which are sizeable due to their
dependence on large log terms.
$\bullet$ The contribution of Higgses which is small mimics that of the
SM with a mass in the vicinity of $\sim 100$ GeV.
$\bullet$ Gauge\footnote{Gauge boson contribution are different for the
different fermion species $l,c,b$. This is due to the fact that their
non-universal corrections depend on the charge and weak isospin
assignments of the external fermions. In the case of the bottom
quark there are additional, significant top quark corrections.}
and Higgs boson contributions
tend to cancel large universal contributions of matter fermions.
$\bullet$ We find that the non-universal Electroweak SUSY
corrections are very small for a ``heavy''
SUSY breaking scale, $M_{1/2}=M_0=A_0=600$ GeV.
Although separetaly vertex and external fermion corrections are
large they cancel each other out yielding contributions almost two
orders of magnitude smaller than the rest of the EW corrections. For a
``light'' SUSY breaking scale, $M_{1/2}=M_0=A_0=100$ GeV this
cancellation does not occur due to the large splittings of the squarks
and sleptons.
$\bullet$ The non-universal SQCD contributions, although a priori
expected to be larger than the electroweak corrections turn out to
be even smaller. This fact is due to cancellations both from its
diagram in vertex or in wave function renormalization, and from
different types of the diagrams. This situation is valid both
in the ``heavy'' and ``light'' SUSY breaking limit and we believe
that this cancellations arise from a QED like Ward identities between
vertex and wave function renormalization corrections.
\section{Z-Boson Observables in the MSSM}
Having obtained the value of the effective weak mixing angle,
we are able to find other Z-observables in the MSSM. For
instance the physical W-boson mass is defined by
\begin{equation}
M_W\ =\ M_Z~\hat{c}~\sqrt{\hat{\rho}} =
M_Z~ c_f~ \sqrt{\frac{\hat{\rho}}{1-
\frac{\hat{s}^2}{\hat{c}^2} \Delta {\hat k}_f}}\;.
\label{eq:10}
\end{equation}
In figure 3 we plot the W-boson mass versus the input values
for the $M_{1/2}$ parameters for two characteristic values
of $M_0$ in the ``light'' SUSY breaking scale ($M_0=100$ GeV)
and in the ``heavy'' SUSY breaking scale ($M_0=600$ GeV).
As one can see the W-mass is in agreement with the presently
experimentally observed value, $M_W=80.427\pm 0.075$($M_W=80.405\pm 0.089$)
GeV obtained from LEP(CDF,UA2,D\O) experiments \cite{booklet} for
rather low(high) values of $M_{1/2}$. Moreover, our results are in
agreement with those of refs.\cite{Bagger,Hollik1,Pok3}.
\begin{figure}
\centerline{\psfig{figure=mw.eps,height=3.5in}}
\caption{Predicted values for the W-boson mass in the MSSM.}
\end{figure}
Having at hand the effective weak mixing angle we can
easily derive the left-right asymmetries
which in term of the $s_f^2$ can defined as
\begin{equation}
A_{LR}^f \ =\ {\cal A}^f \ =\ \frac{2~T_3^f~\left (T_3^f-2~s_f^2~Q^f
\right )}{T_3^{f 2} + \left (T_3^f - 2 s_f^2 Q^f \right )^2} \;,
\label{eq:11}
\end{equation}
where $Q^f$ is the electric charge and $T_3^f$ is the third component
of the isospin of the fermion $f$. As it is depicted in Figure 4, the
MSSM predictions for ${\cal A}_e$ agree with LEP+SLD average value
(${\cal A}_e = 0.1505 \pm 0.0023$) when both $M_{1/2}$ and $M_0$
take on values around $M_Z$. In the heavy SUSY breaking scale, the
MSSM agrees with the LEP value, ${\cal A}_e = 0.1461 \pm 0.0033$. Note
that increasing the value of $M_{1/2}$, ${\cal A}_e$ takes on constant
values which means that large logs have been decoupled from the
expression (\ref{eq:11}). In this case we have recovered
SM results displayed in Particle Data Booklet \cite{booklet}.
\begin{figure}
\centerline{\psfig{figure=ae.eps,height=3.5in}}
\caption{Predicted values for the electron left right asymmetry
in the MSSM.}
\end{figure}
So far we have not considered the constraint resulting from
the experimental value of $\alpha_s(M_Z)$ \cite{dedes2}.
In Figure 5 we have
plotted the acceptable values of the soft breaking parameters
which are compatible with LEP ($\alpha_s(M_Z)=0.119\pm 0.004$,
$s_{eff}^2=0.23152 \pm 0.00023$)\cite{Altarelli}
and the CDF/D\O ($m_t=175 \pm 5$ GeV) data
\cite{top}.
\begin{figure}
\centerline{\psfig{figure=all.eps,height=2.2in}}
\caption{Acceptable values in the $M_{1/2}-M_0$ plane according to the
LEP+SLD data. Only large values of $M_{1/2}$ are acceptable.}
\end{figure}
The trilinear soft couplings as well as the
parameter $\tan\beta (M_Z)$ are taken arbitrarily in the region
(0-900 GeV) and (2-30), respectively.
As we observe from Figure 5,
MSSM with unification of gauge coupling constants, universality
of the soft masses at $M_{GUT}$ and radiative EW symmetry breaking
is valid in the region $M_{1/2}\gtrsim 500$ GeV and $M_0\gtrsim 70$ GeV.
The lower bound on $M_{1/2}$ is put mainly from the experimental value
of $\alpha_s$ and on $M_0$ from the requirement that LSP is neutral.
In this region, the physical gluino mass is above 1 TeV, the LSP is
$\gtrsim 200$ GeV, the squark and slepton masses are $\gtrsim 800$ GeV
and $\gtrsim 210$ GeV and the light Higgs boson mass is greater than
$108$ GeV.
\section{Conclusions}
In ref.\cite{Dedes5} we have included the full one-loop
supersymmetric corrections to the effective weak mixing angle
which is experimentally determined in LEP and SLD experiments.
Our analysis enables one to pass from the ``theoretical'' weak mixing
angle $\hat{s}^2$, which can be predicted from GUT analysis, to
the experimental effective weak mixing angle $s_f^2$.
We conclude that :
$\bullet$ There are no logarithmic corrections of the form $\log
(\frac{M_{1/2}^2}{M_Z^2})$ (or more generally $\log
(\frac{M_{SUSY}^2}{M_Z^2})$) to the effective weak mixing angle.
$\bullet$ Supersymmetric QCD corrections tend to vanish everywhere in
the $M_{1/2}-M_0$ plane.
$\bullet$ The predicted MSSM values for the effective weak mixing
angle are in agreement with the present LEP+SLD average value in the
``heavy'' SUSY breaking scale while they are in agreement with the
SLD data in the ``light'' SUSY breaking scale..
$\bullet$ MSSM predicts values of the W-boson mass which are in
agreement with the new CDF,UA2,D{\O} average value.
$\bullet$ In the ``heavy'' SUSY breaking limit MSSM seems to
prefer the experimental LEP value for the electron left-right asymmetry
${\cal A}_e$.
$\bullet$ Finally, values of $M_{1/2}$ which are greater than
$500$ GeV are favoured by the MSSM if one assumes the present LEP+SLD
and CDF/D{\O} data for $s_{eff}^2$, $\alpha_s$ and $m_t$, respectively.
\acknowledgments
A.D. and K.T. acknowledge financial support from the
research program $\Pi{\rm ENE}\Delta$-95
of the Greek Ministry of Science and Technology. A.B.L. and K. T. acknowledge
support from
the TMR network ``Beyond the Standard Model", ERBFMRXCT-960090.
A. B. L. acknowledges
support from the Human Capital and Mobility program CHRX-CT93-0319.
|
1,108,101,564,596 | arxiv | \section{Introduction}
Poor eating habits have been linked to an increased risk of cardiovascular disease, cancer, obesity, diabetes, cataracts, and possibly dementia \cite{Rosenberg09}.
As such, there is growing interest in recording the food intake of at-risk individuals and exploring diet interventions to prevent long-term health problems \cite{Rosenberg09}. Today's gold standard for recording food intake is the self-administered food diary. This approach depends on an individual's meticulous attention to their daily diet. As such, it is prone to human error and lacks an objective validation check for the self-reported data \cite{Burke11}.
To address these problems, several automated alternatives to the food diary have been proposed. For instance, radio-frequency identification (RFID) equipped devices like the Dietary Intake Monitoring System (DIMS) were introduced in \cite{Ofei14}, but they cannot differentiate between users. Video-based systems are user-specific, but they are bulky, intrusive and require extensive memory and computational resources \cite{Wen09, Cadavid2012}. Wearable devices \cite{Zheng14, Lara13} are another promising approach, but the current wearable implementations have limited computational power, which requires them to transmit raw data for offline processing wirelessly; this continuous wireless streaming severely limits the wearable devices' battery life. In this paper, we introduce a power-efficient shallow GRU neural network for detecting eating episodes that can be implemented directly on a wearable device. Our eating detection model requires no offline processing or wireless communication, thus dramatically improving battery life and helping to make wearable devices a more effective alternative to the food diary.
\section{Eating Detection Overview}
Our system for detecting eating episodes comprises a contact microphone that senses jaw movement \cite{Shengjie18} and a micro-controller that analyzes the microphone output for chewing sounds. The algorithm on the micro-controller is based on the gated recurrent unit (GRU) neural network \cite{Cho14}, which has been applied successfully to various audio event detection problems \cite{Kim17, kusupati2018fastgrnn}. For our algorithm, we modified the GRU activation functions, quantized the weights, and implemented all computations as integer operations. These modifications allow our model to fit within the memory, computational, and latency constraints of low-power embedded processors.
We start by providing details on the neural network architecture in Section \ref{sec:Details}. Next, Section \ref{sec:Results} shows the results of the model implementation and then a discussion in Section \ref{sec:Disc} followed by future work in Section \ref{sec:Fush}. We then summarize the paper in Section \ref{sec:Conc}.
\section{Embedded Neural Network Details}
\label{sec:Details}
\subsection{Arm Cortex M0+}
We implemented our algorithm on the Arm Cortex M0+ because it is the most energy-efficient Arm processor available for constrained embedded applications \cite{Arm}, which makes it ideal for use in small-sized power conservative wearable devices. The challenge of using such a low-power processor is that it has no dedicated floating-point unit, operates at 48 MHz speed, and it has only 32 KB RAM and 256 KB flash memory.
These limitations restrict the size of the neural network that can be implemented and the size of data that can be computed at a given time.
\subsection{GRU Modifications for Arm Cortex M0+}
\label{subsec:GRU}
In order to meet the resource constraints of the Arm Cortex M0+, we modify the traditional GRU cell \cite{Cho14} and design the Tiny Eats GRU described by:
\begin{equation}
\begin{aligned}
\ \textbf{r}_t &= \frac{\varsigma(\textbf{W}_r[\textbf{x}, \textbf{h}_{t-1}]) + 1}{2}\
\end{aligned}
\label{resetgate}
\end{equation}
\begin{equation}
\begin{aligned}
\ \textbf{z}_t &= \frac{\varsigma(\textbf{W}_z[\textbf{x}, \textbf{h}_{t-1}]) + 1}{2}\
\end{aligned}
\label{updategate}
\end{equation}
\begin{equation}
\begin{aligned}
\ \Tilde{\textbf{h}}_t &= \varsigma(\textbf{W}_h[\textbf{x}, (\textbf{r}\odot \textbf{h}_{t-1})]) \
\end{aligned}
\label{cellstate}
\end{equation}
\begin{equation}
\begin{aligned}
\ \textbf{h}_t &= \textbf{z}_t\textbf{h}_{t-1} + (1 - \textbf{z}_t)\Tilde{\textbf{h}}_t, \
\end{aligned}
\label{candidatestate}
\end{equation}
where \textbf{x} represents the input features, \textbf{W} represents the 8 bit quantized weights, \textbf{r} represents the reset gate, \textbf{z} represents the update gate, \begin{math}\Tilde{\textbf{h}}\end{math} represents the candidate state, \textbf{h} represents the new cell state, and \begin{math}\odot\end{math} represents the element-wise multiplication of two vectors.
The Tiny Eats GRU cell diverges from the traditional GRU cell in that it uses a shifted soft-sign (\begin{math}\varsigma\end{math}) activation function in place of sigmoid activation function for the update and reset gates. A regular soft-sign is also used in place of tanh activation functions for candidate state as proposed by eGRU cell \cite{Amoh19}. The sigmoid and tanh activation functions are memory- and resource-intensive since they require iterative accesses of look-up tables. The soft-sign is a more computationally-conservative alternative compared to the sigmoid and tanh. This computation is particularly crucial because the Arm Cortex M0+ lacks a floating-point unit (FPU) and Digital Signal Processing (DSP) instruction set.
The Tiny Eats GRU cell diverges from the eGRU cell \cite{Amoh19} in that it maintains the use of a reset gate as proposed by the traditional GRU cell. This helps extract the portion of the history that is relevant to the current computation, as highlighted in equation \eqref{cellstate}.
\subsection{Embedded Neural Network}
The universal approximation theorem states that simple neural networks can represent a wide variety of interesting functions when given appropriate parameters. Instead of using a standard deep neural network, we use a shallow neural network with fewer layers and fewer neurons to represent the differences in STFT parameters of {\itshape eating} and {\itshape non-eating} data. The shallow neural network is computationally cheaper and conserves memory, which is necessary for implementation on the Arm Cortex M0+. In this paper, we propose the shallow neural network architecture shown in Fig. \ref{fig:NN} that consists of 1 output layer and only 3 hidden layers; 2 gated recurrent units designed as explained in Section \ref{subsec:GRU}, each with 16 neurons, and one fully connected layer with 8 neurons. The fully connected layer consists of a linear pre-activation function and a soft-sign activation. The output layer consists of a linear pre-activation function and a soft-max activation function. This design is very similar to the eGRU suggested by {\itshape Amoh and Odame} \cite{Amoh19}.
\begin{figure}[htbp]
\hspace*{1.0cm}
\includegraphics[width=7cm, height=7cm]{GRU_NN.png}
\caption{Gated Recurrent Unit Neural Network Architecture}
\label{fig:NN}
\end{figure}
\section{Experiments and Results}
\label{sec:Results}
\subsection{Eating Detection Dataset}
\label{subsec:STFT}
The data used for this study was previously collected in a controlled laboratory setting from 20 participants using a contact microphone placed behind the participant's ear \cite{Shengjie17}. The contact microphone was connected to a data acquisition device (DAQ) that collected the activity data in real-time at a 20KHz sampling rate with 24-bit resolution. The contact microphone, used for acoustic sensing, is placed at the tip of the mastoid bone and secured using a headband to ensure contact with the body. The participants were instructed to perform an activity that involves both {\itshape eating} and {\itshape non-eating}. {\itshape Eating} activities included eating carrots, protein bars, crackers, canned fruit, instant food, and yogurt, sequentially for 2 minutes per food type. This resulted in a 4 hour total {\itshape eating} dataset.
{\itshape Non-eating} activities included talking and silence for 5 minutes each and then coughing, laughing, drinking water, sniffling, and deep breathing for 24 seconds each. This resulted in a 7 hour total {\itshape non-eating} dataset. Each activity occurred separately and was classified based on activity type as {\itshape eating} or {\itshape non-eating}.
\begin{figure}[htbp]
\includegraphics[width=8.5cm, height=5.5cm]{filtered_data.png}
\caption{Temporal signature of down-sampled and filtered laboratory collected raw data}
\label{fig:filtered raw}
\end{figure}
\subsection{Data Pre-Processing}
\label{subsec:Data}
In this paper, we down-sampled the raw sound data to 500 Hz to reduce power consumption and memory usage and applied a high pass filter with a 20Hz cutoff frequency to attenuate noise \cite{Shengjie17} as shown in Fig. \ref{fig:filtered raw}. We segmented the positive class data ({\itshape eating data}), and negative class data ({\itshape non-eating data}) into 4-second windows, and computed a Short Time Fourier Transform (STFT), with 128 Fast Fourier Transform bins and no overlap, over the segments to view and extract features as shown in Fig. \ref{fig:STFTa} and Fig. \ref{fig:STFTb}.
\begin{figure}[htbp]
\includegraphics[width=8.5cm, height=5cm]{STFT_Eat.png}
\caption{STFT Results of Eating data}
\label{fig:STFTa}
\end{figure}
\begin{figure}[htbp]
\includegraphics[width=8.5cm, height=5cm]{STFT_NotEat.png}
\caption{STFT Results of Non-Eating data}
\label{fig:STFTb}
\end{figure}
The positive class data, as shown in Fig. \ref{fig:STFTa} has a lower frequency and periodic envelope distribution in comparison to the negative class data shown in Fig. \ref{fig:STFTb} that has a higher frequency and more compact distributions. These STFT features were used to train the neural network with each 4-second window being classified as an {\itshape eating} episode or a {\itshape non-eating} episode.
\subsection{Neural Network Training}
\label{subsec:NNTraining}
The neural network shown in Fig. \ref{fig:NN} is first programmed and trained in Python using TensorFlow Keras v2.0. This model uses floating-point. The training dataset contains more representation of {\itshape non-eating} data than it does {\itshape eating} data as described in Section \ref{subsec:STFT}. This imbalance inherently creates a bias in the training algorithm. In order to debias the algorithm, we normalize the binary cross-entropy loss function with each class's representative weight during training. This ensures that both classes are represented fairly.
In order to guarantee that the best performing model is used, a model checkpoint is implemented to save and retrieve the model with the highest validation accuracy and minimal validation loss. The results of this FPU model are as shown in Table \ref{tab:Acc} and Fig. \ref{fig:fpu_training}.
\begin{figure}[htbp]
\includegraphics[width=8.8cm, height=5.5cm]{fpu_tiny_bite_gru.png}
\caption{Loss and Accuracy Training Graphs for Floating Point Tiny Eats GRU}
\label{fig:fpu_training}
\end{figure}
An integer quantization based model \cite{Amoh19} that uses 8-bit quantization for all weights is then implemented in Pytorch and trained, as shown in Fig. \ref{fig:quantized_accuracy}. The weight normalization applied to the cross-entropy loss function, and the model checkpointing used in the FPU model is implemented here as well.
\begin{figure}[htbp]
\includegraphics[width=8.8cm,height=5.5cm]{quantized_gru_lossnacc.png}
\caption{Quantized (8 bit) Tiny Eats GRU Loss and Accuracy Training Graphs}
\label{fig:quantized_accuracy}
\end{figure}
The cross-validation accuracy and loss from both models are compared, as shown in Table \ref{tab:Acc}.
\begin{table}[htbp]
\caption{Comparison of FPU and Quantized Tiny Eats GRU Cross Validation Accuracy}
\begin{center}
\begin{tabular}{|c|c|c|c|}
\hline
\textbf{Model}&\multicolumn{2}{|c|}{\textbf{Model Designs}} \\
\cline{2-3}
\textbf{Metrics} & \textbf{\textit{FPU}}& \textbf{\textit{Quantized}} \\
\hline
Cross-Validation Accuracy & 96.13\% & 94.41\% \\
Cross-Validation Loss & 0.12 & 0.13 \\
Epochs Used & 100 & 200 \\
\hline
\end{tabular}
\label{tab:Acc}
\end{center}
\end{table}
\begin{table}[htbp]
\caption{Evaluation of Tiny Eats GRU on independent Test set vs other models}
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline
\textbf{Model}&\multicolumn{5}{|c|}{\textbf{Models Used}} \\
\cline{2-6}
\textbf{Measure} & \textbf{\textit{Tiny Eats}}& \textbf{\textit{Model in }} & \textbf{\textit{Ear Bit}}& \textbf{\textit{Splendid}}& \textbf{\textit{EGG}}\\
& & \cite{Shengjie17} & \cite{Bedri2017} & \cite{Papapanagiotou2017} & \cite{Farooq2014} \\
\hline
Model & GRU & LR & RF & SVM & FFNN \\
Test Acc. & 95.15\% & 91.5\% & 90.1\% & 93.8\% & 90.1\% \\
F1-Score & 94.89\% & NA & 90.9\% & 76.1\% & NA \\
Precision & 94.68\% & 95.1\% & 86.2\% & 79.4\% & 88.4\% \\
Recall & 95.12\% & 87.4\% & 96.1\% & 80.7\% & 91.8\% \\
\hline
\end{tabular}
\label{tab:Comparisons}
\end{center}
\end{table}
The quantized model is evaluated on an independent test set, and the test accuracy, loss, precision, recall, and F1-score are reported, as shown in Table \ref{tab:Comparisons}. The precision score represents the ratio of correctly classified positive observations of the total number of observations classified as positive. The recall score (sensitivity) represents the ratio of correctly classified positive observations to the total number of true positive observations, as shown in equation \eqref{precision} and equation \eqref{recall} where $TP$ is True Positive, $FN$ is false negative, and $FP$ is false positive.
\begin{equation}
\begin{aligned}
\ Precision &= \frac{TP}{TP + FP}\
\end{aligned}
\label{precision}
\end{equation}
\begin{equation}
\begin{aligned}
\ Recall &= \frac{TP}{TP + FN}\
\end{aligned}
\label{recall}
\end{equation}
The trained integer quantized model is then implemented on the Arm Cortex M0+.
\subsection{Arm Cortex M0+ Implementation}
\label{subsec:ArmImp}
We perform the feature extraction on the Arm Cortex M0+ by implementing a Fast Fourier Transform which utilizes only 9\% (23 KB) of the memory.
The features generated from the Arm FFT are then transmitted to the implemented Tiny Eats GRU. The implementation of the GRU on the Arm Cortex M0+ takes 6 ms to execute one sample and utilizes only 4\% (12 KB) of the memory.
The accuracy of the GRU implementation is verified by transmitting test data via serial communication and reading the output label classification. This was done with a subset of the test data set, which recorded a label match between the Pytorch quantized model and the Arm Cortex M0+ implementation, as shown in Fig. \ref{fig:quantized_output}.
\begin{figure}[htbp]
\includegraphics[width=8.8cm,height=5.2cm]{quantized_gru_output_new.png}
\caption{Quantized (8bit) Tiny Eats GRU Predicted label vs true label}
\label{fig:quantized_output}
\end{figure}
\section{Discussion}
\label{sec:Disc}
In order to account for the limitation in the size of the neural network used, advanced feature engineering is paramount. The Tiny Eats GRU utilizes STFT features shown in Section \ref{subsec:STFT} as the only class of input features. This is crucial because it can be implemented on the Arm Cortex M0+ at a low cost, as shown in Section \ref{subsec:ArmImp}, while ensuring significant accuracy is achieved. The efficient implementation of the STFT on the Arm Cortex M0+ reduces the burden on the neural network. It consequentially allows for a reduction in the number of layers and the number of neurons required for classification. This minimization of neural network parameters is resource-efficient and allows for implementation on resource-constrained micro-controllers while achieving high accuracy.
In order to reduce memory usage and run time execution performance, we utilize quantized weights instead of floating-point weights and fixed-point arithmetic. Using 8-bit quantized weights reduces the memory footprint of the network, and when coupled with fixed-point arithmetic, reduce the power consumption and increases computational speed, with minimal impact on the accuracy of the overall network, making it efficient for deployment on embedded hardware \cite{Han2016, Vanhoucke2011}. As shown in Table \ref{tab:Acc}, there is only a 2\% decrease in accuracy by switching to 8-bit integer quantization from floating-point. 8-bit weight quantizations also ensure that the Tiny Eats GRU is small enough to fit on the Arm Cortex M0+ and run efficiently, as seen in Section \ref{subsec:ArmImp}.
In order to increase computational speed, the Tiny Eats GRU uses soft-sign activations functions, as explained in Section \ref{subsec:GRU}. The results shown in Section \ref{subsec:NNTraining} affirm that the proposed Tiny Eats GRU architecture is a viable replacement for the traditional GRU \cite{Cho14} in eating episode classification. The GRU network with fast soft-sign activation functions achieves a cross-validation accuracy of 96.13\% and 94.41\% for floating-point and quantized weight implementation, respectively, as shown in Table \ref{tab:Acc}. This soft-sign implementation achieves high accuracy while minimizing computation time.
The proposed Tiny Eats GRU's performance outperforms some of the recently proposed approaches to eating detection, as shown in Table \ref{tab:Comparisons}. The accuracy, precision, recall, and F1-score of the Ear Bit Sensor \cite{Bedri2017} that uses a random forest (RF) model, the SPLENDID device \cite{Papapanagiotou2017} that uses support vector machine (SVM) classification, and the electroglottography (EGG) device \cite{Farooq2014} that utilizes feedforward neural networks (FFNN), are compared to that of the Tiny Eats GRU. The Tiny Eats GRU has better overall performance on all evaluation metrics, recording the highest F1-score of 94.89\%, as shown in Table \ref{tab:Comparisons}. These results are comparable to the model design in \cite{Shengjie17} that uses linear regression (LR).
The Tiny Eats GRU is extremely useful in resource-constrained environments like the Arm Cortex M0+, which can be used in low power wearable devices. It operates with only 6 ms latency, which is significantly smaller than the 250 ms FFT feature extraction time and uses only 4\% of the M0+ memory, which allows for the execution of other processes that could include implementing more features.
\section{Future Work}
\label{sec:Fush}
In the future, we can further improve the accuracy by increasing the number of feature classes to represent more relationships besides the STFT. Feature classes like the zero-crossing rate and number of peaks have defining characteristics that are essential in classifying {\itshape eating} and {\itshape non-eating} data. This would also imply that we could reduce the number of neurons required in the Tiny Eats GRU, and consequentially reducing its computation time and memory requirements, making it even more efficient.
Furthermore, we will implement and train this algorithm on data collected outside the lab to infer its performance in an unconstrained, free-living environment.
We will also explore classifying different food types and inferring the volume of food eaten during an eating session to gain more insight into the user's diet.
\section{Conclusion}
\label{sec:Conc}
This paper proposed the Tiny Eats GRU architecture, an implementation on Arm Cortex M0+ for the detection of eating episodes. The Tiny Eats GRU was demonstrated to have significantly low classification costs by being memory-conservative, computationally efficient with significant accuracy results while consuming minimal power. This allows for the implementation of Tiny Eats GRU on severely resource-constrained devices incapable of handling traditional deep neural networks, that can be used in wearable devices.
\section*{Acknowledgment}
This work was supported by the National Science Foundation under award numbers CNS-1565269 and CNS-1565268. We would like to thank Shengjie Bi and Professor David Kotz at Dartmouth College for their support on the Auracle Project. The views and conclusions contained in this document are those of the authors and should not be interpreted as necessarily representing the official policies, either expressed or implied, of the sponsors.
\bibliographystyle{./bibliography/IEEEtran}
|
1,108,101,564,597 | arxiv | \section{Introduction}
A morphism $\varphi : H \rightarrow G$ between two groups $H$ and $G$ is said to be \emph{elementary} if the following condition holds: for every first-order formula $\theta(x_1,\ldots,x_k)$ with $k$ free variables in the language of groups, and for every $k$-tuple $(h_1,\ldots,h_k)\in H^k$, the statement $\theta(h_1,\ldots,h_k)$ is true in $H$ if and only if the statement $\theta(\varphi(h_1),\ldots,\varphi(h_k))$ is true in $G$. In particular, $\varphi$ is injective. When $H$ is a subgroup of $G$ and $\varphi$ is the inclusion of $H$ into $G$, one says that $H$ is an \emph{elementary subgroup} of $G$. If one only considers a certain fragment $\mathcal{F}$ of the set of first-order formulas (for instance the set of $\forall\exists$-formulas, $\exists\forall\exists$-formulas or $\exists^{+}$-formulas, see paragraph \ref{logic} for definitions), one says that $\varphi$ (or $H$) is \emph{$\mathcal{F}$-elementary}.
It was proved by Sela in \cite{Sel06} and by Kharlampovich and Myasnikov in \cite{KM06} that any free factor of a non-abelian finitely generated free group is elementary. Later, Perin proved that the converse holds: if $H$ is an elementary subgroup of $F_n$, then $F_n$ splits as a free product $F_n=H\ast H'$ (see \cite{Per11}). Recently, Perin gave another proof of this result (see \cite{Per19}). More generally, Sela \cite{Sel09} and Perin \cite{Per11} described elementary subgroups of torsion-free hyperbolic groups.
Our main theorem provides a characterization of $\exists\forall\exists$-elementary subgroups of virtually free groups. Recall that a group is said to be virtually free if it has a free subgroup of finite index. In what follows, all virtually free groups are assumed to be finitely generated and non virtually cyclic (here, and in the remainder of this paper, virtually cyclic means finite or virtually $\mathbb{Z}$). In \cite{And19}, we classified virtually free groups up to $\forall\exists$-elementary equivalence, i.e.\ we gave necessary and sufficient conditions for two virtually free groups $G$ and $G'$ to have the same $\forall\exists$-theory. In this context, we introduced Definition \ref{legal} below. Recall that a non virtually cyclic subgroup $G'$ of a hyperbolic group $G$ normalizes a unique maximal finite subgroup of $G$, denoted by $E_G(G')$ (see \cite{Ol93} Proposition 1).
\begin{de}[\emph{Legal large extension}]\label{legal}Let $G$ be a non virtually cyclic hyperbolic group, and let $H$ be a subgroup of $G$. One says that $G$ is a \emph{legal large extension} of $H$ if there exists a finite subgroup $C$ of $H$ such that the normalizer $N_H(C)$ of $C$ is non virtually cyclic, the finite group $E_H(N_H(C))$ is equal to $C$, and $G$ admits the following presentation: \[G=\langle H,t \ \vert \ [t,c]=1, \ \forall c\in C\rangle.\]More generally, one says that $G$ is a \emph{multiple legal large extension} of $H$ if there exists a finite sequence of subgroups $H=G_0\subset G_1\subset \cdots \subset G_n=G$ such that $G_{i+1}$ is a legal large extension of $G_i$ for every integer $0\leq i\leq n-1$, with $n\geq 1$.
\end{de}
In terms of graphs of groups, $G$ is a multiple legal large extension of $H$ if it splits as a finite graph of groups over finite groups, whose underlying graph is a rose and whose central vertex group is $H$, with additional assumptions on the edge groups. The prototypical example of a multiple legal large extension is given by the splitting of the free group $G=F_k$ of rank $k\geq 3$ as $F_k=\langle F_2,t_1,\ldots,t_{k-2} \ \vert \ \varnothing\rangle$. In this example, $H$ is the free group $F_2$.
In \cite{And19}, we proved the following result (see Theorem 1.10).
\begin{te}\label{legalteplus}Let $G$ be a non virtually cyclic hyperbolic group, and let $H$ be a subgroup of $G$. If $G$ is a multiple legal large extension of $H$, then $H$ is $\exists\forall\exists$-elementary.\end{te}
\begin{rque}We conjectured in \cite{And19} that $H$ is elementary.
\end{rque}
We shall prove that the converse of Theorem \ref{legalteplus} holds, provided that $G$ is a virtually free group.
\begin{te}\label{1}Let $G$ be a virtually free group, and let $H$ be a proper subgroup of $G$. If $H$ is $\forall\exists$-elementary, then $G$ is a multiple legal large extension of $H$.\end{te}
\begin{rque}In particular, Theorem \ref{1} recovers the result proved by Perin in \cite{Per11}: an elementary subgroup of a free group is a free factor.\end{rque}
\begin{rque}In our classification of virtually free groups up to $\forall\exists$-elementary equivalence (see \cite{And19}), another kind of extension, called legal small extension, plays an important role. Theorem \ref{1} above shows that if $G$ is a non-trivial legal small extension of $H$, then $H$ is not an elementary subgroup of $G$. See also \cite{And19} Remark 1.15.\end{rque}
Putting together Theorem \ref{1} and Theorem \ref{legalteplus}, we get the following result.
\begin{te}\label{2dejapris}Let $G$ be a virtually free group, and let $H$ be a proper subgroup of $G$. The following assertions are equivalent:
\begin{enumerate}
\item $H$ is $\exists\forall\exists$-elementary;
\item $H$ is $\forall\exists$-elementary;
\item $G$ is a multiple legal large extension of $H$.
\end{enumerate}
\end{te}
In addition, one gives an algorithm that decides whether or not a finitely generated subgroup of a virtually free group is $\exists\forall\exists$-elementary.
\begin{te}There is an algorithm that, given a finite presentation of a virtually free group $G$ and a finite subset $X \subset G$, outputs `Yes' if the subgroup of $G$ generated by $X$ is $\exists\forall\exists$-elementary, and `No' otherwise.\end{te}
\begin{rque}Note that any $\forall\exists$-elementary subgroup of a virtually free group is finitely generated, as a consequence of Theorem \ref{1}.
\end{rque}
Recall that every virtually free group $G$ splits as a finite graph of finite groups (which is not unique), called a Stallings splitting of $G$. The following result is an immediate consequence of Theorem \ref{1}.
\begin{co}Let $G$ be a virtually free group. If the underlying graph of a Stallings splitting of $G$ is a tree, then $G$ has no proper elementary subgroup.\end{co}
For instance, the virtually free group $\mathrm{SL}_2(\mathbb{Z})$, which is isomorphic to $\mathbb{Z}/4\mathbb{Z}\ast_{\mathbb{Z}/2\mathbb{Z}}\mathbb{Z}/6\mathbb{Z}$, has no proper elementary subgroup.
\smallskip
Last, let us mention another interesting consequence of Theorem \ref{1}: if an endomorphism $\varphi$ of a virtually free group $G$ is $\forall\exists$-elementary, then $\varphi$ is an automorphism. Indeed, note that $\varphi(G)$ is a $\forall\exists$-elementary subgroup of $G$, and let us prove that $\varphi(G)=G$. Assume towards a contradiction that $\varphi(G)$ is a proper subgroup of $G$. It follows from Theorem \ref{1} that $G$ is a multiple legal large extension of $\varphi(G)$. Hence, there exists an integer $n\geq 1$ such that the abelianizations of $G$ and $\varphi(G)$ satisfy $G^{\mathrm{ab}}=\varphi(G)^{\mathrm{ab}}\times\mathbb{Z}^n$. But $\varphi(G)$ is isomorphic to $G$ since $\varphi$ is injective, hence $G^{\mathrm{ab}}\simeq G^{\mathrm{ab}}\times\mathbb{Z}^n$, which contradicts the fact that $n$ is non-zero and $G$ is finitely generated. Thus, one has $\varphi(G)=G$ and $\varphi$ is an automorphism.
In fact, the same result holds for torsion-free hyperbolic groups: if $\varphi : G \rightarrow G$ is $\forall\exists$-elementary, then $G$ is a hyperbolic tower in the sense of Sela over $\varphi(G)\simeq G$ (see \cite{Per11}). By definition of a hyperbolic tower, $\varphi(G)$ is a quotient of $G$. Since torsion-free hyperbolic groups are Hopfian by \cite{SelHopf}, $\varphi(G)=G$ and $\varphi$ is an automorphism.
We shall prove the following result, which generalizes the previous observation. Recall that a group is said to be \emph{equationally noetherian} if every infinite system of equations $\Sigma$ in finitely many variables is equivalent to a finite subsystem of $\Sigma$.
\begin{te}\label{44}Let $G$ be a finitely generated group. Suppose that $G$ is equationally noetherian, or finitely presented and Hopfian. Then, every $\exists^{+}$-endomorphism of $G$ is an automorphism.\end{te}
\begin{rque}Note that $\forall\exists$-elementary morphisms are \emph{a fortiori} $\exists^{+}$-elementary. Note also that, contrary to $\forall\exists$-elementary morphisms, $\exists^{+}$-elementary morphisms are not injective in general.\end{rque}
As a consequence, by Proposition 2 in \cite{MR18}, a finitely generated group $G$ satisfying the hypotheses of Theorem \ref{44} above is (strongly) defined by types, and even by $\exists^{+}$-types, meaning that $G$ is characterized among finitely generated groups, up to isomorphism, by the set $\mathrm{tp}_{\exists^{+}}(G)$ of all $\exists^{+}$-types of tuples of elements of $G$. In particular, Theorem \ref{44} answers positively Problem 4 posed in \cite{MR18} and recovers several results proved in \cite{MR18}.
\section{Preliminaries}
\subsection{First-order formulas}\label{logic}
A \emph{first-order formula} in the language of groups is a finite formula using the following symbols: $\forall$, $\exists$, $=$, $\wedge$, $\vee$, $\Rightarrow$, $\neq$, $1$ (standing for the identity element), ${}^{-1}$ (standing for the inverse), $\cdot$ (standing for the group multiplication) and variables $x,y,g,z\ldots$ which are to be interpreted as elements of a group. A variable is \emph{free} if it is not bound by any quantifier $\forall$ or $\exists$. A \emph{sentence} is a formula without free variables. An \emph{existential formula} (or $\exists$\emph{-formula}) is a formula in which the symbol $\forall$ does not appear. An \emph{existential positive formula} (or $\exists^{+}$\emph{-formula}) is a formula in which the symbols $\forall$ and $\neq$ do not appear. A $\forall\exists$\emph{-formula} is a formula of the form $\theta(\bm{x}):\forall\bm{y}\exists\bm{z} \ \varphi(\bm{x},\bm{y},\bm{z})$ where $\varphi(\bm{x},\bm{y},\bm{z})$ is a quantifier-free formula, i.e.\ a boolean combination of equations and inequations in the variables of the tuples $\bm{x},\bm{y},\bm{z}$. An $\exists\forall\exists$\emph{-formula} is defined in a similar way.
\subsection{Properties relative to a subgroup}
Let $G$ be a finitely generated group, and let $H$ be a subgroup of $G$.
\begin{de}\label{action of the pair}An \emph{action} of the pair $(G,H)$ on a simplicial tree $T$ is an action of $G$ on $T$ such that $H$ fixes a point of $T$. We always assume that the action is \emph{minimal}, which means that there is no proper subtree of $T$ invariant under the action of $G$. The tree $T$ (or the quotient graph of groups $T/G$, which is finite since the action is minimal) is called a \emph{splitting of} $(G,H)$, or a \emph{splitting of }$G$\emph{ relative to }$H$. The action is said to be \emph{trivial} if $G$ fixes a point of $T$.\end{de}
\begin{de}\label{one-ended relative to}We say that $G$ is \emph{one-ended relative to} $H$ if $G$ does not split as an amalgamated product $A\ast_C B$ or as an HNN extension $A\ast_C$ such that $C$ is finite and $H$ is contained in a conjugate of $A$ or $B$. In other words, $G$ is one-ended relative to $H$ if any action of the pair $(G,H)$ on a simplicial tree with finite edge stabilizers is trivial.\end{de}
\begin{de}\label{co-Hopfian relative to}The group $G$ is said to be \emph{co-Hopfian relative to} $H$ if every monomorphism $\varphi : G \hookrightarrow G$ that coincides with the identity on $H$ is an automorphism of $G$.\end{de}
The following result was first proved by Sela in \cite{Sel97} for torsion-free one-ended hyperbolic groups, with $H$ trivial.
\begin{te}[see \cite{And18b} Theorem 2.31]\label{coHopf}Let $G$ be a hyperbolic group, let $H$ be a subgroup of $G$. Assume that $G$ is one-ended relative to $H$. Then $G$ is co-Hopfian relative to $H$.\end{te}
\begin{rque}In \cite{And18b}, this result is stated and proved under the assumption that $H$ is finitely generated. However, Lemma \ref{perinlemme} below shows that this hypothesis is not necessary.\end{rque}
\subsection{Relative Stallings splittings}\label{chap2refchap3}
Let $G$ be a finitely generated group. Under the hypothesis that there exists a constant $C$ such that every finite subgroup of $G$ has order less than $C$, Linnell proved in \cite{Lin83} that $G$ splits as a finite graph of groups with finite edge groups and all of whose vertex groups are finite or one-ended. The group $G$ is virtually free if and only if all vertex groups are finite. Given a subgroup $H$ of $G$, Linnell's result can be generalized as follows: the pair $(G,H)$ splits as a finite graph of groups with finite edge groups such that each vertex group is finite or one-ended relative to a conjugate of $H$. Such a splitting is called a \emph{Stallings splitting of} $G$ \emph{relative to} $H$. Note that if $H$ is infinite, there exists a unique vertex group containing $H$, called the \emph{one-ended factor of} $G$ \emph{relative to} $H$. In particular, if $G$ is infinite hyperbolic and $H$ is $\exists$-elementary, then $H$ is infinite, because it satisfies the sentence $\exists x \ (x^{K_G!}\neq 1)$ where $K_G$ denotes the maximal order of an element of $G$ of finite order. As a consequence, in the context of Theorem \ref{1}, the one-ended factor of $G$ relative to $H$ is well-defined.
\subsection{The JSJ decomposition and the modular group}\label{25}
Let us denote by $\mathcal{Z}$ the class of groups that are either finite or virtually cyclic with infinite center. Let $G$ be a hyperbolic group, and let $H$ be a subgroup of $G$. Suppose that $G$ is one-ended relative to $H$. In \cite{GL16}, Guirardel and Levitt construct a splitting of $G$ relative to $H$ called the canonical JSJ splitting of $G$ over $\mathcal{Z}$ relative to $H$. In what follows, we refer to this decomposition as the $\mathcal{Z}$\emph{-JSJ splitting of} $G$ \emph{relative to} $H$. This tree $T$ enjoys particularly nice properties and is a powerful tool for studying the pair $(G,H)$. Before giving a description of $T$, let us recall briefly some basic facts about hyperbolic 2-dimensional orbifolds.
A compact connected 2-dimensional orbifold with boundary $\mathcal{O}$ is said to be \emph{hyperbolic} if it is equipped with a hyperbolic metric with totally geodesic boundary. It is the quotient of a closed convex subset $C\subset \mathbb{H}^2$ by a proper discontinuous group of isometries $G_{\mathcal{O}}\subset \mathrm{Isom}(\mathbb{H}^2)$. We denote by $p : C \rightarrow \mathcal{O}$ the quotient map. By definition, the orbifold fundamental group $\pi_1(\mathcal{O})$ of $\mathcal{O}$ is $G_{\mathcal{O}}$. We may also view $\mathcal{O}$ as the quotient of a compact orientable hyperbolic surface with geodesic boundary by a finite group of isometries. A point of $\mathcal{O}$ is \emph{singular} if its preimages in $C$ have non-trivial stabilizer. A \emph{mirror} is the image by $p$ of a component of the fixed point set of an orientation-reversing element of $G_{\mathcal{O}}$ in $C$. Singular points not contained in mirrors are \emph{conical points}; the stabilizer of the preimage in $\mathbb{H}^2$ of a conical point is a finite cyclic group consisting of orientation-preserving maps (rotations). The orbifold $\mathcal{O}$ is said to be \emph{conical} if it has no mirror.
\begin{de}\label{FBO222}A group $G$ is called a \emph{finite-by-orbifold group} if it is an extension \[1\rightarrow F\rightarrow G \rightarrow \pi_1(\mathcal{O})\rightarrow 1\]where $\mathcal{O}$ is a compact connected hyperbolic conical 2-orbifold, possibly with totally geodesic boundary, and $F$ is an arbitrary finite group called the \emph{fiber}. We call an \emph{extended boundary subgroup} of $G$ the preimage in $G$ of a boundary subgroup of the orbifold fundamental group $\pi_1(\mathcal{O})$ (for an indifferent choice of regular base point). We define in the same way \emph{extended conical subgroups}.\end{de}
\begin{de}\label{QH222}A vertex $v$ of a graph of groups is said to be \emph{quadratically hanging} (denoted by \emph{QH}) if its stabilizer $G_v$ is a finite-by-orbifold group $1\rightarrow F\rightarrow G \rightarrow \pi_1(\mathcal{O})\rightarrow 1$ such that $\mathcal{O}$ has non-empty boundary, and such that any incident edge group is finite or contained in an extended boundary subgroup of $G$. We also say that $G_v$ is QH.
\end{de}
\begin{de}Let $G$ be a hyperbolic group, and let $H$ be a finitely generated subgroup of $G$. Let $T$ be the $\mathcal{Z}$-JSJ decomposition of $G$ relative to $H$. A vertex group $G_v$ of $T$ is said to be \emph{rigid} if it is elliptic in every splitting of $G$ over $\mathcal{Z}$ relative to $H$.\end{de}
The following proposition is crucial (see Section 6 of \cite{GL16}, Theorem 6.5 and the paragraph below Remark 9.29). We keep the same notations as in the previous definition.
\begin{prop}\label{flexible implique rigideNash}If $G_v$ is not rigid, i.e.\ if it fails to be elliptic in some splitting of $G$ over $\mathcal{Z}$ relative to $H$, then $G_v$ is quadratically hanging.\end{prop}
Proposition \ref{JSJpropNash} below summarizes the properties of the $\mathcal{Z}$-JSJ splitting relative to $H$ that are useful in the proof of Theorem \ref{1}.
\begin{prop}\label{JSJpropNash}Let $G$ be a hyperbolic group, and let $H$ be a subgroup of $G$. Suppose that $G$ is one-ended relative to $H$. Let $T$ be its $\mathcal{Z}$-JSJ decomposition relative to $H$.
\begin{itemize}
\item[$\bullet$]The tree $T$ is bipartite: every edge joins a vertex carrying a maximal virtually cyclic group to a vertex carrying a non virtually cyclic group.
\item[$\bullet$]The action of $G$ on $T$ is acylindrical in the following strong sense: if an element $g\in G$ of infinite order fixes a segment of length $\geq 2$ in $T$, then this segment has length exactly 2 and its midpoint has virtually cyclic stabilizer.
\item Let $v$ be a vertex of $T$, and let $e,e'$ be two distinct edges incident to $v$. If $G_v$ is not virtually cyclic, then the group $\langle G_e,G_{e'}\rangle$ is not virtually cyclic.
\item[$\bullet$]If $v$ is a QH vertex of $T$, every edge group $G_e$ of an edge $e$ incident to $v$ coincides with an extended boundary subgroup of $G_v$. Moreover, given any extended boundary subgroup $B$ of $G_v$, there exists a unique incident edge $e$ such that $G_e=B$.
\item[$\bullet$]The subgroup $H$ is contained in a rigid vertex group.
\end{itemize}
\end{prop}
\begin{rque}The rigid vertex group containing $H$ may be QH.\end{rque}
\begin{de}\label{modul}Let $G$ be a hyperbolic group and let $H$ be a subgroup of $G$. Suppose that $G$ is one-ended relative to $H$. We denote by $\mathrm{Aut}_H(G)$ the subgroup of $\mathrm{Aut}(G)$ consisting of all automorphisms whose restriction to $H$ is the conjugacy by an element of $G$. The \emph{modular group} $\mathrm{Mod}_H(G)$ of $G$ relative to $H$ is the subgroup of $\mathrm{Aut}_H(G)$ consisting of all automorphisms $\sigma$ satisfying the following conditions:
\begin{itemize}
\item[$\bullet$]the restriction of $\sigma$ to each rigid or virtually cyclic vertex group of the $\mathcal{Z}$-JSJ splitting of $G$ relative to $H$ coincides with the conjugacy by an element of $G$,
\item[$\bullet$]the restriction of $\sigma$ to each finite subgroup of $G$ coincides with the conjugacy by an element of $G$,
\item[$\bullet$]$\sigma$ acts trivially on the underlying graph of the $\mathcal{Z}$-JSJ splitting relative to $H$.
\end{itemize}
\end{de}
We will need the following result.
\begin{te}\label{short}Let $G$ be a hyperbolic group, let $H$ be a subgroup of $G$ and let $U$ be the one-ended factor of $G$ relative to $H$. There exist a finite subset $F\subset U\setminus \lbrace 1\rbrace$ and a finitely generated subgroup $H'\subset H$ such that, for every non-injective homomorphism $\varphi : U \rightarrow G$ that coincides with the identity on $H'$ up to conjugation, there exists an automorphism $\sigma\in\mathrm{Mod}_H(U)$ such that $\ker(\varphi\circ\sigma)\cap F\neq \varnothing$.\end{te}
\begin{proof}This result is stated and proved in \cite{And18b} under the assumption that $H$ is finitely generated (see Theorem 2.32), in which case one can take $H'=H$. We only give a brief sketch of how the proof can be adapted if $H$ is not assumed to be finitely generated. In \cite{And18b}, the assumption that $H$ is finitely generated is only used in the proof of Proposition 2.27 in order to ensure that the group $H$ fixes a point in a certain real tree $T$ with virtually cyclic arc stabilizers (namely the tree obtained by rescaling the metric of a Cayley graph of $G$ by a given sequence of positive real numbers going to infinity). Let $\lbrace h_1,h_2,\ldots\rbrace$ be a generating set for $H$, and let $H_n$ be the subgroup of $H$ generated by $\lbrace h_1,\ldots,h_n\rbrace$. If $H$ is not finitely generated, then there exists an integer $n_0$ such that, for all $n\geq n_0$, the subgroup $H_n$ is not virtually cyclic. It follows that all $H_n$ fix the same point of $T$ for $n\geq n_0$, which proves that $H$ is elliptic in $T$. Hence, one can just take $H'=H_{n_0}$.\end{proof}
\subsection{Related homomorphisms and preretractions}\label{322}
We denote by $\mathrm{ad}(g)$ the inner automorphism $h\mapsto ghg^{-1}$.
\begin{de}[Related homomorphisms]\label{reliés222}
Let $G$ be a hyperbolic group and let $H$ be a subgroup of $G$. Assume that $G$ is one-ended relative to $H$. Let $G'$ be a group. Let $\Lambda$ be the $\mathcal{Z}$-JSJ splitting of $G$ relative to $H$. Let $\varphi$ and $\varphi'$ be two homomorphisms from $G$ to $G'$. We say that $\varphi$ and $\varphi'$ are $\mathcal{Z}$-\emph{JSJ-related} or \emph{$\Lambda$-related} if the following two conditions hold:
\begin{itemize}
\item[$\bullet$]for every vertex $v$ of $\Lambda$ such that $G_v$ is rigid or virtually cyclic, there exists an element $g_v\in G'$ such that \[{\varphi'}_{\vert G_v}=\mathrm{ad}({g_v})\circ \varphi_{\vert G_v};\]
\item[$\bullet$]for every finite subgroup $F$ of $G$, there exists an element $g\in G'$ such that \[{\varphi'}_{\vert F}=\mathrm{ad}({g})\circ \varphi_{\vert F}.\]
\end{itemize}
\end{de}
\begin{de}[Preretraction]\label{pre222}
Let $G$ be a hyperbolic group, and let $H$ be a subgroup of $G$. Assume that $G$ is one-ended relative to $H$. Let $\Lambda$ be the $\mathcal{Z}$-JSJ splitting of $G$ relative to $H$. A $\mathcal{Z}$-\emph{JSJ-preretraction} or $\Lambda$\emph{-preretraction} of $G$ is an endomorphism of $G$ that is $\Lambda$-related to the identity map. More generally, if $G$ is a subgroup of a group $G'$, a preretraction from $G$ to $G'$ is a homomorphism $\Lambda$-related to the inclusion of $G$ into $G'$. Note that a $\Lambda$-preretraction coincides with a conjugacy on $H$, since $H$ is contained in a rigid vertex group of $\Lambda$.
\end{de}
The following easy lemma shows that being $\Lambda$-related can be expressed in first-order logic. This lemma is stated and proved in \cite{And18b} (see Lemma 2.22) under the assumption that $H$ is finitely generated, but this hypothesis is not used in the proof.
\begin{lemme}\label{deltarelies222}
Let $G$ be a hyperbolic group and let $H$ be a subgroup of $G$. Assume that $G$ is one-ended relative to $H$. Let $G'$ be a group. Let $\Lambda$ be the $\mathcal{Z}$-JSJ splitting of $G$ relative to $H$. Let $\lbrace g_1,\ldots ,g_n\rbrace$ be a generating set of $G$. There exists an existential formula $\theta(x_1,\ldots , x_{2n})$ with $2n$ free variables such that, for every $\varphi,\varphi'\in \mathrm{Hom}(G,G')$, $\varphi$ and $\varphi'$ are $\Lambda$-related if and only if $G'$ satisfies $\theta (\varphi(g_1),\ldots , \varphi(g_n),\varphi'(g_1),\ldots ,\varphi'(g_n))$.
\end{lemme}
The proof of the following lemma is identical to that of Proposition 7.2 in \cite{And18}.
\begin{lemme}\label{lemmeperin2}Let $G$ be a hyperbolic group. Suppose that $G$ is one-ended relative to a subgroup $H$. Let $\Lambda$ be the $\mathcal{Z}$-JSJ splitting of $G$ relative to $H$. Let $\varphi$ be a $\Lambda$-preretraction of $G$. If $\varphi$ sends every QH vertex group of $\Lambda$ isomorphically to a conjugate of itself, then $\varphi$ is injective.
\end{lemme}
\subsection{Centered graph of groups}\label{centered}
\begin{de}[Centered graph of groups]\label{graphecentre222}A graph of groups over $\mathcal{Z}$, with at least two vertices, is said to be \emph{centered} if the following conditions hold:
\begin{itemize}
\item[$\bullet$]the underlying graph is bipartite, with a particular QH vertex $v$ such that every vertex different from $v$ is adjacent to $v$;
\item[$\bullet$]every stabilizer $G_e$ of an edge incident to $v$ coincides with an extended boundary subgroup or with an extended conical subgroup of $G_v$ (see Definition \ref{FBO222});
\item[$\bullet$]given any extended boundary subgroup $B$, there exists a unique edge $e$ incident to $v$ such that $G_e$ is conjugate to $B$ in $G_v$;
\item[$\bullet$]if an element of infinite order fixes a segment of length $\geq 2$ in the Bass-Serre tree of the splitting, then this segment has length exactly 2 and its endpoints are translates of $v$.
\end{itemize}
The vertex $v$ is called \emph{the central vertex}.
\end{de}
\begin{figure}[!h]
\includegraphics[scale=0.4]{centered.png}
\caption{A centered graph of groups. Edges with infinite stabilizer are depicted in bold.}
\end{figure}
We also need to define relatedness and preretractions in the context of centered graphs of groups.
\begin{de}[Related homomorphisms]\label{reliés2222}
\normalfont
Let $G$ and $G'$ be two groups. Let $H$ be a subgroup of $G$. Suppose that $G$ has a centered splitting $\Delta$, with central vertex $v$. Suppose that $H$ is contained in a non-central vertex of $\Delta$. Let $\varphi$ and $\varphi'$ be two homomorphisms from $G$ to $G'$. We say that $\varphi$ and $\varphi'$ are \emph{$\Delta$-related} (relative to $H$) if the following two conditions hold:
\begin{itemize}
\item[$\bullet$]for every vertex $w\neq v$, there exists an element $g_w\in G'$ such that \[{\varphi'}_{\vert G_w}=\mathrm{ad}(g_w)\circ \varphi_{\vert G_w};\]
\item[$\bullet$]for every finite subgroup $F$ of $G$, there exists an element $g\in G'$ such that \[{\varphi'}_{\vert F}=\mathrm{ad}(g)\circ \varphi_{\vert F}.\]
\end{itemize}
\end{de}
\begin{de}[Preretraction]\label{special}Let $G$ be a hyperbolic group, let $H$ be a subgroup of $G$, and let $\Delta$ be a centered splitting of $G$. Let $v$ be the central vertex of $\Delta$. Suppose that $H$ is contained in a non-central vertex of $\Delta$. An endomorphism $\varphi$ of $G$ is called a \emph{$\Delta$-preretraction} (relative to $H$) if it is $\Delta$-related to the identity of $G$ in the sense of the previous definition. A $\Delta$-preretraction is said to be \emph{non-degenerate} if it does not send $G_v$ isomorphically to a conjugate of itself.\end{de}
\section{Elementary subgroups of virtually free groups}
In this section, we prove Theorem \ref{1}. Recall that this theorem claims that if $G$ is a virtually free group and $H$ is a proper $\forall\exists$-elementary subgroup of $G$, then $G$ is a multiple legal large extension of $H$.
\subsection{Elementary subgroups are one-ended factors}
As a first step, we will prove the following result.
\begin{prop}\label{256}Let $G$ be a virtually free group. Let $H$ be a subgroup of $G$. If $H$ is $\forall\exists$-elementary, then $H$ coincides with the one-ended factor of $G$ relative to $H$. In other words, $H$ appears as a vertex group in a splitting of $G$ over finite groups.\end{prop}
The proof of Proposition \ref{256}, which is inspired from \cite{Per11}, consists in showing that if $G$ is a hyperbolic group and $H$ is strictly contained in the one-ended factor of $G$ relative to $H$, then there exists a centered splitting $\Delta$ of $G$ relative to $H$, and a non-degenerate $\Delta$-preretraction of $G$ (see Lemmas \ref{2} and \ref{lemmelemme} below). However, if $G$ is virtually free, Lemma \ref{cyclic2} below shows that such a preretraction cannot exist.
We shall prove Proposition \ref{256} after establishing a series of preliminary lemmas. The following result is a generalization of Lemma 4.20 in \cite{Per11}. Recall that all group actions on trees considered in this paper are assumed to be minimal (see Definition \ref{action of the pair}). As a consequence, trees have no vertex of valence 1. We say that a tree $T$ endowed with an action of a group $G$ is \emph{non-redundant} if there exists no valence 2 vertex $v$ such that both boundary monomorphisms into the vertex group $G_v$ are isomorphisms.
\begin{lemme}\label{perinlemme}Let $G$ be a finitely generated group, and let $H$ be a subgroup of $G$. Suppose that $G$ is one-ended relative to $H$ and that there is a constant $C$ such that every finite subgroup of $G$ has order at most $C$. Then there exists a finitely generated subgroup $H''$ of $H$ such that $G$ is one-ended relative to $H''$.
\end{lemme}
\begin{proof}
Let $\lbrace h_1,h_2,\ldots \rbrace$ be a generating set for $H$, possibly infinite. For every integer $n\geq 1$, let $H_n$ be the subgroup of $H$ generated by $\lbrace h_1,\ldots,h_n \rbrace$. By Theorem 1 in \cite{Wei12}, there is a maximum number $m_n$ of orbits of edges in a non-redundant splitting of $G$ relative to $H_n$ over finite groups. Let $T_n$ be such a splitting with $m_n$ orbits of edges, and let $G_n$ be the vertex group of $T_n$ containing $H_n$.
We shall prove that $G_{n+1}$ is contained in $G_n$ for all $n$ sufficiently large. First, note that the sequence of integers $(m_n)_{n\in\mathbb{N}}$ is non-increasing, because $T_{n+1}$ is a splitting of $G$ relative to $H_n$. In particular, there exists an integer $n_0$ such that $m_n=m_{n+1}$ for every $n\geq n_0$. We claim that $G_{n+1}$ is elliptic in $T_n$. Otherwise, there exists a non-trivial splitting $G_{n+1}=A\ast_CB$ or $G_{n+1}=A\ast_C$ with $C$ finite and $H_n\subset A$, and one gets a non-redundant splitting of $G$ relative to $H_n$ over finite groups with $m_{n+1}+1=m_n+1$ edges by replacing the vertex group $G_{n+1}$ of the graph of groups $T_{n+1}/G$ with the previous one-edge splitting of $G_{n+1}$, which contradicts the definition of $m_n$.
Hence, for $n\geq n_0$, one has $G_{n}\subset G_{n_0}$. In particular, $G_{n_0}$ contains $H_n$ for every integer $n$. Thus, $G_{n_0}$ contains $H$. Since $G$ is assumed to be one-ended relative to $H$, one has $G=G_{n_0}$ and one can take $H''=H_{n_0}$.
\end{proof}
We will need the following well-known result in the proof of Lemma \ref{2} below.
\begin{prop}[\cite{Bow98}, Proposition 1.2]\label{bowNash}If a hyperbolic group splits over quasi-convex subgroups, then every vertex group is quasi-convex (hence hyperbolic).
\end{prop}
\begin{lemme}\label{2}Let $G$ be a hyperbolic group. Let $H$ be a $\forall\exists$-elementary subgroup of $G$. Let $U$ be the one-ended factor of $G$ relative to $H$. Let $\Lambda$ be the $\mathcal{Z}$-JSJ splitting of $U$ relative to $H$. If $H$ is strictly contained in $U$, then there exists a non-injective $\Lambda$-preretraction $U\rightarrow G$.\end{lemme}
\begin{proof}Let $H'$ be the finitely generated subgroup of $H$ given by Theorem \ref{short} and let $H''$ be the finitely generated subgroup of $H$ given by Lemma \ref{perinlemme} above. Let $H_0$ be the finitely generated subgroup of $H$ generated by $H'\cup H''$.
Let us prove that every morphism $\varphi:U\rightarrow H$ whose restriction to $H_0$ coincides with the identity is non-injective. First, note that $U$ is one-ended relative to $H_0$ (since it is one-ended related to $H''$ which is contained in $H_0$), and that $U$ is hyperbolic by Proposition \ref{bowNash} above. Therefore, by Theorem \ref{coHopf}, $U$ is co-Hopfian relative to $H_0$. Hence, a putative monomorphism $\varphi:U\rightarrow H\subset U$ whose restriction to $H_0$ coincides with the identity is surjective, viewed as an endomorphism of $U$. But $\varphi(U)$ is contained in $H$, which shows that $U=\varphi(U)$ is contained in $H$. It's a contradiction since $H$ is stricly contained in $U$, by assumption.
We proved in the previous paragraph that every morphism $\varphi:U\rightarrow H$ whose restriction to $H_0$ coincides with the identity is non-injective. Therefore, by Theorem \ref{short}, for every morphism $\varphi:U\rightarrow H$ whose restriction to $H_0$ (which contains $H'$) coincides with the identity, there exists an automorphism $\sigma\in\mathrm{Mod}_{H}(U)$ such that $\varphi\circ \sigma$ kills an element of the finite set $F\subset U\setminus \lbrace 1\rbrace$ given by Theorem \ref{short}. In addition, note that the morphisms $\varphi\circ \sigma$ and $\varphi$ are $\Lambda$-related (see Definition \ref{reliés2222}). Hence, for every morphism $\varphi:U\rightarrow H$ whose restriction to $H_0$ coincides with the identity, there exists a morphism $\varphi':U\rightarrow H$ that kills an element of the finite set $F$, and which is $\Lambda$-related to $\varphi$. We will see that this statement $(\star)$ is expressible by means of a $\forall\exists$-sentence with constants in $H$.
Let $U=\langle u_1,\ldots,u_n \ \vert \ R(u_1,\ldots,u_n)=1\rangle$ be a finite presentation of $U$. Let $\lbrace h_1,\ldots,h_p\rbrace$ be a finite generating set for $H_0$. For every $1\leq i\leq p$, the element $h_i$ can be written as a word $w_i(u_1,\ldots,u_n)$. Likewise, one can write $F=\lbrace v_1(u_1,\ldots,u_n),\ldots,v_k(u_1,\ldots,u_n)\rbrace$.
Observe that there is a one-to-one correspondence between the set of homomorphisms $\mathrm{Hom}(U,H)$ and the set of solutions in $H^n$ of the system of equations $R(x_1,\ldots,x_n)=1$. The group $H$ satisfies the following $\forall\exists$-formula, expressing the statement $(\star)$:
\begin{small}
\begin{align*}
&\mu(h_1,\ldots,h_p): \ \forall x_1\ldots\forall x_n \ \left(R(x_1,\ldots,x_n)=1 \ \wedge \ \bigwedge_{i=1}^p w_i(x_1,\ldots,x_n)=h_i\right)\\&\Rightarrow \left(\exists x'_1\ldots\exists x'_n \ R(x'_1,\ldots,x'_n)=1 \ \wedge \ \theta(x_1,\ldots,x_n,x'_1,\ldots,x'_n)=1 \wedge \bigvee_{i=1}^k v_i(x'_1,\ldots,x'_n)= 1 \right)
\end{align*}
\end{small}
where $\theta$ is the formula given by Lemma \ref{deltarelies222}, expressing that the homomorphisms $\varphi$ and $\varphi'$ defined by $h_i\mapsto x_i$ and $\varphi' : h_i \mapsto x'_i$ are $\Lambda$-related, where $\Lambda$ denotes the $\mathcal{Z}$-JSJ splitting of $U$ relative to $H$.
Since $H$ is $\forall\exists$-elementary (as a subgroup of $G$), the group $G$ satisfies $\mu(h_1,\ldots,h_p)$ as well. For $x_i=u_i$ for $1\leq i\leq p$, the interpretation of $\mu(h_1,\ldots,h_p)$ in $G$ provides a tuple $(g_1,\ldots,g_n)\in G^n$ such that the application $p : U \rightarrow G$ defined by $u_i \mapsto g_i$ for every $1\leq i\leq p$ is a homomorphism, is $\Lambda$-related to the inclusion of $U$ into $G$ (see Definition \ref{reliés2222}), and kills an element of $F$. As a conclusion, $p$ is a non-injective $\Lambda$-preretraction from $U$ to $G$ (see Definition \ref{pre222}).\end{proof}
The following easy lemma is proved in \cite{And18b} (see Lemma 4.5).
\begin{lemme}\label{petitlemme}
Let $G$ be a group endowed with a splitting over finite groups. Let $T_G$ denote the Bass-Serre tree associated with this splitting. Let $U$ be a group endowed with a splitting over infinite groups, and let $T_U$ be the associated Bass-Serre tree. If $p:U\rightarrow G$ is a homomorphism injective on edge groups of $T_U$, and such that $p(U_v)$ is elliptic in $T_G$ for every vertex $v$ of $T_U$, then $p(U)$ is elliptic in $T_G$.
\end{lemme}
\begin{lemme}\label{lemmelemme}Let $G$ be a hyperbolic group. Let $H$ be a subgroup of $G$. Let $U$ be the one-ended factor of $G$ relative to $H$. Let $\Lambda$ be the $\mathcal{Z}$-JSJ splitting of $G$ relative to $H$. Suppose that there exists a non-injective $\Lambda$-preretraction $p:U\rightarrow G$. Then there exists a centered splitting of $G$ relative to $H$, called $\Delta$, and a non-degenerate $\Delta$-preretraction of $G$.
\end{lemme}
\begin{proof}First, we will prove that there exists a QH vertex $x$ of $\Lambda$ such that $U_x$ is not sent isomorphically to a conjugate of itself by $p$. Assume towards a contradiction that this claim is false, i.e.\ that each stabilizer $U_x$ of a QH vertex $x$ of $\Lambda$ is sent isomorphically to a conjugate of itself by $p$. We claim that $p(U)$ is contained in a conjugate of $U$. Let $\Gamma$ be a Stallings splitting of $G$ relative to $H$. By definition of $U$, there exists a vertex $u$ of the Bass-Serre tree $T$ of $\Gamma$ such that $G_u=U$. First, let us check that the hypotheses of Lemma \ref{petitlemme} are satisfied:
\begin{enumerate}
\item by definition, $\Gamma$ is a splitting of $G$ over finite groups, and $\Lambda$ is a splitting of $U$ over infinite groups;
\item $p$ is injective on edge groups of $\Lambda$ (as a $\Lambda$-preretraction);
\item if $x$ is a QH vertex of $\Lambda$, then $p(U_x)$ is conjugate to $U_x$ by assumption. In particular, $p(U_x)$ is contained in a conjugate of $U$ in $G$. As a consequence, $p(U_x)$ is elliptic in $T$ (more precisely, it fixes a translate of the vertex $u$ of $T$ such that $G_u=U$). If $x$ is a non-QH vertex of $\Lambda$, then $p(U_x)$ is conjugate to $U_x$ by definition of a $\Lambda$-preretraction. In particular, $p(U_x)$ is elliptic in $T$.
\end{enumerate}
By Lemma \ref{petitlemme}, $p(U)$ is elliptic in $T$. It remains to prove that $p(U)$ is contained in a conjugate of $U$. Observe that $U$ is not finite-by-(closed orbifold), as a virtually free group. Therefore, there exists at least one non-QH vertex $x$ in $\Lambda$. Moreover, since $p$ is inner on non-QH vertices of $\Lambda$, there exists an element $g\in G$ such that $p(U_x)=gU_xg^{-1}$. Hence, $p(U)\cap gUg^{-1}$ is infinite, which proves that $p(U)$ is contained in $gUg^{-1}$ since edge groups of the Bass-Serre tree $T$ of $\Gamma$ are finite.
Now, up to composing $p$ by the conjugation by $g^{-1}$, one can assume that $p$ is an endomorphism of $U$. By Lemma \ref{lemmeperin2}, $p$ is injective. This is a contradiction since $p$ is non-injective by hypothesis. Hence, we have proved that there exists a QH vertex $x$ of $\Lambda$ such that $U_x$ is not sent isomorphically to a conjugate of itself by $p$.
Then, we refine $\Gamma$ by replacing the vertex $u$ fixed by $U$ by the $\mathcal{Z}$-JSJ splitting $\Lambda$ of $U$ relative to $H$ (which is possible since edge groups of $\Gamma$ adjacent to $u$ are finite, ans thus are elliptic in $\Lambda$). With a little abuse of notation, we still denote by $x$ the vertex of $\Gamma$ corresponding to the QH vertex $x$ of $\Lambda$. Then, we collapse to a point every connected component of the complement of $\mathrm{star}(x)$ in $\Gamma$ (where $\mathrm{star}(x)$ stands for the subgraph of $\Gamma$ constituted of $x$ and all its incident edges). The resulting graph of groups, denoted by $\Delta$, is non-trivial. One easily sees that $\Delta$ is a centered splitting of $G$, with central vertex $x$.
The homomorphism $p:U\rightarrow G$ is well-defined on $G_x$ because $G_x=U_x$ is contained in $U$. Moreover, $p$ restricts to a conjugation on each stabilizer of an edge $e$ of $\Delta$ incident to $x$. Indeed, either $e$ is an edge coming from $\Lambda$, either $G_e$ is a finite subgroup of $U$; in each case, $p_{\vert G_e}$ is a conjugation since $p$ is $\Lambda$-related to the inclusion of $U$ into $G$. Now, one can define an endomorphism $\varphi : G \rightarrow G$ that coincides with $p$ on $G_x=U_x$ and coincides with a conjugation on every vertex group $G_y$ of $\Gamma$, with $y\neq x$. By induction on the number of edges of $\Gamma$, it is enough to define $\varphi$ in the case where $\Gamma$ has only one edge. If $G=U_x\ast_C B$ with $p_{\vert C}=\mathrm{ad}(g)$, one defines $\varphi : G \rightarrow G$ by $\varphi_{\vert U_x}=p$ and $\varphi_{\vert B}=\mathrm{ad}(g)$. If $G=U_x\ast_C=\langle U_x,t \ \vert \ tct^{-1}=\alpha(c), \forall c\in C\rangle$ with $p_{\vert C}=\mathrm{ad}(g_1)$ and $p_{\vert \alpha{(C)}}=\mathrm{ad}(g_2)$, one defines $\varphi : G \rightarrow G$ by $\varphi_{\vert U_x}=p$ and $\varphi(t)=g_2^{-1}tg_1$.
The endomorphism $\varphi$ defined above is $\Delta$-related to the identity of $G$ (in the sense of Definition \ref{reliés2222}), and $\varphi$ does not send $G_x$ isomorphically to a conjugate of itself. Hence, $\varphi$ is a non-degenerate $\Delta$-preretraction of $G$ (see Definition \ref{special}).\end{proof}
The following result is proved in \cite{And18b} (Lemma 4.4).
\begin{lemme}\label{cyclic2}Let $G$ be a virtually free group, and let $\Delta$ be a centered splitting of $G$. Then $G$ has no non-degenerate $\Delta$-preretraction.\end{lemme}
We can now prove Proposition \ref{256}.
\medskip
\begin{proofPROP}Let $U$ be the one-ended factor of $G$ relative to $H$. Assume towards a contradiction that $H$ is strictly contained in $U$. Then by Proposition \ref{2}, there exists a non-injective preretraction $U\rightarrow G$ (with respect to the $\mathcal{Z}$-JSJ splitting of $U$ relative to $H$). By Lemma \ref{lemmelemme}, there exists a centered splitting $\Delta$ of $G$ relative to $H$ such that $G$ has a non-degenerate $\Delta$-preretraction. This contradicts Lemma \ref{cyclic2}. Hence, $H$ is equal to $U$.\end{proofPROP}
\subsection{Proof of Theorem \ref{1}}
Recall that Theorem \ref{1} claims that if $H$ is a $\forall\exists$-elementary proper subgroup of a virtually free group $G$, then $G$ is a multiple legal large extension of $H$. Before proving this result, we will define five numbers associated with a hyperbolic group, which are encoded into its $\forall\exists$-theory (see Lemma \ref{5inv} below).
\begin{de}\label{5nombres}Let $G$ be a hyperbolic group. We associate to $G$ the following five integers:
\begin{itemize}
\item[$\bullet$]the number $n_1(G)$ of conjugacy classes of finite subgroups of $G$,
\item[$\bullet$]the sum $n_2(G)$ of $\vert \mathrm{Aut}_G(C_k)\vert$ for $1\leq k\leq n_1(G)$, where the $C_k$ are representatives of the conjugacy classes of finite subgroups of $G$, and \[\mathrm{Aut}_G(C_k)=\lbrace \alpha\in \mathrm{Aut}(C_k) \ \vert \ \exists g\in N_G(C_k), \ \mathrm{ad}(g)_{\vert C}=\alpha\rbrace,\]
\item[$\bullet$]the number $n_3(G)$ of conjugacy classes of finite subgroups $C$ of $G$ such that $N_G(C)$ is infinite virtually cyclic,
\item[$\bullet$]the number $n_4(G)$ of conjugacy classes of finite subgroups $C$ of $G$ such that $N_G(C)$ is not virtually cyclic (finite or infinite),
\item[$\bullet$]the number $n_5(G)$ of conjugacy classes of finite subgroups $C$ of $G$ such that $N_G(C)$ is not virtually cyclic (finite or infinite) and $E_G(N_G(C))\neq C$.
\end{itemize}
\end{de}
The following lemma shows that these five numbers are preserved under $\forall\exists$-equivalence. Its proof is quite straightforward and is postponed after the proof of Theorem \ref{1}.
\begin{lemme}\label{5inv}Let $G$ and $G'$ be two hyperbolic groups. Suppose that $\mathrm{Th}_{\forall\exists}(G)=\mathrm{Th}_{\forall\exists}(G')$. Then $n_i(G)=n_i(G')$, for $1\leq i\leq 5$.\end{lemme}
Theorem \ref{1} will be an easy consequence of the following result.
\begin{prop}\label{concon}Let $G$ be a virtually free group. Let $H$ be a proper subgroup of $G$. Suppose that the following three conditions are satisfied:
\begin{enumerate}
\item $n_i(H)=n_i(G)$ for all $1\leq i\leq 5$,
\item $H$ appears as a vertex group in a splitting of $G$ over finite groups,
\item two finite subgroups of $H$ are conjugate in $H$ if and only if they are conjugate in $G$.
\end{enumerate}Then $G$ is a multiple legal large extension of $H$ (see Definition \ref{legal}).
\end{prop}
\begin{proof}First, note that the equality $n_4(G)=n_4(G)$ implies that $H$ is non virtually cyclic. Indeed, if $H$ is virtually cyclic, then $n_4(H)=0$, whereas $n_4(G)$ is greater than $1$ since $N_G(\lbrace 1\rbrace)=G$ is not virtually cyclic by assumption.
Let $T$ be the Bass-Serre tree of the splitting of $G$ given by the second condition. Up to refining this splitting, one can assume without loss of generality that the vertex groups of $T$ which are not conjugate to $H$ are finite. In other words, $T$ is a Stallings splitting of $G$ relative to $H$, in which $H$ is a vertex group by assumption. Moreover, up to collasping some edges, one can assume that $T$ is \emph{reduced}, which means that if $e=[v,w]$ is an edge of $T$ such that $G_e=G_v=G_w$, then $v$ and $w$ are in the same orbit. We denote by $\Gamma$ the quotient graph of groups $T/G$.
We will deduce from the third condition that the underlying graph of $\Gamma$ has only one vertex. Assume towards a contradiction that the Bass-Serre tree $T$ of $\Gamma$ has at least two orbits of vertices. Hence, there is a vertex $v$ of $T$ which is not in the orbit of the vertex $v_H$ fixed by $H$. By definition of $\Gamma$, the vertex stabilizer $G_v$ is finite. Thus, there exists an element $g\in G$ such that $gG_vg^{-1}$ is contained in $H$. Therefore, $G_v$ stabilizes the path of edges in $T$ between the vertices $v$ and $g^{-1}v_H$. It follows that $G_v$ coincides with the stabilizer of an edge incident to $v$ in $T$, which contradicts the assumption that $T$ is reduced.
Hence, the underlying graph of $\Gamma$ is a rose, and the central vertex group of $\Gamma$ is $H$. Moreover, edge stabilizers of $\Gamma$ are finite. In other words, there exist pairs of finite subgroups $(C_1,C'_1),\ldots,(C_n,C'_n)$ of $H$, together with automorphisms $\alpha_1\in\mathrm{Isom}(C_1,C'_1),\ldots,\alpha_n\in\mathrm{Isom}(C_n,C'_n)$ such that $G$ has the following presentation:\[G=\langle H,t_1,\ldots,t_n \ \vert \ \mathrm{ad}(t_i)_{\vert C_i}=\alpha_i, \ \forall i\in\llbracket 1,n\rrbracket\rangle.\]
By assumption, the integers $n_i(G)$ and $n_i(H)$ are equal, for $1\leq i\leq 5$. From the equality $n_1(G)=n_1(H)$, one deduces immediately that the finite groups $C_i$ and $C'_i$ are conjugate in $H$ for every integer $i\in \lbrace 1,\ldots ,n\rbrace$. Therefore, one can assume without loss of generality that $C'_i=C_i$.
Note that for every finite subgroup $C$ of $H$, the group $ \mathrm{Aut}_{H}(C)$ is contained in $\mathrm{Aut}_{G}(C)$. Thus, the equality $n_2(G)=n_2(H)$ guarantees that $\mathrm{Aut}_{H}(C_i)$ is in fact equal to $\mathrm{Aut}_{G}(C_i)$, for every $1\leq i\leq n$. Hence, since the automorphism $\mathrm{ad}(t_i)_{\vert C_i}$ of $C_i$ belongs to $\mathrm{Aut}_{G}(C_i)$, there exists an element $h_i\in N_H(C_i)$ such that $\mathrm{ad}(h_i)_{\vert C_i}=\mathrm{ad}(t_i)_{\vert C_i}$. Up to replacing $t_i$ with $t_ih_i^{-1}$, the group $G$ has the following presentation:\[G=\langle H,t_1,\ldots,t_n \ \vert \ \mathrm{ad}(t_i)_{\vert C_i}=\mathrm{id}_{C_i}, \ \forall i\in\llbracket 1,n\rrbracket\rangle.\]
In order to prove that $G$ is a multiple legal large extension of $H$ (see Definition \ref{legal}), it remains to prove that the following two conditions hold, for every integer $1\leq i\leq n$:
\begin{enumerate}
\item the normalizer $N_H(C_i)$ is non virtually cyclic (finite or infinite),
\item and the finite group $E_H(N_H(C_i))$ coincides with $C_i$.
\end{enumerate}
The equalities $n_3(G)=n_3(H)$ and $n_4(G)=n_4(H)$ ensure that $N_H(C_i)$ is not virtually cyclic. Indeed, if $N_H(C_i)$ were finite, then $N_{G}(C_i)$ would be infinite virtually cyclic and $n_3(G)$ would be at least $n_3(H)+1$; similarly, if $N_H(C_i)$ were infinite virtually cyclic, then $N_{G}(C_i)$ would be non virtually cyclic and $n_4(G)\geq n_4(H)+1$. Hence, the first condition above is satisfied.
Last, it follows from the equality $n_5(G)=n_5(H)$ that the finite group $E_H(N_H(C_i))$ coincides with $C_i$, otherwise $n_5(G)\geq n_5(H)+1$, since $E_{G}(N_{G}(C_i))=C_i$. Thus, the second condition above holds. As a conclusion, $G$ is a multiple legal large extension of $H$ in the sense of Definition \ref{legal}.
\end{proof}
We can now prove Theorem \ref{1}.
\medskip
\begin{proofTH1}Let $G$ be a virtually free group, and let $H$ be a $\forall\exists$-elementary subgroup of $G$. In particular, $G$ and $H$ have the same $\forall\exists$-theory. It follows from Lemma \ref{5inv} that $n_i(H)$ is equal to $n_i(G)$ for all $1\leq i\leq 5$. Hence, the first condition of Proposition \ref{concon} holds.
By Proposition \ref{256}, $H$ is a vertex group in a splitting of $G$ over finite groups, which means that the second condition of Proposition \ref{concon} is satisfied.
It remains to check the third condition of Proposition \ref{concon}, namely that two finite subgroups of $H$ are conjugate in $H$ if and only if they are conjugate in $G$. First, recall that $H$ and $G$ have the same number of conjugacy classes of finite subgroups, since $n_1(G)=n_1(H)$. Then, the conclusion follows from the following observation: if two finite subgroups $A=\lbrace a_1,\ldots ,a_m\rbrace$ and $B=\lbrace b_1,\ldots,b_m\rbrace$ of $H$ are not conjugate in $H$, then they are not conjugate in $G$. Indeed, $H$ satisfies the following universal formula: \[\theta(a_1,\ldots,a_m,b_1,\ldots,b_m):\forall x \ \bigvee_{i=1}^m\bigwedge_{j=1}^m xa_ix^{-1}\neq b_j.\]Since $H$ is $\forall\exists$-elementary (in particular $\forall$-elementary), $G$ satisfies this sentence as well. Therefore, $A$ and $B$ are not conjugate in $G$.\end{proofTH1}
It remains to prove Lemma \ref{5inv}. First, recall that if $G$ is hyperbolic and $g\in G$ has infinite order, there is a unique maximal virtually cyclic subgroup of $G$ containing $g$, denoted by $M(g)$. More precisely, $M(g)$ is the stabilizer of the pair of fixed points of $g$ on the boundary $\partial_{\infty} G$ of $G$. If $h$ and $g$ are two elements of infinite order, either $M(h)=M(g)$ or the intersection $M(h)\cap M(g)$ is finite; in the latter case, the subgroup $\langle h,g\rangle$ is not virtually cyclic. Let $K_G$ denote the maximum order of an element of $G$ of finite order. One can see that an element $g\in G$ has infinite order if and only if $g^{K_G!}$ is non-trivial, and that if $g$ and $h$ have infinite order, then $M(g)=M(h)$ if and only if the commutator $[g^{K!},h^{K!}]$ is trivial. In other words, the subgroup $\langle g,h\rangle$ is virtually cyclic if and only if $[g^{K!},h^{K!}]=1$.
\medskip
\begin{proofL}Let us denote by $K_G$ the maximal order of a finite subgroup of $G$. Since $G$ and $G'$ have the same existential theory, we have $K_G=K_{G'}$. Let $n\geq 1$ be an integer. If $n_1(G)\geq n$, then the following $\exists\forall$-sentence, written in natural language for convenience of the reader and denoted by $\theta_{1,n}$, is satisfied by $G$: there exist $n$ finite subgroups $C_1,\ldots ,C_n$ of $G$ such that, for every $g\in G$ and $1\leq i\neq j\leq n$, the groups $gC_ig^{-1}$ and $C_j$ are distinct. Since $G$ and $G'$ have the same $\exists\forall$-theory, the sentence $\theta_{1,n}$ is satisfied by $G'$ as well. As a consequence, $n_1(G')\geq n$. It follows that $n_1(G')\geq n_1(G)$. By symmetry, we have $n_1(G)=n_1(G')$.
\smallskip
In the rest of the proof, we give similar sentences $\theta_{2,n},\ldots,\theta_{5,n}$ such that the following series of equivalences hold: $n_i(G)\geq n \Leftrightarrow G$ satisfies $\theta_{i,n} \Leftrightarrow$ $G'$ satisfies $\theta_{i,n}\Leftrightarrow n_i(G')\geq n$.
\smallskip
One has $n_2(G)\geq n$ if and only if $G$ satisfies the following $\exists\forall$-sentence $\theta_{2,n}$: there exist $\ell$ finite subgroups $C_1,\ldots ,C_{\ell}$ of $G$ and, for every $1\leq i\leq \ell$, a finite subset $\lbrace g_{i,j}\rbrace_{1\leq j\leq n_i}$ of $N_G(C_i)$ such that:
\begin{itemize}
\item[$\bullet$]for every $g\in G$ and $1\leq i\neq j\leq n$, the groups $gC_ig^{-1}$ and $C_j$ are distinct;
\item[$\bullet$]the sum $n_1+\cdots+n_{\ell}$ is equal to $n$;
\item[$\bullet$]for every $1\leq i\leq \ell$, and for every $1\leq j\neq k\leq n_i$, the automorphisms $\mathrm{ad}(g_j)_{\vert C_i}$ and $\mathrm{ad}(g_k)_{\vert C_i}$ of $C_i$ are distinct.
\end{itemize}
\smallskip
One has $n_3(G)\geq n$ if and only if $G$ satisfies the following $\exists\forall$-sentence $\theta_{3,n}$: there exist $n$ finite subgroups $C_1,\ldots ,C_n$ of $G$ and $n$ elements $g_1\in N_G(C_1),\ldots,g_n\in N_G(C_n)$ of infinite order (i.e.\ satisfying $g_i^{K_G!}\neq 1$) such that:
\begin{itemize}
\item[$\bullet$]for every $g\in G$ and $1\leq i\neq j\leq n$, the groups $gC_ig^{-1}$ and $C_j$ are distinct;
\item[$\bullet$]for every $1\leq i\leq n$ and $g\in N_G(C_i)$, the subgroup $\langle g,g_i\rangle$ of $N_G(C_i)$ is virtually cyclic, i.e.\ $[g^{K_G!},g_i^{K_G!}]=1$.
\end{itemize}
\smallskip
One has $n_4(G)\geq n$ if and only if $G$ satisfies the following $\exists\forall$-sentence $\theta_{4,n}$: there exist $n$ finite subgroups $C_1,\ldots ,C_n$ of $G$ and, for every $1\leq i\leq n$, a couple of elements $(g_{i,1},g_{i,2})$ normalizing $C_i$ such that:
\begin{itemize}
\item[$\bullet$]for every $g\in G$ and $1\leq i\neq j\leq n$, the groups $gC_ig^{-1}$ and $C_j$ are distinct;
\item[$\bullet$]for every $1\leq i\leq n$, the subgroup $\langle g_{i,1},g_{i,2}\rangle$ is not virtually cyclic (i.e.\ $[g_{i,1}^{K_G!},g_{i,2}^{K_G!}]$ is non-trivial).
\end{itemize}
\smallskip
One has $n_5(G)\geq n$ if and only if $G$ satisfies the following $\exists\forall$-sentence $\theta_{5,n}$: there exist $2n$ finite subgroups $C_1,\ldots ,C_n$ and $C'_1\varsupsetneq C_1,\ldots,C'_n\varsupsetneq C_n$ of $G$ and, for every $1\leq i\leq n$, a couple of elements $(g_{i,1},g_{i,2})$ normalizing $C_i$, such that:
\begin{itemize}
\item[$\bullet$]for every $g\in G$ and $1\leq i\neq j\leq n$, the groups $gC_ig^{-1}$ and $C_j$ are distinct;
\item[$\bullet$]for every $1\leq i\leq n$, the subgroup $\langle g_{i,1},g_{i,2}\rangle$ is not virtually cyclic;
\item[$\bullet$]every element of $G$ that normalizes $C_i$ also normalizes $C'_i$.
\end{itemize}\end{proofL}
\section{Algorithm}
In this section, we shall prove the following theorem.
\begin{te}\label{algoalgoalgoalgo}There is an algorithm that, given a finite presentation of a virtually free group $G$ and a finite subset $X \subset G$, outputs `Yes' if the subgroup of $G$ generated by $X$ is $\exists\forall\exists$-elementary, and `No' otherwise.\end{te}
We shall use the following fact.
\begin{lemme}\label{lemmecomp}A subgroup $H$ of $G$ is $\exists\forall\exists$-elementary if and only if the three conditions of Proposition \ref{concon} are satisfied.\end{lemme}
\begin{proof}
If the conditions of Proposition \ref{concon} are satisfied, then either $H=G$, or $H$ is a proper subgroup and $G$ is a multiple legal large extension of $H$, by Proposition \ref{concon}. In both cases, the subgroup $H$ is $\exists\forall\exists$-elementary by Theorem \ref{legalteplus}. Conversely, if $H$ is $\exists\forall\exists$-elementary, then either $H=G$ or $H$ is a proper subgroup of $G$ and $G$ is a multiple legal large extension of $H$, by Theorem \ref{1}.\end{proof}
The proof of Theorem \ref{algoalgoalgoalgo} consists in showing that the conditions of Proposition \ref{concon} can be decided by an algorithm.
\subsection{Algorithmic tools} First, we collect several algorithms that will be useful in the proof of Theorem \ref{algoalgoalgoalgo}.
\subsubsection{Solving equations in hyperbolic groups}The following theorem is the main result of \cite{DG10}.
\begin{te}\label{DG10}There exists an algorithm that takes as input a finite presentation of a hyperbolic group $G$ and a finite system of equations and inequations with constants in $G$, and decides whether there exists a solution or not.
\end{te}
\subsubsection{Computing a finite presentation of a subgroup given by generators}The following result is a particular case of Theorem 20 in \cite{BM16}.
\begin{te}\label{compu}There is an algorithm that, given a finite presentation of a hyperbolic and locally quasiconvex group $G$, and a finite subset $X$ of $G$, produces a finite presentation for the subgroup of $G$ generated by $X$.
\end{te}
Recall that a group is said to be \emph{locally quasiconvex} if every finitely generated subgroup is quasiconvex. Marshall Hall Jr.\ proved in \cite{Mar49} that every finitely generated subgroup of a finitely generated free group is a free factor in a finite-index subgroup, which shows in particular that finitely generated free groups are locally quasiconvex. It follows easily that finitely generated virtually free groups are locally quasiconvex. Thus, Theorem \ref{compu} applies when $G$ is virtually free.
\subsubsection{Basic algorithms}
\begin{lemme}\label{algofini}There is an algorithm that takes as input a finite presentation of a hyperbolic group and computes a list of representatives of the conjugacy classes of finite subgroups in this hyperbolic group.\end{lemme}
\begin{proof}There exists an algorithm that computes, given a finite presentation $\langle S \ \vert \ R\rangle$ of a hyperbolic group $G$, a hyperbolicity constant $\delta$ of $G$ (see \cite{Pa96}). In addition, it is well-known that the ball of radius $100\delta$ in $G$ contains at least one representative of each conjugacy class of finite subgroups of $G$ (see \cite{Bra00}). Moreover, two finite subgroups $C_1$ and $C_2$ of $G$ are conjugate if and only if there exists an element $g$ whose length is bounded by a constant depending only on $\delta$ and on the size of the generating set $S$ of $G$, such that $C_2=gC_1g^{-1}$ (see \cite{BH05}).\end{proof}
\begin{lemme}[\cite{DG11}, Lemma 2.5]\label{algon}There is an algorithm that computes a set of generators of the normalizer of any given finite subgroup in a hyperbolic group.\end{lemme}
\begin{lemme}[\cite{DG11}, Lemma 2.8]\label{algon2}There is an algorithm that decides, given a finite set $S$ in a hyperbolic group, whether $\langle S\rangle$ is finite, virtually cyclic infinite, or non virtually cyclic (finite or infinite).\end{lemme}
\begin{lemme}\label{algofini2}There is an algorithm that takes as input a finite presentation of a hyperbolic group $G$ and a finite subgroup $C$ of $G$ such that $N_G(C)$ is non virtually cyclic (finite or infinite), and decides whether or not $E_G(N_G(C))=C$.\end{lemme}
\begin{proof}By Lemma \ref{algofini}, one can compute some representatives $A_1,\ldots,A_k$ of the conjugacy classes of finite subgroups of $G$. Given an element $g\in G$, let $\theta_g(x)$ be a quantifier-free formula expressing the following fact: there exists an integer $1\leq i\leq k$ such that the finite set $\lbrace C,g\rbrace$ is contained in $xA_ix^{-1}$. Note that the group $\langle C,g\rangle$ is finite if and only if the existential sentence $\exists x \ \theta_g(x)$ is true in $G$.
One can compute a finite generating set $S$ for $N_G(C)$ using Lemma \ref{algon}. By Theorem \ref{DG10} above, one can decide if the following existential sentence with constants in $G$ is satisfied by $G$: there exist two elements $g$ and $g'$ such that
\begin{enumerate}
\item $g$ does not belong to $C$;
\item $\theta_g(g')$ is satisfied by $G$ (hence, the subgroup $C':=\langle C,g\rangle$ is finite);
\item for every $s\in S$, one has $sC's^{-1}=C'$.
\end{enumerate}
Note that such an element $g$ exists if and only if $C$ is strictly contained in $E_G(N_G(C))$. This concludes the proof of the lemma.\end{proof}
The following lemma is an immediate corollary of Lemmas \ref{algofini}, \ref{algon}, \ref{algon2} and \ref{algofini2} above.
\begin{lemme}\label{algo5}There is an algorithm that takes as input a finite presentation of a hyperbolic group $G$ and computes the five numbers $n_1(G),\ldots,n_5(G)$ (see Definition \ref{5nombres}).
\end{lemme}
\subsection{Decidability of the first condition of Proposition \ref{concon}}
\begin{lemme}\label{condition1}There is an algorithm that, given a finite presentation of a virtually free group $G$ and a finite subset $X \subset G$ generating a subgroup $H=\langle X\rangle$, outputs `Yes' if $n_i(H)=n_i(G)$ for all $i\in \lbrace 1,2,3,4,5\rbrace$ and `No' otherwise.\end{lemme}
\begin{proof}
By Theorem \ref{compu}, there is an algorithm that takes as input a finite presentation $G=\langle S_G \ \vert \ R_G\rangle$ and $X$, and produces a finite presentation $\langle S_H \ \vert \ R_H\rangle$ for $H$. By Lemma \ref{algo5}, one can compute $n_i(G)$ and $n_i(H)$ for every $i\in \lbrace 1,2,3,4,5\rbrace$.\end{proof}
\subsection{Decidability of the second condition of Proposition \ref{concon}}
\begin{lemme}\label{condition2}There is an algorithm that, given a finite presentation of a virtually free group $G$ and a finite subset $X \subset G$ generating a subgroup $H=\langle X\rangle$, outputs `Yes' if $H$ is infinite and coincides with the one-ended factor of $G$ relative to $H$ (well-defined since $H$ is infinite), and `No' otherwise.\end{lemme}
\begin{proof}By Lemma \ref{algon2}, one can decide if $H$ is finite or infinite. By Lemma 8.7 in \cite{DG11}, one can compute a Stallings splitting of $G$ relative to $H$. Let $T$ be the Bass-Serre tree of this splitting. Let $U$ be the one-ended factor of $G$ relative to $H$ and let $u$ be the vertex of $T$ fixed by $U$. By Corollary 8.3 in \cite{DG11}, one can decide if there exists an automorphism $\varphi$ of $G$ such that $\varphi(H)=U$, which is equivalent to deciding if $U=H$. Indeed, if $\varphi(H)=U$, then $H$ fixes the vertex $u$ for the action of $G$ on $T$ twisted by $\varphi$. Thus, by definition of $U$ as the one-ended factor relative to $H$, the pair $(U,H)$ acts trivially on the tree $T$ for the action twisted by $\varphi$. Consequently, $\varphi(U)$ fixes $u$ as well. Therefore, one has $\varphi(U)=U=\varphi(H)$, and it follows that $U=H$ since $\varphi$ is an automorphism of $G$.\end{proof}
\subsection{Decidability of the third condition of Proposition \ref{concon}}
\begin{lemme}\label{condition3}There is an algorithm that, given a finite presentation of a virtually free group $G$ and a finite subset $X \subset G$ generating a subgroup $H=\langle X\rangle$, decides whether or not every finite subgroup of $G$ is conjugate to a subgroup of $H$.\end{lemme}
\begin{proof}
By Theorem \ref{compu}, there is an algorithm that takes as input a finite presentation $G=\langle S_G \ \vert \ R_G\rangle$ and $X$, and produces a finite presentation $\langle S_H \ \vert \ R_H\rangle$ for $H$. By Lemma \ref{algofini}, there is an algorithm that computes two lists $\lbrace A_1,\ldots ,A_n\rbrace$ and $\lbrace B_1,\ldots,B_n\rbrace$ of representatives of the conjugacy classes of finite subgroups of $G$ and $H$ respectively. Then, for every finite subgroup $A_i$ of $G$ in the first list, deciding if $A_i$ is conjugate in $G$ to $B_j$ for some $j\in \lbrace 1,\ldots ,n\rbrace$ is equivalent to solving the following finite disjunction of systems of equations with constants in $G$, which can be done using Theorem \ref{DG10}:
\[\theta(x) : \exists x \ (xA_ix^{-1}=B_1) \vee \ldots \vee (xA_ix^{-1}=B_n).\]
Hence, there is an algorithm that outputs `Yes' if every finite subgroup of $G$ is conjugate to a subgroup of $H$, and `No' otherwise.\end{proof}
\subsection{Proof of Theorem \ref{algoalgoalgoalgo}}
Theorem \ref{algoalgoalgoalgo} is an immediate consequence of Lemma \ref{lemmecomp} combined with Lemmas \ref{condition1}, \ref{condition2} and \ref{condition3}.
\section{$\exists^{+}$-elementary morphisms}
We prove Theorem \ref{44}.
\begin{te}Let $G$ be a group. Suppose that $G$ is finitely presented and Hopfian, or finitely generated and equationally noetherian. Then, every $\exists^{+}$-endomorphism of $G$ is an automorphism.\end{te}
\begin{proof}Let $\langle g_1,\ldots ,g_n\ \vert \ R(g_1,\ldots,g_n)=1\rangle$ be a presentation of $G$, with $R$ eventually infinite. Let $\varphi: G \rightarrow G$ be an $\exists^{+}$-endomorphism. For every integer $1\leq i\leq n$, let $h_i=\varphi(g_i)$.
Note that there is a one-to-one correspondence between the set of homomorphisms $\mathrm{Hom}(G,G)$ and the set of solutions in $G^n$ of the system of equations $R(x_1,\ldots,x_n)=1$. If $G$ is finitely presentable, one can assume without loss of generality that the system of equations $R$ is finite. If $G$ is equationally noetherian, there is a finite subsystem $R_i(x_1,\ldots,x_n)=1$ of $R(x_1,\ldots,x_n)=1$ such that the sets $\mathrm{Hom}(G,G)$ and $\mathrm{Hom}(G_i,G)$ are in bijection, where $G_i$ denotes the finitely presented group $\langle g_1,\ldots ,g_n\ \vert \ R_i(g_1,\ldots,g_n)=1\rangle$. Hence, one can always assume without loss of generality that the system $R$ is finite.
Every element $h_i$ can be written as a word $w_i(g_1,\ldots,g_n)$, and the group $G$ satisfies the following existential positive formula:\[\mu(h_1,\ldots,h_n):\exists x_1 \ldots \exists x_n \ R(x_1,\ldots,x_n)=1 \wedge h_i=w_i(x_1,\ldots,x_n).\]Indeed, one can just take $x_i=g_i$ for every $1\leq i\leq n$, which shows that the formula $\mu(h_1,\ldots,h_n)$ is satisfied by $G$. Since this formula is existential positive and since the morphism $\varphi$ is $\exists^{+}$-elementary and $h_i=\varphi(g_i)$, the statement $\mu(g_1,\ldots,g_n)$ is true in $G$ too. As a consequence, there exist some elements $k_1,\ldots,k_n$ in $G$ such that $R(k_1,\ldots,k_n)=1$ and $g_i=w_i(k_1,\ldots,k_n)$ for every $1\leq i\leq n$. Let us define an endomorphism $\psi$ of $G$ by $\psi(g_i)=k_i$ for every $1\leq i\leq n$.
Recall that $h_i=w_i(g_1,\ldots,g_n)$. Thus, one has \[\psi(h_i)=w_i(\psi(g_1),\ldots,\psi(g_n))=w_i(k_1,\ldots,k_n)=g_i.\]As a consequence, the composition $\psi\circ \varphi$ maps $g_i$ to itself, i.e.\ is the identity of $G$. It follows that $\psi$ is surjective.
Last, recall that equationally noetherian groups are Hopfian. It follows that $\psi$ is an automorphism of $G$. Hence, $\varphi$ is an automorphism of $G$.\end{proof}
\renewcommand{\refname}{Bibliography}
\bibliographystyle{plain}
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1,108,101,564,598 | arxiv | \section{Introduction}
\IEEEPARstart{I}{n} cities, parking lots are costly in terms of space, construction and maintenance costs. Parking lots take up 6.57\% of urban land use \cite{davis2010environmental} and collectively make up nearly 7000 $\text{km}^2$ of land use in the United States \cite{jakle2004lots}. In U.S. cities, the area of parking lots take up more than 3 times the area of urban parks \cite{davis2010environmental}. Government-set requirements for private developers\footnote{An example of government-set requirements on parking spaces can be seen in the municipal code of Placer County, CA: \url{https://qcode.us/codes/placercounty/view.php?topic=17-2-vii-17_54-17_54_060}. For example, the municipal code of Placer County states that restaurants are required to have 1 parking spot per 100 square feet of floor area, and shopping centers are required to have 1 parking spot per 200 square feet of floor area.} often require developers to provide parking lots to meet peak parking demand, resulting in an excessive number of parking lots with low utilization \cite{davis2010environmental}. Construction costs (excluding land acquisition) of an average parking lot costs nearly \$20,000 in the United States, while examples of annual maintenance cost of a single parking lot include \$461 in Fort Collins, CO \cite{VTPIreport} and \$729 in San Francisco, CA \cite{natstreetreport}.
Drivers also spend a significant amount of time looking for parking, overpay for what they use, and pay a high amount of parking fines. A recent study extrapolates from surveys taken in 10 American cities, 10 British cities, and 10 German cities, and found that drivers in the United States, United Kingdom, and Germany spend averages of 17, 44 and 41 hours annually respectively to find parking\footnote{The parking search time in the United States, United Kingdom, and Germany translates to an economic cost of \$72.7b, \$29.7b, and \$46.2b \cite{inrixreport}.}\cite{inrixreport}. In larger US cities, the search cost is significantly higher, with an estimated 107 hours annually in New York City and 85 hours in Los Angeles. This additional search time also contributes to congestion and air pollution in cities \cite{shoup2006cruising}. The same survey found that drivers overpay fees for 13 to 45 hours of parking per annum, and were also fined \$2.6b, \$1.53b, and \$434m over the period of a year in the United States, United Kingdom and Germany respectively\cite{inrixreport}.
Cities can benefit tremendously from a large-scale and accurate study of parking behavior. Especially with the possible transformation in land transportation brought on by autonomous vehicles \cite{duarte2018impact}, policymakers and urban planners would benefit from a scalable and accurate measurement and analysis of parking trends and activities. Additionally, real-time parking quantification method can also provide human or computer drivers with relevant parking information to reduce search time and allow for efficient route planning. In addition, a real-time parking measurement system can also enhance parking enforcement in cities; using May 2017 estimates by the Bureau of Labor Statistics, the labor costs in terms of just salary paid to parking enforcement officers amount to more than \$350 million annually \cite{blsreport}. The use of such a technical solution will allow these officers to have "eyes" on the ground, cover more parking lots with less physical effort, and increase municipal revenues from parking fines. A real-time, scalable and accurate parking measurement method is also a critical capability required by future parking system features, such as demand-based pricing \cite{pierce2013getting} and reservation for street parking \cite{geng2013new}, that have the potential to make parking more convenient for drivers, and to incentivize socially beneficial behavior.
\begin{figure*}
\centering
\resizebox{0.623\columnwidth}{!}{%
\includegraphics[scale=0.5]{images/pklot1.png}}
\resizebox{0.623\columnwidth}{!}{%
\includegraphics[scale=0.5]{images/pklot2.png}}
\resizebox{0.623\columnwidth}{!}{%
\includegraphics[scale=0.5]{images/pklot3.png}}\\
\vspace{1mm}
\resizebox{0.623\columnwidth}{!}{%
\includegraphics[scale=0.5]{images/car_coco1.png}}
\resizebox{0.623\columnwidth}{!}{%
\includegraphics[scale=0.5]{images/car_coco2.png}}
\resizebox{0.623\columnwidth}{!}{%
\includegraphics[scale=0.5]{images/truck_coco1.png}}
\caption[Sample images from PKLot dataset and Common Objects in Context dataset]{Top 3 images show the 3 different perspectives that the PKLot dataset by Almeida et al \cite{de2015pklot} is obtained from. Bottom 3 images show 3 sample images that is in the COCO dataset, reflecting the diverse contexts that object instances are identified in.}
\label{fig:cocopklot}
\end{figure*}
\section{Existing quantification methods}
Existing parking utilization methods can be divided into three types: counter-based, sensor-based and image-based \cite{de2015pklot}. Counter-based methods are restricted to deployment in gated parking facilities, and they work by counting the number of vehicles that enter and exit the parking facility. Sensor-based methods rely on physical detection sensors that are placed above or below parking lots, but are constrained by the significant capital costs of the large number of physical sensors required to cover large parking facilities \cite{de2015pklot}. Image-based methods rely on camera systems and are able to cover large outdoor or indoor parking lots when there are suitably high and unobstructed vantage points. Image-based methods also contain richer but less structured information than counter-based and sensor-based methods; for example, it is possible to identify specific vehicle characteristics from image-based methods but it is difficult to do so using counter-based or sensor-based methods.
Huang et al \cite{huang2010hierarchical} further divide image-based methods into car-driven and space-driven methods. Car-driven methods primarily detects and tracks cars, and use car-detection results to quantify parking usage. Traditional object detection methods such as Viola et al \cite{viola2004robust} that rely on "simple" image representations and learning from training examples have been applied to identify vehicles in videos taken of parking lots by Lee et al \cite{lee2005automatic} and Huang et al \cite{huang2010hierarchical}.
\subsection{Space-driven methods}
However, due to potential occlusions and perspective distortions of camera systems \cite{amato2017deep}, existing studies have instead focused on space-driven methods. Space-driven methods primarily observe changes in highlighted parking lots in an image frame. Past studies have used methods ranging from texture classifiers \cite{de2015pklot}, support vector machines \cite{bong2008integrated} and even recent deep learning-based methods \cite{amato2017deep,vu2018parking} to classify whether a parking space is occupied. These methods however rely on extensive, manual and relatively niche task labelling of the occupancy status of parking spots. For example, de Almeida et al \cite{de2015pklot} manually labelled 12,417 images of parking lots across multiple parking lots on the campus of Federal University of Parana (UFPR) and the Pontifical Catholic University of Parana (PUCPR) located in Curitiba, Brazil.
Besides the extensive effort required in obtaining image datasets and labelling them, the data collection process of parking spaces requires individual parking facilities to agree to data sharing and distribution. Fundamentally, space-based methods are not highly scalable, as they require extensive labelling and re-training of models for every distinct parking facility.
\subsection{Car-driven methods}
On the other hand, recent advancement in generic object detection through large-scale community projects led by organizations such as Microsoft \cite{lin2014microsoft} have allowed for access to large open datasets with more than 200,000 labelled images with more than 1.5 million object instances
\cite{cocowebsite}. The labelled instances include labels for different motor vehicles, including trucks, buses, cars, and motorcycles, and are taken in a variety of contexts and image quality.
In Figure \ref{fig:cocopklot}, we see that the extensive dataset labelled by de Almeida et al \cite{de2015pklot} consists of images taken from 3 spots, while the popular and open Common Objects in Context (COCO) dataset used for object detection takes images from a diverse range of perspectives.
Past work in car-based parking quantification relies on traditional object detection \cite{huang2010hierarchical}. An example of such a method is proposed by Tsai et al \cite{tsai2007vehicle} that uses color to identify vehicle candidates, and trains a model that uses corners, edge maps and wavelet transform coefficients to verify candidates. While traditional computer vision techniques are able to achieve good levels of accuracy, they rely heavily on feature selection by researchers and hence may be sub-optimal. On the contrary, deep learning based computer vision techniques are able to automatically select and identify features in a hierarchical manner\cite{bengio2009learning,zhou2014learning}.
\section{Experiment and Data Collection}
We collected 3 days of video footage of street parking around the MIT campus area in the City of Cambridge, USA. The locations of the studied parking lots can be seen in Figure \ref{fig:sites}. A summary of the sites is provided in Table \ref{table:sites}.
\begin{figure}[H]
\centering
\resizebox{1.0\columnwidth}{!}{%
\includegraphics[scale=0.5]{images/sites_crop.jpg}}
\caption[Experiment sites]{Locations of street parking where video data was collected}
\label{fig:sites}
\end{figure}
\begin{table}[h]
\centering
\resizebox{1.0\columnwidth}{!}{%
\begin{tabular}{|c|cl|}
\hline
Site Name & Footage Length & Site Description \\
& (minutes) & \\
\hline
\textit{Facilities} & 1696.5 & Four parking lots without lot boundaries.\\
& & Camera mounted across street with small chance of occlusions. \\
\hline
\textit{IDC} & 3622.5 & Four parking lots with clear lot boundaries. \\
& & Camera mounted at a high vantage point with no chance of occlusion. \\
\hline
\textit{Museum} & 3140.5 & Four parking lots with clear lot boundaries. Camera mounted at \\
& & a low vantage point and across the street with high chance of occlusions. \\
\hline
\end{tabular}}
\caption[Summary of experiment sites]{Descriptive summary of experiment sites}
\label{table:sites}
\end{table}
The video footages were taken in the summer of 2017. The footage was collected from the duration of 04:00:00 to 23:59:59 therefore include footages when lighting conditions are not ideal. This is explained further later in the implementation challenges, and our validation results in Section V differentiates the complete results from results taken during visible and peak hours. We define visible and peak hours as the duration from 07:00:00 to 18:59:59, which overlaps with durations when the studied parking sites require payment, and when natural lighting is substantial at the sites.
\section{Methodology}
\subsection{Frame-wise Instance Segmentation}
In this paper, we use the car-based method on images obtained. Unlike the space-based method, the car-based method depends on having an accurate and generalizable vehicle detector. The space-based method relies on classifying whether a specific parking space is occupied or not; this requires hand-labelling a specific parking facility and training a model that may not be generalizable to parking facilities other than the one that has been labelled.
\begin{figure}
\centering
\resizebox{0.40\columnwidth}{!}{%
\includegraphics[scale=0.5]{images/object_detection.png}}
\resizebox{0.54\columnwidth}{!}{%
\includegraphics[scale=0.5]{images/semantic.png}}\\
\vspace{1mm}
\resizebox{0.95\columnwidth}{!}{%
\includegraphics[scale=0.5]{images/instance.png}}
\caption[Object detection, semantic segmentation, and instance segmentation]{Top left image shows the results of the object detection algorithm, which detects object classes and localizes them with a rectangular bounding box. Top right image shows the result of semantic segmentation algorithm, which labels every pixel in an image with an object class. Bottom image shows the result of the instance segmentation algorithm used in this study, which detects object classes of interests, localizes them with a bounding box, and also provides a pixel-level localization or mask of identified objects.}
\label{fig:cvcomp}
\end{figure}
For the task of accurately quantifying parking space utilization, we use an instance segmentation algorithm as the baseline algorithm. Instance segmentation allows us to simultaneously identify individual instances of vehicles and precisely locate the boundaries of identified vehicle instances.
There has been significant progress in the deep learning-based object detection and semantic segmentation literature that have enabled real-time and accurate performance. In the area of object detection, Ren et al \cite{ren2015faster} overcame a significant bottleneck in past object detection algorithms by replacing time-consuming traditional region proposal methods such as Selective Search \cite{uijlings2013selective} and EdgeBoxes \cite{zitnick2014edge} with a learnable and fast Region Proposal Network (RPN). For semantic segmentation, Long et al \cite{long2015fully} demonstrated that fully convolutional neural networks perform better than past neural net architectures with downsampling and upsampling. Yu et al later \cite{yu2015multi,yu2017dilated} introduced dilated convolutions that widen receptive fields of convolutional units.
The algorithm that we employ for our purposes is based a Tensorflow implementation of He et al's \cite{he2017mask} Mask Region-based Convolutional Neural Network (Mask-RCNN). Based on Ren et al's \cite{ren2015faster} Faster-RCNN algorithm, Mask-RCNN adds a branch that predicts a mask or region-of-interest that serves as the pixel segmentation within the bounding boxes of each identified object instance. The simultaneous training and evaluation processes allow for fast training and real-time evaluation. Our Mask-RCNN implementation is trained using 2 NVIDIA GTX 1080Ti, and our training data is the COCO dataset, which has over 330 thousand training images, 1.5 million object instances and 80 distinct object categories, including cars, trucks, motorcycles and other motor vehicles. We used similar training parameters as reported by He et al \footnote{Training parameters are reported in \cite{he2017mask}, and also provided in the authors' Github repository: \href{https://github.com/facebookresearch/Detectron}{https://github.com/facebookresearch/Detectron}}, except that we used a decaying learning rate schedule with warm restarts that was introduced by Loshchilov et al \cite{loshsgdr2016}. Our Mask-RCNN implementation achieved an Average Precision (AP) score of 39.0 on the COCO test set, which is comparable to published Mask-RCNN metrics.
\subsection{Parking Identification}
\begin{figure}
\centering
\resizebox{\columnwidth}{!}{%
\includegraphics[scale=0.5]{images/facilities_params.png}}
\resizebox{\columnwidth}{!}{%
\includegraphics[scale=0.5]{images/idc_params.png}}\\
\vspace{1mm}
\resizebox{\columnwidth}{!}{%
\includegraphics[scale=0.5]{images/museum_params.png}}
\caption[Experiment sites with drawn parking lots and surrounding road areas]{Experiment sites with drawn parking lots and surrounding road areas. Top left image shows the \textit{Facilities} site, top right image shows the \textit{IDC} site, and bottom image shows the \textit{Museum} site.}
\label{fig:parkingid}
\end{figure}
Parking lots are identified and surrounding road-areas are identified via hand-drawn labels. Using a simple area-based threshold, vehicles with a significant proportion of its area located in road-areas are determined to be either not parked or obstructions. Figure \ref{fig:parkingid} shows the identified parking lots in red and the surrounding road-areas in blue for all 3 studied sites.
\subsection{Implementation Challenges}
We quantify parking utilization of lot $i$ and time $t$ as the ratio of the space utilization in the horizontal or x dimension and the horizontal space of lot $i$:
\[
\text{Utilization}_{i,t} = \frac{\text{Occupied horizontal space}_{i,t}}{\text{Horizontal space}_i}
\]
Using this measure, we find that a straightforward application of Mask-RCNN resulted in a noisy measurement of lot utilization. We directly applied Mask-RCNN to the recorded footages sampled at every 15 seconds, used the method described in 6.2.2 to identify vehicles that are parked, and applied the above definition to obtain parking utilization. For illustrative purposes, we focus on a particular duration of time at the \textit{Museum} site, and provide the utilization measurements and actual car stays during this duration for a single lot in Figure \ref{fig:noisymuseumsample}.
\begin{figure}[h]
\centering
\resizebox{1.0\columnwidth}{!}{%
\includegraphics[scale=0.5]{images/util_raw_museum_labels.png}}
\caption[Sample of noisy raw utilization measurements from \textit{Museum} site]{Sample of noisy raw utilization measurements from lot 3 of \textit{Museum} site. Actual car stay durations are drawn on top.}
\label{fig:noisymuseumsample}
\end{figure}
There are three factors that contribute to the noise seen in the utilization measurements:
\begin{enumerate}
\item \textbf{Occlusion}: A significant challenge for car-based methods is possible occlusion \cite{amato2017deep} of parking spots and observed vehicles. In the data collected, the occlusion problem is particularly severe for the \textit{Museum} site. Data at the \textit{Museum} site was collected via a camera mounted in the building across from the parking lots. This vantage point however is at a relatively low angle relative to the parking lots, hence data collected from this site often obstructed by vehicles in the street between the camera and the street parking lots of interest. Figure \ref{fig:museumocclusion} provides a side-by-side example of an unobstructed and occluded view of street parking lots at the \textit{Museum} site.
\item \textbf{Weather and lighting conditions}: Figure \ref{fig:museumlighting} provides a side-by-side example of the \textit{Museum} site that illustrates the effect of changes in lighting conditions.
\end{enumerate}
\begin{figure}[H]
\centering
\resizebox{1.0\columnwidth}{!}{%
\includegraphics[scale=0.5]{images/occlusion.png}}
\caption[Occlusion of street parking lots studied at the \textit{Museum} Site]{Unobstructed and occluded view of street parking lots at the \textit{Museum} Site}
\label{fig:museumocclusion}
\end{figure}
\vspace{-7mm}
\begin{figure}[H]
\centering
\resizebox{1.0\columnwidth}{!}{%
\includegraphics[scale=0.5]{images/lighting.png}}
\caption[Comparison of lighting conditions at street parking lots studied at the \textit{Museum} Site]{Unobstructed and occluded view of street parking lots at the \textit{Museum} Site}
\label{fig:museumlighting}
\end{figure}
\begin{enumerate}
\setcounter{enumi}{2}
\item \textbf{Random errors}: As we only consider detected cars if they exceed a certain threshold, idiosyncrasies in the video footage may result in random drops in detection, and also random false detections.
\end{enumerate}
\subsection{Smoothing Technique: Intelligent Car Tracking Filter} Signal processing techniques such as mode filters or low-pass frequency filters may not be effective in filtering out failures or noise in detection if they are not restricted to a particular (high) frequency domain. Furthermore, the use of such filters require calibration that is static. For example, the mode filter has a kernel size that would determine the length of maximum signal noise that it can handle.
A major contribution of this paper is to introduce an intelligent car tracking filter. Instead of simply applying the mode filter on raw utilization values extracted using instance segmentation provided by Mask-RCNN, the intelligent car tracking filter maintains a memory of characterizing features of past cars detected, and compares across detected vehicles in the past to smooth detected car locations.
\begin{figure}[H]
\centering
\resizebox{1.0\columnwidth}{!}{%
\includegraphics[scale=0.5]{images/filter_diagram.png}}
\caption[Diagram of intelligent car tracking filter]{Diagram describing the intelligent car tracking filter}
\label{fig:filterdiagram}
\end{figure}
\noindent As illustrated in Figure \ref{fig:filterdiagram}, the filter repeats the following steps: \begin{enumerate}
\item Run instance segmentation on sampled images to extract car masks
\item Vehicle features are extracted from detected vehicles\\
(a) A car classification model is applied on extracted vehicle masks to obtain a vector of features\\
(b) Color histograms of the two vehicles are computed as a feature vector
\item Car model feature vector and color histogram feature vectors are stored in memory
\item Past identified cars are compared and matched based on car model, color histogram feature vectors, and locations of cars in memory, to filter car locations
\item Cars from earliest frame is deleted
\end{enumerate}
\noindent For step 2 (a), we train the car classification model using the Stanford Cars dataset, which contains 196 different make and models of cars in a dataset of 16,185 images \cite{cardataset}. We use the ResNet-50 architecture for the car classification model due to its residual network structure that allows for effective learning for deep neural networks \cite{he2016deep}. By running the trained car classification model on the detected car instances, we obtain a feature vector $X_c\in\mathbb{R}^K,X_c\in(0,1)^K,\;K=196$. For step 2 (b), we obtain a feature vector $X_h\in\mathbb{R}^M,X_c\in(0,1)^M,\;M=24$ that characterizes the color histogram of the extracted vehicle mask.
We consider the case that the intelligent car tracking filter is applied to a video input streamed at a consistent frame rate of 1 frame every $S$ seconds, and the memory of the filter keeps track of all cars detected in the past $n$ frames. Without loss of generality, we assume that $n$ is odd. Consider that at time $t$, we should optimally infer the locations of cars at time $t-\text{round}(\frac{n}{2})\cdot S$, or $\text{round}(\frac{n}{2})$ frames ago.
Figure \ref{fig:extremecase} describes this extreme case with a scenario where $n=11$. In general, with a memory of $n$ frames and at present time $t$, inferring the locations of cars in the past at time $t-\text{round}(\frac{n}{2})\cdot S$, or $\text{round}(\frac{n}{2})$ frames ago would handle the maximum duration of occlusion. Step 4 does this by matching cars in its memory and inferring that the car is present at frame $\text{round}(\frac{n}{2})$ if (1) the matched car is found before and after frame $\text{round}(\frac{n}{2})$, and/or (2) the car is detected at frame $\text{round}(\frac{n}{2})$. Figure \ref{fig:singlecase} illustrates the scenario when the car is only detected at frame $\text{round}(\frac{n}{2})$, while Figure \ref{fig:usualcase} describes the typical scenario when the matched car is detected multiple times.
\begin{figure}[h]
\centering
\resizebox{1.0\columnwidth}{!}{%
\includegraphics[scale=0.5]{images/extremecomp.png}}
\caption[Maximum occlusion that can be handled by filter]{Diagram describing the maximum occlusion scenario that the intelligent filter can handle}
\label{fig:extremecase}
\end{figure}
\begin{figure}[h]
\centering
\resizebox{1.0\columnwidth}{!}{%
\includegraphics[scale=0.5]{images/singlecomp.png}}
\caption[Single detection at frame $\text{round}(\frac{n}{2})$]{Diagram describing the single detection case}
\label{fig:singlecase}
\end{figure}
\vspace{-3mm}
\begin{figure}[h]
\centering
\resizebox{1.0\columnwidth}{!}{%
\includegraphics[scale=0.5]{images/usualcomp.png}}
\caption[Typical scenario for matching cars]{Diagram describing the typical scenario when the matched car is detected multiple times}
\label{fig:usualcase}
\end{figure}\\
A pair of cars $A$ and $B$ have model feature vectors $X_c$, color histograms $X_h$ and horizontal pixel locations of the detected cars $X_l$, where $X_l\in\mathbb{R}^2$. We consider this pair of cars as matched if $||X_c^A-X_c^B||_1<T_c$, $D_B(X_h^A-X_h^B)<T_b$ and $||X_l^A-X_l^B||_1<T_l$, where $T_c,T_b,T_l$ are threshold levels for matches calibrated by manually looking at pairs of extracted vehicle masks, and $D_B$ is the Bhattacharyya distance function that measures similarity between two distributions \cite{bhattacharyya1943measure}. Once the detected car instances are matched and the filter infers that the car is present at frame $\text{round}(\frac{n}{2})$, the system returns the mode of all detected $X_l$.
In summary, the intelligent car tracking filter acts as a mode filter with a dynamic kernel that adjusts to matched vehicle instances in order to smooth the identified car locations. As a brief visual illustration of its effectiveness, we applied the intelligent car tracking filter on the same duration for a single lot in the \textit{Museum} site as seen in Figure \ref{fig:noisymuseumsample}, and provide the results in Figure \ref{fig:filterdemo}.
\begin{figure}
\centering
\resizebox{1.0\columnwidth}{!}{%
\includegraphics[scale=0.5]{images/filterdemo.png}}
\caption[Intelligent filter applied to sample of utilization measurements from \textit{Museum} site]{Intelligent filter applied to sample of utilization measurements from \textit{Museum} site}
\label{fig:filterdemo}
\end{figure}
\section{Validation Data and Metrics}
With actual deployment in a smart parking enforcement or payment in mind, we evaluate the filter based on its detection, spatial and time accuracy, as well as processing speed. To validate these metrics, we went through the recorded videos and manually labelled randomly selected frames or for randomly selected vehicles. Table \ref{table:labels} describes the validation datasets that we constructed.
\begin{table}[h]
\centering
\resizebox{1.0\columnwidth}{!}{%
\begin{tabular}{|c|cl|}
\hline
Validation Metrics & Dataset Size & Label Description \\
\hline
Detection and & 138 frames & Randomly selected 138 frames\\
Spatial Accuracy & 309 vehicles & across all 3 sites and labelled bounding \\
& & boxes for parked vehicle instances \\
\hline
Time & 40 vehicles & Randomly selected 40 vehicles across \\
Accuracy & 3410 mins & all 3 sites and recorded the time that the\\
& & vehicle entered and left its parking lot\\
\hline
\end{tabular}}
\caption[Summary of validation datasets]{Descriptive summary of validation datasets}
\label{table:labels}
\end{table}
For the first dataset in Table \ref{table:labels}, we only used frames where all parking lots of interest are unobstructed. Furthermore, both datasets were obtained via random sampling from the entire timeframe of 04:00:00 to 23:59:59, which includes periods of poor visibility. As mentioned earlier, we further used a subset of the validation set where all data was restricted to the duration between 07:00:00 to 18:59:59. The size of the first dataset reduces to 113 frames and 253 vehicles, while the size of the second dataset reduces to 30 vehicles with total duration of 1626 mins of validation footage.
\subsection{Detection Accuracy}
Using the first dataset described in Table \ref{table:labels}, we use a simple ratio of vehicles that are detected and labelled to be present (denoted as $TP$), over the sum of vehicles that are detected and labelled to be present, vehicles that are detected but not labelled (denoted as $FP$), and vehicles that are not detected and labelled, (denoted as $FN$). Denoting $TP_i,FP_i,FN_i$ as the evaluation metrics for the $i$th frame in the validation dataset, we first sum these metrics across all 150 validation frames before taking the ratio:
\[
\text{Detection Accuracy} = \frac{\sum\limits_{i=1}^{150}TP_i}{
\sum\limits_{i=1}^{150}(TP_i+FP_i+FN_i)}
\]
\subsection{Spatial Accuracy}
We further use the first dataset described in Table \ref{table:labels} to validate the spatial accuracy of this method. For each $j$th detected vehicle instance, we have a manually labelled bounding box and a detected vehicle mask. We extract the left and right horizontal pixel coordinates of the bounding box (denoted as $X_{l,\text{true}}^j$ and $X_{r,\text{true}}^j$), and also the leftmost and rightmost horizontal pixel coordinates of the vehicle mask (denoted as $X_{l,\text{output}}^j$ and $X_{r,\text{output}}^j$). Denoting $N_{TP}=\sum\limits_{i=1}^{150} TP_i$ as the total number of all detected and labelled vehicle instances, we define spatial accuracy as the average ratio of horizontal area of intersection between box and mask over the horizontal area of union between box and mask. Note that the following definition follows the convention where $X_{r,\text{true}}^j\geq X_{l,\text{true}}^j$ and $X_{r,\text{output}}^j\geq X_{l,\text{output}}^j$.
\begin{equation*}
\resizebox{\columnwidth}{!}{%
$\text{Spatial Accuracy} = \frac{1}{N_{TP}}\cdot\sum\limits_{j=1}^{N_{TP}}\frac{\min(X_{r,\text{true}}^j,X_{r,\text{output}}^j)-\max(X_{l,\text{true}}^j,X_{l,\text{output}}^j)}{\max(X_{r,\text{true}}^j,X_{r,\text{output}}^j) - \min(X_{l,\text{true}}^j,X_{l,\text{output}}^j)}
$}
\end{equation*}
\subsection{Time Accuracy}
We use the second dataset described in Table \ref{table:labels} to evaluate the time accuracy of the filter. The time accuracy metric measures the ratio of detected occupancy of a particular lot over the actual occupancy of a particular lot. Denoting the number of frames when a vehicle is detected as being in the lot for the $i$th validated vehicle as $F_{i,\text{output}}$, and the total number of frames when the $i$th validated vehicle as actually labelled being parked in the lot as $F_{i,\text{true}}$, we define the first time accuracy metric as:
\small
\[
\text{Time Accuracy} = \frac{\sum\limits_{i=1}^{42} F_{i,\text{output}}}{\sum\limits_{i=1}^{42}F_{i,\text{true}}}
\]
\normalsize
The San Francisco Metropolitan Transport Authority defines this metric as the occupancy accuracy metric, and uses this metric to evaluate potential vendors for SFPark, a project aimed at managing demand for parking spaces in San Francisco \cite{sfparkreport}.
\subsection{Processing Efficiency}
We measure the average processing time per sampled frame as the benchmark for processing efficiency, and also the total processing time for the entire duration of videos taken at all 3 sites.
\section{Validation Results}
\subsection{Accuracy Results: Pure Detection and Memory Filter}
\begin{table}[h]
\centering
\begin{tabular}{|c|ccc|}
\hline
Memory Size & Detection & Spatial & Time\\
(\# of frames) & Accuracy& Accuracy & $\text{Accuracy}$\\
& (\%) & (\%) & (\%) \\
\hline
Mask-RCNN Only & 89.2 & 92.1 & 64.9 \\
5 & 90.2 & 91.2 & 71.3\\
25 & 94.0 & 91.0 & 80.0 \\
50 & 94.6 & 88.9 & 83.2 \\
100 & 93.5 & 88.7 & 87.5 \\
150 & 93.0 & 86.1 & 87.9 \\
\hline
\end{tabular}
\caption[Accuracy results from validation datasets sampled from full timeframe]{Accuracy results from validation datasets sampled from full timeframe}
\label{table:fullresults}
\end{table}
\begin{table}[h]
\centering
\begin{tabular}{|c|ccc|}
\hline
Memory Size & Detection & Spatial & Time \\
(\# of frames) & Accuracy& Accuracy & $\text{Accuracy}$\\
& (\%) & (\%) & (\%) \\
\hline
Mask-RCNN Only & 90.0 & 92.2 & 72.2 \\
5 & 89.7 & 91.6 & 78.3 \\
25 & 93.1 & 91.6 & 88.2\\
50 & 93.5 & 90.0 & 91.2 \\
100 & 92.2 & 89.4 & 95.8 \\
150 & 91.6 & 88.7 & 96.1\\
\hline
\end{tabular}
\caption[Accuracy results from validation datasets sampled from 07:00:00 to 18:59:59]{Accuracy results from validation datasets sampled from 07:00:00 to 18:59:59}
\label{table:peakresults}
\end{table}
We compare the results of simply running the Mask-RCNN instance segmentation algorithm (Mask-RCNN Only in Tables \ref{table:fullresults} and \ref{table:peakresults}) against applying our filter on top of Mask-RCNN with different memory sizes on the video footages sampled at 1 frame every 15 seconds. Validation results in both the full timeframe (Table \ref{table:fullresults}) and the peak timeframe (Table \ref{table:peakresults}) demonstrate that the filter significantly increases $\text{Time Accuracy}$, and slightly increases Detection Accuracy. The significant improvements in the Time Accuracy metric can be attributed to the inferences that the intelligent filter are able to make.
Comparing results between in the peak and full timeframes, we see that accuracy is generally higher in the peak timeframes. This reflects the sensitivity of image-based methods to poor lighting during nighttime, especially since the \textit{Museum} and \text{Facilities} sites are not well-lit in the night.
\subsection{Accuracy Results: Comparison to Industry Benchmarks}
In 2014, SFPark evaluated trial parking sensor systems including image recognition systems, radar sensors and infrared cameras. One key metric that SFPark tracked was the occupancy accuracy metric that is defined identically with $\text{Time Accuracy}$ \cite{sfparkdataguide}, and charts\footnote{Pages 12 and 18 have charts that suggest that the validation was conducted during parking meter operation hours that are contained within our peak timeframe definition \cite{sfparkreport}.} in the SFPark evaluation report suggests that the evaluation was conducted in daytime. The results show that our intelligent filter significantly outperforms the industry benchmark image method provided by Cysen. Furthermore, the image recognition benchmark has similar performance to simply applying Mask-RCNN during the peak time period (see Detection Only - Peak in Table \ref{table:industrycomp}), suggesting that existing commercial vendors utilize an image-based rather than a video-based approach. In addition, the performance of the intelligent filter in the peak time period (see Intelligent Filter - Peak in Table \ref{table:industrycomp}) is comparable to the performances of the best sensor systems evaluated by SFPark.
\begin{table}[h]
\centering
\begin{tabular}{|c|c|}
\hline
Sensor System (Vendor) & $\text{Time Accuracy}$ \\
\hline
Radar/Magnetometer (Fybr) & 78\% \\
Radar (Sensys) & 98\% \\
Infrared (CPT) & 92\% \\
Image Recognition (Cysen) & 77\% \\
Magnetometer (StreetSmart) & 81\% \\
\hline
Detection Only - Full & 65\% \\
Detection Only - Peak & 72\% \\
Intelligent Filter - Full & 88\% \\
Intelligent Filter - Peak & 96\%\\
\hline
\end{tabular}
\caption[Occupancy accuracy comparison with industry standards]{Comparison of occupancy accuracy of intelligent filter with industry standards. Top half of table contains industry benchmarks while bottom half shows our validation results}
\label{table:industrycomp}
\end{table}
\vspace{-8mm}
\subsection{Processing Speed}
We used a single computer with an Intel i7-7700K CPU, 16GB of RAM and a NVIDIA GTX-1080Ti GPU that has a retail market price of \$1,500 to evaluate the average processing time and total processing time. Table \ref{table:speedcomp} shows the processing speed in terms of average processing time per frame, and total processing time for the entire duration of our recorded footage..
\begin{table}[H]
\centering
\begin{tabular}{|c|cc|}
\hline
Memory Size & Average Processing Time & Total Processing Time\\
(\# of frames) & Per Frame (seconds) & (seconds) \\
\hline
Mask-RCNN Only & 0.399 & 13501\\
5 & 0.413 & 13975\\
25 & 0.435 & 14720 \\
50 & 0.469 & 15870 \\
100 & 0.556 & 18814 \\
150 & 0.673 & 22773 \\
\hline
\end{tabular}
\caption[Processing speed comparison]{Comparison of processing speed of filter with different memory sizes}
\label{table:speedcomp}
\end{table}
\section{Conclusion}
\begin{figure*}[h]
\centering
\resizebox{1.0\textwidth}{!}{%
\includegraphics[scale=0.5]{images/ecg.png}}
\caption[Vehicle location information generated from intelligent filter]{Vehicle location information for the \textit{Museum} site generated from intelligent filter. Grey lines mark boundaries between parking lots. Blue lines, red lines, green lines, and yellow lines mark the boundaries of car parked in lot 0, 1, 2, and 3 respectively.}
\label{fig:ecg}
\end{figure*}
The validation results demonstrate that our method significantly improves accuracy by treating the parking measurement problem as a \textit{video} problem rather than an \textit{image} problem. By combining information from video frames before and after, our system is able to better infer vehicle occupancy as compared to pure image-based methods. Furthermore, evaluation results by SFMTA supports the claim that our system is comparable in performance to the advanced commercial systems that employ more expensive sensors. For further verification of our system's relative performance to other methods, future studies should compare different sensor systems on identical parking sites and at identical periods of time.
\subsection{Feasibility as a city-wide system}
The fast average processing time per frame shows that our system can be financially feasible in a real-world deployment. Using a memory size of 100 frames, and a video sample rate of 1 frame every 15 seconds, the computer that we used for evaluation would be able to handle up to 89 parking lots. This translates to a cost of around \$17 per parking lot for processing ability, and we estimate that a camera, a single board computer and other associated costs would cost a further \$15 per parking lot. The eventual cost of around \$32 per parking lot is highly competitive as compared to existing sensors such as ground-based parking sensors that costs up to \$200 per parking lot\footnote{For example, PNI Corp prices its PlacePod ground sensors at \$200 per parking sensor. Price obtained from \url{ https://www.pnicorp.com/product-category/smart-parking/}.}. In demonstrating that our method is accurate and competitive at a per parking lot level basis, our paper opens up the opportunity for further research into the scalability of a camera and video based street parking monitoring system at a city-wide scale, especially in comparison to existing projects and technologies such as SFPark.
\subsection{Tradeoff between immediacy and accuracy}
An important qualification is that a feature of our implementation of the intelligent filter is that the filtered car location information is almost real-time. This is due to the inference that the system needs to make, which utilizes both information from "future" and "past" frames. Depending on the system's application in parking enforcement, quantification or real-time information to drivers, the system can make inferences about frames closer to the present time, $t$, and incur losses in accuracy. This change can be done by switching the inference frame from the current $t-\text{round}(\frac{n}{2})$ frame to a frame that is closer to the present time $t$.
\subsection{Generalizability to other sites}
We experienced little difficulty in extending our system to the three sites that we studied. Unlike space-based methods that require labelling and training for every distinct parking facility, we only need to mark out parking lot boundaries and surrounding road areas once to configure our system for a new parking facility. Labelling was only required to validate our results.
\subsection{Information beyond binary occupancy}
Our system can provide richer information than traditional binary occupancy information, especially in detecting space-based information and car make and color information. Figure \ref{fig:ecg} plots the left and right boundaries of filtered car locations for the \textit{Museum} site. Furthermore, our video-based method is also capable of recognizing specific type of vehicles, including conventional taxis and delivery trucks. Hence, our method provides the technical foundation for richer ways to understand curbside and parking occupancy that exceeds beyond traditional binary parking statistics.
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\section*{Acknowledgment}
\noindent The authors would like to thank Philips Lighting for supporting this project. In addition, the authors would also like to thank Allianz, Amsterdam Institute for Advanced Metropolitan Solutions, Brose, Cisco, Ericsson, Fraunhofer Institute, Liberty Mutual Institute, Kuwait-MIT Center for Natural Resources and the Environment, Shenzhen, Singapore- MIT Alliance for Research and Technology (SMART), UBER, Victoria State Government, Volkswagen Group America, and all the members of the MIT Senseable City Lab Consortium for supporting the lab's research.
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1,108,101,564,599 | arxiv | |
1,108,101,564,600 | arxiv | \section{#1}\setcounter{equation}{0}}
\def\thesection.\arabic{equation}{\thesection.\arabic{equation}}
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\begin{document}
\begin{titlepage}
\bigskip
\bigskip\bigskip\bigskip
\centerline{\Large \bf Analytic Study of Small Scale Structure on Cosmic Strings}
\bigskip\bigskip\bigskip
\centerline{\large Joseph Polchinski}
\bigskip
\centerline{\em Kavli Institute for Theoretical Physics}
\centerline{\em University of California}
\centerline{\em Santa Barbara, CA 93106-4030} \centerline{\tt [email protected]}
\bigskip
\bigskip
\centerline{\large Jorge V. Rocha}
\bigskip
\centerline{\em Department of Physics}
\centerline{\em University of California}
\centerline{\em Santa Barbara, CA 93106} \centerline{\tt [email protected]}
\bigskip
\bigskip
\bigskip\bigskip
\begin{abstract}
The properties of string networks at scales well below the horizon are poorly understood, but they enter critically into many observables. We argue that in some regimes, stretching will be the only relevant process governing the evolution.
In this case, the string two-point function is determined up to normalization: the fractal dimension approaches one at short distance, but the rate of approach is characterized by an exponent that plays an essential role in network properties.
The smoothness at short distance implies, for example, that cosmic string lensing images are little distorted. We then add in loop production as a perturbation and find that it diverges at small scales. This need not invalidate the stretching model, since the loop production occurs in localized regions, but it implies a complicated fragmentation process. Our ability to model this process is limited, but we argue that loop production peaks a few orders of magnitude below the horizon scale, without the inclusion of gravitational radiation. We find agreement with some features of simulations, and interesting discrepancies that must be resolved by future work.
\end{abstract}
\end{titlepage}
\baselineskip = 16pt
\sect{Introduction}
The evolution of cosmic string networks is a challenging problem. The need to consider large ratios of length and time scales makes a complete numerical analysis impossible, while the nonlinearity of the process defeats a purely analytic treatment. Thus a full understanding of most of the signatures of cosmic strings will require a careful combination of analytic and numerical approaches. This is needed both to establish precisely the current bounds on the dimensionless cosmic string tension $G\mu$, and to anticipate what will be the most sensitive future measurements. Also, if cosmic strings are one day discovered, a precise understanding of the network properties will be needed in order to distinguish different microscopic models.
On scales close to the horizon size, the networks are reasonably well understood from simulations~\cite{Albrecht:1989mk,Bennett:1989yp,Allen:1990tv}.\footnote{For reviews of network properties and other aspects of cosmic strings see Refs.~\cite{VilShell,Hindmarsh:1994re}.}
In particular, there are a few dozen long strings crossing any horizon volume. On shorter scales, however, the situation is far less clear. There is a nontrivial short distance structure on the strings, which arises because the intercommutation process produces kinks~\cite{Bennett:1987vf}. There have been many previous analytic and numerical studies of this, but there is no general consensus as to its nature.
In particular, the size at which typical loops form is uncertain to tens of orders of magnitude. One widely-held assumption has been that gravitational radiation is necessary in order for cosmic string networks to scale~\cite{ACK1993}, and that it determines the size of loops. If so, the loops will be parametrically smaller than the horizon scale, by a power of $G\mu$~\cite{Siemens:2002dj} (even shorter scales have been suggested~\cite{Vincent:1996rb}). More recent work appears to be converging on loop sizes at a fixed fraction of the horizon scale, but even here there are significant differences. Ref.~\cite{Vanchurin:2005pa} suggests that loops form at around $0.1$ times the horizon scale, whereas Refs.~\cite{RSB2005,MarShell2005} (and some portion of the conventional wisdom) suggest a scale several orders of magnitude smaller. The larger loops would lead to enhanced signatures of several types, and tighter bounds on $G\mu$~\cite{Olum:2006at}.
In a system with a large hierarchy of scales, one might hope that analytic methods would be particularly useful in separating the processes occurring at different scales,
while numerical methods would be needed to understand the nonlinearities at a given scale. This is the philosophy that we will attempt to implement. We focus on a microscopic description, similar in spirit to Ref.~\cite{Embacher:1992zk} and in particular Ref.~\cite{ACK1993}. An important difference from Ref.~\cite{ACK1993} is that we are less ambitious: that work was largely directed at understanding the horizon-scale structure quantitatively, whereas we are only interested in shorter scales. Also, we do not attempt to reduce the dynamics to a parameterized model but rather focus on the full two-point function; this two-point function appears to be characterized by a critical exponent, which did not enter into previous work. Finally, our (tentative) conclusions are opposite to Ref.~\cite{ACK1993}, in that we believe that the network scales even without gravitational radiation.
The full microscopic equations for the string network~\cite{Embacher:1992zk,ACK1993} appear to be too complicated to solve, and so we model what we hope are the essential physical processes. We have had some success in matching results from simulations, but there are also some discrepancies which may indicate additional processes that must be included.
In section~2 we consider the evolution of a small segment on a long string. If the rate of string intercommutations is fixed in horizon units, then over a range of scales the only important effect would be stretching due to the expansion of the universe. We are then able to determine the two-point functions characterizing the small scale structure. We find that the string is actually rather smooth, in agreement with recent simulations~\cite{MarShell2005}: its fractal dimension approaches one as we go to smaller scales. There is a nontrivial power law, but this appears in the {\it approach} of the fractal dimension to one; that is, the critical exponent $\chi$ is related to the power spectum of perturbations on the long string. The two-point functions depend on two parameters that must be taken from simulations. One of these is the mean $v^2$ of an element of the long string; this determines the exponent $\chi$. The second parameter is the normalization of the two-point function. Our results for the two-point function match reasonably well with the simulations~\cite{MarShell2005} over a range of scales, but there is substantial deviation at the shortest scales. It remains to be seen whether this is due to transient effects in the simulations, or real effects that we have omitted.
In section~3 we study the effects of the small scale structure on string lensing. Because the string is fairly smooth, these effects are not as dramatic as has been considered in some previous work. However, there are calculable deviations from perfect lensing. We also consider, for lensing by a long string, the likely trajectory of the string relative to the axis of a given lensed object. We discuss some deviations from gaussianity in the small scale structure, due to kinks.
In section~4 we add in loop production as a perturbation. Even though the strings are relatively smooth at short distance, we find that the total rate of loop production diverges for small loops; the rate of divergence appears to agree with recent simulations~\cite{RSB2005}.
This divergence does not invalidate our stretching model, because the loop production is localized to regions where the left- and right-moving tangents ${\bf p}_{\pm}$ are approximately equal, but it points to a complicated fragmentation process, cascading to smaller and smaller loops. We are not able to follow the fragmentation process analytically at present, but we give an analytic argument as to why the cascade should terminate, leaving loops at some small but fixed multiple of the horizon scale. We also point out an interesting puzzle related to loop velocities, which again points to a complicated fragmentation process.
In section~5 we discuss various applications and future directions. The behavior of string networks is notoriously complicated, and it appears to remain so even when one focusses on small scales. Thus we view our work as part of an ongoing dialogue between analytic calculations and simulations, which we hope will lead to a more complete and precise picture.
\vspace{2pc}
\sect{The stretching regime}
\subsection{Assumptions}
We consider ``vanilla'' cosmic strings, a single species of local string without superconducting or other extra internal degrees of freedom. The evolution of a network of such strings is dictated by three distinct processes. First, the expansion of the universe stretches the strings: on scales larger than the horizon scale $d_H$ irregularities on a string are just conformally amplified, but on smaller scales the string effectively straightens~\cite{Vilenkin1981}. This is described by the Nambu action in curved spacetime. Second, gravitational radiation also has the effect of straightening the strings, but it is significant only below a length scale proportional to $d_H$ and to a positive power of the dimensionless string coupling $G\mu$~\cite{BB1989,Sakellariadou:1990ne,Hindmarsh:1990xi,Quashnock:1990wv,Siemens:2002dj}. Since this is parametrically small at small $G\mu$, we will ignore this effect for the present purposes. Finally, intercommutations play an important role in reaching a scaling solution, in particular through the formation of closed loops of string. At first we shall neglect this effect but we will be forced to return to this issue in section~4.
Let us consider the evolution of a small segment on a long string. We take the segment to be very short compared to the horizon scale, but long compared to the scale at which gravitational radiation is relevant. The scaling property of the network implies that the probability per Hubble time for this segment to be involved in a long string intercommutation event is proportional to its length divided by $d_H$, and so for short segments the intercommutation rate per Hubble time will be small. Formation of a loop much larger than the segment might remove the entire segment from the long string, but this should have little correlation with the configuration of the segment itself, and so will not affect the probability distribution for the ensemble of short segments. Formation of loops at the size of the segment and smaller could affect this distribution, and the results of section~4 will indicate that the production of small loops is large, but this process takes place only in localized regions where the left- and right-moving tangents are approximately equal. Thus, there is a regime where stretching is the only relevant process.
If we follow a segment forward in time, its length increases but certainly does so more slowly than the horizon scale $d_H$, which is proportional to the FRW time $t$. Thus the length divided by $d_H$ decreases, and therefore so does the rate of intercommutation. If we follow the segment backward in time, its length eventually begins to approach the horizon scale, and the probability becomes large that we encounter an intercommutation event. Our strategy is therefore clear. For the highly nonlinear processes near the horizon scale we must trust simulations. At a somewhat lower scale we can read off the various correlators describing the behavior of the string, and then evolve them forward in time using the Nambu action until we reach the gravitational radiation scale. The small probability of an intercommutation involving the short segment can be added as a perturbation. This approach is in the spirit of the renormalization group, though with long and short distances reversed.
\subsection{Two-point functions at short distance}
In an FRW spacetime,
\begin{eqnarray}
ds^2 &=& -dt^2 + a(t)^2 d{\bf x} \cdot d{\bf x} \nonumber\\
&=& a(\tau)^2 (-d\tau^2 + d{\bf x} \cdot d{\bf x})\ ,
\end{eqnarray}
the equation of motion governing the evolution of a cosmic string is~\cite{TurokBhat1984}
\begin{equation}
\ddot{\bf x} + 2\,\frac{\dot{a}}{a}\,(1-\dot{\bf x}^2)\, \dot{\bf x} = \frac{1}{\epsilon} \left( \frac{{\bf x}'}{\epsilon} \right)' \ .
\label{EOM}
\end{equation}
Here $\epsilon$ is given by
\begin{equation}
\epsilon \equiv \left( \frac{{\bf x}'^2}{1-\dot{\bf x}^2} \right)^{1/2}.
\label{epsilon}
\end{equation}
These equations hold in the transverse gauge, where $\dot{\bf x}\cdot{\bf x}'=0$. Dots and primes refer to derivatives relative to the conformal time $\tau$ and the spatial parameter $\sigma$ along the string, respectively. The evolution of the parameter $\epsilon$ follows from equation (\ref{EOM}),
\begin{equation}
\frac{\dot{\epsilon}}{\epsilon} = -2\,\frac{\dot{a}}{a} \, \dot{\bf x}^2.
\label{epsev}
\end{equation}
From the second derivative terms it follows that signals on the string propagate to the right and left with $d\sigma = \pm d\tau/\epsilon$. Thus the structure on a short piece of string at a given time is a superposition of left- and right-moving segments, and it is these that we follow in time. In an expanding spacetime the left- and right-moving waves interact --- they are not free as in flat spacetime.
From Eq.~(\ref{epsev}) it follows that the time scale of variation of $\epsilon$ is the Hubble time, and so to good approximation we can replace $\dot{\bf x}^2$ with the time-averaged $\bar v^2$
(bars will always refer to RMS averages), giving $\epsilon \propto a^{-2\bar v^2}$ as a function of time only.\footnote{The transverse gauge choice leaves a gauge freedom of time-independent $\sigma$ reparameterizations. A convenient choice is to take $\epsilon$ to be independent of $\sigma$ at the final time, and then $\epsilon$ will be $\sigma$-independent to good approximation on any horizon length scale in the past.}
From the definition of $\epsilon$ it then follows that the energy of a segment of string of coordinate length $\sigma$ is
$E = \mu a(\tau) \epsilon(\tau) \sigma$.
For simplicity we will refer to $E/\mu$ as the length $l$ of a segment,
\begin{equation}
l = a(\tau) \epsilon(\tau) \sigma\ ,
\end{equation}
though this is literally true only in the rest frame. The scale factor is
\begin{equation}
a \propto t^\nu \propto \tau^{\nu'}\ ,\quad \nu' = \nu/(1 - \nu)\ ,
\end{equation}
where
\begin{eqnarray}
\mbox{radiation domination:}&& \nu = 1/2\ ,\quad \nu' = 1\ ,\quad \bar v^2 \approx 0.41 \ ,\nonumber\\
\mbox{matter domination:}&& \nu = 2/3\ ,\quad \nu' = 2\ ,\quad \bar v^2 \approx 0.35 \ .
\end{eqnarray}
The mean RMS velocities for points on long strings are taken from simulations~\cite{MarShell2005}. It follows that
\begin{equation}
l \propto \tau^{\zeta'} \propto t^{\zeta}\ , \quad \zeta' = (1 - 2\bar v^2) \nu' \ , \quad
\zeta = (1 - 2\bar v^2) \nu \ . \label{stretch}
\end{equation}
In the radiation era, $\zeta_r \sim 0.1$ and in the matter era $\zeta_m \sim 0.2$. Thus the physical length of the segment grows in time, but more slowly than the comoving length~\cite{Vilenkin1981}, and much more slowly than the horizon length $d_H \propto t$.
For illustration, consider a segment of length $(10^{-6}\; {\rm to}\; 10^{-7}) d_H$, as would be relevant for lensing at a separation of a few seconds and a redshift of the order of $z\simeq 0.1$. According to the discussion above, $l / d_H$ depends on time as $t^{\zeta - 1} \sim t^{-0.8}$ in the matter era. Thus the length of the segment would have been around a hundredth of the horizon scale at the radiation-to-matter transition. In other words, it is the nonlinear horizon scale dynamics in the radiation epoch that produces the short-distance structure that is relevant for lensing today, in this model. This makes clear the limitation of simulations by themselves for studying the small scale structure on strings, as they are restricted to much smaller dynamical ranges.
In terms of left- and right-moving unit vectors $\bf{p}_\pm \equiv \dot{\bf{x}} \pm \frac{1}{\epsilon}\bf{x}'$, the equation of motion~(\ref{EOM}) can be written as~\cite{Albrecht:1989mk}
\begin{equation}
\dot{\bf p}_\pm \mp \frac{1}{\epsilon} {\bf p}'_\pm = - \frac{\dot{a}}{a} \left[ {\bf p}_\mp - ({\bf p}_+ \cdot {\bf p}_-)\, {\bf p}_\pm \right].
\label{EOM2}
\end{equation}
We will study the time evolution of the left-moving product ${\bf p}_+(\tau,\sigma) \cdot {\bf p}_+(\tau,\sigma')$. For this it is useful to change variables from $(\tau,\sigma)$ to $(\tau,s)$ where $s$ is constant along the left-moving characteristics,
$\dot s - s'/\epsilon = 0$.
Then
\begin{eqnarray}
\partial_\tau ({\bf p}_+ (s,\tau) \cdot {\bf p}_+(s',\tau)) &=&
- \frac{\dot a}{a} \biggl( {\bf p}_-(s,\tau)\cdot{\bf p}_+(s',\tau)+ {\bf p}_+(s,\tau)\cdot{\bf p}_-(s',\tau) \nonumber\\
&&\hspace{-60pt} + \alpha (s,\tau)\,{\bf p}_+(s,\tau)\cdot{\bf p}_+(s',\tau) + \alpha (s',\tau)\,{\bf p}_+(s,\tau)\cdot{\bf p}_+(s',\tau) \biggr)\ , \label{bilinear}
\end{eqnarray}
where $\alpha = - {\bf p}_+ \cdot{\bf p}_- = 1 - 2v^2$.
The equations of motion~(\ref{EOM2}, \ref{bilinear}) are nonlinear and do not admit an analytic solution, but they simplify when we focus on the small scale structure. If ${\bf p}_+ (s,\tau)$ were a smooth function on the unit sphere, we would have $1 - {\bf p}_+ (s,\tau) \cdot {\bf p}_+(s',\tau) = O([s-s']^2)$ as $s'$ approaches $s$. We are interested in any structure that is less smooth than this, meaning that it goes to zero more slowly than $[s-s']^2$. For this purpose we can drop any term of order $[s-s']^2$ or higher in the equation of motion (smooth terms of order $s-s'$ cancel because the function is even).
Consider the product ${\bf p}_-(s,\tau) \cdot {\bf p}_+(s',\tau)$. The right-moving characteristic through $(s,\tau)$ and the left-moving characteristic through $(s',\tau)$ meet at a point $(s,\tau-\delta)$ where $\delta$ is of order $s - s'$.\footnote
{Explicitly, for given $\tau$ we could choose coordinates where $\epsilon(\tau) = 1$ and $s(\tau',\sigma) = \sigma + \tau' - \tau+ O( [\tau' - \tau]^2)$, and then $\delta = (s-s') /2$.} Eq.~(\ref{EOM2}) states that ${\bf p}_+$ is slowly varying along left-moving characteristics (that is, the time scale of its variation is the FRW time $t$), and ${\bf p}_-$ is slowly varying along right-moving characteristics. Thus we can approximate their product at nearby points by the local product where the two geodesics intersect,
\begin{equation}
{\bf p}_-(s,\tau) \cdot {\bf p}_+(s',\tau) = - \alpha(s',\tau - \delta) + O(s - s') \ . \label{ppa}
\end{equation}
Then
\begin{eqnarray}
\partial_\tau ({\bf p}_+ (s,\tau) \cdot {\bf p}_+(s',\tau)) &=&
\frac{\dot a}{a} \biggl( \alpha(s',\tau - \delta) + \alpha(s,\tau + \delta) \nonumber\\
&&\hspace{-90pt} - \alpha (s,\tau)\,{\bf p}_+(s,\tau)\cdot{\bf p}_+(s',\tau) - \alpha (s',\tau)\,{\bf p}_+(s,\tau)\cdot{\bf p}_+(s',\tau) \biggr) + O([s - s']^2)\ .
\end{eqnarray}
When we integrate over a scale of order the Hubble time, the $\delta$ shifts in the arguments have a negligible effect $O(\delta)$ and so we ignore them. Defining
\begin{equation}
h_{++}(s,s', \tau) = 1 - {\bf p}_+ (s,\tau) \cdot {\bf p}_+(s',\tau)\ ,
\end{equation}
we have
\begin{equation}
\partial_\tau h_{++}(s,s', \tau) =
- \frac{\dot a}{a} h_{++}(s,s', \tau) [ \alpha(s',\tau ) + \alpha(s,\tau) + O([s - s']^2) ] \ .
\end{equation}
Thus
\begin{equation}
h_{++}(s,s', \tau_1) = h_{++}(s,s', \tau_0) \exp\Biggl\{ - \nu' \int_{\tau_0}^{\tau_1} \frac{d\tau}{\tau}
[ \alpha(s',\tau ) + \alpha(s,\tau) + O([s - s']^2) ] \label{hppint}
\Biggr\} \ .
\end{equation}
Averaging over an ensemble of segments, and integrating over many Hubble times (and therefore a rather large number of correlation times) the fluctuations in the exponent average out and we can replace $\alpha(s,\tau )$ with $\bar\alpha = 1 - 2\bar v^2$,
\begin{equation}
\langle h_{++}(s,s', \tau_1) \rangle \approx
\langle h_{++}(s,s', \tau_0) \rangle (\tau_1/\tau_0)^{- 2 \nu' \bar\alpha}\ . \label{avexp}
\end{equation}
Note that in contrast to previous equations the approximation here is less controlled. We do not at present have a good means to estimate the error. It depends on the correlation between the small scale and large scale structure (the latter determines the distribution of $\alpha$), and so would require an extension of our methods. We do expect that the error is numerically small; note that if we were to consider instead $\langle \ln h_{++}(s,s', \tau) \rangle$ then the product in Eq.~(\ref{hppint}) would become a sum and the averaging would involve no approximation at all.
Averaging over a translationally invariant ensemble of solutions, we have
\begin{equation}
\langle h_{++}(\sigma - \sigma', \tau_1) \rangle \approx
(\tau_1/\tau_0)^{- 2 \nu' \bar\alpha}\langle h_{++}(\sigma - \sigma', \tau_0) \rangle \ .
\end{equation}
We have used the fact that to good approximation (again in the sense of Eq.~(\ref{avexp})), $\epsilon$ is only a function of time, and so we can choose $\sigma = s - \int d\tau/\epsilon$ and $\sigma-\sigma' = s - s'$. Equivalently,
\begin{equation}
\langle h_{++}(\sigma - \sigma', \tau) \rangle \approx \frac{ f(\sigma - \sigma') }{\tau^{2 \nu' \bar\alpha} }\ . \label{soln}
\end{equation}
The ratio of the segment length to $d_H = (1+\nu') t$ is
\begin{equation}
\frac{l}{d_H} \propto \frac{a \epsilon (\sigma - \sigma')}{t} \propto \frac{\sigma - \sigma'} {\tau^{1 + 2 \nu'\bar v^2 }}\ .
\end{equation}
The logic of our earlier discussion is that we use simulations to determine the value of $h_{++}$ at $l/d_H$ somewhat less than one, and then evolve to smaller scales using the Nambu action. That is,
\begin{equation}
h_{++}(\sigma - \sigma', \tau) = h_0 \ \mbox{when}\ \sigma - \sigma' = x_0 \tau^{1 + 2 \nu'\bar v^2 }\ , \label{match}
\end{equation}
for some constants $x_0$ and $h_0$. We assume scaling behavior near horizon scale, so that $h_0$ is independent of time. Using this as an initial condition for the solution~(\ref{soln}) gives
\begin{equation}
\langle h_{++}(\sigma - \sigma', \tau) \rangle \approx h_0 \Biggl( \frac{\sigma - \sigma'}{x_0 \tau^{1 + 2 \nu'\bar v^2 }}\Biggr)^{2\chi} \approx A (l/t)^{2\chi}\ ,\quad \chi = \frac{\nu'\bar\alpha}{1 + 2 \nu'\bar v^2 }
= \frac{\nu\bar\alpha}{1 - \nu\bar\alpha}
\ . \label{hpp}
\end{equation}
In the last form we have expressed the correlator in terms of physical quantities, the segment length $l$ defined earlier and the FRW time $t$.\footnote{We have not yet needed to specify numerical normalizations for $\sigma$ and $\tau$, or equivalently for $\epsilon$ and $a$. The value of $h_0$ depends on this choice, but the value of $A$ does not.}
Eq.~(\ref{hpp}) is our main result. Equivalently (and using $\sigma$ parity),
\begin{equation}
\langle {\bf p}_+ (\sigma,\tau) \cdot {\bf p}_+(\sigma',\tau) \rangle
= \langle {\bf p}_- (\sigma,\tau) \cdot {\bf p}_-(\sigma',\tau) \rangle
\approx 1 - A (l/t)^{2\chi}
\ .
\label{pppp}
\end{equation}
In the radiation era $\chi_r \sim 0.10$ and in the matter era $\chi_m \sim 0.25$. There can be no short distance structure in the correlator ${\bf p}_+ \cdot {\bf p}_-$, because the left- and right-moving segments begin far separated, and the order $\dot a/a$ interaction between them is too small to produce significant nonsmooth correlation. Thus, from~(\ref{ppa}) we get
\begin{equation}
\langle {\bf p}_+ (\sigma,\tau) \cdot {\bf p}_-(\sigma',\tau) \rangle
= -\bar\alpha + O(\sigma - \sigma')
\ . \label{pppm}
\end{equation}
\subsection{Small fluctuation approximation}
Before interpreting these results, let us present the derivation in a slightly different way. The exponent $\chi$ is positive, so for points close together the vectors $ {\bf p}_+ (\sigma,\tau)$ and ${\bf p}_+(\sigma',\tau) $ are nearly parallel. Thus we can write the structure on a small segment as a large term that is constant along the segment and a small fluctuation:
\begin{equation}
{\bf p}_+ (\sigma,\tau) = {\bf P}_+ (\tau) + {\bf w}_+ (\tau,\sigma) - \frac{1}{2} {\bf P}_+ (\tau)
w_+^2 (\tau,\sigma) + \ldots\ , \label{smallf}
\end{equation}
with $P_+(\tau)^2 = 1$ and ${\bf P}_+ (\tau) \cdot {\bf w_+} (\tau,\sigma) = 0$. Inserting this into the equation of motion~(\ref{EOM2}) and expanding in powers of ${\bf w}_+$ gives
\begin{eqnarray}
\dot{\bf P}_+ &=& - \frac{\dot{a}}{a} \left[ {\bf P}_- - ({\bf P}_+ \cdot {\bf P}_-)\, {\bf P}_+ \right]\ ,
\\[6pt]
\dot{\bf w}_+ - \frac{1}{\epsilon} {\bf w}'_+ &=& \frac{\dot{a}}{a} \left[ ({\bf w}_+ \cdot {\bf P}_-)\, {\bf P}_+ + ({\bf P}_+ \cdot {\bf P}_-)\, {\bf w}_+ \right] \nonumber\\
&=& - ({\bf w}_+ \cdot \dot{\bf P}_+)\, {\bf P}_+ + \frac{\dot{a}}{a} ({\bf P}_+ \cdot {\bf P}_-)\, {\bf w}_+\ .
\label{EOM3}
\end{eqnarray}
Since the right-moving ${\bf p}_-$ is essentially constant during the period when it crosses the short left-moving segment, we have replaced it in the first line of (\ref{EOM3}) with a $\sigma$-independent ${\bf P}_-$. In the second line of (\ref{EOM3}) we have used ${\bf w}_+ \cdot {\bf P}_+ = 0$. In the final equation for ${\bf w}_+$, the first term is simply a precession: ${\bf P}_+$ rotates around an axis perpendicular to both ${\bf P}_+$ and ${\bf P}_-$, and this term implies an equal rotation of ${\bf w}_+$ so as to keep ${\bf w}_+$ perpendicular to ${\bf P}_+$. Eq.~(\ref{EOM3}) then implies that in a coordinate system that rotates with ${\bf P}_+$,
${\bf w}_+$ is simply proportional to $a^{-\alpha}$.
It follows that
\begin{equation}
\left\langle {\bf p}_+ (\sigma,\tau) \cdot {\bf p}_+(\sigma',\tau) \right\rangle - 1 = - \frac{1}{2} \left\langle [{\bf w}_+ (\sigma,\tau) - {\bf w}_+(\sigma',\tau)]^2 \right\rangle
\label{upup}
\end{equation}
scales as $a^{-2\bar\alpha}$ as found above (again we are approximating as in eq.~(\ref{avexp}), and again this statement would be exact if we instead took the average of the logarithm). Similarly the four-point function of ${\bf w}_+$ scales as $a^{-4\bar\alpha}$. We have not assumed that the field ${\bf w}_+$ is gaussian; the
$n$-point functions, just like the two-point function, can be matched to simulations near the horizon scale. We can anticipate some degree of nongaussianity due to the kinked structure; we will discuss this further in section~3.2.
\subsection{Discussion}
Now let us discuss our results for the two-point functions. We can also write them as
\begin{eqnarray}
{\rm corr}_x(l,t) \equiv
\frac{ \left<{\bf x}'(\sigma,\tau)\cdot{\bf x}'(\sigma',\tau)\right> }{\left<{\bf x}'(\sigma,\tau)\cdot{\bf x}'(\sigma,\tau)\right>} &\approx& 1 - \frac{A}{2(1 - \bar v^2)} (l/t)^{2\chi}\ ,
\nonumber\\
{\rm corr}_t(l,t) \equiv
\frac{\left< \dot{\bf x}(\sigma,\tau)\cdot \dot{\bf x}(\sigma',\tau)\right>}
{\left< \dot{\bf x}(\sigma,\tau)\cdot \dot{\bf x}(\sigma,\tau)\right>} & \approx&
1 - \frac{A}{2 \bar v^2} (l/t)^{2\chi}\ . \label{corrs}
\end{eqnarray}
These are determined up to two parameters $\bar v^2$ and $A$ that must be obtained from simulations.
A first observation is that these scale, they are functions only of the ratio of $l$ to the horizon scale. This is simply a consequence of our assumptions that the horizon scale structure scales and that stretching is the only relevant effect at shorter scales. We emphasize that these results are for segments on long strings; we will discuss loops in Sec.~4.
It is natural to characterize the distribution of long strings in terms of a fractal dimension.
The RMS spatial distance between two points separated by coordinate distance $\sigma$ is
\begin{eqnarray}
\langle r^2(l,t) \rangle &=& \left<{\bf x}' \cdot{\bf x}' \right> \int_0^l dl' \int_0^l dl''\,
{\rm corr}_x(l' - l'',t) \nonumber\\
&\approx& (1 - \bar v^2) l^2 \left[ 1 -
\frac{A (l/t)^{2\chi}}{(2\chi+1)(2\chi+2)(1 - \bar v^2)} \right]\ .
\label{r2}
\end{eqnarray}
We can then define the fractal dimension $d_f$ (which is 1 for a straight line and 2 for a random walk),
\begin{eqnarray}
d_f &=& \frac{2 d \ln l}{d\ln \langle r^2(l,t) \rangle} \nonumber\\
&\approx& 1 + \frac{A \chi (l/t)^{2\chi}}{(2\chi+1)(2\chi+2)(1 - \bar v^2)} + O((l/t)^{4\chi})\ .
\label{fracdim}
\end{eqnarray}
The fractal dimension approaches 1 at small scales: the strings are rather smooth.
There is a nontrivial scaling property, not in the fractal dimension but rather in the deviation of the string from straightness,
\begin{equation}
1 - {\rm corr}_x(l,t) \propto (l/t)^{2\chi}\ , \quad
1 - {\rm corr}_t(l,t) \propto (l/t)^{2\chi}\ . \label{crit}
\end{equation}
We define the scaling dimension $d_s = 2\chi$. Note that $d_s$ is not large, roughly $0.2$ in the radiation era and $0.5$ in the matter era, so the approach to smoothness is rather slow.
\begin{figure}
\center \includegraphics[width=25pc]{model_vs_data_rad.eps}
\caption[]{Comparison of the model (dashed line) with the data provided by~\cite{MarShell2005} (solid red line) in the radiation-dominated era, for which the correlation length is $\xi \simeq 0.30 t$. } \label{mvd rad}
\end{figure}
Our general conclusions are in agreement with the recent simulations of Ref.~\cite{MarShell2005}, in that the fractal dimension approaches 1 at short distance. To make a more detailed comparison it is useful to consider a log-log plot of $1 - {\rm corr}_x(l,t)$ versus $l$, as suggested by the scaling behavior~(\ref{crit}); we thank C. Martins for replotting the results of Ref.~\cite{MarShell2005} in this form. The comparison is interesting. At scales larger than $d_{\rm H}$ (which is $\sim 6.7 \xi$ in the radiation era and $\sim 4.3 \xi$ in the matter era)
the correlation goes to zero. Rather abruptly below $d_{\rm H}$ the slope changes and agrees reasonably well with our result. It is surprising to find agreement at such long scales where our approximations do not seem very precise. On the other hand, at shorter scales where our result should become more accurate, the model and the simulations diverge; this is especially clear at the shortest scales in the radiation-dominated era (Fig.~\ref{mvd rad}).
\begin{figure}
\center \includegraphics[width=25pc]{model_vs_data_mat.eps}
\caption[]{Comparison of the model (dashed line) with the data provided by~\cite{MarShell2005} (solid blue line) in the matter-dominated era, for which the correlation length is $\xi \simeq 0.69t$.} \label{mvd mat}
\end{figure}
One possible explanation for the discrepancy is transient behavior in the simulations. We have argued that the structure on the string is formed at the horizon scale and `propagates' to smaller scales (in horizon units) as the universe expands. In Ref.~\cite{MarShell2005} the horizon size increases by a factor of order 3, and so even if the horizon-scale structure forms essentially at once, the maximum length scale over which it can have propagated is $3^{1 - \zeta}$, less than half an order of magnitude. At smaller scales, the small scale structure seen numerically would be almost entirely determined by the initial conditions. On the other hand, the authors of Ref.~\cite{MarShell2005} (private communication) argue that their result appears to be an attractor, independent of the initial conditions, and that loop production may be the dominant effect.
Motivated by this we will examine loop production Sec.~4. Indeed, we will find that this is in some ways a large perturbation, but we are still unable to identify a mechanism that would produce the two-point function seen in the simulations. This is an important issue to be resolved in future work.
Thus far we have discussed ${\rm corr}_x$. Our result~(\ref{corrs}) implies a linear relation between ${\rm corr}_x$ and ${\rm corr}_t$. In fact, this holds more generally from the argument that there is no short-distance correlation between $ {\bf p}_+$ and ${\bf p}_-$, Eq.~(\ref{pppm}):
\begin{eqnarray}
(1-\bar v^2 ) {\rm corr}_x(l,t) - \bar v^2 {\rm corr}_t(l,t) &=& -\frac{1}{2}
\Bigl\langle {\bf p}_+ (\sigma,\tau) \cdot {\bf p}_-(\sigma',\tau) + {\bf p}_- (\sigma,\tau) \cdot {\bf p}_+(\sigma',\tau) \Bigr\rangle \nonumber\\
&\to& \bar\alpha\ .
\end{eqnarray}
Inspection of Fig.~2 of Ref.~\cite{MarShell2005} indicates that this relation holds rather well at all scales below $\xi$.
The small scale structure on strings is sometimes parameterized in terms of an effective tension~\cite{Vilenkin:1990mz,Carter:1990nb}. For a segment of length $l$ the effective tension is given by
\begin{eqnarray}
\frac{ \mu_{\rm eff} }{ \mu } = \frac{\sqrt{1 - \overline{v}^2} l}{\langle r(l) \rangle }
\approx 1 + \frac{A}{2(1 - \overline{v}^2)(2\chi+1)(2\chi+2)}\left( \frac{l}{t} \right)^{2\chi} \ ,
\end{eqnarray}
where we have made use of result~(\ref{r2}). Note that this is strongly dependent on the scale $l$ of the coarse-graining.
In conclusion, let us emphasize the usefulness of the log-log plot of $1 - {\rm corr}_x$. In a plot of the fractal dimension, all the curves would approach one at short distance, though at slightly different rates. The difference is much more evident in Figs.~1 and~2, and gives a clear indication either of transient effects or of some physics omitted from the model.
\subsection{The matter-radiation transition}
In this subsection we will assume that our stretching model is actually valid down the scale where gravitational radiation sets in. If loop production or other relatively rapid processes are actually determining the small scale structure then this subsection is moot.
We have noted that at very short scales we see structure that actually emerged from the horizon dynamics during the radiation era. Thus we should take the radiation-to-matter transition into account in our calculation of the small-scale structure. At the time of equal matter and radiation densities,
\begin{equation}
1 - \langle {\bf p}_+ (\sigma,\tau_{\rm eq}) \cdot {\bf p}_+(\sigma',\tau_{\rm eq}) \rangle
\approx A_{\rm r} (l_{\rm eq}/t_{\rm eq})^{2\chi_{\rm r}}
\ , \label{teq}
\end{equation}
where $l_{\rm eq}$ is the length of the segment between $\sigma$ and $\sigma'$ at $t_{\rm eq}$.
Assuming that the transition from radiation-dominated to matter-dominated behavior is sharp (which is certainly an oversimplification), we evolve forward to today using the result~(\ref{soln}). The right-hand side of Eq.~(\ref{teq}) is then multiplied by a factor $(t/t_{\rm eq})^{-2\nu_{\rm m}\bar\alpha_{\rm m}}$. In terms of the length today, $l = l_{\rm eq} (t/t_{\rm eq})^{\zeta_{\rm m}}$, we have
\begin{equation}
1 - \langle {\bf p}_+ (\sigma,\tau) \cdot {\bf p}_+(\sigma',\tau) \rangle
\approx A_{\rm r} (l/t)^{2\chi_{\rm r}} (t/t_{\rm eq})^{-2\zeta_{\rm m}
- 2\chi_{\rm r}\zeta_{\rm m} + 2\chi_r}
\ . \label{radmat}
\end{equation}
This expression applies to scales $l(t)$ that, evolved backward in time, reached the horizon scale $d_H$ before the transition occurred, i.e. at a time $t_*$ defined by $l(t_*) \sim d_H$ such that $t_* < t_{\rm eq}$. For longer scales, which formed during the matter era (for which $t_* > t_{\rm eq}$), we have simply
\begin{equation}
1 - \langle {\bf p}_+ (\sigma,\tau) \cdot {\bf p}_+(\sigma',\tau) \rangle
\approx A_{\rm m} (l/t)^{2\chi_{\rm m}}
\ . \label{radmat2}
\end{equation}
\begin{figure}
\center \includegraphics[width=25pc]{transition.eps}
\caption[]{Structure on the string, $\langle h_{++} \rangle$, as a function of the length $l$ at present time (solid curve). On scales larger than the critical length $l_{\rm c} \sim 3\times 10^{-5} d_H$ the structure is determined by the matter era expression. On scales below $l_{\rm c}$ the transition result~(\ref{radmat}) gives an enhanced effect. The dashed curves show the extrapolations of the two relevant expressions: on small scales the actual structure is enhanced relative to the pure matter era result.} \label{transition}
\end{figure}
The transition between the two forms occurs along the curve determined by the intersection of the two surfaces~(\ref{radmat}) and~(\ref{radmat2}). This determines the critical length at the time of equal matter and radiation densities, $l_{\rm c}(t_{\rm eq}) = (A_{\rm r}/A_{\rm m})^{1/(2\chi_{\rm m}-2\chi_{\rm r})}t_{\rm eq}$. In terms of the length at some later time $t$, the transition occurs at
\begin{equation}
\frac{l_{\rm c}(t)}{t} \approx \left(\frac{A_{\rm r}}{A_{\rm m}}\right)^{1/(2\chi_{\rm m}-2\chi_{\rm r})} \left(\frac{t}{t_{\rm eq}}\right)^{\zeta_{\rm m}-1}
\ , \label{radmat3}
\end{equation}
so that the transition scale at the present time is $l_{\rm c} \sim 3\times 10^{-5} d_H$ (Fig.~\ref{transition}). The transition result~(\ref{radmat}) implies more structure at the smallest scales than would be obtained from the matter era result, by a factor $(l_{\rm c}/l)^{2(\chi_{\rm m} - \chi_{\rm r})} \sim (l_{\rm c}/l)^{0.3}$.
Of course, precise studies of the small scale structure must include also the effect of the recent transition to vacuum domination; this period has been brief so the effect should be rather small.
\sect{Lensing}
Let us now consider the effect of the small scale structure on the images produced by a cosmic string lens. Previous work has discussed possible dramatic effects~\cite{DeLaix1997,Bernardeau:2000xu}, including multiple images and large distortions. We can anticipate that the rather smooth structure that we have found, which again we note is subject to our assumptions, will produce images with only small distortion. We will use our stretching model of the two-point function. If this proves incorrect one could apply the analysis using phenomenological values of $\chi$ and $A$; for example, the extrapolation of the results of
Ref.~\cite{MarShell2005} give a smoother string, and even less distorted images.
\subsection{Distortion of images}
We quote the result of ref.~\cite{DeLaixVach1996} for the angular deflection of a light ray by a string,
\begin{equation}
\mbox{\boldmath $\gamma$}_\perp({\bf y}_{\perp}) = 4G\mu \int d\sigma \Biggl[ \frac{F_{\mu\nu} \gamma^\mu_{(0)} \gamma^\nu_{(0)}}{1 - \dot x_\parallel} \frac{ {\bf x}_\perp - {\bf y}_{\perp} }{
| {\bf x}_\perp - {\bf y}_{\perp} |^2 } \Biggr]_{t = t_0(\sigma)}\ .
\end{equation}
Here $\gamma_{(0)}^\mu$ is the four-velocity of the unperturbed light ray, which we take to be $(1,0,0,1)$ as shown in Fig.~\ref{schematic}, $y^\mu$ is a reference point on this ray, and subscripts $\perp$ and $\parallel$ are with respect to the spatial direction of the ray. Also, $x^\mu(\sigma,t)$ is the string coordinate,\footnote{
The region where the the light ray passes the string is small on a cosmological scale, so to use our earlier results we can locally set $a = \epsilon = 1$, $dl = d\sigma$, $\partial_t = \partial_\tau$.}
in terms of which
\begin{equation}
F^{\mu\nu} = \dot x^\mu \dot x^\nu - x^{\mu\prime} x^{\nu\prime} - \frac{1}{2}\eta^{\mu\nu}
(\dot x^\rho \dot x_\rho - x^{\rho\prime} x_\rho^{\prime})\ ,
\end{equation}
and $t_0(\sigma)$ is defined by $t_0(\sigma) = x_3(\sigma,t_0) - y_3$.
\begin{figure}
\center \includegraphics[width=25pc]{schematic.eps}
\caption[]{Schematic representation of the system considered, with the string lens displayed along the $x_1$-axis and the distant source and the observer located at points $S$ and $O$ respectively.} \label{schematic}
\end{figure}
In keeping with Sec.~2.3 we separate the string locally into a straight part and a fluctuation; we will keep the deflection only to first order in the fluctuation. We consider here only the simplest geometry, in which the straight string is perpendicular to the light ray and at rest, so that ${\bf P}_+ = -{\bf P}_- = (1,0,0)$. To first order in the fluctuation, $x^\mu(\sigma,t) = (t,\sigma,x_2(\sigma,t),x_3(\sigma,t))$. One then finds
\begin{eqnarray}
\gamma_1({\bf y}_{\perp}) &=& 4 G \mu \int d\sigma\, \frac{- \dot x^{\vphantom2}_3(\sigma,t_0) (\sigma - y_1) + x_2^{\prime}(\sigma,t_0) y_2}{(\sigma - y_1)^2 + y_2^2} \ ,
\nonumber\\
\gamma_2({\bf y}_{\perp}) &=& - {\rm sgn}(y_2) \beta + 4 G \mu \int d\sigma\, \frac{\dot x^{\vphantom2}_3(\sigma,t_0) y_2 + x_2^{\prime}(\sigma,t_0) (\sigma - y_1)}{(\sigma - y_1)^2 + y_2^2}
\ ,
\end{eqnarray}
with $\beta = 4\pi G\mu$ being half the deficit angle of the string.
To the order that we work $t_0$ is a constant, corresponding to the time when the light ray is perpendicular to the straight string.
We can use our results for the small scale structure to calculate the two-point functions of the deflection. Let us focus on the local magnifications parallel and perpendicular to the string, given by basic lensing theory as
\begin{eqnarray}
M_1 ({\bf y}_{\perp}) &=& 1 - \frac{D_l (D_o - D_l)}{D_o} \frac{\partial\gamma_1}{\partial y_1} ({\bf y}_{\perp})\ ,\nonumber\\
M_2 ({\bf y}_{\perp}) &=& 1 - \frac{D_l (D_o - D_l)}{D_o} \frac{\partial\gamma_2}{\partial y_2}
({\bf y}_{\perp})\ .
\end{eqnarray}
Here $D_o$ and $D_l$ are the distances of the source from the observer and the lens respectively (these would be the angular diameter distances on cosmological scales). It is particularly interesting to consider the differential magnifications for the two images produced by a string,
\begin{equation}
\delta M_1 = M_1 ({\bf y}_{\perp}) - M_1 ({\bf y}'_{\perp})
\ ,\quad
\delta M_2 = M_2 ({\bf y}_{\perp}) - M_2 ({\bf y}'_{\perp})\ .
\end{equation}
We take for simplicity ${\bf y}_{\perp} = (0, b)$ and ${\bf y}'_{\perp} = (0, -b)$ for $b = \beta D_l (D_o - D_l)/D_o$; this corresponds to the symmetric images of an object directly behind the string. Then
\begin{equation}
\delta M_1 = - \delta M_2 = -\frac{2b^2}{\pi} \int d\sigma\, x_2^{\prime}(\sigma,t_0) \partial_\sigma \left( \frac{1}{\sigma^2 + b^2} \right)\ .
\end{equation}
From Sec.~2 we obtain
\begin{eqnarray}
\left\langle {\bf w}_+ (\sigma,t) \cdot {\bf w}_+(\sigma',t) \right\rangle
&=& \left\langle {\bf w}_- (\sigma,t) \cdot {\bf w}_-(\sigma',t) \right\rangle
= 4 \left\langle x_2'(\sigma,t) x_2'(\sigma',t) \right\rangle
\nonumber\\
&=& A \Bigl\{ (\sigma/t)^{2\chi} + (\sigma'/t)^{2\chi} - ([\sigma - \sigma']/t)^{2\chi} \Bigr\}\ .
\label{u2p}
\end{eqnarray}
Note that Eq.~(\ref{upup}) does not fully determine the two-point functions~(\ref{u2p}), and in fact that the latter cannot be translation invariant. We have fixed the ambiguity by defining the expectation value to vanish when $\sigma$ or $\sigma'$ vanishes (that is, at the point on the string nearest to the light ray); this amounts to a choice of how one splits ${\bf p}_\pm$ into ${\bf P}_\pm$ and ${\bf w}_\pm$.
As is stands, equation (\ref{u2p}) applies only when $|\sigma -\sigma'|$ is larger than the critical length $l_{\rm c}$. If this is not the case, one must take into account the radiation-to-matter transition as discussed in section~2.5. According to equation (\ref{radmat}), the RHS of (\ref{u2p}) gets multiplied by a power of $t/t_{\rm eq}$. From $\left\langle x_2' x_2' \right\rangle$ we obtain\footnote{The lensing scale, which is provided by $b$, is typically of the order of $10^{-7}d_H$, thus much smaller than the critical length $l_{\rm c}$. We can then safely use the formula for $\left\langle x_2' x_2' \right\rangle$ valid for the smallest scales over the full range of integration. Corrections from the longest scales will increase the final result, but not significantly.}
\begin{equation}
\left\langle \delta M_1^2 \right\rangle
= \left\langle \delta M_2^2 \right\rangle
= \frac{\chi_{\rm r}(1-2\chi_{\rm r})}{2 \cos (\pi\chi_{\rm r})} A_{\rm r} (2b/t_0)^{2\chi_{\rm r}}(t_0/t_{\rm eq})^{-2\zeta_{\rm m}
- 2\chi_{\rm r}\zeta_{\rm m} + 2\chi_r}\ .
\end{equation}
Plugging in the numeric values for $\chi$, $A$ and $\zeta$ and using a representative value for the dimensionless parameter $G\mu \sim 10^{-7}$, we obtain a RMS differential magnification slightly below $1\%$:
\begin{equation}
\left\langle \delta M^2 \right\rangle^{1/2}
\simeq 0.009\ .
\end{equation}
\subsection{Alignment of lenses and nongaussianity}
Another question related to short-distance structure is the alignment of lenses. Suppose we see a lens due to a long string, with a certain alignment. Where should we look for additional lens candidates? Previous discussions~\cite{Huterer:2003ze,Oguri:2005dt} have considered the two extreme cases of a string that is nearly straight, and a string that is a random walk on short scales; clearly the networks that we are considering are very close to the first case.
We keep the frame of the previous section, with the lens at the origin in ${\bf x}_\perp$ and aligned along the 1-axis. Then as we move along the string, the RMS transverse deviation is
\begin{eqnarray}
\langle x_2^2(l) \rangle &=& \int_0^l d\sigma' \int_0^l d\sigma''\, \langle x'_2(\sigma)
x'_2(\sigma') \rangle \nonumber\\
&=& \frac{A l^2}{4(\chi + 1)} (l/t)^{2\chi}\ .
\end{eqnarray}
The extension in the $x$ direction is just $l$, so the RMS angular deviation is
\begin{equation}
\delta\varphi \sim
\sqrt{A / 4(\chi + 1)} (l/t)^{\chi} \equiv {\overline{\delta\varphi}}\ .
\end{equation}
If we put in representative numbers, looking at an apparent separation on the scale of arc-minutes for a lens at a redshift of order $0.1$, we obtain with the matter era parameters a deviation ${\delta\varphi} \sim 0.05$ radians. That is, any additional lenses should be rather well aligned with the axis of the first. If the string is tilted by an angle $\psi$ to the line of sight, then projection effects increase $\delta\theta$ and $\delta\varphi$ by a factor $1/\cos\psi$. Of course, for lensing by a {\it loop}, the bending will be large at lengths comparable to the size of the loop.
Lens alignment provides an interesting setting for discussing the nongaussianity of the structure on the string. If the fluctuations of $x_2'$ were gaussian, then the probability of finding a second lens at an angle $\delta\varphi$ to the axis of the first would be proportional to $e^{-\delta\varphi^2/2\overline{\delta\varphi}^2}$, and therefore very small at large angles. However, we have considered thus far a typical string segment, which undergoes only stretching. There will be a small fraction of segments that contain a large kink, and one might expect that it is these that dominate the tail of the distribution of bending angles.
Let us work this out explicitly. Consider a left-moving segment of coordinate length $\sigma$, and let $P(\sigma,\tau,k)\, dk$ be the probability that it contain a kink for which the discontinuity $|{\bf p}_+ - {\bf p}_+'|$ lies between $k$ and $k+dk$ ($0 < k < 2$). There are two main contributions to the evolution of $P$. Intercommutations introduce kinks at a rate that we assume to scale, so that it is proportional to the world-sheet volume in horizon units, $a^2 \epsilon \sigma \tau^{-2\nu'/\nu} d\tau$, and to some unknown function $g(k)$. Also, the expansion of the universe straightens the kink, $k \propto a^{-\bar\alpha}$~\cite{BB1989}. Then\footnote{Other effects are often included in the discussion of kink density, such as the removal of kinky regions by loop formation~\cite{AllenCald1990}, but these have a small effect.}
\begin{equation}
\frac{\partial P}{\partial \tau} = \tau^{2\nu' (1 - \bar v^2 - 1/\nu)}
\sigma g(k) + \bar\alpha \frac{\dot a}{a} \frac{\partial}{\partial k}(kP)\ . \label{pevol}
\end{equation}
We set $P$ to zero at the matching time $\tau_0$ defined by
\begin{equation}
\sigma= x_0 \tau_0^{1 + 2 \nu'\bar v^2 }\ ,
\end{equation}
as in Eq.~(\ref{match}): earlier kinks are treated as part of the typical distribution, while $P$ identifies kinks that form later. For simplicity we assume that $x_0$ is small enough that the probability of more than one kink can be neglected.
To solve this, define
\begin{equation}
\kappa = k \tau^{\zeta'}\ , \quad
Q(\sigma,\tau,\kappa) = k P(\sigma,\tau,\kappa)\ .
\end{equation}
Then
\begin{equation}
\tau \partial_\tau Q = \sigma g(\kappa \tau^{-\zeta'}) \kappa \tau^{-1-\nu'}\ .
\end{equation}
This can now be integrated to give
\begin{eqnarray}
P(\sigma,\tau,\kappa) &=&
\sigma \tau^{\zeta'} \int_{\tau_0}^\tau \frac{d\tau'}{\tau'} \tau'^{-1-\nu'}
g(k \tau^{\zeta'} /\tau'^{\zeta'}) \nonumber\\
&=& \frac{\tilde\sigma}{\zeta'} k^{-1/\zeta}
\int_k^{k_0} \frac{dk'}{k'} k'^{1/\zeta} g(k')\ . \label{psol}
\end{eqnarray}
Here $k_0 = k (x_0 /\tilde\sigma)^{\zeta'/(1+2\nu'\bar v^2)}$ and $\tilde\sigma = \sigma/\tau^{1 + 2 \nu'\bar v^2}$. Note that $\tilde\sigma$ is just a constant times $l/t$, so the probability distribution scales. The source $g(k)$ vanishes by definition for $k > 2$, so $k_0 > 2$ is equivalent to $k_0 = 2$.
Rather than the angle $\delta\varphi$ between the axis of the first lens and the position of the second lens, it is slightly simpler to consider the angle $\delta\theta$ between the two axes. In the small fluctuation approximation this is just $x_2'(\sigma)$, and so the RMS fluctuation is
\begin{equation}
\overline{\delta\theta} = \sqrt{A/2}(l/t)^\chi\ .
\end{equation}
In the gaussian approximation, the probability distribution is $e^{-\delta\theta^2/2\overline{\delta\theta}^2}$ and so is very small for large angles. On the other hand, a large angle might also arise from a segment that happens to contain a single recent kink. Treating the segment as straight on each side of the kink, the probability density is then precisely the function $P(\sigma,\tau,k)\, dk$ just obtained, with $k = \delta\theta$. If we consider angles that are large compared to $\overline{\delta\theta}$ but still small compared to 1, the range of integration in the solution~(\ref{psol}) extends essentially to the full range $0$ to $2$ and so the integral gives a constant. Then
\begin{equation}
P(\sigma,\tau,k)\, dk \propto k^{-1/\zeta} dk = \delta\theta^{-1/\zeta} d\delta\theta \ .
\end{equation}
Thus the tail of the distribution is not gaussian but a power law, dominated by segments with a `recent' kink. One finds the same for the distribution of $\delta\varphi$. It is notable however that the exponent in the distribution is rather large, roughly 4 in the matter era and 10 in the radiation era. Thus the earlier conclusion that the string is rather straight still holds. The sharp falloff of the distribution also suggests that a gaussian model might work reasonably well, and indeed we will employ this in the next section.
\sect{Loop formation}
The model presented thus far assumes that stretching is the dominant mechanism governing the evolution of the small scale structure. Motivated by the discrepancy with Ref.~\cite{MarShell2005} (and at the suggestion of those authors), we now consider the production of small loops. Given the smoothness of the strings on short distances as found above, one might have thought that loop production on small scales would be suppressed, but we will see that this is not the case. We will ignore loop reconnection, based on the standard argument that this is rare for small loops~\cite{Bennett:1989yp}.
We will treat the stretching model as a leading approximation, and add in the loop production as a perturbation. However, we will find that the loop production diverges at small scales, and so the problem becomes nonlinear and our analytic approach breaks down. Ultimately we would expect that some sort of improved scaling argument can be used to follow the structure to still smaller scales, but this is beyond our current methods; it is necessary first to identify the relevant processes.
\subsection{The rate of loop production}
In this section we aim to determine the rate of loop production, which occurs when a string self-intersects. That is, we want to compute the number of loops $d\cal N$ with invariant length between $l$ and $l+dl$ originating from a self-intersection at a string coordinate between $\sigma$ and $\sigma + d\sigma$ during a time interval $(t,t+dt)$. Loops much smaller than the horizon size evolve in an essentially flat space so we are able to use null coordinates $u = t + \sigma$ and $v = t - \sigma$. In this case the left- and right-moving vectors ${\bf p}_\pm$ depend only on $u$ or $v$ respectively and the condition for the formation of a loop of length $l$ is ${\bf L}_+ = {\bf L}_-$ with
\begin{eqnarray}
{\bf L}_+(u,l) &\equiv& \int_u^{u+l} du'\, {\bf p}_+(u') \ , \nonumber \\
{\bf L}_-(v,l) &\equiv& \int_{v-l}^v dv'\, {\bf p}_-(v') \ .
\label{loopcondition}
\end{eqnarray}
Hence the number of loops formed is given by
\begin{equation}
d{\cal N} = \delta^{(3)}({\bf L}_+(u,l) - {\bf L}_-(v,l)) \left| \det {\bf J} \right| du\, dv\, dl \ ,
\label{dN}
\end{equation}
where ${\bf J}$ is the Jacobian for the transformation $(u,v,l) \rightarrow {\bf L}_+ - {\bf L}_-$. This formalism is as in Refs.~\cite{Embacher:1992zk,ACK1993}.
Expanding around the stretching result in the form of section~2.3, we have the functional probability distribution
\begin{equation}
{\cal P}[{\bf p}_+, {\bf p}_-] \approx {\cal P}_0({\bf P}_+, {\bf P}_-)
{\cal P}_+[{\bf w}_+] {\cal P}_-[{\bf w}_-]\ .
\end{equation}
That is, with stretching alone the left- and right-movers cannot be correlated on small scales. We will further assume that ${\cal P}_\pm$ are gaussian with variance given by the two-point function. As discussed in Sec.~3.1 this two-point function is not completely determined by Eq.~(\ref{upup}), and in general we have
\begin{eqnarray}
\left\langle {\bf w}_+ (u') \cdot {\bf w}_+(u'') \right\rangle
&=& - A |u' - u''|^{2\chi}/t^{2\chi} + f(u') + f(u'') \nonumber\\
&\equiv& 2G(u' ,u'') \ .
\label{wpwp}
\end{eqnarray}
The function $G$ is defined for later reference, with the factor of 2 representing the sum over transverse directions. To specify the function $f$ one needs to choose how to split ${\bf p}_\pm$ into ${\bf P}_\pm$ and ${\bf w}_\pm$. The natural choice here is to make ${\bf P}_\pm$ parallel to ${\bf L}_\pm$, i.e.\ ${\bf P}_\pm = {\bf L}_\pm/L_{\pm} $. The transverse fluctuations along the segment that forms the loop, say $0 < \sigma < l$, then have vanishing average,
\begin{equation}
\int_u^{u+l} du'\, {\bf w}_+(u') = \int_{v-l}^v dv'\, {\bf w}_-(v') = {\bf 0} \ . \label{vanav}
\end{equation}
Integrating Eq.~(\ref{wpwp}) $\int_u^{u+l} du'$ and using the condition~(\ref{vanav}) to set this to zero gives $f(u') = \phi(u'-u)$ where
\begin{equation}
\phi(\sigma) = \frac{A}{(2\chi+1) l t^{2\chi}} \left[ \sigma^{2\chi+1} + (l-\sigma)^{2\chi+1} - \frac{l^{2\chi+1}}{2\chi+2} \right] \ .
\label{fsigma}
\end{equation}
Similarly, $\left\langle {\bf w}_- (v') \cdot {\bf w}_-(v'') \right\rangle = 2G(v',v'')$ with $f(v') = \phi(v-v')$.
The three-dimensional $\delta$-function in Eq.~(\ref{dN}) can be expressed in spherical coordinates as
\begin{equation}
\delta^{(3)}({\bf L}_+ - {\bf L}_-) = \frac{1}{L_+^2} \delta(L_+ - L_-)\, \delta^{(2)}({\bf P}_+ -{\bf P}_-)\ .
\label{deltas}
\end{equation}
We take the $z$-direction to be aligned along ${\bf P}_+ = {\bf P}_-$,
in which case the Jacobian matrix is given by
\begin{equation}
{\bf J} = \left(
\begin{array}[tcb]{ccc}
w^x_+(l) - w^x_+(0) & \; w^y_+(l) - w^y_+(0) \; & \frac{1}{2}\!\left[ w^2_+(0) - w^2_+(l) \right] \\ \\
w^x_-(0) - w^x_-(l) & \; w^y_-(0) - w^y_-(l) \; & \frac{1}{2}\!\left[ w^2_-(l) -w^2_-(0) \right] \\ \\
w^x_+(l) - w^x_-(0) & \; w^y_+(l) - w^y_-(0) \; & \frac{1}{2}\!\left[ w^2_-(0) -w^2_+(l) \right]
\end{array}
\right) \ .
\end{equation}
For notational simplicity we have, after differentiating, set $u=0$ and $v=l$.
The fluctuations ${\bf w}_\pm$ effectively live in the $x$-$y$ plane due to the orthogonality condition ${\bf P}_\pm \cdot {\bf w}_\pm = 0$.
Let us first estimate the scaling of $d{\cal N}$ with $l$. We have
\begin{equation}
L_+ - L_- = \frac{1}{2} \int_0^l d\sigma \left(w_+^2(\sigma) - w_-^2(\sigma) \right) =
O(l^{2\chi + 1} / t^{2\chi})\ . \label{lpm}
\end{equation}
The width of the distribution of $L_+ - L_-$ is of order $l^{1+2\chi}$ and so the average of $\delta(L_+ - L_-)$ is a density of order $l^{-1-2\chi}$. The $\delta$-function~(\ref{deltas}) then scales as $l^{-3-2\chi}$ while the Jacobian scales as $l^{4\chi}$ (the columns are respectively of orders $l^{\chi}$, $l^{\chi}$, $l^{2\chi}$) , giving in all
\begin{equation}
d{\cal N} \propto \frac{dl}{l^{3 - 2\chi}}\ ,\quad
l \,d{\cal N} \propto \frac{dl}{l^{2 - 2\chi}}\ .
\label{dNscale}
\end{equation}
Thus, for $2\chi \leq 1$ (as found in both the matter and radiation eras), the total rate of string loss diverges at small $l$: the calculation breaks down.
We will discuss this cutoff further in the next subsection, but first we estimate the numerical coefficient. We see no reason for a strong correlation between the $\delta$-function and Jacobian factors in $d{\cal N}$, so we take the product of their averages. For the radial part of the $\delta$-function, a gaussian average gives
\begin{eqnarray}
\left< \delta(L_+ - L_-) \, \right> &=& \int_{-\infty}^\infty \frac{dy}{2\pi} \,
\left< e^{iy(L_+ - L_-)}\, \right> \nonumber\\
&=& \int_{-\infty}^\infty \frac{dy}{2\pi} \,
e^{ iy \left< (L_+ - L_-) \right>_c - \frac{y^2}{2} \left< (L_+ - L_-)^2 \right>_c
- i \frac{y^3}{6} \left< (L_+ - L_-)^3 \right>_c + \ldots }
\nonumber\\
&=& \int_{-\infty}^\infty \frac{dy}{2\pi} \,
e^{-y^2 R(\chi) + O(y^3) } \nonumber\\
&\approx& \frac{1}{\sqrt{4\pi R(\chi)}} \ .
\end{eqnarray}
The subscripts $c$ in the second line refer to the connected expectation values, obtained by contracting the gaussian fields ${\bf w}_\pm$ with the propagator~(\ref{wpwp}). We have defined
\begin{eqnarray}
R(\chi) &=& \int_0^l d\sigma
\int_0^l d\sigma'\,G^2(\sigma,\sigma',t) = \frac{l^{2 + 4\chi}}{t^{4\chi}} A^2 C(\chi)\ ,
\\
C(\chi) &=& \textstyle \frac{1}{4 (1 + 2\chi)^2}
\left({\frac{1 + 2\chi}{1 + 4\chi} + \frac{1}{(1 + \chi)^2} - \frac{4}{3 + 4\chi} - \frac{ 4 \Gamma^2(2 + 2\chi)}{\Gamma(4 + 4\chi)} } \right)\ .
\end{eqnarray}
Numerically $C(\chi) = (0.0049, 0.0106, 0.0111)$ for $\chi = (0.1, 0.25, 0.5)$.
We now turn to the angular part of the probability distribution function. This must be a function of ${\bf P}_+ \cdot {\bf P}_-$, and we will take it to have exponential form~\cite{ACK1993}:
\begin{equation}
{\cal P}({\bf P}_+ , {\bf P}_-) = \frac{\lambda}{(4\pi)^2 \sinh \lambda} \, e^{-\lambda {\bf P}_+ \cdot {\bf P}_-} \ .
\label{expform}
\end{equation}
The prefactor normalizes the distribution. The parameter $\lambda$ can be determined from the requirement that $ \left< \, {\bf P}_+ \cdot {\bf P}_- \right> = - \overline{\alpha} $ , or equivalently
\begin{equation}
\frac{1}{\lambda} - \frac{\cosh \lambda}{\sinh \lambda} \simeq - \overline{\alpha} \ .
\end{equation}
Thus, $\lambda \sim 3 \overline{\alpha}$ . More precisely, $\lambda$ is $0.55$ in the radiation epoch and $0.95$ in the matter epoch.
We are now able to calculate the average value of (\ref{deltas}). Recognizing that $ \left< L_+^2 \right> \simeq l^2 $ we have
\begin{eqnarray}
\left< \delta^{(3)}({\bf L}_+ - {\bf L}_-) \right> &\approx&
\frac{\lambda \left< \delta(L_+ - L_-) \, \right>}{(4\pi l)^2 \sinh \lambda}
\int d^2 {\bf P}_+\, d^2{\bf P}_-\, e^{-\lambda {\bf P}_+ \cdot {\bf P}_-} \delta^{(2)}({\bf P}_+ -{\bf P}_-) \nonumber \\
&\approx& \frac{\lambda \, e^{-\lambda} }{4\pi A \sqrt{4\pi C(\chi)} \sinh \lambda \, }\frac{t^{2\chi}}{ l^{3 + 2\chi}} \ .
\label{avdelta}
\end{eqnarray}
We now consider $\left< |\det {\bf J}| \right>$. In the gaussian approximation the relevant probability distribution is
\begin{equation}
{\cal P}_\pm({\bf w}_\pm(l) , {\bf w}_\pm(0)) = \frac{\det {\bf M}_\pm}{(2\pi)^2} \,
\exp \left[ -\frac{1}{2} ({\bf V}_\pm^i)^T {\bf M}_\pm {\bf V}^i_\pm \right] \ ,
\end{equation}
where the index $i$ is summed over the two coordinates $x$ and $y$, and
\begin{equation}
{\bf V}^i_\pm \equiv \left( \begin{array}{c} w^i_\pm(l) \\ w^i_\pm(0) \end{array} \right) \ .
\end{equation}
(That is, the columns and rows of the $2\times 2$ matrix ${\bf M}$ correspond to the points $0$ and $l$, not the index $i$.)
As usual, the whole distribution is determined solely by the two-point functions which have already been determined in~(\ref{wpwp}) and~(\ref{fsigma}),
${\bf M}_\pm^{-1} \delta^{ij}= \left< {\bf V}^i_\pm ({\bf V}^j_\pm)^T \right> $. Thus, one finds that
\begin{equation}
{\bf M}_\pm = \frac{2(t/l)^{2\chi}}{A(1-\chi)}
\left( \begin{array}{cc} 1 & \chi \\ \chi & 1 \end{array} \right)
\;\; , \;\;
\det {\bf M}_\pm = \left( \frac{2}{A} \right)^2 \left(\frac{t}{l}\right)^{4\chi} \frac{1+\chi}{1-\chi}
\ .
\end{equation}
We now have all we need to write down an expression for $\left< |\det {\bf J}| \right>$ . Before we do so, let us perform a simplifying change of variables:
\begin{equation}
\begin{array}{lllllll}
X_\pm \equiv \sqrt{\frac{1+\chi}{1-\chi} \frac{1}{2A}} \left({t}/{l}\right)^\chi
\left[ w^x_\pm(l) + w^x_\pm(0) \right] \ , \\ [7pt]
Y_\pm \equiv \sqrt{\frac{1}{2A}} \left({t}/{l}\right)^\chi
\left[ w^x_\pm(l) - w^x_\pm(0) \right] \ , \\ [7pt]
Z_\pm \equiv \sqrt{\frac{1+\chi}{1-\chi} \frac{1}{2A}} \left({t}/{l}\right)^\chi
\left[ w^y_\pm(l) + w^y_\pm(0) \right] \ , \\ [7pt]
W_\pm \equiv \sqrt{\frac{1}{2A}} \left({t}/{l}\right)^\chi
\left[ w^y_\pm(l) - w^y_\pm(0) \right] \ ,
\end{array}
\end{equation}
under which the expectation value of the loop Jacobian takes a compact form:
\begin{equation}
\left< |\det {\bf J}| \right> = \frac{A^2}{2\pi^4} \left(\frac{l}{t}\right)^{4\chi}
\int d^8{\bf X} \: e^{-{\bf X}^2} \: \left| F({\bf X}) + \frac{1-\chi}{1+\chi} \, G({\bf X}) \right| \ .
\label{integral}
\end{equation}
Here we have defined an 8-dimensional vector ${\bf X}~\equiv~(X_+,X_-,Y_+,Y_-,Z_+,Z_-,W_+,W_-)$ and two functions of it:
\begin{eqnarray}
F({\bf X}) &\equiv& ( Y_+ W_- - Y_- W_+ ) \left( Y_-^2 + W_-^2 - Y_+^2 - W_+^2 \right)\ , \nonumber \\
G({\bf X}) &\equiv& 2 ( W_+ W_- - Y_+ Y_- ) (X_+ - X_-) (Z_+ - Z_-) + \\
& & + \: (Y_+ W_- + Y_- W_+) \left[ (X_+ - X_-)^2 - (Z_+ - Z_-)^2 \right] \ . \nonumber
\end{eqnarray}
Numerically the integral is $\simeq 110$ for the radiation epoch and $\simeq 94$ for the matter epoch; analytic estimates agree.
Finally, we can combine results~(\ref{avdelta}) and~(\ref{integral}). Noticing that $du\, dv = 2 d\sigma\, dt$, we get
\begin{equation}
d \langle {\cal N} \rangle = \frac{c}{t^3} \left( \frac{l}{t} \right)^{2\chi-3} d\sigma \, dt \, dl
\label{loopdensity}
\end{equation}
with $c=0.121$ in the radiation epoch and $c=0.042$ in the matter epoch.
\subsection{The small loop divergence and fragmentation}
The total rate of string loss $\int l \, d \langle {\cal N} \rangle $ diverges at small $l$ for $\chi \leq 0.5$, as is the case in both the radiation and matter eras. The fractal dimension approaches 1 at short distance, but the exponent $\chi$, which characterizes the approach to this limit, indicates a relatively large amount of short distance structure. For example, if we consider the functions $p_{\pm}$ on the unit sphere, then $\delta \sigma \propto \delta p_{\pm}^{1/\chi}$: the effective fractal dimension is $1/\chi$, though the path is not continuous, there are gaps due to kinks.
Of course, the total rate of string loss is bounded, and so for small loops we must be multiply counting. A precise account of this effect will be very complicated; see for example Ref.~\cite{ACK1993}, which parameterizes it in terms of a modification of the exponential distribution~(\ref{expform}) taking the form of a `hole' in the forward direction ${\bf p}_+ = {\bf p}_-$. For the present we just determine the scale at which our calculation {\it must} break down.
In a scaling solution, the total amount of long string in a comoving volume scales as $\ell_\infty \propto a^3/t^2$, while stretching alone would give $\ell_\infty \propto a^{\bar\alpha}$. The rate of string lost to loops must be
\begin{equation}
\frac{1}{\ell_\infty} \Biggl( \frac{\partial \ell_\infty}{\partial t} \Biggr)_{loops} = \frac{2 - 2 \nu(1 + \bar v^2)}{t}
\ .
\end{equation}
Our result for the same quantity, with a cutoff $l > l_*$, is
\begin{equation}
\frac{1}{\ell_\infty} \left( \frac{\partial \ell_\infty}{\partial t} \right)_{loops} = \int_{l_*}^\infty dl \, \frac{c l}{t^3} \left( \frac{l}{t} \right)^{2\chi-3}
= \frac{c}{(1-2\chi) t} \left( \frac{l_*}{t} \right)^{2\chi - 1} \ .
\end{equation}
Equating these gives
\begin{equation}
l_* \sim 0.18 t \label{lstar}
\end{equation}
in both eras.
The cutoff~(\ref{lstar}) is rather large, just an order of magnitude below the horizon scale. However, this is just the beginning of the story. The formation of these large loops does not terminate the process of loop production on smaller scales; this continues on the large loop, unaffected at least initially by the separation of the loop from the infinite string.\footnote{By the same token, the fragmentation process begins even before the large loop pinches off. By causality the reconnection that forms the loop cannot immediately have an effect on the fragmentation process in its interior. Thus the large scale~(\ref{lstar}) may not be seen in the
primary loops as usually defined. Rather, we should follow null rays from the two world-sheet points of primary loop formation backwards to the point where they meet, and define the length of the `causal primary' loop as twice the time difference so as to enclose the entire causally disconnected region.}
Thus we must expect a rather dramatic fragmentation process. The divergence of the rate of loop loss implies that at scales $l$ well below $l_*$ the na\"ive number density of loops of size $\sim l$ is greater than $1/l$: these become dense on the string. Thus we might come to the conclusion that the fragmentation process continues until we end up with all loops at the gravitational radiation scale, the opposite of the na\"ive result~(\ref{lstar}) above.\footnote{The previous study~\cite{Scherrer:1989ha} may provide little guidance. The small scale structure there was limited to 10 harmonics, and neither of the two spectra studied corresponds closely to our case: the less noisy was equivalent to $\chi = 0.5$, where the loop production is only logarithmically divergent, while the more noisy is dominated by harmonics near the cutoff and does not resemble our distribution.}
In fact, we believe that this is not precisely the situation either. Rather, a non-self-intersecting loop will occasionally form, likely an order of magnitude or more smaller than the primary loop, and this length of loop will then be lost to the fragmentation process on smaller scales. If a fraction $1/k$ of the string is lost in this way at each scale, then the total length of string in loops of size less than $l$ will scale as $l^{1/k}$. The exponent may be rather small, so there could be a substantial production of small loops. The divergent rate of loop production implies that our initial attempt to use analytic methods to separate scales has been too simple, but we might hope that a more sophisticated scaling model along the lines just indicated will be successful.
Let us briefly describe an analytic model of the formation of non-self-intersecting loops. When the loop forms, we can define the distributions
\begin{equation}
\rho_\pm({\bf p}) = \frac{1}{l} \oint d\sigma\, \delta^2({\bf p} - {\bf p}_{\pm}(\sigma))\ .
\end{equation}
Because of the high fractal dimensions of the curves ${\bf p}_{\pm}(\sigma)$, we might think of these distributions as reasonably smooth functions. The structure on the high harmonics of the loop will tend to smear these distributions into gaussians, but they will be skewed by the particular random values of the lowest harmonics. Thus the distributions $\rho_+$ and $\rho_-$ will differ. The production of the smallest loops is roughly local in ${\bf p}$, and so it can only continue until whichever of the two densities is smaller has been depleted. This leaves residual distributions
\begin{equation}
\rho_+({\bf p}) - \rho_-({\bf p}) = +\rho^r_+({\bf p}) \ {\rm or}\ {-\rho^r_-}({\bf p})
\end{equation}
according to the sign of the difference at each point. This nonvanishing difference implies a cutoff on the production of the smallest loops, and allows non-self-intersecting loops of finite size to form.
\subsection{Comparison with simulations}
The results that we have found have some notable agreements, and disagreements, with simulations. One success is an apparent agreement with the recent simulations of Ref.~\cite{RSB2005} for the distribution of loop sizes: that reference finds a number density of loops per volume and per length $dn /dl \propto l^{-p}$
with $p = 2.5 \pm 0.1$ in the matter era and $p = 3.0 \pm 0.1$ in the radiation era. We have exponents $3 - 2\chi = 2.5$ and 2.8 respectively. Our exponent is for the production rate rather than the density, but in this regime the density is dominated by recently produced loops and so these are the same.
Let us verify this, and also obtain the relative normalizations for the two quantities. The number of loops per comoving volume of length between $l$ and $l+dl$ is unaffected by the expansion of the universe, and we are assuming that we are at scales where gravitational radiation can be neglected. The number then changes only due to production:
\begin{equation}
\frac{d}{dt}\left( a^3 \frac{dn}{dl} \right) = \frac{a^3}{ \gamma^2 t^2} \frac{c}{l^3} \left( \frac{l}{t} \right)^{2\chi}\ ,
\end{equation}
where we have inserted the string scaling density $1/\gamma^2 t^2$. This integral is dominated by recent times as long as $3\nu - 1 - 2\chi > 0$, as holds in both eras (again, $a \propto t^{\nu}$). Integrating over time then gives
\begin{equation}
\frac{dn}{dl} = \frac{c}{(3\nu - 1 - 2\chi) \gamma^2 t^4} \left( \frac{l}{t} \right)^{-3 + 2\chi}\ .
\end{equation}
In the notation of Ref.~\cite{RSB2005},
\begin{equation}
C_\circ = \frac{c}{(3\nu - 1 - 2\chi) \gamma^2} (1 - \nu)^{-1 -2\chi}\ ,
\end{equation}
where the last factor is from the conversion from $d_H$ to $t$. Using $\gamma = 0.59, 0.30$ respectively in the matter and radiation eras~\cite{MarShell2005}, this gives $C_\circ = 1.25$, 10.3. These are larger than found in Ref.~\cite{RSB2005} by factors of 20 and 260 respectively.
A large difference is not surprising, as we have argued that there will be substantial fragmentation which will move the loops to smaller $l$ where the distributions are larger. However, it remains to be seen whether this can account for such an enormous difference. The fragmentation need not change the scaling as long as the long-string correlation functions remains a power law, because the shapes of the produced loops, and the resulting fragmentation, all scale; thus the agreement of the loop production exponents may be real.
As we have noted, the loop scaling must eventually break down due to conservation of string, but with these reduced normalizations this happens at much lower scales, $\bar l_* \sim 4 \times 10^{-4}\, t$ and $2 \times 10^{-4}\, t$ respectively. Below these scales we expect the distribution to fall as discussed in the fragmentation discussion.
Thus we might expect the distribution to peak near these scales, but perhaps to be rather broad in both directions. A possible phenomenological formula for loop production after fragmentation would be
\begin{equation}
\frac{ d {\cal N} }{d\sigma \, dt \, dl} = \frac{\bar c}{t^3} \frac{x^{2\chi-3} }
{1 + (x/x_*)^{2\chi -1 - 1/k} }\ , \quad x = l/t\ .
\end{equation}
The constant $\bar c$ is the reduced value of our $c$, determined by comparison with simulations such as ref.~\cite{RSB2005}, the exponent $1/k$, which is likely to be small, must be determined by improved simulations of fragmentation, and the value $x_* \sim \bar l_*/t$ is determined by conservation of total string. Specific signatures, however, might be dominated by the tails, either at large loops or small.
The relatively large scale~(\ref{lstar}) is similar to that found in the recent paper~\cite{Vanchurin:2005pa}. As discussed in Sec.~4.2, we expect that the final loop size is much smaller because of extensive fragmentation; the divergence that we have found for small loop formation will be equally present in the calculation of fragmentation of a large loop into smaller ones. Ref.~\cite{Vanchurin:2005pa} finds a much less extensive fragmentation. This points up the need for more complete studies of this process. This should be possible within the context of flat spacetime simulations, beginning with long strings having the small scale structure as input (this does depend on resolving first the issue of the long string two-point function, to which we return below).
The production of very small loops takes place when ${\bf p_+} \sim {\bf p_-}$: that is, near cusps on the long strings as proposed in Refs.~\cite{Albrecht:1989mu,SiemOlum2003}. Indeed, all the loops that we consider are rather smaller than the correlation length, so the functions ${\bf p_+}$ and ${\bf p_-}$ are each somewhat localized on the unit sphere, and necessarily in the same region since ${\bf L}_+ = {\bf L}_-$. In this sense all of our loops are produced near cusps. It is possible that there is a second population of loops that form at scales much longer than the correlation length, though most of these will quickly reconnect with long strings.
The picture of a complicated fragmentation process is consistent with the results of Ref.~\cite{MarShell2005}. Now let us return to the two-point functions as found there. The partial agreement with our stretching model might now seem surprising, given the large production of small loops. If the loop formation were distributed on the long strings, there would be at least one additional effect, where the string shortens due to loop emission and more distant (and therefore less correlated) segments are brought closer together. However, the loop production is not distributed: it occurs when the functions ${\bf p}_{\pm}(\sigma)$, as they wander on the unit sphere, come close together. Our picture is that whole segments of order $l_*$ (\ref{lstar}) are then excised. Thus, most of the segments that remain on a long string at given time would have been little affected by loop production.
The discrepancy at short scales, which is particularly evident in the radiation era (Fig.~1), remains a puzzle. Let us first note that at the shortest scales in Figs. 1 and 2 the slope of the curve approaches and possibly even exceeds unity, corresponding to the critical value $\chi = 0.5$: if the two-point function is of this form then the small-loop production converges. Thus it is appealing to assume that it is feedback from production of short loops that accounts for the break in the curve in Fig.~1.\footnote{Note also that $\chi = 0.5$ corresponds to the functions ${\bf p}_\pm$ being random walks on the unit sphere. This again suggests this value might be some dynamical fixed point.}
This is distinctly different from the cutoff via a ${\bf p}_+ - {\bf p }_-$ hole as discussed in Sec.4.2 and and in Ref.~\cite{ACK1993}: the latter is a modification of the $+-$ correlations (though one that would have little effect on the two-point function), whereas Fig.~1 shows a modification of the $++$ correlation.
We are unable to identify a physical mechanism that would account for the observed two-point function. Consider a very short segment on a long string. With time, its size as a fraction of the horizon size decreases, so it moves to the left on Figs.~1 and~2. If it experiences only stretching, it will follow the slope of the dashed line. The loop production must have a strong bias toward removing segments with large fluctuations to account for the simulations. However, the general picture above is that the initial loop formation occurs at the scale $l_*$, and so any small segments will be carried away without any bias on their own internal configuration. Thus we would expect the two-point function to be given by the stretching power law down to arbitrarily small scales, until gravitational radiation enters.\footnote{One effect that may increase $\chi$ slightly is that segments that do not fall into the loop production `hole' at ${\bf p}_+ \sim {\bf p }_-$ may experience on average a larger value of $\alpha = -{\bf p}_+ \cdot {\bf p }_-$ and so have a reduced two-point function --- see Eq.~(\ref{hppint}). However, it seems impossible that this could account for the factor of 5 discrepancy of slopes evident in Fig.~1.}
We will not attempt to connect with the flat spacetime simulations of Refs.~\cite{Vanchurin:2005yb,Vanchurin:2005pa} because our approach requires expansion to drive the fractal dimension to 1 at short distance.
Finally we would like to note a significant puzzle regarding the loop velocities. For a loop of length $l$, the mean velocity is
\begin{equation}
{\bf v} = \frac{1}{2l} ({\bf L}_+(u,l) + {\bf L}_-(v,l)) = \frac{1}{l} {\bf L}_+(u,l) \ .
\end{equation}
Then
\begin{eqnarray}
\langle v^2(l) \rangle &=& \frac{1}{l^2} \int_0^l \!\!\!\int_0^l d\sigma\, d\sigma'\, \langle {\bf p}_+ (\sigma) \cdot {\bf p}_+(\sigma') \rangle \nonumber\\
&=& 1 - \frac{2A}{(2\chi + 1)(2\chi + 2)} (l/t)^{2\chi}\ .
\end{eqnarray}
Looking for example at loops with $l = 10^{-2} t$, we obtain a typical velocity $0.985$ in the matter era and $0.90$ in the radiation era. These are significantly larger than are generally expected; Ref.~\cite{Bennett:1989ch} gives values around $0.75$ and $0.81$ for loops of this size. We emphasize that this is not a consequence of our dynamical assumptions, but can be seen directly by using the two-point functions of Ref.~\cite{MarShell2005} (the discrepancy is then even greater in the radiation era). Assuming that both simulations are correct, we must conclude that the two-point function on loops is very different from that on long strings, in fact less correlated. One possible explanation is that loop formation is biased in this way, but this seems unlikely: loops form when right- and left-moving segments of equal ${\bf L}(l)$ meet, and this will happen most often in the center of the distribution. Rather, this seems to be another indication of the complicated nature of the fragmentation: the final non-self-intersecting loops must contain segments which began on the long string many times further separated than the final loop size.
\sect{Conclusions and future directions}
Our attempt to isolate the physics on different scales has not been completely successful, but it suggests some interesting directions for future work. First of all, the two-point function on long strings must be better understood: is the discrepancy in Fig.~1 a result of omitted physics or numerical transients? This question can be settled numerically, by studying sensitivity to initial conditions, and by studying the time-dependence of an ensemble of small segments. A good understanding of the two-point function would then allow extrapolation to much smaller scales, for application to lensing (as in section~3), gravitational radiation~\cite{Vanchurin:2005pa}, and possible interference between short distance structure and cusps~\cite{SiemOlum2003}.
Second, our work points to correlations among the two-point function, loop production, fragmentation, and loop velocity. In particular, our conjecture of a very complicated fragmentation process can be tested --- for example, the argument that non-self-intersecting loops must contain segments that originated far apart on the long string. Also, the normalization difference between our loop production and that in Ref.~\cite{RSB2005} must presumably be understood in terms of fragmentation.
In conclusion, cosmic string networks are probably not as complicated as turbulence, but they share the property that a rather simple set of classical equations leads to a complicated dynamics that challenges both numerical and analytical attacks. We hope that our work is a step toward a more unified understanding of the properties of these networks.
\section*{Acknowledgments}
We are grateful to Carlos Martins and Paul Shellard for providing the data from their numerical simulations and for valuable comments and suggestions. We also thank Ed Copeland and Thibault Damour for discussions and Xavier Siemens for comments.
JVR acknowledges financial support from {\it Funda\c{c}\~ao para a Ci\^encia e a Tecnologia}, Portugal, through grant SFRH/BD/12241/2003. This work was supported in part by NSF grants PHY99-07949 and PHY04-56556.
|
1,108,101,564,601 | arxiv | \section{Introduction}
In the standard model of particle physics \cite{SM} there are open questions
which have not yet found an answer. Chief among these is the fermion family or generation
puzzle as to why the first generation of quarks and leptons (up quark, down quark,
electron and electron neutrino) are replicated in two families or generations
of increasing mass (the second generation consisting of charm quark, strange quark, muon and muon
neutrino; the third generation consisting of top quark , bottom quark, tau and tau neutrino)
In addition to explaining why there are heavier copies of the first generation
of fermions one would like to explain the mass hierarchy of the generations and
the mixings between the generations characterized by the CKM (Cabbibo-Kobayashi-Maskawa) matrix.
Several ideas have been suggested such as a horizontal family symmetry \cite{FS}.
Recently theories with extra dimensions have been used in a novel way to
try and explain some of the open questions in particle physics and cosmology.
In \cite {ADD} \cite{RS} \cite{Gog} the hierarchy problem (i.e.
why the gravitational interaction is many orders of magnitude weaker than the
strong and electroweak interactions of particle physics) was addressed using large or
infinite extra dimensions. Early versions of these extra dimensional models were
investigated by several researchers \cite{Ak} \cite{Ru} \cite{Vi} \cite{Gib}.
In contrast to extra dimensions in the usual Kaluza-Klein picture,
in the models with large or infinite extra dimensions
gravity acts in all the spacetime dimensions, while the other particles and fields
are confined, up to some energy scale, to a $3+1$ dimensional brane.
These recent extra dimensional models have also been applied to answer other
questions of particle physics and cosmology. Non-supersymmetric string models \cite{crem}
have been constructed which have been able to reproduce the standard model particles
from intersecting D5-branes. More phenomenological brane models \cite{Mass} \cite{neronov} \cite{Mi}
\cite{tait} have been constructed to explain the hierarchy of masses and/or the CKM elements of
the fermions. In \cite{DE} \cite{DM} brane world models were used to explain
dark energy and dark matter. General studies of higher dimensional cosmologies can be
found in \cite{Mel1} \cite{Mel2}.
In this paper we attempt to give a toy model for the generation problem using a brane world
model in 6D. We obtain three 4D fermion families from the
zero modes of a single 6D spinor field. A mass hierarchy and mixings between
the three zero modes are obtained by introducing a Yukawa type interaction
between the 6D spinor field and a 6D scalar field. This gives a common origin
(i.e. the higher dimensional Yukawa interaction) for both the mass
spectrum and the mixings of the fundamental fermions.
\section{6D gravitational background}
In \cite{Mi} \cite{GoSi} \cite{GoSi1} \cite{Ma} a 6D brane world model was
investigated which gave universal gravitational trapping of fields of
spins $0, \frac{1}{2}, 1, 2$, to the brane. The system considered was 6D gravity with
a cosmological constant and some matter field energy-momentum. The action for this
system was
\begin{equation} \label{action}
S = \int d^6x\sqrt{- ^6g}\left[\frac{M^4}{2}(^6R + 2 \Lambda) + ^6L \right] ~,
\end{equation}
where $\sqrt{-^6g}$ is the determinant, $M$ is the fundamental
scale, $^6R$ is the scalar curvature, $\Lambda$ is the
cosmological constant and $^6L$ is the Lagrangian of the matter
fields. All of these quantities are six dimensional.
The ansatz for the 6D metric was taken as
\begin{equation}\label{ansatzA}
ds^2 = \phi ^2 (r)\eta _{\alpha \beta } (x^\nu )dx^\alpha
dx^\beta - \lambda (r)(dr^2 + r^2d\theta ^2)~ ,
\end{equation}
where the Greek indices $\alpha, \beta,... = 0, 1, 2, 3$ refer to
4-dimensional coordinates. The metric of ordinary 4-space,
$\eta_{\alpha \beta }(x^\nu)$, has the signature $(+,-,-,-)$. The
functions $\phi (r)$ and $\lambda (r)$ depend only on the extra
radial coordinate, $r$, and thus are cylindrically symmetric in
the transverse polar coordinates ($0 \le r < \infty$, $0 \le
\theta < 2\pi$). The ansatz for the energy-momentum tensor of the matter
fields was taken to have the form
\begin{equation} \label{source}
T_{\mu\nu} = - g_{\mu\nu} F(r), ~~~~~ T_{ij} = - g_{ij}K(r),
~~~~~ T_{i\mu} = 0 ~.
\end{equation}
Other than satisfying energy-momentum conservation i.e.
\begin{equation}\label{energy-con}
\nabla^A T_{AB} = \frac{1}{\sqrt{-^6 g}}
\partial _A (\sqrt{-^6 g}T^{AB}) + \Gamma ^B _{CD} T^{CD} =
K^{\prime} + 4\frac{\phi^\prime}{\phi} \left(K - F \right) = 0~,
\end{equation}
the energy-momentum tensor was unrestricted, although it was desirable for
it to satisfy physical requirements such as being everywhere finite, and
being peaked near the brane. In \eqref{energy-con} and in the rest of the paper
a prime indicates derivative with respect to $r$
In \cite{Mi} \cite{GoSi1} it was found that the above system had the following
non-singular solution
\begin{equation} \label{phi}
\phi (r) = \frac{c^b+ a r^b}{c^b+ r^b}~,
\end{equation}
where $a, b, c$ are constants and $a>1$. All other ansatz functions were given in
terms of $\phi (r)$.
\begin{equation}
\label{g}
\lambda (r)= \frac{\rho^2 \phi ^{\prime}}{r} = \rho ^2 (a-1) b c^b \frac{r^{b-2}}{(c^b+r^b)^2} ~,
\end{equation}
where $\rho$ is an integration constant with units of length, which was related to
the constants $a$ and $b$ by
\begin{equation}
\label{ab}
\frac{\rho^2 \Lambda}{10 M^4} = \frac{b}{a-1} ~.
\end{equation}
This solution in terms of $\phi (r)$ and $\lambda (r)$ represents a non-singular, thick brane.
The brane thickness is proportional to $c$.
Recently the matching conditions for a general thick brane of codimension 2 was given
\cite{jose}. This provides a general framework in which to study gravitational
phenomena and particle trapping in codimension 2 braneworlds.
The source functions are also determined by $\phi (r)$.
\begin{equation} \label{FK}
F(r) = \frac{f_1}{2 \phi^2} +\frac{3 f_2}{4 \phi} ~, ~~~~~ K(r) =
\frac{f_1}{\phi^2} +\frac{f_2}{\phi} ~,
\end{equation}
where $f_1$ and $f_2$ are constants given by
\begin{equation} \label{parameters}
f_1 = -\frac{3\Lambda}{5} a~, ~~~~~ f_2=\frac{4 \Lambda}{5}(a+1)~,
\end{equation}
In \cite{Mi} and \cite{GoSi1} and it was shown that for $b=2$ this solution gave a
universal, gravitational trapping for fields with spins $0, \frac{1}{2} , 1, 2$ within the
brane width ($\approx c$) of $r=0$.
For $b > 2$ \eqref{g} shows that the scale factor for
the extra dimensions, $\lambda (r) = 0$ at $r=0$, raising the possibility of having a
singularity on the brane and making the solutions with $b > 2$ unphysical. However,
by looking at invariants such as the Ricci scalar, $R$, one finds that they are
non-singular at $r=0$. This indicates that the zero of $\lambda (r)$ at $r=0$ for $b >2$ solutions
is not a physical singularity. The Ricci scalar for the above solution is
\begin{equation}
\label{ricci}
R = \frac{2b (-5 c^{2 b} + 4 a c^{2 b} + 10 c^b r^b -
22 a c^b r^b + 10 a^2 c^b r^b + 4 a r^{2 b} - 5a^2 r^{2 b})}{(a-1)\rho^2 (c^b + a r^b)^2} ~.
\end{equation}
It is easy to see that this is finite at $r=0$. Other invariants such as the fully contracted
Riemann tensor, $R_{AB} ~^{CD} R_{CD} ~^{AB}$, or the square of the Ricci tensor, $R_{AB} R^{AB}$
also turn out to be finite at $r=0$. Thus we take the zero in the scale factor $\lambda (r)$ to
be a coordinate rather than physical singularity. Note that $\lambda (r)$ goes to zero both at
$r=0$ and $r= \infty$ so that the metric \eqref{ansatzA} essentially becomes 4D at these
locations. Thus the solution given in \eqref{phi} is like the 2 brane model of \cite{RS}
where two 4D branes sandwich the higher dimensional, bulk spacetime.
The weak point in the above is that the source ansatz functions have no clear physical
interpreation. It would be desirable to show that an energy-mometum tensor of the
general form given by $F(r), K(r)$ could arise from some realistic source
such as a scalar field. Because $\phi (r)$ increases as $r \rightarrow \infty$
\eqref{FK} \eqref{parameters} indicate that $F(r), K(r)$ have their maximum near $r=0$
and decrease as $\rightarrow \infty$. This behavior is similar to soliton solutions of classical
field theory. Thus $F(r), K(r)$ might be considered as modeling some solitonic field
configuration which forms the brane. As specific examples references \cite{Br} \cite{Br1}
investigate solitonic scalar field configurations which form branes. Another possible physical
interpretation of the source ansatz functions can be given by transforming the metric given by
\eqref{ansatzA} \eqref{phi} \eqref{g} via the transformation $r=c \tan^{2/b} (\beta /2)$
and setting $c^2 /4 =10 M^4 / \Lambda =\rho ^2 (a-1)/ b$. With this the metric takes
the form
\begin{equation}
\label{cylinder}
ds^2 = \phi ^2 (\beta)\eta _{\alpha \beta } (x^\nu )dx^\alpha
dx^\beta - \frac{c^2}{4} \left( d\beta^2 + \frac{b^4}{4} \sin ^2 \beta d\theta ^2 \right)~ ,
\end{equation}
where $\phi (\beta) = \frac{1}{2} ((a+1) +(1-a) \cos \beta)$. When $b=2$ one can see
(after renaming the angles in a standard way as $\beta \rightarrow \theta$ and
$\theta \rightarrow \varphi$) that the geometry of the extra dimensions is that of a sphere.
However when $b>2$ one has zeroes in the 2D scale function, $\lambda (r)$,
at $r=0$ and $r=\infty$ and a conical deficit angle of $\delta = 2 \pi - b \pi$.
This is essentially the ``football''-shaped geometry for the extra dimensions
considered in \cite{carroll} (see in particular equation (24) of \cite {carroll}).
The 6D solution in \cite{carroll} involved realistic sources: an electromagnetic field in the
form of a magnetic flux and a bulk cosmological constant. Thus the source ansatz functions,
$F(r) , K(r)$ may be acting as effective magnetic flux plus bulk cosmological constant.
\section{Three families from three zero modes}
We now study the motion of a 6D fermion field in the gravitational background
given by \eqref{ansatzA}. We make the identification that different fermion zero-mass modes
(i.e. solutions for which the 4D part of the fermion wavefunction satisfies
$\gamma _\mu \partial ^\mu \psi (x _\nu ) = 0$) correspond to different families.
Thus we want to see if it is possible to obtain three zero modes. The picture that
we present here is a toy model in the sense that the effective 4D fermion fields do not have
the full $SU(3) \times SU(2) \times U(1)$ charges of the real Standard
Model fields. As will be shown below the fermions considered here carry only a $U(1)$
charge associated with the rotational symmetry around the brane i.e. the symmetry
associated with the extra dimensional variable $\theta$. This is the same as the first
example in \cite{neronov}, with the $U(1)$ quantum number being associated with
family number. Reference \cite{neronov} gives other more
complex and realistic examples where the fermions carry $U(1) \times U(1)$ or
$SU(2) \times U(1)$ charges. In the $U(1) \times U(1)$ example one of the
$U(1)$'s is associated with a horizontal, family symmetry \cite{FS} which distinguishes between
fermions of different generations, while the other $U(1)$ is then an ordinary gauge symmetry.
In the present case the single $U(1)$ is associated with the family symmetry.
Other authors \cite{hung} also have studied 6D brane models with more realistic standard model charges.
One can gauge \cite{neronov} any of the above examples using the Kaluza-Klein approach.
For example, with the simple $U(1)$ model considered here the metric in \eqref{ansatzA} has
a Killing vector, $\partial _\theta$, which via the standard Kaluza-Klein mechanism implies an
associated gauge boson in the effective 4D theory. The gauge field arises from the off-diagonal
components of the higher dimensional metric as
\begin{equation}
\label{kk-gauge}
ds^2 = \phi ^2 (r)\eta _{\alpha \beta } (x^\nu )dx^\alpha
dx^\beta - \lambda (r)(dr^2 + (r d\theta + A_\mu dx^\mu)^2 )~ .
\end{equation}
This $A_\mu$ is analogous to the horizontal gauge boson of the family symmetry models
\cite{FS}. In \cite{neronov2} it was shown that the zero mode of the
gauge field $A_\mu$ was localized to the brane. Since horizontal gauge
bosons have an experimentally fixed lower mass limit a more realistic model would
need to have some symmetry breaking, Higgs mechanism in order to give $A_\mu$ an acceptably
large mass.
From the previous section we found that in 6D one has the freedom to
let the exponent $b$ take values other than $b=2$. The motion of fermions in
the 6D brane solution of \eqref{ansatzA} for the $b=2$ case was
studied in \cite{Ma}. It was found that in this case only one
zero mode occurred, and thus only one family. Therefore we want to
consider the $b>2$ case and show that for a certain range of $b$ it is possible
to get three zero modes. The constant $b$ controls the steepness of the scale functions
$\phi (r)$ and $\lambda (r)$. Therefore it is reasonable that larger $b$ should give a stronger
confinement of the fermions to the vicinity of the brane at $r=0$, and thus to a larger number of
zero modes.
The 6D action and resulting equations of motion for a spinor field are
\begin{equation}
\label{fermion}
S_\Psi = \int d^6 x \sqrt{-^6g}\bar{\Psi} i \Gamma^A D_A \Psi
~~, \qquad \Gamma ^A D_A \Psi = \Gamma ^{\mu} D_{\mu} \Psi + \Gamma ^r D_r \Psi +
\Gamma ^{\theta} D_{\theta} \Psi = 0 ~.
\end{equation}
In the above $\Gamma ^A = h^A _{\bar B} \gamma ^B$ are the 6D curved spacetime
gamma matrices, and $h^A _{\bar B}$ are the sechsbiens defined via
$g_{AB} = h ^{\bar A} _A h^{\bar B} _B \eta _{\bar A \bar B}$. In order to
evaluate the 6D Dirac equation in \eqref{fermion} we need to calculated the
spin connections
\begin{equation} \label{spin1}
\omega^{\bar{M}\bar{N}}_M = \frac{1}{2} h^{N\bar{M}} (\partial_M
h^{\bar{N}}_N - \partial_N h^{\bar{N}}_M) - \frac{1}{2}
h^{N\bar{N}}(\partial_M h^{\bar{M}}_N - \partial_N h^{\bar{M}}_M)
- \frac{1}{2} h^{P\bar{M}}h^{Q\bar{N}}(\partial_P h_{Q\bar{R}} -
\partial_Q h_{P\bar{R}})h^{\bar{R}}_M ~.
\end{equation}
The non-zero spin connections are
\begin{equation} \label{spin2}
\omega^{\bar{r}\bar{\nu}}_\mu = \delta ^{\bar{\nu}} _{\mu} \frac{\sqrt{r \phi '}}{\rho} ~ , ~~~~~
\omega^{\bar{r}\bar{\theta}}_\theta = \sqrt{\frac{r}{\phi '}} \partial _r
\left( \sqrt{r \phi' } \right) ~.
\end{equation}
With these one can explicitly calculate the various covariant derivatives in \eqref{fermion}
\begin{equation} \label{covariant}
D_\mu\Psi = \left( \partial_\mu + \frac{1}{2} \omega^{\bar{r}\bar{\nu}}_\mu
\gamma _r \gamma _\nu \right) \Psi ~ , ~~~~~ D_r\Psi = \partial_r \Psi ~ , ~~~~~
D_\theta \Psi = \left(\partial_\theta + \frac{1}{2} \omega^{\bar{r}\bar{\theta}}_\theta
\gamma _r \gamma _\theta \right) \Psi ~.
\end{equation}
We now assume that the 6D fermion spinor can be decomposed as $\Psi (x^A) = \psi (x_\mu ) \otimes
\zeta (r, \theta)$ into 4D and 2D parts. We are interested in the zero-mass modes so the 4D fermion
part is taken to satisfy $\gamma _\mu \partial ^\mu \psi (x _\nu ) = 0$. The 2D spinor can be expanded
as
\begin{equation}
\label{2Dspinor}
\zeta (r, \theta ) =
\left(
\begin{array}
[c]{c}
f_l (r) \\
g_l (r)
\end{array}
\right)
e^{i l \theta}
\end{equation}
We take the gamma matrices of the extra space as in \cite{neronov} \cite{Mi}
\begin{equation}
\label{2dgamma}
\gamma ^r =\left(
\begin{array}
[c]{cc}%
0 & 1\\
1 & 0
\end{array}
\right) \qquad
\gamma ^\theta =\left(
\begin{array}
[c]{cc}%
0 & -i\\
i & 0
\end{array}
\right) .
\end{equation}
Combining equations \eqref{fermion} - \eqref{2dgamma} we arrive at the following
equations for $f_l (r)$ and $g_l (r)$
\begin{equation}
\label{fg-fermion}
\left[ \partial_r + 2 \frac{\phi '}{ \phi} + \frac{1}{2}
\frac{\partial_r \left( \sqrt{r \phi ' } \right)}{\sqrt{r \phi'}}
+ \frac{l}{r} \right] g_l(r) = 0 ~, \qquad
\left[ \partial_r + 2 \frac{\phi '}{ \phi} + \frac{1}{2}
\frac{\partial_r \left( \sqrt{r \phi ' } \right)}{\sqrt{r \phi'}}
- \frac{l}{r} \right] f_l(r) = 0 ~.
\end{equation}
The solutions for $f_l (r)$ and $g_l (r)$ are
\begin{equation}
\label{fg-solution}
f_l (r) = a_l \phi (r) ^{-2} (r \phi ' (r) )^{-\frac{1}{4}} r^l ~ , ~~~~
g_l (r) = b_l \phi (r) ^{-2} (r \phi ' (r) )^{-\frac{1}{4}} r^{-l} ~.
\end{equation}
Because of the $l$ dependence in both $f_l (r)$ and $g_l (r)$, different
$l$ values give fermion fields with different profiles in the bulk.
This will result in different masses and mixings for different $l$ values
when the fermion field is coupled to a scalar field in the next section.
One criteria for the trapping of the fermion field is that it should be
normalizable with respect to the extra dimensions $r, \theta$
\begin{equation}
\label{norm}
1 = \int \sqrt{-^6 g} {\bar \zeta (r, \theta)} \zeta (r, \theta) dr d \theta = \int _0 ^{2 \pi} d \theta
\int _0 ^\infty dr \phi ^4 \rho ^2 \phi '\frac{\phi ^{-4}}{\sqrt{r \phi '}}
(a_l^2 r^{2l} + b_l ^2 r^{-2l}) =
2 \pi \rho ^2 \int _0 ^\infty dr \sqrt{\frac{\phi '}{r}} (a_l ^2 r^{2l} + b_l ^2 r^{-2l})
\end{equation}
From \eqref{g} one finds $\sqrt{\phi ' /r} = (a-1)^{\frac{1}{2}} b^{\frac{1}{2}} c^{\frac{b}{2}}
r^{\frac{b}{2} -1} (c^b + r^b)^{-1}$.
Thus in order for \eqref{norm} to be normalizable and for the particular fermion $l$-mode
to be trapped, we want the integral
\begin{equation}
\label{norm2}
2 \pi \rho ^2 \sqrt{(a-1) b c^b}
\int _0 ^\infty \frac{r^{\pm 2 l + \frac{b}{2} -1}}{c^b + r^b} dr ~,
\end{equation}
to be finite. If \eqref{norm2} diverges the particular $l$-mode will not
be trapped. This requirement that \eqref{norm2} be finite leads to restrictions
on $b$ for particular values of $l$. Evaluating the integral \eqref{norm2}
gives
\begin{equation}
\label{norm3}
2 \pi^2 \rho ^2 c^{\pm 2l} \sec\left(\frac{2l\pi}{b} \right) \sqrt{\frac{a-1}{b}}
\qquad \text{if} \qquad b > 4|l|,
\end{equation}
and \eqref{norm2} diverges if $b \le 4 |l|$.
Thus in order to have three normalizable $l$-modes we require that
$4 < b \le 8$. Under these conditions the $l=0$, and $|l|=1$ modes are normalized
and trapped, while $|l| \ge 2$ modes are not. Since the integrand in \eqref{norm2} is positive definite
and only has possible divergences at $r=0$ and $r=\infty$ one can come to this conclusion
by investigating the $r \rightarrow 0$ and $r \rightarrow \infty$ behavior of this
integrand. For $|l| =1$ one finds that for $l=+1$ the integrand behaves as
$^{lim} _{r \rightarrow 0} \simeq r^{1 + \frac{b}{2}}$ ,
$^{lim} _{r \rightarrow \infty} \simeq r^{1 - \frac{b}{2}}$ ; for $l=-1$, it
behaves as $^{lim} _{r \rightarrow 0} \simeq r^{-3 + \frac{b}{2}}$,
$^{lim} _{r \rightarrow \infty} \simeq r^{-3 - \frac{b}{2}}$. The $r \rightarrow 0$
limit of $l=+1$ and $r \rightarrow \infty$ limit of $l=-1$ give convergent results.
On the other hand the $r \rightarrow 0$ limit of $l=-1$ and $r \rightarrow \infty$ limit of $l=+1$
give convergent results only if $b>4$. One can see the for $b>4$ the $l=0$ mode
is normalized. For $|l|=2$ one finds that for $l=+2$ the integrand behaves as
$^{lim} _{r \rightarrow 0} \simeq r^{3 + \frac{b}{2}}$ ,
$^{lim} _{r \rightarrow \infty} \simeq r^{3 - \frac{b}{2}}$ ; for $l=-2$, it
behaves as $^{lim} _{r \rightarrow 0} \simeq r^{-5 + \frac{b}{2}}$,
$^{lim} _{r \rightarrow \infty} \simeq r^{-5 - \frac{b}{2}}$.
The $l=+2$ integral diverges at $r \rightarrow \infty$ and the $l=-2$ diverges at
$r \rightarrow 0$ if $b \le 8$. This analysis again shows that one has three normalizable modes
(i.e. three fermion families) when the $4 < b \le 8$.
One could also consider using the criteria for trapping that the fermion action be finite when integrated over
the extra dimensions.
\begin{equation}
\label{f-action}
S_\Psi = \int d^6 x \sqrt{-^6g}\bar{\Psi} i \Gamma^A D_A \Psi =
2 \pi \rho ^2 \int _0 ^\infty dr \frac{1}{\phi} \sqrt{\frac{\phi '}{r}} (a_l ^2 r^{2l} + b_l ^2 r^{-2l}) ~.
\int d^4 x \sqrt{-\eta} {\bar \psi} i \gamma ^\nu \partial _\nu \psi
\end{equation}
The fermions are trapped if the integral over $r$ in the last expression is convergent. This integral
is almost the same as the last integral in \eqref{norm}. It differs only by a factor of $1 / \phi$ which
comes from the sechsbien that modifies the gamma matrices, $\gamma ^\nu$. The explicit expression
for $\phi ^{-1} \sqrt{\phi ' /r}$ can be read off from \eqref{g} and \eqref{phi}. From
this one finds that $\phi ^{-1} \sqrt{\phi ' /r} \propto r^{\frac{b}{2} -1} (c^b + a r^b)^{-1}$.
The only change with respect to the normalization condition \eqref{norm} is that $r^b \rightarrow a r^b$
in the denominator. Thus the integral of the action over the extra coordinates will have
the same convergence properties as the normalization condition \eqref{norm} , thus giving the same conclusion
that three zero-mass modes will be trapped if $4< b \le 8$.
In \cite{Ma} only the $b=2$ case in \eqref{phi} was considered and only one zero-mass mode occurred.
Thus the existence of three zero modes is the result of allowing the exponent in \eqref{phi}
to take values $b>2$. In \cite{Si} it was shown that the solution of \eqref{phi} \eqref{g} \eqref{FK}
could be generalized to spacetimes of dimension greater than 6D. For these greater than 6D spacetimes the
exponent, $b$, in \eqref{phi} was not free, but fixed to $b=2$. This would seem to imply that
only in 6D can one have more than one fermion generation for the background solution given by
\eqref{phi} \eqref{g} \eqref{FK}. However in the case where spacetime greater than 6D
one could consider taking the higher generations as non-zero mass modes. Also one could try to
generalize the other 6D brane solution given in \cite{Ma} to spacetime dimensions greater than 6D.
In discussing the masses and mixing between the different families (i.e. different $l$) we will
need the normalization relationship between $a_l$ and $b_l$. From \eqref{norm} \eqref{norm2} and \eqref{norm3}
we find
\begin{equation}
\label{norm4}
a_l ^2 c^{2l} + b_l^2 c^{-2l} = \frac{\cos \left(\frac{2 l \pi}{b} \right)}{2 \pi ^2 \rho ^2} \sqrt{\frac{b}{a-1}} ~.
\end{equation}
This condition allows us to write $b_l$ in terms of $a_l$ or visa versa.
\section{Mixings and Masses}
By adjusting the exponent $b$ in our gravitational background solution we have three zero mass
modes which can be taken as a toy model for three generations of fermions.
There are two problems: first there is no mixing between the different
generations due to the orthogonality of the angular parts of the
higher dimensional wave functions. Overlap integrals like
$\int_0 ^\infty \int _0 ^{2 \pi} {\bar \zeta _l} \zeta _m dr d \theta$, which characterize the mixing
between the different states, vanish since $\int _0 ^{2 \pi} e^{-il\theta} e^{im\theta} d \theta =0$
if $l \ne m$. Second, all the states are massless, whereas the fermions
of the real world have masses that increase with each succeeding family.
Following \cite{neronov} we address both of these issues
by introducing a coupling between the 6D fermions and a 6D scalar field of the form
$H_p (x ^A ) {\bar {\Psi}} _l (x^B) \Psi _{l'} (x^C )$. This adds to the action a scalar-fermion
interaction of the form
\begin{equation}
\label{FFS}
S_{sf} = f \int d^4 x dr d\theta \sqrt{- ^6 g} H_p \bar {\Psi} _l \Psi _{l'} ~,
\end{equation}
$f$ is a constant which gives the scalar-fermion coupling strength.
We now take the scalar field to be of the form
\begin{equation}
\label{higgs}
H_p (x ^A) = H_p (r) e^{ip \theta}
\end{equation}
i.e. only depending on the bulk coordinates $r , \theta$, but not on the brane
coordinates $x ^\mu$. In \cite{neronov} the same form as in \eqref{higgs} was
taken for the scalar field, but certain simplifying assumptions were made about the form
of $H(r)$ -- it was assumed to be either a constant or a delta-function.
In \cite{tait} other forms for the scalar field profile were used.
In the following we will determine the form of $H(r)$ by studying the field
equations for a scalar field in the background provided by \eqref{phi} \eqref{g}.
Note that in the form \eqref{higgs} the scalar field is only dynamical with respect to the
extra dimensions, $r, \theta$, but not with respect to the brane spacetime dimensions,
$x^\mu$. Thus one has a scalar field mechanism for fermion mass generation without a
dynamical 4D scalar particle.
Substituting \eqref{higgs} into \eqref{FFS} we find
\begin{equation}
\label{FFS2}
S_{sf} = U_{l l'} \int d^4 x \bar{ \psi} _l (x ^\mu ) \psi _{l'} ( x ^\mu) \qquad \text{where} \qquad
U_{ll'} = f \int dr d \theta \sqrt{- ^6 g} H_p (r) e^{i (p-l+l') \theta} \bar{ \zeta _l }(r) \zeta _{l'}
(r ) ~.
\end{equation}
$U_{ll'}$ will be non-zero when $p-l+l' = 0$. When $l=l'$ this gives a mass term and when
$l \ne l'$ this gives a mixing term between the $l$ and $l'$ modes.
To get explicit results for $U_{ll'}$ one needs an explicit form for $H_p (r)$. This is done
by solving the field equations for a test scalar field in the background given by $\phi (r)$ and
$\lambda (r)$ for the different $p-$modes. The equation for the scalar field in the background given
by \eqref{phi} \eqref{g} is
\begin{equation}
\label{eqn-sca}
\frac{1}{\sqrt{- ^6 g}} \partial _A \left(\sqrt{- ^6 g} g^{AB} \partial _B H_p (x^A ) \right) = 0 ~.
\end{equation}
Inserting \eqref{higgs} into \eqref{eqn-sca} we get
\begin{equation}
\label{eqn-sca2}
H_ p '' (r) + \left( \frac{1}{r} + \frac{4 \phi ' (r)}{\phi (r)} \right) H_p ' (r) -
\frac{p^2}{r^2} H_p (r) = 0 ~.
\end{equation}
From \eqref{phi} we find that
\begin{equation}
\label{4pp}
\frac{4 \phi ' (r)}{\phi (r)}=
\frac{4 (a-1) \frac{b}{c} \left( \frac{r}{c}\right)^{b-1}}{\left(1 + \left( \frac{r}{c}\right)^b \right)
\left(1 + a\left( \frac{r}{c}\right)^b \right)} ~.
\end{equation}
Inserting this expression back into \eqref{eqn-sca2} we were not able to find a closed
form solution, but were only able to solve it numerically.
From the form of $4 \phi ' (r) / \phi (r)$ in \eqref{4pp} one can
see that when $ 4 < b \le 8$ that the $-p^2/r^2$ and $1/r$ terms dominate in the limits
$r \rightarrow 0$ and $r \rightarrow \infty$. If one drops the $4 \phi ' (r) / \phi (r)$ term
then one finds that the asymptotic ($r \rightarrow 0$ and $r \rightarrow \infty$) solutions to
\eqref{eqn-sca2} are
\begin{eqnarray}
\label{scalar-soln}
H_p (r) = A_{+ p} r^{ |p|} ~~~ \text{or} ~~~ A_{-p} r^{-|p|} ~~~ \text{for} ~~ p \ne 0 \\
H_0 (r) = A_0 ~~ \text{or} ~~ B_0 \ln (r) ~~~ \text{for} ~~ p=0 ~,
\end{eqnarray}
where $A_{\pm p}$, $A_0$, and $B_0$ constants. These asymptotic solutions gave a
fair representation to the numerical solution even for intermediate values of $r$.
The $p=0$ solutions can be written in combined
form as $H_0 (r) = B_0 \ln (r/c_0)$ where $A_0 = - B_0 \ln (c_0)$. This form will be used
the next subsection to give masses to the three zero modes. The singularities in $H_0 (r)$, at
$r=0$ and $r= \infty$, are not a problem since the combination of the fermion ``wave function"
and the metric ansatz functions go to zero fast enough at $r=0$ and $r= \infty$ to negate
these singularities in the Yukawa coupling integral \eqref{FFS2}.
\subsection{Masses}
From \eqref{FFS2} one can sees that the mass terms are those for which $l=l'$ and thus
we want to consider couplings to the $p=0$ scalar mode in \eqref{scalar-soln}. We will use
the combined form of the two solutions namely $H_0 (r) = B_0 \ln (r/ c_0)$. With this
\eqref{FFS2} becomes
\begin{eqnarray}
\label{mass3}
m_l = U_{ll} &=& f \int dr d \theta \sqrt{- ^6 g} H_0 (r) \bar{ \zeta _l }(r)
\zeta _{l} (r )
= 2 f B_0 \pi \rho ^2 \sqrt{(a-1) b c^b} \int _0 ^\infty
\frac{ \ln \left( \frac{r}{c_0} \right)r^{b/2 -1}}{c^b+r^b} \left[a_l^2r^{2l} + b_l^2 r^{-2l} \right] dr \nonumber \\
& = & f B_0 \left( \ln \left( \frac{c}{c_0} \right) + \frac{\pi K_l}{b} \tan \left( \frac{2 l \pi}{b} \right) \right)
\qquad \text{where} \qquad
K_l = \frac{4 a_l^2 c^{2l} \pi ^2 \rho ^2}{\cos \left(\frac{2 \pi l }{b} \right)} \sqrt{\frac{a-1}{b}} -1 ~.
\end{eqnarray}
In arriving at the final line in \eqref{mass3} where have used \eqref{norm4} to replace $b_l$
in terms of $a_l$. Looking at only the $\tan (2 l \pi / b)$ term and taking $f B_0 K_l \pi / b >0$ gives a
hierarchy of masses of $m_{-1} < m_0 < m_{+1}$. However $m_{-1} <0$ and $m_0 = 0$ which is phenomenologically
wrong. Taking $\ln (c/ c_0)$ as positive (i.e. $c>c_0$) and sufficently large can shift
the mass spectrum so that all masses are positive and while still
maintaining the hierarchy $m_{-1} < m_0 < m_{+1}$. Because
our fermions only carry a $U(1)$ charge the above hierarchy is a toy model.
Here, as an example, we take the three fermions as the ``down" sector of quark generations where $l=-1$
is the down quark, $l=0$ is the strange quark, and $l=+1$ is the bottom quark. Taking
the masses of the down, strange and bottom quarks as $m_{-1}= m_d =4$ MeV , $m_0 = m_s = 100$ MeV and
$m_{+1} = m_b = 4400$ MeV we find from \eqref{mass3}
\begin{equation}
\label{mass4}
\frac{m_d}{m_s} = \frac{m_{-1}}{m_0} = 0.04 = \left( \frac{-K _{-1} \pi \tan ( 2 \pi / b )}{b \ln ( c/ c_0 )}
+1 \right) ~ , ~~~~ \frac{m_b}{m_s} = \frac{m_{+1}}{m_0} = 44.00 = \left( \frac{ K_{+1} \pi \tan ( 2 \pi /b )}{b \ln ( c/ c_0 )}
+1 \right) ~.
\end{equation}
Solving these equations for $a_1$ and $a_{-1}$ (which are embedded in the definition of
$K_{+1}$ and $K_{-1}$) gives
\begin{equation}
\label{a1}
a_1 = \frac{D}{c^2} \sqrt{(1 + 43 x)\cos \left(\frac{2 \pi}{b} \right)} ~~~, ~~~
a_{-1} = D \sqrt{(1 + 0.96 x)\cos \left(\frac{2 \pi}{b} \right)} ~,
\end{equation}
where $x=\frac{b \ln (c/c_0)}{\pi \tan (2 \pi /b)}$ and $D ^2 = \frac{c^2}{4 \pi^2 \rho ^2} \sqrt{\frac{b}{a-1}}$. If
$a_1$ and $a_{-1}$ are chosen as in \eqref{a1} then the mass ratios in \eqref{mass4} are
obtained.
\subsection{Mixings}
A similar analysis can be carried out with the mixings between the different ``families"
characterized by different $l$ number. The mixings are delineated by $U_{0,1} , U_{1,0} ,
U_{1, -1} , U_{-1, 1} , U_{0, -1}$ and $U_{-1, 0}$. In the case of mixings the scalar field
must have a non-zero angular eigenvalue i.e. $H_p (r , \theta) = H_p (r) e^{i p \theta}$
with $p \ne 0$) which satisfies $p-l+l' = 0$. Thus for $U_{-1, 0}$ and $U_{0,1}$ one needs $p=-1$;
for $U_{1, 0}$ and $U_{0, -1}$ one needs $p=1$; for $U_{1, -1}$ one needs $p=2$; for $U_{-1 , 1}$
one needs $p=-2$. We will require the following relationship between the mixings: $U_{0,1} = U_{1,0}$,
$U_{1, -1} = U_{-1, 1}$ and $U_{0, -1} = U_{-1, 0}$. This in turn implies that $H_p (r)$ should
depend only on $|p|$ ({\it e.g.} $H_1 (r) = H_{-1} (r)$). Looking at the first line
in \eqref{scalar-soln} this means that we can take either the first solution or the
second but not the sum in general unless $A_{+1} = A_{-1}$. In what follows we will take
$H_p = A_{+p} r^{|p|}$. The conclusions are not qualitatively different if we make
the other choice $H_p = A_{-p} r^{-|p|}$. With this we find
\begin{equation}
\label{ckm}
U_{ll'} = f \int dr d \theta \sqrt{- ^6 g} H_p (r) \bar{ \zeta _l }(r)
\zeta _{l'} (r ) = 2 \pi \rho ^2 f \sqrt{(a-1) b c^b} \int _0 ^{\infty} H_p (r)
\frac{r^{\frac{b}{2} -1}}{c^b+r^b} \left[a_l ^* a_{l'} r^{l+l'} + b_l ^* b_{l'} r^{-l-l'} \right] ~,
\end{equation}
where in \eqref{ckm} we have carried out the $d \theta$ integration, and the condition
$p-l+l'$ holds. Inserting $H_p (r) = A_{+p} r^{|p|}$ in \eqref{ckm} and assuming all $a_l$ and
$b_l$ are purely real we get
\begin{eqnarray}
\label{ckm2}
U_{ll'} &=& 2 \pi \rho ^2 f \sqrt{(a-1) b c^b} A_{+p} \int _0 ^{\infty} \frac{r^{\frac{b}{2}+p-1}}{c^b +r^b}
\left[ a_l a_{l'} r^{l+l'} + b_l b_{l'} r^{-l-l'} \right] \nonumber \\
&=& 2 \pi^2 \rho ^2 f \sqrt{\frac{a-1}{b}} A_{+p} c^{|p|} \left[
a_l a_{l'} c^{l+l'} \sec \left( \frac{(l+l'+|p|) \pi}{b} \right) +
b_l b_{l'} c^{-l-l'} \sec \left( \frac{(l+l'-|p|) \pi}{b} \right) \right]~.
\end{eqnarray}
Now the four cases ($l=0, ~ l'=-1$), ($l=-1,~ l'=0$), ($l=0, ~ l'=1$) and ($l=1,~ l'=0$) involve $|p|=1$
and from \eqref{ckm2} yield
\begin{eqnarray}
\label{ckm3}
U_{0,-1} &=& U_{-1,0} = \frac{f D A_{+1} c^2}{2} \left( \frac{a_0 a_{-1}}{c^2} + b_0 b_{-1} \sec \left( \frac{2 \pi}{b} \right)
\right) \\
\label{ckm3a}
U_{0,1} &=& U_{1,0} = \frac{f D A_{+1} c^2}{2} \left( \sec \left( \frac{2 \pi}{b} \right) a_0 a_{-1} + \frac{b_0 b_{-1}}{c^2}
\right)
\end{eqnarray}
For the two cases $l=1,~ l'=-1$, $l=-1,~ l'=1$ one has $|p|=2$
\begin{equation}
\label{ckm4}
U_{1,-1} = U_{-1,1} = \frac{f D A_{+2} c^2}{2} \left(a_1 a_{-1} + b_1 b_{-1} \right) \sec \left( \frac{2 \pi}{b} \right)~.
\end{equation}
where as in the previous subsection $D ^2 = \frac{c^2}{4 \pi^2 \rho ^2} \sqrt{\frac{b}{a-1}}$.
We now show that parameters (i.e. $a, b, c, a_{0}$) can be chosen so the
ratios of the above mixings match the ratios of the CKM mixing matrix elements. If this
can be done then taking $a_{\pm 1}$ as in \eqref{a1} will yield the correct mass
ratios. From \cite{pdb} we take $U_{0, -1} = V_{us} \approx 0.224$,
$U_{0, 1} = V_{cb} \approx 0.040$ and $U_{-1, 1} = V_{ub} \approx 0.0036$. Again note that since
in our model we only have one flavor in each family (here taken as the ``down''
flavor or sector) these associations between $U_{i , j}$ and $V_{ij}$ are to be taken as
representing generic inter-family mixing. Now combining \eqref{ckm3} \eqref{ckm3a}
\eqref{ckm4} gives
\begin{eqnarray}
\label{ckmratio}
\frac{U_{0, -1}}{U_{0,1}} = \frac{a_0 a_{-1} + c^2 \sec \left( \frac{2 \pi}{b} \right) b_0 b_{-1}}
{b_0 b_1 + c^2 \sec \left( \frac{2 \pi}{b} \right) a_0 a_1} \approx 5.6 \\
\label{ckmratioa}
\frac{U_{0, -1}}{U_{-1,1}} = \frac{ A_{+1} \left( \frac{a_0 a_{-1}}{c^2} + b_0 b_{-1} \sec \left( \frac{2 \pi}{b} \right)
\right)}{A_{+2}\left(a_1 a_{-1} + b_1 b_{-1} \right) \sec \left( \frac{2 \pi}{b} \right)}
\approx 60.5 ~.
\end{eqnarray}
To determine the last ratio one needs to determine the normalization constants, $A_{+1}$ and $A_{+2}$ of
the $p=1$ and $p=2$ scalar field modes. For this one needs to explicitly evaluate the
integral $1= A_p ^2 \int \sqrt{- ^6g} H_p ^2 dr d \theta$ for $p=1$ and $p=2$, with $H_p (r)$ given
by \eqref{scalar-soln}. Explicitly the ratio $A_{+1} / A_{+2}$ is
\begin{equation}
\label{a+1a+2}
\frac{A_{+1}}{A_{+2}} = c \sqrt{ \frac{\int _0 ^\infty \frac{(1+ay^b)^4}{(1+y^b)^6} y^{b+3}dy}
{\int _0 ^\infty \frac{(1+ay^b)^4}{(1+y^b)^6} y^{b+1}dy}} ~.
\end{equation}
The two mass ratios have already been fixed by choosing $a_{\pm 1}$ (this also
fixes $b_{\pm 1}$ because of \eqref{norm4}) as in \eqref{a1}. Next the mixing
ratio, $\frac{U_{0, -1}}{U_{0,1}}$, in \eqref{ckmratio} can be fixed by choosing $a_0$
(this also fixes $b_0$ because of \eqref{norm4}). In particular let us parameterize
$a_0$ as $a_0 =\frac{D}{c} \eta$ so that $b_0 = \frac{D}{c} \sqrt{2 - \eta ^2}$ where
$\eta$ is arbitrary, and $D$ and $c$ are previously defined. In this way
one can see that \eqref{ckmratio} and also \eqref{a1} are independent of $c$. Finally one
can fix \eqref{ckmratioa} by choosing $a$ in \eqref{a+1a+2} to give the
ratio $A_{+1}/A_{+2}$. As an example
choose $a=7.5$, $b=4.041$, $c=1.0$, $D=1.0$ (this can be done by adjusting $\rho$), and $x=0.001$ (this
can be done by adjusting $c_0$). In this way one finds $a_1 \approx 0.1289$ and $a_{-1} \approx 0.1263$.
The associated $b_l$'s are $b_1 \approx 0.1234$ and $b_{-1} \approx 0.1262$. Using these
one find that the mass ratios in \eqref{mass4} are satisfied.
Next choosing $\eta \approx 0.223$ so that $a_0 \approx 0.223$ and $b_0 \approx 1.397$
one finds that the mixings in \eqref{ckmratio} and \eqref{ckmratioa} are satisfied.
Other values of $a$, $b$, $c$, $D$ and $x$ in this general range worked as well. However
in general the various relationships worked best when $b$ was close to $4$.
\section{Summary and Conclusions}
In this paper we studied the field equations of fermions in the
background of the non-singular, 6D brane solution of \cite{Mi} \cite{GoSi} .
By allowing the exponent, $b$, in the 4D scale function, $\phi (r)$, to
take values $b >2$ we found that we could get multiple zero mass
modes which were identified with different fermion generations.
In particular for $4< b \le 8$ we obtained three zero mass modes corresponding
to different $l$ eigenvalues: $l=-1, 0, 1$. The charge $l$ played the role
of the family number. When one fixes the value
of $b=2$ as in \cite{GoSi} one has only one zero mode \cite{Ma}.
For $b>2$ the 2D scale function, $\lambda$ has
a zero both at $r=0$ and $r= \infty$. However the scalar invariants such as the
Ricci scalar are well behaved and non-singular over the entire range of $r$,
indicating these points are coordinate rather than physical singularities.
The masses and mixings between the different generations
was given by a common mechanism -- the introduction of a
scalar field with a Yukawa coupling to the fermions.
An interesting extension of the above scenario is to see if
the scalar field could play a dual role: (i) as the mechanism for
generating the masses and mixings and (ii) as the matter source
for forming the brane. In \cite{Br} \cite{Br1} it was shown that
a scalar field could be used as a source to construct a thick brane. Thus
it might be possible to replace the phenomenological matter sources,
$F(r)$ and $K(r)$, by a scalar field source. Such a scenario would be more
economical since the scalar would serve the dual role of forming the brane
and giving masses and mixings to the fermions.
\begin{flushleft}
{\bf Acknowledgments} DS acknowledges the CSU Fresno College of
Science and Mathematics for a sabbatical leave during the period
when part of this work was completed. DS also thanks Prof. Vitaly Melnikov
for the invitation to work at VNIIMS and the People's Friendship
University of Russia where part of this work was completed.
\end{flushleft}
|
1,108,101,564,602 | arxiv | \section{Introduction}
With an exponential population growth rate, it has become critical to produce sufficient food to meet global needs. However, food production is a complex process involving many issues such as climate change, soil pollution, plant diseases, etc. Plant diseases are not only a significant threat to food safety on a global scale but also a potential disaster for the health and well-being of consumers \cite{singh2018pesticide} \cite{hoppin2017pesticides}. Due to the difficulties involved in proper disease identification, farmers apply a mixture of various pesticides, which in turn causes vegetation loss, subsequently leading to either monetary loss or affecting health. In the USA alone, food allergy cases have increased by approximately 18 percent since 2003. \cite{hoppin2017pesticides} By leveraging the increase in computing power we can take advantage of deep learning methodologies for disease detection. Nevertheless, in many scenarios, it is almost impossible to collect a large volume of data concerning a particular disease in a plant species. For example, sugarcane fungal infections such as red rot (Colletotrichum falcatum) and smut (Sporisorium scitamineum) are predominant in the South Indian peninsula. This can lead to a highly unbalanced dataset.
Considerable work \cite{badage2018crop}\cite{Saleem2019}\cite{Nagasubramanian2019} \cite{Bhatia2020} has been done to address the problem of plant disease identification \cite{maniyath2018plant} \cite{Wang2017} using machine learning, but no approach has thus far been proposed to tackle it in low data regimes.
\begin{figure}[!ht]
\begin{center}
\includegraphics[width=1.0\linewidth]{leaves.png}
\end{center}
\caption{Sample Images of types of Leaves Diseases in Mini-Leaves dataset \cite{aicrowd} by AI Crowd}
\label{fig:0}
\end{figure}
However, Mrunalini et al.\cite{badnakhe2011application} used traditional vision based feature extraction techniques followed by clustering using K-means to accurately classify leaf diseases. Similarly, Piyush et al. \cite{chaudhary2012color} has proposed an ensemble image filter followed by spot segmentation, which doesn't require huge amount of data for training. The only drawback of using traditional vision based filters for feature extraction is that there's no universal approach, unlike Convolutional neural network architectures, which works for various objectives. Few-shot learning concept has recently grabbed attention of a lot of AI applications such as imitation learning, medical disease detection etc. Few-shot as concept have various implementation ranging from non-parametric approaches such as K-Nearest Neighbor to complex deep learning algorithms.
In this paper, we propose SSM-Net, a few-shot metrics-based deep learning architecture for plants disease detection in an imbalanced and scarce data scenario. We have compared our proposed SSM-Net with other well known metric-based few-shot approaches in terms of accuracy and F1-score metric. Furthermore, we have showcased that by using proposed SSM Net (Stacked Siamese Matching), we have been able to learn better feature embeddings, achieve an accuracy of 94.3\% and F1 score of 0.90. The code is available on Github:https://github.com/shruti-jadon/PlantsDiseaseDetection.
\begin{figure}[t]
\begin{center}
\includegraphics[width=1.0\linewidth]{vgg16.jpg}
\end{center}
\caption{Architecture of VGG16 \cite{simonyan2014very} Network. We took advantage of Transfer Learning(VGG16) to extract better differentiating features up till convolutional layer.}
\label{fig:2}
\end{figure}
\section {Approaches}
Few-shot learning is a method to train an efficient machine learning model to predict any objective with less amount of data. Few-Shot Learning approaches \cite{jadon2019hands} can be widely categorized into 3 cases: Metrics based methods, Models based methods, and Optimization-based methods. In this paper, we have decided to tackle the problem of plant disease detection using metrics-based approaches and compared it with the widely-used transfer learning approach in scarce data cases. Metrics-based methods, as the name suggests, are based upon metrics such as feature embeddings, objective function, evaluation metric, etc. A metric plays a very important role in any Machine Learning model, If we are able to somehow extract proper features in initial layers of a neural network, we can optimize any network using only a few-examples. In this paper, we have taken advantage of two such metrics-based approaches: Siamese Network and Matching Network to create SSM Net. We have also taken the widely used Transfer Learning approach into account and showcased the comparison among all methods in Experiments Section. Before proceeding to Experiments, Let's first understand existing and proposed approaches.
\subsection{Transfer Learning}
Transfer learning \cite{weiss2016survey} refers to the technique of using knowledge gleaned from solving one problem to solve a different problem. Generally, we use the help of well-known networks such as Alex Net, VGG 16, Inception, Exception, etc., trained on the ImageNet dataset. For our case, we have extracted middle layer features of the VGG16 network(refer fig \ref{fig:2}) and fine-tuned by adding a linear layer using a cross-entropy loss function. We have also taken into account that Transfer Learning extracted features can be helpful in learning more advanced objectives and therefore used them to improve upon other approaches as shown in Experiments and Results Section.
\subsection{Siamese Networks}
A Siamese network \cite{koch2015siamese}, as the name suggests, is an architecture with two parallel layers. In this architecture, instead of a model learning to classify its inputs using classification loss functions, the model learns to differentiate between two given inputs. It compares two inputs based on a similarity metric and checks whether they are the same or not. This network consists of two identical neural networks, which share similar parameters, each head taking one input data point. In the middle layer, we extract similar kinds of features, as weights and biases are the same. The last layers of these networks are fed to a contrastive loss function layer, which calculates the similarity between the two inputs.
\begin{figure}[t]
\begin{center}
\includegraphics[width=1.0\linewidth]{siamese.png}
\end{center}
\caption{Architecture of Siamese Network. We took advantage of Transfer Learning(VGG16) to extract better differentiating features using Contrastive Loss Function.}
\label{fig:3}
\end{figure}
The whole idea of using Siamese architecture \cite{Jadon2019ImprovingSN}\cite{jadon2019hands} is not to classify between classes but to learn to discriminate between inputs. So, it needed a differentiating form of loss function known as the contrastive loss function. For our case, we have leveraged transfer learning as shown in Fig \ref{fig:3}, to extract complex embeddings which were not possible to learn with less amount of data-set.
\begin{figure}[t]
\begin{center}
\includegraphics[width=1.0\linewidth]{matchingnetworks.png}
\end{center}
\caption{Architecture of Matching Networks. We took advantage of Transfer Learning(VGG16) in process of creating full-contextual embeddings}
\label{fig:4}
\end{figure}
\subsection{Matching Networks}
Matching networks \cite{vinyals2016matching}, in general, propose a framework that learns a network that maps a small training dataset and tests an unlabeled example to the same embeddings space. Matching networks aim to learn the proper embeddings representation of a small training dataset and use a differentiable kNN with a cosine similarity measure to ensure whether a test data point is something ever to have been seen or not. Matching networks(refer fig \ref{fig:4}) are designed to be two-fold: Modeling Level and Training Level. At the training level, they maintain the same technique of training and testing. In simpler terms, they train using sample-set, switching the task from minibatch to minibatch, similar to how it will be tested when presented with a few examples of a new task.
At the modeling level, Matching networks takes the help of full-contextual embeddings in order to extract domain-specific features of the support set and query image. For our case, to extract better features from the support set and query image, we have leveraged transfer learning by pre-training on Matching networks on miniImageNet \cite{russakovsky2015imagenet}.
\subsection{SSM(Stacked-Siamese-Matching) Net}
Even for Matching Networks to train and learn better features, we need a decent amount of data to avoid overfitting. Using Siamese Networks we were able to extract good discriminative features. We then decided to leverage these extracted features to learn further about differences among diseases. Therefore, we have proposed a Siamese Head Plugin on top Matching Networks(refer fig \ref{fig:5}) to extract more focused features as shown in Figure 4. In this Network Architecture, instead of extracting features directly from the Transfer learning head, first, we fine-tune them using Siamese Network Architecture(With Transfer Learning Extracted Features) and once the Siamese Network is trained. We extract features using Siamese Networks for our Sugarcane Disease data and feed into pre-trained Matching Networks Architecture on miniImageNet \cite{russakovsky2015imagenet}. Using this approach, we were able to further improve the classification accuracy by ~2\%.
\begin{figure}[t]
\begin{center}
\includegraphics[width=1.0\linewidth]{matchingnetworks2.png}
\end{center}
\caption{Framework of SSM(Stacked-Siamese-Matching) Net. Here, fine-tuned Siamese Network is being used as discriminative feature extractor plugin on top of Matching Network Architecture.}
\label{fig:5}
\end{figure}
\section{Experiments and Results}
In this section, we describe our experimental setup. An implementation of Siamese Network, Matching Network, and SSM-Net is publicly available at Github: https://github.com/shruti-jadon/PlantsDiseaseDetection.
\subsection{Datasets}
For this work, we have experimented on two datasets: Mini-Leaves data-set \cite{aicrowd} by AI Crowd and our collected sugarcane data-set. Mini-Leaves dataset consists of 43525 training and 10799 test images of plant leaves at 32x32 pixels. These images belongs to 38 types of leaves disease classes. One of the major challenge is the leaves images are distorted, making it complex to extract features. On the other hand, Sugracane dataset is collected with the help of farmers in India. Overall, the aim of this work is to provide farmers (using drones) ability to mass detect disease and spray accurate pesticides in that region. Our data consist of a total of ~700 images of 11 types of sugarcane disease, as shown in Table \ref{table1}.
\begin{table}[!ht]
\begin{center}
\begin{tabular}{|l|c|}
\hline
Disease Type & No. of Images \\
\hline\hline
grassy Shoot & 90\\
leaf spot & 80\\
leaf scald & 46 \\
red rot & 93 \\
nitrogen abundance & 84\\
orange rust & 57\\
pyrilla & 66 \\
smut & 38\\
woolly aphid & 59\\
wilt & 45\\
yellow leaf disease & 42\\
\hline
\end{tabular}
\end{center}
\caption{Categories \& Number of Images of Sugarcane Diseases Dataset}
\label{table1}
\end{table}
\begin{table*}
\begin{center}
\begin{tabular}{|l|c|c|c|}
\hline
Dataset & Method & Augmented & Silhouette-Score \\
\hline\hline
Mini-Leaves Dataset & Transfer Learning & N & 0.064 \\
& Siamese Networks & N & \textbf{0.528} \\
& Siamese Networks[Modified]&N & 0.41 \\
\hline
Sugarcane Dataset & Transfer Learning &N & 0.108 \\
& Transfer Learning &Y & 0.103 \\
& Siamese Networks &N & 0.335 \\
& Siamese Networks &Y & 0.428\\
& Siamese Networks[Modified]&N & 0.341\\
& Siamese Networks[Modified] &Y & \textbf{0.556} \\
\hline
\end{tabular}
\end{center}
\caption{Decision Boundary(Discriminative Features) evaluation using Silhouette Score on Siamese Network and Transfer Learning approach.}
\end{table*}
\subsection{Implementation Details}
As part of our experiments, we needed to implement four Networks: Transfer Learning (VGG16), Siamese Network, Matching Network, and SSM-Net(proposed). For Transfer Learning, we have used VGG16 Net \cite{simonyan2014very} pretrained on Image-Net dataset. Though VGG16 Net consists of 16 layers, but for our experiments we have extracted features till convolutional layers to avoid learning Image-Net specific features. We have implemented four conv-layered \textbf{Siamese Network} following \cite{Jadon2019ImprovingSN} with contrastive loss function, It has been modified to take features from VGG16 based transfer learning. We also implemented \textbf{Matching Network} following \cite{jadon2019hands} with LSTM-based embeddings extraction. To ensure the stability of implemented architectures, we have trained Matching Networks and SSM-Net on miniImageNet dataset \cite{russakovsky2015imagenet} and fine-tuned on above mentioned datasets. We use the standard split on miniImageNet dataset of 64 base, 16 validation and 20 test classes.
For mini-leaves dataset, we had ~43525 images, we split the data into 25 base classes and 13 test classes, with images of size 32 $\times$ 32 $\times$ 3. Similarly,
as part of Sugarcane dataset, we had ~700 images, and with data augmentation \cite{wong2016understanding} \cite{cap2020leafgan}, we increased it to ~850 images. The sugarcane dataset is split into 6 base classes and 5 test classes, with images of size 224 $\times$ 224 $\times$ 3.
\subsection{Evaluation Protocol}
We compared our proposed SSM-Net outcomes in terms of decision boundaries, accuracy, and F1-Score. To assess decision boundaries, we have used Silhouette score, widely used in unsupervised learning approaches to evaluate clustering. Silhouette score measures the similarity of an point to its own cluster (intra-cluster) compared to other clusters (inter-cluster). It ranges from $-1$ to $+1$, where a high value indicates better clusters.
\begin{equation}
silhouette-score=\frac{(b - a)}{max(a, b)}
\end{equation}
Here, $a$ is mean intra-cluster distance and $b$ is the mean nearest-cluster distance for each sample.
Similarly, to assess classification outcomes, we have used F1-Score apart from accuracy to assess the quality of our experiments in terms of true positives and false negatives.
\begin{equation}
f1-score = \frac{2*(precision*recall)}{(precision+recall)}
\end{equation}
For training purposes, we have trained our Matching Networks and SSM-Net as 5 Shot-5 Way i.e; each batch consists of 5 classes per set, and each class has 5 examples.
\subsection{Results}
We compared the original Siamese network and Matching network results with baselines(Transfer learning using VGG16), to validate the effectiveness of metrics-based learning approaches method for classification/identification in the extremely low-data regime.
\textbf{Note:} We have used Siamese Network for embeddings extraction, not for classification. For classification, we have compared Matching Networks and our proposed SSM-Net.
\textbf{Comparison of Decision Boundaries with Strong Baseline}
We showcased the results of Siamese Networks in comparison to the fine-tuning of VGG16 Network in terms of decision boundaries in Table 2. Note that for decision boundary evaluation, we have extracted last layer embeddings and cluster them to the number of classes i.e; eleven. To evaluate our cluster strength, we have used the silhouette score which calculate inter-cluster vs intra-cluster distance. It is known that the silhouette score of close to 1 means better-defined clusters. For data augmentation techniques we used brightness, random scaling, rotation, and mirror flipping. It is observed that Siamese Network performs well on defining better decision boundaries in comparison to Transfer Learning+Fine-Tuning approach even with data-augmentation. Our modified Siamese Network is able to achieve 0.55 Silhouette score an increase of ~0.45 from Transfer learning with Augmentation on sugarcane dataset. Similarly using mini-leaves dataset, we observed Siamese Network based features resulted in 0.52 Silhouette score a drastic improvement from 0.06 obtained using Transfer learning. We have also noticed that mini-leaves dataset perform better without help of transfer learning features, e.g; in case of Siamese Networks with VGG 16, we obtained 0.41 Silhouette score whereas simple Siamese Networks obtained best outcome of 0.52.
\textbf{Comparison of Accuracy and F1-Score with Strong Baseline}
Here, we showcased the outcomes of 5 Shot-5 Way SSM Net, Matching Networks, and Transfer learning(VGG16) in terms of Accuracy and F1-Score listed in Table 3. Similar to our last experiment, we have used Data Augmentation techniques. We have observed that our proposed Matching Networks with Siamese Network Head performs better than other approaches. It is able to achieve an accuracy of 94.3\% and an F1-Score of 0.90 on sugarcane dataset. Similarly, on mini-leaves dataset we obtained accuracy of 91\% and F1 score of 0.85 using SSM-Net.
\begin{table*}
\begin{center}
\begin{tabular}{|l|c|c|c|c|}
\hline
Datset & Method & Augmentation & Accuracy & F1-Score \\
\hline\hline
Mini-Leaves Dataset & Transfer Learning[VGG]& N & 77.5\% & 0.693 \\
& \textbf{Matching Networks}& N & \textbf{84.5\%} &0.83 \\
& Matching Networks[miniImageNet]& N &82.7\%&0.80 \\
& \textbf{SSM-Net}& N&\textbf{92.7\%}&\textbf{0.91} \\
\hline
Sugarcane Dataset & Transfer Learning[VGG]& N & 57.4\% & 0.39 \\
& Transfer Learning[VGG]& Y & 89.3\%\% & 0.83 \\
& Matching Networks& N & 80.5\% &0.3 \\
& Matching Networks& Y & 85.5\% &0.80 \\
& Matching Networks[miniImageNet]& N &84.7\%&0.63 \\
& Matching Networks[miniImageNet]& Y &91.4\%&0.80 \\
& SSM-Net& N&85.4\%&0.72\\
& \textbf{SSM-Net }& Y&\textbf{94.3\%}&\textbf{0.90} \\
\hline
\end{tabular}
\end{center}
\caption{Accuracy and F1-Score performance of SSM Net, Matching Networks and Transfer Learning variants on Sugarcane disease data-set. Each
result is obtained over ~250 epochs. \textbf{Note:} All Matching Networks and SSM-Net Experiments outcomes are from 5-Way 5-Shot setup}
\end{table*}
\begin{figure*}[!ht]
\begin{center}
\includegraphics[width=0.85\linewidth]{output2.jpg}
\end{center}
\caption{Feature Embeddings visualization of Transfer Learning vs Modified Siamese Network Embeddings on Sugarcane datset.}
\label{fig:6}
\label{fig:onecol}
\end{figure*}
\section{Conclusion and Future Work}
Crop disease detection plays a crucial role in improving agricultural practices. If we can successfully automate early detection of crop disease, it will help us save on the amount of pesticides used and reduce crop damage. Here, we have proposed a custom metrics-based few-shot learning method, SSM Net. In this, we leveraged transfer learning and metrics-based few-shot learning approaches to tackle the problem of low data disease identification. We showcased that:
\begin{enumerate}
\item With the help of combined transfer learning and Siamese networks, we can obtain better feature embeddings.
\item Using SSM-Net we can achieve better accuracy in plants disease identification even with less amount of data.
\end{enumerate}
We envision that the proposed workflow might be applicable to other datasets \cite{Jadon_2020} that we will explore in the future. Our code implementation is available on Github: https://github.com/shruti-jadon/PlantsDiseaseDetection.
{\small
\bibliographystyle{ieee_fullname}
|
1,108,101,564,603 | arxiv | \section{Introduction}
\allowdisplaybreaks
Dynamic autocatalysis mechanisms are inherent to several real-world dynamical networks including most of the planet's cells from bacteria to human, engineered networks as well as economic systems \cite{chandra11, Autocatalytic, buzi11, buzi2010control}. In an interconnected control system with autocatalytic structure, the system's product (output) is necessary to power and catalyze its own production. The destabilizing effects of such ``positive'' autocatalytic feedback can be countered by negative regulatory feedback. There have been some recent interest to study models of glycolysis pathway as an example of an autocatalytic dynamical network in biology that generates adenosine triphospate (ATP), which is the cell's energy currency and is consumed by different mechanisms in the cell \cite{chandra11,motee6}. Other examples of autocatalytic networks include engineered power grids whose machinery are maintained using their own energy product as well as financial systems which operate based on generating monetary profits by investing money in the market. Recent results show that there can be severe theoretical hard limits on the resulting performance and robustness in autocatalytic dynamical networks. It is shown that the consequence of such tradeoffs stems from the autocatalytic structure of the system \cite{chandra11,BuziD10,motee6}.
The recent interest in understanding fundamental limitations of feedback in complex interconnected dynamical networks from biological systems and physics to engineering and economics has created a paradigm shift in the way systems are analyzed, designed, and built. Typical examples of such complex networks include metabolic pathways \cite{Goldbeter96}, vehicular platoons \cite{Jovanovic05,Jovanovic08,Raza96,Seiler04,Swaroop99}, arrays of micro-mirrors \cite{Neilson01}, micro-cantilevers \cite{Napoli99}, and smart power grids. These systems are diverse in their detailed physical behavior, however, they share an important common feature that all of them consist of an interconnection of a large number of systems that affect each others' dynamics. There have been some progress in characterization of fundamental limitations of feedback for some classes of dynamical networks. For example, only to name a few, reference \cite{Middleton10} gives conditions for string instability in an array of linear time-invariant autonomous vehicles with communication constraints, \cite{Vinay07} provides a lower bound on the achievable quality of disturbance rejection using a decentralized controller for stable discrete time linear systems with time delays, \cite{Padmasola06} studies the performance of spatially invariant plants interconnected through a static network, \cite{Leong} studies the time domain waterbed effect for single state linear systems and shows time domain analysis is useful for understanding the waterbed effect with respect to $l_1$-norm optimal control, and \cite{Siami14arxiv} investigates performance deterioration in linear dynamical networks subject to external stochastic disturbances and quantifies several explicit inherent fundamental limits on the best achievable levels of performance and show that these limits of performance are emerged only due to the specific interconnection topology of the coupling graphs. Furthermore, \cite{Siami14arxiv} characterizes some of the inherent fundamental tradeoffs between notions of sparsity and performance in linear consensus networks.
Most of the above cited research on fundamental limitations of feedback in interconnected dynamical systems have been focused on networks with linear time-invariant dynamics. The main motivation of this paper stems from a recent work presented in \cite{chandra11} that presents that glycolysis oscillation can be an indirect effect of fundamental tradeoffs in this system. The results of this work is based on a linearized model of a two-state model of glycolysis pathway and tradeoffs are stated using Bode's results. In this paper, our approach to characterize hard limits is essentially different in the sense that it uses higher dimensional and more detailed nonlinear models of the pathway. We interpret fundamental limitations of feedback by using hard limits (lower bounds) on $\mathcal L_2$-gain disturbance attenuation of the system \cite{Middleton03, Schwartz96, Schaft}, and $\mathcal L_2$-norm squared of the output of the system \cite{motee6,seron99}.
In this paper, our goal is to build upon our previous results \cite{motee6, siami12} and develop methods to characterize hard limits on performance of autocatalytic pathways. First, we study the properties of such pathways through a two-state model, which obtained by lumping all the intermediate reactions into a single intermediate reaction (Fig. \ref{fig_glycolysis-1}). Then, we generalize our results to autocatalytic pathways that are composed of a chain of enzymatically catalyzed intermediate reactions (Fig. \ref{fig_glycolysis-2}). We show that due to the existence of autocatalysis in the system (which is a biochemical necessity
, a fundamental tradeoff between a notion of fragility and net production of the pathway emerges. Also, we show that as the number of intermediate reactions grows, the price for better performance increases.
\section{Minimal Autocatalytic Pathway Model}
\subsection{Two-State Model}
\label{section2}
{We consider autocatalysis mechanism in a glycolysis pathway. The central role of glycolysis is to consume glucose and produce adenosine triphosphate (ATP), the cell's energy currency. Similar to many other engineered systems whose machinery runs on its own energy product, the glycolysis reaction is autocatalytic. The ATP molecule contains three phosphate groups and energy is stored in the bonds between these phosphate groups. Two molecules of ATP are consumed in the early steps (hexokinase, phosphofructokinase/PFK) and four ATPs are generated as pyruvate is produced. PFK is also regulated such that it is activated when the adenosine monophosphate (AMP)/ATP ratio is low; hence it is inhibited by high cellular ATP concentration \cite{Goldbeter96,Selkov75}. This pattern of product inhibition is common in metabolic pathways. We refer to \cite{chandra11} for a detailed discussion.
Experimental observations in Saccharomyces cerevisiae suggest that there are two synchronized pools of oscillating metabolites \cite{Hynne01}. Metabolites upstream and downstream of phosphofructokinase (PFK) have $180$ degrees phase difference, suggesting that a two-dimensional model incorporating PFK dynamics might capture some aspects of system dynamics \cite{Betz65}, and indeed, such simplified models qualitatively reproduce the experimental behavior \cite{Goldbeter96,Selkov75}. }
We assume that a lumped variable $x$ can encapsulate relevant information of all intermediate metabolites and consider a minimal model with three biochemical reactions as follows
\begin{eqnarray}
\begin{cases}
\begin{matrix}
\text{PFK Reaction:}& s ~+~ \alpha y \xrightarrow{~R_{\text{PFK}}~} ~ x, \\
\text{PK Reaction:}& x~ \xrightarrow{~R_{\text{PK}}~} ~ (\alpha + 1) y ~+~ x',\\
\text{Consumption:}& y ~ \xrightarrow{~R_{\text{CONS}}~} ~ \varnothing.
\end{matrix}
\end{cases}
\label{reaction-two}
\end{eqnarray}
In the PFK reaction, $s$ is some precursor and source of energy for the pathway with no dynamics associated, $y$ denotes the product of the pathway (ATP), $x$ is intermediate metabolites, $x'$ is one of the by-products of the second biochemical reaction (pyruvate kinase/PK). $\varnothing$ is a null state, $\alpha>0$ is the number of $y$ molecules that are invested in the pathway, and $\alpha + 1$ is the number of $y$ molecules produced. $A \xrightarrow{~k~} B$ denotes a chemical reaction that converts the chemical species $A$ to the chemical species $B$ at rate $k$. The PFK reaction consumes $\alpha$ molecules of ATP with allosteric inhibition by ATP.
In the second reaction, pyruvate kinase (PK) produces $\alpha+1$ molecules of ATP for a net production of one unit\footnote{For the sake of simplicity of notations, we normalize the reactions such that consumption of one molecule of $y$ produces two molecules of $y$, which is equivalent to $\alpha=1$. }. The third reaction models the cell's consumption of ATP. We refer to Fig. \ref{fig_glycolysis-1} for a schematic diagram of biochemical reactions in the minimal model.
A set of ordinary differential equations that govern the changes in concentrations $x$ and $y$ can be written as
\begin{eqnarray}
\begin{cases}
\dot{x} \, = \, R_{\text{PFK}}(y) \,-\, R_{\text{PK}}(x,y),\\
\dot{y} \, = \, -\alpha \, R_{\text{PFK}}(y) \,+\, (\alpha+1) \, R_{\text{PK}}(x,y) \,-\, R_{\text{CONS}}(y).
\end{cases}
\label{model-2-s}
\end{eqnarray}
The reaction rates are chosen according to the following steps. For the PFK reaction, we have
\begin{equation}
R_{\text{PFK}}(y)~=~\frac{2 y^{a}}{1+ y^{2h}},
\label{eq-150}
\end{equation}
where $a$ models cooperativity of ATP binding to PFK and $h$ is the feedback strength of ATP on PFK. For the PK reaction, we use
\begin{equation}
R_{\text{PK}}(x,y)~=~\frac{2 k x}{1+ y^{2g}},
\end{equation}
where $k$ is intermediate reaction rate and $g$ is the feedback strength of ATP on PK. The coefficients 2 in the numerator and feedback coefficient of the reaction rates come from the normalization. Finally, the product $y$ is consumed by basal consumption rate of $1+\delta$, {i.e., }
\begin{equation}
R_{\text{CONS}}~=~1~+~ \delt
\label{model-dist}
\end{equation}
in which $\delta$ is the perturbation in ATP consumption{\footnote{~In Example \ref{ex-2} of Section \ref{sec2}, the case of $R_{\rm CONS}~=~ k_y y + \delta$ is also studied.}. In Section \ref{sec2}, we consider more general reaction rates which are suitable for a broad class of chemical kinetics models such as Michaelis-Menten and mass-action. Reaction rates (\ref{eq-150})-(\ref{model-dist}) are consistent with biological intuition and experimental data in the case of the glycolysis pathway \cite{chandra11}.
In the final reaction, the effect of an external time--varying disturbance $\delta$ on ATP demand is considered.
The product of the pathway, ATP, inhibits the enzyme that catalyzes the first and second reactions, and the exponents $h$ and $g$ capture the strength of these inhibitions, respectively. By combing all steps, the nonlinear dynamics of (\ref{model-2-s})-(\ref{model-dist}) can be cast as
\begin{equation}
\begin{cases}
\dot{x}_1 ~ = ~ \frac{2 x_2^{a}}{1+ x_2^{2h}} ~-~ \frac{2 k x_1}{1+ x_2^{2g}},\\
\\
\dot{x}_2 ~ = ~ -\alpha \frac{2 x_2^{a}}{1+ x_2^{2h}} ~+~ (\alpha+1)\frac{2 k x_1}{1+ x_2^{2g}} ~-~ \left ({1+\delta}\right ),
\end{cases}
\label{gly-two}
\end{equation}
with output variable
\begin{equation}
y=x_2
\end{equation}
for $x_1,x_2 \geq 0$.
In order to make several comparisons possible, we normalize all concentrations such that the equilibrium point of the unperturbed system (i.e., when $\delta = 0$) becomes
\begin{equation}
\left[\begin{array}{c}
x_1^* \\
x_2^*
\end{array}\right]
\,=\, \left[\begin{array}{c}
\frac{1}{k} \\
1
\end{array}\right].
\label{fixed-point}
\end{equation}
This can be achieved by nondimensionalizing the model.
\begin{figure}
\begin{center}
\scalebox{.8}{
\begin{tikzpicture}[auto, thick, node distance=2cm, >=triangle 45]
\draw
node at (0,0)[right=-3mm]{}
node [input, name=input] {}
node [draw, thick, rectangle, minimum height = 3em,
minimum width = 5em, right of=input] (PFK) {PFK}
node [sum, right of=PFK] (inter) {6C-P}
node [draw, thick, rectangle, minimum height = 3em,
minimum width = 5em, right of=inter] (PK) {PK}
node [sum, right of=PK] (ATP) {ATP}
node [output, right of=ATP] (output) {} ;
\draw[->](input) -- node {} (PFK);
\draw[->](PFK) -- node {} (inter);
\draw[->](inter) -- node {} (PK);
\draw[->](PK) -- node {} (ATP);
\draw[->](ATP) -- node {$1+\delta$} (output);
\draw [-] (PK) -- (6,1.2)--(2,1.2)--(PFK);
\draw [-] (1.5,-.7) -- (2.5,-.7);
\draw [-] (5.5,-.7)--(6.5,-.7);
\draw [dashed] (6,-.7)--(6,-1.2)--node {$g$} (8,-1.2)--(ATP);
\draw [dashed] (2,-.7)--node {$u$} (2,-1.2);
\end{tikzpicture}}
\end{center}
\caption{A schematic diagram of the minimal glycolysis model. The constant glucose input along with $\alpha$ ATP molecules produce a pool of intermediate metabolites, which then produces $\alpha+1$ ATP molecules.}
\label{fig_glycolysis-1}
\end{figure}
In the minimal glycolysis model (\ref{gly-two}) expression $\frac{2}{1+ x_2^{2h}}$ can be interpreted as the effect of the regulatory feedback control mechanism employed by nature, which captures inhibition of the catalyzing enzyme. This observation suggests the following control system model for the minimal model of the glycolysis pathway
\begin{eqnarray}
\hspace{-.5cm}\begin{bmatrix}
\dot{x}_1\\
\dot{x}_2
\end{bmatrix} &=& \begin{bmatrix}
{1}\\
{-\alpha}
\end{bmatrix} x_2^a u+
\begin{bmatrix}
{-1}\\
{\alpha+1}
\end{bmatrix} \frac{2 k x_1}{1+ x_2^{2g}} \label{cont-gly1} - \begin{bmatrix}
{0}\\
{1+\delta}
\end{bmatrix},
\label{cont-gly-1}
\end{eqnarray}
where $u$ is the control input and captures the effect of a general feedback control mechanism. Our primary motivation behind development and analysis of such control system models for this metabolic pathways is to rigorously show that existing fundamental tradeoffs in such models are truly unavoidable and independent of control mechanisms used to regulate such pathways. For glycolysis autocatalytic pathways, the results of the following sections assert that the existing fundamental limits on performance of the pathway depend only on the autocatalytic structure of the underlying network.
{\it Stability properties of this model:}
According to \cite{chandra11}, the equilibrium point (\ref{fixed-point}) of two-state glycolysis model (\ref{gly-two}) is stable if
{\[ 0~<~ h-a~<~\frac{k+g(1+\alpha)}{\alpha}.\]
Our aim is to show that for any stabilizing control input there is a fundamental limit on the best achievable performance by the closed-loop pathway.
\subsection{Performance Measures}
We quantify fundamental limits on performance of the glycolysis pathway via two different approaches.
\subsubsection{$\mathcal{L}_2$-Gain from Exogenous Disturbance Input to Output}
{In order to quantify lower bounds on the best achievable closed-loop performance of the two-state model (\ref{cont-gly1}), we need to solve the corresponding regional state feedback $\mathcal {L}_2$-gain disturbance attenuation problem with guaranteed stability. This problem consists of determining a control law $u$ such that the closed-loop system has the following properties: (i) the zero equilibrium of the system (\ref{cont-gly1}) with $\delta(t) = 0$ for all $ t \geq 0$ is asymptotically stable with region of attraction containing $\Omega$ (an open set containing the equilibrium point), (ii)
for every $\delta \in {L}_2 (0, T )$ such that the trajectories of the system remain in $\Omega$, the $\mathcal {L}_2$-gain of the system from $\delta$ to $y$ is less than or equal to $\gamma$, {i.e., }
\begin{equation}
\int_{0}^{ T}(y(t)-y^*)^2 dt ~\leq ~\gamma^2 \int_{0}^{T}\delta^2(t) dt
\label{problem}
\end{equation}
for all $T\geq 0$ and zero initial conditions.
It is well-known that there exists a solution to the static state feedback $\mathcal {L}_2$-gain disturbance attenuation problem with guaranteed stability, in some neighborhood of the equilibrium point, if there exists a smooth positive definite solution of the corresponding Hamilton-Jacobi inequality; we refer to \cite{Middleton03, Schaft} for more details.}
{The simplest robust performance requirement for model (\ref{cont-gly-1}) is that the concentration of $y$ (i.e., ATPs) remains nearly constant when there is a small constant disturbance in ATP consumption $\delta$ (see \cite{motee6,chandra11}). However, even temporary ATP depletion can result in cell death. Therefore, we are interested in a more complete picture of the transient response to external disturbances. We show that there exists a hard limit on the best achievable disturbance attenuation, which we denoted it by $\gamma ^{*}$, for system (\ref{cont-gly1}) such that the problem of disturbance attenuation (\ref{problem}) with internal stability is solvable for all $\gamma > \gamma^{*}$, but not for all $\gamma < \gamma^{*}$.
For a linear system, it is known that the optimal disturbance attenuation can be calculated using zero-dynamics of the system \cite{Middleton03,seron99}. There is no fundamental limit on performance if and only if exogenous disturbance $\delta$ does not influence the unstable part of the zero-dynamics of the system (as it is defined in \cite{Schwartz96} for nonlinear systems).}
\subsubsection{$\mathcal{L}_2$-Norm or Total Energy of the Output}
{We characterize fundamental limitations of feedback for system (\ref{cont-gly1}) with initial condition $x(0)=x_0$ and zero external disturbances (i.e., $\delta(t)=0$) by considering the corresponding cheap optimal control problem. This case consists of finding a stabilizing state feedback control which minimizes the functional
\begin{equation}
J_{\epsilon}(x_0;u)~=~\frac{1}{2}~\int_{0}^{\infty}~\big[~\left( y(t)-y^* \right)^2 ~+~ \epsilon^2 \left(u(t)-u^*\right)^2 ~\big]~dt, \label{cheap-cost}
\end{equation}
when $\epsilon$ is a small positive number. As $\epsilon \rightarrow 0$, the optimal value $J^*_{\epsilon}(x_0)$ tends to $J^*_0(x_0)$, the ideal performance of the system. It is well-known (e.g., see \cite{sepulchre97}, page $91$) that this problem has a solution if there exists a positive semidefinite optimal value function which satisfies the corresponding Hamilton--Jacobi-Bellman equation (HJBE). The interesting fact is that the ideal performance is indeed a hard limit on performance of system (\ref{cont-gly1}). It is known that for a specific class of systems the ideal performance is the optimal value of the minimum energy problem for the zero-dynamics of the system (see \cite{seron99} for more details). The ideal performance (hard limit function) is zero if and only if the system has an asymptotically stable zero-dynamics subsystem.}
\subsection{Fundamental limits on the Performance Measures}
\subsubsection{$\mathcal L_2$-Gain Disturbance Attenuation}
In the following, it is shown that there exists a hard limit on the best achievable degrees of disturbance attenuation for system (\ref{cont-gly-1}).
\begin{theorem}\label{theorem-01}
Consider the optimal $\mathcal{L}_2$-gain disturbance attenuation problem for the minimal glycolysis model (\ref{cont-gly-1}). Then, the best achievable disturbance attenuation gain $\gamma ^{*}$ for system (\ref{cont-gly-1}) satisfies the following inequality
\begin{equation}
\gamma^* ~\ge ~\mathbf{\Gamma}(\alpha,k,g)
\label{roboust}
\end{equation}
and the hard limit function is given by
\begin{equation}
\mathbf{\Gamma}(\alpha,k,g)~=~\frac{\alpha}{k+g\alpha}.
\end{equation}
\end{theorem}
\begin{proof}
We recall that the optimal value of the achievable disturbance attenuation level $\gamma^*$ is a number with the property that the problem of disturbance attenuation with internal stability is locally solvable for each prescribed level of attenuation $\gamma > \gamma^*$ and not for $\gamma < \gamma^*$.
In the first step, we introduce a new auxiliary variable $z=x_1+\frac{1}{\alpha}x_2$. By transforming the dynamics of the system using the following change of coordinates
\begin{equation}
\left[\begin{array}{c}
y \\
z
\end{array}\right]=\left[\begin{array}{cc}
0 & 1 \\
1 & \frac{1}{\alpha}
\end{array}\right] \left[\begin{array}{c}
x_1 \\
x_2
\end{array}\right],
\end{equation}
we obtain the following form
\begin{eqnarray}
\begin{cases}
\dot{y}~=~ - \frac{\alpha+1}{\alpha}\frac{2ky}{1+y^{2g}} +
(\alpha+1)\frac{2kz}{1+y^{2g}}-\alpha y^au-(1+\delta) \label{zero-gly0} \\
\dot{z}~=~\frac{1}{\alpha}\frac{2 k z}{1+y^{2g}}-\frac{1}{\alpha^2}\frac{2 k y}{1+y^{2g}}-\frac{1}{\alpha}(1+\delta).
\end{cases}
\label{zero-gly}
\end{eqnarray}
Note that the optimal $\mathcal{L}_2$-gain disturbance attenuation of transformed system (\ref{zero-gly}) and the original system are the same. Based on \cite[Section 8.4]{Schaft2000} the optimal disturbance level for the linearized problem will provide a lower bound for the optimal disturbance of the nonlinear system.
Furthermore, for the linear system this problem reduces to a disturbance attenuation problem for the zero dynamics with cost on the control input. Thus we consider the linearized zero dynamics of (\ref{zero-gly}) as follows
\begin{equation}
\dot{\bar z}~=~\frac{k}{\alpha}\bar z-\frac{g\alpha+k}{\alpha^2}\bar y-\frac{1}{\alpha}\delta, \label{linearized-1}
\end{equation}
where
\begin{eqnarray}
\begin{cases}
\bar z ~ = ~ z-z^* ~=~ z-(x^*+\frac{1}{\alpha}y^*) \\
\bar y ~ = ~ y - y^*
\end{cases}
\end{eqnarray}
We now calculate optimal disturbance attenuation problem (from $\delta$ to $y$) for the zero dynamics with cost on its control input $y$. For system (\ref{linearized-1}), the optimal value of $\gamma$ is given by (see \cite{Scherer92, Schwartz96} for more details)
\begin{equation}
\gamma_L^*~=~ \frac{\alpha}{k+g\alpha}.
\label{gamma}
\end{equation}
Thus, we can conclude that
\[\gamma^* ~\ge~ \gamma_L^*~=~\mathbf{H}(\alpha,k,g)~=~\frac{\alpha}{k+g\alpha}.\]
\end{proof}
Theorem \ref{theorem-01} illustrates a tradeoff between robustness and efficiency (as measured by complexity and metabolic overhead). From (\ref{roboust}) the glycolysis mechanism is more robust efficient if $k$ and $g$ are large. On the other hand, large $k$ requires either a more efficient or a higher level of enzymes, and large $g$ requires a more complex allosterically controlled PK enzyme; both would increase the cell's metabolic load.
The hard limit function $\mathbf{\Gamma}(\alpha,k,g)$ in Theorem \ref{theorem-01} is an increasing function of $\alpha$. This implies that increasing $\alpha$ (more energy investment for the same return) can result in worse performance.
It is important to note that these results are consistent with results in \cite{chandra11}, where a linearized model with a different performance measure is used.
\subsubsection{Total Output Energy}
It is shown that there exists a hard limit on the best achievable ideal performance ($\mathcal{L}_2$-norm of the output) of system (\ref{cont-gly-1}). One can see that some minimum output energy ({i.e., } ATP) is required to stabilize the unstable zero-dynamics (\ref{zero-gly}). This output energy represents the energetic cost of the cell to stabilize it to its steady-state. In the following theorem, we show that the minimum output energy is lower bounded by a constant which is only a function of the parameters and initial conditions of the glycolysis model. This hard limit is independent of the feedback control strategy used to stabilize the system.
\begin{theorem}\label{theorem-03}
Suppose that the equilibrium of interest is given by (\ref{fixed-point}) and ${u^*}=1$. Then, there is a hard limit on the performance measure of the unperturbed ($\delta = 0$) system (\ref{cont-gly-1}) in the following sense
\begin{equation}
\int_{0}^{\infty}~(y(t;u_0)-\bar{y})^2~dt~\geq~ \frac{\alpha^3k}{(g\alpha+k)^2}~z_0^2 +J(z_0;\alpha,k,g),
\end{equation}
where $z_0=\left(x(0)-{x^*}\right)+\frac{1}{\alpha}\left(y(0)-{y^*}\right)$, $u_0$ is an arbitrary stabilizing feedback control law for system (\ref{cont-gly-1}), $J(0;\alpha,k,g)=J(z;\alpha,k,0)=0$ and $|J(z;\alpha,k,g)| \leq c|z|^3$ on an open set $\Omega$ around the origin in ${\mathbb{R}}$.
\end{theorem}
\begin{proof}
By introduction of a new variable $z=x_1+\frac{1}{\alpha}y$, we rewrite (\ref{cont-gly-1}) in the canonical form (\ref{zero-gly}).
We denote by $\pi(y,z;\epsilon)$ the solution of the HJB PDE corresponding to the cheap optimal control problem to (\ref{cont-gly-1}). We apply the power series method \cite{Albrecht61,Lukes69} by first expanding $\pi(y,z;\epsilon)$ in series as follows
\begin{equation}
\pi(y,z;\epsilon)~=~ \pi^{[2]}(y,z;\epsilon) ~+~ \pi^{[3]}(y,z;\epsilon) ~+~ \ldots \label{power_series}
\end{equation}
in which $k$th order term in the Taylor series expansion of $\pi(y,z;\epsilon)$ is denoted by $\pi^{[k]}(y,z;\epsilon)$. Then (\ref{power_series}) is plug into the corresponding HJB equation of the optimal cheap control problem. The first term in the series is
\begin{equation*}
\pi^{[2]}(y,z;\epsilon) ~=~ \left[
\begin{array}{cc}
y-y^* & z-z^* \\
\end{array}
\right] P(\epsilon) \left[
\begin{array}{c}
y-y^* \\
z-z^* \\
\end{array}
\right],
\end{equation*}
where $P(\epsilon)$ is the solution of algebraic Riccati equation to the cheap control problem for the linearized model $(A_0,B_0)$. It can be shown that $P(\epsilon)$ can be decomposed in the form of a series in $\epsilon$ (see \cite{KwakernaakS72} for more details)
\begin{equation*}
P(\epsilon)~=~\left[
\begin{array}{cc}
\epsilon P_1 & \epsilon P_2 \\
\epsilon P_2 & P_0+\epsilon P_3 \\
\end{array}
\right] + \mathcal{O}(\epsilon^2).
\end{equation*}
Since the pole of the zero-dynamics of the linearized model is located at the $\frac{k}{\alpha}$, we can verify that $P_0 =\frac{2\alpha^3k}{(g\alpha+k)^2}$. Therefore, it follows that $\pi^{[2]}(y,z;\epsilon)=\frac{\alpha^3k}{(g\alpha+k)^2}z_0^2+\mathcal{O}(\epsilon)$.
We only explain the key steps. One can obtain governing partial differential equations for the higher-order terms $\pi^{[k]}(y,z;\epsilon)$ for $k \geq 3$ by equating the coefficients of terms with the same order. It can be shown that $\pi^{[k]}(y,z)=\pi_0^{[k]}(z) + \epsilon \pi_1^{[k]}(y,z)+\mathcal{O}(\epsilon^2)$ for all $k \geq 3$.
Then, by constructing approximation of the optimal control feedback by using computed Taylor series terms, one can prove that $\pi(y,z;\epsilon) \rightarrow \frac{\alpha^3k}{(g\alpha+k)^2}z_0^2+(\text{higher order terms in } z_0)$ as $\epsilon \rightarrow 0$. Thus, the ideal performance cost value is $\frac{\alpha^3k}{(g\alpha+k)^2}z_0^2+J(z_0;\alpha,k,g)$.
\end{proof}
According to Theorems \ref{theorem-01} and \ref{theorem-03}, a fundamental tradeoff between a notion of fragility and net production of the pathway emerges as follows: increasing $\alpha$ (number of ATP molecules invested in the pathway), increases fragility of the network to small disturbances (based on Theorem \ref{theorem-01}) and it can result in undesirable transient behavior (based on Theorem \ref{theorem-03}).
The large fluctuation in the level of ATP is not desirable, if the level of ATP drops below some threshold, there will not be sufficient supply of ATP for different pathways in the cell and that can result to cell death.
\begin{figure*}
\begin{center}
\scalebox{1}{
\begin{tikzpicture}[auto, thick, node distance=2cm, >=triangle 45]
\draw
node at (0,0)[right=-3mm]{}
node [input, name=input] {}
node [draw, thick, rectangle, minimum height = 3em,
minimum width = 5em, right of=input] (PFK) {PFK}
node [sum, right of=PFK] (x1) {$x_1$}
node [right of=x1](noghte){\ldots}
node [sum, right of=noghte] (xn) {$x_n$}
node [draw, thick, rectangle, minimum height = 3em,
minimum width = 5em, right of=xn] (PK) {PK}
node [sum, right of=PK] (ATP) {ATP}
node [output, right of=ATP] (output) {};
\draw[->](input) -- node {} (PFK);
\draw[->](PFK) -- node {$K_1$} (x1);
\draw[->](x1)-- node {$K_2$} (noghte);
\draw[->](noghte)-- node {$K_{n}$} (xn);
\draw[->](xn)-- node {} (PK);
\draw[->](PK) -- node {} (ATP);
\draw[->](ATP) -- node {$1+\delta$} (output);
\draw [-] (PK) -- (10,1.2)--(2,1.2)--(PFK);
\draw [-] (1.5,-.7) -- (2.5,-.7);
\draw [-] (9.5,-.7)--(10.5,-.7);
\draw [dashed] (10,-.7)--(10,-1.2)--node {$g$}(12,-1.2)--(ATP);
\draw [dashed] (2,-.7)--node {$u$} (2,-1.2);
\end{tikzpicture}}
\end{center}
\caption{ A schematic diagram of a glycolysis pathway model with intermediate reactions. The constant glucose input along with $\alpha$ ATP molecules produce a pool of intermediate metabolites, which then produces $\alpha+1$ ATP molecules.}
\label{fig_glycolysis-2}
\end{figure*}
\section{Autocatalytic Pathways With Multiple Intermediate Metabolite Reactions}
In Subsection {\ref{section2}}, we studied the property of such pathways with a two-state model (\ref{cont-gly-1}), which is obtained by lumping all the intermediate reactions into a single intermediate reaction. In the next step, we consider autocatalytic pathways with multiple intermediate metabolite reactions as shown in Fig. \ref{fig_glycolysis-2}:
\begin{eqnarray}
\begin{cases}
\begin{matrix}
\text{PFK Reaction:}& s ~+~ \alpha y \xrightarrow{~R_{\text{PFK}}~} ~ x_1, \\
\text{Intermediates:}& x_1 \xrightarrow{~R_{\text{IR}}~} x_2 ~ \cdots ~\xrightarrow{~R_{\text{IR}}~}x_{n},\\
\text{PK Reaction:}& x_{n}~ \xrightarrow{~R_{\text{PK}}~} ~ (\alpha + 1) y ~+~ x',\\
\text{Consumption:}& y ~ \xrightarrow{~R_{\text{CONS}}~} ~ \varnothing.
\end{matrix}
\end{cases}
\label{reaction-n}
\end{eqnarray}
A set of ordinary differential equations that govern the changes in concentrations of $x_i$ for $i=1, \ldots, n$ and $y$ can be obtained as follows
\begin{eqnarray}
\begin{cases}
\dot{x}_1 ~=~ R_{\text{PFK}}(y) \,-\, R_{\text{IR}}(x_1),
\\ \label{cont-gly3}
\dot{x}_2~ =~ R_{\text{IR}}(x_1)\,-\,R_{\text{IR}}(x_2),
\\
~~~~~ \vdots \\
\dot{x}_n~=~ R_{\text{IR}}(x_{n-1})\, - \, R_{\text{PK}}(x_n,y), \\
~\dot{y} ~ = ~(\alpha+1)R_{\text{PK}}(x_n,y)\, -\, \alpha R_{\text{PFK}}(y) \,-\, {R_{\text{CONS}}}
\end{cases}
\label{general-model}
\end{eqnarray}
for $x_i \geq 0$ and $y \geq 0$. Our notations are similar to those of the two-state pathway model (\ref{reaction-two}). The reaction rates are given as follows
\begin{eqnarray}
\begin{cases}
R_{\text{PFK}}(y)~=~\frac{2 y^{a}}{1+ y^{2h}},\\
R_{\text{PK}}(x_n,y)~=~\frac{2 K_n x_n}{1+ y^{2g}},\\
R_{\text{IR}}(x_i)~=~K_i x_i ~~\text{for}~~n=1,2,\ldots,n,\\
R_{\text{CONS}}~=~1+\delta
\label{rates}
\end{cases}
\end{eqnarray}
Furthermore, in the glycolysis model (\ref{general-model}), similar to the minimal model (\ref{cont-gly-1}), expression $\frac{2}{1+x^{2h}}$ can be interpreted as the effect of the regulatory feedback control mechanism employed by nature that captures inhibition of the catalyzing enzyme. Hence, we can derive a control system model for the autocatalytic pathway with multiple intermediate metabolite reactions as follows
\begin{small}
\begin{equation}
\begin{cases}
\dot{x}_1 ~=~ y^a u~-~K_1x_1,\\
\dot{x}_2~ =~ K_1{x_1}~ - ~ K_2x_2,\\
~~~~~ \vdots\\
\dot{x}_n~=~ K_{n-1}{x_{n-1}}~ - ~ \frac{2K_n x_n}{1+y^{2g}}, \\
\dot{x}_{n+1} ~ = ~(\alpha+1)\frac{2K_n x_n}{1+x_{n+1}^{2g}} ~ - ~ \alpha x_{n+1}^a u ~- ~ {(1+\delta)},\\
~y~=~x_{n+1},
\end{cases}
\label{cascade-model}
\end{equation}
\end{small}for $x_i \geq 0$ and $y \geq 0$.
In order to simplify our analysis and be able to calculate explicit formulae, we assume that $K:=K_1=\dots=K_n>0$. We normalize all concentrations such that unperturbed steady states become
\begin{eqnarray}
y^*=x^*_{n+1}=1 ~~~\textrm{and}~~~ x_i=K^{-1}
\label{eq-point}
\end{eqnarray}
for all $i=1,\ldots,n$.
\subsection{$\mathcal L_2$-Gain Disturbance Attenuation}
We extend our results in Theorem \ref{theorem-01} to higher dimensional model of autocatalytic pathways. In the following theorem, we show that there exists a size-dependent hard limit on the best achievable disturbance attenuation for system (\ref{cascade-model}).
\begin{theorem}\label{theorem-02}
Consider the optimal $\mathcal{L}_2$-gain disturbance attenuation problem for glycolysis model (\ref{cascade-model}). Then, the best achievable disturbance attenuation gain $\gamma ^{*}$ for system (\ref{cascade-model}) satisfies the following inequality
\begin{equation}
\gamma^* ~ \ge ~ \mathbf{\Gamma}(\alpha,K,g,n),
\label{roboust-2}
\end{equation}
where the hard limit function is given by
\begin{eqnarray}
&& \mathbf{\Gamma}(\alpha,K,g,n) = \nonumber \\
&&~~~~~~~\left[\left( K+g\alpha \left(\frac {\alpha+1}{\alpha}\right)^{\frac{n-1}{n}} \right) \left( \left(\frac {\alpha+1}{\alpha} \right)^{\frac{1}{n}}-1\right) \right]^{-1}.
\nonumber
\end{eqnarray}
\end{theorem}
\begin{proof}
First, by introducing a new variable $z_1=x_1+\frac{1}{\alpha}y$, we can cast the zero-dynamics of (\ref{cascade-model}) in the following form
\begin{small}
\begin{eqnarray}
\begin{cases}
\dot{z}_1 ~=~ -Kz_1~+~ \frac{\alpha+1}{\alpha}\frac{2Kx_n}{1+y^{2g}}~+~\frac{K}{\alpha}y-\frac{1}{\alpha}(\delta+1),
\\
\dot{x}_2 ~=~ K{z_1}~-~\frac{K}{\alpha}y~-~Kx_2,
\\
~~~\cdots\\
\dot{x}_n~=~ K{x_{n-1}} ~-~ \frac{2Kx_n}{1+y^{2g}}.
\end{cases}
\label{zero-gly-cascade}
\end{eqnarray}
\end{small}
Let us define
\begin{equation}
z := \begin{bmatrix}z_1& x_2& \dots& x_n\end{bmatrix}^{\text{T}},
\label{eq:667}
\end{equation}
and
\begin{equation}
z^*:=\begin{bmatrix}\frac{1}{K}+\frac{1}{\alpha}& \frac{1}{K}& \ldots& \frac{1}{K}\end{bmatrix}^{\text T}.
\label{eq:672}
\end{equation}
Then, we rewrite (\ref{zero-gly-cascade}) in the following form
\begin{equation}
\dot{\bar z}~=~A\bar z~+~B\bar y~+~C\delta~+~\bar f(\bar z,\bar y),
\label{nonlinear-zero-cas}
\end{equation}
where
\begin{eqnarray}
&&A~=~\left[
\begin{smallmatrix}
-K & 0& 0& ~\ldots~ & (1+\frac{1}{\alpha})K \\
K &-K& 0& ~\ldots~ & 0 \\
0 & K& -K&~\ldots~&0\\
& \vdots& & ~\ddots~ & \vdots \\
0 & 0 & 0 & \ldots & -K \\
\end{smallmatrix}
\right],\nonumber \\
&&B~=~\left[
\begin{smallmatrix}
-\frac{\alpha+1}{\alpha}g+\frac{K}{\alpha} \\
-\frac{K}{\alpha} \\
\vdots \\
g\\
\end{smallmatrix}\right],~C~=~\left[
\begin{smallmatrix}
-\frac{1}{\alpha} \\
0\\
\vdots \\
0\\
\end{smallmatrix}\right],
\label{A-B-C}
\end{eqnarray}
$\bar z=z-z^*$, $\bar y= y - y^*$, $\bar f(0,0) = 0$ and
\begin{equation}
\Big {\|} \frac{\partial \bar f(\bar z,\bar y)}{\partial (\bar z,\bar y)} \Big{\|} ~\leq~ c |(\bar z,\bar y)|,
\end{equation}
near the origin in ${\mathbb{R}}^n$ for $c > 0$.
Now, according to \cite{Schaft} we know that if the system (\ref{nonlinear-zero-cas}) has $\mathcal L_2$-gain $\leq \gamma$, then the linearized system has $ \mathcal L_2$-gain $\leq \gamma$.
Hence, we only consider the linearized system, {i.e., }
\begin{equation}
\dot{\bar z}~=~A\bar z~+~B \bar y~+~C\delta.
\label{zero-cas}
\end{equation}
Note that $\lambda=K\left[(\frac{\alpha+1}{\alpha})^{\frac{1}{n}}-1\right]$ is the eigenvalue of $A$ with the greatest real part. And the corresponding left eigenvector of $\lambda$, is $v=\begin{bmatrix}1& (\frac{\alpha+1}{\alpha})^{\frac{1}{n}}& \ldots& (\frac{\alpha+1}{\alpha})^{\frac{n-1}{n}}\end{bmatrix}^{\text T}$. Now, we consider the following subsystem of (\ref{zero-cas})
\begin{equation*}
\dot{\tilde z} = \lambda \tilde z+\Big[\big((1+\frac {1}{\alpha})^{\frac{n-1}{n}}-(1+\frac{1}{\alpha})\big)g-\frac{K}{\alpha}\big( (1+\frac {1}{\alpha})^{\frac{1}{n}} -1\big )\Big ]\bar y-\frac{1}{\alpha}\delta.
\end{equation*}
Based on the result of \cite{Scherer92} and \cite{Schwartz96}, the formula to compute the optimal value of $\gamma$ reduces to
\begin{equation*}
\gamma_L^*~\geq~\frac{1}{\big( K+g\alpha (1+\frac {1}{\alpha})^{\frac{n-1}{n}} \big )\big((1+\frac {1}{\alpha})^{\frac{1}{n}}-1\big)}.
\end{equation*}
Note that according to Proposition $6$ of \cite{Schaft}, $\gamma_L^*$ is a lower bound for the optimal $\gamma^*$ of the nonlinear system (\ref{cascade-model}).
\end{proof}
\subsection{Total Output Energy}
It is proven that there exists a size-dependent hard limit on the best achievable ideal performance of system (\ref{cascade-model}).
\begin{theorem}\label{theorem-04}
Suppose that the equilibrium of interest is given by (\ref{eq-point}) and ${u^*}=1$. Then, the $\mathcal{L}_2$-norm of the output of the unperturbed system (\ref{cascade-model}) cannot be made arbitrarily small, which implies that there is a fundamental limit on performance in the following sense
\begin{eqnarray}
&& \hspace{-1.3cm} \int_{0}^{\infty}\big(y(t;u_0)-{y^*}\big)^2~dt~\\ \nonumber
&&~~~~~~~~~~~~\geq~ \mathbf H(z_0;\alpha,K,g,n)~+J(z_0;\alpha,K,g,n),
\end{eqnarray}
where
\begin{eqnarray}
&&\hspace{-.5cm}\mathbf H(z_0;\alpha,K,g,n)= \nonumber\\
&&\frac{\alpha^2K\big ( \frac{1}{\alpha}(y(0)-y^*)+\sum_{i=1}^{n}(\frac{\alpha+1}{\alpha})^{\frac{i-1}{n}}(x_i(0)-x_i^*)\big)^2}{\big((\frac {\alpha+1}{\alpha})^{\frac{1}{n}}-1\big)\big( K+g\alpha (\frac {\alpha+1}{\alpha})^{\frac{n-1}{n}} \big )^2},\nonumber
\end{eqnarray}
$u_0$ is an arbitrary stabilizing feedback control law for system (\ref{cascade-model}), $z_0= z(0)-z^*$ where $z$ and $z^*$ are defined by \eqref{eq:667} and \eqref{eq:672} respectively, $J(0;\alpha,K,g,n)=J(z;\alpha,K,0,n)=0$, and $|J(z;\alpha,K,g,n)| \leq c|z|^3$ on an open set $\Omega$ around the origin in ${\mathbb{R}}^n$
\end{theorem}
\begin{proof}
The proof of this theorem based on results from \cite{Albrecht61,Lukes69} and Theorem \ref{theorem-03}. Similar to the proof of Theorem \ref{theorem-02}, one can cast the zero-dynamics of the unperturbed system \eqref{cascade-model} as follows
%
\begin{equation}
\dot{\bar z}~=~A\bar z~+~B\bar y~+~\bar f(\bar z,\bar y),
\label{nonlinear-zero-cas}
\end{equation}
where $A$ and $B$ are given by \eqref{A-B-C},
$\bar z=z-z^*$, $\bar y= y - y^*$, $\bar f(0,0) = 0$ and
\begin{equation*}
\Big {\|} \frac{\partial \bar f(\bar z,\bar y)}{\partial (\bar z,\bar y)} \Big{\|} ~\leq~ c |(\bar z,\bar y)|
\end{equation*}
near the origin in ${\mathbb{R}}^n$ for $c > 0$. We denote by $\pi(y,z;\epsilon)$ the solution of the HJB PDE corresponding to the cheap optimal control problem to the unperturbed system (\ref{cascade-model}). We apply the power series method \cite{Albrecht61,Lukes69} by first expanding $\pi(y,z;\epsilon)$ in series as in \eqref{power_series}, where $\pi^{[k]}(y,z;\epsilon)$ denotes $k$'th order term in the Taylor series expansion of $\pi(y,z;\epsilon)$. Then, (\ref{power_series}) is plugged into the corresponding HJB equation of the optimal cheap control problem. The first term in the series is
\begin{equation*}
\pi^{[2]}(y,z;\epsilon) ~=~ \left[
\begin{array}{cc}
y-y^* & z-z^* \\
\end{array}
\right] P(\epsilon) \left[
\begin{array}{c}
y-y^* \\
z-z^* \\
\end{array}
\right],
\end{equation*}
where $P(\epsilon)$ is the solution of algebraic Riccati equation to the cheap control problem for the linearized model. It can be shown that $P(\epsilon)$ can be decomposed in the form of a series in $\epsilon$ (see \cite{KwakernaakS72} for more details)
\begin{equation*}
P(\epsilon)~=~\left[
\begin{array}{cc}
\epsilon P_1 & \epsilon P_2 \\
\epsilon P_2 & P_0+\epsilon P_3 \\
\end{array}
\right] + \mathcal{O}(\epsilon^2)
\end{equation*}
in which $P_0$ is the positive solution of the associated algebraic Riccati equation for $(A,B)$, {i.e., }
\[ A^{\text{T}} P_0 + P_0 A~=~ P_0 BB^{\text{T}}P_0.\]
It follows that
\[\pi^{[2]}(y,z;\epsilon)=\frac{1}{2}z_0^{\text{T}} P_0 z_0+\mathcal{O}(\epsilon).\]
One can obtain governing partial differential equations for the higher-order terms $\pi^{[k]}(y,z;\epsilon)$ for $k \geq 3$ by equating the coefficients of terms with the same order. It can be shown that
\[ \pi^{[k]}(y,z)=\pi_0^{[k]}(z) + \epsilon \pi_1^{[k]}(y,z)+\mathcal{O}(\epsilon^2)\] for all $k \geq 3$. Then, by constructing approximation of the optimal control feedback by using computed Taylor series terms, one can prove that $\pi(y,z;\epsilon) \rightarrow \frac{1}{2}z_0^{\text{T}} P_0 z_0+(\text{higher order terms in } z_0)$ as $\epsilon \rightarrow 0$. Thus, the ideal performance cost value can be written as
\begin{equation}
\lim_{\epsilon \rightarrow 0}\pi(y,z;\epsilon)~=~\frac{1}{2}z_0^{\text{T}} P_0 z_0+J(z_0;\alpha,K,g,n).
\label{eq:835}
\end{equation}
Next, we obtain a lower bound on $\frac{1}{2}z_0^{\text{T}} P_0 z_0$. The characteristic equation of matrix $A$ is characterized by
\[ (x+K)^{n}-\frac{\alpha +1}{\alpha}K^n~=~0.\]
Therefore, one can see that $\lambda=K\left[(\frac{\alpha+1}{\alpha})^{\frac{1}{n}}-1\right]$ is the eigenvalue of $A$ with the greatest real part and its corresponding left eigenvector is $v=\begin{bmatrix}1& (\frac{\alpha+1}{\alpha})^{\frac{1}{n}}& \ldots& (\frac{\alpha+1}{\alpha})^{\frac{n-1}{n}}\end{bmatrix}^{\text T}$. Now, let us consider the subsystem associated to this mode as follows
\begin{equation*}
\dot{\tilde z} = \lambda \tilde z+\Big[\big((\frac {\alpha+1}{\alpha})^{\frac{n-1}{n}}-(\frac{\alpha+1}{\alpha})\big)g-\frac{K}{\alpha}\big( (\frac {\alpha+1}{\alpha})^{\frac{1}{n}} -1\big )\Big ] \bar y,
\end{equation*}
where
\begin{equation}
\tilde z~=~ v^{\text{T}} \bar z ~=~ \frac{1}{\alpha} \bar y + \sum_{i=1}^n \left( \frac{\alpha+1}{\alpha}\right)^{\frac{i-1}{n}} \bar x.
\label{eq:847}
\end{equation}
The corresponding cost value for this subsystem is given by
\begin{equation}
\frac{1}{2}z_0^{\text{T}} P_0 z_0 ~\geq~ \frac{\alpha^2K \tilde z(0)^2}{\big((\frac {\alpha+1}{\alpha})^{\frac{1}{n}}-1\big)\big( K+g\alpha (\frac {\alpha+1}{\alpha})^{\frac{n-1}{n}} \big )^2}
\label{eq:853}
\end{equation}
which is a lower bound for the linearized cost $ \frac{1}{2}z_0^{\text{T}} P_0 z_0$.
Finally, using \eqref{eq:835}, \eqref{eq:847} and \eqref{eq:853}, we get the desired result.
\end{proof}
In the case that the number of intermediate reactions is one ({i.e., } $n=1$) the results of Theorems \ref{theorem-02} and \ref{theorem-04} reduce to the results of Theorems \ref{theorem-01} and \ref{theorem-03}, respectively.
Through a straightforward analysis, one can argue that $\mathbf H(z_0;\alpha,K,g,n) \in \mathcal{O}(n)$ and $\mathbf{\Gamma}(\alpha,K,g,n) \in \mathcal{O}(n)$, and they can be approximated by
\begin{eqnarray}
&&\hspace{-1cm}\mathbf H(z_0;\alpha,K,g,n) \approx \nonumber \\
&&~\frac{\alpha^2K\big ( \frac{1}{\alpha}(y(0)-y^*)+{\sum_{i=1}^{n}(\frac{\alpha+1}{\alpha})^{\frac{i-1}{n}}(x_i(0)-x_i^*)}\big)^2}{\big( K+g(\alpha+1) \big )^2 \ln(\frac{\alpha+1}{\alpha})}n\nonumber
\end{eqnarray}
and
\begin{equation}
\mathbf{\Gamma}(\alpha,K,g,n) ~\approx~ \frac{n}{ \big (g(\alpha+1)+K \big )\ln (1+\frac{1}{\alpha})}.
\label{33}
\end{equation}
This implies that as the number of intermediate reactions $n$ grows, the price paid for robustness for both $\mathbf H(z_0;\alpha,K,g,n)$ and $\mathbf{\Gamma}(\alpha,K,g,n)$ increases linearly by network size $n$. In general, the larger the number of intermediate reactions involved in the breakdown of a metabolite, the less complex the enzymes involved in the individual reactions need to be. On the other hand, increasing the number of intermediate metabolites results in larger $\mathbf {\Gamma}$ and $\mathbf H$ which means less robustness to disturbances and having undesirable transient behavior.
\section{General Autocatalytic Pathways}
\label{sec2}
In the final step, we turn our focus on networks with autocatalytic structures (as shown in Fig. \ref{fig_3}) that belong to a class of nonlinear dynamical networks with cyclic feedback structures driven by {disturbance}. Each network consists of a group of nonlinear subsystems with state-space dynamics
\begin{eqnarray}
&&\left\{ \begin{array}{rcl}
\dot x_i &=&- f_i (x_i) + u_i \\
y_i& =& g_i(x_i)
\label{eq7}
\end{array}\right.
\end{eqnarray}
for $x_i \geq 0$, $y_i \geq 0$, $1\leq i \leq n$, and
\begin{eqnarray}
&&\left\{ \begin{array}{rcl}
\dot x_{n+1} &=& -f_{n+1}(x_{n+1}) + u_{n+1} - \alpha u, \\
y_{n+1} &=& u,
\end{array}\right.
\label{eq8}
\end{eqnarray}
where $f_i(\cdot)$ and $g_i(\cdot)$ for $i=1,\ldots,n$ are increasing functions. Moreover, $u_i(t)$, $y_i(t)$ and $x_i(t)$ are input, output and state variables of each subsystem, respectively.
These assumptions are suitable for a broad class of chemical kinetics models such as Michaelis-Menten and mass-action. The state-space representation of the nonlinear cyclic interconnected network shown in Fig. \ref{fig_3} is given by
\begin{eqnarray}
\begin{cases}
\dot x_1 ~=~ -f_1 (x_1) + y_{n+1}, \\
\dot x_2 ~=~ -f_2 (x_2) + y_1,\\
~~~~\cdots~ \\
\dot x_{n+1} ~=~ -f_{n+1} (x_{n+1})+ y_{n} - \alpha u +\delta,\\
y ~=~ x_{n+1}.
\end{cases}
\label{eq88}
\end{eqnarray}
\begin{assumption}
We assume that $x_i^*$ for $i=1,\ldots,n$ and $y^*$ are equilibrium points of the unperturbed system (\ref{eq88}). Moreover, it is assumed that
\begin{equation}
a:=f^{\prime}_1(x_1^*)=f^{\prime}_2(x_2^*)= \cdots =f^{\prime}_{n}(x_n^*),
\label{a}
\end{equation}
where $f^{\prime}_i(x_i^*):=\left .\frac{\mathrm d f_i}{\mathrm d x_i} \right |_{x_{i}=x_i^*}$.
\end{assumption}
\begin{figure}[t]
\begin{center}
\scalebox{1.1}{
\begin{tikzpicture}
\draw [ thick] (-2.5,0) circle [radius=0.4];
\draw [ thick] (-1,0) circle [radius=0.4];
\draw [ thick] (.5,0) circle [radius=0.4];
\draw [ thick] (2.5,0) ellipse (.5 and .4);
\draw [->, thick] (-2,0) -- (-1.5,0);
\draw [->, thick] (-.5,0) -- (0,0);
\draw [fill] (1.3,0) circle [radius=.04];
\draw [fill] (1.5,0) circle [radius=.04];
\draw [fill] (1.7,0) circle [radius=.04];
\draw [ thick, dashed] (3.1,0) -- (3.5,0);
\draw [ thick, dashed] (3.5,0) -- (3.5,.8);
\draw [ thick, dashed] (3.5,.8) -- (-3.5,.8);
\draw [ thick, dashed] (-3.5,.8) -- (-3.5,0);
\draw [ ->,thick, dashed] (-3.5,0) -- (-3,0);
\node[] at (-2.5,0) {$x_1$};
\node[] at (-1,0) {$x_2$};
\node[] at (.5,0) {$x_3$};
\node[] at (2.5,0) {$x_n,u$};
\draw [ ->,thick] (2.5,-.8) -- (2.5,-.5);
\node[] at (2.7,-.8) {$\delta$};
\end{tikzpicture}}
\end{center}
\caption{{\small The schematic diagram of the nonlinear network (\ref{eq88}) with a cyclic feedback structure with an output disturbance $\delta$ and control input $u$.}}
\label{fig_3}
\end{figure}
\begin{theorem}
\label{theorem-5}
For cyclic networks (\ref{eq88}), if
\begin{equation}
r:= \left (\frac{g^{\prime}_1(x_1^*) g^{\prime}_2(x_2^*) \cdots g^{\prime}_{n}(x_n^*)}{\alpha}\right )^{\frac{1}{n}} > a,
\label{eqr}
\end{equation}
then there exists a hard limit on the best achievable disturbance attenuation (i.e., $\gamma ^{*} >0$) for system (\ref{eq88}) such that the regional state feedback $\mathcal{L}_2$--gain disturbance attenuation problem with stability constraint is solvable for all $\gamma > \gamma^{*}$ and is not solvable for all $\gamma < \gamma^{*}$.
Furthermore, the hard limit function is given by
\begin{equation}
\gamma^* \ge \mathbf{\Gamma}(f^{\prime}_{n+1}(y^*),r,a)=\frac{1}{f^{\prime}_{n+1}(y^*)+r-a}.
\label{roboust-3}
\end{equation}
\end{theorem}
\begin{proof}
In the first step, we introduce a new auxiliary variable $z_1=x_1+\frac{1}{\alpha} x_{n+1}$. We can cast the linearized zero-dynamics of (\ref{eq8}) in the following form
\begin{equation}
\hspace{-0.65cm} \dot{z} ~= ~A_0 z~+~B_0 y~ +~ C_0 \delta,\label{zero-gly-cascade-2}
\end{equation}
where $z=[z_1,x_2,\cdots,x_{n}]^{{\text{T}}}$,
\begin{eqnarray}
&&A_0=\left[
\begin{matrix}
-a & 0 & \ldots & 0 & {\alpha}^{-1} g^{\prime}_{n}(x_n^*) \\
g^{\prime}_1(x_1^*) & -a & \ldots & 0 & 0 \\
\vdots & \vdots & \ddots &\vdots & \vdots & \\
0 & 0 & \ldots & -a & 0 \\
0 & 0 & \ldots & g^{\prime}_{n-1} (x_{n-1}^*)& -a
\end{matrix}
\right],\nonumber \\
&&B_0=\left[
\begin{matrix}
\frac{a-f^{\prime}_{n+1}(y^*)} {\alpha} \\
-\frac{g^{\prime}_1(x_1^*)} {\alpha} \\
\vdots \\
0\\
0
\end{matrix}
\right],~\text{and}~~C_0=\left[
\begin{matrix}
{\alpha}^{-1} \\
0\\
\vdots \\
0\\
0
\end{matrix}
\right].
\end{eqnarray}
Then, we consider the characteristic equation of matrix $A_0$ which is given by
\begin{equation}
(\lambda+a)^{n}-r^{n}~=~0.
\label{charc}
\end{equation}
From (\ref{eqr}) and (\ref{charc}), it follows that $\lambda_1=r-a$ is the eigenvalue of $A_0$ with the largest real-part value with left eigenvector
\[v_1=\Big [\,1~,~ \frac{r}{g^{\prime}_1(x_1^*)}~, ~\ldots~,~ \frac{r^{n-1}}{g^{\prime}_1(x_1^*)g^{\prime}_2(x_2^*)\cdots g^{\prime}_{n-1}(x_{n-1}^*)}\,\Big]^{{\text{T}}}.\]
The unstable subsystem of (\ref{zero-gly-cascade-2}) is characterized by
\begin{equation}
\dot{z}~=~\lambda_1 z
\, + \, {\alpha}^{-1} \left (a-f^{\prime}_{n+1}(y^*)\, - \, r \right )y \, + \, {\alpha}^{-1} \delta.
\label{zero-case}
\end{equation}
From the results of \cite{Scherer92} and \cite{Schwartz96}, the formula to compute the optimal value of $\gamma$ reduces to
\begin{eqnarray}
\gamma^*_L = \frac{1}{f^{\prime}_{n+1}(y^*)+r-a}.
\end{eqnarray}
We emphasize that according to \cite[Proposition $6$]{Schaft}, $\gamma_L^*$ is a lower bound for the optimal $\gamma^*$ for the nonlinear system (\ref{eq88}).
\end{proof}
\section{Examples}
We apply our results to metabolic pathway \eqref{reaction-two} and quantify its existing hard limits. We assume that the second reaction in \eqref{reaction-two} has no ATP feedback ATP on PK, {i.e., } $g=0$. We consider two scenarios for the consumption rate $R_{\rm CONS}$; in the first example, we assume the product $y$ is consumed by basal consumption rate $1+\delta$, and then, in the second example, we consider the case where the consumption rate depends on $y$.
\begin{example}
Let us consider the minimal representation of autocatalytic glycolysis pathway given by \eqref{reaction-two}. It is assumed that the second reaction in \eqref{reaction-two} has no ATP feedback ATP on PK, {i.e., } $g=0$. Then, we can rewrite \eqref{gly-two} as follows
\begin{eqnarray}
\dot{x}_1 & = & \frac{2 y^{a}}{1+ y^{2h}} ~-~ k x_1,\\
\dot{y} & = &-\alpha \frac{2 y^{a}}{1+ y^{2h}} ~+~ (\alpha+1) k x_1 ~-~ (1+\delta),
\label{gly-two-ex}
\end{eqnarray}
for $x_1 \geq 0$ and $y \geq 0$.
By considering expression $\frac{2 y^{a}}{1+ y^{2h}}$ as the regulatory feedback control employed by nature that captures inhibition of the catalyzing enzyme, a control system model for glycolysis can be obtained as follows
\begin{eqnarray}
\dot{x}_1 & = & -k \,x_1 + u, \label{cont-glyex1}\\
\dot{y} & = & (\alpha+1) k \, x_1 - {\alpha}\, u -1 -\delta, \label{cont-glyex2}
\end{eqnarray}
where $u$ is the control input. Using \eqref{cont-glyex1}-\eqref{cont-glyex2} and Theorem \ref{theorem-5}, it follows that
\begin{equation}
\gamma ~>~ \frac{\alpha}{k},
\label{ex-gain}
\end{equation}
where the equilibrium point of the unperturbed system is given by $x_1=1/k$ and $y=1$. As we expected \eqref{ex-gain} is consistent with the result of Theorem \ref{theorem-01}.
\end{example}
\begin{example}
\label{ex-2}
Let us now consider the minimal representation of autocatalytic glycolysis pathway represented by \eqref{reaction-two} with consumption rate depending on $y$ that is given by
\[ R_{\rm CONS}~=~ k_y y + \delta.\]
We refer to \cite{BuziD10} for a complete discussion.
Then, a set of ordinary differential equations that govern the changes in concentrations $x_1$ and $y$ can be written as
\begin{eqnarray*}
\dot{x}_1 & = &-k\, x_1 ~+~ \frac{2 y^{a}}{1+ y^{2h}}, \\
\dot{y} & = & -\alpha \frac{2 y^{a}}{1+ y^{2h}} ~+~ (\alpha+1)k \, x_1~-~ \left ({k_y y+\delta}\right ),
\end{eqnarray*}
for $x_1 \geq 0$ and $y \geq 0$. The exogenous disturbance disturbance input is assumed to be $\delta \in \mathcal{L}_2([0,\infty))$. To highlight fundamental tradeoffs due to autocatalytic structure of the system, we normalize the concentration such that {steady-states become
\begin{equation}
{y}^*=1~~\text{and}~~~{x_1}^*=\frac{k_y}{ k}.
\label{eq:1062}
\end{equation}}
As we discussed earlier, one may consider expression $\frac{2 y^{a}}{1+ y^{2h}}$ as the regulatory feedback control employed by nature that captures inhibition of the catalyzing enzyme. Hence, we can derive a control system model for glycolysis as follows
\begin{eqnarray}
\dot{x}_1 & = & -k \,x_1 + u, \label{cont-gly12}\\
\dot{y} & = & (\alpha+1) k \, x_1 - {\alpha}\, u -k_y \, y -\delta, \label{cont-gly22}
\end{eqnarray}
where $u$ is the control input.
Now, applying Theorem \ref{theorem-5} to this model, it follows that
\begin{equation}
\gamma ~>~ \frac{\alpha}{k+\alpha k_y}.
\label{trade}
\end{equation}
Equation (\ref{trade}) illustrates a tradeoff between robustness and efficiency (as measured by complexity and metabolic overhead). From (\ref{trade}) the glycolysis mechanism is more robust efficient if $k$ and $k_y$ are large. On the other hand, large $k$ requires either a more efficient or a higher level of enzymes, and large $k_y$ requires a more complex allosterically controlled PK enzyme; both would increase the cell's metabolic load.
We note that the existing hard limit is an increasing function of $\alpha$. This implies that increasing $\alpha$ (more energy investment for the same return) can result in worse performance.
It is important to note that these results are consistent with results in \cite{BuziD10}, where a linearized model with a different performance measure is used.
\end{example}
\section{Conclusion}\label{sec:conclusion}
The primary goal of this paper is to characterize fundamental limits on robustness and performance of a class of dynamical networks with autocatalytic structures. A simplified model of Glycolysis pathway is considered as the motivating application. We explicitly derive hard limits on the best achievable performance of the autocatalytic pathways with intermediate reactions which are characterized as $\mathcal L_2$-norm of the output as well as $\mathcal L_2$-gain of disturbance attenuation.
Then, we explain how these resulting hard limits lead to some fundamental tradeoffs. For instance, due to the existence of autocatalysis in the system, a fundamental tradeoff between a notion of fragility (e.g., cell death) and net production of the pathway emerges. Moreover, it is shown that as the number of intermediate reactions grows, the price paid for robustness increases. On the other hand, the larger the number of intermediate reactions involved in the breakdown of a metabolite, the less complex the enzymes involved in the individual reactions need to be. This illustrates a tradeoff between robustness and efficiency as measured by complexity and metabolic overhead.
\begin{spacing}{1}
\bibliographystyle{IEEEtran}
|
1,108,101,564,604 | arxiv | \section{Introduction}
There are two significances in General Relativity (GR) which cause it not to be a fundamental theory of gravity and needs to be modified or extended. The idea of short range or Ultra Violet (UV) modification of GR, which arises from the non-renormalizability of the theory \cite{Stelle:1976gc}, and the other is large scale or Infra Red (IR) modification of GR, which might be the explanation of the observed late-time universe acceleration\cite{Capozziello:2011et} or of the inflationary stage \cite{Nojiri:2003ft}. A large amount of research has been devoted to both implications but here we focus on the UV modification. Though the addition of higher curvature terms to the Einstein-Hilbert action establishes renormalizability, but this leads to the ghosts problem, that is the equations of motion involve higher-order time derivatives\cite{Stelle:1976gc,Stelle:1977ry}. Another approach including higher spatial derivatives only, and motivated by the Lifshitz theory was constructed by Horava \cite{Horava:2008ih,Horava:2009uw}. The Horava-Lifshitz gravity includes different anisotropic scaling for the space and time and is regarded as a UV completion of GR. \footnote{Horava-Lifshitz gravity is known to suffer from perturbative instability in the IR, ultimately due to an extra mode which comes from the explicit breaking of general covariance (because of higher spatial derivatives but not higher time derivatives). One of the ways to fix this is to have Horava gravity emerging dynamically in the UV while preserving Lorentz invariance (or rather, not having a preferred foliation) in the IR \cite{Cognola:2016gjy}.}
However, there is an alternative UV completion of GR in which instead of modifying the action, the spacetime metric is deformed. This construction named Rainbow Gravity (RG) \cite{Magueijo:2001cr}. This deformation exhibits a different treatment between space and time in the UV limit, nearly the Planck scale, depending on the energy of the particle probing the spacetime, while at low energy limit one recovers the standard form of the metric in GR. This deformation has been shown to cure divergences avoiding any renormalization scheme \cite{Garattini:2011kp,Garattini:2012ec}. As the standard energy-momentum dispersion relation depends on the Lorentz symmetry, it is expected that the standard energy-momentum dispersion relation will also get modified in the ultraviolet limit. The modification of the standard energy-momentum dispersion relation has motivated the development of doubly special relativity (DSR)\cite{AmelinoCamelia:1996pj,AmelinoCamelia:1997gz,AmelinoCamelia:2000ge,Bruno:2001mw}. In fact, DSR is an extension of special relativity which has two upper bounds; the velocity of light and Planck energy.
The generalization of this theory to general relativity is RG \cite{Magueijo:2002am,Magueijo:2002xx,Magueijo:2004vv}.
On the other hand, thermodynamics of black holes \cite{Bekenstein:1972tm,Bardeen:1973gs} has fundamental connections among the classical thermodynamics, general relativity, and quantum mechanics. More specifically, due to the development of AdS/CFT conjecture\cite{Maldacena:1997re,Gubser:1998bc}, this connection has been deepened and a lot of attention has been attracted to AdS black holes. The relation between geometrical properties of the event horizon and thermodynamic quantities provides a clear indication that there is a relation between properties of the spacetime geometry and some kind of quantum physics\cite{Hawking:1974rv,Hawking:1974sw,Hawking:1982dh}.
Chamblin and et al. have shown in Refs. \cite{Chamblin:1999tk,Chamblin:1999hg} that charged AdS black holes have rich phase structures. This phase transition is analogous to a van der Waals liquid-gas phase transition \cite{Banerjee:2011raa} where the cosmological constant is identified as pressure \cite{Kastor:2009wy}. Kubiznak and et al. studied the P-V critical behavior and critical exponents in Refs. \cite{Kubiznak:2012wp,Hansen:2016ayo,Kubiznak:2016qmn} that corroborates deep analogy between a charged AdS black hole and a van der Waals fluid\footnote{Studies about the reentrant phase transition and triple point in the extended phase space have been done in Refs. \cite{Altamirano:2013uqa,Frassino:2014pha,Wei:2014hba,Hennigar:2015wxa}.}, such that a lot of attention has been turned into the study of extended phase space thermodynamics and P-V criticality of black holes in different theories such as Born-Infeld theory \cite{Gunasekaran:2012dq}, Non-linear Maxwell theory \cite{Hendi:2012um,Dehyadegari:2016nkd}, Gauss-Bonnet theory \cite{Cai:2013qga}, and higher order gravities \cite{Sherkatghanad:2014hda,Hendi:2018xuy}.
Another process comes from this analogy is the Joule-Thomson (JT) expansion. In JT expansion a gas at a high pressure passes through a valve or porous plug to a low pressure section such that during the process enthalpy is unchanged and the process is an adiabatic expansion. The interest of study this process for black holes returns to small rate of black hole Hawking radiation which can be regarded as an adiabatic expansion though there is no porous plug. Recently, the JT expansion process in the case of charged AdS black holes has been studied in Ref. \cite{Okcu:2016tgt} and a few later the authors also considered this effect for the Kerr-AdS black holes \cite{Okcu:2017qgo}. The computations for Kerr-Newman AdS black holes have been done in Ref. \cite{Zhao:2018kpz} and the generalizations to charged AdS solutions in the quintessence and monopole black holes have been done in Refs. \cite{Ghaffarnejad:2018exz,AhmedRizwan:2019yxk}. Higher dimensional charged AdS black holes have been argued in Ref. \cite{Mo:2018rgq}. JT effect has been also considered in modified theories such as Lovelock gravity \cite{Mo:2018qkt}, $f(R)$ gravity \cite{Chabab:2018zix}, Gauss-Bonnet gravity \cite{Lan:2018nnp}, nonlinear electrodynamics\cite{Kuang:2018goo}, Einstein-Maxwell-Axion and massive gravity\cite{Cisterna:2018jqg}, and Bardeen theory \cite{Li:2019jcd}.
In this letter, we are curious to study the JT expansion for charged AdS black holes in RG. This generalization is of physical significance since varieties of
intriguing thermodynamic properties have been disclosed for this solution in RG \cite{Ling:2005bp,Galan:2006by,Ali:2014yea,Gim:2015zra,Kim:2016qtp,Feng:2017gms,Dehghani:2018svw,Hendi:2018sbe,EslamPanah:2018ums}. It is naturally expected that this research may give rise to novel findings concerning the JT expansion. The main motivation for studying black holes in RG is to consider the quantum corrections on the classical perspectives. The
idea of energy dependent spacetime is one of those quantum corrections. Considering
this energy dependent spacetime, we can venture on the effects it brings about on the
JT expansion of charged AdS black hole. Due to modification of the original surface gravity in RG, the rainbow Hawking temperature is very sensitive to the concrete expression of rainbow functions and consequently the entropy from the first law. We study the inversion temperature by using the equation of state in which the cooling-heating transition occurs in throttling process. In this treatment, the mass of black hole plays the role of enthalpy in the extended phase space \cite{Kastor:2009wy,Kubiznak:2012wp,Kubiznak:2016qmn,Hansen:2016ayo,Dolan:2011xt,Caceres:2016xjz}. We plot the inversion and isenthalpic curves for different values of parameters and discuss some of the relevant issues in RG.
\section{Charged Black Holes in Rainbow Gravity}
Lorentz symmetry is one of the most important symmetries in nature which might be violated in the UV limit\cite{Adams:2006sv,Gripaios:2004ms,Iengo:2009ix}. Since the standard energy-momentum dispersion relation depends on the Lorentz symmetry, it is expected that it will also get modified in the ultraviolet limit named doubly special relativity(DSR) \cite{AmelinoCamelia:1996pj,AmelinoCamelia:1997gz,AmelinoCamelia:2000ge,Bruno:2001mw}. On the other hand, RG is a generalization of DSR applied to curved spacetime \cite{Magueijo:2002am,Magueijo:2002xx,Magueijo:2004vv}. In DSR theory there is a maximum energy scale in the nature which is the Planck energy in accordance to modification in UV limit. The energy-momentum dispersion relation for a test particle of mass $m$ is given by
\begin{equation}} \def\ee{\end{equation}\label{dr} E^2\, \mathcal{F}^2 \left(E/E_{p},\eta\right)-p^2 \,\mathcal{G}^2 \left(E/E_{p},\eta \right) = m^2,\ee
where the two energy dependent rainbow functions satisfy in
\begin{equation}} \def\ee{\end{equation}\label{rbf1} \lim_{E/E_{p}\rightarrow 0} \mathcal{F}\left(E/E_{p},\eta\right)=1, \qquad \lim_{E/E_{p}\rightarrow 0} \mathcal{G}\left(E/E_{p},\eta\right)=1\,, \ee
and $\eta$ is a dimensionless constant which we called it rainbow parameter. The relation (\ref{dr}) goes to the standard form when the energy of the test particle is much lower than the Planck scale. The rainbow metrics lead to a one parameter family of connections and curvature tensors such that Einstein’s equation becomes \cite{Ling:2005bp,Peng:2007nj}
\begin{equation}} \def\ee{\end{equation}\label{Eeq} G_{\mu\nu}\left(E/E_{p} \right)+\Lambda \left(E/E_{p} \right)\,g_{\mu\nu} \left(E/E_{p} \right)=8\pi G\left(E/E_{p} \right) T_{\mu\nu}\left(E/E_{p} \right),\ee
where $G_{\mu\nu}(E/E_{P})$ and $T_{\mu\nu}(E/E_{P})$ are energy-dependent Einstein and energy-momentum tensors, and $\Lambda(E/E_{P})$ and $G(E/E_{P})$ are energy-dependent cosmological and gravitational constants. In this work, we choose the RG functions discussed in Refs. \cite{AmelinoCamelia:1997gz,AmelinoCamelia:1996pj,Jacob:2010vr,Ali:2014qra,Hendi:2015hja} which are phenomenologically important,
\begin{equation}} \def\ee{\end{equation}\label{rbf2} \mathcal{F}\left(E/E_{p},\eta\right)=1, \qquad \mathcal{G}\left(E/E_{p},\eta\right)=\sqrt{1-\eta\,\left(E/E_{p}\right)^n }\,.\ee
The modified charged AdS black hole in RG is described by the following line element analogous to Schwarzschild black hole in Ref. \cite{Magueijo:2002xx}
\begin{equation}} \def\ee{\end{equation}\label{rgm} ds^2=-\frac{f(r)}{ \mathcal{F}^2} dt^2+\frac{1}{ f(r) \mathcal{G}^2} dr^2+\frac{r^2}{\mathcal{G}^2} d\Omega^2\,,\ee
where
\begin{equation}} \def\ee{\end{equation} f(r)=1-\frac{2 G M}{r}+\frac{Q^2}{r^2}+\frac{r^2}{l^2}\,.\ee
The parameters $M$ and $Q$ are the mass and charge of the black hole which to avoid the singularity we should have $M\ge Q$, and $l$ is the radius of AdS space related to the cosmological constant as $\Lambda=-\frac{3}{l^2}$. The location of event horizon is given by the largest real root of $f(r)=0$.
\section{Thermodynamics of Charged AdS black hole}
The study of thermodynamic properties of asymptotically AdS black holes dates back to the seminal work of Hawking and Page\cite{Hawking:1982dh} about the phase transition in Schwarzschild-AdS black holes and for more complicated backgrounds in \cite{Cvetic:1999ne,Cvetic:1999rb}. The thermodynamics of charged black holes which is our interest in this paper, have been considered extensively in Refs. \cite{Chamblin:1999tk,Chamblin:1999hg}. In particular, in the case of an asymptotically AdS black hole in four dimensions the cosmological constant $\Lambda$ is identified with the pressure by \cite{Kastor:2009wy}
\begin{equation}} \def\ee{\end{equation}\label{pl} P=-\frac{\Lambda(0)}{8\pi}=\frac{3}{8\pi l^2}\,,\ee
and its conjugate variable in black hole thermodynamics is the volume
\begin{equation}} \def\ee{\end{equation}\label{vol} V=\left(\frac{\prt M}{\prt P}\right)_{S,Q}=\frac43 \pi r_{+}^3\,,\ee
where $r_{+}$ is the event horizon of black hole. The temperature of the modified black hole in (\ref{rgm}) can be calculated from the surface gravity $\kappa$ on the horizon \cite{Ling:2005bp}, namely
\begin{equation}} \def\ee{\end{equation}\label{temp1} T=\frac{\kappa}{2\pi}=-\frac{1}{4\pi} \lim_{r\rightarrow r_{+}} \sqrt{\frac{-g^{11}}{g^{00}}} \,\frac{(g^{00})'}{g^{00}}\,, \ee
where prime is the derivative with respect to the $r$. According to the uncertainty principle, $\Delta p\ge \frac{1}{\Delta x}$ can be translated to a lower bound on the energy of a test particle \cite{Adler:2001vs,AmelinoCamelia:2004xx}
\begin{equation}} \def\ee{\end{equation}\label{up} E\ge \frac{1}{\Delta x}\sim \frac{1}{r_{+}}.\ee
Without lose of generality and for later convenience we take $G=1$ and $n=2$ in (\ref{rbf2}), so by substituting dispersion relation (\ref{dr}) and uncertainty relation (\ref{up}) in (\ref{rbf2}) we obtain the rainbow function as
\begin{equation}} \def\ee{\end{equation}\label{rbf3} \mathcal{G}=\sqrt{1-\eta G_{p} m^2}\sqrt{\frac{r_{+}^2}{r_{+}^2+\eta G_{p}}}=\frac{1}{k} \sqrt{\frac{r_{+}^2}{r_{+}^2+\eta G_{p}}}\,,\ee
where $G_{p}=1/E_{p}^2$, $k=[1-\eta G_{p} m^2]^{-1/2}$, and $m$ is the mass of test particle. Thus the temperature from (\ref{temp1}) is given by
\begin{equation}} \def\ee{\end{equation}\label{temp2} T=\frac{1}{4\pi k \sqrt{r_{+}^2+\eta G_{p}}}\,\frac{(8\pi P\,r_{+}^4+r_{+}^2-Q^2)}{r_{+}^2}\,.\ee
The black hole's mass can be easily calculated from the condition $f(r)=0$ in terms of horizon radius
\begin{equation}} \def\ee{\end{equation}\label{mass} M=\frac{3Q^2+3r_{+}^2+8\pi P r_{+}^4}{6r_{+}}\,,\ee
in which we have used the relation (\ref{pl}). In the previous section we assert that the area of horizon will be corrected when we use the modified metric (\ref{rgm}) \cite{Kim:2016qtp}. So, the modified entropy is as follows
\begin{equation}} \def\ee{\end{equation}\label{ent} S=\pi k\, r_{+} \sqrt{r_{+}^2+\eta G_{p}}+\pi k \,\eta \,G_{p} \ln{(\sqrt{r_{+}^2+\eta G_{p}}+r_{+})}\,,\ee
and we have checked that the quantities given in (\ref{temp2})-(\ref{ent}) satisfy the first law of black hole thermodynamic
\begin{equation}} \def\ee{\end{equation}\label{FL} T=\left(\frac{\prt M}{\prt S}\right)_{Q,P},\ee
for constant $Q$ and $P$. The pressure of charged AdS black holes in RG has been calculated in \cite{Li:2018gwf}
\begin{equation}} \def\ee{\end{equation}\label{press} P=\frac{k}{2}\,\sqrt{\frac{r_{+}^2+\eta G_{p}}{r_{+}^4}}\,T+\frac{Q^2-r_{+}^2}{8\pi r_{+}^4}\,,\ee
where in the limit $\eta=0$ it reduces to (3.10) in Ref. \cite{Kubiznak:2012wp}. We can also rewrite the relation (\ref{press}) as the equation of state by substituting $r_{+}$ in terms of volume $V$ from (\ref{vol}).
\section{Joule-Thomson Expansion}
The JT expansion, which is also known as the throttling process, occurs when a gas of high pressure section penetrates to a low pressure section through a porous plug and is a fundamentally irreversible process. In this process, the enthalpy is kept constant and the gas undergoes an adiabatic expansion, so an isenthalpic process can be applied to calculate the temperature change. This temperature change in the throttling process is encoded in the JT coefficient. That is, the numerical value of the slope of an isenthalpic curve on a $T\!-\!P$ diagram, at any point, is called the JT coefficient and is denoted by
\begin{equation}} \def\ee{\end{equation}\label{jtc1} \mu =\left(\frac{\prt T}{\prt P}\right)_{H}\,.\ee
The locus of all points for zero JT coefficient is known as the inversion curve. An important feature of JT coefficient is that the region in $T\!-\!P$ diagram where $\mu>0$, it is called the cooling region and where $\mu<0$, is called heating region.
From the first law of black hole thermodynamics, the change in the mass is given by
\begin{equation}} \def\ee{\end{equation}\label{fl} dM=TdS+\Phi dQ+VdP\,,\ee
where $\Phi$ is the electric potential of the black hole at the horizon
\begin{equation}} \def\ee{\end{equation}\label{elpot} \Phi=\left(\frac{\prt M}{\prt Q}\right)_{S,P}=\frac{Q}{r}\,,\ee
for constant $S$ and $P$. Taking into account $dM=dQ=0$ in the throttling process, then
\begin{equation}} \def\ee{\end{equation}\label{HQ} T\left(\frac{\prt S}{\prt P}\right)_{M}+V=0\,,\ee
where by using thermodynamic variation $dS\!=\!\left(\frac{\prt S}{\prt P}\right)_{T} dP+\left(\frac{\prt S}{\prt T}\right)_{P} dT$ and Maxwell relation $\left(\frac{\prt S}{\prt P}\right)_{T}\!=\!-\left(\frac{\prt V}{\prt T}\right)_{P}$ we obtain
\begin{equation}} \def\ee{\end{equation}\label{jtc2} \mu=\frac{1}{C_{P}}\left[T\left(\frac{\prt V}{\prt T}\right)_{P}-V\right]\,,\ee
so if we substitute the relations (\ref{vol}), (\ref{temp2})-(\ref{ent}) in (\ref{jtc2}), and that $C_{P}=T\left(\frac{\prt S}{\prt T}\right)_{P,Q}$ it gives
\begin{equation}} \def\ee{\end{equation}\label{JT} \mu=\frac{2}{3 k} \left(\frac{r^2}{\eta G_{p}+r^2}\right)^{3/2} \frac{16 \pi P r^6+ (8 \pi \eta G_{p} P+4)r^4+3(\eta G_{p}-2Q^2) r^2- 5 \eta G_{p} Q^2}{ r \left(8 \pi P r^4+r^2-Q^2\right)}\,.\ee
The inversion curves $T_{i}$ versus $P_{i}$ are obtained by solving $\mu=0$, as a result of that we compute $r_{+}$ in terms of $P_{i}$ and then substitute the largest root in (\ref{temp2}). We have plotted the inversion curves for different values of black hole charge in Fig. (\ref{fig1}). As illustrated in figure, the graphs are monotonically increasing, so there is no maximum inversion temperature for charged AdS black holes in RG. This behavior is essentially different from the real gases in thermodynamics such as van der Waals gas \cite{Okcu:2016tgt,Okcu:2017qgo,Zhao:2018kpz}. Each curve has a minimum value of inversion temperature $T_{i}^{min}$, which corresponds to zero inversion pressure, $P_{i}=0$. Since $T_{i}(P)$ is a monotonically increasing function, there is only a minimum inversion temperature and the cooling and heating regions lie above and below the inversion curve, respectively.
Though the inversion curves have the same behavior for the large pressures, but as depicted in Fig. (\ref{fig2}) in the low pressure limit they have different behavior. That is by increasing the charges they fall to low temperatures.
\begin{figure}[H]
\centering
\begin{subfigure}{0.45\textwidth}
\includegraphics[width=\textwidth]{inversions1.pdf}
\caption{Large pressures}
\label{fig1}
\end{subfigure}
%
\hspace{5mm}
\begin{subfigure}{0.45\textwidth}
\includegraphics[width=\textwidth]{inversions2.pdf}
\caption{Low pressures}
\label{fig2}
\end{subfigure}
\caption{\small{The inversion curves for charged AdS black hole in RG with $G_{p}=\eta=k=1$ but different charges. }}
\label{f1}
\end{figure}
From the relation (\ref{JT}) if we use $P_{i}=0$ in $\mu=0$ equation and then solve it for $r_{+}$, we obtain four values for it. By substituting the largest one in (\ref{temp2}) we obtain the minimum inversion temperature
\begin{equation}} \def\ee{\end{equation}\label{tmin1}T_{i}^{min}\!=\!\!\frac{\left(18 Q^2+11 \eta G_{p}-3 \sqrt{9 \eta ^2 G_{p}^2+44 \eta G_{p} Q^2+36 Q^4}\right) \sqrt{\sqrt{9 \eta ^2 G_{p}^2+44 \eta G_{p} Q^2+36 Q^4}+5 \eta G_{p}+6 Q^2}}{40 \sqrt{2} \pi \eta ^2 G_{p}^2 k},\ee
where if we expand it for small $\eta$
\begin{equation}} \def\ee{\end{equation}\label{tmin2}T_{i}^{min}=\frac{1}{6\sqrt{6} k \pi Q}-\frac{G_{p} \eta}{24\sqrt{6}k\pi Q^3}+\mathcal{O}(\eta^2),\ee
the leading term for $k=1$ gives the minimum inversion temperature in Ref. \cite{Okcu:2016tgt}.
We have compared the inversion curves for the charged AdS black holes in rainbow gravity with the one in general relativity in Fig. (\ref{fig3}) and it shows that the slope of the RG's plot is less than the GR case. We have also plotted the inversion temperature curves for different values of the test particle's mass in Fig. (\ref{fig4}).
\begin{figure}[h]
\centering
\begin{subfigure}{0.43\textwidth}
\includegraphics[width=\textwidth]{inversions3.pdf}
\caption{}
\label{fig3}
\end{subfigure}
%
\hspace{5mm}
\begin{subfigure}{0.43\textwidth}
\includegraphics[width=\textwidth]{mass.pdf}
\caption{}
\label{fig4}
\end{subfigure}
\caption{\small{(a) The inversion curve for RG with solid red line ($\eta=1$) has lower slope than the GR with dashed blue line ($\eta=0$). (b) The inversion curve of different masses for $Q=1$, $G_{p}=1$, and $\eta=1$. }}
\label{f2}
\end{figure}
To better understand the behavior of this thermodynamic system we calculate the critical points from inflection point of $P=P(r_{+})$ \cite{Kubiznak:2012wp,Li:2018gwf}, i.e.,
\begin{equation}} \def\ee{\end{equation} \frac{\prt P}{\prt r_{+}}=0\,,\qquad \frac{\prt^2 P}{\prt r_{+}^2}=0\,.\ee
These equations lead to the critical values
\begin{eqnarray}} \def\eea{\end{eqnarray}\label{critr}
r_{c}&\!\!\!\!=\!\!\!\!&\sqrt{2} \left[Q^2+\left(\eta G_{p} Q+Q^3\right)^{2/3}+\frac{\left(\left(\eta G_{p} Q+Q^3\right)^2\right)^{2/3}}{\eta G_{p}+Q^2}\right]^{\frac12}\,,\\
\label{critt} T_{c}&\!\!\!\!=\!\!\!\!&\frac{1}{2\pi k \,r_{c}^2}\frac{r_{c}^2-2Q^2}{r_{c}^2+2\eta G_{p}} \sqrt{r_{c}^2+\eta G_{p}}\,,\eea
where in the limit $\eta\rightarrow 0$ they give the critical values in \cite{Okcu:2016tgt}.
It has been shown in \cite{Okcu:2016tgt} that the ratio of minimum inversion temperature to the critical temperature for the charged AdS black holes in GR is equal one-half, but here we show in the case of RG this value will correct. As shown in Fig.(\ref{fig5}) this ratio is descending by growing $\eta$ while the rate of changes decreases by increasing the charge of black hole. A similar behavior occurs in Lovelock \cite{Mo:2018qkt} or in higher dimensional charged AdS black holes \cite{Mo:2018rgq} when we increase the Lovelock parameter or the dimensions. The Fig. (\ref{fig6}) describes that for a particular value of $\eta=1$ the ratio will deviate significantly for small $Q$ but it does not change for larger values. This also can be seen by expanding the ratio for small and large limit of the charge. While the leading term is 2/5 for small Q
\begin{equation}} \def\ee{\end{equation} \frac{T_{i}^{min}}{T_{c}}=\frac25+\frac{3Q^{4/3}}{5}-\frac{13Q^2}{15}+\mathcal O \left(Q^{7/3}\right),\ee
it is 1/2 for large Q, i.e.,
\begin{equation}} \def\ee{\end{equation} \frac{T_{i}^{min}}{T_{c}}=\frac12-\frac{1}{108 Q^4}+\mathcal O\left(Q^{-6}\right).\ee
\begin{figure}[H]
\centering
\begin{subfigure}{0.45\textwidth}
\includegraphics[width=\textwidth]{mintempeta.pdf}
\caption{different charges}
\label{fig5}
\end{subfigure}
%
\hspace{5mm}
\begin{subfigure}{0.45\textwidth}
\includegraphics[width=\textwidth]{mintempq.pdf}
\caption{ $\eta=1$}
\label{fig6}
\end{subfigure}
\caption{\small{The ratio of minimum inversion temperature to critical temperature. }}
\label{f3}
\end{figure}
Since the JT expansion is an isenthalpic process, it is also of interest to consider the isenthalpic (constant mass) curves. We can obtain the isenthalpic curves in the $T-P$ plane by inserting the value of the event horizon in the equation of state (\ref{press}). The isenthalpic curves for different values of black hole charge have plotted in Figs.(\ref{f4}). The inversion temperature curve intersects them at their maximum points in each diagram. As seen from the figures the inversion curve intersects the isenthalpic curves at lower pressures as the charge increases and vice versa for the mass.
\begin{figure}[H]
\centering
\begin{subfigure}{0.4\textwidth}
\includegraphics[width=\textwidth]{Q1.pdf}
\caption{\small{$Q=1$}}
\label{fig7}
\end{subfigure}
%
\hspace{5mm}
\begin{subfigure}{0.4\textwidth}
\includegraphics[width=\textwidth]{Q2.pdf}
\caption{\small{$Q=2$}}
\label{fig8}
\end{subfigure}
%
\begin{subfigure}{0.4\textwidth}
\includegraphics[width=\textwidth]{Q10.pdf}
\caption{\small{$Q=10$}}
\label{fig9}
\end{subfigure}
%
\hspace{5mm}
\begin{subfigure}{0.4\textwidth}
\includegraphics[width=\textwidth]{Q20.pdf}
\caption{\small{$Q=20$}}
\label{fig10}
\end{subfigure}
\caption{\small{The isenthalpic (red) and inversion (black) curves for $\eta=1$, $G_{p}=1$, and $k=1$. }}
\label{f4}
\end{figure}
We can also have the curves that do not intersect with the inversion curve or the ones that intersect with each other. In the former, the temperature for zero pressure ($P=0$) is denoted by $T_{\circ}$, so by equalizing it to the minimum inversion temperature $T_{i}^{min}$, since the inversion curve is monotonically increasing, we can find a particular value of black hole's mass which we name it $M=M^{*}$. This curve is illustrated in Fig.(\ref{fig11}) by a red dashed dotted graph. As seen in this figure, the isenthalpic curve with $T_{\circ}>T_{i}^{min}$ or $M>M^{*}$ first rises and then after intersecting at extreme point with inversion curve descends, as well as plots in Figs.(\ref{f4}). On the other hand, for the curves with $T_{\circ}<T_{i}^{min}$ or $M<M^{*}$ there is no inversion point in heating region. In the latter curves, in the zero pressure limit, there is also a maximum value for the temperature $T_{\circ}^{max}$ which is correspond to a particular mass $M=\tilde{M}$. We calculate these masses for the case $Q=1$ as $M^{*}=1.024$ and $\tilde{M}=1.208$. For any isenthalpic curve with a mass greater than $\tilde{M}$ the curve should intersect with the others in the $T-P$ plane, as shown in Fig.(\ref{fig12}) by the doted curve for $M>\tilde{M}$.
\begin{figure}[ht]
\centering
\begin{subfigure}{0.45\textwidth}
\includegraphics[width=\textwidth]{intersection1.pdf}
\caption{Intersection with inversion curve}
\label{fig11}
\end{subfigure}
%
\hspace{5mm}
\begin{subfigure}{0.45\textwidth}
\includegraphics[width=\textwidth]{intersection2.pdf}
\caption{Intersection with each other}
\label{fig12}
\end{subfigure}
\caption{\small{The behavior of isenthalpic curves for different values of black hole mass.}}
\label{f5}
\end{figure}
\section{Conclusion}
In this letter we have mainly studied the Joule-Thomson (JT) expansion process in the rainbow gravity (RG) for charged AdS black holes. In this process, which describes the expansion of gas from a high pressure section to a low pressure section through a porous plug, we obtained an exact expression for the JT coefficient $\mu$ by an analytical recipe from the first law of black hole thermodynamics and investigated its behavior intuitively by plotting the inversion and isenthalpic curves. The mass of black hole is kept constant during this expansion as an isenthalpic process. The inversion curves ($\mu=0$) depicted in Figs.(\ref{fig1},\ref{fig2}) show that they have different behavior in low and large pressures and the slope of curves increases by growing the charge of black hole. We have also shown that the slope of inversion curves slows down in RG by increasing both of the rainbow parameter $\eta$ and the test particle's mass $m$ in Figs.(\ref{fig3},\ref{fig4}) in contrast to the effect of charge Q. One of the remarkable by-product studies in this paper is the ratio between the minimum inversion temperature and critical temperature where in the case of charged AdS black holes is equal one-half. Moreover, we have indicated in Figs.(\ref{fig5},\ref{fig6}) that this ratio will change from this value by considering $\eta$ and $Q$.
One of the significant subjects in this work was the analyzing of the isenthalpic curves during the JT expansion. For each curve, the highest value of temperature occurs at maximum point where $\mu=0$ and is called the inversion point, discriminating the cooling from heating process. According to Figs. (\ref{f4}) the higher inversion temperature for JT expansion corresponds with larger enthalpy (mass) and for larger charges the inversion point appears in lower temperature or pressure. The difference between these curves with the ones in \cite{Okcu:2016tgt} shows how the rainbow parameter affect the exhausting/absorbing heat process to the reservoir in cooling/heating regions, respectively. We inferred from the isenthalpic curves in Fig.(\ref{fig11}) that there is an upper bound for the mass of black hole for which we could have an inversion point, that is, for $M<M^{*}$ there is no inversion point and the expansion is always in the regime of heating process, where $M^{*}$ is determined from $T_{i}^{min}=T_{\circ}(p=0)$. We have also shown that the curves with $M>\tilde{M}$, where $\tilde{M}$ comes from the extremum of $T_{\circ}(p=0)$, intersect with the other isenthalpic curves before crossing the inversion curve in the cooling region.
|
1,108,101,564,605 | arxiv | \section{An example on $\mathbb P^4$}
\section*{Acknowledgments} We thank Robert Lazarsfeld for raising the question and for useful discussions.
|
1,108,101,564,606 | arxiv | \section{Introduction}
The discovery of cosmic acceleration \citep{Riess1998,Perlmutter1999} is a phenomenon in search of a theoretical explanation. On one hand, the acceleration can be explained by introducing dark energy which provides negative pressure \citep{Peebles2002}. On the other hand, on cosmological scales, the dynamics of gravity are untested and may deviate from general relativity (GR). Various models with modified gravity \citep{Dvali2000,Carroll2004} are degenerate with dark energy models and can also explain the cosmic acceleration. One of the major interests of large-scale structure (LSS) surveys is to break the degeneracy between these two explanations. \par
An additional gravitational process that can break this degeneracy is the growth of structure. The growth rate, measured through redshift space distortions (RSD) \citep{Kaiser1987b,Hamilton1997}, has been constrained by various analyses \citep{Beutler2012,Samushia2013,Alam2015,2018arXiv180102891R,2018MNRAS.477.1639Z,2018arXiv180102656H,2018MNRAS.477.1604G,2018arXiv180103043Z}. However, the growth rate estimated from its effect on the power spectrum suffers from a degeneracy with clustering bias and the scalar amplitude $A_s$, or equivalently $\sigma_8$. Since the clustering bias is very galaxy sample-specific and not simple to measure, an estimate of the growth rate independent of clustering bias is advantageous. \par
To remedy this, \citet{Zhang2007} introduce the $E_G$ statistic to probe gravity. $E_G$ is a statistic constructed from the ratio between Laplacian of the sum of two curvature fields $\nabla^2(\Phi+\Psi)$ and the velocity divergence field $\theta=\nabla\cdot\vec{v}/H(z)$, where $\vec{v}$ is the comoving peculiar velocity. While $\nabla^2(\Phi+\Psi)$ is probed through lensing measurements, originally through galaxy lensing, $\theta$ is probed through the RSD measurements of the growth rate. Not only is this statistic independent of clustering bias on large scales, but also GR predicts this ratio to be scale-independent. If $E_G$ is measured to be scale-dependent, and not in agreement with the prediction from $\Lambda$CDM cosmogony, then modified gravity would become a promising solution for cosmic acceleration. In addition, various factors which may contribute to uncertainty of $E_G$ were analyzed in \cite{Leonard2015}. $E_G$ was first measured by \citet{Reyes2010a} at redshift $z=0.32$ using galaxy-galaxy lensing information traced by the Luminous Red Galaxy (LRG) sample \citep{Eisenstein2001} from the Sloan Digital Sky Survey (SDSS) \citep{York2000}. \cite{Blake2015} then measured $E_G$ at redshifts $z=0.32$ and $z=0.57$ using galaxy lensing information from multiple datasets. \cite{Giannantonio2016} measure $D_g$, a modified $E_G$ in which the growth rate is not used in the estimator. \cite{Pullen2015} discuss that $\nabla^2(\Phi+\Psi)$ can be estimated through CMB lensing cross-correlated with galaxies to allow measurement of $E_G$ at higher redshifts, a measurement which was first demonstrated in \cite{Pullen2016}.\par
All previous measurements of $E_G$ have assumed that the galaxy sample is only correlated with the lensing perturbations \emph{at that redshift}. However, \cite{MoradinezhadDizgah2016a}, hereafter D16, point out that the magnification bias, in which foreground density perturbations both magnify the regions around galaxies and distort the selection of galaxies near the survey flux limit, can bias the galaxy-convergence and galaxy-galaxy angular power spectra. It was predicted in D16 that $E_G$ could deviate by around 2\% for current surveys but up to $40\%$ from the theoretical value for upcoming high-redshift surveys. Recently, \cite{Singh2018} consider magnification bias for an $E_G$ measurement using galaxy lensing and CMB lensing in real space with data from CMASS and LOWZ \citep{2015ApJS..219...12A}, also finding an expected deviation $\sim2$\%.
It is worth mentioning that since surface brightness is conserved, an $E_G$ measurement is free from magnification bias if the galaxy survey is replaced by an intensity mapping survey in the $E_G$ estimator \cite{2016MNRAS.461.1457P}. However, measuring $E_G$ through 21 cm emission is unlikely to be possible because continuum foregrounds and calibration errors may prevent the measurement of the global intensity signal to the accuracy necessary to measure the growth rate. It may still be possible to measure $E_G$ through CO rotation line and CII line since they are less contaminated by foreground.\par
In this paper, we test how significantly magnification bias will affect upcoming measurements of $E_G$, specifically $E_G$ estimated from CMB lensing.
We start by giving the derivation for the magnification bias effect on $E_G$, finding that the bias is very sensitive to the assumed clustering and magnification biases. We then apply redshift and scale-dependent clustering bias calibrations proposed by \cite{Pullen2016} based on calibrations from \citet{Reyes2010a} to study how $E_G$ is biased for CMB lensing maps from Planck \citep{Lawrence2015} and Advanced ACT (Adv.~ACT) \citep{Henderson2015} cross-correlated with both spectroscopic and photometric galaxy surveys, including the Baryon Oscillation Spectroscopic Survey (BOSS) \citep{2013AJ....145...10D} and the Extended Baryon Oscillation Spectroscopic Survey (eBOSS) \citep{2016AJ....151...44D}, the Dark Energy Spectroscopic Instrument (DESI) \citep{Levi2013}, the Dark Energy Survey (DES) \citep{2005astro.ph.10346T}, the Large Synoptic Survey Telescope (LSST) \citep{2012arXiv1211.0310L}, and Euclid \citep{Laureijs2011}. We confirm that the proposed CMB ``Stage 4'' survey (CMB-S4) \citep{Abazajian2016} does not significantly decrease the errors on $E_G$ further, and thus do not display specific results from this survey. We find through theoretical analysis and simulations that the deviation of $E_G$ should be less than 5\% in quasi-linear scales for spectroscopic and photometric galaxy surveys. We find that most spectroscopic galaxy surveys, except for the DESI luminous red galaxy (LRG) and emission line galaxy (ELG) surveys cross-correlated with CMB lensing maps from Adv.~ACT, do not have magnification biases large enough to significantly contaminate $E_G$. On the other hand, we find that the photometric galaxy surveys, including DES, LSST, and Euclid, will have large magnification biases that will contaminate $E_G$. In these cases, we propose a method to clean the higher-order terms contributed by the lensing magnification effect with observable angular power spectrum, finding it calibrates $E_G$ effectively. We believe this calibration will work well in future surveys with higher precision.\par
The plan of this paper is as follows: In section \ref{S:form} we briefly review the $E_G$ estimator, present our formalism for the magnification bias of $E_G$, and construct our calibrations to remove the bias. Section \ref{S:results} uses simulations to test its performance of our calibration method as well as its stability relative to errors in modeling the calibration. We conclude in section \ref{S:conclu}. In this work we assume the following cosmological parameters: $\Omega_m=0.309$, $\Omega_bh^2=0.02226$, $\Omega_ch^2=0.1193$, $\Omega_k=0$, $h=0.679$, $n_s=0.9667$, and $\sigma_8=0.8$ given in \cite{PlanckCollaboration2015}, \cite{Alam2016}.
\section{$E_G$ Formalism and Calibration Method}\label{S:form}
The first part of this section derives the formalism for the distortion of $E_G$ due to magnification bias. Although this is mostly derived in D16, we do make some small corrections to their derivation that do not on their own affect the final results for $\Delta E_G/E_G$. In the second part of this section we present our method to calibrate the magnification bias in $E_G$, which we test in Section \ref{S:results}.
\subsection{$E_G$ statistic review} \label{S:review}\par
The expression for $E_G$ without magnification bias was derived previously in the literature, so we just review the main points here; see \citet{Pullen2015} and \citet{Pullen2016} for a complete review. Note that we assume flat space for all $E_G$ analyses. $E_G$ is defined in Fourier space for any metric theory of gravity \citet{Zhang2007} as
\begin{eqnarray}
E_G\equiv\frac{c^2k^2(\Phi+\Psi)}{3H_0^2(1+z)\theta(k)}\, ,
\end{eqnarray}
where $H_0$ is the Hubble expansion rate today, $\Phi$ and $\Psi$ are the weak-field potentials in the space and time metric components, respectively, and $\theta$ is the velocity divergence perturbation. Both the numerator and denominator can be related to the matter overdensity $\delta_m=\delta\rho/\bar{\rho}$: the potentials through the field equations of the theory and $\theta=f(z)\delta_m$, where $f(z)$ is the growth rate. For GR, we write
\begin{eqnarray}
E_G=\frac{\Omega_{m0}}{f(z)}\, ,
\end{eqnarray}
where $\Omega_{m0}$ is the relative matter density today, while for other theories of gravity the expression for $E_G$ will be altered and potentially even attain scale-dependence on large scales.
We state that $E_G$ can be estimated \citep{Pullen2015} as
\begin{eqnarray}
\hat{E}_G(\ell,\bar{z})=\Gamma(\bar{z})\frac{C_\ell^{\kappa g}}{\beta(\bar{z}) C_\ell^{gg}}\, ,
\end{eqnarray}
where $\bar{z}$ is the mean redshift of the galaxy survey, $C_\ell^{\kappa g}$ is the measured lensing convergence-galaxy angular cross-power spectrum, $C_\ell^{gg}$ is the measured galaxy angular auto-power spectrum, $\beta=f(z)/b(z)$, where $b(z)$ is the clustering bias as a function of redshift $z$ and $\beta$ is usually fitted from measurements of the monopole and quadrupole moments of the anisotropic correlation function $\xi(r,\mu)$, and $\Gamma(\bar{z})$ is a calibration factor. The expressions for $C_\ell^{\kappa g}$ and $C_\ell^{gg}$ can be written as
\begin{eqnarray}\label{E:clab}
C_\ell^{AB}=\frac{2}{\pi}\int_0^{\infty}k^2dkP(k,z=0)W_\ell^A(k)W_\ell^B(k)\, ,
\end{eqnarray}
where $P(k,z)$ is the matter power spectrum and $W_\ell^A$ is a window function for observable $A$. The window functions for both lensing convergence $\kappa$ and galaxy overdensity $\delta_g$ (assuming no magnification bias) are given by
\begin{eqnarray}
W_\ell^{\kappa}(k,z)&=&\frac{3\Omega_{m0}H_0^2}{2c^2}\int_0^zdz'\frac{c}{H(z')}W(z,z')D(z')\nonumber\\
&&\times(1+z')j_\ell(k\chi(z'))\, ,
\end{eqnarray}
and
\begin{eqnarray}
W_\ell^g(k)=\int dz\,f_g(z)b(z)D(z)j_\ell(k\chi(z))\, ,
\end{eqnarray}
where $\chi$ is the comoving distance given by
\begin{equation}
\chi=\int_0^z\dfrac{cdz'}{H(z')}\, ,
\end{equation}
$f_g$ is the galaxy redshift distribution, $D(z)$ is the normalized growth factor, and $W(z,z')$ is the lensing kernel for a lens redshift $z'$ and a source redshift $z$ written as
\begin{eqnarray}
W(z,z')=\chi(z')\frac{\chi(z)-\chi(z')}{\chi(z)}\, .
\end{eqnarray}
Note that unlike most treatments we keep the source redshift explicit, which will be helpful later. The calibration factor $\Gamma$ is given by
\begin{eqnarray}
\Gamma(z)=C_\Gamma C_b\frac{2c}{3H_0}\left[\frac{H(z)f_g(z)}{H_0(1+z)W(z_{CMB},z)}\right]\, ,
\end{eqnarray}
where $C_\Gamma$ and $C_b$ are extra calibrations for the broad redshift distribution and lensing kernel and for the scale-dependent bias due to nonlinear clustering, respectively, given by
\begin{eqnarray}\label{eq:22}
C_{\Gamma}(\ell,z)&=&\frac{W(z_{CMB},z)(1+z)}{2f_g(z)}\frac{c}{H(z)}\frac{C^{mg}_\ell}{Q^{mg}_\ell}\nonumber\\
C_b(\ell)&=&\frac{C_\ell^{gg}}{b(\bar{z})C_\ell^{mg}}\, ,
\end{eqnarray}
where
\begin{eqnarray}
C_\ell^{mg}&=&\int_{0}^{\infty}dz\frac{H(z)}{c}f_g^2(z)\chi^{-2}(z)P_{mg}\left(k=\frac{\ell}{\chi(z)},z\right)\nonumber\\
Q^{mg}_\ell&=&\frac{1}{2}\int_{0}^{\infty}dzW(z_{CMB},z)f_g(z)\chi^{-2}(z)(1+z)\nonumber\\
&&\times P_{mg}\left(k=\frac{\ell}{\chi(z)},z\right)\, ,
\end{eqnarray}
and $P_{mg}(k,z)=b(z)P(k,z)$.
\subsection{$E_G$ distortion due to magnification bias} \label{S:bias}\par
The effect of magnification bias on $C_\ell^{\kappa g}$ and $C_\ell^{gg}$ are given in D16, so we do not give a full derivation here. Instead, we just state the main points. The correct expression for the observed galaxy overdensity $\Delta_g$ in terms of the matter overdensity $\delta_m$, including only magnification bias and not other relativistic effects, is given by
\begin{eqnarray}
\Delta_g(\mathbf{\hat n},z)=b(z)\delta_m(\mathbf{\hat n},z)+(5s-2)\kappa(\mathbf{\hat n},z)\, ,
\end{eqnarray}
where $\kappa(\mathbf{\hat n},z)$ is the lensing convergence map with a source redshift $z$ and $s$ is the slope of the cumulative magnitude function, also called the \emph{magnification bias}, given by
\begin{eqnarray}\label{eq:13}
s=\left.\frac{d\log_{10} n_g(m<m_*)}{dm}\right|_{m=m_*}
\end{eqnarray}
where $n_g(m<m_*)$ is the number density of galaxies with an apparent magnitude $m$ less than $m_*$. Note that the magnification bias term differs from the same in Eq. 2.13 in D16 by a minus sign, although it does not affect the magnitude of $\Delta E_G/E_G$ because the minus cancels with a second minus sign error in D16 for $C_\ell^{\kappa g}$. Also, we see here the well-known result that the magnification bias vanishes for $s=0.4$ which will be important in our analysis. Substituting $\delta_g$ for $\Delta_g$, we find that the window function for $W_\ell^g$ gains an extra term such that $W_\ell^g=W_\ell^{g1}+W_\ell^{g2}$ where $W_\ell^{g1}$ is the expression without magnification bias and
\begin{eqnarray}
W_\ell^{g2}(k)=(5s-2)\int dz\,f_g(z)W_\ell^\kappa(k,z)\, .
\end{eqnarray}
Putting them back to the general expression of angular power spectrum gives
\begin{eqnarray}\label{eq:17}
C_\ell^{gg}=C_\ell^{g1g1}+2C_\ell^{g1g2}+C_\ell^{g2g2}
\end{eqnarray}
where
\begin{eqnarray}\label{E:gg}
C_\ell^{gigj}=\frac{2}{\pi}\int_0^{\infty}k^2dkP(k,z=0)W_\ell^{gi}(k)W_\ell^{gj}(k)\, ,
\end{eqnarray}
and
\begin{eqnarray}\label{eq:18}
C_\ell^{\kappa g}&=&C_\ell^{\kappa g1}+C_\ell^{\kappa g2}
\end{eqnarray}
where
\begin{eqnarray}\label{E:kg}
C_\ell^{\kappa gi}&=&\frac{2}{\pi}\int_0^{\infty}k^2dkP(k,z=0)W_\ell^{gi}W_\ell^{\kappa}(k,z*)
\end{eqnarray}
Note that to compute these $C_\ell$s we will use the Limber approximation \citep{Loverde2008a}, which works well on scales $l+1/2\gtrsim\bar{r}/\sigma$, where $\bar{r}$ and $\sigma$ are the peak and width of radial selection function. Since $\bar{r}/\sigma$ for all the redshift distributions and lensing kernels we consider in this work is less than 10, to avoid expensive spherical Bessel function integration we apply the Limber approximation to compute angular power spectra at scales $l>10$. Note that scales $l<10$ will not be used in this work. We present the Limber approximation expressions in Appendix \ref{S:cllimber}.
These expressions now allow us to predict how significantly magnification bias will distort our measurement. While we will consider this for specific surveys in the next section, here we consider a more general case. D16 performed a similar analysis, and found that $C^{gg}_{12}$, $C^{gg}_{22}$ and $C^{\kappa g}_2$ contributed by magnification bias could considerably distort $E_G$ and ruin its independence from clustering bias.\par
We consider redshift distributions identical to those in D16:
\begin{eqnarray}\label{eq:20}
f(z)\propto\left(\frac{z}{z_g}\right)^2\exp\left[-\left(\frac{z}{z_g}\right)^2\right], z_g=0.57 \nonumber\\
f(z)=\frac{1}{\sigma\sqrt{2\pi}}\exp\left(-\frac{(z-z_g)^2}{2\sigma^2}\right), z_g=1.5, \sigma=0.68\, .
\end{eqnarray}
where we limit the redshift range to $z\in[0.47-0.67]$ for the first distribution and $z\in[1.4,1.6]$ for the second one as analogs of BOSS+eBOSS LRG and DESI ELG survey, respectively. In both cases, we assume a clustering bias $b=1$ and $s=0$ for simplicity as in D16.
We use the notation:
\begin{equation}\label{eq:19}
\begin{split}
&E_{G0}=\Gamma(\bar{z})\frac{C^{\kappa g}_1}{\beta(\bar{z})C^{gg}_{11}}, E_{G1}=\Gamma(\bar{z})\frac{C^{\kappa g}}{\beta(\bar{z})C^{gg}}\\
&\Delta C^{\kappa g}_l=C^{\kappa g}_{l1}-C^{\kappa g}_l\\
&\Delta C^{gg}_l=C^{gg}_{l11}-C^{gg}_{l}\\
&\Delta E_G=E_{G0}-E_{G1}
\end{split}
\end{equation}\par
The deviations for $C^{gg}_\ell$ and $C^{\kappa g}_\ell$ are shown in Figure \ref{fig:1}. While these results are not exactly the same as in D16, they are broadly similar. One source of the small deviations between our results and those from D16 is the different cosmological parameters we use. Also, D16 does not specify the redshift distribution range it uses. In this section we integrate for angular power spectrum over redshift range $z\in[\bar{z}-0.1,\bar{z}+0.1]$, which might be different to what D16 uses and could cause disagreement between our results. As mentioned in D16, we expect choosing different $b$ and $s$ values will change the deviations in $C_\ell^{\kappa g}$ and $C_\ell^{gg}$ by the factor $(2-5s)/(2b)$, which we will consider further in regards to the fractional deviation of $E_G$.\pa
\begin{figure*}
\centering
\includegraphics[width = 7cm, height=6cm]{ckg057.eps}
\includegraphics[width = 7cm, height=6cm]{ckg15.eps}\\
\includegraphics[width = 7cm, height=6cm]{dckg057.eps}
\includegraphics[width = 7cm, height=6cm]{dckg15.eps}\\
\caption{$C^{\kappa g}_l$ deviation at redshifts centered at $z=0.57$ (left) and $z=1.5$ (right). The first row shows the absolute deviation and the second row shows the fractional deviation. Dashed line denotes negative correlations. Galaxy bias and magnification bias parameter are set as $b=1$ and $s=0$ in the calculation.}
\label{fig:1}
\end{figure*}\par
\begin{figure*}
\centering
\includegraphics[width = 7cm, height=6cm]{cgg057.eps}
\includegraphics[width = 7cm, height=6cm]{cgg15.eps}\\
\includegraphics[width = 7cm, height=6cm]{dcgg057.eps}
\includegraphics[width = 7cm, height=6cm]{dcgg15.eps}\\
\caption{Same as Figure \ref{fig:1} except for the $C_l^{gg}$ deviation.}
\end{figure*}\par
The deviation in $E_G$ due to magnification bias is shown in Figure \ref{fig:2}.
Using the fiducial values $s=0$ and $b=1$, we see significant deviations $\langle\Delta E_G/E_{G0}(\bar{z}=0.57)\rangle=4.37\%$ and $\langle\Delta E_G/E_{G0}(\bar{z}=1.5)\rangle=39.35\%$. However, more common values for the magnification bias for upcoming surveys are $s=0.2$ and 0.48, as shown in Table \ref{t:1}. Setting $s=0.2$ and $b=2.3$ decreases the deviations significantly to $\langle\Delta E_G/E_{G0}(\bar{z}=0.57)\rangle=0.91\%$ and $\langle\Delta E_G/E_{G0}(\bar{z}=1.5)\rangle=8.27\%$. Setting $s=0.48$ and $b=2.3$ changes the deviations to $\langle\Delta E_G/E_{G0}(\bar{z}=0.57)\rangle=-0.36\%$ and $\langle\Delta E_G/E_{G0}(\bar{z}=1.5)\rangle=-3.23\%$.
This indicates that the magnification effect is unlikely to cause a significant problem for $E_G$ measurements from galaxy surveys with large clustering bias and magnification factor $s$ close to 0.4. We do expect that for emission line galaxies, which tend to have lower clustering biases, the magnification bias effect will be larger and potentially significant. We will test $E_G$ deviations under various galaxy and CMB survey cross-correlations using realistic redshift ranges and galaxy clustering and magnification biases in section \ref{S:calitest}.\par
\begin{figure}
\centering
\includegraphics[width=0.45\textwidth]{dEg057.eps}
\includegraphics[width=0.45\textwidth]{dEg15.eps}
\caption{$E_G$ fractional deviation at redshifts centered at $z=0.57$ (top) and $z=1.5$ (bottom). $E_G$ is significantly biased when the galaxy bias and magnification bias parameter are set as $b=1$ and $s=0$, while for more realistic values, the $E_G$ bias is much smaller.}
\label{fig:2}
\end{figure}
\subsection{Calibration method for magnification bias} \label{S:cali}
The true $E_G$ should be estimated using $C_\ell^{\kappa g1}$ and $C_\ell^{g1g1}$ which are not contaminated by magnification bias. Since we cannot estimate these directly from data, we have to instead find a way to subtract the higher order terms in Eq. \ref{eq:17} and Eq. \ref{eq:18} for each $C_\ell$ from magnification bias. Since we are testing cosmology, we cannot just compute them theoretically; however, by looking at the expressions in Eqs.~\ref{E:gg} and \ref{E:kg}, we see that $C_\ell^{\kappa\kappa}$, given by Eq.~\ref{E:clab} for $A=B=\kappa$, shares a very similar form with $C_\ell^{\kappa g2}$ and $C_\ell^{g2g2}$ while $C_\ell^{\kappa g1}$ has a similar form to $C_\ell^{g1g2}$. This implies that these spectra should be highly correlated, such that we can use the measured $C_\ell^{\kappa\kappa}$ to predict the correlations to $C_\ell^{\kappa g}$ and $C_\ell^{gg}$.
In order to predict these correlations, we propose the following scaling formulae:
\begin{eqnarray}\label{eq:30}
\tilde{C}_\ell^{\kappa g2}=\left(\frac{C_\ell^{\kappa g2}}{C_\ell^{\kappa\kappa}}\right)\hat{C}_\ell^{\kappa\kappa}
\end{eqnarray}
\begin{eqnarray}\label{eq:31}
\tilde{C}_\ell^{g2g2}=\left(\frac{C_\ell^{g2g2}}{C_\ell^{\kappa\kappa}}\right)\hat{C}_\ell^{\kappa\kappa}
\end{eqnarray}
\begin{eqnarray}\label{eq:32}
\tilde{C}_\ell^{g1g2}&=&\left(\frac{C_\ell^{g1g2}}{C_\ell^{\kappa g1}}\right)\tilde{C}_\ell^{\kappa g1}\nonumber\\
&=&\left(\frac{C_\ell^{g1g2}}{C_\ell^{\kappa g1}}\right)\times\left[\hat{C}_\ell^{\kappa g}-\left(\frac{C_\ell^{\kappa g2}}{C_\ell^{\kappa\kappa}}\right)\hat{C}_\ell^{\kappa\kappa}\right]
\end{eqnarray}
Here we use hats to denote quantities estimated from data. We use tildes to denote the scaled correlations. Quantities without a marker are computed from theory. These estimated terms will then be subtracted from measured $\hat{C}_\ell^{\kappa g}$ and $\hat{C}_\ell^{gg}$ to estimate $C_\ell^{\kappa g1}$ and $C_\ell^{g1g1}$ to compute $E_G$. In principle, there should be better ways to reduce magnification bias. For example, we could instead perform an Monte Carlo Markov Chain (MCMC) analysis where we fit for $E_G$ and the magnification bias terms in the $C_\ell$s simultaneously. However, the goal of our analysis is to find the simplest way to calibrate magnification bias. The next section uses simulations to determine how well these calibrations work. Also, we expect these correlations to vary slightly with cosmological parameters, which must be set to construct simulations to compute the correlations. Thus, we also test how sensitive the calibrations are to the assumed cosmologies, and specifically how much bias is introduced by using cosmological parameters that are not the true values but are still within known uncertainties.
\section{Results}\label{S:results}
In this section we compute the deviation of $E_G$ due to magnification bias for multiple spectroscopic and photometric surveys cross-correlated with CMB lensing maps from Planck and Adv.~ACT. We find that most of the cases where there are significant deviations are from photometric surveys, although DESI can also exhibit a deviation. We also test our calibration method, finding that it is efficient in removing magnification bias and it is stable with regard to small deviations in the parameters used to construct the calibration.
\subsection{Testing $E_G$ calibrations using simulations} \label{S:calitest}
We use theoretical angular power spectra of the CMB lensing convergence and galaxy overdensity to construct simulated maps. We use these maps for two purposes. The first purpose is to construct errors on $E_G$. $E_G$, being a ratio between 3 measured quantities, is inherently non-Gaussian. While we can use error propagation to get a relatively accurate estimate of the statistical error on $E_G$, simulations can naturally capture the additional contribution to the statistical fluctuations due to this non-Gaussianity. The second purpose of these simulations is to test our proposed calibration method to remove significant magnification bias. In particular, we expect the measured $C_\ell^{\kappa g}$ and calibration terms scaled to $C_\ell^{\kappa\kappa}$ to be correlated, and simulations allow us to capture this effect naturally.
The pipeline of the simulation process we use is described in \cite{Serra2014}. In short, we compute all the correlation terms listed in Eqs.~\ref{eq:24}-\ref{eq:29} and follow Eqs.~\ref{eq:17}-\ref{eq:18} to obtain theoretical values of observables $C_\ell^{gg}$, $C_\ell^{\kappa g}$ and $C_\ell^{\kappa\kappa}$. In each simulation, we construct the covariance matrix $C$ and decompose it using Cholesky decomposition as $C=LL^T$ with $L$, a lower triangular matrix, given by
\begin{eqnarray}\label{eq:34}
C=\begin{pmatrix}C^{\kappa\kappa}_l+N^{\kappa}_l&C^{\kappa g}_l\\C^{\kappa g}_l&C^{gg}_l+N^{g}_l\end{pmatrix}=LL^T
\end{eqnarray}
Here $N^{\kappa}_l$ and $N^g_l$ are lensing noise and shot noise of CMB and galaxy surveys cross-correlated in the simulation.
Spherical harmonic coefficients are then obtained through
\begin{eqnarray}
\begin{pmatrix}a^{\kappa}_{lm}\\a^g_{lm}\end{pmatrix}=L\begin{pmatrix}\xi_1\\\xi_2\end{pmatrix}
\end{eqnarray}
Here $\xi_1$ and $\xi_2$ are arrays of random numbers with zero mean and unit variance.\par
We use $a_{lm}^{\kappa}$ and $a_{lm}^g$ to compute full sky galaxy maps and CMB lensing maps through functions provided by the HEALPix package \citep{Gorski2005}. We apply $N_{\rm side}=1024$ for the galaxy maps and $N_{\rm side}=2048$ for the lensing maps, after which we then add a mask onto the full sky maps. We assume Planck and Adv.~ACT survey overlap completely with galaxy surveys well except that Adv.~ACT only overlaps with 75\% of the DESI survey area. We correlate masked galaxy map and CMB lensing map with each other to get angular power spectra of the partial sky maps $D_\ell^{gg}$, $D_\ell^{\kappa g}$ and $D_\ell^{\kappa\kappa}$ and then estimate $\hat{C}^{gg}_\ell$, $\hat{C}^{\kappa g}_\ell$ and $\hat{C}^{\kappa\kappa}_\ell$ from simulated data using the naive prescription \citep{Hivon2002}
\begin{eqnarray}
\hat{C}_\ell=\frac{\langle D_\ell\rangle}{f_{\rm sky}\omega_2}
\end{eqnarray}
where $f_{\rm sky}\omega_i$ is the $i$-th momentum of the mask $W(\hat{n})$
\begin{eqnarray}
f_{sky}\omega_i=\frac{1}{4\pi}\int_{4\pi}d\hat{n}W^i(\hat{n})
\end{eqnarray}\par
Following \cite{Pullen2015}, we assume that scales dominated by nonlinear clustering will not be used to estimated $E_G$. Thus, we set the maximum wavenumber comprising linear to quasi-linear scales to $k_{nl}$, where $k^3_{nl}P(k_{nl},\bar{z})/(2\pi^2)=1$ and set the maximum scale $\ell_{max}=\chi(\bar{z})k_{nl}$. Throughout this work we use the non-linear matter power spectrum calculated by CAMB (HALOFIT) \cite{2000ApJ...538..473L}, but removing nonlinear scales from the estimator do not significantly affect the results. We then bin the simulated angular power spectrum into 10 bins with $\Delta_\ell=\ell_{max}/10$, after which we remove the lowest and highest $\ell$-bins, leaving us with 8 bins. Note that since the galaxy surveys each have different median redshifts, the relevant scales will vary for each survey.\par
In our analysis we do include scales that could be contaminated by systematic effects other than magnification bias and nonlinear clustering. These effects include redshift space distortions and stellar contamination at large scales and lensing Gaussian and point-source bias at small scales \citep{Pullen2016}. However, the main focus of this paper is on magnification bias, and we leave the mitigation of these other systematic effects to future work.\par
Since we are assessing the $E_G$ calibration using simulations, we consider the total error in the simulated $E_G$, $\sigma_{tot}$, given by
\begin{equation}
\begin{split}
\sigma_{tot}&\sim\sqrt{\sigma^2_{stat}+\sigma^2_{dev}+\sigma^2_{sim}}\\
&=\sigma_{stat}\sqrt{1+\left(\dfrac{\sigma_{dev}}{\sigma_{stat}}\right)^2+\left(\dfrac{\sigma_{sim}}{\sigma_{stat}}\right)^2}\\
&\sim\sigma_{stat}\sqrt{1+\left(\dfrac{\sigma_{dev}}{\sigma_{stat}}\right)^2+\dfrac{1}{N_{\rm sim}}}
\end{split}
\end{equation}
where $\sigma_{stat}$ is the statistical $E_G$ error, $\sigma_{dev}$ is the deviation of $E_G$ caused by magnification bias, and $\sigma_{sim}$ is the error due to using a finite number of simulations $N_{\rm sim}$, which decreases as $\sigma_{sim}\sim\dfrac{\sigma_{stat}}{\sqrt{N_{\rm sim}}}$. Our goal is to find $E_G$ deviations that are at least 50\% of the statistical error, which would increase the total $E_G$ error by 10\%. According to our expression, using 100 simulations should be enough such that the total error should not increase significantly; thus we choose to use 100 simulations in our analysis.
\par
The CMB surveys we consider are the Planck satellite and Adv.~ACT. It was shown recently that non-Gaussian clustering in the lenses can bias CMB lensing estimators dominated by the temperature maps at the 3\% level compared to the signal \citep{2018arXiv180601157B,2018arXiv180601216B}, while the bias can be as high as 7\% for cross-correlations with large-scale structure tracers \citep{2018arXiv180601157B}, though the latter prediction is still preliminary (B\"{o}hm et al. in prep). These biases should affect CMB lensing estimates from both Planck and Adv.~ACT. However, it was also shown in \citet{2018arXiv180601157B} that polarization-based estimates of CMB lensing do not have this bias. Thus, although the sensitivity of CMB-S4 is not demonstrably better than Adv.~ACT, its lensing estimate would be more accurate for an $E_G$ analysis since its polarization maps would dominate its CMB lensing estimate.\par
The galaxy surveys include both spectroscopic and photometric surveys listed in Table \ref{t:1}. Since estimating $E_G$ requires a measurement of $\beta$, photometric galaxy surveys would normally not be considered because the lower redshift precision washes out the RSD effect. However, as argued in \citet{Pullen2015}, the larger number of galaxies make photometric surveys so much more precise in measuring $C_\ell^{\kappa g}$ that although the precision in $\beta$ from a photometric survey is lower \citep{2011MNRAS.415.2193R,2014MNRAS.445.2825A}, the overall error in $E_G$ could still be lower generally than in spectroscopic surveys, particularly if catastrophic redshift errors are kept to a minimum.
\par In all simulations in this subsection we assume a $\Lambda$CDM cosmogony. We assume an eBOSS LRG redshift distribution similar to \cite{2016AJ....151...44D} Table 1 but in the range $z\in[0.4,0.9]$. The eBOSS QSO redshift distribution we use is similar to that in \cite{Parisa} but in the redshift range $z\in[0.9,3.5]$, and we use an expression for the QSO clustering bias from Table 2 in \cite{Zhao2016}. The redshift distributions for DESI survey are from \cite{DESICollaboration2016} Table 2.5. For the Euclid spectroscopic survey we use a redshift distribution estimated from the H$\alpha$ luminosity function from \citet{2013ApJ...779...34C} with a flux limit of $4\times 10^{-16}$ erg/s/cm$^2$. For both DES and LSST, we use the toy redshift distribution model from \citet{2014JCAP...05..023F}, given by
\begin{eqnarray}
f_g(z)=\frac{\eta}{\Gamma\left(\frac{\alpha+1}{\eta}\right)z_0}\left(\frac{z}{z_0}\right)^\alpha\exp^{-(z/z_0)^\eta}\, ,
\end{eqnarray}
where the parameter values are given by $(\alpha,\eta,z_0)=(1.25,2.29,0.88)$ for DES and $(2.0,1.0,0.3)$ for LSST. For the Euclid photometric survey we apply estimates of redshift distributions based on \cite{Hsu2014a,Guo2013}.\par
It can be shown that $s=-(1+\alpha_{LF})/2.5$, where $s$ is the magnification bias factor we define in Eq. \ref{eq:13}, and $\alpha_{LF}$ is the faint-end slope parameter in the Schechter luminosity function \citep{1976ApJ...203..297S}. We present the proof in Appendix \ref{S:mag_fac}. Our $s$ factor sources for BOSS LRG, eBOSS ELG, BOSS QSO, DESI (LRG, ELG, QSO) and Euclid spectroscopic surveys are in sequence \cite{2012ApJ...746..138T}, \cite{Comparat:2014xza}, \cite{2013ApJ...773...14R}, \cite{Aghamousa:2016zmz}, and \cite{2013ApJ...779...34C}. We use the $s$ factor from BOSS or eBOSS survey as BOSS+eBOSS combined magnification bias factor. We use (A.5) in \cite{Montanari:2015rga} to estimate the $s$ factor for all the three photometric surveys. \par
Since the eBOSS/BOSS areas overlap for both the LRGs and the QSOs, we use the eBOSS area as the total area. We use $\bar{N}$ to denote galaxy number density per steradian, and we estimate galaxy shot noise for $C_\ell^{gg}$ as $N_\ell^g=1/\bar{N}$. We assume $\bar{N}_{BOSS+eBOSS}=\bar{N}_{BOSS}+\bar{N}_{eBOSS}$ for BOSS$+$eBOSS survey. Since the galaxy bias we use here is no longer a constant, we use $C_b=C^{g1g1}_{\ell}/[b(\bar{z})C^{mg}_\ell]$ to calibrate $E_G$ bias due to galaxy bias evolution \citep{Reyes2010a,Pullen2016}. Details we apply for different galaxy surveys including $k_{nl}$, $\ell_{max}$ and biases $s$ and $b(z)$ are summarized in Table \ref{t:1}.
\begin{table*}
\centering
\begin{tabular}{l c c c c c c c}
\hline\hline &$b(z)$&s&z&Area($deg^2$)&$N_{gal}$&$k_{nl}(h/Mpc)$&$l_{max}$ \\\hline
BOSS LRG&$1.7/D(z)$&&0.43-0.7&7900&704,000&0.275&380\\\hline
eBOSS LRG&$1.7/D(z)$&&0.6-0.8&7500&300,000&0.299&500\\\hline
BOSS+eBOSS LRG&$1.7/D(z)$&0.2&0.43-0.8&7500&968,000&0.288&450\\\hline
eBOSS ELG&$1/D(z)$&0.48&0.6-1.0&1000&189,000&0.318&620\\\hline
BOSS QSO&$0.53+0.29(1+z)^2$&&2.1-3.5&10,200&175,000&0.611&1000\\\hline
eBOSS QSO&$0.53+0.29(1+z)^2$&&0.9-3.5&7500&573,000&0.450&1000\\\hline
BOSS+eBOSS QSO&$0.53+0.29(1+z)^2$&0.2&0.9-3.5&7500&701,000&0.476&1000\\\hline
DESI LRG&$1.7/D(z)$&0.87&0.6-1.2&14,000&$4.1\times10^6$&0.312&580\\\hline
DESI ELG&$0.84/D(z)$&0.1&0.6-1.7&14,000&$1.8\times10^7$&0.352&810\\\hline
DESI QSO&$1.2/D(z)$&0.29&0.6-1.9&14,000&$1.9\times10^6$&0.400&1000\\\hline
Euclid (spectro)&$0.9+0.4z$&0.16
&0.5-2.0&20,000&$3.0\times10^7$&0.326&660\\\hline
DES&$0.9+0.4z$&0.29&0.0-2.0&5,000&$2.16\times10^8$&0.312&580\\\hline
LSST&$0.9+0.4z$&0.324&0.0-2.5&20,000&$3.6\times10^9$&0.330&680\\\hline
Euclid (photo)&$0.9+0.4z$&0.326&0.0-3.7&20,000&$1.86\times10^9$&0.330&690\\\hline
\end{tabular}
\caption{Properties of spectroscopic and photometric surveys in our analysis.}
\label{t:1}
\end{table*}
\par
The standard deviation of multiple simulations give error estimation about $\hat{C}^{gg}_\ell$, $\hat{C}^{\kappa g}_\ell$ and $\hat{C}^{\kappa\kappa}_\ell$ as well as their covariance. We estimate $E_G$ errors following error propagation as
\begin{eqnarray}
\frac{\sigma^2(\hat{E}_g)}{\hat{E}_g^2}&=&\frac{\sigma^2(\hat{C}_\ell^{\kappa g})}{(\hat{C}_\ell^{\kappa g})^2}+\frac{\sigma^2(\beta)}{\beta^2}+\frac{\sigma^2(\hat{C}_\ell^{gg})}{(\hat{C}_\ell^{gg})^2}\nonumber\\
&&-2Cov(\beta,\hat{C}_\ell^{\kappa g})\frac{1}{\hat{C}_\ell^{\kappa g}\beta}\nonumber+2Cov(\beta,\hat{C}_\ell^{gg})\frac{1}{\beta \hat{C}_\ell^{gg}}\nonumber\\
&&-2Cov(\hat{C}_\ell^{\kappa g},\hat{C}_\ell^{gg})\frac{1}{\hat{C}_\ell^{\kappa g}\hat{C}_\ell^{gg}}
\end{eqnarray}
In our analysis we ignore the term $-2Cov(\beta,\hat{C}_\ell^{\kappa g})/[\hat{C}_\ell^{\kappa g}\beta]$ since the CMB lensing convergence field, unlike the galaxy overdensity, includes perturbations from LSS at all redshifts. $\hat{C}_\ell^{\kappa g}$ therefore should be much less correlated to $\beta$ than $\hat{C}^{gg}_\ell$. In order to compute the fifth term, we apply $Cov(\beta,\hat{C}_\ell^{gg})=r\sigma(\hat{C}_\ell^{gg})\sigma(\beta)$ where the correlation coefficient $r$ is computed using simulations of the galaxy distribution in redshift space implemented in \cite{Pullen2016}. We check that the $2r\sigma(\Hat{C}_\ell^{gg})\sigma(\beta)/(\beta \hat{C}^{gg}_\ell)$ term only contributes less than $8\%$ of $\sigma(\hat{E}_g)$ in all cases we consider, so we also ignore this term. A more precise analysis would use full 3D simulations along light cones in order to simulate CMB lensing, galaxy clustering, and RSD simultaneously; however, we save this for future work.\par
For spectroscopic surveys, we use error predictions of the $\beta$ forecasted by the surveys themselves. We use $\sigma(\beta)=0.025\beta$, $\sigma(\beta)=0.029\beta$ and $\sigma(\beta)=0.035\beta$ for the whole range for BOSS+eBOSS LRG, BOSS+eBOSS QSO, and eBOSS ELG, respectively \citep{2016AJ....151...44D,Zhao2016}; 4\% error in $\beta$ within $\Delta z=0.1$ for the DESI survey \citep{DESICollaboration2016}, and 3\% error in $\beta$ within $\Delta z=0.1$ for the Euclid spectroscopic survey \citep{Amendola2012}. For photometric surveys, we assume errors for $\beta$ based on work by \citet{2011MNRAS.415.2193R} and \citet{2014MNRAS.445.2825A}. For DES, we use $\sigma(\beta)/\beta=0.17\sqrt{0.1(1+1)/(z_2-z_1)}$, while for LSST and photometric Euclid we expect a volume 4 times larger and thus $\sigma(\beta)/\beta=0.085\sqrt{0.1(1+1)/(z_2-z_1)}$, where $z_2$ and $z_1$ are the upper and lower limit of the redshift distribution.\par
We now present results for the performance of our calibration method assuming that the parameters we use to construct the calibrations are known perfectly with no errors. We use 100 simulations in this analysis, and our results are shown in Figures \ref{fig:3} and \ref{fig:4b} and Tables \ref{t:2} and \ref{t:3}. Each error bar is the standard deviation of 100 simulations in one multipole bin in order to model the error for one realization. We find that for all spectroscopic and photometric survey cases the $E_G$ fractional deviation is less than $5\%$, while photometric survey cases suffer from a larger deviation on average. Ignoring shot noise from galaxy surveys and lensing noise from CMB surveys, the $E_G$ deviation in all spectroscopic surveys except DESI LRG and DESI ELG surveys cross-correlated with Adv.~ACT is much less than the error estimated by simulation. In particular, DESI LRG and ELG may exhibit a magnification bias in $E_G$ that is greater than twice the statistical error. This appears to be caused by DESI's high galaxy density and low clustering bias while maintaining a magnification bias parameter $s\neq0.4$. For all the photometric surveys, the very high number density causes the magnification bias in $E_G$ to be approximately 4\%, much greater than statistical errors, even when cross-correlated with the Planck CMB lensing map. For these cases $E_G$ after calibration is corrected from about 2-4 times the simulation error to well within the errors. We believe this calibration will work increasingly well in future surveys with lower shot noise and lensing noise.\par
\begin{figure*}
\begin{center}
\begin{tabular}{ccc} &\Large{Planck}&\Large{Adv.~ACT}\\%&CMB-S4\\
\rotatebox[origin=c]{90}{\Large{DESI LRG}}&\raisebox{-0.5\height}{\includegraphics[width = 7cm, height=6cm]{vsDESILRG_Planck.eps}}&\raisebox{-0.5\height}{
\includegraphics[width = 7cm, height=6cm]{vsDESILRG_ACTPol.eps}}\\
\rotatebox[origin=c]{90}{\Large{DESI ELG}}&\raisebox{-0.5\height}{\includegraphics[width = 7cm, height=6cm]{vsDESIELG_Planck.eps}}&\raisebox{-0.5\height}{
\includegraphics[width = 7cm, height=6cm]{vsDESIELG_ACTPol.eps}}\\
\end{tabular}
\end{center}
\caption{Simulated $E_G$ measurements for spectroscopic surveys with and without magnification bias calibration. The first column includes CMB lensing noise from the Planck survey while the second column assumes Adv. ACTPol. The top row is for DESI LRG, while the bottom row is for DESI ELG. We shift data after calibration by $\Delta l=10$ for reading convenience.}
\label{fig:3}
\end{figure*}
\begin{figure*}
\begin{center}
\begin{tabular}{ccc}
&\Large{Planck}&\Large{Adv.~ACT}\\%&CMB-S4\\
\rotatebox[origin=c]{90}{\Large{DES}}&\raisebox{-0.5\height}{\includegraphics[width = 7cm, height=6cm]{vsDES_Planck.eps}}&
\raisebox{-0.5\height}{\includegraphics[width = 7cm, height=6cm]{vsDES_ACTPol.eps}}\\
\rotatebox[origin=c]{90}{\Large{LSST}}&\raisebox{-0.5\height}{\includegraphics[width = 7cm, height=6cm]{vsLSST_Planck.eps}}&
\raisebox{-0.5\height}{\includegraphics[width = 7cm, height=6cm]{vsLSST_ACTPol.eps}}\\
\rotatebox[origin=c]{90}{\Large{Euclid (photo)}}&\raisebox{-0.5\height}{\includegraphics[width = 7cm, height=6cm]{vsEuclid_photo_Planck.eps}}&
\raisebox{-0.5\height}{\includegraphics[width = 7cm, height=6cm]{vsEuclid_photo_ACTPol.eps}}\\
\end{tabular}
\end{center}
\caption{Same as Figure \ref{fig:3} except for the photometric surveys. The rows from top to bottom corresponds to DES, LSST and Euclid (photo) surveys respectively. We shift data after calibration by $\Delta l=10$ for reading convenience.}
\label{fig:4b}
\end{figure*}
\begin{table*}
\centering
\begin{tabular}{c c c c c}
\hline\hline
&Theoretical $E_G$&no noise& Planck &Adv.~ACT \\\hline
BOSS+eBOSS LRG&$0.390,0.89\%$&$-1.21\%,2.28\%$& $-0.64\%, 7.13\%$&$-0.38\%,2.77\%$\\[1ex]\hline
BOSS+eBOSS QSO&$0.325,1.22\%$& $-0.69\%, 1.08\%$&$2.52\%,13.71\%$&$2.98\%,8.09\%$\\[1ex]\hline
eBOSS ELG&$0.369,-1.12\%$& $0.80\%,4.13\%$&$4.02\%, 19.35\%$&$2.05\%,7.14\%$\\[1ex]\hline
&&& $3.28\%, 3.92\%$&$2.71\%,1.64\%$\\[-1ex]
\raisebox{1.5ex}{DESI LRG}&\raisebox{1.5ex}{$0.372,-3.26\%$}&\raisebox{1.5ex}{$2.95\%,1.21\%$}&$0.64\%,4.01\%$&$0.0032\%,1.67\%$\\[1ex]\hline
&&& $-4.13\%, 3.21\%$&$-3.95\%,1.15\%$\\[-1ex]
\raisebox{1.5ex}{DESI ELG}&\raisebox{1.5ex}{$0.353,4.43\%$}&\raisebox{1.5ex}{$-4.02\%,0.68\%$}&$-0.12\%,3.06\%$&$0.10\%,1.08\%$\\[1ex]\hline
DESI QSO&$0.339,0.628\%$&$-0.39\%,0.52\%$& $0.10\%, 7.88\%$&$0.035\%,3.76\%$\\[1ex]\hline
Euclid&$0.365,0.88\%$&$-0.17\%,0.68\%$& $0.13\%, 3.05\%$&$-0.52\%,1.02\%$\\[1ex]\hline
\end{tabular}
\caption{Summary of simulation results for spectroscopic surveys. Theoretical $E_G$ column shows the average of $E_G$ from theory in all ranges on the left and averaged $\Delta E_G/E_G$ on the right. $E_G$ values using cross-correlations between galaxy surveys and CMB surveys are averaged over 8 bins in the ranges displayed in Figure \ref{fig:3}. The remaining columns display $(\hat{E}_g-E_G^{theory})/E_G^{theory},\sigma(\hat{E}_g)/E_G^{theory}$ for each case. The upper line in each row corresponds to the averaged simulation result before calibration, the lower line presents averaged simulation result after calibration using Eqs.~\ref{eq:30}-\ref{eq:32}. Among all the simulation results, except for DESI LRG and DESI ELG columns, $E_G$ fractional deviation is much smaller than $1\sigma$ before calibration, in which case the advantage of calibration is not obvious, so we don't show calibrated results for the purpose of conciseness. The ``no noise'' column corresponds to averaged results from simulations without considering galaxy shot noise and CMB instrumental noise.}
\label{t:2}
\end{table*}
\begin{table*}
\centering
\begin{tabular}{c c c c c}
\hline\hline
&Theoretical $E_G$&no noise& Planck &Adv.~ACT \\\hline
&&& $3.97\%, 4.05\%$&$3.86\%,2.34\%$\\[-1ex]
\raisebox{1.5ex}{DES}&\raisebox{1.5ex}{$0.372,-4.09\%$}&\raisebox{1.5ex}{$4.14\%,2.32\%$}&$-0.086\%,3.69\%$&$-0.12\%,2.22\%$\\[1ex]\hline
&&& $4.42\%, 1.83\%$&$4.20\%,1.05\%$\\[-1ex]
\raisebox{1.5ex}{LSST}&\raisebox{1.5ex}{$0.362,-4.26\%$}&\raisebox{1.5ex}{$4.22\%,1.02\%$}&$0.18\%,1.66\%$&$0.023\%,0.99\%$\\[1ex]\hline
&&& $4.14\%, 1.68\%$&$4.40\%,0.86\%$\\[-1ex]
\raisebox{1.5ex}{Euclid (photo)}&\raisebox{1.5ex}{$0.356,-4.40\%$}&\raisebox{1.5ex}{$4.33\%,0.84\%$}&$-0.18\%,1.52\%$&$0.062\%,0.81\%$\\[1ex]\hline
\end{tabular}
\caption{Same as Table \ref{t:2} except for photometric surveys. We see that these surveys generally have magnification biases over 4\% while the calibration removes it efficiently.}
\label{t:3}
\end{table*}
\subsection{Calibration tests under parameter uncertainties}\label{S:test2}
Our calibration method depends on the parameters that construct the calibration matching the true values in nature perfectly. In reality, statistical and systematic errors will prevent this; thus, it is important to test how well our calibration method works when any parameters deviate from the true values under reasonable assumptions. The parameters that we consider for this test are cosmological parameters, GR modifications, redshift distribution parameters, and clustering and magnification biases. Although several parameters can deviate at once, biasing our calibration in unforeseen ways, we will only consider cases where one parameter deviates at a time to get a sense of the effect of each deviation.\par
There are multiple ways to test if our calibration method is stable with respect to these uncertainties. Intuitively, one would assume one set of theoretical parameters to calibrate $C_l$ and multiple sets of possible true parameters to do simulations. However, we will perform the equivalent yet less time-consuming method of generating simulations of $\hat{C}_l$ from one set of true parameters for our universe, then assume multiple sets of mis-identified theoretical parameters to construct calibrations $C_l$. We consider various parameter mis-identification cases by setting a baseline with all parameters as the true values then shifting them to the upper and lower bound of its uncertainty in order to produce a biased calibration for $C_l$. In this section we use cosmological parameters with uncertainties $\Omega_m=0.309\pm 0.007$, $\Omega_bh^2=0.02226\pm0.00016$, $\Omega_ch^2=0.1193\pm0.0014$, $h=0.679\pm0.007$, $\Omega_k=0.0000\pm0.0020$ according to \cite{Alam2016}. The upper and lower limit of each parameter is for the Planck + BOSS BAO survey. Assuming standard deviations for each parameter is proportional to $1/\sqrt{V}$, where $V$ is the survey volume, we apply $\sigma_{DESI ELG}=0.22\sigma_{BAO}$, $\sigma_{LSST}=0.11\sigma_{BAO}$, where $\sigma_{BAO}$ represents parameter uncertainties for Planck + BOSS BAO survey. We assume general relativity for a baseline along with the redshift distributions from DESI ELGs and LSST we use in section \ref{S:calitest} and expectation values of these cosmological parameters are the true parameters and use them to produce $\hat{C}_\ell$ in simulation.
We choose $f(R)$ gravity and Chameleon gravity to test how modified gravity can influence the calibration performance. It was shown in \citet{Pullen2015} that for functions $\mu(k,z)$ and $\gamma(k,z)$ that modify the Poisson equation and introduce large-scale anisotropic stress, respectively, $E_G$ is modified according to
\begin{eqnarray}
E_G=\frac{\Omega_{m0}\mu(k,a)[1+\gamma(k,a)]}{2f}\, ,
\end{eqnarray}
We set general relativity (GR) as the base parameters, \emph{i.e.} $\gamma=\mu=1$. For $f(R)$ gravity \citep{Carroll2004,Song2007,Tsujikawa2007,Hojjati2011}), the relevant values for the parameters are
\begin{equation}
\begin{split}
&\mu^{fR}(k,a)=\dfrac{1}{1-B^{f(R)}_0a^{s-1}/6}\cdot\left[\dfrac{1+(2/3)B^{f(R)}_0a^s\bar{k}^2}{1+(1/2)B^{f(R)}_0a^s\bar{k}^2}\right]\\
&\gamma^{fR}(k,a)=\dfrac{1+(1/3)B^{f(R)}_0a^s\bar{k}^2}{1+(2/3)B^{f(R)}_0a^s\bar{k}^2}\\
&\bar{k}=\dfrac{c}{H_0}k
\end{split}
\end{equation}
In this work we use $B_0^{f(R)}=1.36\times10^{-5}$ \citep{2015PhRvD..91j3503B,2015PhRvD..91f3008X,Alam2015a}, $s=4$ (not the magnification bias).\par
For Chameleon gravity \citep{Khoury2004,Bertschinger2008,Hojjati2011} we set $\mu$ and $\gamma$ to
\begin{equation}
\begin{split}
&\mu^{Ch}(k,a)=\dfrac{1+\beta_1\lambda_1^2k^2a^s}{1+\lambda_1^2k^2a^s}\\
&\gamma^{Ch}(k,a)=\dfrac{1+\beta_2\lambda_2^2k^2a^s}{1+\lambda_2^2k^2a^s}\\
&\lambda_2^2=\beta_1\lambda_1^2\\
&\beta_2=\dfrac{2}{\beta_1}-1
\end{split}
\end{equation}
In this work we use $\beta_1=1.2$, $\lambda_1=\sqrt{c^2B_0^{Ch}/(2H_0^2)}$, where $B_0^{Ch}=0.4$.
We consider $\Omega_k\neq0$ cases to test the stability of the calibration under $\Omega_k$ mis-identification. When the universe is not flat, the coordinate distance can be computed generally as:
\begin{equation}\label{eq:34}
r=\left\{
\begin{array}{@{}ll@{}}
\chi, & \text{if}\ \Omega_k=0 \\
\dfrac{c}{H_0\sqrt{\Omega_k}}\sinh\left[\dfrac{\sqrt{\Omega_k} H_0\chi}{c}\right], & \text{if}\ \Omega_k>0 \\
\dfrac{c}{H_0\sqrt{-\Omega_k}}\sin\left[\dfrac{\sqrt{-\Omega_k}H_0\chi}{c}\right], & \text{if}\ \Omega_k<0 \\
\end{array}\right.\, .
\end{equation}
\par
To test how redshift distribution parameters affect $E_G$ measurements, we shift center of the redshift distribution $\mu$ and FWHM $\sigma$ by 1\%, 5\% and 10\% respectively to compute theoretical $C_l$ for mis-identified galaxy survey redshift distribution. Specifically speaking, we define
\begin{eqnarray}
\mu=\int f_g(z)z dz\left[\int f_g(z)dz\right]^{-1}\, ,
\end{eqnarray}
and apply $f_g(z)\rightarrow f_g(z\pm r\mu)$ where $r=1\%$,$5\%$ and $10\%$. Likewise, we apply $f_g(z)\rightarrow f_g(\mu+(z-\mu)(1+r))$ to study impact of FWHM $\sigma$.\par
In this discussion we will use shorthand 'A cross B' to refer to 'survey A cross-correlated with survey B'. Taking DESI ELG cross Adv.~ACT and LSST cross Adv.~ACT as examples, we apply our calibration for all parameter mis-identification cases and the results are shown in Figure \ref{fig:4} and \ref{fig:5}. Data marked by triangle is $E_G$ estimated from simulations where our calibration is not applied. Data marked by circles are calibrated estimation under mis-identified parameters. Data in black are results from true parameters.\par
For DESI, we find that our calibration method is stable with respect to shifts in most cosmological parameters, as well as for shifts in both the magnification and clustering biases. However, the parameters for the redshift distribution will need to be known at the 1\%-level. LSST has a similar story, except that the magnification bias parameter $s$ will also need to be know with 1\% precision. The redshift distribution and the magnification bias parameter can be measured directly from the galaxy sample, thus knowing these values precisely enough should be feasible. \par
Gravity determines redshift space distortion parameter $\beta$ and brings $\mu(k,a)(\gamma(k,a)+1)/2$ factors to angular power spectra $C_\ell$, so it has significant influence on $E_G$. However, since $\beta$ is a measurable quantity and $\mu(k,a)(\gamma(k,a)+1)/2$ factors are very close to unity for $f(R)$ and Chameleon gravity, our calibration method is not sensitive to the tested GR modification.\par
\begin{figure*}
\centering
\includegraphics[width=0.45\textwidth]{DESIELG_Ok_plus.eps}
\includegraphics[width=0.45\textwidth]{DESIELG_Om_plus.eps}\\
\includegraphics[width=0.45\textwidth]{DESIELG_mu_plus.eps}
\includegraphics[width=0.45\textwidth]{DESIELG_sigma_plus.eps}\\
\includegraphics[width=0.45\textwidth]{DESIELG_s_plus.eps}
\includegraphics[width=0.45\textwidth]{DESIELG_b_plus.eps}\\
\caption{Calibration under cosmological parameters, redshift distribution parameters and GR modification mis-identification for DESI ELG cross Adv.~ACT. The dark line is the true $E_G$ value and dark data points are estimated from true parameters. Data marked by triangle are simulation results before calibration. Data marked with circle are simulation results after calibration. In the legend, '+/-' sign means we use upper/lower bound of parameter to compute theoretical angular power spectrum. We shift different data sets by certain amount in order to see them clearly.}
\label{fig:4}
\end{figure*}\par
\begin{figure*}
\centering
\includegraphics[width=0.45\textwidth]{LSST_Ok_plus.eps}
\includegraphics[width=0.45\textwidth]{LSST_Om_plus.eps}\\
\includegraphics[width=0.45\textwidth]{LSST_mu_plus.eps}
\includegraphics[width=0.45\textwidth]{LSST_sigma_plus.eps}\\
\includegraphics[width=0.45\textwidth]{LSST_s_plus.eps}
\includegraphics[width=0.45\textwidth]{LSST_b_plus.eps}\\
\caption{Calibration under cosmological parameters, redshift distribution parameters and GR modification mis-identification for LSST cross Adv.~ACT. Notation conventions are same with Figure \ref{fig:4}.}
\label{fig:5}
\end{figure*}\par
\section{Conclusions}\label{S:conclu}
The $E_G$ statistic is a promising tool to probe GR deviation. \cite{MoradinezhadDizgah2016a} questions that $E_G$ may be biased up to $40\%$ due to the lensing magnification effect in surveys collecting high redshift information and therefore loses its interest.\par
In this work, we show that the lensing magnification is not as significant as \cite{MoradinezhadDizgah2016a} proposed. We also show that for BOSS+eBOSS and upcoming DESI, Euclid survey the $E_G$ fractional bias is less than $5\%$. However, this bias is not necessarily small compared to measurement error due to low shot noise and lensing noise in future surveys. We therefore propose a new calibration method -- using the convergence auto power spectrum $C_{\kappa\kappa}$ to estimate the higher order terms contributed by lensing magnification effect. We find the advantage of this calibration is not obvious for surveys with large noise and large growth rate uncertainty. This method calibrates $E_G$ significantly in DESI LRG and ELG cross correlate with Adv.~ACT CMB lensing since lensing noise in these surveys are much lower. Since photometric surveys contain greater galaxy number density compared to spectroscopic surveys, which largely suppresses the shot noise, this method can calibrates $E_G$ effectively even for photometric surveys cross correlate with Planck CMB lensing. Of course the calibration performance gets better for Adv. ~ACT CMB lensing cases. Moreover, this method works stably under variance of $H_0$, $\Omega_k$ and GR modification, but is sensitive to errors in the redshift distribution and the magnification bias parameter $s$ which are feasible to maintain at a minimal level. This calibration method will be a promising solution to $E_G$ bias due to lensing magnification effect as the precision of future surveys keep increasing.
\section*{Acknowledgments}
We wish to thank A.~Dizgah for discussion about the formalism, as well as M.~Blanton, S.~Ho, and J.~Tinker for comments on our manuscript.
|
1,108,101,564,607 | arxiv | \section{Introduction}
The spectral theory of self-adjoint and unitary operators is a well established topic in mathematics with a rich structure revealed by numerous important results, and which has found many applications, particularly in mathematical physics. See for example the textbooks \cite{Ka, RS, DS, D4, Ku} selected from the abundant literature on the topic. By contrast, the general spectral theory of operators enjoying less symmetry, that is non-normal operators, is more vast, technically more involved and less well understood. However, the spectral theory of non self-adjoint operators has been the object of many works, in various setups of regimes, as can be seen from the works \cite{GoKr, SNF, D1, D2, TE, D3, Sj, CL, CCL, CD} and references therein.
In particular, several analyses of non self-adjoint operators focus on tri-diagonal operators, when expressed in a certain basis, see \cite{D1, D2, CL, CD}. Since Jacobi matrices provide generic models of self-adjoint operators, it is quite natural to deal with non self-adjoint tri-diagonal matrices which are deformations of Jacobi matrices. Moreover, certain models of this sort are physically relevant, see e.g. \cite{HN, GK, FZ}.
In this paper, we introduce and analyze the spectral properties of another set of non-normal operators possessing a band structure in a certain basis, which share similarities with the tri-diagonal non-self-adjoint operators mentioned above. Our operators have a five-diagonal structure and are obtained as deformations of certain unitary operators called CMV matrices, see \cite{Si} for a detailed account. The role played by CMV matrices for unitary operators is similar to that played by Jacobi matrices for self-adjoint operators: they provide generic models of unitary operators; hence we call our models non-unitary operators.
The non-unitary operators considered in this paper arise naturally in the study of random quantum walks on certain infinite graphs, which provide unitary dynamical systems of interest for physics, computer science and probability theory, see for example the reviews \cite{Ke, Ko, V-A, J4}. In particular, random quantum walks defined on ${\mathbb Z}$ are given by special cases of CMV matrices. The study of the spectral properties of random unitary operators and quantum walks defined on trees or lattices, see e.g. \cite{bhj, HJS, JM, ASWe, J3, HJ}, may lead to the analysis of certain autocorrelation functions. We show in Section \ref{rqw} below that in certain cases, the analysis of these autocorrelation functions reduces to the study of iterates of our non-unitary operators, which provides a direct link between spectral properties of non-unitary operators and random quantum walks. Moreover, the structure of our non-unitary operators allows us to determine the spectral nature of the corresponding random quantum walks they are related to.
While the non-unitary operators we study correspond to deformations of random CMV matrices of a special type, and consequently are rather sparse, we show in Section \ref{sextension} that due to certain symmetries they possess, our main results also apply to deformations of random unitary CMV type matrices of a much more general form. Those random unitary operators appear as models in condensed matter physics and can be considered as natural unitary analogs of Anderson type models, see \cite{BB, bhj, HJS}. The corresponding non-unitary deformations they give rise to are thus of a quite general form, displaying generically non zero elements at all entries of the familiar $5$-diagonal structure CMV type matrices possess. In that sense, our spectral analysis applies to non-unitary deformations of typical random CMV type matrices addressed in the literature, which corresponds in this richer framework to the analyses of the non self-adjoint Anderson or Feinberg-Zee models addressed e.g. in \cite{D1, D2, CD}.
\subsection{Main results}
The non-unitary operators $T_\omega$ addressed here are random operators on the Hilbert space $l^2({\mathbb Z})$ with the following structure: In the canonical basis of $l^2({\mathbb Z})$, denoted by $\{e_j\}_{j\in {\mathbb Z}}$, $T_\omega$ is defined as the infinite matrix
\begin{equation}\label{matrixt1}
T_\omega=\begin{pmatrix}
\ddots & e^{i\omega_{2j-1}}\gamma &e^{i\omega_{2j-1}} \delta & & & \cr
&0 &0 & & &\cr
&0 &0 & e^{i\omega_{2j+1}}\gamma &e^{i\omega_{2j+1}}\delta & \cr
& e^{i\omega_{2j+2}}\alpha &e^{i\omega_{2j+2}}\beta & 0& 0 & \cr
& & & 0 &0 & \cr
& & & e^{i\omega_{2j+4}}\alpha &e^{i\omega_{2j+4}}\beta & \ddots
\end{pmatrix},
\end{equation}
where the dots mark the main diagonal and the first column is the image of the vector $e_{2j}$. The phases $\{e^{i\omega_j}\}_{j\in {\mathbb Z}}$ are iid random variables and the deterministic coefficients, when arranged in a matrix $C_0\in M_2({\mathbb C})$, are constrained by the requirement that $C_0$ be a projection on ${\mathbb C}^2$ of a unitary matrix on ${\mathbb C}^3$:
\begin{equation}\label{consti}
C_0=\begin{pmatrix} \alpha & \beta \cr \gamma & \delta
\end{pmatrix} \ \mbox{s.t.}\ \tilde C= \begin{pmatrix}
\alpha & r & \beta \cr
q & g & s \cr
\gamma & t & \delta
\end{pmatrix} \in U(3), \ \mbox{with $0\leq g \leq 1$.}
\end{equation}
When $C_0$ itself is unitary, which corresponds to $g=1$, $T_\omega$ is a unitary random CMV matrix describing a random quantum walk, the spectral properties of which are known, see \cite{JM, ASWe}. In general, however, $C_0$ is a contraction, and $T_\omega$ is a non-normal contraction, i.e. a non-unitary operator. We note here that, in general, $T_\omega$ is not a seminormal operator, i.e. $[T_\omega^*,T_\omega]$ is not definite, see \cite{C}.
Non-unitary operators $T_\omega$ constrained by condition (\ref{consti}) appear as a natural objects in the study of the spectral properties of random quantum walks defined on the lattice ${\mathbb Z}^2$ or on $\cT_4$, the homogeneous tree of coordination number $4$, as explained in Section \ref{qwnuo}. This provides us with an independent motivation to focus on the characterization (\ref{consti}) here, although other choices of deformations of CMV matrices are obviously possible.
Actually, Section \ref{sextension} shows that our spectral results extend to operators of the form $\widetilde T_\omega$ defined in the same basis as that used for (\ref{matrixt1}) by the random infinite matrix
\begin{eqnarray}\label{matrixtt}
&&\widetilde T_\omega= \\ \nonumber
&& \begin{pmatrix}
\ddots & e^{i(\omega_{4j-1}+\omega_{4j-3})}\gamma\delta &e^{i(\omega_{4j-1}+\omega_{4j-3})}\delta^2 & & & \cr
& e^{i(\omega_{4j-1}+\omega_{4j})}\gamma\beta &e^{i(\omega_{4j-1}+\omega_{4j})}\delta\beta & & &\cr
&e^{i(\omega_{4j+1}+\omega_{4j+2})}\gamma\alpha &e^{i(\omega_{4j+1}+\omega_{4j+2})}\gamma\beta & e^{i(\omega_{4j+3}+\omega_{4j+1})}\gamma\delta &e^{i(\omega_{4j+3}+\omega_{4j+1})}\delta^2 & \cr
& e^{i(\omega_{4j+2}+\omega_{4j+4})}\alpha^2 &e^{i(\omega_{4j+2}+\omega_{4j+4})}\alpha\beta & e^{i(\omega_{4j+3}+\omega_{4j+4})}\gamma\beta &e^{i(\omega_{4j+3}+\omega_{4j+4})}\delta\beta & \cr
& & & e^{i(\omega_{4j+5}+\omega_{4j+6})}\gamma\alpha &e^{i(\omega_{4j+1}+\omega_{4j+2})}\gamma\beta & \cr
& & & e^{i(\omega_{4j+6}+\omega_{4j+8})}\alpha^2 &e^{i(\omega_{4j+6}+\omega_{4j+8})}\alpha\beta & \ddots
\end{pmatrix},
\end{eqnarray}
with entries characterised by (\ref{consti}). When $g=1$, the CMV type random operator $\widetilde T_\omega$ is unitary. The extension of our spectral analysis to the non-unitary deformation $\widetilde T_\omega$ is provided by the identity $\sigma (\widetilde T_\omega)=\sigma(T^2_\omega)$ and the spectral mapping theorem.
Our main spectral results about $T_\omega$ read as follows. After dealing with some special cases and with the translation invariant situation where $e^{i\omega_j}=1$, $j\in {\mathbb Z}$, we show in Theorem \ref{t1} that the polar decomposition of $T_\omega=V_\omega K$ has the following structure: the isometric part $V_\omega$ is actually unitary and has the same matrix structure as $T_\omega$, i.e. $V_\omega$ a one dimensional random quantum walk. Moreover, the self-adjoint part $K$ is deterministic with spectrum consisting in two infinitely degenerate eigenvalues $\{g,1\}$ only. One consequence of this fact is that $T_\omega$ is a completely non-unitary contraction operator for $g<1$, so that the random quantum walk operator it comes from has no singular spectrum, see Proposition \ref{Hsing}. This special structure also allows us to get informations on the spectrum of $T_\omega$ in terms of properties on $\sigma(V_\omega)$ and $\sigma(K)$, by applying a general result stated as Theorem \ref{gensigvk} and Corollary \ref{simplif}. This result
determines parts of the resolvent set of a bounded operator of the form $T=AB$ with $A$, $B$ bounded, invertible and normal, in terms of the spectra of $A$ and $B$. A direct consequence is that the disc of radius $g>0$ centered at 0 is always contained in the resolvent set of $T_\omega=V_\omega K$ and, when $V_\omega$ contains a gap in its spectrum, other non-trivial explicitly determined sets also belong to $\rho(T_\omega)$, see Lemmas \ref{form} and \ref{form2}.
Then, we take advantage of the fact that the two spectral projectors of $K$ induce a natural bloc structure for $T_\omega$ which suggests the use of the Schur-Feshbach map. It turns out the blocs of the decomposition of $V_\omega$ are tridiagonal operators. This fact allows us to provide conditions on the parameter $g\in ]0, 1[$ in Theorem \ref{suff} which ensure that the spectrum of $T_\omega$ is contained in a centered ring with inner radius $g$ and outer radius strictly smaller than one. It also allows us to show in Lemma \ref{lcnu} that the circles of radii $1$ and $g$ cannot support any eigenvalues of $T_\omega$. These results are deterministic, but we further show that they hold for any realization of the random phases $\{e^{i\omega_j}\}_{j\in {\mathbb Z}}$. Finally, we take a closer look at the case $g=0$, the farthest to the unitary case, in some sense. Assuming the random phases are uniformly distributed and making use of ergodicity, we show that the almost sure spectrum of $T_\omega$ consists in the origin and a centered ring whose inner and outer radii we determine. Also, in case the peripheral spectrum of $T_\omega$ coincides with the unit circle, we get that it contains no eigenvalue, whereas the spectrum of $V_\omega$ is pure point, and that of the corresponding random quantum walk operator is absolutely continuous, see Proposition \ref{g=0}.
The rest of the paper is organized as follows. Section \ref{rqw} provides a short summary of the relevant informations needed to make connection between the non-unitary operators $T_\omega$ considered in this paper and random quantum walks on $\cT_4$ and ${\mathbb Z}^2$. The link is made explicit in Section \ref{qwnuo}. The spectral properties of non-unitary operators is developed in the following two sections, together with the consequences which can be drawn for the random quantum walks they are related to and the explicit link between $T_\omega$ and $\widetilde T_\omega$. The last section is devoted to the case $g=0$.
{\bf Acknowledgments } This work was supported in part by the French Government through a fellowship granted by the French Embassy in Egypt ( Institut Francais d'Egypte). E. H. thanks Universit\'e Grenoble-1 and the Institut Fourier where this project was started, for support and hospitality. A.J. would like to thank J. Asch, Th. Gallay and S. Nonnenmacher for useful discussions.
\section{Random Quantum Walks on ${\mathbb Z}^2$ and $\cT_4$}\label{rqw}
We provide here the basics on simple random quantum walks defined on the lattice ${\mathbb Z}^2$ and the homogeneous tree $\cT_4$, of coordination number $4$. Such quantum walks naturally depend on a $U(4)$-matrix valued parameter $C$ which drives the walk and monitors the effects of the disorder at the same time. In the next section, we focus on certain families of matrix valued parameters of interest which directly lead to the non-unitary operators $T_\omega$ considered in this paper. We also explain the consequences of our analysis of $T_\omega$ for the corresponding random quantum walks.
For more about random quantum walks and their spectral properties, we refer the reader to the reviews \cite{Ko, V-A, J4} and papers \cite{bhj, HJS, JM, J3, HJ} and references therein.
We describe random quantum walks on the graph $\cT_4$ only according to \cite{HJ}, and will simply mention the occasional changes necessary to deal with the lattice case, as in \cite{J3}.
\subsection{Random quantum walks on $\cT_4$}
Let $\cT_4$ be a homogeneous tree of degree $4$, that we will consider as the tree of the free group generated by
$
A_4=\{a,b, a^{-1}, b^{-1}\},
$
with $a a^{-1}=a^{-1}a=e=b b^{-1}=b^{-1}b$, $e$ being the identity element of the group; see Figure (\ref{tree4}). We choose a vertex of $\cT_4$ to be the root of the tree, denoted by $e$. Each vertex $x=x_{1}x_{2}\dots x_{n}$, $n\in{\mathbb N}$ of $\cT_4$ is a reduced word of finitely many letters from the alphabet $A_4$ and an edge of $\cT_4$ is a pair of vertices $(x,y)$ such that $xy^{-1}\in A_4$.
The number of nearest neighbors of any vertex is thus $4$ and any pair of vertices $x$ and $y$ can be joined by a unique set of edges, or path in $\cT_4$.
\begin{figure}[htbp]
\begin{center}
\includegraphics[scale=.4]{spiral4.eps}
\end{center}\vspace{-.5cm}
\caption{\footnotesize construction of $\cT_4$}\label{tree4}
\end{figure}
We identify $\cT_4$ with its set of vertices, and define the configuration Hilbert space of the walker by
$
l^2(\cT_4)=\Big\{\psi=\sum_{x\in \cT_4}\psi_x |x\rangle \ \mbox{s.t.} \ \psi_x\in {\mathbb C}, \ \sum_{x\in \cT_4}|\psi_x|^2<\infty\Big\},
$
where $|x\rangle$ denotes the element of the canonical basis of $l^2(\cT_4)$ which sits at vertex $x$.
The coin Hilbert space (or spin Hilbert space) of the quantum walker on $\cT_4$ is ${\mathbb C}^4$. The elements of the ordered canonical basis of ${\mathbb C}^4$ are labelled by the letters of the alphabet $A_4$ as $\{|a\rangle, |b\rangle, |a^{-1}\rangle, |b^{-1}\rangle\}$.
The total Hilbert space is
\begin{equation}\label{canbas}
\cK = l^2(\cT_4)\otimes {\mathbb C}^4 \ \mbox{ with canonical basis }\ \big\{x\otimes \tau\equiv |x\rangle\otimes |\tau\rangle ,\ \ x\in \cT_4, \tau\in A_4\big\}.
\end{equation}
The quantum walk on the tree is characterized by the dynamics defined as the composition of a unitary update of the coin (or spin) variables in ${\mathbb C}^4$ followed by a coin (or spin) state dependent shift on the tree.
Let $C\in U(4)$, $U(4)$ denoting the set of $4\times 4$ unitary matrices on ${\mathbb C}^4$. The unitary update operator given by ${\mathbb I}\otimes C$ acts on the canonical basis of $\cK$ as
\begin{equation}\label{reshuffle}
({\mathbb I}\otimes C) x\otimes \tau=|x\rangle\otimes C|\tau\rangle=\sum_{\tau'\in A_4} C_{\tau'\tau}\, x\otimes \tau',
\end{equation}
where $\{C_{\tau'\tau}\}_{(\tau',\tau)\in A_4^2}$ denote the matrix elements of $C$.
The coin state dependent shift $S$ on $\cK$ is defined by
\begin{eqnarray}\label{dirsumshift}
S={\sum}_{\tau\in A_4} S_\tau\otimes |\tau\rangle\bra \tau| ,
\end{eqnarray}
where for all $\tau\in A_4$ the unitary operator $S_\tau$ is a shift that acts on $l^2(\cT_4)$ as
$
S_\tau|x\rangle=|x\tau\rangle, \forall x\in \cT_4
$,
with $S_\tau^{-1}=S_\tau^*=S_{\tau^{-1}}$.
A quantum walk on $\cT_4$ is then defined as the one step unitary evolution operator on $\cK=l^2(\cT_4)\otimes {\mathbb C}^4$ given by
\begin{equation}\label{defeven}
U(C)=S({\mathbb I}\otimes C)
={\sum_{\tau\in A_4\atop x\in \cT_4} |x\tau\rangle\bra x |\otimes |\tau\rangle\bra \tau| C},
\end{equation}
where $C\in U(4)$ is a parameter.
A random quantum walk is defined via the following natural generalization. Let $\cC=\{C(x)\in U(4)\}_{x\in\cT_4}$ be a family of coin matrices indexed by the vertices $x\in\cT_4$. A quantum walk with site dependent coin matrices is defined by
\begin{eqnarray}\label{explicitsitedep}
U(\cC)&=&\sum_{\tau\in A_4,
x\in \cT_4} |x\tau\rangle\bra x |\otimes |\tau\rangle\bra \tau| C(x).
\end{eqnarray}
Consider $\Omega={\mathbb T}^{\cT_4\times A_4}$, ${\mathbb T}={\mathbb R}/2\pi {\mathbb Z}$ the torus, as a probability space with $\sigma$ algebra generated by the cylinder sets and measure $\P=\otimes_{x\in\cT_4\atop \tau\in A_4}d\nu$ where $d\nu(\theta)=l(\theta)d\theta$, $l\in L^\infty({\mathbb T})$, is a probability measure on ${\mathbb T}$.
Let $\{\omega^\tau_x\}_{x\in \cT_4, \tau\in A_4 }$ be a set of i.i.d. random variables on the torus ${\mathbb T}$ with common distribution $d\nu$. We will note $\Omega\ni \omega=\{\omega^\tau_x\}_{x\in \cT_4, \tau\in A_4 }$.
Our random quantum walks are constructed by means of the following families of site dependent random coin matrices: Let $\cC_\omega=\{C_\omega(x)\in U(4)\}_{x\in \cT_4}$ be the collection of random coin matrices depending on a fixed matrix $C\in U(4)$, where, for each $x\in \cT_4$, $C_\omega(x)$ is defined by its matrix elements
$
C_\omega(x)_{\tau \tau'}=e^{i\omega^{\tau}_{x\tau}}C_{\tau\tau'}, \ \ \tau,\tau'\in A_4^2.
$
The site dependence appears in the random phases only of the matrices $C_\omega(x)$, which have a fixed skeleton $C\in U(4)$. We consider random quantum walks
defined by the operator
\begin{equation}
U_\omega(C):=U(\cC_\omega) \ \mbox{on} \ \cK=l^2(\cT_4)\otimes {\mathbb C}^4
\end{equation}
depending on $C\in U(4)$.
Defining a random diagonal unitary operator on $\cK$ by
\begin{equation}\label{dd}
{\mathbb D}_\omega x\otimes \tau = e^{i\omega^\tau_x} x\otimes \tau, \ \ \forall (x,\tau)\in \cT_4\times A_4,
\end{equation}
we get that $U_\omega(C)$ is manifestly unitary thanks to the identity
\begin{equation}\label{iddef}
U_\omega(C)={\mathbb D}_\omega U(C) \ \ \mbox{on } \cK.
\end{equation}
\subsection{Random quantum walks on ${\mathbb Z}^2$}
The definition of a random quantum walk
of the same type on ${\mathbb Z}^2$ instead of $\cT_4$ is the same, {\em mutatis mutandis}: the sites $x\in\cT_4$ are replaced by $x\in{\mathbb Z}^2$ so that the configuration space $l^2(\cT_4)$ is replaced by $l^2({\mathbb Z}^2)$ but the coin space remains ${\mathbb C}^4$ in the definition of $\cK$. Thus the update operator ${\mathbb I}\otimes C$ is the same on $l^2({\mathbb Z}^2)\otimes {\mathbb C}^4$ and on $l^2(\cT_4)\otimes {\mathbb C}^4$. Only the definition of the shifts $S_\tau$ in $S=\sum_{\tau\in A_4}S_\tau \otimes |\tau\rangle\bra\tau|$, see (\ref{dirsumshift}), needs to be slightly changed.
We associate the letters $\tau$ of the alphabet $A_4$ with the canonical basis vectors $\{e_1, e_2\}$ of ${\mathbb R}^2$ as follows
$
a \leftrightarrow e_1,\ a^{-1} \leftrightarrow -e_1, \ b \leftrightarrow e_2,\ b^{-1} \leftrightarrow -e_2
$
and define the action of $S_\tau$ on $l^2({\mathbb Z}^2)$ accordingly: for any $x=(x_1,x_2)\in {\mathbb Z}^2$,
$
S_a|x\rangle=|x+e_1\rangle, \ S_{a^{-1}}|x\rangle=|x-e_1\rangle, \ S_{b}|x\rangle=|x+e_2\rangle,\ S_{b^{-1}}|x\rangle=|x-e_2\rangle.
$
The random quantum walk is then defined by $U_\omega(C)$, as in (\ref{iddef}).
\begin{rem}
All the results concerning $U_\omega(C)$ proven below for random quantum walks defined on $\cT_4$ hold for walks defined on ${\mathbb Z}^2$ as well, with the adaptations given above.
\end{rem}
\subsection{Spectral Criteria}
The main issue about random quantum walks concerns the long time behavior of the discrete random unitary dynamical system on the Hilbert space $\cK$ they give rise to by iteration of $U_\omega(C)$. The resulting dynamics is related to the spectral properties of $U_\omega(C)$ studied in the papers \cite{HJS, JM, ASWe, J3, HJ} on ${\mathbb Z}^d$ and $\cT_d$, as a function of $d\in {\mathbb N}$ and of the unitary matrix valued parameter $C$. We recall here well known spectral criteria which make a direct link between random quantum walks $U_\omega(C)$ on $\cT_4$ and ${\mathbb Z}^2$ and $T_\omega$ defined in (\ref{matrixt1}).
For a unitary operator $U$ on a separable Hilbert space $\cH$, the spectral measure $d\mu_\phi$ on the torus ${\mathbb T}$ associated with a normalized vector $\phi\in \cH$ decomposes as
$
d\mu_\phi=d\mu^{p}_\phi+d\mu^{ac}_\phi+d\mu^{sc}_\phi
$
into its pure point, absolutely continuous and singular continuous components. The corresponding orthogonal spectral subspaces are denoted by $\cH^{\#}(U)$, with $\#\in\{p, ac, sc\}$.
Then, see {\em e.g.} \cite{RS}, Wiener or RAGE Theorem relates the autocorrelation function ${n\mapsto\bra \phi | U^n \phi \rangle}$ to the spectral properties of $U$:
\begin{equation}
\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{n=0}^{N}|\bra \phi | U^n \phi\rangle|^2=\sum_{\theta\in{\mathbb T}}(\mu^{p}_\phi\{\theta\})^2,
\end{equation}
whereas the absolutely continuous spectral subspace of $U$, $\cH^{ac}(U)$, is given by
\begin{equation}\label{critac}
\cH^{ac}(U)
=\overline{\Big\{\phi \ | \ \sum_{n\in {\mathbb N}}|\bra \phi | U^n \phi\rangle|^2<\infty \Big\}}.
\end{equation}
For example, consider $U(C)$ on $\cK$ given by (\ref{defeven}). For any $C\in U(4)$, $\bra x\otimes\tau | U(C)^{2n+1} x\otimes~\tau \rangle=0$ for any $n\in{\mathbb Z}$ and $x\otimes \tau \in \cK$, because $U(C)$ is off-diagonal.
Moreover, if $C={\mathbb I}$, $S=U({\mathbb I})$ further satisfies $\bra x\otimes \tau | S^{2n} x\otimes\tau \rangle=\delta_{0,n}$, for all $x\otimes \tau \in \cK$, so that $d\mu_{x\otimes\tau}=\frac{d\theta}{2\pi}$ and $\sigma(S)=\sigma_{ac}(S)={\mathbb S}$, the whole unit circle. The same holds for $U({\mathbb I})=S$ defined on ${\mathbb Z}^2$.
\section{Quantum Walks and Non-Unitary Operators}\label{qwnuo}
We consider here random quantum walks on $\cT_4$ characterized by coin matrices $C$ with a diagonal element of modulus one. As explained below, the non-trivial part of the dynamics they give rise to induces a systematic drift in one space direction. In other words, the dynamics induces a leakage of the wave vectors in one direction that is associated with a purely absolutely continuous part of spectrum of the corresponding evolution operator.
We approach this spectral question by analysing the restriction of $U_\omega(C)$ to a one-dimensional subspace that defines the random contractions $T_\omega$ we study in this paper. The consequences for such quantum walks of our results about the contractions $T_\omega$, namely the proof that the evolution operator is purely absolutely continuous for all realisations of the disorder, are spelled out in Lemma \ref{lemspr} and Proposition \ref{Hsing}.
Finally, we note that from the perspective of the determination of the spectral phase diagram for random quantum walks on $\cT_4$, the corresponding set of coins matrices is not covered by the work \cite{HJ}.
Without loss, we assume that the coin matrix $C$ with a diagonal element of modulus one takes the following form in the ordered basis $\{|a\rangle, |b\rangle,|a^{-1}\rangle,|b^{-1}\rangle\}$,
\begin{equation}\label{lcm}
C=\begin{pmatrix}
\alpha & r & \beta & 0 \cr
q & g & s & 0 \cr
\gamma & t & \delta & 0 \cr
0 & 0 & 0 & e^{i\theta} \cr
\end{pmatrix}\equiv
\begin{pmatrix} \tilde C & \vec 0 \cr \vec 0 ^T & e^{i\theta}\end{pmatrix}\in U(4),
\ \ \mbox{where}\ \
\tilde C= \begin{pmatrix}
\alpha & r & \beta \cr
q & g & s \cr
\gamma & t & \delta
\end{pmatrix}\in U(3),
\end{equation}
with $\theta\in {\mathbb T}$ and $1\geq g\geq 0$.
The assumption $g\geq 0$ always holds at the price of a multiplication of $C$, and thus of $U_\omega(C)$, by a global phase which does not affect the spectral properties.
By construction, $U_\omega(C)$ admits $\cK^{b^{-1}}$, the subspace characterized by a coin variable equal to $|b^{-1}\rangle$, as an invariant subspace on which it acts as the shift $S_{b^{-1}}$, up to phases. Hence
\begin{equation}
\sigma\left(U_\omega(C)|_{\cK^{b^{-1}}}\right)=\sigma_{ac}\left(U_\omega(C)|_{\cK^{b^{-1}}}\right)={\mathbb S}.
\end{equation}
Let $\cK^{\perp}$ be the complementary invariant subspace
\begin{equation}
\cK^{\perp}=\overline{\mbox{span }}\Big\{x\otimes \tau \ | \ x\in \cT_4, \tau\in \{a,b,a^{-1}\}\Big\},
\end{equation}
where the notation $\overline{\mbox{span }}$ means the closure of the span of vectors considered.
On $\cK^{\perp}$ the action of $U_\omega(C)$ on the quantum walker makes it move horizontally back and forth, but it only makes it go up vertically, see Figure (\ref{tree4}). In a sense, the dynamics induces a leakage of the vectors in the direction corresponding to the coin state $|b\rangle$.
In order to assess that $U_\omega(C)|_{\cK^{\perp}}$ has purely absolutely continuous spectrum, an application of criterion (\ref{critac}) leads us to consider $\bra \psi | U_\omega(C)^n\psi\rangle$, $n\geq 0$, with normalized vector $\psi\in\cK^{\perp}$. Note that by construction, for all $x\in \cT_4$, all $\tau\in \{a,b,a^{-1}\}$
\begin{equation}\label{escape}
\bra x\otimes b | U_\omega(C)^n x\otimes \tau\rangle = \delta_{n,0}\delta_{b,\tau}, \ \ \forall \ n\in {\mathbb N}, \forall \ x\in \cT_4.
\end{equation}
In particular, all spectral measures $d\mu_{x\otimes b}(\theta)=\frac{d\theta}{2\pi}$ on ${\mathbb T}$ and $\sigma\left(U_\omega(C)|_{\cK^{\perp}}\right)={\mathbb S}$ as well. We thus have,
\begin{equation}\label{hbdef}
\cH_b=\overline{\mbox{span }}\Big\{x\otimes \tau \ | \ x\in \cT_4, \tau\in \{b,b^{-1}\}\Big\}\subset \cH^{ac}(U_\omega(C)).
\end{equation}
\subsection{Reduction to One Space Dimension}
To this end we introduce the horizontal subspace associated with the direction $a$
\begin{equation}\label{p0}
\cH_0=\overline{\mbox{span }}\Big\{x\otimes \tau \ | \ x=a^m\in \cT_4, m\in{\mathbb Z}, \tau\in \{a,a^{-1}\}\Big\}\subset \cK^\perp\subset \cK,
\end{equation}
and
$P_0: \cK\rightarrow \cK$, the orthogonal projector onto $\cH_0$. All vectors in this subspace live on the horizontal one dimensional lattice passing through the root of $\cT_4$. We can actually consider vectors on any other horizontal one dimensional lattice by attaching $\cH_0$ to any other vertex.
To study $P_0U_\omega(C)^nP_0$, $n\geq 0$ we first note the following simple lemma which allows us to focus on the restriction of $U_\omega(C)$ to $\cH_0$.
\begin{lem}\label{contraction} Let $T_\omega: \cH_0\rightarrow \cH_0$ be defined by
$
T_\omega=P_0U_\omega(C)P_0|_{\cH_0}
$ and $T=T_\omega|_{\omega=(\cdots, 0, 0, 0,\cdots)}$.
Then, $T_\omega$ is a contraction,
\begin{equation}\label{deft}
T_\omega={\mathbb D}^0_\omega T,\ \mbox{where } \ {\mathbb D}^0_\omega=\mbox{\em diag }(e^{i\omega_x^\tau}),
\end{equation}
is the restriction of (\ref{dd}) to $\cH_0$,
and, for any $n\in {\mathbb N}$,
$
P_0U_\omega(C)^nP_0|_{\cH_0}=T_\omega^n.
$
\end{lem}
\begin{proof} First, we have $\|T_\omega\|=\|P_0U_\omega(C)P_0\|\leq 1$ and $[{\mathbb D}_\omega,P_0]=0$ proves the second statement.
Set $Q_0={\mathbb I} -P_0$ and let us show that for all $k\geq 1$, $P_0U_\omega(C)^kQ_0U_\omega(C)P_0=0$. Indeed, for any basis vector $x\otimes\tau$ of $\cH_0$, $Q_0U_\omega(C)x\otimes\tau$ is proportional to $xb\otimes b$, where $xb\neq a^m$, for all $m\in {\mathbb Z}$. Consequently, $P_0U_\omega(C)^kxb\otimes b=0$, for any $k\geq 1$, which yields the result. \hfill {\vrule height 10pt width 8pt depth 0pt}
\end{proof}
\begin{rems}\label{tbloc}
i) The contraction $T$ can be written according to (\ref{defeven}) as
\begin{equation}\label{quasiwalk}
T=S({\mathbb I}\otimes C_{0})=S_a\otimes |a\rangle\bra a| C_{0}+S_{a^{-1}}\otimes |a^{-1}\rangle\bra a^{-1}| C_{0},
\end{equation}
where $C_0=\Pi_0 C \Pi_0|_{\Pi_0{\mathbb C}^4}$ with $\Pi_0=|a\rangle\bra a|+ |a^{-1}\rangle\bra a^{-1}|$ is a contraction which takes the form
\begin{equation}\label{czero}
C_0=\begin{pmatrix} \alpha & \beta \cr \gamma & \delta
\end{pmatrix} \ \mbox{ in the ordered basis $\{|a\rangle,|a^{-1}\rangle\}$.}
\end{equation}
We will say that $C_0$ characterizes the operator $T$.\\
ii) Such an operator, or its higher dimensional analogs, define {\em contractive quantum walks}.\\
\end{rems}
Since $T_\omega$ is not normal in general, the inequalities
$
\mbox{spr }(T_\omega)\leq \|T_\omega\|\leq 1
$
are not necessarily saturated. Actually, we prove below, Corollary \ref{normt}, that $\|T_\omega\|=1$, so that we need to extract spectral informations about $T_\omega$ in order to get decay as $n\rightarrow \infty$ of the autocorrelation function $|\bra \psi | U_\omega(C)^n \psi\rangle|$, $\psi\in \cH_0$. Hence,
\begin{lem}\label{lemspr} With the notations above,
$
\mbox{\em spr }(T_\omega)<1 \ \Rightarrow \ U_\omega(C) \ \mbox{is purely ac}, \ \forall \omega\in \Omega.
$
\end{lem}
\begin{proof}
If the spectral radius of $T_\omega$ satisfies $\mbox{spr }(T_\omega)<1$, then, for any
$\epsilon>0$ s.t.
$|\ln (\mbox{spr }(T_\omega))|-\epsilon>0$, $\|T^n_\omega\|\leq (\mbox{spr }(T_\omega)e^{\epsilon})^n$, if $n$ is large enough. Thus, for any normalized $\psi\in \cH_0$, we have
$
|\bra \psi | U_\omega^n \psi\rangle| = |\bra \psi | T_\omega^n \psi\rangle|\leq e^{-n(|\ln (\mbox{\scriptsize spr }(T_\omega))|-\epsilon)}, \ \mbox{if $n$ is large enough}.
$
Thus $\cH_0\subset \cH^{ac}(U_\omega)$. Since $\cH_0$ can be attached to any vertex of the tree, we get the result.\hfill {\vrule height 10pt width 8pt depth 0pt}
\end{proof}
\begin{rem} We show below in Proposition \ref{Hsing} that a finer analysis of the structure of $T_\omega$ implies that $U_\omega(C)$ is purely ac for all $\omega$, if $g<1$.\end{rem}
\section{One-Dimensional Contractive Quantum Walk}
We turn to the analysis of the random contractive quantum walk defined by
(\ref{deft}) and (\ref{quasiwalk}) with parameters
\begin{equation}\label{const}
C_0=\begin{pmatrix} \alpha & \beta \cr \gamma & \delta
\end{pmatrix} \ \mbox{s.t.}\ \tilde C= \begin{pmatrix}
\alpha & r & \beta \cr
q & g & s \cr
\gamma & t & \delta
\end{pmatrix}\in U(3) \ \mbox{and}\ 0\leq g \leq 1. \end{equation}
We view this problem as a question of independent interest in the spectral analysis of non self-adjoint or, more adequately in the present context, non-unitary operators.
We start by the following simple property relating $C_0$ to $\tilde C$.
\begin{lem}\label{gmg}
Let $C_0=\begin{pmatrix} \alpha & \beta \cr \gamma & \delta
\end{pmatrix}$ be a contraction on ${\mathbb C}^2$ which is not unitary. Then, there exists $\tilde C\in U(3)$ such that (\ref{const}) holds.
\end{lem}
\begin{proof} By exchanging the basis vectors, we can look for $\tilde C$ in the bloc form $\tilde C=\begin{pmatrix} C_0 & u \cr \bar v^T & g
\end{pmatrix}$, where $u, v$ denote vectors in ${\mathbb C}^2$ and $g\in [0,1]$. Imposing that $\tilde C\in U(3)$, we get,
\begin{eqnarray}
C_0^*C_0&=& {\mathbb I}_{{\mathbb C}^2} - | v\rangle\bra v|, \ \
\|v\|^2=1-g^2, \ \
C_0v=-gu\\ \nonumber
C_0C_0^*&=& {\mathbb I}_{{\mathbb C}^2} - | u\rangle\bra u|, \ \
\|u\|^2=1-g^2, \ \
C_0^*u=-gv.
\end{eqnarray}
It follows that $\sigma(C_0^*C_0)=\{1,g^2\}$, which determines $0\leq g<1$ and the norm of the corresponding eigenvector $v$ of $C_0^*C_0$. If $g\neq 0$, then $u=-C_0v/g$. In case $g=0$, $u$ is a normalized eigenvector of $\ker C_0^*$. \hfill {\vrule height 10pt width 8pt depth 0pt}
\end{proof}
Identifying the subspace $\cH_0$ with $l^2({\mathbb Z})$, we get a representation of $T_\omega$ by a 5-diagonal doubly infinite matrix.
Let $\{e_j\}_{j\in{\mathbb Z}}$, resp. $\{a^m\otimes \tau\}_{m\in {\mathbb Z}}^{\tau\in\{a, a^{-1}\}}$, be the canonical orthonormal basis of $l^2({\mathbb Z})$, resp. $\cH_0$. We map the latter to the former according to the rule
\begin{equation}\label{rule}
e_{2j} = a^j\otimes a, \ \ e_{2j+1} = a^j\otimes a^{-1}, \ \ j\in {\mathbb Z}
\end{equation}
and relabel the random phases $\omega_x^\tau$ accordingly,
so that we can identify $T_\omega$ with the matrix
\begin{equation}\label{matrixt}
T_\omega={\mathbb D}_\omega^0T=\begin{pmatrix}
\ddots & e^{i\omega_{2j-1}}\gamma &e^{i\omega_{2j-1}} \delta & & & \cr
&0 &0 & & &\cr
&0 &0 & e^{i\omega_{2j+1}}\gamma &e^{i\omega_{2j+1}}\delta & \cr
& e^{i\omega_{2j+2}}\alpha &e^{i\omega_{2j+2}}\beta & 0& 0 & \cr
& & & 0 &0 & \cr
& & & e^{i\omega_{2j+4}}\alpha &e^{i\omega_{2j+4}}\beta & \ddots
\end{pmatrix},
\end{equation}
where the dots mark the main diagonal and the first column is the image of the vector $e_{2j}$.
We note three special cases which allow for a complete description of the spectrum of $T_\omega$.
\begin{lem}\label{sc}
If $\alpha=\delta=0$, the subspaces $\mbox{\em span }\{e_{2j+1}, e_{2j+2}\}$ reduce $T_\omega$. We have
\begin{equation}
T_\omega=\oplus_{j\in{\mathbb Z}}T_\omega^{(j)}, \ \mbox{where } \ T_\omega^{(j)}=\begin{pmatrix}
0 & \gamma e^{i\omega_{2j+1}}\cr \beta e^{i\omega_{2j+2}} & 0
\end{pmatrix},\ j\in {\mathbb Z},
\end{equation}
$\sigma(T_\omega)=\cup_{j\in {\mathbb Z}}\{\pm g^{1/2} e^{i\theta/2}e^{i(\omega_{2j+1}+\omega_{2j+2})/2}\}$, and $g=\min{(|\beta|,|\gamma|)}$, $\theta=\arg (\beta\gamma)$.
If $\beta=\gamma=0$, the subspaces $\cH_+=\overline{\mbox{span }}\{e_{2j}\}_{ j\in {\mathbb Z}}$ and $\cH_-=\overline{\mbox{span }}\{e_{2j+1}\}_{ j\in {\mathbb Z}}$ reduce $T_\omega$. We have, with $S_\pm$ the standard shifts on $\cH_\pm$,
\begin{equation}
T_\omega=T^{(+)}_\omega\oplus T^{(-)}_\omega,
\end{equation}
where, $\ T^{(+)}_\omega=T_\omega|_{\cH_+}$ is unitarily equivalent to $|\alpha|S_+$, similarly $\ T^{(+)}_\omega=T_\omega|_{\cH_+}$ is unitarily equivalent to $|\delta|S_-$.
$\sigma(T_\omega)={\mathbb S}\cup g{\mathbb S}$, and $g=\min(|\alpha|, |\delta|)$.
If $g=1$, $T_\omega$ is unitary with $\sigma_c(T_\omega)=\emptyset$, almost surely,
unless $C_0\in U(2)$ is diagonal, in which case $\sigma(T_\omega)=\sigma_{ac}(T_\omega)={\mathbb S}.$
\end{lem}
\begin{proof}
The decompositions of $T_\omega$ under the assumptions made is straightforward. The only point is the determination of the spectral radius when the coefficients are constrained by (\ref{const}). We consider $\alpha=\delta=0$ only, the other case being similar. In such a case (\ref{const}) implies $\bar qs=0$ so that either $q=t=0$ or $s=r=0$. In which case $|\gamma|=1$, or $|\beta|=1$. In the first case, $g^2+|r|^2=1=|r|^2+|\beta|^2$, so that $g=|\beta|=\min(|\beta|, |\gamma|)$. The case $|\beta|=1$ is similar. Finally, the case $g=1$ implies that $C_0$ is unitary, so that $T_\omega$ is a one dimensional random quantum walk, and \cite{JM} applies to yield the result. \hfill {\vrule height 10pt width 8pt depth 0pt}
\end{proof}
\begin{rem}\label{highdim}
Quantum walks of the general form (\ref{iddef}) can be defined on ${\mathbb Z}^d$ or $\cT_{2d}$, with $d\in {\mathbb N}$, using the obvious extension to higher dimensions, see \cite{HJ}. When reduced to a one dimensional lattice of the form $\cH_0$, they give rise to a contractive quantum walk which has the form of a CMV type matrix of the kind (\ref{matrixt}). In general, $U(\cC)$ is not a dilation of the corresponding contractive quantum walk. However, if the quantum walk $U_\omega(C)$ defined on $\cT_{2d}$, say, with coin matrix $C\in U(2d)$
having similar properties as for $d=2$, this property is still true: let us denote the coin states basis by $\{ |a_j\rangle, |a_j^{-1}\rangle\}_{j=1,\dots, d}$ and assume $C |a_j^{-1}\rangle=e^{-i\theta_j}|a_j^{-1}\rangle$, for $j=2,\dots,d$. Consider the subspace $\cH_0$ associated with the direction $a_1$ and $P_0$ the corresponding orthogonal projection onto $\cH_0$; then $U_\omega(C)$ is a dilation of the contraction $T_\omega= P_0 U_\omega(C) P_0$, i.e. Lemma \ref{contraction} holds.
\end{rem}
\subsection{Translation invariant case}
The deterministic, translation invariant case characterized by ${\mathbb D}_\omega={\mathbb I}$, i.e. $T_\omega=T$, is best tackled by Fourier methods. We map $l^2({\mathbb Z})$ unitarily onto $ L^2({\mathbb T};{\mathbb C}^2)$ via the identification
\begin{equation}
\psi=\sum_{j\in{\mathbb Z}}c_j|j\rangle\in l^2({\mathbb Z}) \ \leftrightarrow \ \ f(x)=\begin{pmatrix}f_+(x)\cr f_-(x)\end{pmatrix}\in L^2({\mathbb T};{\mathbb C}^2),
\end{equation} where $f_+(x)=\sum_{j}c_{2j}e^{i2jx}$, $f_-(x)=\sum_{j}c_{2j+1}e^{i(2j+1)x}$, $x\in {\mathbb T}$. Then $T$ is unitarily equivalent on $L^2({\mathbb T};{\mathbb C}^2)$ to the multiplication operator by the analytic matrix valued function
\begin{equation}\label{matt}
T\simeq T(x)=\begin{pmatrix} \alpha e^{i2x} & \beta e^{ix} \cr \gamma e^{-ix} & \delta e^{-i2x}
\end{pmatrix}.
\end{equation}
The following criteria for more symmetries hold true.
\begin{lem}\label{symlem}
i) $T$ is self-adjoint $\Leftrightarrow$ $C_0=\begin{pmatrix} 0 & e^{i\nu} \cr e^{-i\nu} & 0
\end{pmatrix}$, $\nu\in {\mathbb R}$. This implies $g=1$, $T$ is unitary and $\sigma(T)=\{-1,1\}$.\\
ii) $T_\omega$ is unitary $\Leftrightarrow$ $|\det C_0|=\left|\det \begin{pmatrix} \alpha & \beta \cr \gamma & \delta
\end{pmatrix}\right|=1$.
\end{lem}
\begin{proof}
We have $T$ is sef-adjoint if and only if $T(x)$ is self-adjoint for all $x\in {\mathbb T}$, which
together with (\ref{const}) readily implies the first statement. The second statement
is a consequence of the general simple lemma
\begin{lem}\label{lemgen}
Let $W\in M_d({\mathbb C})$ be a contraction. Then, $W$ is unitary $\Leftrightarrow$ $|\det(W)|=1$.
\end{lem}
Indeed, $T_\omega$ is unitary if and only if $T$ is unitary, which is true, see (\ref{quasiwalk}) if and only if $C_0$ is unitary, and the lemma applies to the last matrix valued contraction. \\
\begin{proof}
The direct implication is trivial.
Assume $|\det(W)|=1$ and consider the spectral decomposition
\begin{equation}
W=\sum_{k=1}^m\lambda_kP_k+D_k,
\end{equation}
where $\sigma(W)=\{\lambda_k\}_{1\leq k\leq m}$, and $\{P_k\}_{1\leq k\leq m}$, resp. $\{D_k\}_{1\leq k\leq m}$, are the eigenprojectors, resp. eigennilpotents of $W$. Since $W$ is a contraction the condition on the determinant implies $|\lambda_k|=1$, $k=1, 2, \dots, m$. Moreover, $\|W^n\|\leq 1$ for all $n\geq 0$, so that all eigennilpotents are equal to zero, since
\begin{equation}
W^n=\sum_{k=1}^m\lambda_k^nP_k+\sum_{r=0}^KD_k^r\lambda_k^{n-r}\begin{pmatrix} n \cr r
\end{pmatrix}, \ \ \mbox{$n\geq K$},
\end{equation}
where $K$ is the maximal index of nilpotency of the $D_k's$. Eventually, the general property $\|P_k\|\geq 1$ together with $\sigma(W)\subset {\mathbb S}$ imply that $\|P_k\|=1$ for $W$ to be a contraction, so that $P_k=P_k^*$ for all $k=1, 2, \dots, m$. \hfill {\vrule height 10pt width 8pt depth 0pt}
\end{proof}
\end{proof}
As $T$ is unitarily equivalent to a multiplication operator, its spectrum is readily obtained in the generic case. For all $x\in {\mathbb T}$, consider the eigenvalues of $T(x)$
\begin{equation}\label{evti}
\lambda_{\pm}(x)=\frac12\left(\alpha e^{i2x}+\delta e^{-i2x}\pm\{(\alpha e^{i2x}+\delta e^{-i2x})^2-4(\alpha\delta-\beta\gamma)\}^{1/2}\right).
\end{equation}
Assume that ${\mathbb T}\cap Z=\emptyset$, where $Z=\{x\in {\mathbb C} \ | \ \lambda_-(x)=\lambda_+(x)\}$ is the finite set of exceptional points $T(x)$, see \cite{Ka}. Then, with $P_\pm(x)$ the eigenprojectors of the diagonalizable matrix $T(x)$, we get that $(T-z)^{-1}$ is given for $z\in\rho(T)$ by the multiplication operator
$
R_z(x)=\frac{P_-(x)}{\lambda_-(x)-z}+\frac{P_+(x)}{\lambda_+(x)-z},
$
on $L^2({\mathbb T}; {\mathbb C}^2)$
and $\sigma (T)=\mbox{Ran }\lambda_-\cup \mbox{Ran }\lambda_+$.
\subsection{Polar decomposition of $T_\omega$}
In case the contractive quantum walk $T_\omega$ is random, we cannot use Fourier transform methods to determine $\mbox{spr} (T_\omega)$ but, instead, we resort to the properties of its polar decomposition.
Let us come back to the general case (\ref{matrixt}) and consider the unique decomposition $T_\omega=V_\omega K_\omega$, where $K_\omega$ is a non negative operator on $l^2({\mathbb Z})$ and $V_\omega$ is an isometry on $l^2({\mathbb Z})$. We note that due to (\ref{deft}), $K_\omega$ is independent of the randomness since $T_\omega^*T_\omega = T^*T=K^2$.
\begin{thm}\label{t1}
The contraction $T_\omega$ defined on $l^2({\mathbb Z})$ by (\ref{matrixt}) with the constraint (\ref{const}) admits the polar decomposition $T_\omega=V_\omega K$, where $0\leq K\leq {\mathbb I}$ is given by
\begin{equation}\label{specdeck}
K=P_1+gP_2, \ \ \mbox{with} \ \ \sigma(K)=\sigma_{ess}(K)=\{1,g\} \ \ \mbox{and} \ \ \|K\|=1,
\end{equation}
and with infinite dimensional spectral projectors $P_j$, $j=1,2$ given in (\ref{projk}) below.\\
The isometry $V_\omega$ is unitary on $l^2({\mathbb Z})$ and takes the form $V_\omega={\mathbb D}_\omega^0 V$, with
\begin{equation}\label{matv}
V=\frac{1}{1+g}\begin{pmatrix}
\ddots & \gamma(1+g)-qt &\delta(1+g)-st & & & \cr
&0 &0 & & &\cr
&0 &0 & \gamma(1+g)-qt &\delta(1+g)-st & \cr
& \alpha(1+g)-qr &\beta(1+g)-sr & 0& 0 & \cr
& & & 0 &0 & \cr
& & & \alpha(1+g)-qr &\beta(1+g)-sr & \ddots
\end{pmatrix},
\end{equation}
where the dots mark the main diagonal and the first column is the image of the vector $e_{2j}$.
\end{thm}
\begin{cor}\label{normt} for all $\omega\in \Omega$, $T_\omega$ satisfies:
$
\|T_\omega\|=1 \ \mbox{and } \ T_\omega\, \mbox{is unitary} \Leftrightarrow g=1.
$
\end{cor}
\begin{rems}\label{detg} i) Condition (\ref{const}) implies $g=\left|\det \begin{pmatrix} \alpha & \beta \cr \gamma & \delta
\end{pmatrix}\right|$.\\
ii) The unitary operator $V$ corresponds to a one-dimensional quantum walk with unitary coin matrix $\frac{1}{1+g}\begin{pmatrix} \alpha(1+g)-qr & \beta(1+g)-sr\cr \gamma(1+g)-qt & \delta(1+g)-st \end{pmatrix}$, according to Remark \ref{tbloc}. \\ iii) The random quantum walk $V_\omega$ displays dynamical localization for all values of the parameters in (\ref{const}), unless the coin matrix is diagonal, in which case it is absolutely continuous, see \cite{JM}.\\
iv) When $g=1$, the original random quantum walk characterized by (\ref{lcm}) decouples into one-dimensional problems the solutions of which are known, \cite{JM}. Thus, we assume $0\leq g<1$.\\
v) We have $0\in \sigma(K)$ iff \ $0\in \sigma(T)$, and $\ker \, K=\ker \, T$, since $V$ is unitary.
\end{rems}
The proof of Theorem \ref{t1} entails explicit computations of $K$ and $V_\omega$ which are detailed in the next two propostions.
\begin{prop}\label{k2} Assume $0\leq g<1$. The two-dimensional orthogonal subspaces $\cH^{(k)}=\mbox{span}\{e_{2k}, e_{2k+1}\}$ reduce the operator $K=(T^*T)^{1/2}$ which takes the form
\begin{equation}\label{redk}
K=\bigoplus_{k\in {\mathbb Z}}\kappa_k \ \ \mbox{with respect to } \ \ \cH_0=\bigoplus_{k\in{\mathbb Z}}\cH^{(k)}.
\end{equation}
The bloc $ \kappa_k$ acts in the ordered basis $\{e_{2k}, e_{2k+1}\}$ as
\begin{equation}
\kappa_k= \frac{1}{|q|^2+|s|^2}\begin{pmatrix} g|q|^2+|s|^2 & \bar q s(g-1) \cr q \bar s (g-1) & g|s|^2+|q|^2 \end{pmatrix}, \ \ \forall k\in {\mathbb Z},
\end{equation}
see (\ref{const}).
The spectral decomposition of $\kappa_k$ reads
\begin{equation}
\kappa_k=Q^{(k)}_1+g Q^{(k)}_2, \ \ \mbox{where } \ \ Q^{(k)}_1=\frac{1}{|q|^2+|s|^2}\begin{pmatrix} |s|^2 & -\bar q s \cr -q \bar s & |q|^2 \end{pmatrix}={\mathbb I}_{2}-Q^{(k)}_2.
\end{equation}
\end{prop}
We deduce the spectral decomposition of $K$ given in Theorem \ref{t1} immediately:
\begin{equation}\label{projk}
\sigma(K)=\{1,g\}, \ \ K=P_1+gP_2, \ \ \mbox{where } \ \ P_j=\bigoplus_{k\in {\mathbb Z}} Q_j^{(k)}, \ j=1,2.
\end{equation}
\begin{proof}
A straightforward computation based on definition (\ref{matrixt}) yields
\begin{equation}
K^2=\bigoplus_{k\in {\mathbb Z}}\begin{pmatrix} |\alpha|^2+|\gamma|^2 & \delta \bar \gamma +\beta \bar \alpha \cr \gamma \bar \delta + \alpha \bar \beta & |\beta|^2+|\delta|^2 \end{pmatrix}\equiv \bigoplus_{k\in {\mathbb Z}} \kappa_k^2
\end{equation}
with the decomposition of $\cH_0$ given by (\ref{redk}). Condition (\ref{const}) allows us to rewrite the blocs $\kappa_k^2$ of this decomposition as
\begin{equation}
\kappa_k^2=\begin{pmatrix} 1-|q|^2 & - s \bar q \cr - q \bar s & 1-|s|^2 \end{pmatrix}, \ \ \mbox{ with } \left\{\begin{matrix} \hspace{-.5cm}\det \kappa_k^2=1-(|q|^2+|s|^2)=g^2 \cr \tr \kappa_k^2=2-(|q|^2+|s|^2)=1+g^2.\end{matrix}\right.
\end{equation}
Hence, $\sigma(\kappa_k^2)=\{1, g\}$ with corresponding normalized eigenvectors
\begin{eqnarray}\label{evek}
v^{(k)}_1=\frac{1}{\sqrt{|q|^2+|s|^2}}\begin{pmatrix} s \cr -q \end{pmatrix}, \ \
v^{(k)}_2=\frac{1}{\sqrt{|q|^2+|s|^2}}\begin{pmatrix} \bar q \cr \bar s \end{pmatrix}.
\end{eqnarray}
Explicit computations yield the spectral projectors $Q^{(k)}_1=|v^{(k)}_1\rangle\bra v^{(k)}_1|$ and $Q^{(k)}_2={\mathbb I}_2-Q^{(k)}_2$, and, in turn, $\kappa_k={(\kappa_k^2)}^{1/2}$. The spectral decomposition of $K$ follows immediately.\hfill {\vrule height 10pt width 8pt depth 0pt}
\end{proof}
We now turn to the computation of the isometry $V_\omega={\mathbb D}_\omega^0 V$. Recall that translation invariant operators with the same band structure matrix as $T$ are characterized by a $2\times 2$ matrix, in the same way as $T$ is characterized by $\begin{pmatrix} \alpha & \beta \cr \gamma & \delta
\end{pmatrix}$, see Remark \ref{tbloc}.
\begin{prop}
For $1>g>0$, $V=TK^{-1}$ where $K^{-1}=\bigoplus_{k\in {\mathbb Z}}\kappa_k^{-1}$ and
\begin{equation}
\kappa_k^{-1}=\frac{1}{g(1+g)}\begin{pmatrix}
1-|s|^2+g & s\bar q\cr q \bar s & 1-|q|^2+g
\end{pmatrix}.
\end{equation}
The operator $V$ has the same band structure as $T$ and is characterized by the unitary matrix
\begin{equation}\label{simv}
\begin{pmatrix} \alpha & \beta \cr \gamma & \delta
\end{pmatrix}\kappa_k^{-1}=\frac{1}{1+g}\begin{pmatrix} \alpha(1+g)-qr & \beta(1+g)-sr\cr \gamma(1+g)-qt & \delta(1+g)-st\end{pmatrix}.
\end{equation}
\end{prop}
\begin{rem}
The unitary operator $V$ is well defined in the limit $g\rightarrow 0$, with the constraint (\ref{const}), even though $K^{-1}$ is not.
\end{rem}
\begin{proof} The first statement is a consequence of Proposition \ref{k2} and of the spectral theorem. The invariance of the subspaces $\mbox{span}\{e_{2k}, e_{2k+1}\}$ under $K^{-1}$ and the matrix structure of $T$ imply that $V$ has the same structure as $T$. It is a matter of computation to check statement (\ref{simv}), systematically using constraint (\ref{const}) to simplify the factor $g$ in the denominator.\hfill {\vrule height 10pt width 8pt depth 0pt}
\end{proof}
\subsection{ Structure of the Contraction $T_\omega$}\label{scnu}
Recall that a contraction is said to be completely non-unitary, cnu for short, if it possesses no non-trivial closed invariant subspace on which it is unitary, see {\em e.g.} \cite{SNF}.
\begin{lem}\label{lcnu} Let $0\leq g <1$. Then, for all $\omega\in \Omega$, the operator $T_\omega$ is either cnu or it is unitarily equivalent to the direct sum of a shift and of $g$ times a shift.
Consequently,
\begin{equation}
\sigma_p(T_\omega)\cap {\mathbb S} =\emptyset, \ \ \mbox{and for $0<g<1$,}\ \ \sigma_p(T_\omega)\cap g{\mathbb S}=\emptyset.
\end{equation}
\end{lem}
\begin{proof}
Assume there is a closed subspace ${\mathfrak{h}}_0$ such that $T_\omega|_{{\mathfrak{h}}_0}$ is unitary.
For $\psi\in {\mathfrak{h}}_0$, we have $\|T_\omega\psi\|=\|\psi\|$. This implies with $T_\omega=V_\omega(P_1+gP_2)$, that
\begin{equation}\label{cnu}
({\mathbb I}-T_\omega^*T_\omega)^{1/2}\psi=\sqrt{1-g^2}P_2\psi=0.
\end{equation}
Hence, ${\mathfrak{h}}_0\subset P_1 \cH_0$, and, ${\mathfrak{h}}_0$ being invariant under $T_\omega$, ${\mathfrak{h}}_o\subset \ker P_2 V_\omega P_1.$
The operator $ P_2 V_\omega P_1$ is studied in Lemmas \ref{qlem} and \ref{loffdiag} below, where it is shown that $ \ker P_2V_\omega P_1\neq \{0\} \Leftrightarrow P_2V_\omega P_1=0$ and that this is
equivalent to
\begin{eqnarray}\label{speci} \tilde{C}\in \left\{
\begin{pmatrix}
\alpha & r & 0 \cr
q & g & 0 \cr
0 & 0 & \delta
\end{pmatrix},
\begin{pmatrix}
\alpha & 0 & 0 \cr
0 & g & s \cr
0 & t & \delta
\end{pmatrix}
\right\}\subset U(3).
\end{eqnarray} Hence if (\ref{speci}) doesn't hold, $T_\omega$ is cnu, whereas in case
(\ref{speci}) holds, Lemma \ref{sc} finishes the proof of the first statement. The fact that eigenvalues cannot sit on the unit circle is thus immediate, whereas, for $g>0$, a similar argument applied to the contraction
$
(gT_\omega^{-1})^*=V_\omega(gP_1+P_2)
$
yields the last statement. \hfill {\vrule height 10pt width 8pt depth 0pt}
\end{proof}
\begin{rem}\label{err} The operator $T_\omega$ is cnu if and only if $0\leq g<1$, $|\alpha|<1$ and $|\delta|<1$. Moreover, in case (\ref{speci}) holds, the corresponding random quantum walk operator $U_\omega(C)$ is purely ac by a general argument, see eq. (66) \S 5.4 of \cite{HJ}.
\end{rem}
The fact that $T_\omega$ is completely non-unitary has immediate consequences on the spectrum of $U_\omega(C)$. In particular, the following result extends the description of the spectral diagram discussed in paragraph 5.6 of \cite{HJ}.
\begin{prop}\label{Hsing}
If $0\leq g<1$, then
$
\sigma(U_\omega(C))=\sigma_{ac}(U_\omega(C)),
$
for all $\omega\in \Omega$.
\end{prop}
\begin{proof} We drop the dependence on $\omega$ and $C$ in the notation for this proof, for simplicity.
By Lemma \ref{lcnu}, we can assume $T$ is completely non-unitary. Let $P_{sing}$ be the spectral projection onto the subspace $\cH^{sing}=\cH^{pp}(U)\cap \cH^{sc}(U)$ and recall that $P_0$ is the orthogonal projection onto $\cH_0$. We first show that the subspace
$\cH_0\cap \cH^{sing}$ reduces the operator $U$.
Let $\psi \in \cH_0\cap \cH^{sing}$,
\begin{equation}
U\psi=U P_{sing} \psi=P_{sing} U\psi=P_{sing}\big(P_0 U\psi+({\mathbb I}-P_0)U\psi\big),
\end{equation}
where $({\mathbb I}-P_0)U\psi\in\cH_b$, see (\ref{hbdef}). Using $P_{sing}\cH_b=0$, we get that
$U\psi=P_{sing}P_0U\psi.$
But then $\|U\psi\| \leq \|P_0 U\psi\|\leq \|U\psi\|$ implies $U\psi=P_0U\psi=P_0P_{sing}U\psi$ as well. Hence $\cH_0\cap \cH^{sing}$ is invariant under $U$. By a similar argument, this subspace is invariant under $U^*$ as well. Consequently, $\cH^{sing}$ reduces $T=P_0 U|_{\cH_0}$, which shows that $\cH^{sing}\cap \cH_0=\{0\}$ since $T$ is cnu and $g<1$. Repeating the argument with $\cH_0$ replaced by the horizontal subspace attached to $x\in \cT_4$ arbitrary eventually yields $\cH^{sing}=\{0\}$. \hfill {\vrule height 10pt width 8pt depth 0pt}
\end{proof}
\begin{rem}
In view of Lemma \ref{gmg}, one sees that Lemma \ref{lcnu} and Proposition \ref{Hsing} carry over to the cases described in Remark \ref{highdim}, in case $T_\omega$ is cnu..
\end{rem}
\subsection{ Extensions to Further Contractive Quantum Walks}\label{sextension}
We make use of a symmetry of the contractive quantum walk $T_\omega={\mathbb D}_\omega^0 T$ with $T$ given by (\ref{quasiwalk}) in order relate it to $\widetilde T_\omega$ given by (\ref{matrixtt}).
Let
\begin{eqnarray}
\cH_{\bf e}&=&\overline{\mbox{span}}\{a^m\otimes\tau, \ \ m\in 2{\mathbb Z}, \tau\in\{\pm1\}\}, \nonumber \\
\cH_{\bf o}&=&\overline{\mbox{span}}\{a^m\otimes\tau, \ \ m\in 2{\mathbb Z}+1, \tau\in\{\pm1\}\}
\end{eqnarray}
denote the supplementary subspaces of $\cH_0$ consisting in even and odd sites only in configuration space. The definition (\ref{quasiwalk}) of $T$ makes it clear that $T \cH_{\bf e}\subset \cH_{\bf o}$ and $T \cH_{\bf o}\subset \cH_{\bf e}$, and since ${\mathbb D}^0_\omega$ is diagonal, the same is true for $T_\omega$. Therefore $\cH_{\bf e}$ is invariant under $T^2_\omega$ and by Lemma 2 in \cite{CD}, $\sigma (T^2_\omega)\setminus \{0\}=\sigma(T^2_\omega |_{\cH_{\bf e}})\setminus \{0\}$. Actually we have
\begin{prop}
For all $0\leq g \leq 1$, and with definitions (\ref{matrixt1}) and (\ref{matrixtt}),
\begin{equation} \widetilde T_\omega \simeq T^2_\omega|_{\cH_{\bf e}} \ \Rightarrow \
\sigma(\widetilde T_\omega)=\sigma(T^2_\omega).
\end{equation}
Moreover,
\begin{equation}\label{sfi}
\widetilde T_\omega = \bigoplus_{k\in {\mathbb Z}} S_\omega({2k+1}) \bigoplus_{k\in {\mathbb Z}} S_\omega({2k})
\end{equation}
where, for all $k\in {\mathbb Z}$, we have in the basis $\{e_{2k}, e_{2k+1}\}$, resp. $\{e_{2k+1}, e_{2k+2}\}$
\begin{equation}
S_\omega({2k})=\mbox{\em diag}(e^{i\omega_{4k-1}}, e^{i\omega_{4k+2}})\begin{pmatrix}
\gamma & \delta \cr
\alpha & \beta
\end{pmatrix}, \mbox{\em resp.} \
S_\omega({2k+1})=\mbox{\em diag}(e^{i\omega_{4k+1}}, e^{i\omega_{4k+4}})\begin{pmatrix}
\gamma & \delta \cr
\alpha & \beta
\end{pmatrix}.
\end{equation}
\end{prop}
\begin{proof}
With the convention (\ref{rule}), $\cH_{\bf e}$ is spanned by $\{e_{4k}, e_{4k+1}, \ k\in {\mathbb Z}\}$. Relabelling these basis vectors according to $e_{4k}\mapsto e_{2k}$, $e_{4k+1}\mapsto e_{2k+1}$, explicit computations yield $\widetilde T_\omega \simeq T^2_\omega|_{\cH_{\bf e}}$, as well as (\ref{sfi}). Observe that $ g\neq 0 $ iff $\widetilde T_\omega$ and $T_\omega$ are boundedly invertible and that if $g=0$, we have $0\in \sigma (\widetilde T_\omega)\cap \sigma (T_\omega^2) $. This yields isospectrality of $T_\omega^2$ and $\widetilde T_\omega$. \hfill {\vrule height 10pt width 8pt depth 0pt}
\end{proof}
\begin{rems}
i) The restriction $T^2_\omega|_{\cH_{\bf o}}$ has an explicit form similar to $\widetilde T_\omega$ given by the composition (\ref{sfi}) in the reversed order.\\
ii) In particular, we deduce from the above that $\widetilde T_\omega$ is unitary iff $g=1$, and that it is pure point for $\beta\gamma\neq 0$, whereas it is absolutely continuous if $\beta=\gamma= 0$, \cite{JM}. \\
iii) All the spectral results we derive for $T_\omega$ hold for $\widetilde T_\omega$ via the spectral mapping theorem.
\end{rems}
\section{Spectral Analysis of $T_\omega$}
We use the following notations: $\sigma_p(A)$ denotes the set of eigenvalues of a bounded operator $A$ on $\cH$ and $\sigma_{app}(A)$ denotes its approximate point spectrum. By definition, $\lambda\in \sigma_{app}(A)$ if and only if there exists a sequence of normalized vectors $\{\varphi_n\}_{n\in{\mathbb N}}$ such that $A\varphi_n-\lambda\varphi_n\rightarrow 0$, as $n\rightarrow\infty$. Recall that $\sigma_p(A)\subset \sigma_{app}(A)$ and $\sigma(A)=\sigma_{app}(A)\cup {\overline{\sigma_p(A^*)}}$, where $\overline{X}=\{\bar x ,\ | \ x\in X\}$, for any $X\subset {\mathbb C}$. Also, $\sigma_{app}(A)$ is a nonempty closed set of ${\mathbb C}$ such that $\partial \sigma(A)\subset \sigma_{app}(A)$ and one has the disjoint union $\sigma(A)=\sigma_{app}(A)\cup {\overline{\sigma_{p_1}(A^*)}}$, where ${\sigma_{p_1}(A^*)}=\{\lambda \in {\mathbb C} \ | \ \mbox{s.t.} \ \ker (A^*-\lambda)\neq \{0\}\ \mbox{and}\ \mbox{Ran}(A^*-\lambda)=\cH\}$ is open in ${\mathbb C}$, see \cite{Ku}.
The starting point of analysis of the contraction $T_\omega$ is Theorem 4.4 showing that $T_\omega$ admits a polar decomposition the components of which are bounded normal operators. We are thus naturally lead to the study of spectral properties of products of such operators. The only general result we are aware of in this direction, \cite{W}, provides estimates on the position of the spectrum of such products in terms of the numerical ranges of the components, which is however not strong enough for our purpose. We will use instead
\begin{thm}\label{gensigvk}
Let $T=AB$, where $A$, $B$ are bounded normal operators on $\cH_0$ and let $B_c(r)$ denote the open disc of radius $r>0$ and center $c\in {\mathbb C}$. Then,
\begin{eqnarray} \label{gensetres}
B^{-1}\in \cB(\cH_0)&\Rightarrow & \bigcup_{\tau\in \rho(A)}\bigcap_{b \in \sigma(B)}B_{\tau b}(|b|\; \mbox{\em dist}(\tau, \sigma(A)) )\subset \rho(AB),
\nonumber \\
A^{-1}\in \cB(\cH_0)&\Rightarrow &\bigcup_{\tau\in \rho(B)}\bigcap_{a \in \sigma(A)}B_{\tau a}(|a|\; \mbox{\em dist}(\tau, \sigma(B)) )\subset \rho(AB).
\end{eqnarray}
\end{thm}
\begin{proof}
Under our assumption on $\tau$, and since $B$ is invertible, we have
\begin{equation}
T-z=(A-\tau)B+\tau B-z=(A-\tau)\left({\mathbb I} + (A-\tau)^{-1}(\tau B-z)B^{-1}\right)B,
\end{equation}
which shows that $T-z$ is boundedly invertible if $\|(A-\tau)^{-1}(\tau B-z)B^{-1}\|<1$, thanks to Neumann's series. By the spectral theorem for normal operators applied to the continuous function $x\rightarrow \frac{|\tau x -z|}{|x|}$ defined on the compact set $\sigma (B)$, and using $\|(A-\tau)^{-1}\|=1/\mbox{ dist}(\tau, \sigma(A))$, this condition is met if
\begin{equation}
\max_{b\in \sigma(B)}\frac{|z-\tau b|}{|b|}<\mbox{ dist}(\tau, \sigma(A)).
\end{equation}
Therefore, given $\tau\in \rho(A)$, if $z\in \bigcap_{b\in \sigma(B)}B_{\tau b}(b\, \mbox{dist}(\tau, \sigma(A)) )$, then $z\in\rho(AB)$. Taking the union over $\tau \in \rho(A)$ yields (\ref{gensetres}). The second inclusion is proven analogously, using $A$ invertible and identity for $\tau\in \rho(B)$
\begin{equation}
T-z=A(B-\tau)+\tau A-z=A\left({\mathbb I} + A^{-1}(\tau A-z)(B-\tau)^{-1}\right)(B-\tau).
\end{equation}
\hfill {\vrule height 10pt width 8pt depth 0pt}
\end{proof}
\begin{rem}\label{discrho}
In case $A$ and $B$ have bounded inverses, we get for $\tau=0$ that $B_0(r_{AB})\subset \rho(AB)$, where $r_{AB}=\mbox{\em dist}(0, \sigma(A))\mbox{\em dist}(o, \sigma(B))>0$.
\end{rem}
Applied to our case $T=VK$ with $\sigma(K)=\{g,1\}$, $0< g<1$,
(\ref{gensetres}) simplifies and yields more specific estimates on $\rho(T)$ as a function of the spectrum of the unitary operator $V$.
\begin{cor}\label{simplif} Let $T=VK$ with $V$ unitary and $0<K=(P_1+gP_2)$, $0< g<1$. Then
\begin{eqnarray}\label{setres}
&&\bigcup_{\tau\in \rho(V)}B_{\tau g}(g\, \mbox{\em dist}(\tau, \sigma(V)) )\cap B_{\tau }(\mbox{\em dist}(\tau, \sigma(V)) )\subset \rho(T),\\
&&\label{cor2}
\bigcup_{\tau\in \rho(K)}\bigcap_{v\in \sigma(V)}B_{\tau v}(\mbox{\em dist}(\tau, \sigma(K)) )\subset \rho(T).
\end{eqnarray}
In particular, \vspace{-.5cm}
\begin{equation}
B_0(g)\subset \rho(T). \label{87}
\end{equation}
Moreover, assume the arc $(-\theta,\theta)$ belongs to $\rho(V)$, with $0< \theta <\pi$. Then,
\begin{eqnarray}\label{symset}
&&\bigcup_{\tau\in {\mathbb R}_+ \atop \alpha \in [-\theta, \theta] }B_{e^{i\alpha}\tau}(d_{e^{i\alpha}\tau})\cap B_{ge^{i\alpha}\tau}(gd_{e^{i\alpha}\tau} )
\subset \rho(T),\\ \label{symset2}
&&\bigcup_{\tau\in {\mathbb R}_- \atop \alpha \in [-\pi/2, \pi/2] }\bigcap_{e^{i\nu}\in\sigma(e^{i\alpha}V)}B_{e^{i\nu}\tau}(\delta_{e^{i\alpha}\tau}) \subset \rho(T), \ \ \mbox{where }\\
\label{distalph}
d_{e^{\pm i\alpha}\tau} &=& \mbox{\em dist}(e^{\pm i\alpha}\tau, \sigma(V)) = \sqrt{\tau^2-2\tau\cos(\theta-\alpha)+1} \ \mbox{with } \ \tau>0, \ \alpha \in [0, \theta], \\
\label{deltalph}
\delta_{e^{\pm i\alpha}\tau}&=& \mbox{\em dist}(e^{\pm i\alpha}\tau, \sigma(K))= \sqrt{\tau^2+2|\tau|g\cos(\alpha)+g^2} \ \mbox{with } \ \tau<0, \ \alpha \in [0, \pi/2].
\end{eqnarray}
\end{cor}
\begin{rems}\label{remset} i) The points $\tau\in \rho(V)$ in (\ref{setres}) such that $\mbox{dist}(\tau, \sigma(V))=|1-\tau|$ do not yield more information than (\ref{87}): $\tau<1$ implies $\bigcap_{k\in \sigma(K)}B_{\tau k}(k (1-|\tau|) )\subset B_0(g)$ and $\tau>1$ implies $\bigcap_{k\in \sigma(K)}B_{\tau k}(k (|\tau|-1) )\subset {\mathbb C}\setminus \overline{B_0(1)}$. This is the case when $\sigma(V)={\mathbb S}$.\\
ii) At the expense of a rotation, we can associate to any arc in $\rho(V)$ two sets (\ref{symset}) and (\ref{symset2}) that belong to $\rho(T)$. The corresponding sets are both symmetrical with respect to the bisector of that arc. \\
iii) Lemma \ref{sc} or
Remark \ref{remopt} shows that (\ref{87}) is optimal.
\end{rems}
\begin{proof}
The first statements are mere rewritings of (\ref{gensetres}) and
Remark \ref{discrho} implies (\ref{87}). For (\ref{symset}), we note that $w\in {\mathbb C}$ is such that $\mbox{dist}(w, \sigma(V))=|w-e^{\pm i\theta}|$ if $w=\tau e^{\pm i\alpha}$, with $\alpha \in [0, \theta]$ and $\tau\geq 0$, which establishes (\ref{distalph}). Whereas for (\ref{symset2}), $w=-|\tau|e^{\pm i\alpha}$ with $\alpha \in [0, \pi/2]$ satisfies $\mbox{dist}(w, \sigma(K))=|w-g|=||\tau|e^{\pm i\alpha}+g|$ which yields (\ref{deltalph}). Then a change of variables allows us to express (\ref{cor2}) as (\ref{symset2}) under our assumptions.
\hfill {\vrule height 10pt width 8pt depth 0pt}
\end{proof}
Without attempting to provide a complete analysis, we describe (\ref{symset}) and (\ref{symset2}) in some more details and show that (\ref{symset2}) provides less information in case $\sigma(V)$ displays one gap only. The proofs of the statements are provided in an Appendix. Let $C_c(r)$ denote the circle of center $c\in {\mathbb C}$ and radius $r>0$ and $\partial S$ denote the boundary of a set $S$.
First consider (\ref{symset}) for $\alpha=0$.
Because the intersection of discs can be non-empty when the intersection of their boundary is empty, there is a difference between (\ref{symset}) and the set $D(\theta)$ such that
\begin{equation}\label{deltad}
\partial D(\theta)=\bigcup_{\tau\in {\mathbb R}_+ }C_{\tau}(d_{\tau})\cap C_{g\tau}(gd_{\tau} ),
\end{equation}
and $D(\theta)$ contains the vertical segment between the intersection of two circles. We also set
$R_\gamma(\theta)=\{z\in {\mathbb C} \, | \, \Re z>\gamma\cos(\theta)\}$.
\begin{lem}\label{form} With the notations above, and assuming $\alpha=0$, the LHS of (\ref{symset}) is given by
\begin{equation}\label{set1}
\bigcup_{\tau\in {\mathbb R}_+ }B_{\tau}(d_{\tau})\cap B_{g\tau}(gd_{\tau} )=D(\theta)\cup B_0(g)\cup R_1(\theta), \ \ \mbox{for $\theta\in ]0,\pi/2[$ },
\end{equation}
see Fig. \ref{gaps}, where
$\partial D(\theta)$ is given by the cubic curve
\begin{eqnarray}\label{cubicurve}
y^2&=&\frac{x(x^2-x(1+g)\cos(\theta) +g)}{(1+g)\cos(\theta)-x} \ \mbox{with} \nonumber\\ \label{xtau}
x&=&-\frac{1+g}{2\tau}+(1+g)\cos(\theta)\in [0, (1+g)\cos(\theta)[, \ \ \mbox{for}\ \tau\in [1/(2\cos(\theta)), \infty[.
\end{eqnarray}
For $\pi/2\leq \theta<\pi$,
\begin{equation}\label{set11}
\bigcup_{\tau\in {\mathbb R}_+ }B_{\tau}(d_{\tau})\cap B_{g\tau}(gd_{\tau} )= B_0(g)\cup R_g(\theta).
\end{equation}
Moreover, for fixed $0<\alpha<\theta$, assuming $0<\theta<\pi$, we have
\begin{equation}\label{alphaind}
\bigcup_{\tau\in {\mathbb R}_+ }B_{e^{ i\alpha}\tau}(d_{e^{ i\alpha}\tau})\cap B_{ge^{ i\alpha}\tau}(gd_{e^{ i\alpha}\tau} )\subset\bigcup_{\tau\in {\mathbb R}_+ }B_{\tau}(d_{\tau})\cap B_{g\tau}(gd_{\tau} ).
\end{equation}
\end{lem}
\begin{figure}[htbp]
\begin{center}
\includegraphics[scale=.21]{gapless.eps} \includegraphics[scale=.21]{critical.eps} \includegraphics[scale=.21]{gap.eps}
\end{center}\vspace{-.5cm}
\caption{\footnotesize The sets $D\cup B_0(g)\cup R_1(\theta)$ for $0<\theta<\pi/2$ fixed and increasing values of $g$. The unit circle $\mathbb S$ and $g\mathbb S$ are indicated in red, whereas the black curves denote $\partial D$. The vertical red line corresponds to $\partial R_1(\theta)$.}\label{gaps}
\end{figure}
\begin{rems}\label{alpha1} i) In particular, under our assumptions, the segment $[0,1]\subset \rho(VK)$ if $\cos^2(\theta)<\frac{4g}{(1+g)^2}\in ]0,1[$, see Figure \ref{gaps}.That this condition is necessary in general can be seen on the matrix case
\begin{equation}
V=\begin{pmatrix}e^{i\theta} & 0\cr 0& e^{-i\theta} \end{pmatrix}, \ K=\frac12\begin{pmatrix}1+g & 1-g \cr 1-g & 1+g \end{pmatrix}
\end{equation}
such that $\sigma(VK)=\{\frac12(\cos(\theta)(1+g)\pm\sqrt{\cos^2(\theta)(1+g)^2-4g}) \}\subset {\mathbb R}_+^*$, if $\cos^2(\theta)\geq \frac{4g}{(1+g)^2}$.
\\
ii) The points $0, ge^{i\theta}$ and $e^{i\theta}$ belong to $\partial D(\theta)$ and correspond to the values of $\tau$ given by $1/(2\cos(\theta)), (1+g)/(2\cos(\theta))$ and $(1+g)/(2g\cos(\theta))$ respectively.
\end{rems}
To discuss the set (\ref{symset2}), we need some notations. For $\rho,\rho'>0$, we define, see Figure \ref{gamma},
\begin{equation}
\Gamma_{\rho,\rho'}(\theta)=(B_{-e^{+i\theta}\rho}(\rho+\rho')\cap B_{-e^{-i\theta}\rho}(\rho+\rho')\cap R_{\rho'}(\theta))\cup B_0(\rho').
\end{equation}
\begin{figure}[htbp]
\begin{center}
\includegraphics[scale=.2]{gamma.eps}
\end{center}\vspace{-.5cm}
\caption{\footnotesize The set $\Gamma_{\rho,\rho'}(\theta)$.}\label{gamma}
\end{figure}
where the two discs $B_{-e^{\pm i\theta}\rho}(\rho+\rho')$ tangent to $B_0(\rho')$ at $\rho'e^{\pm i\theta}$.
We prove the following in an Appendix.
\begin{lem}\label{form2} Assume $\sigma(V)=\{e^{i\nu} \ \mbox{s.t.} \ \nu \in [\theta,\pi]\cup[-\pi,-\theta] \}$, with $\theta\in ]0,\pi[$. We have
\begin{equation}\label{set2}
\bigcup_{\tau\in {\mathbb R}_-}\bigcap_{e^{i\nu}\in\sigma(V)}B_{e^{i\nu}\tau}(\delta_{\tau}) = B_0(g)\cup \Delta_g(\theta),
\end{equation}
where $\Delta_g(\theta)$ denotes either the triangle defined by the points $ge^{i\theta}, ge^{-i\theta}, g/\cos(\theta)$ whenever $\theta<\pi/2$, or $\Delta_g(\theta)$ denotes the set delimited by the two non-vertical lines passing by these points and the condition $\Re z\geq g\cos(\theta)$ whenver $\theta\in [\pi/2, \pi[$.
Then, for each $\alpha\in ]0,\pi/2[$ fixed,
\begin{eqnarray}\label{reduct}
\bigcup_{\tau \in {\mathbb R}_-}\bigcap_{e^{i\nu}\in\sigma(V)}B_{e^{i\nu}e^{i\alpha}\tau}(\delta_{e^{i\alpha}\tau}) &=& e^{i\alpha}\bigcup_{|\tau| \in {\mathbb R}_+}\Gamma_{|\tau| , \delta_{\tau e^{ i\alpha}}-|\tau|}(\theta).
\end{eqnarray}
For any $\theta \in [0,\pi[$, and all $\alpha\in ]0,\pi/2[$,
\begin{equation}\label{included}
\bigcup_{\tau \in {\mathbb R}_-}\bigcap_{e^{i\nu}\in\sigma(V)}B_{e^{i\nu}e^{i\alpha}\tau}(\delta_{e^{i\alpha}\tau}) \subset \bigcup_{\tau\in {\mathbb R}_+ }B_{\tau}(d_{\tau})\cap B_{g\tau}(gd_{\tau} ).
\end{equation}
\end{lem}
\begin{exple}\label{ex1}{\em
Let us illustrate the use of Theorem \ref{gensigvk}. Consider
\begin{equation}\label{exple}
\tilde C(\xi,\eta)=
\begin{pmatrix} \cos(\eta) & \cos(\xi)\sin(\eta) &-\sin(\xi)\sin(\eta) \cr
0 &\sin(\xi) & \cos(\xi)\cr
\sin(\eta) & -\cos(\xi)\cos(\eta) & \sin(\xi)\cos(\eta) \end{pmatrix}\in O(3), \ \ \
\xi, \eta\in [0,\pi/2],
\end{equation}
where $(\xi,\eta)$ is restricted to $[0,\pi/2]^2$ for simplicity. We thus compute that
\begin{eqnarray}
T, \ \mbox{resp.} \ V, \ \mbox{is characterized by} \ \begin{pmatrix} \cos(\eta) & -\sin(\eta)\sin(\xi) \cr
\sin(\eta) & \cos(\eta)\sin(\xi) \end{pmatrix},
\mbox{resp.} \ \begin{pmatrix} \cos(\eta) & -\sin(\eta) \cr
\sin(\eta) & \cos(\eta) \end{pmatrix}.
\end{eqnarray}
Moreover, Fourier methods yield
\begin{equation}
\sigma(V)=\{z\in {\mathbb S} \ | \ \arg z \in [\eta, \pi-\eta]\cup [-\pi+\eta, -\eta]\}.
\end{equation}
Assuming the common distribution $d\nu$ of phases has support given by
\begin{equation}\label{suppdnu}
\mbox{supp }d\nu=[-\epsilon, \epsilon], \ \mbox{with}\ \epsilon<\eta,
\end{equation} we have thanks to the general almost sure relation $\sigma(V_\omega)=\sigma(V)e^{i\, \mbox{\scriptsize supp}(d\nu)}$ which holds for products of unitary operators of that sort, see Section 5.1 of \cite{J1}, for example,
\begin{equation}
\sigma(V_\omega)=\{z\in {\mathbb S} \ | \ \arg z \in [\eta-\epsilon, \pi-\eta+\epsilon]\cup [-\pi+\eta-\epsilon, -\eta+\epsilon]\}, \ \mbox{a.s.}
\end{equation}
Hence, Corollary \ref{simplif} applies with $\theta=\eta-\epsilon$ and $g=\sin(\xi)$, and gives rise to two regions of $\rho(T_\omega)$: one described in Lemma \ref{form}, and its symmetric image with respect to the vertical axis. In particular, the spectrum of the corresponding $T_\omega$ is separated into two disjoint parts if
\begin{equation}\label{splits}
\cos^2(\eta-\epsilon)\leq\frac{4\sin(\xi)}{(1+\sin(\xi))^2}.
\end{equation}
}
\end{exple}
Let us continue with some general links between the spectral properties of $T_\omega$ and $U_\omega(C)$.
\begin{lem} \label{kerzero}
Let $U$ be unitary on $\cH$ and $P_0$ be an orthogonal projector. For any $\varphi\in \cH$
\begin{eqnarray}\label{evpu}
UP_0\varphi&=&e^{i\theta}\varphi \Rightarrow \varphi=P_0\varphi \ \mbox{and }\ e^{i\theta}\varphi=U\varphi=P_0UP_0 \varphi,
\\
\label{evpup}
P_0U\varphi&=&e^{i\theta}\varphi \Rightarrow \varphi=P_0\varphi \ \mbox{and }\ e^{i\theta}\varphi=U\varphi=P_0UP_0 \varphi.
\end{eqnarray}
Moreover, writing $Q_0={\mathbb I} -P_0$, we get
\begin{equation}
\ker Q_0UP_0 =\{0\} \Rightarrow \sigma_p(UP_0)\cap {\mathbb S} =\sigma_p(P_0U)\cap {\mathbb S}=\sigma_p(P_0UP_0)\cap {\mathbb S}=\emptyset.
\end{equation}
Furthermore, let $T=P_0U P_0|_{P_0\cH}$. If $e^{i\theta}\in \sigma_{app}(T)\setminus \sigma_p(T)$, then $e^{i\theta}\in \sigma_{app}(U)$.
\end{lem}
\begin{proof} Taking the norm of the left hand side of (\ref{evpu}) yields
$P_0\varphi=\varphi$, $Q_0UP_0\varphi=0$ and the first identities follow. For (\ref{evpup}),
$P_0U\varphi=e^{i\theta}\varphi =P_0e^{i\theta}\varphi$ gives the results directly. Now,
$
P_0U\varphi=e^{i\theta}\varphi \ \Leftrightarrow \ UP_0\psi=e^{i\theta}\psi \ \mbox{where} \ \psi=U\varphi
$
shows with (\ref{evpu}) that (\ref{evpup}) implies $Q_0UP_0\psi=0$. Similarly, $P_0UP_0\varphi=e^{i\theta}\varphi$ implies $Q_0UP_0\varphi=0$. Thus, if $\ker Q_0UP_0 =\{0\} $, we get the absence of eigenvalue of modulus one for $UP_0$, $P_0U$ and $P_0UP_0$.
Finally, let $e^{i\theta}\in \sigma_{app}(T)\setminus \sigma_p(T)$ and $\varphi_n\in P_0\cH$ s.t. $\| \varphi_n \|=1$ and $T\varphi_n-e^{i\theta}\varphi_n\rightarrow 0.$ By assumption,
$
\|U\varphi_n\|^2=\|e^{i\theta}\varphi_n+(P_0U\varphi_n-e^{i\theta}\varphi_n)\|^2+\|Q_0U\varphi_n\|^2,
$
where the parenthesis in the right hand side tends to zero, as $n\rightarrow \infty$. As $U$ is unitary, we have $\lim_{n\rightarrow \infty}Q_0U\varphi_n=0$. Consequently, $e^{i\theta}\in \sigma_{app}(U)$ since
$
U\varphi_n-e^{i\theta}\varphi_n = T\varphi_n-e^{i\theta}\varphi_n+Q_0U\varphi_n\rightarrow 0, \ \mbox{as} \ n\rightarrow \infty.
$
\hfill {\vrule height 10pt width 8pt depth 0pt}
\end{proof}
\begin{rem} i) The same result holds with $T^*$ and $U^*$ in place of $T$ and $U$.\\
ii) If $\ker (Q_0UP_0)=\{0\}$, $\lim_{n\rightarrow \infty}Q_0U\varphi_n=0$ implies that the operator $[Q_0UP_0]^{-1}:{\rm Ran~} Q_0UP_0\subset Q_0\cH\rightarrow P_0\cH$ is not bounded.
\end{rem}
Let us also recall the following properties.
\begin{lem}\label{easyspect} Let $T=V(P_1+gP_2)$ and
$\varphi \in \cH_0$ such that $T \varphi =\lambda \varphi$. Then for all $0<g<1$,
\begin{eqnarray}
|\lambda|=1& \Rightarrow & \begin{matrix} \varphi=P_1\varphi
\ \mbox{and } \ V \varphi=P_1V P_1\varphi=\lambda \varphi \end{matrix},\nonumber \\
|\lambda|=g& \Rightarrow & \begin{matrix} \varphi=P_2\varphi
\ \mbox{and } \ V \varphi =P_2V P_2 \varphi =(\lambda/g) \varphi \end{matrix}.
\end{eqnarray}
Consequently, \vspace{-.80cm}
\begin{eqnarray}
\ker P_2V P_1 =\{0\} & \Rightarrow & \sigma_p(T)\cap {\mathbb S} =\emptyset, \ \mbox{and} \nonumber \\
\ker P_1V P_2 =\{0\} & \Rightarrow & \sigma_p(T)\cap g{\mathbb S} =\emptyset.
\end{eqnarray}
If $g=0$, \vspace{-.80cm}
\begin{eqnarray}
\sigma(T)=\sigma(P_1 V P_1|_{P_1\cH_0})\cup \{0\}.\label{88}
\end{eqnarray}
\end{lem}
\begin{proof} All statements except the last one are consequences of the proof of Lemma \ref{lcnu}.
If $g=0$, $T=V P_1$, so that $\ker \ T=P_2\cH_0$. Statement (\ref{88}) is a consequence of (\ref{scco}) and (\ref{critschur}) in the proof of Theorem \ref{suff} below.
\hfill {\vrule height 10pt width 8pt depth 0pt}
\end{proof}
\begin{rems}
i) Analogous statements hold when $T$ is replaced by $(P_1+gP_2)V=V^*T V$. In particular, the results hold for $T^*$.
\end{rems}
Next, we come back to our random setting and make further use of the structure of $K$ to apply the Feschbach-Schur method in order to obtain conditions on the coefficients of $\tilde C$ (\ref{const}) that ensure that for all realizations $\omega\in \Omega$, $\mbox{spr }(T_\omega)<\|T_\omega\|=1$, in case $g<1$.
\begin{thm} \label{suff} Let $T_\omega=V_\omega(P_1+gP_2)$, where $P_j$ are defined in (\ref{specdeck}) and $0\leq g<1$. Consider $P_jVP_k=V_{jk}$, $j,k\in \{1,2\}$, as operators on $P_k\cH$. If $\|V_{11}\|<1$,
then, for all realizations $\omega\in\Omega$
\begin{equation}\label{nonper}
g<\frac{1-\|V_{11} \|}{\|V_{21} \|\|V_{12} \|+\|V_{22} \|(1-\|V_{11} \|)}\ \ \Rightarrow \ \ \mbox{\em spr }(T_\omega)<1.
\end{equation}
Moreover, the set $\{|z|<g\}\cup\{r(V)<|z|\leq 1\}\subset \rho(T_\omega)$ for all $\omega\in\Omega$, where
\begin{equation}\label{annulus}
r(V)=\frac{1}{2}\left(\|V_{11} \|+g\|V_{22}\| +\sqrt{(\|V_{11} \|-g\|V_{22} \|)^2+4g\|V_{21} \|\|V_{12} \|}\|\right).
\end{equation}
\end{thm}
\begin{rems}
i) The result is deterministic and holds for any operator $T=V(P_1+gP_2)$, where $V$ is unitary and $\{P_j\}_{j=1,2}$ are supplementary orthogonal projectors. \\
ii) In case $V_\omega$ is given by Theorem \ref{t1}, (\ref{nonper}) yields a somehow implicit condition since the norms $\|V_{jk}\|$ depend on $g$, see Lemma \ref{qlem} and Example \ref{ex2} below. \\
iii) Remark \ref{remopt} below shows that $r(V)$ is optimal.\\
iv) This infinite dimensional result is reminiscent of the works \cite{WF, B}, which consider matrices of the form $T_\omega=V_\omega K$ where $V_\omega$ is a unitary, Haar distributed matrix and $K>0$ is given. It is shown under various assumptions that a density of eigenvalues of $T_\omega$ can be defined, which is supported in a deterministic ring.
\end{rems}
\begin{proof} It is enough to prove the second statement. We start with the deterministic case.
Given $K=P_1+gP_2$, we split $\cH_0$ as $\cH_0=\cH_1\bigoplus \cH_2$ where $\cH_j=P_j\cH_0$. Writing $T=VK$ as a bloc structure according to this decomposition, we have for any $z\in {\mathbb C}$
\begin{equation}
T-z{\mathbb I}=\begin{pmatrix}V_{11} -z{\mathbb I}_1& g V_{12} \cr
V_{21} & g V_{22}-z{\mathbb I}_2
\end{pmatrix},
\end{equation}
where ${\mathbb I}_j=P_j|_{\cH_j}$ is the identity operator in $\cH_j$ and $V_{jk}=P_jVP_k$ are understood as operators from $\cH_k$ to $\cH_j$, $j,k\in\{1,2\}$.
For any $z\in \rho(g V_{22})$, we consider the Schur complement $F(z)\in \cB(\cH_1)$ defined by
\begin{equation}\label{scco}
F(z)=(V_{11} -z{\mathbb I}_1)-gV_{12} (g V_{22}-z{\mathbb I}_2)^{-1}V_{21},
\end{equation}
such that
\begin{equation}\label{critschur}
z\in \rho(T)\cap \rho(g V_{22}) \Leftrightarrow 0\in \rho(F(z)).
\end{equation}
As $V$ is unitary, we have $g \|V_{22}\|\leq g<1$, so that $F: \{|z|>g\}\rightarrow \cB(\cH_1)$ is well defined.
If $z\in \rho(V_{11})\cap {\mathbb S}$, we can write
\begin{equation}
F(z)=(V_{11} -z{\mathbb I}_1)\left({\mathbb I}_1-g(V_{11} -z{\mathbb I}_1)^{-1}V_{12} (g V_{22}-z{\mathbb I}_2)^{-1}V_{21}\right),
\end{equation}
which has a bounded inverse if
$
g\|(V_{11} -z{\mathbb I}_1)^{-1}V_{12} (g V_{22}-z{\mathbb I}_2)^{-1}V_{21}\|<1.
$
Assuming that $\|V_{11} \|<1$, we have $\{|z|>\|V_{11} \|\}\subset \rho(V_{11})$ and for $|z|>\max{(g \|V_{22} \|, \|V_{11} \|)}$,
\begin{eqnarray}
g\|(V_{11} -z{\mathbb I}_1)^{-1}V_{12} (g V_{22}-z{\mathbb I}_2)^{-1}V_{21}\| \leq
\frac{g\|V_{12} \|\|V_{21} \|}{(|z|-\|V_{11} \|)(|z|-g\|V_{22} \|)}.
\end{eqnarray}
The inner radius $r(V)$ of the ring (\ref{annulus}) is defined so that the right hand side above is strictly smaller than one and it satisfies $\max{(g\|V_{22} \|, \|V_{11} \|)}\leq r(V) < 1$ whenever
$g<\frac{1-\|V_{11} \|}{\|V_{21} \|\|V_{12} \|+\|V_{22} \|(1-\|V_{11} \|)}$. Thus, according to (\ref{critschur}), this implies that the ring (\ref{annulus}) belongs to the resolvent set of $T$, which yields the result for $T$ in place of $T_\omega$.
To get the result for the random case with $V$ replaced by $V_\omega$, it is enough to show that
\begin{equation}
\|P_jV_\omega P_k\|=\|P_j V P_k\|=\|V_{jk}\|, \ \ \forall \ j,k\in \{1,2\}.
\end{equation}
This is a consequence of the following lemma, which ends the proof of the theorem. \hfill {\vrule height 10pt width 8pt depth 0pt}
\end{proof}
\begin{lem}\label{qlem} Let $\{v_j^{(k)}\}_{k\in {\mathbb Z}}$
be the orthonormal basis of $\cH_j$, $j=1,2$ given by (\ref{evek}). Then
\begin{eqnarray}�\label{vjk}
P_1V_\omega P_1v_1^{(k)}
&=&\frac{1}{1-g^2}\left(- e^{i\omega_{2k-1}}\bar q(s\gamma-q\delta)v_1^{(k-1)}+e^{i\omega_{2k+2}}\bar s(s\alpha-q\beta)v_1^{(k+1)}\right)\\ \nonumber
P_2V_\omega P_2v_2^{(k)}&=&\frac{1}{1-g^2}\left( -e^{i\omega_{2k-1}}st v_2^{(k-1)}-e^{i\omega_{2k+2}}qrv_2^{(k+1)}\right)\\ \nonumber
P_2V_\omega P_1 v_1^{(k)}&=&\frac{1}{1-g^2}\left(s(s\gamma-q\delta)e^{i\omega_{2k-1}}v_2^{(k-1)}+
q(s\alpha-q\beta)e^{i\omega_{2k+2}}v_2^{(k+1)}\right)\\ \nonumber
P_1V_\omega P_2 v_2^{(k)}&=&\frac{1}{1-g^2}\left( e^{i\omega_{2k-1}}\bar q t v_1^{(k-1)}-\bar s r e^{i\omega_{2k+2}}v_1^{(k+1)} \right). \nonumber
\end{eqnarray}
Defining coefficients $w_\pm^{(ij)}$ by
\begin{equation}\label{coefvjk}
P_iV_\omega P_jv_j^{(k)}=e^{i\omega_{2k-1}}w_+^{(ij)}v_i^{(k-1)}+e^{i\omega_{2k+2}}w_-^{(ij)}v_i^{(k+1)},
\end{equation}
we have \vspace{-.8cm}
\begin{eqnarray}\label{normvjk}
\|P_jV_\omega P_k\|&=&|w_+^{(jk)}|+|w_-^{(jk)}|=\|V_{jk}\|
\end{eqnarray}
and, for all $i,j\in \{1,2\}$,
\begin{equation}\label{eqkernul}
\ker P_iV_\omega P_j \neq \{0\} \Leftrightarrow P_iV_\omega P_j=0.
\end{equation}
Let ${\mathbb D}_\eta^{(j)}$ and ${\mathbb D}_\xi^{(j)}$ be defined in
the orthonormal basis $\{v_j^{(k)}\}_{k\in {\mathbb Z}}$ of $\cH_j$ by
$
{\mathbb D}_\eta^{(j)}=\mbox{ \em diag }(e^{i\eta^{(j)}_k})$, and ${\mathbb D}_\xi^{(j)}=\mbox{\em diag }(e^{i\xi^{(j)}_k}),
$
where, for $p\geq 1$
\begin{eqnarray}
\eta^{(j)}_{2p}&=&\sum_{l=0}^p \omega_{4l}-\sum_{l=0}^{p-1}\omega_{4l+1}, \ \ \ \
\eta^{(j)}_{2p+1}= \sum_{l=0}^p \omega_{4l+2}-\sum_{l=0}^{p-1}\omega_{4l+3}\\
\xi^{(j)}_{2p}&=&\sum_{l=0}^{p-1} \omega_{4l+3}-\sum_{l=0}^{p-1}\omega_{4l+2}, \ \
\xi^{(j)}_{2p+1}= \sum_{l=0}^p \omega_{4l+1}-\sum_{l=0}^{p}\omega_{4l},
\end{eqnarray}
and, for $p\leq 0$
\begin{eqnarray}
\eta^{(j)}_{2p}&=&-\sum_{l=p+1}^1 \omega_{4l}+\sum_{l=p}^{1}\omega_{4l+1}, \ \ \ \
\eta^{(j)}_{2p+1}= -\sum_{l=p+1}^1 \omega_{4l+2}+\sum_{l=p}^{0}\omega_{4l+3}\\
\xi^{(j)}_{2p}&=&-\sum_{l=p}^{0} \omega_{4l+3}+\sum_{l=p}^{1}\omega_{4l+2}, \ \
\xi^{(j)}_{2p+1}= -\sum_{l=p+1}^1 \omega_{4l+1}+\sum_{l=p+1}^{1}\omega_{4l}.
\end{eqnarray}
Then, \vspace{-.55cm}
\begin{equation}\label{rel}
P_jV_\omega P_k= {\mathbb D}_\eta^{(j)} V_{jk} {\mathbb D}_\xi^{(k)} \simeq {\mathbb D}_\xi^{(k)} {\mathbb D}_\eta^{(j)} V_{jk}.
\end{equation}
\end{lem}
\begin{proof} The expressions of $P_iV_\omega P_j$ in the bases $\{v_j^{(k)}\}_{k\in {\mathbb Z}}$ are obtained by
explicit computations making use of (\ref{evek}),
\begin{equation}
e_{2k}=\frac{(\bar s v_1^{(k)}+q v_2^{(k)})}{\sqrt{|q|^2+|s|^2}}, \ \ e_{2k+1}=\frac{-\bar q v_1^{(k)}+s v_2^{(k)}}{\sqrt{|q|^2+|s|^2}},
\end{equation}
and of the constraint (\ref{const}). Identity (\ref{normvjk}) is established by a classical argument and (\ref{eqkernul}) is a direct consequence of this identity. Relation (\ref{rel}) is also a matter of verification.
\hfill {\vrule height 10pt width 8pt depth 0pt}
\end{proof}\\
\begin{rem} \label{nicexp} With $\det\begin{pmatrix}\alpha & \beta \cr \gamma & \delta \end{pmatrix}=ge^{i\chi}$, see Remark \ref{detg}, and constraint (\ref{const}), we have
\begin{equation}
\|V_{11}\|=\frac{|\delta-\bar \alpha ge^{i\chi}|+|\alpha -\bar \delta ge^{i\chi}|}{1-g^2},
\end{equation}
where the first / second term is the modulus of the coefficient of $v_1^{(k-1)}$ / $v_1^{(k+1)}$ in (\ref{coefvjk}).
\end{rem}
We establish further properties of $V_{jk}$ and $P_jV_\omega P_k$ as operators from $\cH_k$ to $\cH_j$, that we present in an abstract form.
\begin{prop}\label{spw} Let $W$ be an operator that takes a tridiagonal form in an orthonormal basis of $l^2({\mathbb Z})$ whose sole non zero coefficients satisfy
\begin{equation}
|W_{j, j+1}|=W_-, \ \mbox{and }\ |W_{j, j-1}|=W_+, \ \forall j\in {\mathbb Z}.
\end{equation}
Assume, without loss, that $W_+\geq W_->0$. Then, $\|W\|=W_++W_-$ and
\begin{eqnarray}
\mbox{If \ } W_+\leq 1,&& \{|z|< W_+-W_-\}\subset \rho(W) \nonumber \\
\mbox{If \ } W_+ > 1,&& \{|z|< (W_+-W_-)/(2W_+-1)\}\subset \rho(W).
\end{eqnarray}
If $W$ is further translation invariant,
$
W_{j, j+1}=w_-, \ \mbox{and }\ W_{j, j-1}=w_+, \forall j\in {\mathbb Z},
$
then $W$ is normal and
$\mbox{\em spr }(W)=\|W\|=|w_+|+|w_-|$.
\end{prop}
\begin{rem}
The radius of both disks contained in $\rho(W)$ is smaller than one.
\end{rem}
\begin{proof} The norm of $W$ was already mentioned above.
The structure of $W$ is such that we can write $W=W^+ S_+ + W^- S_-$, where the non zero matrix elements of the operator $S_+/S_-$ lie on the diagonal immediately above/below the main diagonal, and all have modulus one; $S_\pm$ are unitarily equivalent to standard shifts. Thus, for any $|z|\neq 1$, we can write
\begin{eqnarray}
W-z&=&W^+ (S_+-z) + W^- S_- -z(1-W^+)\nonumber\\
&=&W^+ (S_+-z)\Big({\mathbb I} + \frac{(S_+-z)^{-1}}{W^{+}}\Big(W^- S_- -z(1-W^+)\Big)\Big).
\end{eqnarray}
Since
\begin{equation}\label{rsha}
\left\| \frac{(S_+-z)^{-1}}{W^{+}}\Big(W^- S_- -z(1-W^+)\Big) \right\| \leq \frac{W^-+|z||1-W^+|}{W^+|1-|z||},
\end{equation}
the Neumann series implies that $W-z$ admits a bounded inverse if the right hand side of (\ref{rsha}) is bounded above by one. Considering small values of $|z|$ and dealing with the different cases for $W^+$, we get the result.
In case $W$ is translation invariant, we obtain by Fourier methods that $W$ is unitarily equivalent to a scalar multiplication operator
\begin{equation}
W\simeq W(x)=e^{ix}w_+ + e^{-ix}w_- \ \ \mbox{on $L^2({\mathbb T}; {\mathbb C})$}.
\end{equation}
This operator is obviously normal, which ends the proof.
\hfill {\vrule height 10pt width 8pt depth 0pt} \end{proof}
Hence, the translation invariant contractions $P_jVP_j|_{\cH_j}=V_{jj}$ with tri-diagonal representations in the orthonormal basis of $\cH_j$ given by $\{v_j^{(k)}\}_{k\in {\mathbb Z}}$, $j=1,2$, for $0\leq g<1$, with coefficients $w_\pm^{(jj)}$ defined by (\ref{coefvjk})
is normal and satisfies
$
\mbox{spr }(V_{jj})=\|V_{jj}\|=| w_+^{j(j)}|+|w_-^{(jj)}|.
$
\begin{exple}\label{ex2}
{\em
Let us apply the results above to Example \ref{ex1} where $\tilde C$ defined by equation (\ref{exple}). Recall that in this case $g=\sin(\xi)$, and $\xi,\eta\in [0,\pi/2]$. We get
\begin{equation}
\|V_{11}\|=\cos(\eta), \ \ \|V_{21}\|=\sin(\eta), \ \ \|V_{22}\|=\cos(\eta),\ \ \|V_{12}\|=\sin(\eta).
\end{equation}
Thus, for $\eta,\xi \in ]0,\pi/2[$ so that $g>0$, $\|V_{11}\|<1$ and for $\xi$ small enough so that
\begin{equation}\label{crxi}
\sin(\xi)<\frac{1-\cos(\eta)}{\sin^2(\eta)+\cos(\eta)(1-\cos(\eta))},
\end{equation}
condition (\ref{nonper}) holds and we get
\begin{equation}\label{rxi}
r(V)=\frac12\left(\cos(\eta)(1+\sin(\xi))+\sqrt{\cos^2(\eta)(1-\sin(\xi))^2+4\sin(\xi)\sin^2(\eta)}\right).
\end{equation}
Actually, all corresponding operators $P_jV_\omega P_k$ in this case map the basis vector $v_k^{(n)}$ to one of $v_j^{(n\pm 1)}$ only.
In particular, $P_1V_\omega P_1|_{\cH_1}$ and $P_2V_\omega P_2|_{\cH_2}$ are unitarily equivalent to $\cos(\eta)S_1$ and $\cos(\eta)S_2$ respectively, where $S_j$ is the standard shift on $P_j\cH_j$. Hence,
\begin{equation}
\sigma(P_1V_\omega P_1|_{\cH_1})=\cos(\eta){\mathbb S} \ \ \mbox{and} \ \
\sigma(P_2V_\omega P_2|_{\cH_2})=\cos(\eta){\mathbb S}.
\end{equation}
Thus, assuming a phase distribution satisfying (\ref{suppdnu}) and parameters such that condition (\ref{crxi}) holds, we have excluded the presence of spectrum of the corresponding non-unitary operator $T_\omega$ in the union of the ring of inner radius (\ref{rxi}) and of the symmetric sets characterized by Lemma \ref{form}. Moreover, for suitable values of the parameters condition (\ref{splits}) holds as well and $\sigma(T_\omega)$ is contained in two disjoint sets separated by the real axis.
}
\end{exple}
The following more specific properties hold.
\begin{lem}\label{loffdiag} We have
\begin{eqnarray}\label{offdiag}
\|V_{11}\|=0 \Leftrightarrow \ \|V_{22}\|=0 \
\Leftrightarrow \ \tilde C\in \left\{
\begin{pmatrix}
0 & 0 & \beta \cr
q & g & 0 \cr
\gamma & t & 0
\end{pmatrix},
\begin{pmatrix}
0 & r & \beta \cr
0 & g & s \cr
\gamma & 0 & 0
\end{pmatrix}
\right\}\subset U(3),
\end{eqnarray}
and, \vspace{-.8cm}
\begin{eqnarray}\label{diag}
V_{jk}=0 \ \mbox{for some } k\neq j &\Leftrightarrow & V_{jj}\ \mbox{unitary for all $j\in\{1, 2\}$} \\ \nonumber
\Leftrightarrow\ \mbox{$V_{jj}\simeq S_j$, $S_j$ a shift on $\cH_j$
}& \Leftrightarrow &
\tilde{C}\in \left\{
\begin{pmatrix}
\alpha & r & 0 \cr
q & g & 0 \cr
0 & 0 & \delta
\end{pmatrix},
\begin{pmatrix}
\alpha & 0 & 0 \cr
0 & g & s \cr
0 & t & \delta
\end{pmatrix}
\right\}\subset U(3).
\end{eqnarray}
\end{lem}
\begin{rems} i) In case $V_\omega$ is off-diagonal with respect to $\cH_0=\cH_1\bigoplus\cH_2$, so that (\ref{offdiag}) and Lemma \ref{sc} hold, we saw that for all $0\leq g<1$ and all $\omega$,
$\sigma(T_\omega)\subset \{z\in {\mathbb C}\ | |z|=\sqrt{g}\}$. We recover this result by noting that $V_\omega$ off-diagonal implies for $z\neq 0$
\begin{equation}
F(z)=-z\left({\mathbb I}_1-P_1V_\omega^2P_1g/z^2\right),
\end{equation}
where $P_1V_\omega^2 P_1|_{\cH_1}$ is unitary. Hence $F(z)$ is boundedly invertible iff $z^2\in \sigma(gP_1V_\omega^2 P_1|_{\cH_1})$.
\\
ii) In case $V_\omega$ is diagonal with respect to $\cH_0=\cH_1\bigoplus\cH_2$, so that (\ref{diag}) and Lemma \ref{sc} hold, we saw that for all $0\leq g<1$ and all $\omega$,
$\sigma(T_\omega)={\mathbb S}\cup g{\mathbb S}. $
\end{rems}
\begin{proof}
The tridiagonal matrix representation of $V_{jj}$ stems from (\ref{vjk}),
which yields the first statement.
The last statements are obtained by discussing the conditions $w_-^{(jj)}=w_+^{(jj)}=0$
depending on the fact that $q,s$ are zero or not. We first note that the condition $g<1$ forbids $q=s=0$ or $r=t=0$.
For $\|V_{11}\|=0$, the case $qs\neq 0$, is impossible: the expansion of $\det(\tilde C)$ with respect to the second column and $w_-=w_+=0$ imply $\det(\tilde C)=g(\alpha \delta-\gamma \beta)$, which is of modulus 1. This implies $g=|(\alpha \delta-\gamma \beta)|=1$ and $q=s=0$, a contradiction. If $qs=0$, one gets that $\alpha$ or $\delta$ equals 1, which with condition
(\ref{lcm}) yield the result. Similarly, $\|V_{22}\|=0$ imply $q=t=0$ or $s=r=0$ and condition
(\ref{lcm}) again yields the result. The assertions regarding the off diagonal parts of $V_\omega$ are readily obtained by the same type of considerations and the fact that $V_\omega$ is unitary. \hfill {\vrule height 10pt width 8pt depth 0pt}
\end{proof}
\subsection{Ergodicity}
We briefly recall here a spectral consequences of our hypothesis on the way the randomness enters the operator $T_\omega$. Ergodicity provides a tool to estimate from below the spectrum of $T_\omega$, almost surely. Our setup actually enters the more general theory of pseudo-ergodic operators, as developed in \cite{D1, D2}, of which ergodic operators are special cases.
The definition (\ref{deft}) of ${\mathbb D}^0_\omega$ makes the operator ergodic under 2-shifts with respect to the matrix representation (\ref{matrixt}). If $\Sigma$ denotes both the map from $\Omega \rightarrow \Omega$ such that $(\Sigma \omega)_j=\omega_{j+2}$, and the operator defined on $\cH_0$ by $\Sigma e_j =e_{j+2},\ \forall j\in {\mathbb Z}, $ we have
\begin{equation}
T_{\Sigma ^k\omega}=\Sigma^{-k}T_\omega \Sigma^k, \ \forall k\in {\mathbb Z}.
\end{equation}
Following \cite{D1, D2} in making use of independence of the random phases and Borel-Cantelli Lemma, we get
\begin{prop}\label{xxx} Let $l\in 2{\mathbb N}$ and $\theta^{(l)}=(\theta_1, \theta_2, \cdots,\theta_l)\in (\mbox{supp }d\nu)^l\subset{\mathbb T}^l$. Set
$
T_{\theta^{(l)}}:={\mathbb D}^0_\omega T, \ \mbox{where} \ \omega= (\dots, \theta^{(l)}, \theta^{(l)}, \dots)\in \Omega.
$
Then,
\begin{equation}
\cup_{l\in 2{\mathbb N}}\cup_{\theta^{(l)}\in{\mathbb T}^l}\sigma(T_{\theta^{(l)}})\subset \sigma (T_\omega), \ \mbox{almost surely.}
\end{equation}
\end{prop}
\begin{rem}\label{remopt} In particular, if $d\nu(\theta)=d\theta/(2\pi)$, $\cup_{\theta\in [0,2\pi]}e^{i\theta}(\mbox{\em Ran }\lambda_+\cup \mbox{\em Ran }\lambda_-)\subset \sigma(T_\omega)$, where $\lambda_\pm$ are defined in (\ref{evti}). This shows that statements (\ref{87}) and Theorem (\ref{suff}) on the location of $\sigma(T_\omega)$ are optimal, as we argue below.
\end{rem}
Considering Example \ref{ex1}, one checks that when $\xi\rightarrow 0$, condition (\ref{crxi}) holds, $\lambda_+(0)=\frac{1}{2}\left(\cos(\eta)(1+\sin(\xi))+\sqrt{\cos^2(\eta)(1+\sin(\xi))^2-4\sin(\xi)}\right)>0$ and the value $r(V)$ given in (\ref{rxi}) becomes arbitrarily close to $\lambda_+(0)$. Also, when $\cos^2(\eta)<4\sin(\xi)/(1+\sin(\xi))^2$ we have $|\lambda_+(0)|=g=\sin (\xi)$. Since $|\lambda_+(0)|\in \sigma(T_\omega)$ almost surely, Proposition \ref{xxx} shows that statement (\ref{87}) and Theorem (\ref{suff}) on the location of $\sigma(T_\omega)$ are optimal.
\section{Special Case $g=0$}
This section is devoted to a more thorough analysis of the case $g=0$
\begin{equation}\label{kg0}
T_\omega=V_\omega P_1 \ \mbox{corresponding to }\
\tilde C=
\begin{pmatrix}\alpha & r & \beta \cr
q & 0 & s \cr
\gamma & t & \delta \end{pmatrix}\in U(3).
\end{equation}
According to Lemmas \ref{symlem} and \ref{easyspect}, $T_\omega=V_\omega P_1$ is far from being unitary, $\ker \ T_\omega=\cH_2$, for all $\omega\in \Omega$, and $\sigma(T_\omega)=\sigma(P_1V_\omega P_1)\cup\{0\}$. More precisely:
\begin{prop}\label{spwp} If $g=0$, we have for all $\omega\in\Omega$
\begin{equation}
\sigma(T_\omega )\setminus\{0\}\subset \big\{\big| |\alpha|-|\delta| \big|\leq |z|\leq |\alpha|+|\delta|\big\}.
\end{equation}
If $\alpha=0$, resp. $\delta=0$, then $P_1V_\omega P_1|_{\cH_1}$ is unitarily equivalent to $|\delta| S^+$, resp. $|\alpha| S^-$, and
\begin{eqnarray}
&&\sigma(T_\omega)=\max (|\alpha|, |\delta|){\mathbb S} \cup\{0\}\ \mbox{ and }\ \sigma_{p}(T_\omega)=\sigma_{p}(T_\omega^*)=\{0\}.
\end{eqnarray}
Moreover, \vspace{-.8cm}
\begin{eqnarray}
\gamma\neq qt \Leftrightarrow \beta\neq sr \ &\Rightarrow& \ V_\omega \ \mbox{is pure point a.s.} \\
\gamma= qt \Leftrightarrow \beta= sr \ &\Rightarrow& \ V_\omega \ \mbox{is purely ac, } \ \forall \omega\in\Omega.
\end{eqnarray}
\end{prop}
\begin{exple}{\em
Let us consider an explicit parametrization of a $\tilde C\in O(3)$ of the kind (\ref{kg0})
\begin{equation}\label{ctino3}
\tilde C(\xi,\eta)=
\begin{pmatrix}\cos(\xi)\sin(\eta) & \cos(\eta) & -\sin(\xi)\sin(\eta) \cr
\sin(\xi) & 0 & \cos(\xi)\cr
-\cos(\xi)\cos(\eta) & \sin(\eta) & \sin(\xi)\cos(\eta) \end{pmatrix}\in O(3), \ \xi, \eta\in [0,\pi/2],
\end{equation}
where $(\xi,\eta)$ is restricted to $[0,\pi/2]^2$ for simplicity. Then, $|\alpha|+|\delta|<1$ is equivalent to $\sin(\xi+\eta)\neq 1$, i.e. $\xi+\eta\neq \pi/2$, and $\gamma= at$ is equivalent to $\cos(\xi-\eta)=0$, i.e. $(\xi, \eta)=(\pi/2, 0)$, or $(\xi,\eta)=(0,\pi/2)$.
}
\end{exple}
\begin{proof}
Remark \ref{nicexp} implies for $g=0$ that the modulus of the coefficients of the tridiagonal operator $P_1V_{\omega}P_1$ are $|\alpha|$ and $|\delta|$, so Proposition \ref{spw} yields the first statement. We know that $0\in \sigma_p(T_\omega)$. Further assuming that $\alpha\delta=0$, the same remark yields that $P_1V_\omega P_1$ is unitarily equivalent to a shift and consequently, Lemma \ref{easyspect} yields the spectrum of $T_\omega$. Finally, the eigenvalue equation $T_\omega\varphi=\lambda \varphi$, $\lambda\neq 0$, implies that $\varphi_1=P_1\varphi$ satisfies $P_1V_{\omega}P_1\varphi_1=\lambda \varphi_1$, which cannot hold for a shift. The same argument applies to $T_\omega^*$. Then one checks on the unitary operator (\ref{matv}) that $\gamma = qt $ is equivalent to $\beta=sr $. In turn, this implies that $V_\omega$ is unitarily equivalent to a direct sum of two shifts. In all other cases, $V_\omega$ is pure point almost surely as shown in \cite{JM}. \hfill {\vrule height 10pt width 8pt depth 0pt}
\end{proof}
From the foregoing we know that when $g=0$,
$P_1V_\omega P_1={\mathbb D}_\eta^{(1)}V_{11}{\mathbb D}_\xi^{(1)}$, where
\begin{eqnarray}
&&V_{11}=\begin{pmatrix}
\ddots & \delta & & \cr
\alpha & 0 & \delta & \cr
& \alpha & 0 & \delta \cr
& &\alpha & \ddots
\end{pmatrix} \ \simeq \ e^{i(\arg \alpha - \arg \beta)/2}(e^{iy}|\alpha|+e^{-iy}|\delta|), \ \mbox{on }\ L^2({\mathbb T}),
\end{eqnarray}
and
\begin{equation}
e^{-i(\arg \alpha - \arg \delta)/2}\sigma(V_{11})= E(|\alpha|,|\delta|),
\end{equation}
where $E(|\alpha|,|\delta|)$ denotes the ellipse centered at the origin, with horizontal major axis of length $|\alpha|+|\delta|$ and vertical
minor axis of length $||\alpha|-|\delta||$. When the random phases are iid and uniform, we have a complete description of the spectral properties of $T_\omega$ when $g=0$.
\begin{prop}\label{g=0}
Assume $g=0$ and $d\nu(\theta)=d\theta/2\pi$. Then, $T_\omega=V_\omega P_1$ satisfies
\begin{equation}
\sigma(T_\omega)=\{0\}\cup\big\{\big| |\alpha|-|\delta| \big|\leq |z|\leq |\alpha|+|\delta|\big\}, \ \mbox{a.s.}
\end{equation}
When $|\alpha|+|\delta|=1$, the peripheral spectra of the relevant operators coincide with ${\mathbb S}$,
\begin{equation}
\sigma(T_\omega)\cap{\mathbb S}=\sigma(P_1V_\omega P_1|_{\cH_1})\cap{\mathbb S}=\sigma(V_\omega)={\mathbb S}, \ \mbox{a.s.}
\end{equation}
However, the nature of the peripheral spectra of $T_\omega$ and $V_\omega$ differs for $\gamma\neq qt$,
\begin{equation}
\sigma_p(T_\omega)\cap{\mathbb S}=\sigma_p(T_\omega^*)\cap{\mathbb S}=\emptyset, \ \mbox{whereas} \ \sigma_c(V_\omega)=\emptyset \mbox{ a.s.}
\end{equation}
\end{prop}
\begin{rem} This result shows in a sense that the spectral localization of $V_\omega$ does not carry over to the boundary of the spectrum of $T_\omega=V_\omega P_1$. Note that the original operator $U_\omega(C)$ is purely ac when $g<1$, for all $\omega\in\Omega$.
\end{rem}
\begin{proof}
The first consequence of our assumption on the distribution of the random phases is that
$
P_1V_\omega P_1={\mathbb D}_\omega^{(1)}V_{11},
$
where the random phases of the diagonal operator
${\mathbb D}_\omega^{(1)}$ are independent and uniformly distributed, see e.g. Lemma 4.1 in \cite{ABJ}. Hence proposition \ref{xxx} with $\mbox{supp } d\nu(\cdot)=2\pi$, together with Proposition \ref{spwp} show that
\begin{equation}
\big\{\big||\alpha|-|\delta|\big|\leq |z|\leq |\alpha|+|\delta|\big\}=\bigcup_{\theta\in [0,2\pi[} e^{i\theta}E(|\alpha|,|\delta|) = \sigma({\mathbb D}_\omega^{(1)} V_{11}), \ \mbox{almost surely.}
\end{equation}
When $|\alpha|+|\delta|=1$, the peripheral spectra equals ${\mathbb S}$ almost surely by Lemma \ref{easyspect}.
Finally, the nature of the peripheral spectra stems from Lemmas \ref{kerzero} and \ref{lcnu}. \hfill {\vrule height 10pt width 8pt depth 0pt}
\end{proof}
\begin{rem} In case $|\alpha|=|\delta|=1/2$, $V_{11}=\Delta_1$, the discrete Laplacian on $\cH_1$. With $d\nu(\theta)=d\theta/2\pi$,
\begin{equation}
\sigma({\mathbb D}_\omega^{(1)} \Delta_1)=\sigma(T_\omega)=\{|z|\leq 1\}, \ \mbox{almost surely},
\end{equation}
where ${\mathbb D}_\omega^{(1)} \Delta_1$ is a version of the random hopping model of Feinberg and Zee \cite{FZ}.
\end{rem}
|
1,108,101,564,608 | arxiv | \section{Introduction}
In recent years, the advent of new millimeter (mm) and submillimeter (submm) facilities, such as the Atacama Large Millimeter/submillimeter Array (ALMA), with unprecedented sensitivity and frequency coverage, has improved the detectability of cool gas in high-redshift star-forming galaxies \citep[e.g.,][]{Wagg12, Vieira13, Wang13}. \citet{Swinbank12} reported two chance detections of submm galaxies \citep[SMGs,][for a review]{Blain02} in \textsc{[C~ii]} 158-$\micron$
line emission during 870-$\micron$ continuum follow-up observations of 126 SMGs using ALMA, which places the first constraint on the \textsc{[C~ii]} luminosity function at $z = 4.4$. Furthermore, \citet{Hatsukade13} made the deepest unlensed number counts of SMGs using sources incidentally detected in the same fields of view (FoVs) toward near-infrared-selected star-forming galaxies at $z = 1.4$. Such ``incidental'' searches for high-$z$ star-forming galaxies, especially in emission lines, offer a unique opportunity to investigate the line luminosity functions or cool gas mass functions, which are important for constraining galaxy formation models.
Here we report the serendipitous detection of a dusty starburst galaxy, ALMA~J010748.3$-$173028 (hereafter ALMA-J0107), with a significant emission line at 99.75 GHz, which is likely a redshifted $^{12}$CO line. At this position, we find a hard X-ray source, which strongly suggests the presence of a buried active galactic nucleus (AGN).
We assume a cosmology with $\Omega_\mathrm{m} = 0.3$, $\Omega_{\Lambda} = 0.7$, and $H_0 = 70$ km s$^{-1}$ Mpc$^{-1}$ ($h = 0.7$).
\begin{figure*}[htbp]
\begin{center}
\includegraphics[angle=0,width = 0.98\textwidth,keepaspectratio,clip]{f1.eps}
\caption{
(Top left) Image of ALMA J010748.3$-$173028 in 880-$\micron$ continuum emission (contours) overlaid on
\textcolor{red}{the} integrated intensity map of
\textcolor{red}{the} emission line (background). Images are not corrected for primary-beam attenuation. The contours start at $5\sigma$ with a $5\sigma$ step, where $\sigma = 0.14$ mJy beam$^{-1}$ at the ALMA-J0107 position. Crosses mark positions of near-infrared peaks shown in Figure~2 of \citet{Imanishi07}. Partial large circle represents primary beam size at 99.75~GHz. Filled ellipses in bottom-left corner indicate synthesized beam sizes of emission line image (gray) and 880-$\micron$ images (black).
(Top right) 100 GHz spectrum of ALMA J010748.3$-$173028 across the 1.875~GHz spectral window of ALMA Band~3. Flux densities are measured with a $4''$ aperture and corrected for primary beam attenuation ($0.79\times$). Inset shows closeup of the spectrum; solid curve is
\textcolor{red}{the} best-fit Gaussian.
(Bottom) $15'' \times 15''$ multiwavelength (millimeter to X-ray) images of ALMA J010748.3$-$173028. Contours show the ALMA 880-$\micron$ image and start at $4 \sigma$ with a separation of $4 \sigma$, where $\sigma = 0.5$~mJy beam$^{-1}$ is the noise level corrected for primary beam attenuation ($0.28\times$). Contours of SMA 1.3-mm image are drawn at $2\sigma$, $3\sigma$, and $4\sigma$, where $\sigma = 1.21$~mJy beam$^{-1}$.
}
\label{fig:almaimage}
\end{center}
\end{figure*}
\section{ALMA Observations and Results}
ALMA 3-mm and 880-$\micron$ observations toward VV114 (program ID: 2011.0.00467.S) were conducted using the compact configuration in November 2011 and the extended configuration in May 2012 (3-mm observation only). The correlator was configured to cover 98.53--102.35 GHz/110.77--113.90 GHz (3~mm) and 323.51--327.25 GHz/335.68--339.31 GHz (880~$\micron$) with a 0.488 MHz resolution. The 3-mm and 880-$\micron$ primary beam sizes are $62''$ and $19''$, respectively; for the 880-$\micron$ observations, we mosaicked seven pointings to compensate for the small FoV of Band 7. At both 3 mm and 880 $\micron$, Uranus, J1924$-$292, and J0132$-$169 ($6\arcdeg$ away from VV114) were used to calibrate the absolute flux, bandpass, and complex gain, respectively.
We used \textsc{casa} \citep{McMullin07} to calibrate the visibility data and to image them with a robust weighting of 0.5. Note that we made the continuum images using only spectral channels that are free from $^{12}$CO emission from VV114, which leaves 6.7-GHz and 6.6-GHz bandwidths at 3 mm and 880 $\micron$, respectively. We
\texttt{clean}ed the resulting dirty images down to the 1$\sigma$ level. The synthesized beam sizes at 3~mm and 880~$\micron$ are $2\farcs 37 \times 1\farcs 57$ (PA = $94\arcdeg$) and $1\farcs 33 \times 1\farcs 12$ (PA = $120\arcdeg$), respectively. The resulting rms noise levels at 3 mm before correcting for primary beam attenuation were 0.92 mJy~beam$^{-1}$ for a cube with a resolution of 30 km s$^{-1}$ and 50 $\mu$Jy~beam$^{-1}$ for the continuum image. The 880-$\micron$ noise levels for a 30 km~s$^{-1}$ resolution cube and continuum are 1.5 mJy~beam$^{-1}$ and 0.11 mJy~beam$^{-1}$, respectively.
The flux calibration accuracies in both bands are estimated to be 10\%.
We serendipitously detect a 13$\sigma$ line-emitting object, ALMA-J0107, at 99.753 GHz at $\mathrm{(\alpha,\,\delta)_{J2000} = (01^{h} 07^{m}48\fs 32,\, -17\arcdeg 30' 28\farcs 1)}$, as shown in Figure~\ref{fig:almaimage}. The image is marginally resolved, and the beam-deconvolved source size measured using a \textsc{casa} task \texttt{imfit} is $1\farcs 6 \pm 0\farcs 2$, although no velocity structure is found. Figure~\ref{fig:almaimage} (upper right) shows the spectrum of the emission line. A single Gaussian fit to the spectrum shows $S_\mathrm{peak} = 12.9 \pm 1.0$~mJy, $\Delta v = 210 \pm 18$~km~s$^{-1}$. The integrated intensity is $3.14 \pm 0.15$ Jy~km~s$^{-1}$.
This line does not correspond to a CO line or any other line of VV114 itself, even if we search a wide velocity range of $\pm 10000$ km s$^{-1}$ around the systemic velocity of VV114 (6100 km~s$^{-1}$). Moreover, we find no emission line feature in the other spectral windows of Bands 3 and 7 at this position.
Thus, it is natural to consider a redshifted emission line, especially a $^{12}$CO line, arising from a background galaxy. We list the possible redshifts, as well as the corresponding CO luminosities and molecular masses, for up to the $J=6$--5 transitions of $^{12}$CO in Table~\ref{table:id}. The $J=7$--6 ($z=7.087$) transition is ruled out because no [C~\textsc{i}](2--1) at $\nu_\mathrm{obs} = 100.08$~GHz is found. Higher transitions at $z \ge 8.2$ are not plausible. We will identify the line in \S~\ref{sect:redshiftid}.
The 880-$\micron$ continuum emission in Band~7 is detected at the same position as the Band~3 line peak, close to the edge of the ALMA FoV (Figure~\ref{fig:almaimage}, top left). The 880-$\micron$ flux density is $11.2 \pm 0.4$ mJy after correcting for primary beam attenuation, whereas we fail to detect the 3-mm continuum in Band 3 down to the 3$\sigma$ upper limit of $< 0.19$ mJy. The flux density is typical of SMGs \citep{Blain02}, and the inferred far-infrared (FIR) luminosity is $L_\mathrm{FIR} \simeq 1 \times 10^{13} L_\sun$ for $1 < z < 10$ if we assume a dust temperature of $T_\mathrm{dust} = 40$~K and an emissivity index of $\beta = 1.5$.
If the FIR luminosity is powered by starburst activities, a star formation rate is estimated to be $\sim 2\times 10^3 M_\sun$~yr$^{-1}$ following \citet{Kennicutt98}. We also find 1.3-mm continuum emission in published Submillimeter Array (SMA) data \citep{Wilson08}. The 1.3-mm flux density is $5.2 \pm 1.3$~mJy. The Rayleigh--Jeans slope is constrained primarily by the ALMA observations, and the lower limit of the spectral index\footnote{The spectral index $\alpha$ is defined such that $S_{\nu} \propto \nu^{\alpha}$.} is $\alpha = 3.5$, consistent with those found in dusty star-forming galaxies.
At the ALMA position, many ancillary data are available from the radio to the X-ray bands. Table~\ref{table:photometry} lists the results of multiwavelength photometry, and Figure~\ref{fig:almaimage} (bottom) shows multiwavelength images. Unfortunately, heavy blending by VV114 affects the infrared to optical images, but we clearly see a \emph{Chandra} X-ray counterpart and marginally detect it in the \emph{Spitzer}/IRAC bands. It is not clear from the current data whether the source is gravitationally lensed.
\begin{deluxetable}{cccccc}
\tablewidth{0.48\textwidth}
\tablecaption{Line Identification\label{table:id}}
\tablehead{
\colhead{$^{12}$CO} & \colhead{$z$} &
\colhead{$L'_\mathrm{CO}$\tablenotemark{a}} &
\colhead{$M(\mathrm{H_2})$\tablenotemark{b}} &
\multicolumn{2}{c}{Predicted $S_\mathrm{CO}\Delta v$\tablenotemark{c}} \\
\colhead{Transition} & & & &
\colhead{M82\tablenotemark{d}} & \colhead{BR1202\tablenotemark{e}}
}
\startdata
$J = 1\rightarrow 0$ & 0.1556 & 0.37 & 0.29 & 0.6 & 1.1 \\
$J = 2\rightarrow 1$ & 1.311 & 7.1 & 5.7 & 0.8 & 1.6 \\
$J = 3\rightarrow 2$ & 2.467 & 9.9 & 7.9 & 1.1 & 2.4 \\
$J = 4\rightarrow 3$ & 3.622 & 10.5 & 8.4 & 1.4 & 3.5 \\
$J = 5\rightarrow 4$ & 4.777 & 10.4 & 8.3 & 1.7 & 4.9 \\
$J = 6\rightarrow 5$ & 5.932 & 10.0 & 8.0 & 2.0 & 6.6
\enddata
\tablenotetext{a}{CO line luminosity in units of $10^{10}$~K~km~s$^{-1}$ pc$^2$}
\tablenotetext{b}{Molecular gas mass in units of $10^{10}~M_{\sun}$, derived using a conversion factor of $0.8\,M_\sun$ (K km s$^{-1}$ pc$^2$)$^{-1}$ \citep{Downes98} and assuming thermally excited lines}
\tablenotetext{c}{Integrated intensity in units of Jy km s$^{-1}$, predicted from the 880-$\micron$ flux. See details in \S~\ref{sect:lineid}.}
\tablenotetext{d}{$T_\mathrm{dust} = 40$~K, $\beta = 1.5$, and the CO excitation ladder of M82 \citep{Weiss05} are assumed.}
\tablenotetext{e}{The same as (b), but a dust temperature of 50 K and the CO excitation ladder of BR~1202$-$0725 SE \citep{Salome12} are assumed.}
\end{deluxetable}
\section{Redshift Identification}
\label{sect:redshiftid}
\subsection{Photometric Redshift Estimates}
\label{sect:photoz}
To obtain rough estimates of the redshift, we use four template spectral energy distributions (SEDs) of well-studied starburst galaxies: Arp~220, M82 \citep{Silva98}, SMM~J2135$-$0102 \citep{Swinbank10}, and a composite of radio-identified SMGs \citep{Michalowski10}. Then we fit the submm to radio data to the templates. Figures~\ref{fig:photoz}a, b, and c show the minimum $\chi^2$ and FIR luminosity as a function of the redshift and the best-fit SEDs, respectively. The resulting redshifts and the 90\% confidence intervals are $z = 1.53^{+2.31}_{-0.95}$ (Arp 220), $1.67^{+2.12}_{-0.79}$ (M82), $1.90^{+2.25}_{-1.35}$ (SMM~J2135), and $1.74^{+1.28}_{-1.16}$ (mean SMG). Thus, the likely redshift range is $0.6 < z < 4.1$ for all the templates, suggesting that the $^{12}$CO redshift can be $z = 1.31$, 2.47, or 3.62. The FIR luminosity, $\log(L_\mathrm{FIR}/L_\sun) \simeq 12.5$--13, is mostly insensitive to the redshift.
The constraint could become tighter when we use the IRAC photometry in addition to the submm to radio data, although the use of IRAC data should be considered cautiously because IRAC observes the rest-frame optical component, which depends strongly on the stellar population model. The dashed curves in the $\chi^2$ plot of Figure~\ref{fig:photoz}a show the minimum $\chi^2$ values from SED fits using the constraints at 3.6~$\micron$ and 4.5~$\micron$. The best-fit SEDs are shown in Figure~\ref{fig:photoz}d. The resulting redshifts are $z = 2.17^{+2.31}_{-0.70}$ (Arp 220), $4.27^{+1.96}_{-0.89}$ (M82), $2.14^{+2.37}_{-0.63}$ (SMM~J2135), and $2.10^{+1.46}_{-0.54}$ (mean SMG), although we note that the M82 fit might not be reliable in this case because of poor $\chi^2$ values. The plausible redshift range in this case is therefore $1.5 < z < 4.5$.
Consequently, the photometric redshift analysis favors a redshift of $z = 2.467$, whereas $z = 3.622$ is within an acceptable range in terms of the $\chi^2$ values.
\begin{deluxetable}{llcc}
\tablewidth{0.48\textwidth}
\tablecaption{Multiwavelength Counterparts to ALMA-J0107 \label{table:photometry}}
\tablehead{
\colhead{Instrument} & \colhead{Band} & \colhead{Flux Density} & \colhead{Unit}
}
\startdata
VLA\tablenotemark{a}& 3.0 cm & $< 0.3$ (3$\sigma$) & mJy \\
ALMA/Band 3 & 3.0 mm & $< 0.19$ (3$\sigma$) & mJy \\
SMA & 1.3 mm & $5.2 \pm 1.3$ & mJy \\
ALMA/Band 7 & 880 $\micron$ & $11.2 \pm 0.4$ & mJy \\
\emph{Spitzer}/IRAC & 8.0 $\micron$ & $< 0.1$ & mJy \\
\emph{Spitzer}/IRAC & 5.8 $\micron$ & $< 0.1$ & mJy \\
\emph{Spitzer}/IRAC & 4.5 $\micron$ & $0.06 \pm 0.01$\tablenotemark{b} & mJy \\
\emph{Spitzer}/IRAC & 3.6 $\micron$ & $0.04 \pm 0.01$\tablenotemark{b} & mJy \\
\emph{Chandra}/ACIS & 0.5--10 keV & $2.73 \times 10^{-15}$ & erg s$^{-1}$ cm$^{-2}$ \\
\emph{Chandra}/ACIS & 0.5--2 keV & $8.34 \times 10^{-16}$ & erg s$^{-1}$ cm$^{-2}$ \\
\emph{Chandra}/ACIS & 2--10 keV & $1.90 \times 10^{-15}$ & erg s$^{-1}$ cm$^{-2}$
\enddata
\tablenotetext{a}{The data were retrieved from the VLA archive.}
\tablenotetext{b}{The flux density is strongly affected by contamination from VV114 and thus should be regarded as an upper limit.}
\end{deluxetable}
\subsection{Line Identification}
\label{sect:lineid}
To confirm that this is a $^{12}$CO line, we estimate the $^{12}$CO intensities using the 880-$\micron$ continuum intensity and some empirical relations and quantities found in a local starburst galaxy, M82.
From the 880-$\micron$ flux density, we obtain $L_\mathrm{FIR} \sim 9 \times 10^{12} L_\sun$ for $z > 1$ and $\sim 1 \times 10^{12} L_\sun$ for $z = 0.156$ if $T_\mathrm{dust} = 40$~K and $\beta = 1.5$. The inferred FIR luminosity is almost independent of the redshift at $z > 1$. Although these are very crude estimates, we use the $L_\mathrm{FIR}$-to-$L'_\mathrm{CO(3-2)}$ correlation \citep{Iono09} to obtain the $^{12}$CO(3--2) luminosity and then the intensities at the possible redshifts.
We find that $S_\mathrm{CO(3-2)} \Delta v \simeq 3.9$, 1.6, 1.1, 1.0, 0.98, and 1.0 Jy km s$^{-1}$ at $z = 0.156$, 1.31, 2.47, 3.62, 4.78, and 5.93, respectively. Then we assume the CO excitation ladder found in M82 \citep{Weiss05} to obtain the CO intensities at other transitions, which allows us to estimate those at the possible redshifts.
We repeat this procedure for a higher dust temperature ($T_\mathrm{dust} = 50$~K) and the CO excitation found for BR1202$-$0725 SE \citep{Salome12}, in which the heating of the interstellar medium is dominated by a powerful AGN.
The results are given in Table~\ref{table:id}.
The line intensities are on the order of 1 Jy km s$^{-1}$ and are in good agreement with the observed ones, strongly suggesting that the line is $^{12}$CO because non-$^{12}$CO lines, such as $^{13}$CO and HCN, are $\ge 1$ order(s) of magnitude weaker than $^{12}$CO. The atomic carbon [CI](1--0) line is possible (at $z = 3.93$) but less likely because the intensity is typically 1/3 to 1/10 that of $^{12}$CO(3--2) \citep[e.g.,][]{Weiss05a}.
Other possible line attributions include the H$_2$O molecule, which is known to have emission lines as bright as those of $^{12}$CO in the submm band. The major transitions exhibiting strong emission at $\nu_\mathrm{rest} < 1000$~GHz are $J_{K_a,K_c} = 2_{11}$--$2_{02}$ and $2_{02}$--$1_{11}$ at $\nu_\mathrm{rest} = 752.0$ and 987.9 GHz, respectively. However, the redshifts inferred from the H$_2$O lines would be 6.54 and 8.90, which are outside of the photometric redshift range. Therefore, these results in combination with the photometric redshift indicate that the line is most likely a redshifted $^{12}$CO transition at $z = 2.467$ or 3.622.
\begin{figure}[htbp]
\begin{center}
\includegraphics[angle=0,width = 0.48\textwidth,keepaspectratio,clip]{f2.eps}
\caption{Photometric redshift estimates using broadband photometry and templates of spectral energy distributions (SEDs) of Arp~220 (red), M82 (green), composite SMG (blue), and SMM~J2135$-$0201 (orange). (a) Reduced-$\chi^2$ and (b) far-infrared luminosity $L_\mathrm{FIR}$ as a function of redshift. Solid curves are obtained using only the photometric data and constraints at $\lambda \ge 880~\micron$; dashed curves are computed using the 3.6- and 4.5-$\micron$ limits as well.
(c) Broadband photometry and best-fit SED templates obtained using the photometry at $\lambda \ge 880~\micron$. (d) Same as (c), but the IRAC 3.6 and 4.5-$\micron$ constraints are also used.}
\label{fig:photoz}
\end{center}
\end{figure}
\section{Hard X-ray Detection}
We used the primary package of \emph{Chandra}/ACIS-I data (sequence No.\ 600501, observation ID: 7063). The X-ray source at the ALMA position also appears $10''$ east of VV114E in Figure 6 of \citet{Grimes06}. The effective exposure time at the position of ALMA-J0107 was 59~ks. We count the X-ray events with a $4''$-diameter aperture and measure the background level over a $2'$-diameter circular region centered at ALMA-J0107 while masking VV114 and ALMA-J0107. We find $\simeq 100$ counts within the aperture, and background subtraction leaves $\simeq 35$ counts, which is poor statistics but just enough to simply model the spectrum. We use \textsc{xspec} \citep[version 12.8.1,][]{Arnaud96} for spectral modeling and assume a power-law spectrum for the intrinsic spectrum. For simplicity, we consider only an absorbed spectrum with the intrinsic photon index ($\Gamma$) and obscuring column density ($N_\mathrm{H}$) as free parameters, and eliminate other components such as the scattered spectrum. Note that this simple assumption may underestimate the obscuring column.
The results for the likely redshifts of $z = 2.47$ and 3.62, as well as for $z = 1.31$ for reference, are shown in Table~\ref{table:xrays}. The inferred unabsorbed luminosity, $L_\mathrm{X}$, covers the range $43.5 \le \log(L_\mathrm{X}/\mathrm{erg~s^{-1}}) \le 44.6$ for $1.31 \le z \le 3.62$, which is comparable to that of $z \sim 2$ AGN-classified SMGs \citep{Alexander05}. Using the method described in \citet{Tamura10}, we estimate an AGN bolometric luminosity of $L_\mathrm{bol} \sim (0.8$--$34) \times 10^{11} L_\sun$ for $1.31 \le z \le 3.62$. Despite a large uncertainty, this gives a mass of $\sim (0.1$--$5) \times 10^{8} (\eta_\mathrm{Edd}/0.2)^{-1} M_{\sun}$ for the accreting supermassive black hole, where $\eta_\mathrm{Edd}$ is the Eddington ratio, which is typically 0.2--0.6 in SMGs \citep{Alexander08}. The bolometric luminosity is comparable to those found in the most luminous AGNs in the local Universe (e.g., Mrk~231), but it is even less than the FIR luminosity of the host galaxy ALMA-J0107 ($L_\mathrm{FIR} \simeq 9 \times 10^{12} L_\sun$ for $T_\mathrm{dust} = 40$~K), suggesting that the large FIR luminosity of ALMA-J0107 is not dominated by the AGN but can be attributed to massive star formation activities.
\begin{deluxetable}{cccccc}
\tablewidth{0.48\textwidth}
\tablecaption{X-ray Properties of ALMA-J0107 at Plausible Redshifts \label{table:xrays}}
\tablehead{
\colhead{$z$} & \colhead{$\log{N_\mathrm{H}}$} & \colhead{$\Gamma$} & \colhead{$E$\tablenotemark{a}} & \colhead{$L_\mathrm{X}$\tablenotemark{b}} & \colhead{$L_\mathrm{bol}$} \\
& \colhead{(cm$^{-2}$)} & & (keV) & \colhead{(ergs~s$^{-1}$)} & \colhead{($10^{11}L_\sun$)}
}
\startdata
1.31 & $22.8^{+0.5}_{-1.3}$ & $2.6^{+3.0}_{-1.4}$ & 1.2--23 & $3.0 \times 10^{43}$ & 0.8 \\
2.47 & $23.2^{+0.6}_{-1.5}$ & $2.5^{+2.6}_{-1.3}$ & 1.7--35 & $1.4 \times 10^{44}$ & 7.4 \\
3.62 & $23.6^{+0.8}_{-23.6}$ & $2.4^{+3.4}_{-1.3}$ & 2.3--46 & $3.9 \times 10^{44}$ & 34
\enddata
\tablenotetext{a}{Rest-frame energy band corresponding to the observed-frame 0.5--10 keV band.}
\tablenotetext{b}{Unabsorbed luminosity in energy band $E$.}
\end{deluxetable}
\section{Number Counts of CO Emitters}
How frequently is a $\simeq$3 Jy km s$^{-1}$ $^{12}$CO emitter observed by chance? We use a mock galaxy catalog from the \textsc{s$^3$~sax} simulation \citep{Obreschkow09a,Obreschkow09b} to estimate the expected number of detections of redshifted $^{12}$CO, regardless of redshift, with a single pointing/tuning of ALMA at 100 GHz. This is a semi-analytic simulation of neutral atomic (H~\textsc{i}) and molecular (H$_2$) hydrogen in galaxies and the associated CO lines; it is based on the Millennium Simulation \citep{Springel05}, which reliably recovers galaxies with cold hydrogen masses $M$(\textsc{H~i}+H$_2) > 10^8 M_\sun$. It reproduces the local CO(1--0) luminosity function well \citep{Keres03}, whereas those at high-$z$ are not fully verified by observations;
It may underpredict the number density of $\sim 0.6$ Jy km s$^{-1}$ CO emitters at $z \sim 1.5$ by a factor of several, in comparison with CO observations of $z \sim 1.5$ BzK galaxies \citep{Daddi08, Daddi10}. The cumulative number counts of CO emitters, $\mathcal{N}(>\!S\Delta v)$, expected in a bandwidth $d\nu_\mathrm{obs}$ and a primary beam solid angle $d\Omega$ are described as
\begin{eqnarray}
\mathcal{N}(>\!S\Delta v) &=& \sum_{J=1}^{\infty} \left( \frac{dN}{dz} \right)_{z_J} dz\,d\Omega \\
&=& \frac{d\nu_\mathrm{obs}\,d\Omega}{\nu_\mathrm{obs}}
\sum_{J=1}^{\infty} \left( \frac{dN}{dz} \right)_{z_J} (1+z_J),
\end{eqnarray}
where $z_J$ is the redshift at which the $J\rightarrow J-1$ transition of CO is observed at $\nu_\mathrm{obs}$, and $(dN/dz)_{z_J}$ is the surface number density of galaxies observed in the $J\rightarrow J-1$ transition per redshift interval with line fluxes above a certain threshold, $S\Delta v$. To estimate $dN/dz$ at each $z_J$, we extract sources with an integrated intensity higher than 1 Jy km s$^{-1}$ from the simulated volumes defined by an area of $62.5 \times 62.5\,h^{-2}$ Mpc$^2$ with a depth of $\Delta z = 0.50$ at $z = 1.31$, 2.47, 3.62, 4.78, and 5.93 (we choose $\Delta z = 0.10$ for $z = 0.156$).
Consequently, we expect $\simeq 0.011$ source with $>1$ Jy~km~s$^{-1}$ per ALMA FoV (2800 arcsec$^2$) and bandwidth (7.5 GHz). Most of the sources ($\simeq 90$\%) are CO(1--0) at $z = 0.156$. The remaining 10\% are almost evenly distributed at $z = 1.31$, 2.47, and 3.62, but no source is found at $z \ge 4.78$. The SED analysis (\S~\ref{sect:photoz}) rules out the lowest redshift ($z = 0.156$), even though the probability of a chance detection appears to be highest for $z = 0.156$. Although it should be properly tested whether the \textsc{s$^3$~sax} simulation reproduces the brightest ($> 1$ Jy~km~s$^{-1}$) population of CO emitters at $z > 1$, this result implies that ALMA-J0107, likely at $z = 2.47$ or 3.62, is a very rare galaxy that falls within the ALMA bandwidth by chance (one out of $\sim 1000$ FoVs).
\section{Summary}
We presented the detection of a $^{12}$CO-emitting galaxy, ALMA-J0107, beyond the nearby merging galaxies VV114. The integrated intensity of CO and the 880-$\micron$ flux density are $3.14 \pm 0.15$~Jy km s$^{-1}$ and $11.2 \pm 0.5$ mJy, respectively. The photometric redshift analysis favors $z = 2.467$, but $z = 3.622$ is acceptable. The molecular mass and FIR luminosity at the plausible redshifts are $M(\mathrm{H_2}) \sim 8 \times 10^{10} M_{\sun}$ and $L_\mathrm{FIR} \sim 1 \times 10^{13} L_{\sun}$, respectively, which correspond to a star formation rate of $\sim 2000 M_{\sun}$ yr$^{-1}$.
We identified a hard X-ray source at the ALMA position, suggesting the presence of a luminous ($L_\mathrm{X} \sim 10^{44}$ erg s$^{-1}$) AGN behind a large hydrogen column ($23.2 \lesssim \log{[N_\mathrm{H}/\mathrm{cm^{-2}}]} \lesssim 23.6$ for the likely redshifts).
However, the intrinsic properties of the AGN (e.g., the bolometric luminosity) depend strongly on the redshift, although the FIR luminosity and molecular mass are rather insensitive to the redshift. This fact makes it difficult to investigate the power source of ALMA-J0107 and the evolutionary status of black hole growth in ALMA-J0107. It is obviously quite important to confirm the redshift through observations of the other transitions of $^{12}$CO.
This serendipitous detection of a CO-emitting galaxy demonstrates that ALMA is capable of identifying an emission-line galaxy such as ALMA-J0107. We have shown that the likelihood of stumbling across such a source is not high, and redshift determination remains a challenge even when one line and the continuum are clearly detected. Nevertheless, $\sim 1000$ pointings of ALMA Band 3 will offer an additional detection of a $> 1$~Jy km s$^{-1}$ CO source at high redshift ($z > 1$). A CO emitter at this flux level can routinely be detected at 100~GHz in only a few minutes with the full ALMA if the line happens to fall in the observing band. A complete census of background high-$z$ CO emitters in Band 3 archival cubes, as well as \textsc{[C~ii]} emitters in Band 6/7, is encouraged.
\acknowledgments
We acknowledge the anonymous referee for useful comments. We thank C.~D.\ Wilson for providing the SMA image. Y.T.\ thanks T.\ Kawaguchi for fruitful discussions.
This work was supported by JSPS KAKENHI Grant Number 25103503.
This paper makes use of the following ALMA data: ADS/JAO.ALMA\#2011.0.00467.S. ALMA is a partnership of ESO (representing its member states), NSF (USA), and NINS (Japan), together with NRC (Canada) and NSC and ASIAA (Taiwan), in cooperation with the Republic of Chile. The Joint ALMA Observatory is operated by ESO, AUI/NRAO, and NAOJ. NRAO is a facility of the NSF operated under cooperative agreement by Associated Universities, Inc.
{\it Facilities:} \facility{VLA}, \facility{ALMA}, \facility{SMA}, \facility{\emph{Spitzer} (IRAC)}, \facility{CXO (ASIS)}.
|
1,108,101,564,609 | arxiv | \section{Introduction}
|
1,108,101,564,610 | arxiv | \section{Introduction}
\setcounter{equation}{0}
We have seen how string/M theory can be exploited to study
non-perturbative dynamics of low energy supersymmetric gauge theories in
various dimensions.
One of the main motivations is to understand the D
brane dynamics where the gauge theory is realized on the worldvolume of D branes.
This work was initiated by Hanany and Witten \cite{hw} where
the mirror symmetry of $N=4$ gauge theory in three dimensions was described
by changing the position of the NS5 branes ( See,
for example, \cite{bo1} ).
As one changes the relative orientation of the two NS5 branes \cite{bar} while keeping
their common four spacetime dimensions intact, the $N=2$ supersymmetry is
broken to $N=1$ supersymmetry \cite{egk}.
By analyzing this brane configuration they \cite{egk} described and checked a stringy
derivation of Seiberg's duality for $N=1$ supersymmetric $SU(N_c)$ gauge theory with
$N_f$ flavors.
This result was generalized to the brane configurations with orientifolds which
explain $N=1$ supersymmetric theories with gauge group $SO(N_c)$ or
$Sp(N_c)$ \cite{eva} ( See also \cite{ov} for a relevant geometrical approach ).
Both the D4 branes
and NS5 branes used in type IIA string theory
originate from the fivebrane \cite{w1} of M theory. That is, D4 brane is an M theory
fivebrane wrapped over $\bf{S^1}$ and NS5 brane is the one on
$ \bf{R^{10} \times S^1} $. In order to insert D6 branes
one studies a multiple
Taub-NUT space \cite{town}
whose metric is complete and smooth.
The singularities
are removed in eleven dimensions where the brane configuration becomes smooth, the
D4 branes and NS5 branes being the unique M theory fivebrane and
the D6 branes being the Kaluza-Klein monopoles.
The property of $N=2$ supersymmetry
in four dimensions requires that the worldvolume of M theory fivebrane is
${\bf{R^{1,3}}} \times \Sigma$ where $\Sigma$ is uniquely identified with the
curves that
occur to the solutions for Coulomb branch of the four dimensional field theory.
Further generalizations of this configuration with orientifolds were
studied in \cite{lll}.
The exact low energy description of $N=1$ supersymmetric $SU(N_c)$ gauge
theories with $N_f$ flavors in four dimensions have been found in \cite{w2} ( See
also \cite{aotaug,aotsept} for theories with orientifolds ).
This approach has been developed further and
used to study the moduli space of vacua of confining phase of $N=1$ supersymmetric
$SU(N_c)$ gauge theories in four dimensions \cite{dbo}.
In terms of brane configuration of IIA string
theory, this was done by taking
multiples of NS'5 branes rather than a single NS'5 brane. In field theory, we
regard this as taking the superpotential
$\Delta W = \sum_{k=2}^{N_{c}} \mu_{k} \mbox{Tr} (\Phi^{k})$. This
perturbation lifts the non singular locus of the $N=2$ Coulomb branch while
at singular locus there exist massless dyons that can condense due to the perturbation.
In the present work we extend the results of \cite{dbo,aotdec} to
$N=1$ supersymmetric theories
with gauge group $Sp(N_{c})$ and also generalize the previous
work \cite{aotaug} which dealt with a single NS'5 brane
in the sense that we are considering {\it multiple copies} of NS'5 branes.
We will describe how the field theory analysis \cite{ty}
obtained in the low energy superpotential
gives rise to the geometrical structure in $(v, t, w)$ space.
The minimal form for the effective
superpotential obtained by ``integrating in" is not exact \cite{intril},
in general, for several massless dyons.
Note that
the intersecting branes in string/M theory
have been studied to obtain much information about
supersymmetric gauge theories with different gauge groups and in various
dimensions \cite{ah}.
\section{Field Theory Analysis}
\setcounter{equation}{0}
$\bullet$ $N=2$ Theory
Let us consider $N=2$ supersymmetric $Sp(N_{c})$ gauge theory
with matter in the ${\bf 2 N_c}$
dimensional representation of the gauge group $Sp(N_c)$. In terms
of $N=1$ superfields, $N=2$ vector multiplet consists of a field strength
chiral multiplet $W_{\alpha}^{ab}$ and a scalar chiral multiplet
$\Phi_{ab}$, both in the
adjoint representation. The quark hypermultiplets
are made of a chiral multiplet $Q^{i}_{a}$ which couples to the
Yang-Mills fields where
$i = 1,\cdots ,2N_{f}$ are flavor indices
and $a = 1, \cdots , 2N_{c}$ are color indices.
The
$N=2$ superpotential takes the form,
\begin{equation}
\label{super}
W_{N=2} = \sqrt{2} Q^{i}_{a} \Phi^a_b J^{bc} Q^{i}_{c}
+ \sqrt{2} m_{ij} Q^{i}_{a} J^{ab} Q^{j}_{b},
\end{equation}
where $J_{ab}$ is the symplectic metric
$( {0 \atop -1 }{ 1 \atop 0} ) \otimes {\bf 1_{N_c \times N_c}} $
used to raise and lower
$Sp(N_{c})$ color indices ( $
{\bf 1_{N_c \times N_c}}$ is the $N_c \times N_c$ identity matrix )
and $m_{ij}$ is an antisymmetric quark mass matrix
$\label{mass}( { 0 \atop 1 }{ -1 \atop 0 } )
\otimes \mbox{diag} ( m_{1}, \cdots, m_{N_f} ) $.
Classically, the global symmetries are the flavor
symmetry $O(2N_{f})$ when there are no quark masses, in addition to
$U(1)_{R}\times SU(2)_{R}$ chiral R-symmetry.
The theory is asymptotically free for the region $N_{f} < 2N_{c}+2$ and generates
dynamically a strong coupling scale $\Lambda_{N=2}$.
The instanton factor
is proportional to $\Lambda_{N=2}^{2N_{c}+2-N_{f}}$. Then the
$U(1)_{R}$ symmetry is anomalous and is broken down to a discrete
$\bf{Z_{2N_{c}+2-N_{f}}}$ symmetry by instantons.
The $N_c$ complex dimensional moduli space of vacua
contains the Coulomb and Higgs branches.
The Coulomb branch is parameterized by the gauge invariant order parameters
\begin{eqnarray}
u_{2k}=<\mbox{Tr}(\phi^{2k})>, \;\;\;\;\; k=1, \cdots, N_c,
\label{u2k}
\end{eqnarray}
where $\phi$ is the scalar field in $N=2$ chiral multiplet.
Up to a gauge transformation $\phi$ can be diagonalized to a
complex matrix,
$<\phi>=\mbox{diag} ( A_1, \cdots, A_{N_c} )$ where $A_i=
( { a_i \atop 0 }{ 0 \atop -a_i} )$.
At a generic point the vevs of
$\phi$ breaks the $Sp(N_c)$ gauge symmetry
to $U(1)^{N_c}$ and the dynamics of the theory is that of an
Abelian Coulomb phase. The Wilsonian effective Lagrangian in the low
energy can be made of the multiplets of $A_i$ and $W_i$ where
$i=1, 2, \cdots, N_c$.
If $k$ $a_i$'s are equal and nonzero then there
exists an enhanced $SU(k)$ gauge symmetry. When they are also zero,
an enhanced $Sp(k)$ gauge symmetry appears.
The quantum moduli space is described by a family of
hyperelliptic spectral curves \cite{aps}
with associated
meromorphic one forms,
\begin{eqnarray}
y^2 = \left( v^2 C_{2N_c}(v^2)+\Lambda_{N=2}^{2N_c+2-N_f}
\prod_{i=1}^{N_f} m_i \right)^2-
\Lambda_{N=2}^{4N_c+4-2N_f} \prod_{i=1}^{N_f}(v^2-m_i^2),
\label{curve}
\end{eqnarray}
where $C_{2N_c}(v^2)$ is a degree $2N_c$ polynomial in $v$ with coefficients
depending on the moduli $u_{2k}$ appearing in (\ref{u2k}) and
$m_i ( i=1, 2, \cdots, N_f )$ is the
mass of quark\footnote{
Note that the polynomial $C_{2N_c}(v^2)$ is an even function of $v$ which will be identified
with a complex coordinate $( x^4, x^5 ) $ directions in next section and
is given by
$C_{2N_c}(v^2)= v^{2N_c} +\sum_{i=1}^{N_c} s_{2i} v^{2(N_c-i)}=
\prod_{i=1}^{N_c} (v^2-a_i^2) $
where $s_{2k}$ and $u_{2k}$ are related each other by so-called Newton's formula
$2k s_{2k}+ \sum_{i=1}^{k} s_{2k-2i} u_{2i} =0, ( k=1, 2, \cdots, N_c )$
with $s_0=1$. From this recurrence relation,
we obtain $ \partial s_{2j}/\partial u_{2k}=- s_{2(j-k)}/2k
\;\;\; \mbox{for} \;\;\; j \geq k. $}.
$\bullet$ Breaking $N=2$ to $N=1$ ( Pure Yang-Mills Theory )
We are interested in a microscopic $N=1$ theory mainly in a phase
with a single confined photon
coupled to the light dyon hypermultiplet while the photons for the rest are free.
By taking a tree level superpotential perturbation
$\Delta W$ of \cite{ty} made out of the adjoint fields in the vector
multiplets to the $N=2$
superpotential (\ref{super}), the $N=2$ supersymmetry
can be broken to $N=1$ supersymmetry. That is,
\begin{eqnarray}
W = W_{N=2}+\Delta W, \;\;\;
\Delta W = \sum_{k=1}^{N_c-1} \mu_{2k} \mbox{Tr} (\Phi^{2k}) +
\mu_{2N_c} s_{2N_c},
\label{superpo}
\end{eqnarray}
where\footnote{ Our $\mu_{2k}$ is the same as their $g_{2k}/2k$ in \cite{ty}. }
$\Phi$ is the adjoint $N=1$ superfields in the $N=2$ vector multiplet and
Note that the $\mu_{2N_c}$ term is not associated with $u_{2N_c}$ but
$s_{2N_c}$ which is proportional to the sum of $u_{2N_c}$ and the polynomials
of other $u_{2k} ( k < N_c )$ according to the recurrence relation.
Then microscopic $N=1$ $Sp(N_c)$ gauge theory is obtained from $N=2$ $Sp(N_c)$
Yang-Mills theory perturbed by $\Delta W$.
Let us first study $N=1$ pure $Sp(N_c)$ Yang-Mills theory with tree level
superpotential (\ref{superpo}).
Near the singular points where dyons become massless,
the macroscopic superpotential of the theory is given by
\begin{eqnarray}
W = \sqrt{2} \sum_{i=1}^{N_c-1} M_i A_i M_i + \sum_{k=1}^{N_c-1}
\mu_{2k} U_{2k} +\mu_{2N_c} S_{2N_c}.
\label{supereven}
\end{eqnarray}
We denote by $A_i$ the $N=2$ chiral superfield of $U(1)$
gauge multiplets, by $M_i$ those of $N=2$ dyon hypermultiplets,
by $U_{2k}$ the chiral superfields corresponding to $\mbox{Tr} (\Phi^{2k}) $ ( and
by $S_{2k}$ the chiral superfields which are related to $U_{2k}$ ),
in the low energy theory. The vevs of the lowest
components of $A_i, M_i, U_{2k}, S_{2N_c} $ are written as
$a_i, m_{i, dy}, u_{2k}, s_{2N_c}$
respectively.
Recall that $N=2$ configuration is invariant under the group $U(1)_R$
and $SU(2)_R$ corresponding to the chiral R-symmetry of the field theory.
However,
in $N=1$ theory $SU(2)_R$ is broken to $U(1)_J$.
The equations of motion obtained by varying the superpotential with respect to
each field read as follows\footnote{ $
\mu_{2k}=
-\sqrt{2} \sum_{i=1}^{N_c-1} m_{i,dy}^2 \partial a_i/\partial u_{2k},
\mu_{2N_c} =
-\sqrt{2} \sum_{i=1}^{N_c-1} m_{i,dy}^2 \partial a_i/\partial s_{2N_c} $
and $ a_i m_{i, dy}=0.$}.
At a generic point in the moduli space, no massless fields occur (
$a_i\neq 0$ for $ i=1, \cdots, N_c-1 $ ) which
implies $m_{i, dy} =0$ and so $\mu_{2k}$ and $ \mu_{2N_c}$
vanish. Then we obtain
the moduli space of vacua of $N=2$ theory.
On the other hand, we are considering a singular point in the moduli space where
$l$ mutually local dyons are massless.
This means that $l$ one cycles
shrink to zero.
The right hand side of (\ref{curve}) becomes,
\begin{eqnarray}
y^2 =
\left( v^2 C_{2N_c}(v^2)+\Lambda_{N=2}^{2N_c+2-N_f} \right)^2-
\Lambda_{N=2}^{4N_c+4-2N_f} =
\prod_{i=1}^{l}(v^2-p_i^2)^2
\prod_{j=1}^{2N_c+2 -2l}(v^2-q_j^2),
\label{curve1}
\end{eqnarray}
with $p_i$ and $q_j$ distinct.
A point in the $N=2$ moduli space of vacua is characterized by $p_i$ and $q_j$.
The degeneracy of this curve is checked by explicitly
evaluating both $y^2$ and $\partial y^2/ \partial v^2$
at the point $v= \pm p_i,$ leading to vanish.
Since $a_i =0$ for $i=1, \cdots, l$ and $a_i\neq 0$ for
$ i=l+1, \cdots, N_c-1 $,
we get $
m_{i, dy}=0, ( i = l+1, \cdots , N_c-1 ) $
while $m_{i, dy} ( i=1, \cdots, l )$ are not constrained.
We will see how the vevs $m_{i, dy}$ originate from the information about
$N=2$ moduli space of vacua which is encoded by $p_i$ and $q_j$.
We assume that the matrix $\partial a_i/ \partial u_{2k}$ is nondegenerate.
In order to calculate $\partial a_i/ \partial u_{2k}$,
we need the relation
between $ \partial a_i/ \partial s_{2k}$ and the period integral
on a basis of holomorphic one forms on the curve, $
\partial a_i/\partial s_{2k}= \oint_{\alpha_i} v^{2(N_c-k)}dv/y $.
We can express the generating function\footnote{
By plugging $y$ of (\ref{curve1}) and integrating along one cycles
around $v=\pm p_i $, it turns out $
\partial a_i/\partial s_{2k} = p_i^{2(N_c-k)}/
\prod_{j \neq i}^l (p_i^2-p_j^2)\prod_t^{2N_c+2-2l}(p_i^2-q_t^2)^{1/2}, $
since the $l$ one cycles shrink to zero.
Then,
we arrive at the following relation,
$ 2k \mu_{2k} =
\sum_{i=1}^l \sum_{j=1}^{N_c} s_{2(j-k)} p_i^{2(N_c-j)}
\omega_i,$ where $\omega_i = \sqrt{2} m_{i,dy}^2/
\prod_{s \neq i}^l (p_i^2-p_s^2) \prod_t^{2N_c+2-2l} (p_i^2-q_t^2)^{1/2}.$}
for the $\mu_{2k}$ in terms of $\omega_i$
as follows:
\begin{eqnarray}
\sum_{k=1}^{N_c} 2k \mu_{2k} v^{2(k-1)} & = & \sum_{k=1}^{N_c} \sum_{i=1}^l
\sum_{j=1}^{N_c} v^{2(k-1)} s_{2(j-k)} p_i^{2(N_c-j)} \omega_i \nonumber \\
& = &
\sum_{k=-\infty}^{N_c}
\sum_{i=1}^l
\sum_{j=1}^{N_c} v^{2(k-1)} s_{2(j-k)} p_i^{2(N_c-j)} \omega_i +
{\cal O}(v^{-2}) \nonumber \\
& = & \sum_{i=1}^l \frac{ C_{2N_c}(v^2)}{(v^2-p_i^2)} \omega_i
+ {\cal O}(v^{-2}).
\label{mu2k}
\end{eqnarray}
Therefore we can find the parameter $\mu_{2k}$ by reading off the right hand side
of (\ref{mu2k}).
This result determines whether a point in the $N=2$ moduli space of vacua
classified by the set of $p_i$ and $ q_j$ in (\ref{curve1}) remains as an $N=1$ vacuum
after the perturbation, for given a set of perturbation parameters
$\mu_{2k}$ and $ \mu_{2N_c}$.
We will see in section 3 that this corresponds to one of the boundary conditions on
a complex coordinate in $( x^8, x^9 )$ directions as
$v$ goes to infinity.
In order to make the comparison with the brane picture,
it is very useful to define the polynomial
$H (v^2)$ of degree $2l-4$ by
\begin{eqnarray}
\sum_{i=1}^l \frac{\omega_i}{v^2(v^2-p_i^2)} =
\frac{2H (v^2)}{ \prod_{i=1}^l (v^2-p_i^2)}.
\label{heven}
\end{eqnarray}
At a given point $p_i$ and $q_j$ in the $N=2$ moduli space of vacua,
$H (v^2)$ determines the dyon vevs, in other words,
\begin{eqnarray}
m_{i,dy}^2 = \sqrt{2} p_i^2 H (p_i^2) \prod_m (p_i^2-q_m^2)^{1/2},
\label{dyoneven}
\end{eqnarray}
which will be described in terms of the geometric brane picture in next section.
Therefore, all the vevs of dyons $m_{i, dy} ( i=1, \cdots, l )$ are found as well as
$m_{i, dy} ( i=l+1, \cdots, N_c-1 )$ which are zero.
$\bullet$ The Meson Vevs
Let us discuss the vevs of the meson field along the singular locus of the Coulomb
branch. This is due to the nonperturbative effects of $N=1$ theory and obviously
was zero before the perturbation (\ref{superpo}). We will see the property of exactness
in field theory analysis in the context of M theory fivebrane in section 4. Equivalently,
the exactness of superpotential for any values of the parameters is to assume
$W_{\Delta}=0$.
We will follow the method presented in \cite{efgir}.
Let us consider the vacuum where one massless dyon exists
with unbroken $SU(2) \times U(1)^{N_c-1}$ where
\begin{eqnarray}
J \Phi^{cl}= \left( { 0 \atop 1 }{ 1 \atop 0 } \right) \otimes
\mbox{diag} (a_1, a_1, a_2, \cdots, a_{N_c-1}),
\label{vacuaeven}
\end{eqnarray}
and the chiral multiplet $Q=0$ and $J$ is the symplectic metric as before.
These eigenvalues of $\Phi$ can be obtained by differentiating the superpotential
(\ref{superpo}) with respect to $\Phi$ and setting the chiral multiplet $Q=0$.
The vacua with classical $SU(2) \times U(1)^{N_c-1}$ group are those with
two eigenvalues equal to $a_1$ and the rest given by $a_2, a_3, \cdots, a_{N_c-1}$.
It is known from \cite{ty} that, if using $s_{2N_c}$ in the superpotential
perturbation rather than $u_{2N_c},$
the degenerate eigenvalue of $\Phi$
is obtained to be
\begin{eqnarray}
a_1^2=\frac{(N_c-1) \mu_{2(N_c-1)}}{N_c \mu_{2N_c}}.
\label{a1}
\end{eqnarray}
We will see in section 3 that
the asymptotic behavior of a complex coordinate in $( x^8, x^9 )$ directions for large $v$
determines this degenerate eigenvalue by using the condition for generating function of
$\mu_{2k}$ (\ref{mu2k}).
The scale matching condition between the high energy $Sp(N_c)$ scale
$\Lambda_{N=2}$ and the low energy $SU(2)$ scale $\Lambda_{SU(2), N_{f}}$
is related each other.
After integrating out $SU(2)$ quarks we obtain the scale matching
between $\Lambda_{N=2}$ and $\Lambda_{SU(2)}$ for pure $N=1$ $SU(2)$
gauge theory. That is,
\begin{eqnarray}
\Lambda_{SU(2)}^6=\left( \frac{2N_c^2 \mu_{2N_c}^2}{(N_c-1) \mu_{2(N_c-1)}} \right)^2
\Lambda_{N=2}^
{4(N_c+1)-2N_f} \mbox{det} (a_1^2-m^2),
\end{eqnarray}
where matrix $a_1$ means
$( { 0 \atop 1 }{ 1 \atop 0 } ) \otimes a_1$ and
matrix $m$ being
$( { 0 \atop 1 }{ -1 \atop 0 } )
\otimes \mbox{diag} ( m_{1}, \cdots, m_{N_f} ) $.
Then the full exact low energy effective superpotential is given by
\begin{eqnarray}
W_L =
\sum_{k=1}^{N_c-1} \mu_{2k} \mbox{Tr} (\Phi_{cl}^{2k}) +
\mu_{2N_c} s^{cl}_{2N_c}
\pm 2 \Lambda_{SU(2)}^3,
\label{lowsuperpo}
\end{eqnarray}
where the last term is generated by
gaugino condensation in the low energy $SU(2)$ theory ( the sign reflects the vacuum
degeneracy ).
In terms of the original $N=2$ scale, we can write it as
\begin{eqnarray}
W = \sum_{k=1}^{N_c-1} \mu_{2k} \mbox{Tr} (\Phi_{cl}^{2k})
+\mu_{2N_c} s^{cl}_{2N_c} \pm \frac{4N_c^2 \mu_{2N_c}^2}{(N_c-1) \mu_{2(N_c-1)}}
\Lambda_{N=2}^{2(N_c+1)-N_f} \mbox{det} (a_1^2-m^2)^{1/2}.
\end{eqnarray}
Therefore,
one obtains the vevs of meson
$M_i=Q^i_a J^{ab} Q^i_b$ which gives
\begin{eqnarray}
M_i=\frac{\partial W}{\partial m_i^2}=\pm \frac{4 \Lambda_{N=2}^
{2(N_c+1)-N_f}}{\sqrt{2}(a_1^2-m_i^2)}
\frac{N_c^2 \mu_{2N_c}^2}{(N_c-1) \mu_{2(N_c-1)}} \mbox{det} (a_1^2-m^2)^{1/2},
\label{meson}
\end{eqnarray}
where $a_1$ is given by ({\ref{a1}).
We will see in section 3 that the finite value of a complex coordinate
in $( x^8, x^9 )$ directions corresponds to the above vevs of meson
when $v \rightarrow \pm m_i$ and the other complex coordinate related to $( x^6, x^{10} )$
directions vanishes.
\section{ Brane Configuration from M Theory }
\setcounter{equation}{0}
$\bullet$ Type IIA Brane Configuration
We study the theory with the superpotential perturbation
$\Delta W$ (\ref{superpo}) by analyzing M theory fivebranes.
Let us first describe them in the type IIA brane configuration.
Following the procedure of \cite{egk}, the brane
configuration in $N=2$ theory consists of
three kind of branes: the two parallel NS5
branes extend in the directions $(x^0, x^1, x^2, x^3, x^4, x^5)$, the D4 branes
are stretched between two NS5 branes and extend over $(x^0, x^1, x^2, x^3)$ and
are finite in the direction of $x^6$, and the D6 branes extend in the directions
$(x^0, x^1, x^2, x^3, x^7, x^8, x^9)$.
In order to study
symplectic gauge groups, we consider an O4
orientifold which is parallel to the D4 branes in order to keep the
supersymmetry and is not of finite extent in $x^6$ direction. The D4 branes
is the only brane which is not intersected by this O4 orientifold.
The orientifold
gives a spacetime reflection as $(x^4, x^5, x^7, x^8, x^9) \rightarrow
(-x^4, -x^5, -x^7, -x^8, -x^9)$, in addition to the gauging of worldsheet
parity $\Omega$.
The fixed points of the spacetime symmetry define this O4 planes.
Each object which does not lie at the fixed points,
must have its mirror. Thus NS5 branes have a mirror in $(x^4,
x^5)$ directions and D6 branes do a mirror in $(x^7, x^8, x^9)$ directions.
In order to realize the $N=1$ theory with a perturbation (\ref{superpo})
we can think of a single NS5 brane and {\it multiple copies } of NS'5 branes
which are orthogonal to a NS5 brane
with worldvolume, $(x^0, x^1, x^2, x^3, x^8, x^9)$ and between them there
exist D4 branes intersecting D6 branes. The number of NS'5 branes is
$N_c-1$ by identifying the power
of adjoint field appearing in the superpotential (\ref{superpo}).
The brane description for $N=1$ theory with $\mu_{2N_c}=0$
has been studied in the paper of \cite{egk} in type IIA brane configuration. In this case,
all the couplings, $\mu_{2k}$ can be regarded as tending uniformly to infinity. On the other
hand, in M theory configuration there will be no such restrictions.
$\bullet$ M Theory Fivebrane Configuration
Let us describe how the above brane configuration can be embedded in
M theory in terms of
a single M theory fivebrane whose worldvolume is
${\bf R^{1,3}} \times \Sigma$ where $\Sigma$ is identified with Seiberg-Witten
curves \cite{aps} that determine the solutions to Coulomb branch
of the field theory.
As usual, we write $s=(x^6+i x^{10})/R, t=e^{-s}$
where $x^{10}$ is the eleventh coordinate of M theory which is compactified
on a circle of radius $R$. Then the curve $\Sigma$, describing
$N=2$ $Sp(N_c)$ gauge theory with $N_f$ flavors,
is given \cite{lll} by an equation in $(v, t)$ space
\begin{eqnarray}
t^2 - 2 \left( v^2 C_{2N_c}(v^2, u_{2k})+\Lambda_{N=2}^{2N_c+2-N_f} \prod_{i=1}^{
N_f} m_i \right) t +
\Lambda_{N=2}^{4N_c+4-2N_f}
\prod_{i=1}^{N_f} (v^2 -m_i^2) = 0.
\label{cur}
\end{eqnarray}
It is easy to check that this description is the same as (\ref{curve})
under the identification $
t= y + v^2 C_{2N_c}(v^2, u_{2k})+\Lambda_{N=2}^{2N_c+2-N_f} \prod_{i=1}^{N_f} m_i. $
By adding $\Delta W$ which corresponds to the adjoint chiral multiplet,
the $N=2$ supersymmetry will be broken to $N=1$.
To describe the corresponding brane configuration in M theory,
we introduce complex coordinates
\begin{eqnarray}
v= x^4 +i x^5, \;\;\; w=x^8+i x^9.
\end{eqnarray}
To match the superpotential perturbation $\Delta W $ (\ref{superpo}),
we propose the following boundary conditions
\begin{eqnarray}
\begin{array}{ccl}
w^{2} & \rightarrow & \sum_{k=2}^{N_c} 2k \mu_{2k} v^{2(k-1)}\;\;\;\mbox{as}~
v \rightarrow \infty,~
t \sim \Lambda_{N=2}^{2(2N_c+2-N_f)}v^{2N_f-2N_c-2},\\
w & \rightarrow & 0\;\;\;\mbox{as}~
v \rightarrow \infty,~
t \sim v^{2N_c+2}. \\
\end{array}
\label{boundary}
\end{eqnarray}
After deformation, $SU(2)_{7,8,9}$ is broken to $U(1)_{8,9}$ if $\mu_{2k}$
has the charges $(4-4k, 4)$ under $U(1)_{4,5} \times U(1)_{8,9}$.
When we consider now only $k=2$, we obtain that
$w^2 \sim \mu_{4} v^{2}$ as $v \rightarrow \infty$ which is
the same as the relation $w \rightarrow \mu v$ in
\cite{aotaug} after we identify $\mu_{4}$ with $\mu^{2}$. This
identification comes also from the consideration of $U(1)_{4,5}$ and $U(1)_{8,9}$
charges of $\mu$ and $\mu_{4}$.
After perturbation, only the singular part of the $N=2$ Coulomb branch with
$l$ or less mutually local massless dyons remains in the moduli space of vacua.
Let us construct the M theory fivebrane configuration satisfying
the above boundary conditions and assume
that $w^{2}$ is a rational function of $v^{2}$ and $t$.
Our result is really similar to the case of \cite{dbo,aotdec} and we
will follow their notations.
We write $w^{2}$ as follows
\begin{eqnarray}
w^{2}(t,v^2) = \frac{a(v^2) t + b(v^2)}{c(v^2) t + d(v^2)},
\end{eqnarray}
where $a, b, c, d$ are arbitrary polynomials of $v^2$ and $t$ satisfying
(\ref{cur}).
Now we calculate the following two quantities\footnote{ Using the two
solutions of $t$, denoted by $t_+$ and $t_-$ satisfying (\ref{cur}), $
w^{2}(t_+(v^2),v^2)+w^{2}(t_-(v^2),v^2) = (2acG+2adC+2bcC+2bd)/( c^2 G + 2 cdC + d^2)
$
and $
w^{2}(t_+(v^2),v^2)-w^{2}(t_-(v^2),v^2) = 2(ad-bc)S\sqrt{T}/(c^2 G +2 cdC + d^2)
$
where we define $
C \equiv v^2 C_{2N_c}(v^2, u_{2k})+\Lambda_{N=2}^{2N_c+2-N_f}
\prod_{i=1}^{N_f} m_i, $
and $ G \equiv \Lambda_{N=2}^{4N_c-4-2N_f}
\prod_{i=1}^{N_f} (v^2 -m_i^2) $
implying that $
C^2(v^2) - G(v^2) \equiv S^2(v^2) T(v^2) $
where $
S(v^2)=\prod_{i=1}^{l} (v^2-p_i^2), T(v^2)=\prod_{j=1}^{2N_c+2-2l} (v^2-q_j^2) $
with all $p_i $ and $ q_j$'s different.}.
Since $w^{2}$ has no poles for finite value of $v^2$,
$w^{2}(t_+(v^2),v^2) \pm w^{2}(t_-(v^2),v^2)$ also does not have poles which leads to
arbitrary polynomials $H(v^2)$ and $N(v^2)$ given by
\begin{eqnarray}
\frac{acG + adC + bcC +bd}{c^2 G + 2 cdC + d^2} = N, \;\;\;
\frac{(ad-bc)S}{(c^2 G +2 cdC + d^2)} = H.
\end{eqnarray}
It will turn out that the function $H(v^2)$ is exactly
the same as the one (\ref{heven})
defined in field theory analysis.
By making a shift of $a \rightarrow a +Nc, b \rightarrow b +Nd$ due to the
arbitrariness of the polynomials $a$ and $b$ and
combining all the information for $b$ and $d$ ( See, for details, \cite{dbo,aotdec} ), we get
the most general rational function $w^2$ which has no poles for finite
value of $v^2$,
\begin{eqnarray}
w^{2} = N + \frac{at + cHST - aC}{ct - c C + aS/H},
\end{eqnarray}
where $N, a, c, H$ are arbitrary polynomials.
As we choose two $w^2$'s , each of them possessing different polynomials $a$ and $c$
and subtract them, then the numerator of it will be proportional to $t^2-2 C t+G$ which
vanishes according to (\ref{cur}). This means $w^2$ does not depend on $a$ and $c$.
Therefore,
when $c=0$, the form of $w^2$
is very simple.
The general solution for $w^2$ is
\begin{eqnarray}
w^{2} = N(v^2) + \frac{H(v^2) }{\prod_{i=1}^{l}(v^2-p_i^2)}
\left( t- \left( v^2 C_{2N_c}(v^2, u_{2k})+\Lambda_{N=2}^{2N_c+2-N_f}
\prod_{i=1}^{N_f} m_i \right) \right),
\end{eqnarray}
where $H(v^2)$ and $N(v^2)$ are arbitrary polynomials of $v^2$.
Now we want to impose the boundary conditions on $w^2$ in
the above general solution. From the relation,
\begin{eqnarray}
w^{2}(t_{\pm}(v^2),v^2) = N \pm H\sqrt{T},
\label{nht}
\end{eqnarray}
when we impose the boundary condition
$ w \rightarrow 0 $ for $v \rightarrow \infty,~
t = t_-(v) \sim v^{2N_c+2}$
the polynomial $N(v^{2})$ is determined as follows,
\begin{eqnarray}
N(v^2)=\left[ H(v^2) \sqrt{T(v^2)} \right]_+=
\left[ H(v^2) \prod_{j=1}^{2N_c+2-2l}(v^2-q_j^2)^{1/2} \right]_+,
\end{eqnarray}
where $\left[ H(v^2) \sqrt{T(v^2)} \right]_+$ means only nonnegative power of $v^2$
when we expand around $v = \infty$.
The other boundary condition tells that $w^2$ behaves
as
$ w^{2} \rightarrow \sum_{k=1}^{N_c} 2k \mu_{2k} v^{2(k-1)} $ from (\ref{boundary}).
Then by expanding $w^2$ in powers of $v^2$ we can identify $H(v^2)$ with
parameter $\mu_{2k}$. Using
$T^{1/2}= (t-( v^2 C_{2N_c}(v^2, u_{2k})+\Lambda_{N=2}^{2N_c+2-N_f} \prod_{i=1}^{
N_f} m_i ) )/S$ and $t=2 ( v^2 C_{2N_c}+ \cdots )$ from (\ref{cur}) we get
\begin{eqnarray}
w^{2} & =& \left[2 H(v^2) \sqrt{T(v^2)} \right]_+ + {\cal O}(v^{-2}) \nonumber \\
& = & \frac{2 H(v^2)}{\prod_{i=1}^{l}(v^2-p_i^2)}
\left( v^2 C_{2N_c}(v^2, u_{2k})+\Lambda_{N=2}^{2N_c+2-N_f}
\prod_{i=1}^{N_f} m_i \right)
+ {\cal O}(v^{-2}) \nonumber \\
& = &
\sum_{i=1}^l \frac{ C_{2N_c} (v^2) \omega_i}{(v^2-p_i^2)} + {\cal O}(v^{-2}) =
\sum_{k=1}^{N_c} 2k \mu_{2k} v^{2(k-1)},
\end{eqnarray}
where we used the definition of $H $ in (\ref{heven})
and the generating function of $\mu_{2k}$ in (\ref{mu2k}). From this result
one can find the explicit form of $H(v^2)$ in terms of $\mu_{2k}$ by comparing both sides
in the above relation. This is an explanation for field theory results
of (\ref{mu2k}) and (\ref{heven}) which determine the $N=1$ moduli space of vacua
after the perturbation, from the point of view of M theory fivebrane.
It reproduces the equations
which determine the vevs of massless dyons along the singular locus.
The dyon vevs $m_{i, dy}^2$, given by (\ref{dyoneven}) $
m_{i,dy}^2 = \sqrt{2} p_i^2 H (p_i^2) \sqrt{T(p_i^2)}, $
are nothing but the difference between the two finite values of $v^2 w^2$.
This can be seen by taking $v=\pm p_i$ in
(\ref{nht}).
The $N=2$ curve of (\ref{cur}) and (\ref{curve1}) contains double points
at $v=\pm p_i$
and $t=C(p_i^2)$. The perturbation $\Delta W$ of (\ref{superpo}) splits
these into
separate points in $(v, t, w)$ space and the difference in $v^2w^2$ between
these points becomes the dyon vevs.
This is a geometric interpretation of dyon vevs
in M theory brane configuration.
By noting that $w^{2}$ satisfies $
w^4 -2 N w^2 + N^2 - T H^2 = 0,$
and restricting the form of $N, T$ and $H$ like as $N \sim c_{1}v^2 + c_{2},
T \sim c_{3} v^8 + c_{4} v^{6} + c_{5} v^{4} + c_{6} v^2 +c_7,
H \sim \frac{c_{8}}{v^{2}}$, it leads to $
w^4 + (c_9 + c_{10} v^2) w^2 +c_{11} =0 $
for some constants $c_{i} ( i = 1,\cdots, 11 )$.
Then we can solve for $v^2$ in terms of $w^2$ to reproduce the result of
\cite{aotaug}.
As all the couplings $\mu_{2k}$ are becoming very large, $H(v^2)$ and $N(v^2)$
go to infinity. The term of $N^2-T H^2$ goes to zero as we take the limit
of $\Lambda_{N=2} \rightarrow 0$. This tells us that $w^2$ becomes
$( N^2-T H^2 )/2N$ and as $N(v^2)$ goes to zero, $w^2 \rightarrow \infty$
showing the findings in \cite{egk}.
\section{The Meson Vevs in M Theory }
\setcounter{equation}{0}
We continue to study for the meson vevs from the singularity structure
of $N=2$ Riemann surface. The vevs of meson will depend on the moduli structure
of $N=2$ Coulomb branch ( See, for example, (\ref{wi2}) ).
Also, the finite values of $w^2$ can be determined fully by using the property of
boundary conditions of $w^2$ when $v$ goes to be very large.
Let us consider the case of finite $w^2$ at $t=0, v=\pm m_i$ and we want
to compare with the meson
vevs we have studied in (\ref{meson}).
At a point where there exists a single massless dyon ( in other words,
by putting $l=1$ )
and recalling the definition of $T(v^2)$, we have for Yang-Mills with matter
\begin{eqnarray}
\left( v^2 C_{2N_c}(v^2)+\Lambda_{N=2}^{2N_c+2-N_f} \prod_{i=1}^{N_f} m_i \right)^2-
\Lambda_{N=2}^{4N_c+4-2N_f} \prod_{i=1}^{N_f}(v^2-m_i^2) =
(v^2-p_1^2)^2 T(v^2),
\label{onedyon}
\end{eqnarray}
and the function $w^2$ according to (\ref{nht}) reads
\begin{eqnarray}
w^2 = \left[ \frac{h}{v^2} \sqrt{T(v^2)} \right]_+ \pm \frac{h}{v^2} \sqrt{T(v^2)},
\end{eqnarray}
where in this case $l=1$ means that the polynomial $v^2 H(v^2)$ has the degree of zero and
we denote it by a constant $h$\footnote{ From (\ref{onedyon}) we see
for $N_f < 2N_c+2,
\sqrt{T(v^{2})}/v^{2} = C(v^{2})/v^{2}(v^{2}-p_1^{2}) + {\cal O}(v^{-4}) $
and we decompose $C$ as $
C(v^{2})/v^{2} = C(p_1^{2})/p_1^{2} + (v^2-p_1^2)\tilde{C}(v^2)$
for some polynomial $v^2 \tilde{C}(v^2)$ of degree $2N_c$.
This means that the coefficients of $\tilde{C}(v^2)$ can be fixed from
the explicit form of the polynomial
$C(v^2)$.
The part with nonnegative powers
of $v^2$ in
$\sqrt{T(v^{2})}/v^{2}$ becomes $\tilde{C}(v^2)$
as follows $
\sqrt{T(v^{2})}/v^{2} = \tilde{C}(v^2) +
{\cal O}(v^{-2}) \longrightarrow \left[\sqrt{T(v^{2})}/v^{2} \right]_+ =
\tilde{C}(v^2) $.}.
Thus as $v \rightarrow \pm m_i$ the finite value of $w^2$,
denoted by $w_i^2$ can be written as
\begin{eqnarray}
w^2_i = w^2(v^2 \rightarrow m_i^2) =
h \tilde{C} (m_i^2) \pm \frac{h}{m_i^2} \sqrt{T(m_i^2)}.
\label{wi}
\end{eqnarray} From (\ref{onedyon}), the relation
$\sqrt{T(m_i^2)}/m_i^2 = C(m_i^2)/m_i^2(m_i^2-p_1^2)+
{\cal O}(m_i^{-4})$
holds
and the decomposition of $C$ yields $
\sqrt{T(m_i^2)}/m_i^2 = C(p_1^2)/p_1^2(m_i^2-p_1^2) +
\tilde{C}(m_i^2). $
By plugging this value into (\ref{wi}) and taking the minus sign
which corresponds to
$t \rightarrow 0$,
we end up with
\begin{eqnarray}
w_i^2 = \frac{h}{p_1^2} \frac{C(p_1^2)}{(p_1^2-m_i^2)} =
\frac{h}{p_1^2} \Lambda_{N=2}^{ 2N_c+2 - N_f} \frac{\det
( p_1^2 - m^2)^{1/2}}{(p_1^2-m_i^2)},
\label{wi2}
\end{eqnarray}
where we evaluated $ C(p_1^2)$ from (\ref{onedyon})
at $v^2=p_1^2$.
In the above expression we need to know the values of $h$ and $p_1$.
The boundary condition for $w^2$ for large $v$ leads to
\begin{eqnarray}
w^2 \sim 2 h \frac{ C_{2N_c}(v^2)}{v^2-p_1^2}
\sim 2 h v^{2(N_c-1)} + 2 h p_1^2 v^{2(N_c-2)} +
\cdots,
\end{eqnarray}
which should be equal to
$ \sum_{k=1}^{N_c} 2k \mu_{2k} v^{2(k-1)}$.
Now we can read off the values of $h$ and $p_1$ by comparing both sides,
\begin{eqnarray}
h = N_c \mu_{2N_c} ,
\qquad p_1^2 =\frac{(N_c-1) \mu_{2(N_c-1)}}{N_c \mu_{2N_c}}.
\end{eqnarray}
Finally, the finite value for $w^2$ can be written as
\begin{eqnarray}
w_i^2 =\frac{N_c^2 \mu_{2N_c}^2}{(N_c-1) \mu_{2(N_c-1)} }
\Lambda_{N=2}^{2N_c+2-N_f}
\frac{\det ( a_1^2 - m^2)^{1/2}}{(a_1^2-m_i^2)},
\end{eqnarray}
which is exactly, up to constant, the same expression for
meson vevs (\ref{meson}) obtained from field theory analysis
in the low energy superpotential (\ref{lowsuperpo}). This illustrates the fact that
at vacua with enhanced gauge group $SU(2)$ the effective superpotential by integrating
in method with the assumption of $W_{\Delta}=0$ is really exact.
\section{Discussions }
\setcounter{equation}{0}
It is straightforward to deal with Yang-Mills theory with massless matter.
When some of branch points of (\ref{curve}) collide as one changes the moduli,
the Riemann surface degenerates and gives a singularity in the theory corresponding
to an additional massless field. By redefining $y$ we get the $2r+1$ branch points of
Riemann surface and one may expect that an unbroken $Sp(r)$ gauge symmetry.
It is easy to generalize the case of several massless dyons. The classical moduli
space is given by several eigenvalues of $a_i$'s. After integrating out the adjoint fields
in each $SU(r_i)$, we obtain the scale matching condition between the high energy scale
and the low energy scale. Then the meson vevs can be written as the differentiation
of the low energy effective superpotential with respect to $m_i^2$ in field
theory approach. On the other hand, in the M theory fivebrane
configuration we can proceed the method done in a single massless dyon.
That is, $\sqrt{T(v^2)}/v^2$ should be expressed for the several dyons and
the decomposition of $C$ will be given also.
One realizes that a mismatch is found between
field theory results and brane configuration results with $W_{\Delta}=0$. This tells us
that the minimal form for the effective superpotential obtained by integrating in method
is not exact. That is, $W_{\Delta}$ is not zero for the enhanced gauge group $SU(r), r > 2$.
It will be interesting to find the corrections in the future.
|
1,108,101,564,611 | arxiv | \section{Introduction}
Class IV-VI narrow-gap compound semiconductors have been of great interest for many
decades not only for their scientific interest but also for their use in novel
technological instrumentation such as infrared optoelectronics and thermoelectric devices.\cite{Nimtz}
Crystalizing in the rock salt structure, materials such as PbTe, PbSe, SnTe, and their mixed alloys
have been found to have high dielectric constants and quite unusual infrared and electronic properties.\cite{Otfried}
PbSe and SnSe, the two parent compounds of Pb$_{1-x}$Sn$_{x}$Se alloy system, have what is commonly known as band inversion.\cite{Dimmock}
For $x$ = 0, $i.e.$, PbSe has the conduction band in L${}_{6}^{+}$ symmetry and the valence band symmetry is denoted by L${}_{6}^{-}$.
As one increases $x$ in the alloy system, Pb$_{1-x}$Sn$_{x}$Se, the band gap initially reduces and then closes, with a linear
dispersion around the Fermi level. Subsequent increase in $x$ reopens the gap but now the valence band symmetry gets inverted
to L${}_{6}^{+}$, whereas symmetry of the conduction band changes to L${}_{6}^{-}$, identical to SnSe, $i.e.$, $x$ = 1.
Recent theoretical and experimental interest in these alloys has occurred because they have been suggested to represent
a new, non-trivial topological phase called a topological crystalline insulator\cite{Fu,Hsieh}(TCIs).
As in most topological insulators, observations are complicated because the crystals are usually found to be either
p-doped or n-doped due to non-stoichiometry so that surface-state features get overshadowed by an overwhelming bulk-carrier contribution.
This is the case in the data reported here. Nevertheless, the temperature dependence of the free-carrier effective mass and of the
valence-to-conduction band absorption edge could be a suitable point of inquiry to explore the band inversion and temperature-driven
phase transition in such materials. So, apart from determining the optical and transport properties, this study also attempts to
investigate such a possibility in a Pb$_{0.77}$Sn$_{0.23}$Se single crystal.
Studies of PbSe have reported the rock salt crystal structure at ambient temperature and pressure with a lattice parameter
of $a = 6.13~${\AA} and a direct minimum energy band gap of around 0.28~eV at the L point in the Brillouin zone.\cite{Delin,Otfried,Harman}
In contrast, SnSe has been stabilized as an orthorhombic crystal with layered symmetry and an indirect minimum energy band gap of 0.9~eV.\cite{Otfried,Albers}
On the basis of X-ray diffraction studies of annealed powders, it has been established\cite{Littlewood,Dixon}
that the mixed alloy Pb$_{1-x}$Sn${}_{x}$Se stabilizes in the rock salt structure for $0 \leq x \leq 0.43$ and that the minimum band gap
remains at the L point. Infrared absorption measurements\cite{Strauss1967} find that the gap is a complex function of
temperature and stoichiometric ratio $x$. An ARPES study\cite{Dziawa} of Pb$_{0.77}$Sn${}_{0.23}$Se finds that the band gap
is temperature-dependent, with a minimum around 100~K. The bulk band gap reopens at the L point with an inverted symmetry
of the valence band L${}_{6}^{+}$ and the conduction band L${}_{6}^{-}$ as the temperature is further increased.\cite{Dziawa}
\section{EXPERIMENTAL PROCEDURES}
Compositional and structural defects play significant role in determination of chemical and electronic properties of
lead chalcogens. Local compositional disorder significantly affects the band gap formation for the alloy compositions.
In addition, short-range disorder breaks the degeneracy near the band edge\cite{Xing} because PbSe and SnSe represent
two different crystal structures. Therefore, controlled crystal growth is an essential requirement.
A vapor-phase growth technique was employed to prepare single crystals of Pb${}_{0.77}$Sn${}_{0.23}$Se. The crystals
had the rock salt structure with lattice parameter $a = 6.07~${\AA}. To ensure good quality samples, quality-control
conditions were used, including super-saturation of gaseous phase and a seed crystal with the proper structure and orientation.
Lead chalcogen crystals are opaque and have a metallic luster. They are brittle and easily cleave along the (100) plane.
Chemical binding in these systems has both ionic and covalent components. A single crystal of size $4\times4\times2$ mm${}^{3}$
with smooth (100) crystal plane as the exposed surface was selected for optical measurements.
Temperature dependent (10--300~K) reflectance measurement were conducted using a Bruker 113v Fourier-transform interferometer.
A helium-cooled silicon bolometer detector was used in the 40--650 cm${}^{-1}$ spectral range and a DTGS detector was used
from 600--7000 cm${}^{-1}$. Room temperature measurement up to 15,500 cm$^{-1}$ used a Zeiss microscope photometer.
Because the rock salt structure of the material implies isotropic optical properties, all optical measurements were performed
using non-polarized light at near-normal incidence on the (100) crystal plane. To achieve higher accuracy during reflectance
measurements, a small evaporation device incorporated in the metal shroud of the cryostat was used to coat the crystal surface with gold,
minimizing changes of experimental conditions during sample and reference single-beam spectral measurements. The gold coating was easy
to remove since the material is easily cleaved along the (100) plane.
Optical measurements were followed by Hall-effect measurements, using a physical properties measurement system (PPMS) from Quantum Design
which allows transport measurement over 10--300~K. Magnetic fields up to 7~T were applied perpendicular to the (100) plane and ramped
in the upward and downward directions while the Hall voltage was measured transverse to current and field. The Hall-effect measurements
gave the sign (p-type) and value for the carrier density $n$ at 10~K and at room temperature. Errors in the carrier density estimate
originate from the non-uniform thickness of the sample, non-rectangular cross section, and contact size. The errors are of course the same
at each temperature. These errors do affect the discussion in a later section, where we compare Hall measurements and the infrared plasma
frequency $\omega_{p} = \sqrt{4\pi n e^2/m_h^*}$, where $e$ is the electronic charge and $m^*$ is the effective mass of the holes,
Our goal in this comparison is to obtain an estimate of the effective mass. In our analysis, the carrier concentration $n$ has been linearly
extrapolated over intermediate temperatures for the estimation of effective number of carriers $N_{\mbox{\it eff}}$ at various temperatures of
interest. The volume expansion of the crystal with temperature is also included in this calculation, although it is not very significant.
Because of the similarity in the crystal structure and lattice parameter, the linear thermal expansion coefficient for this material is taken
to be the same as PbSe.\cite{Novikova}
\section{EXPERIMENTAL RESULTS}
\subsection{Hall-effect studies}
Previous Hall studies\cite{Martinez,Maier} on PbSe and SnSe reported p-type carriers in these materials. A slight increase in p-type carrier
density at room temperature compared with 70~K was observed in SnSe.\cite{Julian} The origin of the p-type extrinsic carrier in the crystal
is due primarily to the presence of ionized lattice defects associated with deviations from stoichiometry, possibly due to excess of selenium
or vacancies occupied by acceptor impurities.\cite{Strauss1968,Anasabe,Goldberg} Our Hall measurements show p-type carriers with a concentration
of $n = 6\times10^{19}$~cm${}^{-3}$ at room temperature. This value increases to $n = 9\times10^{19}$~cm${}^{-3}$ at 10~K. The relatively high value
for $n$ suggests that our sample is a degenerate semiconductor, in which the chemical potential is pushed down into the valence band due to the
carrier density $n$ exceeding the valence band edge density of states.
\subsection{Reflectance spectra}
The temperature dependence of the reflectance of Pb${}_{0.77}$Sn${}_{0.23}$Se between 40 and 7000 cm${}^{-1}$ (5~meV--870~meV) is shown in Fig. 1.
In the far-infrared range, a high reflectance is observed, decreasing somewhat as temperature increases. The effect of an optically-active transverse
phonon may be seen around 40--200~cm${}^{-1}$. This feature weakens as temperature increases.\cite{Habinshuti} A rather wide valence intraband
transition centered around 650 cm${}^{-1}$ and shifting to lower frequency with increasing temperature affects the reflectance; this is evident
at low temperatures causing a rise in the reflectance level around the central frequency.\cite{Brian} There is a plasma minimum around 930 cm${}^{-1}$
at 10~K; this feature shifts to around 885 cm${}^{-1 }$ as temperature increases to 300~K and becomes slightly less deep. As we shall see, it is
in good agreement with the zero crossing of the Kramers-Kronig derived real part of the dielectric function $\varepsilon_1$.
\begin{figure}[H]
\centering
\includegraphics[width=3.4 in,height=3.4 in,keepaspectratio]{Refl.pdf}
\caption{\label{fig:Refl} (Color online) Temperature-dependent reflectance spectra of a Pb$_{0.77}$Sn$_{0.23}$Se single crystal.}
\end{figure}
In the midinfrared, we observe interband transitions from the occupied states in the valence band to the conduction band.
These transitions become weaker, leading to decreased reflectance, as the temperature is increased. As this is a p-type system,
the chemical potential lies in the valence band; therefore, the experimentally-observed minimum absorption edge in the midinfrared
range is an overestimate of the direct band gap at the L point of the Brillouin zone, as would be observed in an intrinsic Pb$_{0.77}$Sn$_{0.23}$Se crystal.
The top panel of Fig. 2 shows the 300~K reflectance up to 1.92 eV. There is a strong high-energy interband transition at 10,600~cm${}^{-1 }$ (1.3~eV).
\section{ANALYSIS}
\subsection{Kramers-Kronig analysis and optical conductivity}
We used the Kramers-Kronig relations to analyze the bulk reflectance $R(\omega ) $ and then to estimate the real and
imaginary parts of the dielectric function.\cite{Wooten} Before calculating the Kramers-Kronig integral, the low frequency
reflectance data were extrapolated to zero using the reflectance-fit parameters. (The fit is discussed later in this section.)
Reflectance data above the highest measured frequency were extrapolated between 80,000 and $2\times10^{8}$ cm${}^{-1 }$ with the
help of X-ray-optics scattering functions; from the scattering function for every atomic constituent in the chemical formula
and the volume/molecule (or the density) one may calculate the optical properties in the X-ray region.\cite{Henke} A power-law in $1/\omega$
was used to bridge the gap between the data and the X-ray extrapolation. Finally, an $\omega^{-4}$ power law was used above
$2\times10^{8}$~cm${}^{-1 }$. The optical properties were derived from the measured reflectance and the Kramers-Kronig-derived phase
shift on reflection.
\begin{figure}[H]
\centering
\includegraphics[width=4.5 in, height=4.5 in, keepaspectratio]{300K.pdf}
\caption{\label{fig:300K} (Color online) (a) Reflectance spectrum from 40--15,500~cm${}^{-1}$ (5 meV--~1.92 eV) at 300K. (b) Kramers-Kronig
calculated optical conductivity from 40--15,500~cm${}^{-1}$ (5 meV--~1.92 eV) at 300K.}
\end{figure}
The lower panel of Fig.~2 shows the Kramers-Kronig-derived optical conductivity at 300~K. The narrow Drude contribution and
the contributions to the conductivity from interband transitions around 3500, 5100, and 10,600 cm$^{-1}$ are consistent with the reflectance.
The temperature dependence of the Kramers-Kronig derived real part of the optical conductivity, $\sigma_{1}(\omega)$ over 40--7000 cm${}^{-1}$
is shown in Fig. 3. The far-infrared range shows the Drude contribution to $\sigma_{1}(\omega)$. Between 10--100~K, the Drude
relaxation rate $1/\tau$ is below the measured frequency range, explaining the non-constant area under the displayed conductivity
spectrum as temperature changes. The relaxation rate increases as temperature is increased until, by 200 K, most of the Drude spectral
weight is seen in the figure.
\begin{figure}[H]
\centering
\includegraphics[width=3.5 in,height=3.5 in,keepaspectratio]{conductivity.pdf}
\caption{\label{fig:conductivity} (Color online) Temperature-dependent optical conductivity of Pb${}_{0.77}$Sn${}_{0.23}$Se single crystal
obtained by Kramers-Kronig analysis.}
\end{figure}
The Drude conductivity partially overlaps with a small and dispersive contribution from the valence intraband transition at
650 cm$^{-1}$. This feature can be seen in the spectra at 10--100~K but becomes masked by the Drude spectrum as temperature increases.
The valence intraband transition is an excitation from an occupied state below the Fermi level to an empty state (in another valence subband)
above the Fermi level. It would only be observed if the Fermi level were to lie somewhere within the valence band.
In the midinfrared range, the optical conductivity decreases as temperature is increased. At most temperatures, two overlapping conductivity
peaks may be discerned around 3500 and 5100 cm$^{-1}$; these features, arising from interband transition, are also consistent with the reflectance.
The onset of the interband transitions is estimated by extrapolating linearly the conductivity to $\sigma_1(\omega) = 0$. This intercept is
about 2300 cm$^{-1}$ at 10~K, then decreases to around 2200 cm$^{-1}$ at 100~K, and then increases back to 2300 cm$^{-1}$ at 300~K.
The temperature dependence of all parameters is discussed in a following subsection.
\subsection{The real part of the dielectric function}
The Kramers-Kronig result also allows us to estimate the real part of dielectric function $\varepsilon_1$ shown in Fig. 4.
At low frequencies, $\varepsilon_1$ is negative, a defining property of a metal. The temperature dependence of the free-carrier
scattering rate is very evident in this dielectric function plot. For temperatures above 150~K, $\varepsilon_1$ becomes almost flat at low
frequencies, implying that the scattering rate is on the order of a few hundred cm$^{-1}$. In contrast, at low temperatures
$\varepsilon_1 \sim 1/\omega^{-2}$, sharply decreasing, indicating a very small scattering rate, one below the minimum measured frequency.
\begin{figure}[H]
\centering
\includegraphics[width=3.5 in,height=3.5 in,keepaspectratio]{epsilon.pdf}
\caption{\label{fig:epsilon} (Color online) Temperature dependence of the real part of the dielectric function of Pb${}_{0.77}$Sn${}_{0.23}$Se.
The inset shows the zero crossing of $\varepsilon_1(\omega)$, illustrating the temperature dependence of the screened plasma frequency.}
\end{figure}
The inset graph shows the zero crossing of $\varepsilon_1$ (representing the screened plasma frequency) for different temperatures.
At 10~K the screened plasma frequency is around 900 cm$^{-1}$; this value is maintained to about 150~K, after which it decreases to
around 815 cm$^{-1}$ by 300~K. This Kramers-Kronig-derived screened plasma frequency is in good agreement with the reflectance plasma minimum.
\subsection{Sum rule analysis and \boldsymbol{${N_{\mbox{\it eff}}\frac{m}{m_h^{*}}}$} }
Once we have the real part of the conductivity, we can calculate the partial sum using
\begin{equation}
N_{\mbox{\it eff}}\frac{m}{m^{*}}=\frac{2mV_{c}}{\pi e^{2}}\int^{\omega}_{0}\sigma_{1}(\omega^{\prime})d\omega^{\prime},
\end{equation}
to obtain the effective number of carriers participating in optical transitions up to frequency $\omega$. The quantities in this equation are $m^*$,
the effective mass, $m$, the free-electron mass, $e$, the electronic charge, and $V_c$, the volume taken up by one ``molecule'' of Pb${}_{0.77}$Sn${}_{0.23}$Se.
Figure 5 shows the temperature-dependent partial sum rule results. The free-carrier contribution saturates around 0.015 at 10~K, decreasing to 0.011
at 300~K. These data suggest that either the carrier density is decreasing or the effective mass is increasing as temperature increases. The Hall
results tell us that it is the former. We use the Hall carrier density to estimate the free carrier effective mass and its temperature dependence.
The results are shown in Fig. 6. A uniform error is added to the data points based on the uncertainty in the $N_{\mbox{\it eff} }$ and the optical
conductivity-dependent partial sum rule derived from Karamers-Kronig relations.
\begin{figure}[H]
\centering
\includegraphics[width=3.4 in,height=3.6 in,keepaspectratio]{Sumrule.pdf}
\caption{\label{fig:Sumrule} (Color online) Temperature-dependent integrated spectral weight of Pb${}_{0.77}$Sn${}_{0.23}$Se.}
\end{figure}
The effective mass changes only slightly over the range of measurements; there is a weak minimum at 100~K where $m_{h }= 0.335 m_{e }$ and then
a rise to $m_{h }= 0.350 m_{e }$ at 300~K. The average effective mass is $\langle m_{h }\rangle_T = 0.341 m_{e}$.
\begin{figure}[H]
\centering
\includegraphics[width=3.4 in,height=3.4 in,keepaspectratio]{Effectivemass.pdf}
\caption{\label{fig:Effectivemass} (Color online) Temperature-dependent hole effective mass of Pb${}_{0.77}$Sn${}_{0.23}$Se.}
\end{figure}
We note that on account of the doping, the Fermi level is far from the putative Dirac point, and, moreover that the optical conductivity
spectrum from which this effective mass is derived is an angular average over the Fermi surface and an energy average over a scale set by
the relaxation rate, $1/\tau$.
\subsection{Energy-loss function}
Finally, the temperature dependence of the longitudinal response of this material, namely the loss function $L(\omega)$, is plotted against
frequency in Fig. 7. The free-carrier peak in the loss function is a good estimate of the screened Drude plasma frequency. This estimate is
in good agreement with the zero crossing of $\varepsilon_1$ in Fig. 4 as well as the value calculated by fitting with the Drude-Lorentz model,
discussed in the next section.
\begin{figure}[H]
\centering
\includegraphics[width=3.5 in,height=3.5 in,keepaspectratio]{lossfunction.pdf}
\caption{\label{fig:lossfunction} (Color online) Temperature-dependent energy loss function of Pb${}_{0.77}$Sn${}_{0.23}$Se.}
\end{figure}
\subsection{Drude-Lorentz model fits}
We used the Drude-Lorentz model to fit the reflectance data. The Drude component characterizes the free carriers and their dynamics at zero frequency
whereas the Lorentz contributions are used for the optically-active phonon in the far-infrared region along with valence intraband and interband
transitions in the mid and near infrared. The dielectric function is:
\begin{equation}
\varepsilon (\omega)=\varepsilon_{\infty}- \frac{\omega _{p}^{2} }{\omega^{2}+i\omega /\tau } +
\sum _{j=1}^6\frac{\omega_{pj}^{2} }{\omega_{j}^{2}-\omega^{2} -i\omega \gamma_{j} }
\end{equation}
where the first term represents the core electron contribution (transitions above the measured range, the second term is free carrier contribution
characterized by Drude plasma frequency $\omega _{p}$ and free carrier relaxation time $\tau $ and the third term is the sum of six Lorentzian
oscillators representing phonons, valence intraband, and interband electronic contributions. The Lorentzian parameters are the $j$th oscillator
plasma frequency $\omega _{pj}$, its central frequency $\omega _{j}$, and its linewidth $\gamma _{j}$. This dielectric function model is used in
a least-squares fit to the reflectance.
\begin{figure}[H]
\centering
\includegraphics[width=3.4 in,height=3.4 in,keepaspectratio]{reflfit.pdf}
\caption{\label{fig:reflfit} (Color online) Drude-Lorentz fit to the reflectance of Pb${}_{0.77}$Sn${}_{0.23}$Se at 300~K and 10~K.}
\end{figure}
\begin{figure}[H]
\centering
\includegraphics[width=3.4 in,height=3.4 in,keepaspectratio]{conductivityfit.pdf}
\caption{\label{fig:conductivityfit} (Color online) A comparison of the Kramers-Kronig-derived optical conductivity and the conductivity calculated
from the Drude-Lorentz fit to the reflectance of Pb${}_{0.77}$Sn${}_{0.23}$Se. Results are shown for 10 and 300~K. The inset shows the Drude and
the valence intraband contributions at the two temperatures.}
\end{figure}
Figure 8 compares the calculated and measured reflectance at 10~K and 300~K; the 300~K fit is shown up to 15,500 cm${}^{-1}$ whereas the 10~K
data are fitted to 7000 cm${}^{-1}$. Similar quality fits were obtained at all other temperatures.
The comparison between the Kramers-Kronig-derived and the Drude-Lorentz-model conductivity spectra at 10~K and 300~K is shown in Fig. 9.
The parameters are the ones used to fit the reflectance. That the agreement is good gives us confidence in both procedures. The inset graph
shows the Drude contribution and the valence intraband contributions at 10~K and 300~K. The valence intraband peak is small and dispersive
but is necessary for a good description of the data. Table I lists the 21 parameters used during the fitting routine to fit the reflectance
and the conductivity spectra at 300~K.
\begin{table}[h]
\caption { Drude-Lorentz parameters for Pb${}_{0.77}$Sn${}_{0.23}$Se at room temperature (300~K). }
\centering
\begin{tabular}{c c c c}
\hline\hline
Modes assignment & Oscillator & Central & Linewidth \\
in fitting & strength $\omega _{p}$ & frequency $\omega _{j }$ & $\gamma _{ j}$ \\
routine & (cm${}^{-1}$) & (cm${}^{-1}$) & (cm${}^{-1}$) \\[1ex]
\hline
Drude & 4160 & 0 & 195 \\
Optical phonon & 240 & 40 & 5.5 \\
Valence intraband & 1230 & 550 & 560 \\
Interband 1 & 4550 & 3500 & 1900 \\
Interband 2 & 11,000 & 5120 & 3950 \\
Interband 3 & 20,630 & 10,620 & 1950 \\
Interband 4& 23,750 & 12,350 & 4150 \\ [1ex]
\hline\hline
\end{tabular}
\end{table}
\begin{figure}[H]
\centering
\includegraphics[width=3.2 in,height=3.2 in,keepaspectratio]{drudeplasmafrequency2.pdf}
\caption{\label{fig:drudeplasmafrequency2} (Color online) Comparison between the Drude-Lorentz model fitted and Hall-derived Drude plasma frequency (10~K-300~K)}
\end{figure}
Figures 10--13 show the temperature dependence of selected parameters from the fits. First, Fig. 10 shows the Drude plasma frequency,
whose strength decreases as temperature increases. We also show a plasma frequency calculated from the Hall measurement
$\omega _{p} = \sqrt{4 \pi n e^2/m_h^*}$ where $n$ comes from the Hall data and $m_h^*$ from the sum rule analysis for the free carriers.
Within the error bars, the two results agree.
\begin{figure}[H]
\centering
\includegraphics[width=3.6 in,height=3.6 in,keepaspectratio]{centralfrequency.pdf}
\caption{\label{fig:centralfrequency} (Color online) Temperature dependence of the central frequency for the fitted Lorentz oscillators.}
\end{figure}
Secondly, the temperature dependence of the central frequencies of the lowest interband transition, the valence intraband transition and the
optical phonon is shown in Fig. 11. The lowest interband transition has its minimum value at $100\pm 25$~K. As temperature increases,
the valence intraband oscillator shifts towards lower frequencies whereas the phonon mode in the far-infrared range shifts slightly towards
higher frequencies.
The linewidth of many of these modes suggests significant overlap amongst them, especially in the midinfrared region.
Figure 12 shows the temperature dependence of the linewidth $\gamma$ of these transitions. The Drude relaxation rate (bottom panel) becomes as
small as 17 cm${}^{-1}$ (well below our experimental low frequency limit) at 10~K, increasing with temperature to 195 cm${}^{-1}$ at 300~K.
The corresponding hole mobility $\mu = e\tau/m_h^*$ decreases from 1630 cm${}^2$/Vs at 10 K to 140 cm${}^2$/Vs at 300 K. Based on these hole mobility
values, the optical conductivity $\sigma = n e \mu$ is estimated to be 23,200 $\Omega$${}^{-1}$cm${}^{-1}$ at 10~K, decreasing to 1320 $\Omega$$^{-1}$cm${}^{-1}$ at 300~K.
These estimates are in good agreement with the conductivity values found from the fitting routine: 21,800 $\Omega$${}^{-1}$cm${}^{-1}$
at 10~K, decreasing to 1490 $\Omega$${}^{-1}$cm${}^{-1}$ at 300~K. The linewidth of every other oscillator shows a similar temperature trend
except for the valence intraband mode which has a weak (and not very significant) minimum at 100 K.
\begin{figure}[H]
\centering
\includegraphics[width=3.9 in,height=3.9 in,keepaspectratio]{linewidth.pdf}
\caption{\label{fig:linewidth} (Color online) Temperature dependence of the linewidth for the fitted Lorentz oscillators.}
\end{figure}
\begin{figure}[H]
\centering
\includegraphics[width=3.8 in,height=3.8 in,keepaspectratio]{Normalizedstrength.pdf}
\caption{\label{fig:Normalizedstrength} (Color online) Temperature dependence of the oscillator strength (normalized by the 10~K value) for the
fitted Lorentz oscillators.}
\end{figure}
Finally, the temperature dependence of the oscillator strength for several oscillators is shown in Fig. 13. The strength has been normalized
by its value at 10~K. (The Drude oscillator strength is shown in Fig. 10.) Three oscillators have their highest strength at the lowest
temperature; it decreases monotonically as temperature increases, with a change of slope around 100~K. The valence intraband mode is different:
its shows a clear minimum at 100~K and is higher at 300~K than at 10~K.
\section{DISCUSSION}
\subsection{ Free carrier characteristics and band inversion }
A scattering rate calculation for a p-type Si${}_{1-x}$Ge${}_{x}$ alloy predicts quite a high scattering rate.\cite{Joyce}
Impurity scattering is expected to be strong on account of the disorder associated with the alloy. In contrast, our Pb${}_{0.77}$Sn${}_{0.23}$Se
single crystal has a quite small free carrier scattering rate at low temperature. Indeed, the free-carrier scattering rate (shown in the bottom
panel of Fig. 12) remains temperature dependent below 50 K. Considering the alloy form of the material with large stoichiometric imbalance,
this result is rather surprising.
Previous studies have shown that the extrinsic carrier density in Pb${}_{1-x}$Sn${}_{x}$Se affects not only the Hall coefficient but also
its temperature dependence. For lightly doped systems, $R_{H}$ changes sign from positive to negative as temperature is increased, mainly
due to the increasing $n$ type contribution from thermally excited carriers. On the other hand, for heavily doped systems, the Hall coefficient
is positive throughout and has weak temperature dependence.\cite{Dixon} As temperature is increased, due to increasing thermal energy,
the number of free carriers increases. However an increase in the band gap due to increasing temperature also reduces the number of thermally
excited carriers. The value of the Hall coefficient depends on the relative dynamics of these two opposing mechanisms. The slight increase in
the Hall coefficient with temperature in our sample could be due to an imbalance between these two competing processes.
Temperature dependent ARPES measurements of the (100) surface of Pb${}_{0.77}$Sn${}_{0.23}$Se monocrystals have been interpreted as finding
band inversion in the material around 100~K.\cite{Dziawa} The band curvature at the L point changes during band inversion and band dispersion
becomes almost linear during gap closure at 100~K. The temperature dependence of the carrier effective mass (shown in Fig. 6) does show a weak minimum,
in agreement with this idea.
\subsection{Minimum band gap and the valence intraband transition}
The phenomenon of first shrinking and then expansion of band gap as temperature is decreased should affect the minimum absorption edge
directly. Fig. 11 shows that the central frequency of the lowest interband transition has a minimum at 100~K. In addition, the onset of
this interband transition, estimated by linear extrapolation of the low-energy edge in the Kramers-Kronig-derived optical conductivity to
zero conductivity suggests a similar trend. An ARPES study also finds a minimum band gap at 100~K.\cite{Dziawa}
We can point to other features in the optical spectrum that present themselves around 100~K: the frequency of the optically-active phonon (Fig. 11)
and the linewidth and the spectral weight of the valence intraband transition (Figs. 12 and 13).
\subsection{Electronic structure calculations}
The energy band structure of Pb${}_{0.75}$Sn${}_{0.25}$Se was calculated within density-functional theory using the VASP
package.\cite{Kresse93,Kresse94,Kresse96,Kresse1996} The input structure consists of 8 atoms cell with 3 Pb, 1 Sn and 4 Se atoms.
The calculation was based on a generalized gradient approximation (GGA) for the exchange-correlation potential using the Perdew, Burke, and
Ernzerhof formalism\cite{John} along with spin-orbit coupling for both geometric and electronic optimization. We used an energy cutoff of 500 eV,
10${}^{-6}$~eV energy minimization, and $8\times8\times8$ $k$-points. The calculation gave a lattice constant of 6.16~{\AA}, an overestimate by
0.09 ~{\AA} compared with the measured lattice constant.\cite{Haas} The electronic band structure was calculated along high-symmetry $k$-points in
the Brillouin zone with the Fermi energy fixed to zero ({\it i.e.,} an undoped crystal). As shown in Fig. 14, the band structure calculation indicates
that the material has a direct band gap at the L-point in the Brillouin zone with an energy gap of 0.20 eV. Fig. 15 shows an enlarged view of the
conduction band minimum and the valence band maximum at the L point.
\begin{figure}[H]
\centering
\includegraphics[width=3.5 in,height=3.5 in,keepaspectratio]{bandstructure.pdf}
\caption{\label{fig:bandstructure} (Color online) Calculated electronic band structure of Pb${}_{0.75}$Sn${}_{0.25}$Se.}
\end{figure}
The spin-orbit interaction is important in determining the actual band ordering, band splitting and the details of the electronic structure
in class IV-VI narrow-gap semiconductors. Fig. 15 shows the splitting of the valence band into 3 valence sub-bands. The splitted valence bands
are named as split-off, heavy-hole and light-hole bands.\cite{Jacques, Braunstein}
\begin{figure}[H]
\centering
\includegraphics[width=3.2 in,height=3.2 in,keepaspectratio]{bandstructure2.pdf}
\caption{\label{fig:bandstructure2} (Color online) Enlarged view at the L point of the low-energy part of the band structure showing the
splitting of the valence band. CBM = conduction band minimum; VBM = valence band maximum; VSB= valence sub-band.}
\end{figure}
This VASP calculation also produced the density of states. Using the carrier density from experiment and this density of states,
we can estimate that the Fermi energy in our doped sample lies around $E_F = -50$~meV. The lowest interband transition takes place
from $E_Fi$ to the empty state in the conduction band at the same $k$-point . The band structure calculation then predicts this absorption
to start around 0.30 eV (2420 cm${}^{-1}$). This prediction is in good agreement with the estimate of 2300 cm$^{-1}$ from the threshold for
the interband transitions seen in the optical conductivity.
In addition to the optical interband transitions between the valence and the conduction band, there are also optical transitions within the
valence band, the valence intraband transition discussed above. These transitions are seen in a number of p-type semiconductors, when the top of
the valence band is populated with holes. It is then possible to make intraband transitions from low-lying sub-bands to states above the Fermi level.
These transitions have been observed in a number of semiconductors, including Ge, GaAs, and InSb.\cite{Kaiser, Gobeli, Grundmann} The single crystal
of Pb${}_{0.77}$Sn${}_{0.23}$Se shows similar valence intraband transitions whose temperature dependence is shown in Figs. 11--13.
\section{CONCLUSIONS}
Hall measurements of Pb${}_{0.77}$Sn${}_{0.23}$Se single crystals disclose the p-type nature of the material and show that the carrier density
decreases from $9\times10^{19}$~cm${}^{-3}$ at 10~K to $6\times10^{19}$~cm${}^{-3}$ at 300~K. Temperature-dependent optical reflectance measurement
shows degenerate semiconductor behavior. At 10~K, the screened Drude plasma minimum lies around 930 cm${}^{-1 }$; it shifts slightly towards lower
frequency as temperature increases.
In the midinfrared and near-infrared region, the spectra show valence intraband transition and valence to conduction band excitations.
The temperature dependence of the lowest-energy interband transition energy shows a minimum at 100~K. This behavior suggest band inversion in the material,
also predicted by many other previous studies. The effective hole mass also has a minimum value at 100~K consistent with linear band dispersion at the
L point in the Brillouin Zone. Theoretical calculations provide the details of electronic band structure, supporting the existence of valence intraband
transition. Our experimental and theoretical studies suggest that material undergoes a temperature-driven band inversion at the L point as also predicted
by previous study.\cite{Dziawa}
\begin{acknowledgments}
The author wish to thank Xiaoxiang Xi for his constructive discussions on material's band structure. This work was supported by the DOE through contract No.
DE-AC02-98CH10886 at Brookhaven National Lab and by the NSF DMR 1305783 (AFH).
\end{acknowledgments}
|
1,108,101,564,612 | arxiv | \section{Introduction}\label{intro}
The variational quantum circuit (VQC, also known as parameterised quantum circuit, PQC) approach, first proposed for solving the ground state energy of molecules \cite{peruzzo2014variational}, have been extended to many open research problems including in the field of quantum machine learning \cite{schuldpetruccione2021}, quantum chemistry \cite{RevModPhys.92.015003}, option pricing \cite{2020optionpricing} and quantum error correction \cite{johnson2017qvector, Xu2021-dt}. The performance of VQC methods largely depend on the choice of a suitable ans\"atze, which is not an easy task because generally the search space is very large and it is not well established whether there is a common principle for designing such ans\"atze. For problems involving physical systems such as in quantum chemistry, we can rely on the well-defined properties of molecular systems for ans\"atz designing, like the hardware efficient ans\"atze~\cite{2017hardwareefficientvqe} and physical-inspired ans\"atze, such as $k$-UpCCGSD~ \cite{physicalinspiredansatze1doi:10.1021/acs.jctc.8b01004}. However, this cannot be generalised to other areas such as designing variational error correction circuits or quantum optimisation problems. For example, in \cite{Xu2021-dt}, when developing a variational circuit that can encode logical states for the 5-qubit quantum error correction code, the authors adopted an expensive approach by randomly searching over a large number (order of 10000) of circuits. It is anticipated that, with the increasing number of application areas for VQCs and the need for scalability to tackle large problem sizes without relying on fundamental physical properties, such random search methods or methods based purely on human heuristics will struggle to find suitable ans\"atzes. Therefore, it is important to develop efficient methods for the automated design of variational quantum circuits. Here we focus on the development of algorithms for the automated design of VQCs by leveraging the power of artificial intelligence (AI) which can be deployed for a wide range of applications.
Although modern AI research often focuses on applications of image and natural language processing, the power of AI can also bring new knowledge in many areas, especially scientific discovery.
AlphaFold2 managed to discover new mechanism for the bonding region of the protein and inhibitors \cite{alphafold2Jumper2021-lw} with competitive accuracy on predicting the three-dimensional structure of proteins in the 14th Critical Assessment of protein Structure Prediction (CASP) competition. In 2021, machine learning algorithms helped mathematicians discover new mathematical relationships in two different areas of mathematics \cite{Davies2021-xh}. Like variational quantum circuits, modern deep neural networks (DNN) also face a design problem when composing the network for certain tasks. With the help of AI algorithms, researchers developed techniques to efficiently search suitable network architectures in a large search space. Famous algorithms for neural architecture search (NAS) include the DARTS algorithm \cite{DARTS_DBLP:conf/iclr/LiuSY19}, which models the choice of operations placed in different layers as an independent categorical probabilistic model that can be optimised via gradient descent methods, and the PNAS algorithm \cite{PNAS10.1007/978-3-030-01246-5_2}, which models the search process with sequential model-based optimisation (SMBO) strategy.
Tree-based algorithms were also proposed for NAS, such as AlphaX \cite{AlphaXDBLP:conf/aaai/WangZJTF20}, which models the search process similarly as the search stage of AlphaGo \cite{AlphaGoDBLP:journals/nature/SilverHMGSDSAPL16}. Recently, a new NAS algorithm based on tree search and combinatorial multi-armed bandits, proposed in \cite{huang2021neural}, outperforms other NAS algorithms, including the previously mentioned algorithms.
Based on progress in neural architecture search algorithms, efforts have been made on developing similar approaches for Quantum Ans\"atz (Architecture) Search (QAS) problems. Zhang \textit{et.al}~\cite{zhang2021differentiable} adapted the DARTS algorithm \cite{DARTS_DBLP:conf/iclr/LiuSY19} from NAS for QAS, which models the distribution of different operations within a single layer with the independent category probabilistic model. The search algorithm will update the parameters in the VQC as well as the probabilistic model. However, it has been shown in NAS literature that DARTS tend to assign fast-converge architectures with high probability during sampling \cite{Shu2019-jf, Zhou2020-dg}. Also, the off-the-shelf probabilistic distributions for modelling the architecture space tend to have difficulties when the search space is large. Later, the same group of authors developed a neural network to evaluate the performance of parameterised quantum circuits without actually training the circuits, and incorporated this neural network into quantum architecture search \cite{zhang2021neural}. While NAS algorithms often focus on image related tasks and it has been proved through many experiments that one neural network architecture can act as a backbone feature extractor for many downstream tasks, the structures of variational quantum circuits for different problems often vary a great deal with different problems, casting some doubts on the generalisation abilities of such neural predictor based QAS algorithms. Kuo \textit{et.al} \cite{kuo2021quantum} proposed a deep reinforcement learning based method for tackling QAS. The reinforcement learning agent is optimised by the advantage actor-critic and proximal policy optimisation algorithms.
However, NAS algorithms based on policy gradient reinforcement learning have been shown to get easily stuck in local minimal, producing less optimal solutions \cite{ENASpmlr-v80-pham18a, Sutton1999-nj}. Also, the data size for training a reinforcement learning agent will explode when the number of actions the agent can choose from is large. He \textit{et.al} \cite{chen2021quantum} applied meta-learning techniques to learn good heuristics of both the architecture and the parameters. Du \textit{et al.} \cite{du2020quantum} proposed a QAS algorithm based on the one-shot neural architecture search, where all possible quantum circuits are represented by a supernet with a weight-sharing strategy and the circuits are sampled uniformly during the training stage. After finishing the training stage, all circuits in the supernet are ranked and the best performed circuit will be chosen for further optimisation. Later Linghu \textit{et.al}~\cite{Linghu2022-yy} applied similar techniques on search to a classification circuit on a physical quantum processor. Meng \textit{et.al}~\cite{9566740mctsqas} applied Monte-Carlo tree search to ans\"atz optimisation for problems in quantum chemistry and condensed matter physics. However, these studies often restrict their demonstrations within one or two types of problems and small-sized systems.
In order to develop a search technique that can be applied to larger search spaces and different variational quantum problems, we introduce an algorithm for QAS problems based on combinatorial multi-armed bandit (CMAB) model as well as Monte-Carlo Tree Search (MCTS). In order to explore extremely large search spaces compared to previous work in the literature, the working of our strategy is underpinned by a reward scheme which dictates the choices of the quantum operations at each step of the algorithm with the na\"ive assumption \cite{CMAB_RTS}. This enabled our strategy to work on larger systems, more than 7 qubits, whereas the existing examples \cite{zhang2021differentiable, chen2021quantum, kuo2021quantum, zhang2021differentiable, du2020quantum, zhang2021neural} are restricted to typically 3 or 4 qubits, with the largest being 6 qubits. To demonstrate the working of our method, we showed its application to a variety of problems including encoding the logic states for the [[4,2,2]] quantum error detection code, solving the ground energy problem for different molecules as well as linear systems of equations, and searching the ans\"atz for solving optimisations problems. Our work confirms that the automated quantum architecture search based on the MCTS+CMAB approach exhibits great versatility and scalability, and therefore should provide an efficient solution and new insights to the problems of designing variational quantum circuits.
This paper is organised as follows: Section~\ref{methods} introduces the basic notion of Monte-Carlo tree search, as well as other techniques required for our algorithm, including nested MCTS and na\"ive assumptions from the CMAB model. Section \ref{experiments} reports the results based on the application of our search algorithm to various problems, including searching for encoding circuits for the [[4,2,2]] quantum error detection code, the ans\"atz circuit for finding the ground state energy of different molecules, as well as circuits for solving linear system of equations and optimisation. In Section \ref{discussion} we discuss the results and conclusions.
\section{Methods}\label{methods}
\subsection{Problem Formulation}
In this paper, we formulate the quantum ans\"atz search problem, which is aimed to automatically design variational quantum circuits to perform various tasks, as a tree structure. We slice a quantum circuit into layers, and for each layer there is a pool of candidate operations. Starting with an empty circuit, we fill the layers with operations chosen by the search algorithm, from the first to the final layer.
\begin{figure}[H]
\centering
\includegraphics[width=\textwidth]{Figures/MCTS-QAS_Fig_1.png}
\caption{An overview of the algorithmic framework proposed in this paper. The operation pool (c) is obtained by tailoring the basic operations (a) with respect to the device topology (b). After that, we formulate the combinations of different choices of operations at different layer position in the circuit (d) as a search tree (e). In (f), we evaluate our circuit on a quantum processor or quantum simulator to get value of the loss or reward function, and according to the value of the loss/reward function we update the parameters on a classical computer, then use MCTS to search for the current best circuit. We then send the updated circuit structure together with the updated parameters to the quantum processor/simulator to obtain a new set of loss/reward values. The process depicted in (f) will repeat until a circuit that meets the stopping criteria is found. Then, as shown in (g), we will follow the usual process to optimize the parameters in the searched variational quantum circuit by classical-quantum hybrid computing.}
\label{fig:overview}
\end{figure}
A quantum circuit is represented as a (ordered) list, $\mathcal{P}$, of operations of length $p$ chosen from the operation list. The length of this list is fixed within the problem.
The operation pool is a set
\begin{equation}
\mathcal{C} = \{U_0, U_1, \cdots, U_{c-1} \},
\end{equation}
with $\vert \mathcal{C} \vert = c$ the number of elements. Each element $U_i$ is a possible choice for a certain layer of the quantum circuit. Such operations can be parameterised (e.g. the $R_Z(\theta)$ gate), or non-parameterised (e.g. the Pauli gates). A quantum circuit with four layers could, for instance, be represented as:
\begin{equation}
\mathcal{P} = [U_0, U_1, U_2, U_1],
\end{equation}
where, according to the search algorithm, the operations chosen for the first, second, third and fourth layer are $U_0$, $U_1$, $U_2$, $U_1$. In this case, $p=4$ and the size of the operation pool $\vert \mathcal{C} \vert = c$. The search tree is shown in Fig. \ref{fig:treeexample} In this paper, we will only deal with unitary operations or unitary channels. The output state of such a quantum circuit can then be written as:
\begin{equation}
\vert\varphi_{\rm out}\rangle = U_1 U_2 U_1 U_0 \vert\varphi_{\rm init}\rangle\label{eq:U1U3U1U2},
\end{equation}
where $\vert \varphi_{\rm init}\rangle$ is the initial state of the quantum circuit. For simplicity, we will use integers to denote the chosen operations (such operations can be whole-layer unitaries, like the mixing Hamiltonians often seen in typical QAOA circuits, or just single- and two-qubit gates).
\begin{figure}[H]
\centering
\begin{quantikz}[transparent, row sep={0.8cm,between origins}]
\qw & \midstick[wires=3,brackets=right]{$|\varphi_{\rm init}\rangle$} & \gate[2,disable auto height]{U_0} & \qw & \qw & \qw & \qw\\
\qw & & \qw & \gate[2,disable auto height]{U_1} & \gate{U_2} & \gate[2,disable auto height]{U_1} & \qw\\
\qw & & \qw & \qw & \qw & \qw & \qw
\end{quantikz}
\caption{An example of the circuit corresponding to the series of unitaries applied to $\vert \varphi_{\rm init}\rangle$ in Eqn.\ref{eq:U1U3U1U2}.}
\label{fig:U1U3U1U2_circ}
\end{figure}
For example, the quantum circuit from Eqn.~\ref{eq:U1U3U1U2} can be written as:
\begin{equation}
\mathcal{P} = [0, 1, 2, 1]\label{eq:U1U3U1U2_list}
\end{equation}
and the operation at the $i^{th}$ layer can be referred as $k_i$. For example, in the quantum circuit above, we have $k_2=1$.
\begin{figure}[H]
\centering
\includegraphics[width=0.4\textwidth]{Figures/tree_example.png}
\caption{The tree representation (along the arc with blue-shaded circles) of the unitary described in Eqns. \ref{eq:U1U3U1U2} and \ref{eq:U1U3U1U2_list} as well as Fig.~\ref{fig:U1U3U1U2_circ}. The circle with $s_0$ is the root of the tree, which represents an empty circuit. Other circles with $s_i^j$ in it denote the $j^{th}$ node at the $i^{th}$ level of the tree. $i$ can also indicate the number of layers currently in the circuit at state $s_i^j$. For example, on the leftmost branch of the tree, there is a node labelled $s_2^0$, indicating that it is the $0^{th}$ node at level $2$. At $s_2^0$, the circuit would be $\mathcal{P}_{s_2^0}=[U_0, U_0]$, which clearly only has 2 layers. We can also see that some of the possible branches along the blue-node path are pruned, leading to the size of operation pool at some node smaller than the total number of possible choices $c = \vert \mathcal{C}\vert$.}
\label{fig:treeexample}
\end{figure}
The performance of the quantum circuit can be evaluated from the loss $\mathcal{L}$ or reward $\mathcal{R}$, where the reward is just the negative of the loss. Both are functions of $\mathcal{P}$, and the parameters of the chosen operations $\boldsymbol{\theta}$:
\begin{equation}
\mathcal{L}(\mathcal{P},\boldsymbol{\theta})=L(\mathcal{P}, \boldsymbol{\theta})+\lambda
\end{equation}
\begin{equation}
\mathcal{R}(\mathcal{P},\boldsymbol{\theta})=R(\mathcal{P}, \boldsymbol{\theta})-\lambda,
\end{equation}
where $\lambda$ is some penalty function that may only appear when certain circuit structures appear, as well as other kinds of penalty terms, like penalty on the sum of absolute value of weights or the number of certain type of gates in the circuit; $L$ and $R$ are the loss/reward before applying the penalty. The purpose of the penalty term $\lambda$ is to `sway' the search algorithm from structures we do not desire.
Instead of storing all the operation parameters for each different quantum circuit, we share the parameters for a single operation at a certain location. That is, we have a multidimensional array of shape $(p, c, l)$, where $l$ is the maximum number of parameters for the operations in the operation pool. If all the operations in the pool are just the $U3$ gate \cite{nielsen00}:
\begin{equation}
U 3(\theta, \phi, \lambda)=\left[\begin{array}{cc}
\cos \left(\frac{\theta}{2}\right) & -e^{i \lambda} \sin \left(\frac{\theta}{2}\right) \\
e^{i \phi} \sin \left(\frac{\theta}{2}\right) & e^{i(\phi+\lambda)} \cos \left(\frac{\theta}{2}\right)
\end{array}\right]
\end{equation}
as well as its controlled version $CU3$ gate on different (pairs of) qubits, then in this case $l=3$.
To reduce the space required to store the parameters of all possible quantum circuits, for a quantum circuit with operation $k$ at layer $i$, the parameter is the same at that layer for that specific operation is the same for all other circuits with the same operation at the same location, which means we are sharing the parameters of the unitaries in the operation pool with other circuits. For example, in Fig.\ref{fig:treeexample}, besides the blue-node arc $\mathcal{P}=[U_0, U_1, U_2, U_1]$, there are also other paths, such $\mathcal{P}^{'} = [U_0, U_1, U_1, \cdots]$, and since the first two operations in $\mathcal{P}$ and $\mathcal{P}^{'}$ are the same, then we will share the parameters of $U_0$ and $U_1$ between these two circuits by setting the parameters to be the same for the $U_0$ and $U_1$ in both circuits, respectively.
Such a strategy is often called ``parameter-sharing'' or ``weight-sharing'' in the neural architecture search literature.
As shown in Fig~\ref{fig:treeexample} and mentioned earlier, the process of composing or searching a circuit can be formulated in the form of the tree structure. For example, if we start from an empty list $P = [\;]$ with maximal length four and an operation pool with three elements $C = \{U_0, U_1, U_2\}$,
then the state of the root node of our search tree will be the empty list $s_0^0 = [\;]$. The root node will have three possible actions (if there are no restrictions on what kind of operations can be chosen), which will lead us to three children nodes with states $s_1^0 = [U_0]=[0], s_1^1 = [U_1]=[1], s_1^2=[U_2] = [2]$. For each of these nodes, there will be a certain number of different operations that can be chosen to append the end of the list, depending on the specific restrictions. There will always be a ``placeholder'' operation that can be chosen if all other operations fail to meet the restrictions. The penalty resulting from the number of ``placeholder'' operations will only be reflected in the loss (or reward) of the circuit. The nodes can always be expanded with different actions, leading to different children, until the maximum length of the quantum circuit has been reached, which will give us the leaf node of the search tree.
The process of choosing operations at each layer can be viewed as a both a \textit{local} and \textit{global} multi-armed bandit (MAB). A multi-armed bandit, just as its name indicates, is similar to a bandit, or slot machine (in the casino), but has multiple levers, or arms, that can be pulled. Or equivalently, it can be viewed as someone who has multiple arms (maybe Squidward) that can pull the levers on different slot machines. In both cases, the rewards obtained from pulling different arms follow different (often unknown) distributions. The person pulling these arms needs to develop a strategy that can maximise his rewards from the machine(s). If we consider the whole circuit search problem as an MAB (the global MAB, $MAB_g$), then the "arms" are different circuit configurations. Although the rewards of these circuits are relatively easy to obtain based on the value of their cost functions after training of the circuits is finished (which still requires a fair amount of time for training), the exploding number of possible circuit configurations when the size of operation pool and number of layers increase makes it impossible to perform an informed search for suitable solutions while training every circuit we encountered during the search process. Since our circuit is basically a combination of different choice of layer unitaries, we can decompose the whole problem into the choices of unitaries at each layer, which is the local MAB, $MAB_i$, $i$ denoting the MAB problem from choosing the suitable unitary at layer $i$. In the local MAB for a single layer, the "arms" of the MAB are no longer the circuit configuration, instead the (permitted) unitary operations from the operation pool $\mathcal{C}$. Although the number of choices for the local MABs is considerably smaller than the global MAB, the reward for each arm is not directly observable. In next section, we will introduce the na\"ive assumption \cite{CMAB_RTS} to approximate the rewards of the local MABs from the global MAB, which will help us determine the rewards of the actions on each node (state) on the search tree for MCTS.
\begin{itemize}
\item \textit{Local MAB}: The choice of unitary operations at each layer can be considered a \textit{local} MAB. That is, different unitary operations can be treated as different ``arms'' of the bandit;
\item \textit{Global MAB}: We can also treat the composition of the entire quantum circuit as a \textit{global} MAB. That is, different quantum circuits can be viewed as different ``arms'' of the global bandit.
\end{itemize}
\subsection{Monte Carlo tree search (MCTS), nested MCTS and the na\"ive assumption}
Monte Carlo tree search (MCTS) is a heuristic search algorithm for a sequence decision process. It has achieved great success in other areas, including defeating the 18-time world champion Lee Sedol in the game of Go \cite{AlphaGoDBLP:journals/nature/SilverHMGSDSAPL16, AlphaGoZeroDBLP:journals/nature/SilverSSAHGHBLB17}. Generally, there are four stages in a single iteration of MCTS (see Fig.~\ref{fig:mcts}) \cite{MCTS_for_game10.5555/3022539.3022579}:
\begin{figure}[]
\centering
\includegraphics[width=0.8\textwidth]{Figures/mcts.png}
\caption{Four stages of Monte Carlo tree search. From left to right, up to down: Selection: Go down from the root node to a non fully expanded leaf node; Expansion: Expand the selected node by taking an action; Simulation: Simulate the game, which in our case is the quantum circuit, to obtain reward information \textbf{R}; Backpropagation: Back-propagation of the reward information along the path (arc) taken.}
\label{fig:mcts}
\end{figure}
\begin{itemize}
\item \textsc{Selection:}(Fig.\ref{fig:mcts}(a)) In the selection stage, the algorithm will, starting from the root of the tree, find a node at the end of an arc (a path from the root of the tree to the leaf node, the path marked by bold arrows and blue circles in Fig.\ref{fig:mcts}). The nodes along the arc are selected according to some policy, often referred as the ``selection policy'', until a non fully expanded node or a leaf node is reached. If the node is a leaf node, i.e after selecting the operation for the last layer of the quantum circuit, we can directly jump to the simulation stage to get the reward of the corresponding arc. If the node is not a leaf node, i.e the node is not fully expanded, then we can progress to the next stage;
\item \textsc{Expansion:}(Fig.\ref{fig:mcts}(b)) In the expansion stage, at the node selected in the previous stage, we choose a previously unvisited child by choosing a previously unperformed action. We can see from the upper right tree in Fig.\ref{fig:mcts} that a new node has been expanded at the end of the arc;
\item \textsc{Simulation:}(Fig.\ref{fig:mcts}(c)) In the simulation stage, if the node obtained from the previous stages is not a leaf node, we continue down the tree until we have reached a leaf node, i.e finish choosing the operation for the last layer. After we have the leaf node, we simulate the circuit and obtain the loss $\mathcal{L}$ (or reward $\mathcal{R}$). Usually, the loss $\mathcal{L}$ is required to update the parameters in the circuit;
\item \textsc{Backpropagation:}(Fig.\ref{fig:mcts}(d)) In this stage, the reward information obtained from the simulation stage is back-propagated through the arc leading from the root of the tree to the leaf node, and the number of visits as well as the (average) reward for each node along the arc is be updated.
\end{itemize}
The nested MCTS algorithm \cite{nestedmontecarlosearch} is based on the vanilla MCTS algorithm. However, before selecting the best child according to the selection policy, a nested MCTS will be performed on the sub-trees with each child as the root node. Then the best child will be selected according to the selection policy with updated reward information, see Fig.~\ref{fig:nestedmcts}.
\begin{figure}[H]
\centering
\includegraphics[width=0.8\textwidth]{Figures/nmcs.png}
\caption{Nested Monte Carlo tree search. Left: The root node has three possible actions, which in this case are unselected initially. We perform MCTS on all three children nodes (generated by the three possible actions) to update their reward information. After one iteration of MCTS with each child as root node for the search tree that MCTS performed on, the rewards of these three actions leading to the three child nodes are 10, 10, 20, respectively. In this case, the right child has the highest reward. Middle: After selecting the right side child node, we perform the same MCTS on all three possible children nodes as before, which gives updated reward information. In this case the middle child node has the highest reward, meaning that at this level we expand the middle child node. Right: Similar operations as before. If we only perform nested MCTS at the root node level, then it will be a level-1 nested MCTS.}
\label{fig:nestedmcts}
\end{figure}
We denote a quantum circuit with $p$ layers $\mathcal{P} = [k_1,\cdots, k_p]$, with each layer $k_i$ having a search space no greater than $\vert \mathcal{C} \vert = c$ (where $c$ is the number of possible unitary operations, as defined earlier). Then each choice for layer $k_i$ is a \textit{local arm} for the \textit{local MAB}, $MAB_i$. The set of these choices is also denoted as $k_i$. The combination of all $p$ layers in $\mathcal{P}$ forms a valid quantum circuit, which is called a \textit{global arm} of the \textit{global MAB, $MAB_g$}.
Since the global arm can be formed from the combination of the local arms, if we use the na\"ive assumption \cite{CMAB_RTS}, the global reward $R_{\rm global}$ for $MAB_g$ can be approximated by the sum of the reward of local MABs, and each local reward only depends on the choice made in each local MAB. This also means that, if the global reward is more easily accessed than the local rewards, then the local rewards can be approximated from the global reward. With the na\"ive assumption, we can have a linear relationship between the global reward and local rewards:
\begin{equation}
R_{\rm global} = \frac{1}{p}\sum_{i=1}^p R_i
\end{equation}
When searching for quantum circuits, we have no access to the reward distribution of individual unitary operations, however, we can apply the na\"ive assumption to approximate those rewards (``local reward'') with the global reward:
\begin{equation}
R_{i} \approx R_{\rm global}
\end{equation}
where $R_{i}$ is the reward for pulling an arm at \textit{local $MAB_i$} and $R_{\rm global}$ is the reward for the global arm.
Also, if we use the na\"ive assumption, we will not need to directly optimise on the large space of global arms as in traditional MABs. Instead, we can apply MCTS on the local MABs to find the best combination of local arms.
In the original work on nested MCTS~\cite{nestedmontecarlosearch}, a random policy was adopted for sampling. In this paper we will instead change it to the famous UCB policy~\cite{UCB_paper_10.5555/944919.944941}. Given a local $MAB_i$, with the set of all the possible choices $k_i$, the UCB policy can be defined as:
\begin{equation}
UCB: \argmax_{arm_j\in k_i} \Bar{R}(k_i, arm_j) + \alpha \sqrt{\frac{2\ln n_i}{n_j}}
\end{equation}
where $\Bar{R}(k_i, arm_j)$ is the average reward for $arm_j$ (i.e the reward for operation choice $U_j$ for layer $k_i$) in local $MAB_i$, $n_i$ is the number of times that $MAB_i$ has been used and $n_j$ is the number of times that $arm_j$ has been pulled. The parameter $\alpha$ provides a balance between exploration ($\sqrt{\frac{2\ln n_i}{n_j}}$) and exploitation ($\Bar{R}(k_i, arm_j$)). The UCB policy modifies the reward which the selection of action will be based on.
For small $\alpha$, the actual reward from the bandit will play a more important role in the UCB modified rewards, which will lead to selecting actions with previously observed high rewards. When $\alpha$ is large enough, the second term, which will be relatively large if $MAB_i$ has been visited many times but $arm_j$ of $MAB_i$ has only been pulled a small number of times, will have more impact on the modified reward, leading to a selection favoring previously less visited actions.
\subsection{QAS with Nested Na\"ive MCTS}
Generally, a single iteration for the search algorithm will include two steps for non-parameterised circuits, and two more parameter-related steps for parameterised quantum circuits. The set of parameters, which will be referred to as the parameters of the super circuit, or just parameters, in the following algorithms, follow the same parameter sharing strategy as described in Section 2.1. That is, if the same unitary operation (say, $U_2$) appears in the same location (say, layer \#5) across different quantum circuits, then the parameters are the same, even for different circuits. Also, with parameterised quantum circuits (PQC), it is common practice to ``warm-up'' the parameters by randomly sampling a batch of quantum circuits, calculating the averaged gradient, and update the parameters according to the averaged gradient, to get a better start for the parameters during the search process. During one iteration of the search algorithm, we have:
\begin{enumerate}
\item Sample a batch of quantum circuits from the super circuit with Algorithm \ref{alg:sampleArc};
\item (For PQCs) Calculate the averaged gradients of the sampled batch, add noise to the gradient to guide the optimiser to a more ``flat'' minimum if needed;
\item (For PQCs) Update the super circuit parameters according to the averaged gradients;
\item Find the best circuit with Algorithm \ref{alg:exploitArc}.
\end{enumerate}
We could also set up an early-stopping criteria for the search. That is, when the reward of the circuit obtained with Algorithm \ref{alg:exploitArc} meets a pre-set standard, we will stop the search algorithm and return the circuit that meet such standard (and further fine-tune the circuit parameters if there are any).
With the na\"ive assumption, which means the reward is evenly distributed on the local arms pulled for a global MAB, we can impose a prune ratio during the search. That is, given a node that has child nodes, if the average reward of a child node is smaller than a ratio, or percentage, of the average reward of the said node, then this child node will be removed from the set of all children, unless the number of children reached the minimum requirement.
\begin{algorithm}
\caption{SampleArc}\label{alg:sampleArc}
\begin{algorithmic}
\Require sample policy $Policy$, parameters of the super circuit $param$, number of rounds in sampling $N$
\Ensure list representation $\mathcal{P}$ of quantum circuit
\State $curr \gets GetRoot(Tr)$ \Comment{Starting from the root node of the tree $Tr$}
\State $i\gets0$ \Comment{Counter}
\While{$i<N$}
\State $ExecuteSingleRound(curr, Policy, param)$
\State $i\gets i+1$
\EndWhile
\While{$curr$ is not leaf node}
\State $curr\gets SelectNode(curr, Policy)$
\EndWhile
\State $\mathcal{P}\gets GetListRepresentation(curr)$
\end{algorithmic}
\end{algorithm}
\begin{algorithm}[H]
\caption{ExploitArc}\label{alg:exploitArc}
\begin{algorithmic}
\Require exploit policy $Policy$, parameters of the super circuit $param$, number of rounds in exploitation $N$
\Ensure list representation $\mathcal{P}$ of quantum circuit
\State $curr \gets GetRoot(Tr)$ \Comment{Starting from the root node of the tree $Tr$}
\While{$curr$ is not leaf node}
\State $i\gets0$ \Comment{Counter}
\While{$i<N$}
\State $ExecuteSingleRound(curr, Policy, param)$
\State $i\gets i+1$
\EndWhile
\State $curr\gets SelectNode(curr, Policy)$
\EndWhile
\State $\mathcal{P}\gets GetListRepresentation(curr)$
\end{algorithmic}
\end{algorithm}
\begin{algorithm}
\caption{SelectNode}\label{alg:selectChild}
\begin{algorithmic}
\Require current node $n$, selection policy $Policy$
\Ensure selected node $n'$
\If{$n$ is fully expanded}
\State $PruneChild(n)$ \Comment{Prune children nodes according to certain threshold}
\State $n' \gets GetBestChild(n, Policy)$ \Comment{Select the best child}
\Else
\State $n' \gets ExpandChild(n)$ \Comment{Expand the node}
\EndIf
\end{algorithmic}
\end{algorithm}
\begin{algorithm}[H]
\caption{ExecuteSingleRound}\label{alg:executeSingleRound}
\begin{algorithmic}
\Require current node $n$, selection policy $Policy$, parameters of the super circuit $param$
\Ensure leaf node $n'$
\State $n'\gets n$
\While{$n'$ is not leaf node}
\State $n'\gets SelectNode(n', Policy)$
\EndWhile
\State $R \gets Simulation(n', param)$ \Comment{Obtain reward from simulation}
\State $Backpropagate(n', R)$ \Comment{Back-propagate the reward information along the arc}
\end{algorithmic}
\end{algorithm}
\section{Numerical Experiments and Results}\label{experiments}
\subsection{Searching for the encoding circuit of [[4,2,2]] quantum error detection code}\label{422}
The [[4,2,2]] quantum error detection code is a simple quantum error detection code, which needs 4 physical qubits for 2 logical qubits and has a code distance 2. It is the smallest stabilizer code that can detect X- and Z-errors \cite{qec_intro_guide}. One possible set of code words for the [[4,2,2]] error detection code is:
\begin{equation}
\mathcal{E}_{\rm [[4,2,2]]}=\operatorname{span}\left\{\begin{array}{l}
|00\rangle_{L}=\frac{1}{\sqrt{2}}(|0000\rangle+|1111\rangle) \\
|01\rangle_{L}=\frac{1}{\sqrt{2}}(|0110\rangle+|1001\rangle) \\
|10\rangle_{L}=\frac{1}{\sqrt{2}}(|1010\rangle+|0101\rangle) \\
|11\rangle_{L}=\frac{1}{\sqrt{2}}(|1100\rangle+|0011\rangle)
\end{array}\right\}
\end{equation}
The corresponding encoding circuit is shown in Fig.\ref{fig:lit422}.
\begin{figure}[H]
\centering
\begin{quantikz}[transparent, row sep={0.8cm,between origins}]
\qw & \ctrl{0} & \qw & \qw & \qw & \targ{}\vqw{0} & \qw\\
\qw & \qw & \ctrl{0} & \qw & \targ{}\vqw{0} & \qw & \qw\\
\qw & \targ{}\vqw{-2} & \targ{}\vqw{-1} & \targ{}\vqw{0} & \qw & \qw & \qw\\
\qw & \gate{H} & \qw & \ctrl{-1} & \ctrl{-2} & \ctrl{-3} & \qw
\end{quantikz}
\caption{Encoding circuit of the [[4,2,2]] code \cite{qec_intro_guide} to detect X- and Z-errors. It needs 4 physical qubits for 2 logical qubits and has a code distance 2. By our settings, the number of layers equals to the number of operations in the circuit. In this figure, the number of layers is 6.}
\label{fig:lit422}
\end{figure}
Quantum error detection and correction is vital to large-scale fault-tolerant quantum computing. By searching for the encoding circuit of the [[4,2,2]] error detection code, we demonstrate that our algorithm has the potential to automatically find device-specific encoding circuits of quantum error detection and correction codes for future quantum processors.
\subsubsection{Experiment Settings}
When searching for the encoding circuit of the [[4,2,2]] quantum error correction code, we adopted an operation pool consisting of only non-parametric operations: the Hadamard gate on each of the four qubits and CNOT gates between any two qubits. The total size of the operation pool is $4 + \frac{4!}{2!\times2!}\times2=16$. When there are 6 layers in total, the overall size of the search space is $16^6\approx1.67\times10^7$.
The loss function for this task is based on the fidelity between the output state of the searched circuit and the output generated by the encoding circuit from Section 4.3 of \cite{qec_intro_guide} (also shown in Fig.~\ref{fig:lit422}) when input states taken from the set of Pauli operator eigenstates and the magic state $\vert T \rangle$ are used:
\begin{equation}
\mathcal{S}=\{\vert 0 \rangle, \vert 1 \rangle, \vert + \rangle, \vert - \rangle, \vert +i \rangle, \vert -i \rangle, \vert T \rangle\}
\end{equation}
where $\vert T \rangle = \frac{\vert 0 \rangle + e^{i\pi/4}\vert 1 \rangle}{\sqrt{2}}$.
The input states (initialised on all four qubits) are
\begin{equation}
\mathcal{I}_{[[4,2,2]]} = \{\vert \varphi_1 \rangle \otimes \vert \varphi_2 \rangle\otimes\vert 00\rangle \; \vert \; \vert \varphi_1 \rangle, \vert \varphi_2 \rangle \in \mathcal{S}\}
\end{equation}
We denote the unitary on all four qubits shown in Fig.~\ref{fig:lit422} as $U_{[[4,2,2]]}$, and the unitary from the searched circuit as $U_{Searched\;[[4,2,2]]}$, which is a function of the structure $\mathcal{P}_{Searched\;[[4,2,2]]}$. The loss and reward function can then be expressed as:
\begin{equation}
L_{[[4,2,2]]} = 1-\frac{1}{\vert \mathcal{I}_{[[4,2,2]]} \vert}\sum_{\vert \psi_i\rangle\in \mathcal{I}_{[[4,2,2]]}} \langle \psi_i \vert U_{\rm Searched\;[[4,2,2]]}^{\dagger} O_{[[4,2,2]]}(\vert \psi_i\rangle) U_{\rm Searched\;[[4,2,2]]} \vert \psi_i\rangle
\end{equation}
\begin{equation}
R_{[[4,2,2]]} = 1-L_{[[4,2,2]]}
\end{equation}
where
\begin{equation}
O_{[[4,2,2]]}(\vert \psi_i\rangle) = U_{[[4,2,2]]} \vert \psi_i\rangle \langle \psi_i \vert U_{[[4,2,2]]}^{\dagger},\; \vert \psi_i\rangle\in \mathcal{I}_{[[4,2,2]]}
\end{equation}
The circuit simulator used in this and the following numerical experiments is Pennylane \cite{bergholm2020pennylane}.
\subsubsection{Results}
To verify whether the search algorithm will always reach the same solution, we ran the search algorithm twice, and both times the algorithm found an encoding circuit within a small numbers of iterations (Fig.~\ref{fig:422_reward}), although the actual circuit are different from each other, as shown in Fig.~\ref{fig:422_circ}. The search process that gave the circuit in Fig.\ref{fig:422_first_circ} met the early-stopping criteria in four iterations, and the search process that gave the circuit in Fig.\ref{fig:422_second_circ} met the early-stopping criteria in eight iterations, as shown in Fig.~\ref{fig:422_reward}.
\begin{figure}[H]
\centering
\includegraphics[width=0.8\textwidth]{Figures/fig_422_rewards_1_2.pdf}
\caption{Rewards when searching for encoding circuits of the [[4,2,2]] code. We can see that in both cases the algorithm was able find the encoding circuit that generated the required code words in just a few iterations. `Circuit a' refers to the search rewards for the circuit in Fig.\ref{fig:422_first_circ} and `Circuit b' refers to the search rewards for the circuit in Fig.\ref{fig:422_second_circ}.}\label{fig:422_reward}
\end{figure}
\begin{figure}[H]
\centering
\begin{subfigure}[b]{0.48\textwidth}
\centering
\begin{quantikz}[transparent, row sep={0.8cm,between origins}]
\qw & \qw & \ctrl{0} & \targ{}\vqw{0} & \qw & \qw & \ctrl{0} & \qw\\
\qw & \qw & \qw & \qw & \ctrl{0} & \targ{}\vqw{0} & \qw & \qw\\
\qw & \qw & \qw & \qw & \targ{}\vqw{-1} & \qw & \targ{}\vqw{-2} & \qw\\
\qw & \gate{H} & \targ{}\vqw{-3} & \ctrl{-3} & \qw & \ctrl{-2} & \qw & \qw
\end{quantikz}
\caption{}
\label{fig:422_first_circ}
\end{subfigure}
~
\begin{subfigure}[b]{0.48\textwidth}
\centering
\begin{quantikz}[transparent, row sep={0.8cm,between origins}]
\qw & \ctrl{0} & \qw & \qw & \ctrl{0} & \targ{}\vqw{0} & \qw\\
\qw & \targ{}\vqw{-1} & \targ{}\vqw{0} & \ctrl{0} & \targ{}\vqw{-1} & \qw & \qw\\
\qw & \qw & \qw & \targ{}\vqw{-1} & \qw & \qw & \qw\\
\qw & \gate{H} & \ctrl{-2} & \qw & \qw & \ctrl{-3} & \qw
\end{quantikz}
\caption{}
\label{fig:422_second_circ}
\end{subfigure}
\caption{Two different encoding circuits of the [[4,2,2]] code produced by the search algorithm.}\label{fig:422_circ}
\end{figure}
\subsection{Solving linear equations}
The variational quantum linear solver (VQLS), first proposed in \cite{Bravo-Prieto_undated-oq}, is designed to solve linear systems $Ax=b$ on near term quantum devices. Instead of using quantum phase estimation like the HHL algorithm \cite{HHL}, which is unfeasible on near term devices due to large circuit depth, VQLS adopts a variational circuit to prepare a state $\ket{x}$ such that
\begin{equation}
A\ket{x} \propto \ket{b}
\end{equation}
In this section, we will task our algorithm to automatically search for a variantional circuit to prepare a state $\ket{x}$ to solve $Ax = b$ with A in the form of
\begin{equation}
A = \sum_l c_l A_l
\end{equation}
where $A_l$ are unitaries, and $\ket{b} = H^{\otimes n}\ket{\mathbf{0}}$.
\noindent
We will also adopt the local cost function $C_L$ described in \cite{Bravo-Prieto_undated-oq}:
\begin{equation}\label{eqn:vqls_local_loss}
C_{L}=1-\frac{\sum_{l, l^{\prime}} c_{l} c_{l^{\prime}}^{*}\bra{0}V^{\dagger} A_{l^{\prime}}^{\dagger} U P U^{\dagger} A_{l} V\ket{0}}{\sum_{l, l^{\prime}} c_{l} c_{l^{\prime}}^{*}\bra{0}V^{\dagger} A_{l^{\prime}}^{\dagger} A_{l} V\ket{0}}
\end{equation}
where $U=H^{\otimes n}$, $V$ is the (searched) variational circuit that can produce the solution state $V\ket{0} = \ket{x}$, and $P=\frac{1}{2}+\frac{1}{2 n} \sum_{j=0}^{n-1} Z_{j}$ \cite{pennylane_vqls}.
\subsubsection{Experiment Settings}
The linear system to be solved in our demonstration is:
\begin{equation}
A = \zeta I + J X_1 + J X_2 + \eta Z_3 Z_4
\end{equation}
\begin{equation}
\ket{b} = H^{\otimes 4}\ket{0}
\end{equation}
with $J = 0.1, \zeta = 1, \eta = 0.2$.
The loss function we adopted follows the local loss $C_L$ in Eqn.~\ref{eqn:vqls_local_loss}. However, since the starting point of the loss values often has a magnitude of $10^{-2}\sim 10^{-3}$, we will need scaling in the reward function:
\begin{equation}
\mathcal{R} = e^{-10 C_L}-\lambda
\end{equation}
where $\lambda$ is a penalty term depending on the number of Placeholder gates in the circuit. The operation pool consists of CNOT gates between neighbouring two qubits as well as the first and fourth qubits, the Placeholder and the single qubit rotation gate Rot \cite{nielsen00}:
\begin{equation}
Rot(\phi, \theta, \omega)=R_Z(\omega) R_Y(\theta) R_Z(\phi)=\left[\begin{array}{cc}
e^{-i(\phi+\omega) / 2} \cos (\theta / 2) & -e^{i(\phi-\omega) / 2} \sin (\theta / 2) \\
e^{-i(\phi-\omega) / 2} \sin (\theta / 2) & e^{i(\phi+\omega) / 2} \cos (\theta / 2)
\end{array}\right]
\end{equation}
The size of the operation pool $c = \vert \mathcal{C}\vert = 16$, and number of layers $p = 10$, giving us a search space of size $\vert \mathcal{S} \vert = 10^{16}$. There is also an additional restriction of maximum number of CNOT gates in the circuit, which is 8, the number of CNOT gates required to created two layers of circular entanglement.
\subsubsection{Results}
The search rewards as well as fine-tune losses are shown in Fig~\ref{fig:vqls_search_finetune}. We can see that the search algorithm can produce a circuit with high reward (exceeds the threshold) quickly and the loss of the optimised parameters can reach close to 0. Although facing a large search space, our algorithm can still find a circuit (shown in Fig~\ref{fig:vqls_circ}) that minimises the loss function (Fig~\ref{fig:vqls_4q_finetune}) and leads us to results close to the classical solution. A comparison of the results obtained by directly solving the linear equation $Ax=b$ and the results obtained by sampling the state $\ket{x}$ produced by the searched circuit is shown in Fig~\ref{fig:vqls_results_compare}.
\begin{figure}[H]
\centering
\begin{subfigure}[t]{0.48\textwidth}
\includegraphics[width=\textwidth]{Figures/fig_vqls_4q_search_rewards.pdf}
\caption{Search rewards for VQLS. The change of rewards with respect to the iterations is shown. We can see that the reward quickly reached the early stopping threshold at iteration 10. In the VQLS case, the reward is scaled since the initial reward with random sampled circuit structure and parameters is already at the magnitude of $10^{-2}$.}
\label{fig:vqls_4q_search}
\end{subfigure}
~
\begin{subfigure}[t]{0.48\textwidth}
\includegraphics[width=\textwidth]{Figures/fig_vqls_4q_finetune.pdf}
\caption{Fine-tune loss for the VQLS circuit.After the searched stopped at iteration 10 as shown in Fig~\ref{fig:vqls_4q_search}, the structure of the circuit is left unchanged and its parameters are optimised to achieve smaller losses. The final loss of the optimised parameters is very close to 0.}
\label{fig:vqls_4q_finetune}
\end{subfigure}
\caption{The search rewards and fine-tune loss for VQLS experiment. }\label{fig:vqls_search_finetune}
\end{figure}
\begin{figure}[H]
\centering
\begin{quantikz}[transparent, row sep={0.8cm,between origins}]
\qw & \gate{H} & \gate{Rot} & \targ{}\vqw{0} & \qw & \targ{}\vqw{0} & \gate{Rot} & \qw\\
\qw & \gate{H} & \gate{Rot} & \ctrl{-1} & \gate{Rot} & \ctrl{-1} & \gate{Rot} & \qw\\
\qw & \gate{H} & \gate{Rot} & \ctrl{0} & \qw & \qw & \qw & \qw\\
\qw & \gate{H} & \gate{Rot} & \targ{}\vqw{-1} & \qw & \qw & \qw & \qw
\end{quantikz}
\caption{Circuit searched for the VQLS problem. $Rot(\phi, \theta, \omega)=R_Z(\omega) R_Y(\theta) R_Z(\phi)$. The four Hadamard gates at the beginning of the circuit are to put everything in an equal superposition, and not included when constructing the search tree, i.e. the composed circuits will always start with four Hadamard gates placed on the four qubits. When drawing the circuit, the Placeholder gates, which are just identity gates, are removed from searched $\mathcal{P}$, although they were considered when constructing the search tree.}
\label{fig:vqls_circ}
\end{figure}
\begin{figure}[H]
\centering
\includegraphics[width=0.95\textwidth]{Figures/fig_vqls_search_results_compare.pdf}
\caption{Comparison between classical probabilities, which obtained from solving the matrix equation with the classical method, i.e. $x = A^{-1}b$, of the normalised solution vector $\frac{x}{||x||}$ for $Ax = b$ (left), and the probabilities obtained by sampling the state $\ket{x}$ produced by the trained circuit in Fig~\ref{fig:vqls_circ} (right). The number of shots for measurement is $10^6$. We can see that the quantum results is very close to the classically obtained ones, showing that our algorithm can be indeed applied to finding variational ans\"atz for VQLS problems.}
\label{fig:vqls_results_compare}
\end{figure}
\subsection{Search for quantum chemistry ans\"atze}\label{h2}
Recently, there has been a lot of progress made on finding the ground state energy of a molecule on a quantum computer with variational circuit, both on theoretical \cite{li2017efficient,mcclean2016theory,wecker2015progress} and experimental \cite{peruzzo2014variational,o2016scalable, colless2017implementing, kandala2017hardware, colless2018computation, dumitrescu2018cloud} front.
Normally, when designing the ans\"atz for the ground energy problem either a physically plausible or a hardware efficient ans\"atz needs to be found. However, our algorithm provides an approach which can minimise the effort needed to carefully choose an ans\"atz and automatically design the circuit according to the device gate set and topology.
Generally speaking, solving the ground energy problem with quantum computers is an application of the variational principle \cite{sakurai_napolitano_2017}:
\begin{equation}
E_0 \leq \frac{\langle\tilde{0}|H| \tilde{0}\rangle}{\langle\tilde{0} \mid \tilde{0}\rangle} \label{eq:variational}
\end{equation}
where $H$ is the system Hamiltonian, $| \tilde{0}\rangle$ is the ``trail ket'' \cite{sakurai_napolitano_2017}, or ans\"atz, trying to mimic the real wave function at ground state with energy $E_0$, which is the smallest eigenvalue of the system Hamiltonian H. Starting from $\ket{0^{\otimes n}}$ for an $n-$qubit system, the ``trial ket'' can be written as a function of a set of (real) parameters $\theta$:
\begin{equation}
\ket{\tilde{0}} = \ket{\varphi (\theta)} = U(\theta)\ket{0^{\otimes n}}
\end{equation}
Given an ans\"atz, the goal of optimisation is to find a set of parameters $\theta$ that minimises the right hand side of Eqn \ref{eq:variational}. However, in our research, the form of the trail wave function will no longer be fixed. We will not only vary the parameters, but also the circuit structure that represent the ans\"atz.
\subsubsection{Experiment settings}
\paragraph{Search an ans\"atz for finding the ground energy of $\text{H}_2$:}
In this experiment, we adopted the 4-qubit Hamiltonian $H_{hydrogen}$ for the hydrogen molecule $\text{H}_2$ generated by the Pennylane-QChem \cite{bergholm2020pennylane} package, when the coordinates of the two hydrogen atoms are $(0, 0, -0.66140414)$ and $(0, 0, 0.66140414)$, respectively, in atom units. The goal of this experiment is to find an ans\"atz that can produce similar states as the four-qubit Givens rotation for single and double excitation.
The unitary operator \footnote{see \url{https://pennylane.readthedocs.io/en/latest/code/api/pennylane.SingleExcitation.html}} that performs single excitation on a subspace spanned by $\{\ket{01}, \ket{10} \}$ can be written as
\begin{equation}
U(\phi)=\left[\begin{array}{cccc}
1 & 0 & 0 & 0 \\
0 & \cos (\phi / 2) & -\sin (\phi / 2) & 0 \\
0 & \sin (\phi / 2) & \cos (\phi / 2) & 0 \\
0 & 0 & 0 & 1
\end{array}\right]
\end{equation}
And the transformation of the double excitation on the subspace spanned by $\{|1100\rangle,|0011\rangle\}$ is\footnote{see \url{https://pennylane.readthedocs.io/en/latest/code/api/pennylane.DoubleExcitation.html}} :
\begin{equation}
\begin{aligned}
&|0011\rangle \rightarrow \cos (\phi / 2)|0011\rangle+\sin (\phi / 2)|1100\rangle \\
&|1100\rangle \rightarrow \cos (\phi / 2)|1100\rangle-\sin (\phi / 2)|0011\rangle
\end{aligned}
\end{equation}
Following \cite{pennylane_dev_team_2021}, we initialised the circuit with the 4-qubit vacuum state $\vert \psi_0\rangle=\vert 0000\rangle$. We denote the unitary for the searched ans\"atz $U_{\rm SearchedAnsatz}$, which is a function of its structure $\mathcal{P}_{\rm SearchedAnsatz}$ and corresponding parameters. Then the loss and reward functions can be written as:
\begin{equation}
L_{\text{H}_2} =\langle \psi_0 \vert U_{\rm SearchedAnsatz}^{\dagger} H_{\rm Hydrogen} U_{\rm SearchedAnsatz} \vert \psi_0\rangle
\end{equation}
\begin{equation}
R_{\text{H}_2} = -L_{\text{H}_2}
\end{equation}
The operation pool consists of Placeholder gates, Rot gates and CNOT gates with a linear entanglement topology (nearest neighbour interactions). The maximum number of layers is 30, with maximum number of CNOT gates $30/2 = 15$, and no penalty term for the number of Placeholder gates:
\begin{equation}
\mathcal{R}_{\text{H}_2, \rm Pool\;1} = R_{\text{H}_2}
\end{equation}
Such settings of operation pool and number of layers will give us an overall search space of size $14^{30}\approx 2.42\times 10^{34}$. However, the imposed hard limits and gate limits will drastically reduce the size of the search space.
\paragraph{Search an ans\"atz for finding the ground energy of $\text{LiH}$}
The loss and reward functions for the $\text{LiH}$ task are similar to the $\text{H}_2$ one:
\begin{equation}
L_{\text{LiH}} =\langle \psi_0 \vert U_{\rm SearchedAnsatz}^{\dagger} H_{\text{LiH}} U_{\rm SearchedAnsatz} \vert \psi_0\rangle
\end{equation}
\begin{equation}
R_{\text{LiH}} = -L_{\text{LiH}}
\end{equation}
and the initial state is also the vacuum state $\ket{\psi_0} = \ket{0}^{\otimes 10}$. The Hamiltonian is obtained at bond length 2.969280527 Bohr, or 1.5712755873606 Angstrom, with 2 active electrons and 5 active orbitals. The size of the operation pool $c = \vert \mathcal{C} \vert = 38$, including Rot gates, Placeholder and CNOT gates operating on neighbouring qubits on a line topology. The maximum number of layers is 20, giving us a search space of size $\vert \mathcal{S} \vert = 38^{20} \approx 3.94\times 10^{31}$. The `hard limit' on the number of CNOT gates in the circuit is $20/2=10$.
\paragraph{Search an ans\"atz for finding the ground energy of $\text{H}_2\text{O}$} The loss and reward functions of the water molecule are shown as follows:
\begin{equation}
L_{\text{H}_2\text{O}} =\langle \psi_0 \vert U_{\rm SearchedAnsatz}^{\dagger} H_{\text{H}_2\text{O}} U_{\rm SearchedAnsatz} \vert \psi_0\rangle
\end{equation}
\begin{equation}
R_{\text{H}_2\text{O}} = -L_{\text{H}_2\text{O}}
\end{equation}
and the initial state is also the vacuum state $\ket{\psi_0} = \ket{0}^{\otimes 8}$. The Hamiltonian is obtained when the three atoms are positioned at the following coordinates:
\begin{equation}
\text{H}:(0.,0.,0.);
\text{O}:(1.63234543, 0.86417176, 0);
\text{H}:(3.36087791, 0.,0.)
\end{equation}
Units are in Angstrom.
Active electrons is set to 4 and active orbitals is set to 4. The size of the operation pool $c = \vert \mathcal{C} \vert = 30$, including Rot gates, Placeholder and CNOT gates operating on neighbouring qubits on a line topology. The maximum number of layers is 20, giving us a search space of size $\vert \mathcal{S} \vert = 30^{50} \approx 7.18\times 10^{73}$. The `hard limit' on the number of CNOT gates in the circuit is 25.
\subsubsection{Results}
\paragraph{$\text{H}_2$ Results} The search reward when finding the suitable circuit structure is shown in Fig~\ref{fig:h2_search} and the training process for the circuit produced by the search algorithm is shown in Fig~\ref{fig:h2_finetune}. The ans\"atz is presented in Fig~\ref{fig:h2_circ}. We can see from Fig~\ref{fig:h2_circ} that the unitaries are not randomly placed on the four wires, instead there present familiar structures like the decomposition of the SWAP gate and Ising coupling gates. An example of the Ising coupling gates (often appears in quantum optimisation problems) is the $R_{ZZ}$ gate:
\begin{equation}
R_{Z Z}(\theta)=e^{ -i \frac{\theta}{2} Z \otimes Z}=\left[\begin{array}{cccc}
e^{-i \frac{\theta}{2}} & 0 & 0 & 0 \\
0 & e^{i \frac{\theta}{2}} & 0 & 0 \\
0 & 0 & e^{i \frac{\theta}{2}} & 0 \\
0 & 0 & 0 & e^{-i \frac{\theta}{2}}
\end{array}\right] = CNOT_{1,2}RZ_{2}(\theta)CNOT_{1,2}
\end{equation}
Where $CNOT_{1,2}$ is the CNOT gate controlled by the first qubit and target on the second qubit, and $RZ_{2}(\theta)$ is a Z-rotation gate on the second qubit.
However, other parts of the circuit are not familiar, which indicates that the search algorithm can go beyond human intuition. The total number of gates in the circuit is 22, including 13 local CNOT gates.
\begin{figure}[H]
\centering
\begin{subfigure}[t]{0.48\textwidth}
\includegraphics[width=\textwidth]{Figures/fig_h2_vac_init_search_rewards.pdf}
\caption{Search rewards for the $\text{H}_2$ ans\"atz. We can see that for most of the 50 iterations, the reward for the best circuit sampled from the search tree stays over 0.7.}
\label{fig:h2_search}
\end{subfigure}
~
\begin{subfigure}[t]{0.48\textwidth}
\includegraphics[width=\textwidth]{Figures/fig_h2_vac_init_fine_tune_loss.pdf}
\caption{Fine-tune loss for the searched $\text{H}_2$ circuit. At the last iteration of optimisation, the energy is around -1.1359 Ha. The classically computed full configuration interaction result with PySCF \cite{Sun2018-nq, Sun2020-ej}, which is around -1.132 Ha and marked by the red horizontal dashed line. The difference between the energy achieved by the searched circuit and PySCF is close to chemical accuracy.)}
\label{fig:h2_finetune}
\end{subfigure}
\caption{The search rewards and fine-tune loss for $\text{H}_2$ circuit. experiment.}\label{fig:h2_search_finetune}
\end{figure}
\begin{figure}[H]
\centering
\begin{quantikz}[transparent, row sep={0.8cm,between origins}]
\qw & \gate{Rot} & \qw & \qw & \qw & \ctrl{0} & \gate{Rot} & \targ{}\vqw{0} & \ctrl{0} & \targ{}\vqw{0} & \qw & \qw & \qw & \ctrl{0} & \qw & \qw\\
\qw & \qw & \qw & \qw & \targ{}\vqw{0} & \targ{}\vqw{-1} & \qw & \ctrl{-1} & \targ{}\vqw{-1} & \ctrl{-1} & \targ{}\vqw{0} & \gate{Rot} & \targ{}\vqw{0} & \targ{}\vqw{-1} & \qw & \qw\\
\qw & \gate{Rot} & \ctrl{0} & \gate{Rot} & \ctrl{-1} & \ctrl{0} & \gate{Rot} & \targ{}\vqw{0} & \gate{Rot} & \targ{}\vqw{0} & \ctrl{-1} & \qw & \ctrl{-1} & \targ{}\vqw{0} & \gate{Rot} & \qw\\
\qw & \qw & \targ{}\vqw{-1} & \gate{Rot} & \qw & \targ{}\vqw{-1} & \qw & \ctrl{-1} & \qw & \ctrl{-1} & \qw & \qw & \qw & \ctrl{-1} & \qw & \qw
\end{quantikz}
\caption{The circuit for finding the ground energy of the $\text{H}_2$ molecule produced by the search algorithm. We can see that there are already familiar structures emerging, like the SWAP gate between the first two qubits in the middle and the Ising coupling gate-like structure right under the decomposed SWAP gate.}
\label{fig:h2_circ}
\end{figure}
\paragraph{$\text{LiH}$ Results} The search reward when finding the suitable circuit structure for LiH is shown in Fig~\ref{fig:lih_search} and the training process for the circuit produced by the search algorithm is shown in Fig~\ref{fig:lih_finetune}. The ans\"atz is presented in Fig~\ref{fig:lih_circ}. The circuit produced by the search algorithm is simpler compared to the $\text{H}_2$ ans\"atz in Fig~\ref{fig:h2_circ}, indicating that the initial state may be very close to the ground energy state.
\begin{figure}[H]
\centering
\begin{subfigure}[t]{0.48\textwidth}
\includegraphics[width=\textwidth]{Figures/fig_lih_search_rewards.pdf}
\caption{Search rewards for the $\text{LiH}$ ans\"atz. We can see that for most of the 50 iterations, the reward for the best circuit sampled from the search tree stays over 7.7.}
\label{fig:lih_search}
\end{subfigure}
~
\begin{subfigure}[t]{0.48\textwidth}
\includegraphics[width=\textwidth]{Figures/fig_lih_fine_tune_loss.pdf}
\caption{Fine-tune loss for the searched $\text{LiH}$ circuit. At the last iteration of optimisation, the energy is around -7.9526 Ha, close to the chemical accuracy compared to classically computed full configuration interaction energy with PySCF \cite{Sun2018-nq, Sun2020-ej}, which is around -7.8885 Ha}
\label{fig:lih_finetune}
\end{subfigure}
\caption{The search rewards and fine-tune loss for $\text{LiH}$ circuit. experiment.}\label{fig:lih_search_finetune}
\end{figure}
\begin{figure}[H]
\centering
\begin{quantikz}[transparent, row sep={0.8cm,between origins}]
\qw & \gate{Rot} & \qw & \qw & \qw & \qw & \qw & \qw\\
\qw & \gate{Rot} & \qw & \qw & \qw & \targ{}\vqw{0} & \qw & \qw\\
\qw & \gate{Rot} & \qw & \ctrl{0} & \targ{}\vqw{0} & \ctrl{-1} & \ctrl{0} & \qw\\
\qw & \qw & \targ{}\vqw{0} & \targ{}\vqw{-1} & \ctrl{-1} & \gate{Rot} & \targ{}\vqw{-1} & \qw\\
\qw & \gate{Rot} & \ctrl{-1} & \qw & \qw & \qw & \qw & \qw\\
\qw & \gate{Rot} & \qw & \qw & \qw & \qw & \qw & \qw\\
\qw & \qw & \qw & \qw & \qw & \qw & \qw & \qw\\
\qw & \gate{Rot} & \qw & \qw & \qw & \qw & \qw & \qw\\
\qw & \qw & \qw & \qw & \qw & \qw & \qw & \qw\\
\qw & \gate{Rot} & \qw & \qw & \qw & \qw & \qw & \qw
\end{quantikz}
\caption{Circuit structure produced by the search algorithm for LiH. We can see that the structure of the circuit is quite simple, compared to the circuit for $\text{H}_2$ in Fig~\ref{fig:h2_circ}, indicating that the vacuum state $\ket{\psi_0} = \ket{0}^{\otimes 10}$ is already very close to the ground energy state.}
\label{fig:lih_circ}
\end{figure}
\paragraph{$\text{H}_2\text{O}$ Results} The search reward when finding the suitable circuit structure for $\text{H}_2\text{O}$ is shown in Fig~\ref{fig:h2_search} and the training process for the circuit produced by the search algorithm is shown in Fig~\ref{fig:h2o_finetune}. The ans\"atz is presented in Fig~\ref{fig:h2o_circ}, which has 38 gates in total, including 10 local CNOT gates. Although there are still some familiar structures, such as the Ising coupling in the circuit, the heuristics behind most parts of the circuit are already unintuitive for human researchers.
\begin{figure}[H]
\centering
\begin{subfigure}[t]{0.48\textwidth}
\includegraphics[width=\textwidth]{Figures/fig_H2O_search_rewards.pdf}
\caption{Search rewards for the $\text{H}_2\text{O}$ ans\"atz. We can see that for most of the 50 iterations, the reward for the best circuit sampled from the search tree stays over 74.9.}
\label{fig:h2o_search}
\end{subfigure}
~
\begin{subfigure}[t]{0.48\textwidth}
\includegraphics[width=\textwidth]{Figures/fig_H2O_fine_tune_loss.pdf}
\caption{Fine-tune loss for the searched $\text{H}_2\text{O}$ circuit. At the last iteration of optimisation, the energy is around -75.4220 Ha, close to the chemical accuracy compared to classically computed full configuration interaction energy with PySCF \cite{Sun2018-nq, Sun2020-ej}, which is around -75.4917 Ha}
\label{fig:h2o_finetune}
\end{subfigure}
\caption{The search rewards and fine-tune loss for $\text{H}_2\text{O}$ circuit.}\label{fig:h2o_search_finetune}
\end{figure}
\begin{figure}[H]
\centering
\includegraphics[width=0.9\textwidth]{Figures/fig_h2o_circ_quantikz.png}
\caption{Circuit for $\text{H}_2\text{O}$ produced by the search algorithm.}
\label{fig:h2o_circ}
\end{figure}
\subsection{Solving the \textsc{MaxCut} problem}
As a classic and well-known optimisation problem, the \textsc{MaxCut} problem plays an important role in network science, circuit design, as well as physics \cite{Bharti2022-sw}. The objective of the \textsc{MaxCut} problem is to find a partition $z$ of vertices in a graph $G = (V, E)$ which maximises the number of edges connecting the vertices in two disjoint sets $A$ and $B$:
\begin{equation}
C(z) =\sum_{a=1}^m C_a(z)
\end{equation}
where $C_a(z) = 1$ if the $a^{th}$ edge connects one vortex in set $A$ and one vortex in set $B$, and $C_a(z) = 0$ otherwise. To perform the optimisation on a quantum computer, we will need to transform the cost function into Ising formulation:
\begin{equation}
H_C = -\sum_{(i, j)\in E} \frac{1}{2} (I - Z_i Z_j)w_{ij}
\end{equation}\label{qaoa_ham}
where $Z_i$ is the Pauli $Z$ operator on the $i^{th}$ qubit and $w_{ij}$ is the weight of edge $(i, j)\in E$ for weighted \textsc{MaxCut} problem. For unweighted problems, $w_{ij} = 1$. In this formulation, vertices are represented by qubits in computational bases. By finding the wave-function that minimises the cost Hamiltonian $H_C$, we can find the solution that maximises $C(z)$. Previously, the major components of the QAOA (quantum approximate optimisation algorithm) ans\"atz are the cost Hamiltonian encoded by the cost unitary and the mixing Hamiltonians encoded by the mixing unitaries \cite{Farhi2014-ug}. Although this ans\"atz can find all the solutions in a equal superposition form, it is not always effective when the number of layers is small. Also, when the number of qubits (vertices) grows, the required number of layers and the number of shots during measurement to extract all of the solutions will also grow.
Since we have already had a Hamiltonian as our cost function in Sec.~\ref{h2}, we follow similar approach as quantum chemistry to find one of the solutions when the number of vertices is large.
\subsubsection{Experiment Settings}
\paragraph{Unweighted \textsc{MaxCut}}
\begin{figure}[H]
\centering
\includegraphics[width=0.8\textwidth]{Figures/fig_max_cut_large_7q.pdf}
\caption{Problem graph for the unweighted \textsc{MaxCut} experiment}
\label{fig:max_cut_prob}
\end{figure}
The problem graph for the unweighted \textsc{MaxCut} experiment is shown in Fig. \ref{fig:max_cut_prob}. This problem has six equally optimal solutions: 1001100, 0110010, 0111010, 1000101, 1001101 and 0110011, all have $C(z)=7$. The loss function is based on the expectation of the cost Hamiltonian $H_C$:
\begin{equation}
L_{\textsc{MaxCut}} = (\bra{+})^{\otimes 7}U_{\rm SearchedAnsatz}^{\dagger} H_C U_{\rm SearchedAnsatz} (\ket{+})^{\otimes 7}
\end{equation}
The reward function is simply the negative of the loss function:
\begin{equation}
\mathcal{R}_{\textsc{MaxCut}} = -L_{\textsc{MaxCut}}
\end{equation}
We ran the search algorithm twice with the same basic settings, including the operation pool and the maximum number of layers. Since there is a random sampling process during the warm-up stage, the final solutions found by the algorithm are expected to be different. The operation pool consists of CNOT gates between every two qubits, the Placeholder and the single qubit rotation gate \cite{nielsen00}:
\begin{equation}
Rot(\phi, \theta, \omega)=R_Z(\omega) R_Y(\theta) R_Z(\phi)=\left[\begin{array}{cc}
e^{-i(\phi+\omega) / 2} \cos (\theta / 2) & -e^{i(\phi-\omega) / 2} \sin (\theta / 2) \\
e^{-i(\phi-\omega) / 2} \sin (\theta / 2) & e^{i(\phi+\omega) / 2} \cos (\theta / 2)
\end{array}\right]
\end{equation}
The size of the operation pool $c = \vert \mathcal{C} \vert = 28$, and the number of layers $p = 15$, leading to a search space of size $\vert \mathcal{S} \vert = 28^{15} \approx 5 \times10^{21}$. The `hard' restrictions on the maximum number of CNOT gates in a circuit, which is 7, can help reduce the size of the search space.
\paragraph{Weighted \textsc{MaxCut}} For weighted \textsc{MaxCut}, we have a five-node graph, which is shown in Fig~\ref{fig:max_cut_weighted_prob}. The solution for this problem, 00011 (11100) is simpler than the unweighted version. The reward and loss function follow the same principle of the unweighted problem. The size of the operation pool $c = \mathcal{C} = 20$, and the number of layers $p = 10$, leading to a search space of size $\vert \mathcal{S} \vert = 20^{10} \approx 1.02\times 10^{13}$. The `hard' restriction on the maximum number of CNOTs in the circuit is 5.
\begin{figure}[H]
\centering
\includegraphics[width=0.8\textwidth]{Figures/fig_max_cut_weighted_5q.pdf}
\caption{Problem graph for the weighted \textsc{MaxCut} experiment. The weights on edges (0,2), (0,4), (0,1), (2,4), (4,1) and (2,3) are 2, 6, 1, 5, 4 and 3, respectively.}
\label{fig:max_cut_weighted_prob}
\end{figure}
\subsubsection{Results}
\paragraph{Unweighted \textsc{MaxCut}}
The two runs of the search algorithms gave us two circuits (Fig. \ref{fig:qaoa_7q_circ}), leading to two of the six optimal solutions (Fig. \ref{fig:qaoa_7q_solution}). The search rewards and fine-tune losses for both circuits are shown in Figure~\ref{fig:qaoa_7q_search_finetune_both}. During the search stage, since we already know the maximum reward it could reach is 7, and the reward can only be integers, we set the early-stopping limit to 6.5 to reduce the amount of time spent on searching, which means the algorithm will stop searching and proceed to fine-tuning the parameters in the circuit after the reward exceeds 6.5. In a real-world application, we could let the search algorithm run through all of the pre-set number of iterations and record the best circuit structure as well as the corresponding rewards at each iteration at the same time. Then after the search stage finishes, we can choose the best circuit (or top-k circuits) in the search history to fine-tune, increasing our chance to find the optimal solution.
\begin{figure}[H]
\centering
\begin{subfigure}[b]{0.48\textwidth}
\centering
\begin{quantikz}[transparent, row sep={0.8cm,between origins}]
\qw & \gate{H} & \gate{Rot} & \qw & \qw & \qw & \qw & \qw\\
\qw & \gate{H} & \gate{Rot} & \qw & \qw & \qw & \qw & \qw\\
\qw & \gate{H} & \gate{Rot} & \ctrl{0} & \gate{Rot} & \targ{}\vqw{0} & \ctrl{0} & \qw\\
\qw & \gate{H} & \qw & \targ{}\vqw{-1} & \gate{Rot} & \ctrl{-1} & \targ{}\vqw{-1} & \qw\\
\qw & \gate{H} & \gate{Rot} & \qw & \qw & \qw & \qw & \qw\\
\qw & \gate{H} & \gate{Rot} & \qw & \qw & \qw & \qw & \qw\\
\qw & \gate{H} & \gate{Rot} & \qw & \qw & \qw & \qw & \qw
\end{quantikz}
\caption{}
\label{fig:qaoa_7q_first_circ}
\end{subfigure}
~
\begin{subfigure}[b]{0.48\textwidth}
\centering
\begin{quantikz}[transparent, row sep={0.8cm,between origins}, column sep = 0.2cm]
\qw & \gate{H} & \qw & \targ{}\vqw{0} & \gate{Rot} & \qw & \qw & \qw & \qw & \qw & \qw\\
\qw & \gate{H} & \gate{Rot} & \qw & \qw & \qw & \qw & \qw & \qw & \qw & \qw\\
\qw & \gate{H} & \qw & \qw & \gate{Rot} & \qw & \qw & \qw & \qw & \qw & \qw\\
\qw & \gate{H} & \qw & \qw & \gate{Rot} & \qw & \qw & \qw & \qw & \qw & \qw\\
\qw & \gate{H} & \qw & \qw & \gate{Rot} & \ctrl{0} & \targ{}\vqw{0} & \qw & \qw & \qw & \qw\\
\qw & \gate{H} & \qw & \qw & \gate{Rot} & \targ{}\vqw{-1} & \ctrl{-1} & \targ{}\vqw{0} & \ctrl{0} & \gate{Rot} & \qw\\
\qw & \gate{H} & \gate{Rot} & \ctrl{-6} & \qw & \qw & \qw & \ctrl{-1} & \targ{}\vqw{-1} & \qw & \qw
\end{quantikz}
\caption{}
\label{fig:qaoa_7q_second_circ}
\end{subfigure}
\caption{Two different circuits finding two different solutions of the \textsc{MaxCut} problem shown in Fig. \ref{fig:max_cut_prob}. Fig. \ref{fig:qaoa_7q_first_circ} gives the solution 0110010 (see Fig. \ref{fig:qaoa_7q_first_solution}) and Fig. \ref{fig:qaoa_7q_second_circ} gives the solution 0111010 (see Fig. \ref{fig:qaoa_7q_second_solution}).}\label{fig:qaoa_7q_circ}
\end{figure}
\begin{figure}[H]
\centering
\begin{subfigure}[b]{0.48\textwidth}
\includegraphics[width=\textwidth]{Figures/fig_maxcut_1_res_0110010.pdf}
\caption{}
\label{fig:qaoa_7q_first_solution}
\end{subfigure}
~
\begin{subfigure}[b]{0.48\textwidth}
\includegraphics[width=\textwidth]{Figures/fig_maxcut_2_res_0111010.pdf}
\caption{}
\label{fig:qaoa_7q_second_solution}
\end{subfigure}
\caption{Two different optimal solutions found by the circuits in Fig. \ref{fig:qaoa_7q_first_circ} and Fig. \ref{fig:qaoa_7q_second_circ}, respectively.}\label{fig:qaoa_7q_solution}
\end{figure}
\begin{figure}[H]
\centering
\begin{subfigure}[t]{0.48\textwidth}
\includegraphics[width=0.95\textwidth]{Figures/fig_qaoa_7q_1_search_rewards.pdf}
\caption{The change of rewards w.r.t. search iteration during the search for the ans\"atz (in Fig.\ref{fig:qaoa_7q_first_circ}) that gives the solution 0110010 (Fig. \ref{fig:qaoa_7q_first_solution}). To reduce the amount of time for searching, we stopped the algorithm after the search reward exceeded 6.5.}
\label{fig:qaoa_search_reward_1}
\end{subfigure}
~
\begin{subfigure}[t]{0.48\textwidth}
\includegraphics[width=\textwidth]{Figures/fig_qaoa_7q_1_fine_tune_loss.pdf}
\caption{The change of loss w.r.t. optimisation iteration during the fine-tune for the ans\"atz (in Fig.\ref{fig:qaoa_7q_first_circ}) that gives the solution 0110010 (Fig. \ref{fig:qaoa_7q_first_solution}). We can see that the final loss is very close to -7, indicating that the circuit we found can produce an optimal solution.}
\label{fig:qaoa_finetune_1}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.48\textwidth}
\includegraphics[width=0.99\textwidth]{Figures/fig_qaoa_7q_2_search_rewards.pdf}
\caption{The change of rewards w.r.t. search iteration during the search for the ans\"atz (in Fig.\ref{fig:qaoa_7q_second_circ}) that gives the solution 0111010 (Fig. \ref{fig:qaoa_7q_second_solution}). To reduce the amount of time for searching, we stopped the algorithm after the search reward exceeded 6.5.}
\label{fig:qaoa_search_reward_2}
\end{subfigure}
~
\begin{subfigure}[b]{0.48\textwidth}
\includegraphics[width=\textwidth]{Figures/fig_qaoa_7q_2_fine_tune_loss.pdf}
\caption{The change of loss w.r.t. optimisation iteration during the fine-tune for the ans\"atz (in Fig.\ref{fig:qaoa_7q_second_circ}) that gives the solution 0111010 (Fig. \ref{fig:qaoa_7q_second_solution}). We can see that the final loss is very close to -7, indicating that the circuit we found can produce an optimal solution.}
\label{fig:qaoa_finetune_2}
\end{subfigure}
\caption{Search and fine-tune rewards for the circuits in Fig~\ref{fig:qaoa_7q_circ}. }\label{fig:qaoa_7q_search_finetune_both}
\end{figure}
\paragraph{Weighted \textsc{MaxCut}}
The search rewards and fine-tune losses for the weighted \textsc{MaxCut} problem are shown in Fig~\ref{fig:qaoa_5q_search_and_finetune}. We can see that the search converged quickly and the fine-tune loss is very close to -18, indicating that the circuit (see Fig~\ref{fig:qaoa_5q_circ}) produced by our search algorithm can indeed find an optimal solution (see Fig~\ref{fig:qaoa_5q_solution}).
\begin{figure}[H]
\centering
\begin{subfigure}[b]{0.46\textwidth}
\includegraphics[width=\textwidth]{Figures/fig_qaoa_5q_1_search_rewards.pdf}
\caption{Search rewards for the five-node weighted \textsc{MaxCut} problem}
\label{fig:qaoa_5q_search}
\end{subfigure}
~
\begin{subfigure}[b]{0.48\textwidth}
\includegraphics[width=\textwidth]{Figures/fig_qaoa_5q_1_fine_tune_loss.pdf}
\caption{Fine-tune loss for the five-node weighted \textsc{MaxCut} problem}
\label{fig:qaoa_5q_finetune}
\end{subfigure}
\caption{The search rewards and fine-tune losses of for the five-node \textsc{MaxCut} problem.}\label{fig:qaoa_5q_search_and_finetune}
\end{figure}
\begin{figure}[H]
\centering
\begin{subfigure}[b]{0.48\textwidth}
\begin{quantikz}[transparent, row sep={0.8cm,between origins}]
\qw & \gate{H} & \qw & \targ{}\vqw{0} & \gate{Rot} & \targ{}\vqw{0} & \qw & \qw & \qw & \qw\\
\qw & \gate{H} & \qw & \qw & \gate{Rot} & \qw & \ctrl{0} & \qw & \qw & \qw\\
\qw & \gate{H} & \qw & \qw & \qw & \qw & \targ{}\vqw{-1} & \gate{Rot} & \ctrl{0} & \qw\\
\qw & \gate{H} & \qw & \qw & \qw & \qw & \gate{Rot} & \qw & \targ{}\vqw{-1} & \qw\\
\qw & \gate{H} & \gate{Rot} & \ctrl{-4} & \gate{Rot} & \ctrl{-4} & \qw & \qw & \qw & \qw
\end{quantikz}
\caption{The searched circuit for the five-node weighted \textsc{MaxCut} problem}
\label{fig:qaoa_5q_circ}
\end{subfigure}
~
\begin{subfigure}[b]{0.46\textwidth}
\includegraphics[width=\textwidth]{Figures/fig_maxcut_5q_res_00011.pdf}
\caption{The solution sampled, which is 00011, from the circuit shown left.}
\label{fig:qaoa_5q_solution}
\end{subfigure}
\caption{The searched circuit and sampled solution for the five-node \textsc{MaxCut} problem.}\label{fig:qaoa_5q_circ_and_solution}
\end{figure}
\section{Discussion}\label{discussion}
In this paper, we first formulated the circuit search problem as the tree structure. The sampled circuit can be represented as an arc (path from the root to a leaf) on the tree. We also introduced combinatorial multi-armed bandit and na\"ive assumption to model the selection of unitary operators for each layer in the circuit, and approximate the rewards of different unitaries with the reward of a fully constructed circuit. The search process is solved with Monte Carlo tree search (MCTS) algorithm. We demonstrated the effectiveness of our algorithmic framework with various examples, including finding the encoding circuit of the [[4,2,2]] quantum error detection code, developing the ans\"atz for variationally solving system of linear equations, searching the circuit for solving the ground state energy problem of different molecules, as well as circuits for solving optimisation problems on a graph. To our understanding, we are the first to propose such a versatile framework for the automated discovery of quantum circuits with MCTS and combinatorial multi-armed bandits. Results showed that our framework can be applied to many different areas, especially those with problems that can be formulated as finding the ground state energy of a certain Hamiltonian.
From the experiments and results shown in the previous sections, we can see that, by formulating quantum ans\"atz search as a tree-based search problem, one can easily impose various kinds of restrictions (`hard limits') on the circuit structure, leading to the pruning of the search tree and the search space. Also, by introducing Placeholders, one can explore smaller circuit sizes. Since current deep reinforcement learning algorithms struggle when the state space is large but the number of reward states is small. Compared to other research work in quantum ans\"atz search, including the differentiable quantum ans\"atz algorithm proposed in \cite{zhang2021differentiable}, and other QAS algorithms based on meta-learning \cite{chen2021quantum} or reinforcement learning \cite{kuo2021quantum}, which only investigate small-scale problems, like 3- or 4-qubit quantum Fourier transform in \cite{zhang2021differentiable}, 3-qubit classification task and 4-qubit $\text{H}_2$ ground state energy problem in \cite{du2020quantum}. A larger example can be seen in \cite{zhang2021neural}, which is a 6-qubit transversal Ising field model. In our research, we not only looked into 4-qubit systems like the $\text{H}_2$ molecule, but also larger systems like the $\text{LiH}$ and $\text{H}_2\text{O}$ molecule as well as \textsc{MaxCut} on 7-node graphs. Our circuit depth is also often larger than the previous mentioned research. Since our operation pool consist of single-qubit gates on each qubit and CNOT gates either on neighbouring qubits or between every two qubits in the system, resulting a much larger size of operation pool compared to other research. With these two factors combined, our search space size is generally larger than other QAS research.
However, there are still several hyper-parameters that need to be tuned before the search algorithm can produce satisfying results, which leaves us space for improvement for the automation level of the algorithm. In the future, we would like to investigate the performance of our algorithm under noises, as well as improve the scalability of our algorithm by introducing parallelization to the tree search algorithm when using a quantum simulator. We would also like to introduce more flexible value and/or policy functions into the algorithm.
Overall, our research has shown that MCTS enhanced with combinatorial multi-armed bandit is a very efficient approach to search for quantum circuits for a variety of problems, even when the search space is large. Therefore, it took an important leap towards the widespread applications of variational quantum algorithms on many problems.
\paragraph{Acknowledgement} The authors acknowledge the support from Defense Science Institute. We thank useful discussions and advice from Hanxun (Curtis) Huang. The computational resources were provided by the National Computing Infrastructure (NCI) and Pawsey Supercomputing Center through National Computational Merit Allocation Scheme (NCMAS).
\bibliographystyle{unsrt}
\typeout{}
|
1,108,101,564,613 | arxiv | \section{Introduction}
\label{intro}
For massive, hot stars of spectral types OBA, scattering and absorption in spectral
lines transfer momentum from the star's intense radiation field to the plasma, and so provide the force necessary to
overcome gravity and drive a strong stellar wind outflow \citep[see][for an extensive
review]{Puls08}. The first quantitative description of such line-driving was given in the seminal
paper by \citet{Castor75}: hereafter `CAK'. Like many wind models to date,
CAK used the so-called Sobolev approximation \citep{Sobolev60}
to compute the radiative acceleration. This assumes that hydrodynamic flow
quantities\footnote{Or more specifically, occupation number densities and source functions.} are
constant over a few Sobolev lengths $ \ell_{\rm Sob} = \varv_{\rm th}/(d\varv_{\rm n}/dn)$
(for ion thermal speed $\varv_{\rm th}$ and projected velocity gradient $d\varv_{\rm n}/dn$ along a coordinate
direction $\hat{n}$),
allowing then for a \textit{local} treatment of the line radiative transfer.
Such a Sobolev approach ignores the strong `line deshadowing instability' (LDI)
that occurs on scales near and below the Sobolev length \citep{Owocki84};
numerical radiation-hydrodynamic modeling of the non-linear evolution of the
LDI shows that the time-dependent wind develops a very inhomogeneous,
`clumped' structure \citep{Owocki88, Feldmeier97, Dessart03,
Dessart05, Sundqvist13, Sundqvist15}. Such clumpy LDI models
provide a natural explanation for a number of observed
phenomena in OB-stars, such as the soft X-ray emission and
broad X-ray lines observed by orbiting telescopes like {\sc chandra}
and {\sc xmm-newton} \citep{Feldmeier97, Berghofer97, Gudel09,
Cohen10, Martinez17}, the extended regions of zero residual flux typically
seen in saturated UV resonance lines \citep{Lucy83, Puls93,
Sundqvist10}, and the migrating spectral sub-peaks superimposed on
broad optical recombination lines \citep{Eversberg98, Dessart05b,
Lepine08}.
But a severe limitation of most of the above-mentioned models is their assumed spherical
symmetry. The fact that most LDI simulations in the past have been limited to 1D is mainly a
consequence of the computational cost associated with carrying out the non-local integrals
needed to compute the radiation acceleration at each simulation time-step,
while simultaneously resolving length-scales below $\ell_{\rm Sob}$. Specifically, following
the general escape-integral methods
developed by \citet{Owocki96}, some $n_{\rm x} \approx 3 \varv_\infty/\varv_{\rm th}
\approx 1000$ discrete frequency points are typically needed to properly resolve
line profiles and model the expanding flow. In 2D or 3D, a proper treatment of the
multi-dimensional wind further requires integrations along a set of oblique rays in order
to compute the radiative force in the radial and lateral directions. A major issue then
becomes misalignment of nonradial rays with the discrete numerical
grid (i.e. that oblique ray-integrations from any given point
in the mesh in general do not intersect any other point),
requiring that all integrations be repeated for each grid node (and
also then involving complex interpolation schemes to trace the rays).
As an explicit example (see also \citealt{Dessart05}),
for a 2D grid of $n_{\rm r}$ radial and $n_\phi$ azimuthal
points, one needs $n_{\rm r} n_\phi$ integrations of order $n_{\rm r} n_{\rm x}$ operations
for every considered ray; this gives an overall scaling $n_{\rm ray} n_{\rm x} n_{\rm r}^2 n_\phi$,
implying for a typical case of
$n_{\rm ray} \approx 5$, $n_\phi \approx 100$,
and $n_{\rm x} \approx n_{\rm r} \approx 1000$ on order $10^{11-12}$
operations to evaluate the radiative force. Moreover, such a
calculation has to be
carried out at \textit{each time-step of the hydrodynamical simulation},
which for a typical courant time $\sim 5$\,sec in a hot-star wind
outflow, and a total simulation-time of, say, $\sim 50$ dynamical
time scales $t_{\rm dyn} = R_\ast/\varv_\infty \sim10$\,ksec,
requires some $\sim10^5$ repeated evaluations of the radiative force.
This simple example thus illustrates quite vividly the rather daunting
task of constructing multi-dimensional LDI wind models.
Nonetheless, a few previous attempts have been performed. \citet{Dessart03}
carried out `2D hydro+1D radiation' simulations by simply focusing only on the
line force from a single radial ray, thus ignoring lateral influences. This led then
to extensive break-up of the spherical shells seen in 1D simulations,
resulting in lateral incoherence all the way down to the grid-scale. However,
these simulations ignore the lateral component of the diffuse radiative
force, which linear stability analysis \citep{Rybicki90} shows could lead
to damping of velocity variations at scales below the lateral Sobolev
length $\ell_{\rm Sob} = r\varv_{\rm th}/\varv$ and as such to more
lateral coherence than seen in the single-ray 2D simulations. \citet{Dessart05} made
a first attempt to include oblique rays, by using a special, restricted
numerical grid in a 2D plane that forced 3 rays to always intersect
the discrete mesh points \citep{Owocki99b}. But while these simulations did seem to suggest a
somewhat larger lateral coherence than comparable 1-ray models,
the inherent limitations of the method (e.g. in resolving the proper
lateral scales) left results uncertain \citep{Dessart05}.
This paper introduces a `pseudo-planar' modeling approach for a
multi-dimensional wind subject to the LDI. In this
`box-in-a-wind' method, all sphericity effects of
the expanding flow are included in a radial direction $r$, but
some curvature terms are ignored in the lateral direction(s). As discussed in
\S\ref{modeling} (and detailed in Appendix A), for a 2D simulation in the $r,y$ plane this allows us to
consider 5 `long characteristic' rays with a computational cost-scaling
$3 n_{\rm x} n_{\rm r} n_{\rm y}$, thus reducing the general scaling above
with a factor $\sim n_{\rm r} = 1000$ for our standard set-up. Using this method,
\S\ref{results} examines the resulting 2D clumpy wind structure in
much greater detail than possible before, and \S\ref{discussion} discusses the
results, compares to other simulation test-runs, and outlines future work.
\section{Modeling}
\label{modeling}
The simulations here use the numerical PPM
\citep{Colella84} hydrodynamics code {\sc VH-1}\footnote{The
{\sc VH-1} hydrodynamics computer-code package has been developed
by J. Blondin and collaborators, and is available for download at:
http://wonka.physics.ncsu.edu/pub/VH-1/}
to evolve the conservation equations of mass and
momentum for a 2D, isothermal line-driven stellar wind outflow. A key point of this paper
is that while we keep all sphericity effects of an expanding outflow in the radial direction, we
neglect some curvature terms in the lateral direction(s); for details, see Appendix A. Preserving
all properties
of a spherical outflow, this pseudo-planar, box-in-a-wind approach allows us to resolve
laterally the relevant clump-length-scales, as well as implement non-radial rays for the radiative
line-driving in a time-efficient way (see further below and Appendix A).
All presented results adopt the same stellar and wind parameters as in \citet{Sundqvist13, Sundqvist15},
given here in Table 1, which are typical for an O-star in the Galaxy.
The standard set-up uses a spatial grid with 1000 discrete radial ($r$)
mesh-points between $R_\ast \le r \le 2R_\ast$ and 100 lateral ($y$)
ones that cover in total $0.1 R_\ast$. As such, the grid is uniform and
has a constant step-size $\Delta = 0.001 R_\ast$; a small
$\Delta$ is required to resolve both the sub-sonic wind-base with
effective scale height $H = a^2R_\ast^2/(GM_\ast(1-\Gamma_{\rm e})) \approx 0.002 R_\ast > \Delta$
and the resulting small-scale 2D clump structures in the supersonic wind
(the focus of this paper). Each simulation evolves from a smooth, CAK-like initial condition,
computed by relaxing to a steady state a 1D spherically symmetric
time-dependent simulation that uses a CAK/Sobolev form for the line-force.
To prevent artificial structure due to numerical truncation errors
we use an evolution time-step that is the minimum of a fixed 2.5~sec and
a variable 1/3 of the courant time (see discussion in \citealt{Poe90}).
As in previous work, the lower boundary at the assumed stellar
surface fixes the density to a value $\sim 5-10$ times that at the
sonic point. Moreover, since we are interested in structures that are
considerably smaller than the computational box, the lateral boundaries
are simply treated as periodic.
\begin{table}
\centering
\caption{Summary of stellar and wind parameters}
\begin{tabular}{p{2.8cm}ll}
\hline \hline Name & Parameter & Value \\
\hline
Stellar luminosity & $L_\ast$ &
$8\,\times\,10^{5}\,\rm L_\odot$ \\
Stellar mass & $M_\ast$ & 50 \,$\rm M_\odot$ \\
Stellar radius & $R_\ast$ & 20 \,$\rm R_\odot$ \\
Isoth. sound speed & $a$ & 23.4\,km/s \\
Average & & \\
\ - wind speed at $2R_\ast$& $\langle \varv_{\rm max} \rangle$ & 1230\,km/s \\
\ - mass-loss rate & $\langle \dot{M} \rangle$ &
$1.3\,\times\,10^{-6}\,\rm M_\odot/yr$ \\
CAK exponent & $\alpha$ & 0.65 \\
Line-strength & & \\
\ - normalization & $\bar{Q}$ & 2000 \\
\ - cut-off & $Q_{\rm max}$ & 0.004$\bar{Q}$ \\
Ratio of ion thermal & & \\
speed to sound speed & $\varv_{\rm th}/a$ & 0.28 \\
Eddington factor & $\Gamma_{\rm e} = $ & 0.42 \\
& $\kappa_{\rm e} L_\ast/(4 \pi G M_\ast c)$ & \\
\\
\hline
\end{tabular}
\label{Tab:params}
\end{table}
\subsection{Radiative driving}
\begin{figure}
\resizebox{\hsize}{!} {\includegraphics[]{pp_v2.png}}
\centering
\caption{Sketch illustrating the basic idea of the pseudo-planar, box-in-a-wind approach
used in this paper. The upper left illustrates the general situation of non-alignment
between oblique rays and the numerical grid points. The lower panel then shows
how we create a pseudo-planar box in the wind by cutting out a small, but representative,
fraction of the wind volume. For illustration purposes,
it shows projections onto the equatorial plane of rays in the prograde (blue), retrograde (red), and radial (black) directions, for two lateral periods of a simple case with just $n_{\rm y}=2$ zones in lateral direction $y$. The right panel then illustrates how extension out of the equatorial plane involves a total of 5 rays: one radial plus two oblique pairs that extend up/down from the plane.
See Appendix A for a detailed explanation and for further illustrations of the assumed ray geometry.}
\label{Fig:pp}
\end{figure}
The central challenge in these simulations is to compute the
2D radiation line-force in a highly structured, time-dependent wind with a
non-monotonic velocity. This requires non-local integrations of the
line-transport within each time-step of the simulation, in order to
capture the instability near and below the Sobolev length. To meet
this objective, we develop here a multi-dimensional pseudo-planar
extension of the smooth source function \citep[SSF,][]{Owocki91b} method
described extensively in \citet{Owocki96} (see also \citealt{Sundqvist13}).
Appendix A describes in detail this 2D-SSF formulation; below
follows a summary of key features.
Our pseudo-planar 2D-SSF approach allows us to follow the non-linear
evolution of the strong, intrinsic LDI in the radial
direction, while simultaneously accounting for the potentially stabilizing effect of the
scattered, diffuse radiation field, in \textit{both} the radial and lateral directions
\citep{Lucy84, Owocki85, Rybicki90}. SSF further assumes
the line-strength number distribution to be given by an exponentially truncated power-law.
In this formalism, $\alpha$ is the standard CAK power-law index, which can be physically
interpreted as the ratio of the line force due to optically thick lines to the total line
force; $\bar{Q}$ is a line-strength normalization constant, which
can be interpreted as the ratio of the total line force to the
electron scattering force in the case that all lines were optically
thin; $Q_{\rm max}$ is the maximum line-strength cut-off\footnote{Note that we have recast the line force using the
$\bar{Q}$ notation of \citet{Gayley00} rather than the $\kappa_0$
notation of OP96. $\bar{Q}$ has the advantage of being a
dimensionless measure of line-strength that is independent of the
thermal speed. The relation between the two parameter formulations
is given in Appendix A.}.
For typical O-star conditions at solar
metallicity, $Q_{\rm max} \approx \bar{Q} \approx 2000$
\citep{Gayley95, Puls00}. In practice, keeping the nonlinear amplitude
of the instability from exceeding the limitations of the numerical
scheme requires a significantly smaller cut-off \citep{Owocki88, Sundqvist13}.
As noted
in the introduction, including oblique rays
in a multi-dimensional outflow presents severe computational
challenges, largely due to the general misalignment of the
rays with the nodes of the numerical grid. While
earlier attempts of 2D LDI simulations have either used a
`2D-hydro 1D-radiation' approach \citep{Dessart03} or experimented
with a restricted special radial grid set-up \citep{Dessart05}, the pseudo-planar
method introduced here largely circumvents these
issues of grid-misalignment. Namely, while radial ray-integrations are
here calculated identically to the original SSF method, for oblique rays
both the azimuthal radiation angle $\phi$ and the ray's radial directional cosine
$\mu \equiv \cos \theta = \hat{r} \cdot \hat{n}$ become
constant throughout the computational domain.
To this end, we apply a set of 5 rays with $\mu, \phi =
(1,1/\sqrt{3},1/\sqrt{3},1/\sqrt{3},1/\sqrt{3})\,,\,(0,\pi/4,-\pi/4,3\pi/4,-3\pi/4)$
(see simple illustration in Fig. \ref{Fig:pp}, and Appendix A for a detailed explanation).
In addition to the (trivial) radial ray, this thus
considers 4 oblique rays that are also pointing up/down with respect to
the 2D equatorial plane in which the hydrodynamical calculations are carried
out (in order to avoid certain 2D `flat-land' radiation effects, see
\citealt{Gayley00}).
For our assumed grid then, with constant spacings in radial and lateral directions,
information can be used for \textit{all} grid nodes when the
ray-integration for a given $(\mu,\phi)$ pair has been
performed only once over $R_\ast \le r \le 2R_\ast$ for each
of the lateral grid-points. This means that the solid
angle integrations required to compute the
line-force in the radial and lateral directions then can be performed
without the need of any further ray-integrations. In addition, because
of the symmetry of rays pointing up/down from the equatorial plane,
we only have to explicitly carry out the integrations for 3 of our
5 angles. With respect to the general situation, this
means we have effectively reduced the number of required
`long characteristic' ray-integrations at each time-step with a factor of $\sim n_r$
($=10^3$ for our standard set-up here)!
Another attractive feature of this pseudo-planar model is that it preserves
all properties for a 1D purely radial outflow. As detailed in Appendix A,
this is achieved by preserving the general scaling of the flux with radius for a spherical outflow,
by including a sink term for the density to mimic spherical divergence, and by including
in the force equations terms to account for stellar rotation along the lateral axis $y$. As such, our
approach allows for easy testing and benchmarking, and we have
verified that a simulation run with $n_y=1$ and
integration weights for all oblique rays set to zero indeed gives the same results as a
`normal' 1D spherical radial-ray SSF simulation. However, since such radial models
are also subject to the global wind instability associated with nodal topology
\citep{Poe90, Sundqvist15}, they exhibit clumpy structure all the
way down to the lower boundary \citep{Sundqvist13, Sundqvist15}. While there
are strong indications that clumping in hot star winds indeed
extends to very near-photospheric layers \citep[e.g.,][]{Cohen14},
in these first 2D simulations we nonetheless opt to stabilize the wind base by
introducing a small radial increase in $\bar{Q}$ between $R_\ast < r < 1.5 R_\ast$.
This allows us study the emerging clump formation and structure in a somewhat
more controlled environment as compared to simulations
that lie on the nodal topology branch (see \S\ref{discussion}).
\section{Simulation results}
\label{results}
\begin{figure*}
\vspace{1cm}
\begin{minipage}{18.0cm}
\resizebox{\hsize}{!} {\includegraphics[]{rho_count1.png}}
\centering
\end{minipage}
\vspace{1cm}
\begin{minipage}{18.0cm}
\resizebox{\hsize}{!} {\includegraphics[]{rho_count2.png}}
\centering
\end{minipage}
\caption{Spatial and temporal variations of log density relative to the initial, smooth
`CAK' steady-state at $t=0$, with color ranging from densities a decade
below the $t=0$ value (blue) to a decade above (red). The vertical variation extends
from the subsonic wind-base at the stellar surface $R_\ast$ to a height of one $R_\ast$
above. For clarity, the lateral variation is displayed over \textit{twice} the horizontal box
length $0.1 R_\ast$. The upper row shows time evolution over the initial 100 ksec after the
CAK initial condition, in steps of 10 ksec; the bottom row uses the same step-size of 10 ksec
to show the evolution between 300 and 400 ksec, long after the initial condition has developed
into a statistically steady turbulent flow.}
\label{Fig:rho}
\end{figure*}
\begin{figure*}
\vspace{1cm}
\resizebox{\hsize}{!} {\includegraphics[]{rho_count_zoom.png}}
\centering
\caption{As in Fig. \ref{Fig:rho}, spatial and temporal variations of log density relative to the initial, smooth
`CAK' steady-state at $t=0$ are shown, with color ranging from densities a decade
below the $t=0$ value (blue) to a decade above (red). Here the vertical variation only extends between $1.9R_\ast$ and
$2.0R_\ast$ and the lateral variation is displayed over \textit{one} horizontal box of $0.1 R_\ast$; there are thus $100 \times 100$ discrete mesh-points in each of the displayed squares. From left to right are shown a 2 ksec time-evolution long after the initial
condition, in steps of 0.5 ksec.}
\label{Fig:rho_zoom}
\end{figure*}
\begin{figure*}
\vspace{1cm}
\resizebox{\hsize}{!} {\includegraphics[]{vrad_count.png}}
\centering
\caption{Spatial and temporal variations of radial velocity $\varv_{\rm rad}$, with color ranging
from 0 (blue) to 2000\,$\rm km/s$ (red). As in Fig. \ref{Fig:rho}, the vertical variation extends
from the subsonic wind-base at the stellar surface $R_\ast$ to a height of one $R_\ast$
above, and the lateral variation is displayed over \textit{twice} the horizontal box
length $0.1 R_\ast$. The frames from left to right show the time evolution of $v_{\rm rad}$ over
400 ksec after the CAK initial condition, in steps of 50 ksec.}
\label{Fig:vrad}
\end{figure*}
\begin{figure*}
\vspace{1cm}
\resizebox{\hsize}{!} {\includegraphics[]{snaps.png}}
\centering
\caption{Radial cuts through the 2D simulation box of density
$\rho \, \rm [g/cm^3]$ (left) and radial velocity $\varv_{\rm rad} \, \rm [cm/s]$ (right). The red curves
are taken at a snapshot long after the simulation has developed into a statistically quite steady flow; the black
curves compare this to average values.}
\label{Fig:1d-cut}
\end{figure*}
Fig. \ref{Fig:rho} illustrates directly a key result of our simulations, namely
the spatial and temporal variation in $\log \rho$
relative to the initial, smooth `CAK' steady-state. The figure
shows clearly how a radial shell structure first develops, but then quickly
breaks up into laterally complex density variations. The upper
panel displays snapshots during the first 100
$\rm ksec$ of the simulation, illustrating how
already after a few dynamical flow-times $t_{\rm dyn} \approx R_\ast/\langle \varv_{\rm max} \rangle
\approx \rm 11 \, ksec$ the characteristic shells, seen in all
1D LDI simulations, brake up in what initially seem to resemble
Rayleigh-Taylor structures. The lower panel then shows how, as
time passes by, the structures eventually develop into a complex but
statistically quite steady flow, characterized now by localized density
enhancements (`clumps') of very small spatial
scales embedded in larger regions of much lower density.
Fig. \ref{Fig:rho_zoom} zooms in on the same log density in a
small $0.1R_\ast$ square-box over a short time-sequence long
after the initial condition. This illustrates in greater detail
the quite complex 2D density structure, showing a range of
scales as well as high-density clumps with different
shapes. The figure also demonstrates that,
although the structures are small, they are clearly resolved by
our numerical grid.
Fig. \ref{Fig:vrad} displays
temporal and spatial variations in radial velocity,
illustrating essentially the same kind of
outer-wind shock structure and high velocity streams as
corresponding 1D simulations; however, also the velocity
now exhibits extensive lateral variation, reflecting again
the break-up of 1D shells into small-scale 2D clumps.
Fig. \ref{Fig:1d-cut} emphasizes some similarities between these
2D simulations and corresponding 1D ones, by
showing a radial cut through the simulation box at
a time-snapshot (again taken long after the simulation has developed
into a statistically steady flow). The figure demonstrates
how such radial cuts indeed still show the characteristic
structure of the non-linear growth of the LDI, namely
high-speed rarefactions that steepen into strong shocks
and wind plasma compressed into spatially narrow `clumps'
separated by rather large regions of rarified gas. There are some
differences though: In addition to the lateral break-up of shells discussed above, another key
distinction between 1D and 2D simulations is that the radial density variations are a
bit lower in the latter; this occurs because of the lateral `filling in' of radial
rarefactions \citep[see also][]{Dessart03} and is discussed further in the following
section.
\subsection{Statistical properties}
\begin{figure*}
\vspace{1cm}
\resizebox{\hsize}{!} {\includegraphics[]{stats.png}}
\centering
\caption{Selected statistical properties of the 2D simulation, see text. The upper left panel plots
the clumping factor $f_{\rm cl}$; the upper right panel shows the time-dependent
mass-loss rate, $\dot{M} \, \rm [M_\odot/yr]$ vs. sec., computed in two different ways for the red and black curves (see text);
the lower left panel displays lateral (black) and radial (red) density correlation lengths as well as a Gaussian fit to these (blue, dashed); the lower right panel then finally plots radial (left) and lateral (right) velocity dispersions.}
\label{Fig:stat}
\end{figure*}
Fig. \ref{Fig:stat} summarizes some statistical results of the simulations. All averaging
have here started at $t = 250 \, \rm ksec$, in order to separate out any dependence
on the initial conditions and the adjustment to a new radiative
force balance. The upper left panel in Fig. \ref{Fig:stat} shows the clumping factor:
\begin{equation}
f_{\rm cl} = \frac{\langle \rho^2 \rangle}{\langle \rho \rangle^2},
\end{equation}
where angle brackets denote averaging both laterally and
over time in order to separate out $f_{\rm cl}$'s primary dependence on radius.
The plot illustrates how the lateral `filling-in' of radially compressed gas (see above)
decreases the quantitative clumping factor significantly in a 2D simulation as compared to earlier 1D
models where $f_{\rm cl} \ga 10$ \citep[e.g.,][]{Sundqvist13};
this is also consistent with the previous 2D results by \citet{Dessart03}. Note, however,
that the actual values of $f_{\rm cl}$ in our 2D simulation are likely somewhat
underestimated, due to our choice of stabilizing the wind-base against instability caused by
nodal topology (see previous section). As
discussed extensively by \citet{Sundqvist13}, in these near-photospheric layers the
quantitative clumping factor is very sensitive to such choices made for the calculation
of the radiative acceleration, as well as to any variability that may be assumed for the
photospheric lower boundary. Regardless of such caveats, the basic qualitative result
here that 2D simulations yield relatively lower values of $f_{\rm cl}$ than
comparable 1D simulations is quite robust.
The upper right panel of Fig. \ref{Fig:stat} then shows the time-dependent mass-loss rate:
\begin{equation}
\dot{M} \equiv 4 \pi r^2 \rho \varv_{\rm rad}.
\end{equation}
The black line in this plot shows a simple lateral average of the mass-flux escaping the
outermost radial grid-point at a specific time. However, since our simulation box
only covers $0.1 R_\ast$, such an average very likely overestimates the time-dependent
mass loss significantly. To compensate for this, the red curve in the plot instead
uses an average over all grid-points $r \ge 1.5 R_\ast$ at a specific time,
which approximates averaging over a full stellar surface $4 \pi (2 R_\ast)^2 \approx 50 R_\ast^2.$
As expected, this curve shows a drastically lower temporal variation of $\dot{M}$, despite the
highly time-dependent flow. This is consistent e.g. with decade-long observations of spectral lines in
O-stars, which typically indicate that time-variations in the mass-loss rate of such stars are low.
To estimate typical clump length-scales, the lower left panel of Fig. \ref{Fig:stat}
plots a density autocorrelation length:
\begin{equation}
f_{\rm c}(\Delta) = \sum_{\rm time} \sum_{\rm i}
(\rho_i - \langle \rho \rangle) \, (\rho_{i-\Delta} - \langle \rho \rangle),
\end{equation}
where $\langle \rho \rangle$ averages laterally and over time. A lateral correlation
length is calculated at each of the $\Delta = 0-99$ lateral mesh-points
and normalized to its $\Delta = 0$ value. The figure then plots an
average of this lateral correlation length between $r/R_\ast = 1.9-2.0$
(black curve), as well as a radial correlation length (red curve)
defined analogously. The lateral and radial density correlation lengths are very similar,
and as such illustrates how a statistical ensemble of clumps is quite isotropic in
these simulations. This does not imply that
any given clump is isotropic (see Fig. \ref{Fig:rho_zoom}), but rather that, on
average, the well-developed density variations in the simulations do not have a
strong preferred direction.
The Gaussian fit plotted in the blue dashed curve provides an estimate of the
autocorrelation length in terms of the gaussian FWHM $\approx 0.01 R_\ast$.
Such small characteristic scales agree well with the theoretical expectation (see introduction)
that the critical length scale for these clumpy wind simulations is of order the
Sobolev length $\ell_{\rm Sob}$, which for the lateral direction at $2 R_\ast$ is
$\ell_{\rm Sob}/R_\ast = 2\varv_{\rm th}/\varv \approx 0.01$.
Identifying this as a typical clump length scale $\ell_{\rm cl}$, we may further make a
simple estimate of the typical clump mass $\ell_{\rm cl}^3 \rho_{\rm cl} \approx
10^{-6} R_\ast^3 \, 7 \times 10^{-14} \, \rm g/cm^3 \approx 10^{17} \, \rm g$, where
the estimated clump density here simply reads off the output of the simulations
(e.g., Fig. \ref{Fig:1d-cut}). More generally, such a clump
mass-estimate may be obtained using the Sobolev length and mass conservation:
\begin{equation}
m_{\rm cl} \approx \ell_{\rm Sob}^3 \rho_{\rm cl}
\approx \frac{\varv_{\rm th}^3 \dot{M} f_{\rm cl} r}{\varv^4 4 \pi},
\label{Eq:mcl}
\end{equation}
which for the 2D simulation analyzed here indeed
gives $m_{\rm cl} \approx 10^{17} \, \rm g$ for typical
values at $2 R_\ast$. Quite generally, eqn. \ref{Eq:mcl} shows
explicitly how rather low clump masses are expected to emerge
from the LDI.
Finally, the lower right panels in Fig. \ref{Fig:stat} plots the radial and
lateral velocity dispersions:
\begin{equation}
\varv_{\rm disp} = \sqrt{\langle \varv^2 \rangle - \langle \varv \rangle^2},
\end{equation}
where averages are constructed like for the clumping
factor above. These plots show how, as expected (see also \citealt{Dessart03}),
the lateral velocity dispersion is on order the isothermal sound speed,
whereas the radial dispersion is much higher and
expected to rise above several hundreds $ \rm km/s$ in the outer wind.
\section{Summary and future work}
\label{discussion}
We have introduced a pseudo-planar, box-in-a-wind approach suitable for
carrying out radiation-hydrodynamical simulations in situations where the
computation of the radiative acceleration is
challenging and time-consuming. The method is used here to
simulate the 2D non-linear evolution of the strong
line-deshadowing instability (LDI) that causes clumping in the
stellar winds from hot, massive stars. Accounting fully for both
the direct and diffuse radiation components in the calculations of
both the radial and lateral radiative accelerations, we examine in
detail the small-scale clumpy wind structure resulting from our simulations.
Overall, the 2D simulations show that the LDI first manifests itself
by mimicking the typical shell-structure seen in 1-D simulations, but
these shells then quickly break up because of
basic hydrodynamic instabilities like Rayleigh-Taylor and influence
of the oblique radiation rays. This results in a quite
complex 2D density and velocity structure, characterized by
small-scale density `clumps' embedded in larger regions of fast and
rarefied gas.
While inspection of radial cuts through the 2D simulation box confirms
that the typical radial structure of the LDI is intact, quantitatively the
lateral `filling-in' of gas leads to lower values of the clumping factor
than for corresponding 1D models. A correlation-length
analysis further shows that, statistically, density-variations in the well-developed
wind are quite isotropic; identifying then the computed autocorrelation length with a
typical clump size gives $\ell_{\rm cl}/R_\ast \sim 0.01$ at $2 R_\ast$,
and thus also quite low typical clump-masses $m_{\rm cl} \sim 10^{17}$\,g.
This agrees well with the theoretical expectation that the important
length-scale for LDI-generated wind-structure is of order the Sobolev
length $\ell_{\rm Sob}$.
\begin{figure*}
\vspace{1cm}
\resizebox{\hsize}{!} {\includegraphics[]{rotation.png}}
\centering
\caption{Spatial and temporal variations of log density, radial velocity, and lateral velocity for a
model with stellar rotation at the surface $\varv_{\rm y} = 300$\,km/s (see text), with color ranging
as in the earlier Figs. \ref{Fig:rho} and \ref{Fig:vrad}. The vertical variation in this simulation
extends only from 1.0-1.5 $R_\ast$, but the lateral variation is displayed as before
over \textit{twice} the horizontal box length $0.1 R_\ast$. From left to right are shown the time evolution
over 350 ksec after the CAK initial condition, in steps of 50 ksec.}
\label{Fig:rotation}
\end{figure*}
\paragraph{Influence of rotation and topology.} As noted in
\S2 and \S3.1, the level of structure in near photospheric
layers is likely underestimated in the simulation analyzed above,
due to our choice to stabilize the wind base.
To demonstrate this further, Fig. \ref{Fig:rotation} shows a test-run with
identical 2D set-up as before, but now introducing stellar rotation with
a fixed $\varv_{\rm rot} = 300$\,km/s at the surface,
and an initial condition set by steady-state angular momentum
conservation, $\varv_{\rm y}(r) = \varv_{\rm rot} R_\ast/r$. The
figure shows that once the simulation has adjusted to its new
force conditions, radial streaks of high density now appear already
at the surface; in other test-runs, we have found that such
structures are typical for simulations with an unstable base
and nodal topology. The radial streaks in this
rotating model migrate along with the surface rotation, and embedded in
the larger-scale structures are the typical small-scale clumps
discussed previously. As speculated already in
\citet{Sundqvist15}, these tentative first results thus suggest
that rotating LDI models may quite naturally lead to
the type of combined large- and small-scale
structure needed to explain in parallel various observed phenomena
in hot-star winds, like discrete absorption components (DACs)
\citep{Kaper99} and small-scale wind clumping \citep{Eversberg98}. Future work will examine
in detail connections between these rotating LDI models and the
presence of various types of wind sub-structure.
The simulations presented in this paper also lead naturally to a number of follow-up
investigations; already in the pipe-line are the development of a formalism for characterizing
porosity-effects in turbulent media (Owocki \& Sundqvist 2017) and the influence of
the clumpy wind on the accretion properties of an orbiting neutron star in a
so-called high-mass X-ray binary (HMXB) system (el-Mellah et al. 2017). More
directly related to this paper, we also plan to (in addition to further analyzing the effects of rotation and topology)
extend the current simulations to 3D and to higher wind radii, and also develop a
more general radiative transfer scheme (allowing for an arbitrary number of rays) for
the computation of the line acceleration within a pseudo-planar box-in-a-wind.
\begin{acknowledgements}
This work was supported in part by
SAO Chandra grant TM3-14001A awarded to the
University of Delaware, and in part by the visiting
professor scholarship ZKD1332-00-D01 for SPO
from KU Leuven. SPO acknowledges sabbatical leave
support from the University of Delaware, and we
also thank John Castor for helpful discussions on
long-characteristic methods. We finally thank the
referee for useful comments on the paper.
\end{acknowledgements}
\bibliographystyle{aa}
|
1,108,101,564,614 | arxiv | \section{Introduction}
\label{intro}
Semi-inclusive hadron-production processes are important
for investigating properties of quark-hadron matters in
heavy-ion collisions and for finding the origin of the nucleon spin
in lepton-nucleon scattering and polarized proton-proton collisions.
Fragmentation functions (FFs) are key quantities in describing such
hadron-production processes in high-energy reactions.
They indicate hadron-production probabilities from partons.
The FFs have been determined mainly by hadron-production data
of $e^+ e^-$ reaction. Many measurements were done in the $Z^0$-mass
region, whereas lower-energy data are not sufficient.
The determination of the FFs is not in an excellent situation in comparison
with the one of parton distribution functions (PDFs), which is obvious
from the fact that there are huge differences between the FFs of
different analysis groups, especially for disfavored-quark and
gluon FFs \cite{ffs_before_2006}. It led us to investigate
uncertainties of the FFs \cite{hkns07} as it was done
in the PDFs for the nucleon and nuclei \cite{pdf-errors,errors}.
In addition, making global analyses in the leading order (LO) of the
running coupling constant $\alpha_s$ and the next-to-leading
order (NLO) at the same time, we showed that the role of
NLO terms for reducing the uncertainties of the determined
FFs. We provided a useful code in calculating the optimum
FFs for the pion, kaon, and proton from our global analyses
\cite{hkns07,ffs-web}. After our studies, there are works
on related global analyses of the FFs \cite{ffs-summary, ffs-recent}
and also on a hadron-model estimate \cite{ffs-njl}.
Next, we proposed a possible method for exotic-hadron search
by using the FFs \cite{hkos08}. In particular, the FFs are
usually classified into favored and disfavored functions.
The favored means the fragmentation from a quark which exists
in a hadron $h$ as a constituent in a naive quark model.
The disfavored means the fragmentation from a sea quark.
Therefore, internal quark configuration should be reflected
in both FFs. This fact led us to investigate an interesting
suggestion to use the FFs for a possible exotic-hadron search
by looking at differences between the favored and disfavored
functions \cite{hkos08}.
We explain these works in this article.
In Sec. \ref{ffs}, the FFs are defined in $e^+ e^-$ annihilation
processes. Our global analysis method is explained in Sec. \ref{method},
and results are shown in Sec. \ref{results}.
The idea of using the FFs for exotic-hadron search is introduced
in Sec. \ref{exotic}. Our studies are summarized in Sec. \ref{summary}.
\section{Fragmentation functions in $e^+ e^-$ annihilation}
\label{ffs}
The cross section of hadron-$h$ production in the $e^+ e^-$ annihilation
is described by a $q\bar q$-pair production
$e^+ e^- \rightarrow q\bar q$ followed by a fragmentation process
from $q$ ($\bar q$ or gluon emitted from $q$ or $\bar q$) to
the hadron $h$. The FF is defined by the cross section
\begin{equation}
F^h(z,Q^2) = \frac{1}{\sigma_{tot}}
\frac{d\sigma (e^+e^- \rightarrow hX)}{dz} ,
\label{eqn:def-ff}
\end{equation}
where $\sigma_{tot}$ is the total hadronic cross section, and
$Q^2$ is the virtual photon or $Z^0$ momentum squared
in $e^+e^- \rightarrow \gamma, Z^0$. It is equal to the center-of-mass
energy squared $s$ ($=Q^2$). The variable $z$ is defined by the energy
fraction:
\begin{equation}
z \equiv \frac{E_h}{\sqrt{s}/2} = \frac{2E_h}{Q},
\label{eqn:def-z}
\end{equation}
where $E_h$ is the hadron energy. Namely, $z$ is the hadron energy
scaled to the beam energy ($\sqrt{s}/2$). The fragmentation is
described by the summation of all the parton contributions:
\begin{equation}
F^h(z,Q^2) = \sum_i C_i(z,\alpha_s) \otimes D_i^h (z,Q^2).
\label{eqn:def-ffqqbarg}
\end{equation}
Here, $D_i^h(z,Q^2)$ is a fragmentation function of the hadron $h$
from a parton $i$ ($=g, \ u,\ d,\ s,\ \cdot\cdot\cdot$),
$C_i(z,\alpha_s)$ is a coefficient function which is calculated
in perturbative QCD \cite{ffs_before_2006,qqbar-cross}, and
the convolution integral $\otimes$ is defined by
$f (z) \otimes g (z) = \int^{1}_{z} dy / y \, f(y) g (z/y) $.
The measurements of the FFs have been done in various $Q^2$,
whereas they are parametrized at a fixed $Q^2$ point ($\equiv Q_0^2$)
as explained in the next section. The initial functions at $Q_0^2$
are evolved to the experimental $Q^2$ points by the standard DGLAP
evolution equations. The equations are essentially the same as the ones
for the PDFs by exchanging the splitting functions $P_{qg}$ and $P_{gq}$.
There are also some differences between their NLO expressions
\cite{splitting}, for example, in the modified minimal subtraction
($\overline {\rm MS}$) scheme.
\section{Global analysis method}
\label{method}
The FFs are expressed in terms of a number of parameters, which are
determined by a $\chi^2$ analysis of the $e^+ + e^- \rightarrow h+X$
data. The initial functions are provided at $Q_0^2$ as
\begin{equation}
D_i^h(z,Q_0^2) = N_i^h z^{\alpha_i^h} (1-z)^{\beta_i^h} ,
\end{equation}
where $N_i^h$, $\alpha_i^h$, and $\beta_i^h$ are the parameters.
An apparent constraint for the parameters is the energy sum rule:
\begin{equation}
\sum_h M_i^h \equiv \sum_h \int_0^1 dz \, z \, D_i^h (z,Q^2) = 1 ,
\label{eqn:sum}
\end{equation}
where $M_i^h$ is the second moment of $D_i^h (z,Q^2)$.
However, it is almost impossible to confirm this sum since
the summation over all the hadrons cannot be taken practically.
In our analysis, we tried to be careful that the sum does not
significantly exceed 1 even within analyzed hadrons.
\begin{wraptable}{r}{0.50\textwidth}
\vspace{-0.4cm}
\caption{Experiments, center-of-mass energies,
and numbers of data points are listed for used data sets
of $e^+ +e^- \rightarrow \pi^\pm +X$ \cite{hkns07}.}
\label{tab:exp-pion}
\vspace{-0.2cm}
\begin{center}
\begin{tabular}{lcc} \hline \hline
Experiment & $\sqrt{s}$ (GeV) & \# of data \\
\hline
TASSO & 12,14,22,30,34,44 & 29 \\
TPC & 29 & 18 \\
HRS & 29 & \, 2 \\
TOPAZ & 58 & \, 4 \\
SLD & 91.28 & 29 \\
SLD (u,d,s) & 91.28 & 29 \\
SLD (c) & 91.28 & 29 \\
SLD (b) & 91.28 & 29 \\
ALEPH & 91.2 \, & 22 \\
OPAL & 91.2 \, & 22 \\
DELPHI & 91.2 \, & 17 \\
DELPHI (u,d,s) & 91.2 \, & 17 \\
DELPHI (b) & 91.2 \, & 17 \\
\hline
Total & & 264 \, \\
\hline
\end{tabular}
\end{center}
\end{wraptable}
There are two types in the FFs: favored and disfavored functions.
For the FFs of light quarks ($u,d,s$), they are assumed to be equal
if they are favored or disfavored ones. The favored functions are
given by
\begin{align}
D_{u}^{\pi^+} (z, & Q_0^2) = D_{\bar d}^{\pi^+} (z,Q_0^2)
\nonumber \\
& = N_{u}^{\pi^+} z^{\alpha_{u}^{\pi^+}}
(1-z)^{\beta_{u}^{\pi^+}} ,
\label{eqn:favored}
\end{align}
for $\pi^+$.
The $\pi^+$ productions from $\bar u$, $d$, $s$, and $\bar s$ are
disfavored processes so that they are assumed to be equal
at $Q_0^2$:
\begin{align}
D_{\bar u}^{\pi^+} (z, & Q_0^2) = D_{d}^{\pi^+} (z,Q_0^2)
\nonumber \\
& = D_{s}^{\pi^+} (z,Q_0^2)
= D_{\bar s}^{\pi^+} (z,Q_0^2)
\nonumber \\
& = N_{\bar u}^{\pi^+} z^{\alpha_{\bar u}^{\pi^+}}
(1-z)^{\beta_{\bar u}^{\pi^+}} .
\label{eqn:disfavored}
\end{align}
The FFs from a gluon and heavy quarks are defined separately as
\begin{align}
D_{g}^{\pi^+} (z,Q_0^2)
& = N_{g}^{\pi^+} z^{\alpha_{g}^{\pi^+}} (1-z)^{\beta_{g}^{\pi^+}} ,
\nonumber \\
D_{c}^{\pi^+} (z,m_c^2) & = D_{\bar c}^{\pi^+} (z,m_c^2)
= N_{c}^{\pi^+} z^{\alpha_{c}^{\pi^+}} (1-z)^{\beta_{c}^{\pi^+}} ,
\nonumber \\
D_{b}^{\pi^+} (z,m_b^2) & = D_{\bar b}^{\pi^+} (z,m_b^2)
= N_{b}^{\pi^+} z^{\alpha_{b}^{\pi^+}} (1-z)^{\beta_{b}^{\pi^+}} ,
\label{eqn:gcb}
\end{align}
where $m_c$ and $m_b$ are charm- and bottom-quark masses.
The parameters in Eqs. (\ref{eqn:favored}), (\ref{eqn:disfavored}),
and (\ref{eqn:gcb}) are determined so as to fit the data
in Table \ref{tab:exp-pion}, where experimental collaborations,
center-of-mass energies, and the numbers of data are listed
for the charged-pion production.
It is clear that most data are taken at the $Z^0$ mass.
The FFs for the kaon and proton are parametrized in the similar way
by considering favored and disfavored functions. The amounts of
data are almost the same as the ones in Table \ref{tab:exp-pion}.
The detailed should be found in the original article \cite{hkns07}.
The parameters for the kaon and proton are determined
in separate $\chi^2$ analyses.
One of our major purposes is to show the uncertainties of the FFs
as explained in Sec. \ref{intro}. The uncertainties
have been already estimated in the studies of nucleonic and nuclear PDFs
\cite{pdf-errors, errors}. The same Hessian method is used
for the uncertainty estimation. The $\chi^2$ is expanded around
the minimum $\chi^2$ point $\hat \xi$:
$
\Delta \chi^2 (\xi) = \chi^2(\hat{\xi}+\delta \xi)-\chi^2(\hat{\xi})
=\sum_{i,j} H_{ij}\delta \xi_i \delta \xi_j \ ,
$
where $H_{ij}$ is called Hessian which is the second derivative
matrix, $\xi$ indicates a parameter set, and $\hat \xi$ is the set
at the minimum $\chi^2$ point.
The confidence region is given in the parameter space by
supplying a value of $\Delta \chi^2$. Using the Hessian matrix
obtained in a $\chi^2$ analysis, we estimated the uncertainty
of the FF by
$
[\delta D_i^h (z)]^2=\Delta \chi^2 \sum_{j,k}
\left[ \partial D_i^h (z) / \partial \xi_j \right]_{\hat\xi}
H_{jk}^{-1}
\left[ \partial D_i^h (z) / \partial \xi_k \right]_{\hat\xi} .
$
There are some variations among groups on the appropriate
$\Delta \chi^2$ value for showing the uncertainty range
in a global analysis. The details are explained in our article \cite{hkns07}
about our $\Delta \chi^2$ choice.
\section{Determined fragmentation functions for pion, kaon, and proton}
\label{results}
\begin{wrapfigure}{r}{0.42\textwidth}
\vspace{-0.2cm}
\begin{center}
\epsfig{file=pion-q-data.eps,width=0.42\textwidth} \\
\end{center}
\vspace{-0.2cm}
\caption{Charged-pion fragmentation function in $e^+e^-$
annihilation process. Our analysis results are
compared with experimental data \cite{hkns07}.}
\label{fig:pion-q-data}
\end{wrapfigure}
\vspace{-0.3cm}
\begin{figure}[t!]
\vspace{-0.0cm}
\begin{center}
\epsfig{file=pion-ff-q-1.eps,width=0.40\textwidth}
\hspace{0.5cm}
\epsfig{file=pion-ff-comp2.eps,width=0.40\textwidth} \\
\end{center}
\vspace{-0.1cm}
\caption{Determined fragmentation functions for the pion
and their comparison with other parametrizations
\cite{hkns07, inpc07}. The shaded bands indicate
estimated uncertainties.}
\label{fig:pion-ffs-comp}
\vspace{0.5cm}
\begin{center}
\epsfig{file=kaon-ff-q-1.eps,width=0.40\textwidth}
\hspace{0.5cm}
\epsfig{file=proton-ff-q-1.eps,width=0.40\textwidth} \\
\end{center}
\vspace{-0.1cm}
\caption{Determined fragmentation functions for the kaon and
proton \cite{hkns07}.}
\label{fig:kp-ff-comp}
\end{figure}
We show obtained FFs of the pion by the $\chi^2$ analyses
of the $e^+ e^- \rightarrow \pi^{\pm} X$ data
in Fig. \ref{fig:pion-q-data}, where the FF data in the form of
Eq. (\ref{eqn:def-ff}) and our parametrization result
with an uncertainty band are shown at $Q^2=M_Z^2$.
The good agreement with the data indicates that the fit is
successful from small- to large-$z$ regions.
Next, each FF is shown for the pion on the left-hand-side of
Fig. \ref{fig:pion-ffs-comp} with uncertainty bands in both LO and
NLO ($\overline {\rm MS}$). It should be noted that the charm- and
bottom-quark FFs are shown at the scale of their mass thresholds
$Q^2$=$m_c^2$ or $m_b^2$, whereas the others are shown at $Q^2$=1 GeV$^2$.
The uncertainties are generally larger in the LO, which indicates
the FF determination is improved due to NLO terms.
We notice that disfavored-quark and gluon FFs
have large uncertainties in both LO and NLO.
The NLO functions are then compared with other parametrizations
of KKP, Kretzer, AKK, and DSS in Fig. \ref{fig:pion-ffs-comp}.
Some disfavored-quark and gluon functions, for example $s$-quark
functions of Kretzer and AKK, are completely different
between the analysis groups; however,
they are within our uncertainty bands.
There are not much differences between
the groups in favored- ($u$), charm-, and bottom-quark functions
except for the small-$z$ region.
The large discrepancies among the different parametrizations
especially in the disfavored-quark and gluon functions
were not clearly understood before our work.
Therefore, it is important to point out in our studies that
they are not due to an inappropriate analysis of some groups
and that they come from experimental errors and inaccurate
flavor decomposition for light quarks. It is simply impossible
to determine accurate disfavored-quark and gluon FFs from current
experimental data.
For the kaon and proton, our LO and NLO FFs are shown
in Fig. \ref{fig:kp-ff-comp}. We found that both FFs are not
better determined than the pion FFs. There is a similar tendency
with the pion case that the disfavored-quark and gluon FFs are not
determined well. The NLO improvement, namely the uncertainty reduction,
is clear in the kaon, whereas it is not apparent in the proton.
We also found \cite{hkns07} in both kaon and proton that
all the different analyses are consistent with each other
in spite of their large differences in some functions
because they are roughly within our uncertainty bands.
\section{Exotic-hadron search by fragmentation functions}
\label{exotic}
\begin{wraptable}{r}{0.50\textwidth}
\vspace{-0.3cm}
\caption{Second moments of the FFs in the NLO for $\pi^+$, $K^+$, and $p$
at $Q^2$=1 GeV$^2$.}
\label{tab:2nd-moments}
\vspace{-0.0cm}
\begin{center}
\begin{tabular}{lcc} \hline \hline
Type & Function & 2nd moment \\
\hline
Favored & $D_u^{\pi^+}$ & 0.401$\pm$0.052 \\
Disfavored & $D_{\bar u}^{\pi^+}$ & 0.094$\pm$0.029 \\
\hline
Favored & $D_u^{K^+}$ & 0.0740$\pm$0.0268 \\
Favored & $D_{\bar s}^{K^+}$ & 0.0878$\pm$0.0506 \\
Disfavored & $D_{\bar u}^{K^+}$ & 0.0255$\pm$0.0173 \\
\hline
Favored & $D_u^{p}$ \ \ & 0.0732$\pm$0.0113 \\
Disfavored & $D_{\bar u}^{p}$ \ \ & 0.0084$\pm$0.0057 \\
\hline
\end{tabular}
\end{center}
\end{wraptable}
From the analyses of ordinary hadrons, pion, kaon, and proton,
we found characteristic differences between the favored and
disfavored FFs. In Table \ref{tab:2nd-moments},
the second moments are shown for $\pi^+$, $K^+$, and $p$.
It is clear that the moments are larger for the favored functions
than the ones for the disfavored functions. It indicates that
internal quark configuration can be found by looking at
flavor dependence of the FFs. This fact suggested us
to use the FFs for exotic hadron search \cite{hkos08}.
The exotic means a hadron with internal quark configuration
other than ordinary $q\bar q$ and $qqq$. This topic has been
investigated for a long time; however, an undoubted evidence
has not been found yet. In the last several years, there have
been reports on exotic candidates mainly from Belle and BaBar
collaborations in charmed hadrons. In our work, we investigated
a possibility that the FFs can be used for an exotic hadron
search by using differences between the favored and disfavored FFs.
As one of the exotic mesons, we investigated a possibility of
determining quark configuration of $f_0 (980)$, which
structure has been controversial for many years.
According to a simple quark model, it is described
by the configuration $(u\bar u + d\bar d)/\sqrt{2}$.
However, it is known that theoretical strong decay widths are
an order of magnitude larger than the experimental width \cite{f0-decay}.
Therefore, it is considered to be $s\bar s$, tetraquark, or
$K \bar K$ molecule state. It used to be considered as
a glueball candidate, but recent lattice QCD estimates
indicate that the lowest scalar-meson mass is about
1700 MeV. There is an indication that the internal configuration
could be determined by the radiative decay
$\phi \rightarrow f_0 \gamma$ and 2$\gamma$ decay
$f_0 \rightarrow 2\gamma$ \cite{hkos08,f0-e1}; however,
the FF method could become a better way for judging
its internal structure as well as other exotic-meson
configurations.
\begin{table}[t]
\caption{Possible $f_0(980)$ configurations and their features
in FFs at small $Q^2$ \cite{hkos08}.}
\label{tab:f0-config}
\begin{center}
\begin{tabular}{cccc}
\hline
Type & Configuration
& Second moments
& Peak positions \\
\hline
Nonstrange $q\bar q$ & $(u\bar u+d\bar d)/\sqrt{2}$
& $M_s<M_u<M_g$
& $z_u^{\rm max}>z_s^{\rm max}$ \\
Strange $q\bar q$ & $s\bar s$
& $M_u < M_s \lesssim M_g$
& $z_u^{\rm max}<z_s^{\rm max}$ \\
Tetraquark (or $K\bar K$) & $(u\bar u s\bar s+d\bar d s\bar s)/\sqrt{2}$
& $M_u \sim M_s \lesssim M_g$
& $z_u^{\rm max} \sim z_s^{\rm max}$ \\
Glueball & $gg$
& $M_u \sim M_s < M_g$
& $z_u^{\rm max} \sim z_s^{\rm max}$ \\
\hline
\end{tabular}
\end{center}
\end{table}
We summarized in Table \ref{tab:f0-config} how to judge
the structure of the $f_0$ meson, especially
by the second moments and $z$-dependent functional forms.
All the possible configurations are considered in the table.
For example, if $f_0$ is an $s\bar s$ state, $s\bar s$ is formed
from $s$ by creating an $s\bar s$ pair from a radiated gluon.
In the same way, a color neutral $s\bar s$ can
be formed from a gluon by its splitting into an $s\bar s$ pair
and a subsequent gluon radiation for color neutrality
(see Ref.9 for details).
The gluon can be radiated from either $s$ or $\bar s$, so that
$M_g$ could be larger than $M_s$ as indicated in the table.
If $s$ is a favored quark, a significant portion of its
energy (namely large $z$) is transferred to $f_0 (s\bar s)$.
It leads to a functional form which is mainly distributed in
the large-$z$ region. In the table, this is denoted
as $z_u^{max} < z_s^{max}$.
In the same way, the relations in the second moments
and the functional forms are listed for other configurations.
Our suggestions are intended to give an idea on the criteria,
and further details need to be investigated in more sophisticated
hadron models.
There are data on the FFs of $f_0$ in the $e^+ e^-$ annihilation.
We have done a global analysis for determining the FFs in the same way
with the analyses in Sec. \ref{method} \cite{hkos08}.
This is intended to judge the quark configuration
of $f_0$ by using the criteria of Table \ref{tab:f0-config}.
The determined second moments are given by
$M_u = 0.0012 \pm 0.0107$,
$M_s = 0.0027 \pm 0.0183$, and
$M_g = 0.0090 \pm 0.0046$.
Then, the moment ratio becomes
$M_u/M_s=0.43\pm 6.73$. From the ratio $0.43$, the $f_0$ seems to
be mainly an $s\bar s$ state. However, it is obvious from its huge error $6.73$
that a clear determination is currently not possible.
\begin{wrapfigure}{r}{0.40\textwidth}
\vspace{-0.4cm}
\begin{center}
\epsfig{file=f0-ffs-errors.eps,width=0.40\textwidth} \\
\end{center}
\vspace{-0.1cm}
\caption{Determined FFs for $f_0(980)$ \cite{hkos08}.
Note that uncertainty bands are much larger than the FFs.}
\label{fig:f0-ffs}
\end{wrapfigure}
The obtained FFs from the global analysis are shown in Fig. \ref{fig:f0-ffs}.
The up- and strange-quark functions are distributed mainly in the large-$z$
region, which may indicate that $u$ and $s$ could be important constituents
of $f_0$. However, the uncertainties of the determined FFs are huge and
they are an order of magnitude larger than the FFs themselves.
Therefore, it is not possible to draw a conclusion on its structure
from the global analysis at this stage.
Hopefully, much better data will be reported in the near future possibly
from the Belle collaboration \cite{belle} for determining
structure of exotic hadrons including $f_0 (980)$.
\section{Summary}
\label{summary}
Our studies on the fragmentation functions were reported.
First, global analyses have been done for pion, kaon,
and proton for determining their FFs.
In the past, it was not clear why there are large discrepancies
among various parametrizations on disfavored-quark and gluon
FFs. We clarified that they are consistent with each other
by estimating the uncertainties of the FFs by the Hessian method
in the sense that all the distributions are within our error bands.
We also clarified the role of NLO terms in reducing the uncertainties
by comparing the determined FFs and their uncertainties in the LO and NLO.
Our code for calculating the FFs was supplied on our web site \cite{ffs-web}.
Next, we investigated a possibility of using the FFs for finding
internal structure of exotic hadrons by using differences
between the favored and disfavored FFs. We proposed to use the second moments
and $z$-dependent functional forms of the FFs for determining the internal
structure. As an example, the $f_0 (980)$ meson was studied by
considering all the possibilities of $q\bar q$, tetraquark, and glueball
configurations. A global analysis has been done also for the FFs of
$f_0$ by using the current $e^+ e^-$ data; however, they were not accurately
determined to the level of finding the internal structure, namely a difference
between the favored-quark and disfavored-quark (and gluon) FFs.
Much more accurate data are needed to discuss the internal structure
by the FFs.
\section*{Acknowledgements}
The authors would like to thank discussion with T. Nagai, M. Oka,
and K. Sudoh on the fragmentation functions.
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